Download - SPH and -SPH: Applications and Analysis · Both SPH and -SPH are applied to test problems, with solutions analysed within physical and spectral space. In Chapter 2 an extensive review

SPH and α-SPH:

Applications and Analysis

John Anthony Mansour

B.Sc.(Hons)

Thesis submitted for the degree of

Doctor of Philosophy

School of Mathematical Sciences

Monash University

October 2007

for my parents

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiDeclarations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivSummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction 1

2 Smoothed Particle Hydrodynamics 42.1 The SPH discrete approximation . . . . . . . . . . . . . . . . . . 6

2.1.1 The approximation function . . . . . . . . . . . . . . . . . 62.1.2 The first derivative . . . . . . . . . . . . . . . . . . . . . . 82.1.3 The second derivative . . . . . . . . . . . . . . . . . . . . 102.1.4 The kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 SPH applied to fluid dynamics . . . . . . . . . . . . . . . . . . . 132.2.1 The continuity equation . . . . . . . . . . . . . . . . . . . 132.2.2 The momentum equation . . . . . . . . . . . . . . . . . . 152.2.3 The energy equations . . . . . . . . . . . . . . . . . . . . 172.2.4 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.5 Equation of state . . . . . . . . . . . . . . . . . . . . . . . 212.2.6 Integrals of motion . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Variable Resolution Implementations . . . . . . . . . . . . 252.3.3 Neighbouring particle list . . . . . . . . . . . . . . . . . . 272.3.4 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 The α-SPH turbulence model 303.1 LANS-α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 α-SPH: equations of motion . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 The filtered velocity . . . . . . . . . . . . . . . . . . . . . 333.2.2 The momentum equation . . . . . . . . . . . . . . . . . . 35

3.3 Integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.1 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.2 Iteration for filtered velocity . . . . . . . . . . . . . . . . . 43

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

i

4 One-Dimensional Tests 444.1 Burgers’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.1 Colliding shocks . . . . . . . . . . . . . . . . . . . . . . . 454.1.2 The steepening shock front . . . . . . . . . . . . . . . . . 49

4.2 One-dimensional Navier-Stokes . . . . . . . . . . . . . . . . . . . 544.2.1 The Euler system . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Forced Navier-Stokes simulations . . . . . . . . . . . . . . 72

4.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . 80

5 The Kelvin-Helmholtz instability 825.1 Constant velocity fluids in relative motion . . . . . . . . . . . . . 83

5.1.1 Linear results . . . . . . . . . . . . . . . . . . . . . . . . . 845.1.2 Computational configuration . . . . . . . . . . . . . . . . 845.1.3 Determination of mode growth rates . . . . . . . . . . . . 865.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 The hyperbolic tangent velocity profile . . . . . . . . . . . . . . . 925.2.1 Linear results . . . . . . . . . . . . . . . . . . . . . . . . . 925.2.2 Computational configuration . . . . . . . . . . . . . . . . 965.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Two-Dimensional Turbulence 1036.1 Computational configuration . . . . . . . . . . . . . . . . . . . . 107

6.1.1 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.1.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.1.3 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.1.4 Equation of state . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Intermediate scale forcing . . . . . . . . . . . . . . . . . . . . . . 1136.3 Large scale forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3.1 Quasi-steady solutions . . . . . . . . . . . . . . . . . . . . 1216.3.2 Steady solutions . . . . . . . . . . . . . . . . . . . . . . . 1236.3.3 Steady solutions incorporating α-SPH . . . . . . . . . . . 127

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Conclusion 134

A α-SPH: variable-h terms 138

B α-SPH: Resulting differential equations 142

C The spectral method 145C.1 Non-linear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 146C.2 Iteration for filtered velocity . . . . . . . . . . . . . . . . . . . . . 147C.3 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147C.4 Normal modes of linearised energy . . . . . . . . . . . . . . . . . 149

D Fourier mode construction using particle data 151

ii

Acknowledgements

iii

Declarations

This thesis contains no material which has been accepted for the award of anyother degree or diploma in any university or other institution. To the best ofmy knowledge, this thesis contains no material previously published or writtenby another person, except where due reference is made within the text of thethesis.

iv

Summary

In this thesis a study of the Smoothed Particle Hydrodynamics (SPH) methodis undertaken. Furthermore, a recent modification to SPH known as α-SPHis also considered. This variant is designed for application to turbulent fluiddynamics. Both SPH and α-SPH are applied to test problems, with solutionsanalysed within physical and spectral space.

In Chapter 2 an extensive review of the SPH method is undertaken. Firstthe details of SPH as a general numerical method are considered, followed byspecifics of the SPH application to fluid dynamics, including derivation of equa-tions of motion within a Lagrangian framework. Likewise in Chapter 3, theparticulars of α-SPH are considered, including derivation of the α-SPH equa-tions and discussion of conservation properties.

An extensive array of one-dimensional tests are performed in Chapter 4.Where available, results are compared with analytic solutions. Elsewhere, highlyaccurate spectral solutions form benchmarks for SPH and α-SPH simulations.Numerous observations are made by considering SPH solutions in spectral space.In particular, the importance of a variable smoothing length implementation tocorrect non-linear energy cascades and the influence of secondary SPH pressuregradient terms. Simulations of α-SPH demonstrate it to be successful in induc-ing closure of Euler dynamics, though results for small values of the turbulenceparameter are found to be unsatisfactory.

Linear regime studies of the Kelvin-Helmholtz simulations are presented inChapter 5. Expected growth rates from linear stability theory are recoveredfor SPH simulations where sufficient resolution is utilised. For poorly resolvedKelvin-Helmholtz perturbations, incorrect growth rates are shown to be directlyrelated to deficiencies of the SPH pressure gradient. We demonstrate that theα-SPH scheme is successful in reducing growth rates.

Random forcing is used to induce two-dimensional turbulence in Chapter 6.Large scale dynamics are found to compare favourably with theoretical expec-tations, though a number of difficulties are encountered. At the shortest scales,numerical artifacts related to insufficient resolution are apparent, though thesedo not appear to significantly influence large scales. Results for α-SPH methodare given, though simulations are restricted to small values for the turbulenceparameter, and findings are inconclusive.

v

Chapter 1

Introduction

The advent of digital computers in the twentieth century heralded an era ofstrong proliferation in the mathematical sciences. A new avenue of investigationwas opened, and previously intractable problems were now able to be tackledusing the brute numerical force afforded by the digital calculator. Indeed thepotential application of computational techniques has increased in line with theever increasing processing power of the computer. Despite recent pessimism,technological developments continue to yield advancements in line with Moore’slaw, which states that computational speed doubles approximately every twoyears. Even so, for many classes of problems, solution via pure numerical forceis still very much a distant dream, and in some cases a fundamental impossi-bility. Perhaps the most prominent example of such a problem is that of fluidturbulence. The solution to problems of turbulence defies numerical calculationby virtue of the huge range of scales which must be resolved for accurate integra-tion. Indeed the required processing power can be many orders of magnitudegreater than what is currently available. It is perhaps ironic then that mostflows encountered in problems of engineering interest are turbulent in nature.

While the straightforward approach of resolving all relevant dynamics isusually not a possibility, numerous techniques have been developed wherebyapproximations are introduced to reduce the computational costs to practicallevels. Indeed the field of turbulence research is extremely active, with an exten-sive library of publications appearing in the literature. Researchers are drawnby the self-perpetuating complexity of turbulence, the challenge of conqueringan open problem, or simply the visual allure and beauty of turbulent flows.Many of the greatest scientists of our time have put their minds to problemsof turbulence, though not all have met with success. Indeed Horace Lamb, inspeech to the British Association for the Advancement of Science, is reputed tohave quipped, “I am an old man now, and when I die and go to heaven thereare two matters on which I hope for enlightenment. One is quantum electrody-namics, and the other is the turbulent motion of fluids. And about the formerI am rather optimistic.” With such luminaries struggling to gain insight to themysteries of turbulence, it seems there is little hope of developing a full un-derstanding of the physical phenomena at play, nor of being able to accuratelysimulate highly turbulent flows. However turbulence is a field as broad as it iscomplex, and piece by piece, progress is made.

In this thesis we consider the solution to various dynamical problems of a tur-

1

2

bulent nature. To this end, we utilise the numerical method known as SmoothedParticle Hydrodynamics (SPH) (Lucy, 1977; Gingold and Monaghan, 1977). Asinferred by it’s name, SPH divides a flow into a set of discrete particles, withthe evolution of each particle determined by the SPH equations of motion. Onebenefit of SPH is that it does not require a regular grid, instead only relying onthe radial direction and displacement between particles to construct the requiredquantaties. This endows SPH with a geometric generality which allows for it tobe trivially applied to a wide range of problems. In Chapter 2, a comprehensiveoverview of the SPH method is given. We consider the various developments andmodifications to the method, both from the perspective of numerical methods,and also from the view of SPH in application to fluid dynamics.

SPH has seen widespread application in many fields, perhaps none more sothan astrophysics and fluid dynamics. While for most problems in these fieldswe can expect turbulent dynamics to play some role, the significance of turbu-lence is often overlooked or ignored. Indeed the robust nature of SPH can be adouble-edged sword, allowing simulations to proceed without giving indicationof poorly-resolved dynamics. Alternatively, and as is usually the case, the costof resolving all physically relevant scales simply puts direct simulation out of thequestion. In light of these limitations, a new modification of the standard SPHalgorithm has been devised. The new implementation, known as α-SPH (Mon-aghan, 2002), purports to provide a more mathematically rigorous handling ofshort scales than what might be encounter for standard SPH. Based on theLagrangian-averaged Navier-Stokes differential equations of Chen et al. (1998),α-SPH may be considered a minimal model of turbulence, the basic premisebeing that energy propagation to short scales is inhibited, while simultaneouslymaintaining the conservation of key physical quantaties. The details of α-SPHare explored in Chapter 3, and various numerical tests in both one and twodimensions are performed throughout the thesis to evaluate the effectiveness ofthis new model.

However this thesis is only partially concerned with issues of turbulencemodelling. Perhaps the more prominent theme is the behavior of the SPHalgorithm itself, afterall we cannot make a judgement on the effectiveness of anymodelling regime without first knowing the accuracy of the standard algorithm.While modern technology allows the use of large SPH particle populations,with simulations utilising of order one million particles realistic on a currentdesktop machine, important turbulent dynamics occur at all scales, and mayeasily saturate the available bandwidth, potentially introducing error. In thisthesis, we attempted to develop an understanding of the behavior of SPH wheredynamics are marginally resolved, both through direct comparison with highlyaccurate spectral calculations, and with comparison with expected theoreticalresults.

Presented within is the first thorough investigation of SPH dynamical be-havior in spectral space. A novel new method is utilised to take generallydistributed particle data, and recast this data in terms of trigonometric func-tions. In Chapter 4, this allows us to make a very direct comparison betweena one-dimensional SPH algorithm and equivalent spectral algorithm. Whereasprevious investigation primarily consider physical space quantities and proper-ties, here we are also able to quantify SPH dynamics across the entire spectrumof modes. Insight into the significance of secondary terms born of the SPHapproximation is gained, along with the influence of the SPH smoothing length

3 Introduction

parameter, and the cost and benefits of a Lagrangian derivation. A similarinvestigate is performed in two dimensions for SPH simulations of the Kelvin-Helmholtz instability (Chapter 5). Here the interface of two counter-streamingfluids is tracked to accurately determine the growth of perturbations in the lin-ear regime. Application of varying wavelength perturbations allows us to mapthe change in growth rates as resolution limits are approached. The above oneand two-dimensional tests are also performed for the α-SPH algorithm, withresults compared and contrasted with standard SPH.

While one-dimensional results give some indication of the performance ofSPH and α-SPH, we do not expect results to necessarily generalise to higherdimensions given the geometric simplicity and absence of transverse waves. Like-wise, linear stability results only hint at what might be found for broad spectrumturbulent simulations. Bearing this in mind, fully turbulent two-dimensionalsimulations have been performed in Chapter 6, with forcing applied in a peri-odic box to induce a Kraichnan (1967) regime. Through the use of the spectralrecomposition method, comparison is able to be made with theoretical predic-tions and to findings in the literature, the vast majority of which is presentedin the spectral domain. This is the first such simulation using the SPH method,and indeed one of a small number of Kraichnan turbulence simulations wherespectral methods have not been applied. This regime provides a challengingvehicle by which the intricacies of SPH may be studied, along with a trulyturbulent foundation for the evaluation of the α-SPH modification.

Chapter 2

Smoothed ParticleHydrodynamics

Recent advancements in computer technology has brought the practical abilityto perform numerics on a very large scale. This has allowed for the analysisof differential equations through the use of discretisation techniques, and manysuch techniques exists. One such method, and the subject of the current thesis,is the method of Smoothed Particle Hydrodynamics (SPH) (Lucy, 1977; Gingoldand Monaghan, 1977). Here, a physical domain is decomposed into a number ofnodes, or particles, which then interact with other nodes according to discreteapproximations to problem dynamics. While the quality of the decompositionwill to an extent be determined by the arrangement of nodes, SPH does not placeany particular requirements on node configuration. Further, where we use SPHtechniques to simulate hydrodynamics, nodes are advected with the fluid flow.We now make the analogy of nodes as physical particles, carrying with themphysical attributes such as mass and temperature. In this sense, SPH forms anative approximation to Lagrangian fluid dynamics, and allows for constructionof simulations in an intuitive fashion. Unless otherwise explicitly stated, wediscuss SPH with respect to its application to fluid dynamics.

SPH is one of a family of techniques known as methless methods. As the namesuggests, these methods do not rely on a mesh or grid to construct functionapproximations, or derivative approximations by which differential equationsmay be calculated. This relieves simulations of the inherent geometry found inother methods such as finite differences. Such geometric deficiencies can leadto difficulty or errors in a number of situations, such as simulations involvingdiscontinuities not aligned with the numerical grid, with excessive numericaldiffusion potentially resulting. Mesh based methods also do not lend themselvesnaturally to simulations involving complex geometries, with special measuresbeing required, such as the construction of problem specific grids. In contrast,SPH calculations only rely on the radial interactions of particles and as suchshow minimal preference to particular geometry configurations. Furthermore,algorithms may be easily constructed for problems involving interacting objects,such as moving walls or floating bodies, and free surfaces are also handled withminimal difficulty. For techniques requiring grids, such simulations certainlypresent great difficulty and increased complexity.

4

5 Smoothed Particle Hydrodynamics

SPH naturally leads to solutions with a resolution which varies in space andtime. This results from the Lagrangian nature of SPH, where our numericalnodes are considered as particles carrying mass, and therefore concentrate reso-lution in regions of high density. Indeed, on account of this density, such regionsoften hold relatively large proportions of energy, and it follows that accuratecalculations of their dynamics should take precedence. Though grid refinementmethods are certainly capable of achieving similar results, this comes at ad-ditional cost and refinement implementations are often neither straightforwardnor intuitive. SPH provides such capability natively, though certain simulations,such as those of nearly incompressible fluids, may not benefit from this inherentvariable resolution. We also find that SPH algorithms prove robust in simu-lations where turbulent dynamics are under-resolved, tending to redistributeenergy away from poorly represented scales in a non-dissipative fashion (seeChapter 4). This behavior is similar to that required of turbulence modellingschemes, and indeed we may consider it as implicit turbulence modelling. Thisbehavior of SPH allows integrations to continue though under-resolved, andmean dynamics are often still found to be reproduced correctly (see Chapter4). Analogue simulations using spectral methods lead to energy accumulatingat the resolution limit, and solution corruption usually follows. It should benoted that this may be advantageous at times, indicating insufficient resolution.Furthermore, correct dynamical behavior cannot be expected where we rely onimplicit SPH turbulence modelling, though an aim of this thesis is to investi-gate a quantifiable and physical approach to turbulence modelling which mayminimise reliance on this implicit behavior.

The earliest inceptions of smooth particle hydrodynamics (SPH) date tothe work of Lucy (1977) and Gingold and Monaghan (1977) which brings SPHinto its thirtieth year of development. In this time, it has seen applicationto many areas of numerical modelling, most notably perhaps being the vastarray of publications concerning SPH applications to astrophysics: interstellargas dynamics and star formation (Bromm et al., 1999; Lattanzio et al., 1985),magnetohydrodynamics (Price and Monaghan, 2004; Phillips and Monaghan,1985) and cosmology (Springel and Hernquist, 2002; Thacker et al., 2000) toname but a few (see Monaghan (1992) for further references). Applicationhas also been found in areas such as fracture dynamics ((Benz and Asphaug,1995; Bonet and Kulasegaram, 2005), elastic (Gray et al., 2001) and viscoelastic(Ellero et al., 2002) flows, free surface and incompressible flows (Monaghan,1994; Morris et al., 1997; Cummins et al., 1997). Recently SPH has also founduse in the animation industry owing to its robust nature and straightforwardapplication to complex geometries.

This chapter presents an overview of the theory, derivation, and implemen-tation of SPH, much of which is found in the review articles of Monaghan(1992, 2005). It is structured as follows. In the first section we consider SPHfrom a numerical perspective, with formulations of approximation functions andderivatives outlined, along with respective errors, and appropriate forms for ourapproximation kernel. The second section is concerned with application of SPHto fluid dynamics. Here we outline a variational derivation of our constitutiveequations of continuity, momentum and energy. Different forms of equationof state and viscosity are presented, along with consideration of integrals ofmotion. Finally we turn to the practical considerations of computing SPH ap-proximations, such as timestepping and stability, as well as the self-consistent

2.1 The SPH discrete approximation 6

resolution formulation and boundary implementations.

2.1 The SPH discrete approximation

2.1.1 The approximation function

The foundation of the SPH scheme is the weighted average integral, which leadsto an approximation Ah(r) to some function A(r):

Ah(r) =∫R

A(r′)W (r − r′, h) dr′, (2.1)

where W is our kernel (or weight function), r is a position vector, and dr′ isan element of the spatial domain. The action of parameter h will depend onthe choice of kernel, but generally it will determine the domain over which theaveraging occurs. Throughout this work it will be referred to as the smoothinglength parameter. To simplify notation, we will not explicitly state the depen-dence of the kernel on h for the time being, nor state the integration domain R.A Taylor expansion of A(r′) about r gives

Ah(r) = A(r)∫W (r − r′) dr′

+∫W (r − r′) ((r′ − r) · ∇r∗)A(r∗)

∣∣∣r∗=r

dr′ (2.2)

+12

∫W (r − r′)

[((r′ − r) · ∇r∗)

2A(r∗)

∣∣∣r∗=r

+O((r′ − r)3)]

dr′.

An example in two dimensions is given to clarify notation:

((r′ − r) · ∇r∗)A(r∗)∣∣∣r∗=r

= (x′ − x)∂A

∂x(r) + (y′ − y)

∂A

∂y(r),

for r = (x, y). Consistency requires∫W (r − r′) dr′ = 1, (2.3)

and where the kernel is symmetric about its argument, the second term in (2.2)disappears and we can write Ah = A + O(h2). We also note that the aboveexpansions require that our function A be smooth on some scale ∆r = r − r′.

The leap to SPH begins with the discretisation of equation (2.1). In onedimension this can be achieved by simple application of the trapezoidal rule.For a set of nodes at positions rb with respective nodal values Ab we have

AhS(r) =∑b

AbW (r − rb)∆rb ' Ah(r)

with∆rb =

12

(rR − rL).

Here rL and rR are the nodes respectively to the left and right of the node atrb. This sum is performed over all nodes, however since kernels with compact


support are normally chosen, typically a small subset of nodes (those withinclose proximity to position r) are the only non-zero contributions. Steppingup to higher dimensions, this decomposition of the domain into finite elementsceases to become trivial. For problems of fluid dynamics, the method of SPHdeals with this by assigning a mass to each node. Rewriting equation (2.1) as

Ah(r) =∫A(r′)ρ(r′)

ρ(r′)W (r − r′) dr′, (2.4)

where ρ is a material density, we can now make the following interpretation:

ρ(r′)dr′ = dm(r′)

where dm(r′) is an element of mass. Nodes can now carry with them a cer-tain mass, and as such form a good analogy to physical particles. Numericalquadrature is thus achieved, and we can write the following approximation

AhS(r) =∑b

mb

ρbAbW (r − rb) ' Ah(r) (2.5)

where mb and ρb are the mass and density associated with particle b, whileAb = A(rb). This can be recast using the notation of finite elements, where wecan now define a shape function as

φb =mb

ρbW (r − rb)

and the approximation is then written

AhS(r) =∑b

φb(r)Ab.

The discretisations outlined above form the foundation upon which all SPHschemes can be constructed.

Naturally, this secondary step of approximation brings with it some addeddegree of error. Given that there is no prescription on the arrangement ofnodes, quantifying this additional error is no trivial task and will be different foreach simulation. Considering Taylor expansions can yield some insight however.Equation (2.2) is rewritten with summations replacing integrals:

AhS(r) ' A(r)∑b

∆VbW (r − rb)

+∑b

∆VbW (r − rb)[((rb − r) · ∇r∗) A(r∗)

]r∗=r

(2.6)

+12

∑b

∆VbW (r − rb)[((rb − r) · ∇r∗)

2A(r∗)

]r∗=r

for some volume element ∆Vb associated with node b. Inspection of equation(2.6) reveals that our normalisation (2.3) is now only approximate, and firstorder terms no longer vanish exactly. As such, we do not expect to reproduceconstant functions exactly. Analysis of Monaghan (2005) for one-dimensionalequispaced data indicates that a Gaussian kernel (see section 2.1.4) will lead


to errors which diminish exponentially in (h/∆r)2, for particle spacing ∆r.So reasonable accuracy can perhaps be expected where data is approximatelyequispaced, and for appropriately chosen smoothing lengths.

A correction which restores exact constant reproduction is attributed toShepard (1968). It can be derived easily. We rewrite equation (2.5):

AhS(r) =∑b

∆VbAbW (r − rb),

and now for constant Ab = k we have

k =AhS(r)∑

b ∆VbW (r − rb),

so we can now define a new corrected scheme:

Ahshep(r) =AhS(r)∑

b ∆VbW (r − rb),

which is exact where A is constant. The extra computation involved in calcu-lating the above come at negligible cost. Further corrections to the summationapproximation can be constructed to ensure linear completeness, such as themoving least squares method (Krongauz and Belytschko, 1996) or the repro-ducing kernel method (Liu et al., 1995). These methods introduce significantadditional operation count to calculations, though for certain simulations thisextra cost may prove worthwhile.

2.1.2 The first derivative

We may obtain an approximation to a first derivative by exact differentiationof the approximation integral (2.1)

(∇A)h(r) =∫∇A(r′)W (r − r′) dr′. (2.7)

Integration by parts then yields

(∇A)h(r) =∫∂R

A(r′)W (r − r′) ds−∫R

A(r′)∇r′W (r − r′) dr′. (2.8)

Here ds is an element along the domain boundary. In application, we usuallyhave the first term on the left of (2.8) disappearing, either exactly or approxi-mately. We then write

(∇A)h(r) ∼= −∫A(r′)∇r′W (r − r′) dr′

=∫A(r′)∇r W (r − r′) dr′ = ∇Ah(r). (2.9)

We now take Taylor series expansions of A(r′) about r to find

∇Ah(r) = A(r)∫∇rW (r − r′) dr′

+∫∇rW (r − r′) ((r′ − r) · ∇r∗) A(r∗)

∣∣∣r∗=r

dr′ (2.10)

+12

∫∇rW (r − r′)

[((r′ − r) · ∇r∗)

2A(r∗)

∣∣∣r∗=r

+O((r′ − r)3)]

dr′.


The first term in the above equation will vanish as long as the kernel is chosento be symmetric about its argument. Via integration by parts, the second termwill reduce to the required gradient. This however relies on the kernel beingnormalised correctly (2.3). We are left with

∇Ah(r) = ∇A(r) (2.11)

+∫ [

((r′ − r) · ∇r∗)2A(r∗)

∣∣∣r∗=r

+O((r′ − r)3)]∇rW (r − r′) dr′.

In SPH, we approximated equation (2.9) with a summation:

∇AhS(r) =∑b

∆VbA(rb)∇rW (r − rb) ' ∇Ah(r). (2.12)

Unfortunately, in making this approximation, we only approximately find theabove simplifications. Equation (2.10) is written, with truncation of high orderterms,

∇AhS(r) ' A(r)∑b

∆Vb∇rW (r − r′)

+∑b

∆Vb ((r′ − r) · ∇rA(r))∇rW (r − r′) (2.13)

As the first term in (2.13) is not identically zero, this approximation does notyield correct derivatives for constants. We say that it lacks zeroth-order com-pleteness.

For most calculations, we are usually only concerned with evaluating func-tions at the nodes. Equation (2.12) becomes

∇AhS(ra) =∑b

∆VbA(rb)∇rW (r − rb)∣∣r=ra

' A(ra)∑b

∆Vb∇rW (r − rb)∣∣r=ra

(2.14)

+∑b

∆Vb((rb − r) · ∇rA(r)

)∇rW (r − r′)

∣∣r=ra

.

We can ensure that the first term on the right hand side of equation (2.14) van-ishes for constant functions by writing equations in symmetric form (Monaghan,1988):

∇AhS(ra) =∑b

∆Vb(A(rb)−A(ra)

)∇rW (r − rb)

∣∣r=ra

. (2.15)

However, this correction is only applicable when the function is being evalu-ated at a node. A generalised route to all forms of this correction is given inMonaghan (2005), which suggests we write, for any differentiable function Φ,

∇A =1Φ(∇(AΦ)−A∇(Φ)

). (2.16)

Our SPH summation (2.14) then yields

∇AhS(ra) =1

Φa

∑b

∆VbΦb(A(rb)−A(ra)

)∇rW (r − rb)

∣∣r=ra


We find equation (2.15) by setting Φ = 1.A correction leading to equivalent order accuracy may be had by taking

derivatives of the Shepard function (2.1.1). This has the added advantage ofbeing applicable anywhere in the domain. Further corrections to (2.14) can behad by renormalising to remove the coefficient that will appear in front of thegradient term (this coefficient is precisely unity for the integral approximation(2.11), but only approximates this for the summation (2.14)) . Modifications torestore linear completeness at all points in the domain have also been suggested.Johnson and Beissel (1996) provided corrections for calculations performed onaxisymmetric fields, while Krongauz and Belytschko (1997) presented resultsfor general fields. These modifications tend to be computationally cumbersomehowever, and also give unreliable conservation of key physical quantities due toasymmetry (see section 2.2.6). All calculations presented here only make use ofthe symmetrisation correction (2.15).

SPH forms for other first derivative operators can be derived in a similarfashion (Monaghan, 1992).

2.1.3 The second derivative

The second derivative may be obtained in a similar fashion to the first derivative.In analogy to equation (2.7), we write

(∇2A)h(r) =∫∇2A(r′)W (r − r′) dr′. (2.17)

Now we apply integration by parts twice, again assuming that surface termsvanish, to find

∇2Ah(r) =∫A(r′)∇2

rW (r − r′) dr′. (2.18)

As previously, we can now approximate this with summations. However, it turnsout that approximations to second derivatives found in this way are excessivelysensitive to the node configurations. Because of this, a number of approachesexist in the literature whereby the second derivative is determined indirectly,usually requiring only the first derivative of the kernel.

One such approach begins with an integral approximation to the secondderivative (Brookshaw, 1985; Cleary and Monaghan, 1999; Espanol and Re-venga, 2003). Perhaps the most general of these approximation equations isgiven by Espanol and Revenga∫ (

A(r)−A(r′))F (|(r − r′)|)

[5

(r − r′)α(r − r′)β

|(r − r′)|2− δαβ

]dr′

=∂2A(r)∂rα∂rβ

+O(h2) (2.19)

with

F (|(r − r′)|) =(r − r′) · ∇rW (r − r′)

|(r − r′)|2

and where Greek indices indicate Cartesian component of subject. The aboveapplies to integrations performed in three dimensions and assumes kernels which


exhibit spherically symmetry. An SPH equivalent can then be written(∂2A(r)∂rα∂rβ

)a

'∑b

∆Vb(A(ra)−A(rb)

)F (|(ra−rb)|)

[5

(ra − rb)α(ra − rb)β

|(ra − rb)|2−δαβ

].

(2.20)We will be concerned with constructing the Laplacian in later sections. Usingequation (2.19),

∇2A(r) =∂2A

∂x2+∂2A

∂y2+∂2A

∂z2

' 2∫ (

A(r)−A(r′))F (|(r − r′)|) dr′, (2.21)

with (r1, r2, r3) = (x, y, z). This brings us to a similar integral approximant tothat given in Brookshaw (1985) and Cleary and Monaghan (1999). That thisindeed leads to the Laplacian of A(r) is easily verified using Taylor expansionsof A(r′). The SPH equivalent then takes the simpler form(

∇2A(r))a' 2

∑b

∆Vb(A(ra)−A(rb)

)F (|(ra − rb)|). (2.22)

Other approaches which utilises only first order kernel derivatives are givenin Flebbe et al. (1994) and Watkins et al. (1996). Though subtly different, bothmethods largely consist of performing SPH differentiation twice to arrive at asecond derivative. To calculate ∇2A, we can write

F (ra) = (∇A(r))a =∑b

∆Vb(A(rb)−A(ra)

)∇rW (r − rb)

∣∣r=ra

.

Then taking the divergence of the above, we have(∇2A(r)

)a

= (∇ · F (r))a =∑b

∆Vb(F (rb)− F (ra)

)· ∇rW (r − rb)

∣∣r=ra

.

Watkins performed tests comparing the above methods under a number of nodeand vector field configurations. These results suggest that calculating secondderivatives using recursive first derivatives leads to superior results, especiallywhere nodes are arranged randomly, though testing was far from exhaustive. Ingeneral, SPH simulations of hydrodynamics problems will not lead to such de-gree of disorder (Monaghan, 2005). It follows that the significance of Watkins’results for such simulations is questionable. Furthermore, any potential im-provements in accuracy come at a cost, as further summations are now requiredto be calculated separately.

2.1.4 The kernel

Thus far, our definition of the SPH method has not specified the propertiesthe smoothing kernel W should exhibit, except that it is required to meet thenormalisation defined by equation (2.3). Further, the kernel should tend towardsthe Dirac delta function in the limit of diminishing smoothing length:

limh→0

W (r − r′, h) = δ(r − r′). (2.23)


Infinitely many kernels may be constructed to satisfy these properties, andthroughout the literature a large range have been used. While not strictlynecessary, it is usually desirable for any given kernel to be a function of radialdistance alone. This ensures the required even symmetry for our approximationfunction (2.1) to exhibit second order accuracy in h. Furthermore, this symme-try is required for the conservation of key physical quantities such as momentum(see section 2.2.6). A general form for these kernels can be written

W (|∆r|, h) =κ

hνf(q), (2.24)

where ∆r = |r − r′| and q = |∆r|/h. The parameter ν is the kernel’s dimen-sionality, while κ is a constant, determined such that (2.24) satisfies equation(2.3). In the early SPH work of Gingold and Monaghan (1977) the Gaussiankernel was employed. It takes the form

W (|∆r|, h) =κ

hνexp(−q2). (2.25)

For normalisation, we require the constant κ takes the values 1/π1/2, 1/π and1/π3/2 for one, two and three dimensions respectively. An alternate kernel,constructed of cubic splines, is found in Monaghan and Lattanzio (1985):

W (|∆r|, h) =κ

hν

1− 3

2q2 + 3

4q3 0 ≤ q ≤ 1

14 (2− q)3 1 ≤ q ≤ 20 2 ≤ q .

(2.26)

Here κ takes the values 2/3, 10/7π and 1/π for one, two and three dimensionsrespectively. Both the Gaussian and the cubic spline kernel form appropriatekernels for SPH summations. The Gaussian kernel has the added benefit ofbeing infinitely differentiable, which leads to improved stability qualities (Priceand Monaghan, 2004). This however comes at the cost of computational effi-ciency, as the Gaussian kernel does not have compact support. Therefore, allparticles contribute to the summation in equation (2.5), and computations scaleas N2, where N is the number of particles/nodes. In practise though, contri-butions from particles beyond a few smoothing lengths are negligible, and canbe ignored. This truncation allows for efficient computations, but is a trade offwith accuracy. It is preferable to choose kernels with finite supports, such as thecubic spline kernel. No truncations are required for such kernels; summationcontributions are zero beyond their domain of influence. For example the cubicspline falls to zero for distances beyond 2h, allowing for computations whichscale as N . The cubic spline kernel is used for all calculations found in thiswork.

Owing to its construction, the cubic spline kernel only exhibits smooth firstorder derivatives. This may lead to inferior stability traits (Morris, 1996) incomparison with smoother kernels such as the Gaussian, even though our cal-culations do not use kernel second derivatives (see section 2.1.3). Higher orderspline kernels may certainly be constructed to address this issue, such as the


quartic spline kernel (Schoenberg, 1946):

W (|∆r|, h) =κ

hν

( 5

2 − q)4 − 5( 3

2 − q)4 + 10( 1

2 − q)4 0 ≤ q ≤ 1

2

( 52 − q)

4 − 5( 32 − q)

4 12 ≤ q ≤

32

( 52 − q)

4 32 ≤ q ≤

52

0 52 ≤ q .

(2.27)

In practise, we do not expect such kernels to impart significant additional accu-racy, despite improved stability qualities. Most importantly, though many formsare possible which satisfy the conditions above, best accuracy is found when thekernel takes a symmetric bell type profile (Fulk and Quinn, 1996; Hongbin andXin, 2005).

Accurate approximations are found where we take h > ∆x for node seper-ation ∆x. For the cubic spline, smoothing length usually takes the valueh = 1.3 ∆x. Larger smoothing lengths give better discretisations of (2.1.1),owing to the larger number of nodes representing it. Though the effective reso-lution of the approximation will be reduced (approximation is now ‘smoother’),and computational cost will be increased significiantly. Expected superior ac-curacy of large scales is generally outweighed by increased computational cost,and inferior representation of short scales.

For multidimensional modelling, anisotropic kernels may be costructed withthe symmetry W (r) = W (−r), such as those resulting from tensor productsof one-dimensional kernels. This may be desirable where the distribution islargely anisotropic (Shapiro et al., 1996), resulting in insufficient summationcontribution in certain directions, and correspondingly inferior surface repre-sentations. Such kernels will not affect linear momentum conservation, andwhere anisotropy is defined by local particle configuration, angular momentumcan also be conserved.

2.2 SPH applied to fluid dynamics

We now turn to the application of SPH to problems of hydrodynamics, fromwhich the method of SPH was born. While in the previous section we consid-ered SPH from a purely numerical point of view, we now introduce physics tothe problem. We make the interpretation of nodes as physical particles, eachcarrying with them attributes such as mass and velocity. This analogy provesuseful in developing a sense of intuition when working with SPH.

2.2.1 The continuity equation

SPH algorithms may be constructed to satisfy continuity via a number of routes.The simplest is perhaps a straightforward application of equation (2.5) to reveal

ρa =∑b

mbWab. (2.28)

We have made the following notation simplifications in the above: ρa = ρ(ra),Wab = W (ra − rb). The explicit dependence of the kernel on smoothing lengthis omitted for the time being. Alternatively, we can write an equation for the

2.2 SPH applied to fluid dynamics 14

time rate of change density. Beginning with the continuity equation written forthe Lagrangian frame,

dρdt

= −ρ∇ · v, (2.29)

which we recast in SPH summation form:

dρadt

= ρa∑b

mb

ρbvab · ∇aWab. (2.30)

Here the notation ∇a specifies the gradient with respect to coordinate ra, whilevab = va − vb. The particular choice of continuity formulation will largelydepend on the dynamical problem at hand, though perhaps use of the densitytime derivative is usually preferable. In particular, for simulations involvingfluids of different densities, the summation (2.28) will result in false densitygradients at fluid-fluid interfaces. Similarly, at free surfaces, summations willbe incorrectly normalised, leading to lower than expected near surface densities(see Bonet and Rodriguez-Paz (2005) for a formulation which accounts for this).Equation (2.30) can also take a number of forms. For instance, taking a directLagrangian derivative of (2.28) gives us

dρadt

=∑b

mbvab · ∇aWab. (2.31)

In analogy to equation (2.16), Price (2004) writes a general form from which allsuch equations may be found. For some scalar function φ, the divergence (2.29)is given as

dρdt

= φ

[v · ∇

(ρ

φ

)−∇ ·

(ρv

φ

)](2.32)

from which we may now write

dρadt

= φa∑b

mb

φbvab · ∇aWab. (2.33)

If we take φ = ρ2−σ, setting σ = 1 leads to equation (2.30), while for σ = 2we yield (2.31). We note that different forms for the continuity equation will ingeneral lead to different momentum equations where the momentum equationis derived through a variational framework (see section 2.2.2). From this per-spective, some authors have given versions of the momentum equation whichare inconsistent with their choice of continuity equation. The benefits of strictconsistency are not necessarily obvious, though variational derivations may as-sure conservation of various integrals of motion, such as linear and angularmomentum.

Key differences between continuity implementation are often to be foundat the interface between fluids of differing density. It is common practice inSPH to integrate such systems as a single fluid with different mass particles,though in reality the two fluids should be considered separately, with boundaryforces acting on each at the interface. So we then wish to determine whichmethod gives the best approximation to such configurations. Differences arisein the ways with which equations (2.30) and (2.31) calculate the divergence∇ · v (Monaghan, 2005). Calculations of divergence near an interface using


(2.31) result in divergences with dependence on particle mass. However, ∇ · vis not a function of mass. Equation (2.30) on the other hand includes theterm mb/ρb within the summation, which we interpret as the volume elementoccupied by particle b, ie mb/ρb = ∆Vb. Calculations of divergence via (2.30)give no dependence on particle mass, as we require. Improved accuracy is foundfor multi-fluid simulations using (2.30) (Colagrossi, 2004). Monaghan (2005)asserts that either (2.30) or (2.31) provide sufficient accuracy where densityratios are less than 2 : 1.

As discussed previously, using mass summations (2.28) leads to density gra-dients across the interface. This is highly undesirable as such gradients willcause unphysical pressure forces. This may be realised as surface oscillations insimulations involved free surfaces (Monaghan, 1992). Several authors (Ott andSchnetter, 2003; Tartakovsky and Meakin, 2005) have suggested an alternatesummation yielding number densities can be used to avoid density gradients atmulti-fluid interfaces:

na =∑b

Wab, (2.34)

where na is the number density at particle a. Tartakovsky and Meakin used theabove with a modified momentum equation to perform Rayleigh-Taylor simu-lations for miscible flows, with particle masses evolved according to a diffusionequation. Summation equation (2.34) proves advantageous in implementationswhere particle mass may vary in time, with equations of state written in termsof number densities. However such summations will not correct erroneous nor-malisation at free surfaces, and false density gradients ensue.

2.2.2 The momentum equation

Neglecting viscosity, our acceleration equation reduces to the Euler equation,

dv

dt= −1

ρ∇P, (2.35)

for a pressure force P . Application of equation (2.15) gives

dvadt

= − 1ρa

∑b

mb

ρb(Pb − Pa)∇aWab. (2.36)

While the above returns zero acceleration for constant pressure fields, it fails toconserve linear or angular momentum exactly. We instead prefer to constructour momentum equation such that it exhibits symmetry leading to line of sightforcings between particle pairs. A natural path to such formulations begins withthe Lagrangian

L =∫ (

12ρv · v − u(ρ, s)

)dr, (2.37)

for thermal energy per unit mass u, as function of density ρ and entropy s(Eckart, 1960). We now write a discrete approximation to this:

L =∑b

mb

(12vb · vb − u(ρb, sb)

). (2.38)


The equation of motion for particle a is then found through an application ofthe Euler-Lagrange equation:

ddt

(∂L

∂via

)− ∂L

∂ria= 0, (2.39)

where the superscript i refers to a vector component. Where we take (r,v) asour canonical variables, equation (2.39) becomes

madviadt

+∑b

mb

(∂ub∂ρb

)s

∂ρb∂ria

= 0. (2.40)

An application of the first law of thermodynamics yields(∂ub∂ρb

)s

=Pbρ2b

.

We now differentiate our summation density (2.28) to find

∂ρb∂ria

=∑c

mc∂Wbc

∂ria(δab − δac)

and putting all this together we have

madviadt

= −∑b

∑c

mbmcPbρ2b

∂Wbc

∂ria(δab − δac)

= −∑b

∑c

mbmcPbρ2b

δab∂Wbc

∂ria+∑b

∑c

mbmcPbρ2b

δac∂Wbc

∂ria

= −∑c

mamcPaρ2a

∂Wac

∂ria+∑b

mbmaPbρ2b

∂Wba

∂ria.

Where our kernel is radially symmetric we finally have (returning to vectornotation),

dvadt

= −∑b

mb

(Paρ2a

+Pbρ2b

)∇aWab. (2.41)

An alternate route to this equation is demonstrated in Monaghan (1992), wherethe term on the right hand side of (2.35) is rewritten

∇Pρ

= ∇(P

ρ

)+P

ρ2∇ρ,

and then recast using the SPH summation (2.12) to find equation (2.41). Manysymmetric variants are possible, though if variational consistency is desired, careshould be taken in selecting the appropriate form for the continuity equation.Indeed, Bonet and Lok (1999) demonstrate that calculations using mis-matchedcontinuity and momentum equations lead to inferior results.

Where continuity is achieved through the use of rate of change equations(such as (2.30)), a slightly different approach is required for the variationalderivation of momentum. We begin by writing our action (Price and Monaghan,2004)

S =∫ t2

t1

L(r1, . . . , rN ,v1, . . . ,vN )dt,


where we have N particles. We require the action to be stationary for any smallperturbation to the path between the given limits. Taking a small deviation δrato coordinates of particle a, we have to first order

δS =∫ t2

t1

(δra ·

∂L

∂ra+

dδradt· ∂L∂va

)dt,

and integration by parts on the second term gives

δS =∫ t2

t1

δra ·(∂L

∂ra+

ddt

∂L

∂va

)dt.

We note here that the secondary term (which arises due to integration by parts)disappears, since our coordinate deviation goes to zero at the limits. Still tofirst order, we can now write

δS =∫ t2

t1

δra ·

(ma

dvadt

+∑b

mb∂ub∂ρb

δρb∂ra

)dt,

where δρb represents the co-moving variation in ρb, which will depend on ourchoice of continuity equation. As we require the first order change in action tobe zero for any such variable perturbation, we then have

madvadt

= −∑b

mb∂ub∂ρb

δρb∂ra

. (2.42)

As an example, we consider the general form of continuity (equation (2.33))given in Price (2004):

δρb = φb∑c

mc

φc(δrb − δrc)∇bWbc.

Inserting this into (2.42), and making use of the first law of thermodynamics,yields

dvadt

= −∑b

mb

(Paρ2a

φaφb

+Pbρ2b

φbφa

)∇aWab, (2.43)

where we have again exploited the kernel’s symmetry. For density equation(2.30), we set φ = ρ, so we then have

dvadt

= −∑b

mb

(Pa + Pbρaρb

)∇aWab, (2.44)

while letting φ = 1 leads to momentum equation (2.41) found earlier via thedensity summation (2.28).

2.2.3 The energy equations

The equation for rate of change of thermal energy can be easily determinedstarting with the first law of thermodynamics,

du = Tds− PdV

= Tds+P

ρ2,


which then for constant entropy, leads to

dudt

=P

ρ2

dρdt.

From here we may apply our particular choice of SPH continuity equation. Usingequation (2.30), we have

duadt

=Paρa

∑b

mb

ρbvab · ∇aWab. (2.45)

For variationally consistent derivations, the total energy of the system will beequal to the total thermal and kinetic energy, which we may write

e =∑b

mb

(12vb · vb + ub

), (2.46)

and then taking the co-moving derivative:

dedt

=∑b

mb

(vb ·

dvbdt

+dubdt

).

Now using equations (2.44) and (2.45), we arrive at

dedt

= −∑b

mb

∑c

mc

Pcρbρc

vb +Pbρbρc

vc

· ∇bWbc,

and we can now infer that the change of total energy of particle b can be written

debdt

= −∑c

mc

Pcρbρc

vb +Pbρbρc

vc

· ∇bWbc. (2.47)

2.2.4 Viscosity

As with most aspects of SPH, there exists many different approaches by whichviscosity may be implemented. Again, the choice will be largely determinedby the particular problem in consideration. Many methodologies are gearedtowards evolving astrophysical dynamics, often where high Mach number veloc-ities, and correspondingly shocks, may be found. Viscosity is often then addedonly as a tool; a method to tame numerical artifacts born of shocks and discon-tinuities. Monaghan (1992) gives the following such form for viscosity term

Πab =

−αcabζab + βζ2

ab

ρabvab · rab < 0

0 vab · rab ≥ 0(2.48)

whereζab =

hvab · rabrab · rab + η2

. (2.49)

This viscosity enters into our SPH algorithm via the momentum equation, wherewe write

dvadt

= −∑b

mb (F (Pa, Pb, ρa, ρb) + Πab)∇aWab. (2.50)


Here F is some pressure term, taking various forms as outlined earlier. Forthe viscous term (2.48), we have that ρab and cab represent average density andsound speeds of particles a and b 1, and also that rab = ra−rb. The parametersα and β are usually set to α = 1 and β = 2 for astrophysical simulations,though resulting integrations are not overly sensitive to the precise value used(Monaghan, 1992). The term associated with α results in Navier-Stokes typebulk and shear viscosity, as Taylor expansions will reveal. Further, we have theterm linked to our β parameter which mimics a von Neumann-Richtmyer typeviscosity. It will largely come into effect with high Mach number shocks, helpingto prevent particle penetration. Indeed, this artificial viscosity is constructedexplicitly to handle shocks and is not required elsewhere in the domain, hencethe action of (2.48) whereby the viscosity is set to zero where fluid elements aremoving apart. Also, parameter η is usually taken to be some small fraction ofsmoothing length h, and is included simply to prevent singularities.

For this thesis we are concerned with constructing approximations to Navier-Stokes type dynamics. Mainly considered are fluids of a largely incompressiblenature (see section 2.2.5), where we do not expect shocks to present given lowMach numbers. We as such require a more appropriate formulation of viscositythan that given above, and a number of possibilities are suggested in the liter-ature. The straightforward approach of Flebbe et al. (1994) and Watkins et al.(1996) is to directly compute all viscous terms of the Navier-Stokes equations.Both authors similarly use recursive application of the first order derivativesoutlined in section 2.1.2 to arrive at higher order derivatives (see section 2.1.3).An implementation applied to low Reynolds numbers is given by Morris et al.(1997) using an alternate formulation for the SPH velocity derivatives. Cleary(1998) outlines a viscosity using derivatives found in a similar fashion to term(2.49), and allowing for simulation of fluids of differing densities and viscosities:

Πab = − ξ

ρaρb

4µaµbµa + µb

(vab · rab

rab · rab + η2

). (2.51)

Here we have the dynamic viscosities µa and µb for particles a and b. Thefactor ξ is a type of normalisation constant determined such that our dynamicviscosity µ coincides with that of the Navier-Stokes equations. It will depend onthe dimensionality of the problem, as well as the particular choice of kernel andsmoothing length h. It may be determined either empirically with numericalexperiment, or via Taylor expansions.

The viscosity formulation used for most work presented here is written

Πab = − cabαρab

(vab · rab|rab|

)(2.52)

(Monaghan, 1997), where we modulate viscosity strength using the α parameter.Taylor expansions of (2.52) lead us to a relation between the α parameter anda kinematic viscosity:

α =ν

κcabh, (2.53)

with κ being similar to the ξ of Cleary’s viscosity. It’s value will also be de-termined by the choice of kernel and dimensionality. For example, in one-dimension where the cubic spline kernel is used, we have κ = 14/15, while in

1For these averaged values, we have utilised the harmonic mean.


two-dimensions the value κ = 15/224 is appropriate. It is also worth notingthe effect of the density term ρab in (2.52). We find that our viscosity in theone-dimensional continuum limit leads to terms of the form

A1ρa

∂

∂x

(ρ∂v

∂x

)a

,

for some constant A. Now replacing ρab with ρb, we instead find

A

(∂2v

∂x2

)a

.

So somewhat subtle changes to our viscosity can lead to large changes in dy-namics. The difference between the two above terms will be minimal for nearlyincompressible flows, where density will be almost constant. Where large den-sity gradients exist, significant departure is found. This is demonstrated insection 4.1.1 where Burgers’ equation is considered.

Averaged terms such as ρab are often chosen to maintain the symmetry fromwhich conservation stems. However at times compromise is required and sac-rifice of conservation may be preferable to achieve other goals. For instance,Watkins et al. (1996) required a precise specification of the bulk and shear vis-cosities, but this comes at the cost of angular momentum conservation. Likewise,Morris et al. (1997) also sacrifice exact angular momentum for an implementa-tion they state performs better for low Reynolds numbers. The viscosity usedhere, (2.52) conserves both linear and angular momentum, while also havingthe additional desirable attributes of being Galilean invariant, and vanishingfor solid body rotation.

The addition of viscous dynamics necessitates modification of the thermalenergy equation (2.45) to account for energy being leached from kinetic to ther-mal. To determine our modified thermal energy equation, we note that givenconservation of energy, we can write

dEkdt

= −dETdt

,

where EK is a total kinetic energy, while ET is total thermal energy. We nowwrite

dEkdt

=ddt

(∑a

12mava · va

)=

∑a

mava ·dvadt

= −∑a

mava ·∑b

mb

(Pa + Pbρaρb

+ Πab

)∇aWab

= −∑a

ma

Paρa

∑b

mb

ρbvab · ∇aWab +

12

∑b

mbΠabvab · ∇aWab

,

and so we concludeduadt

=Paρa

∑b

mb

ρbvab · ∇aWab +

12

∑b

mbΠabvab · ∇aWab. (2.54)


This is equivalent to equation (2.45) with the addition of a term due to viscosity,which can be shown (Monaghan, 1997) always acts to increase thermal energy.

2.2.5 Equation of state

The SPH method natively leads to approximations of the compressible Navier-Stokes equations. As such, we need to complement the equations outlined abovewith a further equation which will relate our state variables of pressure anddensity. The equation of state may take a number of forms which depend onthe physics of the medium we are modelling.

Most of the work undertaken in this thesis considers almost incompressiblefluid regimes where thermal effects are unimportant, so we consider pressure afunction of density only. In nature, we may find such regimes in fluids such aswater, and quiet often the gaseous dynamics of the earth’s atmosphere. Mostof the literature on fluid dynamics makes an approximation to these systems,considering them to be fully incompressible and thus allowing simplification ofthe Navier-Stokes equations. We are required to make a similar approximation,though we instead construct models which are more compressible than the actualsystem.

Fluids such as water have very large sound speeds; this results in very shorttimesteps (see section 2.3.1), and expensive computations. We hence makeapproximations to these fluids using smaller sounds speeds. While this allowsfor numerically tractable simulations, it leads us to the forementioned increasedcompressibility. We may modulate sound speed through the equation of state.One common choice is the equation of state given in Batchelor (1967):

P = B

(ρ

ρ0

)γ− 1

(2.55)

where we normally take γ = 7, which results in large variations in pressure forsmall density changes. We also have the reference density ρ0, and the factor Bwhich will be used to determine sound speed. It can be shown that for soundspeed cs, we have

c2s =∂P

∂ρ=γB

ρ0

(ρ

ρ0

)γ−1

, (2.56)

so that the unperturbed fluid takes the sound speed c2s = γB/ρ0. A scaleanalysis of the Euler equation (2.35) indicates the following relation:

V 2max

c2s=δρ

ρ,

for some characterising maximum velocity Vmax of our fluid. We wish to restrictdensity fluctuations to mimic incompressible regimes. We set our sound speedto cs = 10Vmax, resulting in density fluctuations of order one percent. The pa-rameter B is then determined according to equation (2.56). Where we considerflows with fluids of different reference densities, it may be desirable for them tohave equivalent sound speeds. We therefore write equation (2.55) as

P = ρ0B

(ρ

ρ0

)γ− 1. (2.57)


2.2.6 Integrals of motion

The derivation given above maintains various symmetries found in the physicalsystem. This leads to conservation of important quantities such as momentum,which therefore imposes constraints on our modelled dynamics which reflectthose of the physical system.

Conservation of linear momentum follows from homogeneity of space, wherebyparallel translations leave closed mechanical systems unchanged (Landau andLifshitz, 1976). It follows that for such homogeneity, we require that the La-grangian be translation invariant. This can be easily shown by taking a newco-ordinate r′ = r + δr, for some constant deviation δr. Clearly we havev′ = dr′/dt = dr/dt = v, so velocity is unchanged by this transformation.Likewise for density

ρa =∑b

mbW (ra − rb)

=∑b

mbW ((r′a − δr)− (r′b − δr))

=∑b

mbW (r′a − r′b) = ρa′ .

Our kernel is a function of the magnitude of it’s argument, which is invariantto translations δr, so we have |ra − rb| = |r′a − r′b|, and we now have thatfor Lagrangian (2.38), L = L′ (assuming constant entropy), and so we havetranslation invariance. As such, if our equations of motion are variationallyconsistent, then they must conserve momentum. That this is the case can beverified by consideration of the momentum equation (either (2.44) or (2.41)).We first note that we can write

∇aWab = rabFab

where we assume that W = W (|rab|) and that therefore F = F (|rab|). Thisis an assumption made to reach momentum equations (2.44) or (2.41) via thevariational framework, so it’s use here is warranted. Now simply noting that

ddt

(∑a

mava) = (∑a

madvadt

)

= −∑a

∑b

mamb

(Pa + Pbρaρb

)rabFab

= 0

where momentum equation (2.44) has been used and we have exploited symme-try. It can also be seen that we have particle pairs applying equal and oppositeforces upon each other, from which momentum conservation follows.

Similarly, conservation of angular momentum follows from isotropy of space,which is realised through the Lagrangian’s invariance to rotations. This in-variance is again trivial. Density invariance follows by the same arguments asabove, and the square of the velocity will clearly not be effected by rotation, soit follows that the Lagrangian is not changed. Now considering the momentum


equation, as above, we find:

ddt

(∑a

mara × va) = (∑a

mara ×dvadt

)

= −∑a

∑b

mamb

(Pa + Pbρaρb

)ra × rabFab

= −∑a

∑b

mamb

(Pa + Pbρaρb

)ra × rb Fab

= 0

where we again exploit symmetries in the summations. While we have notincluded viscosity in the above calculations, where viscosity takes a symmetricform such as (2.52), we expect momentum conservation will not be violated.

A further integral of motion is that of total energy, as given by equation(2.46). Homogeneity in time is the symmetry from which this stems. So givena time independent Lagrangian, such as will be used in this thesis, total energyconservation follows.

2.3 Implementation

2.3.1 Timestepping

The discritisation of spatial derivatives via SPH leaves us with a set of ordinarydifferential equations for which we seek solution. Where we use a summationdensity, they can be represented as

dradt

= va (2.58a)

dvadt

= ga(r1, . . . , rn,v1, . . . ,vn), (2.58b)

for some vector function ga, with the subscript denoting evaluated at coordinatera. Where viscous dissipation is not included, our formulation can be derivedin a variationally consistent manner (as in section 2.2.2). It is then desirable touse integrators which reflect the symmetries of the Hamiltonian system. We usethe second-order Verlet integration scheme, which for system (2.58) is written

r1/2a = r0

a + 12∆tv0

a (2.59a)

v1a = v0

a + ∆t g1/2a (2.59b)

r1a = r1/2

a + 12∆tv1

a, (2.59c)

for fixed timestep ∆t. Superscripts here determine relative time coordinate ofparticles. This iteration scheme is part of a class of schemes known as geometricintegrators which are designed to reproduce various geometric traits of the truesystem. The Verlet scheme reproduces a number of key features of the Hamil-tonian system, such as reversibility and symplecticity, which leads to excellentenergy conservation qualities, as well as conservation of angular momentum(McLachlan and Quispel, 1999; Leimkuhler et al., 1996). Very long time inte-grations are possible using symplectic methods, where standard methods maynot maintain energy conservation to sufficient degree.

2.3 Implementation 24

For the timestep ∆t, we simply use a Courant condition, which we may write

∆t = 0.5 min(

h

vsig

)(2.60)

for smoothing length h and signal speed vsig, which is usually taken to be

vsig = ca + cb (2.61)

for interacting particles a and b. However, where symplecticity is to be pre-served, we require that our timestep remain constant for the entire integration.This requirement stems from artifacts that are introduced at the intermediatesteps of equations (2.59). Where the timestep is constant, these artifacts arecancelled at proceeding steps, but for variable timestep size, symmetry is lostand errors do not cancel (Leimkuhler et al., 1996). It is still possible to con-struct reversible integrators for variable timesteps (Monaghan, 2005). However,though reversible methods retain some of the desirable features of geometricintegrators, energy conservation suffers and can be expected to drift from itsinitial value (McLachlan and Perlmutter, 2004).

Where density is evolved via continuity equation (2.31), instead of (2.58) wehave

dradt

= va (2.62a)

dρadt

= ka(r1, . . . , rn,v1, . . . ,vn) (2.62b)

dvadt

= ga(r1, . . . , rn,v1, . . . ,vn). (2.62c)

Here the function ka is evaluated at the coordinates of particle a, and theequivalent Verlet scheme to (2.59) is

r1/2a = r0

a + 12∆tv0

a (2.63a)

ρ1/2a = ρ0

a + 12∆t k0

a (2.63b)

v1a = v0

a + ∆t g1/2a (2.63c)

r1a = r1/2

a + 12∆tv1

a (2.63d)

ρ1a = ρ1/2

a + 12∆t k1

a. (2.63e)

The above maintains all the desirable attributes of the equivalent for the summa-tion density. Where continuity equation (2.31) is used, the function ka becomesdependent on particle densities. The final density step (2.63e) is now implicitand iteration is required.

We note that equations (2.59) also become implicit where viscosity is in-cluded, as the force function ga will require knowledge of v1/2 for the velocitystep. In practise, convergence may be found with a small number of point iter-ations, at least for simulations encountered here. However, as our system is nolonger Hamiltonian, there is perhaps little reason to use the Verlet integrator,and other choices such as a modified Euler may be preferable.


2.3.2 Variable Resolution Implementations

Where particles remain largely equispaced throughout integration, such as forincompressible fluid flows simulations, constant-h techniques generally performadequately. However where we expect a large variation in particle number den-sities over short scales, constant-h implementations suffer on a number of fronts.

Possibly the foremost limitation of constant-h SPH is that of resolution.By construction, the SPH approximation will only resolve to length scales oforder h. That this is the case can be demonstrated with a simple example. Weconsider the approximation integral (2.1), which effectively takes some averageof the subject function over a subdomain (determined by the kernel’s basis).For simplicity, we choose our kernel to be the top-hat function, which we definein one dimension as

WTH(|x− x′|, h) =1

2h

1 0 ≤ |x− x′| ≤ h0 h < |x− x′| , (2.64)

and consider the function A(x) = sin(kx), for which the approximation integralnow gives

sin(kx) ' sin(kx) =∫R

sin(kx′)WTH(x− x′, h) dx′

=∫ x+h

x−h

12h

sin(kx′) dx′

= sin(kx)[

sin(kh)kh

]= sin(kx)F (kh). (2.65)

We see that the approximation integral reproduces our subject function sin(kx)along with a factor F (kh). The factor F depends on kh = 2πh/λ, for wavelengthλ, and attenuates the function sin(kx) with strength depending on the ratioof smoothing length to wavelength. We see that in the limit h/λ → 0, wehave F → 1, and so (2.65) exactly recovers sin(kr), as we expect. For thevalues h = λ/5, λ/10, λ/100 we have respectfully F ' 0.76, 0.94, 0.99. So forreasonable accuracy, we perhaps require a smoothing length one tenth the sizeof the shortest length scale to be reproduced. Of course here we have useda top-hat kernel which we expect leads to greater smoothing than a Gaussianbased kernel.

So we wish to reduce h and thereby improve reproduction of short lengthscale dynamics. The next question is perhaps how far we can reduce h? Toanswer this we must remember that the approximation integral (2.1) is discre-tised in the SPH technique, and so we must retain sufficient nodes (ie. particles)to reproduce it accurately. This brings us to another deficiency of constant-hSPH, which may be encountered for example in simulations involving expand-ing gasses. In this situation, we may use insufficient particles to accuratelyapproximate (2.1). Monaghan (2005) shows that for equispaced particles ap-proximating linear functions, errors are small where h > ∆, for particle spacing∆. Using variable-h SPH, we may therefore ensure that our smoothing lengthis sufficiently large such that the h > ∆ requirement is met.

At the other end of the spectrum we may have that h ∆. With regardto computational cost, we have the undesirable situation where a large numberof particles fall under a kernel’s umbrella. This leads to inefficient calculations


tending towards operation counts of N2 for an SPH population of N particles.Our scheme may now also suffer loss of accuracy. Firstly, we have intrinsic reso-lution limitation imposed by the approximate function (2.1) as outlined above.This leads to our second and more critical point of failure. Given that the ap-proximation function now smoothes all scales below h, we may fail to capturethe true density profile appropriate to certain ‘dense’ particle configurations.However, our calculations rely on density to determine our numerical quadra-ture, and we now have the situation where our volume element ∆Vb may notcorrespond to mb/ρb. SPH relies on this quadrature to accurately approximatethe integral (2.1). So we now make the requirement

ma

ρa= ∆Va = (∆ra)ν (2.66)

where ∆r gives an indication of typical particle separations at particle a, and νis the dimensionality. For ha = σ∆ra we then have

ha = σ

(ma

ρa

) 1ν

. (2.67)

The value σ will depend on the form of the kernel used. For the cubic spline,1.2 < σ < 2 is usually appropriate. We note that the above leads to an implicitform for summation density (2.28). We restate it here:

ρa =∑b

mbW (ra − rb, ha) (2.68)

=∑b

mbW (ra − rb, σ(ma/ρa)1ν ) (2.69)

This represents a self-consistent formulation for ρa which only relies on param-eter σ for which it is not overly sensitive (as long as σ is chosen within anappropriate range). The above density equation is highly non-linear, the detailsof which depend on the form of kernel used. In practise, we solve equations(2.67) and (2.68) iteratively until some convergence criterion is met. This ap-proach yields values for the density and smoothing length which are consistent.A more computationally efficient method is to simply use density values fromthe previous timestep, thus circumventing the requirement for iteration, thoughsacrificing precise consistency. Bonet and Rodriguez-Paz (2005) report that in-ferior results ensue, and additional costs of iterative approach may possibly beoffset with lower particle numbers. Another possibility is to evolve the smooth-ing length in time

dhadt

= − haνρa

dρadt

though this will also not guarantee consistency of smoothing length and density.To maintain symmetry and the various conserved qualities that follow, its

then desirable to write our kernels as either an average

W (rab, ha, hb) =12(W (rab, ha) +W (rab, hb)

)(2.70)

or to take averages of smoothing lengths

W (rab, ha, hb) = W(rab,

12 (ha + hb)

). (2.71)


We recall our variational derivation of the momentum equation, and in par-ticular the transition from a continuum Lagrangian (2.37) to the discrete form(2.38). The accuracy of this discretisation will depend on our specification ofdensity, or rather the accuracy thereof. The most accurate specification for den-sity can be expected to follow from the implicit form (2.69), though our use ofthis density in the Lagrangian (2.38) leads to an alternate momentum equation.We follow the derivation of Monaghan (2002) starting from (2.40), which wereproduce here:

madviadt

+∑b

mb

(∂ub∂ρb

)s

∂ρb∂ria

= 0. (2.72)

Our density derivative now becomes

Ωb∂ρb∂ria

= ma∂Wab(hb)∂ria

+ δab∑c

mc∂Wac(ha)

∂ria, (2.73)

and we have

Ωb = 1− ∂hb∂ρb

∑c

mc∂Wbc(hb)∂hb

(2.74)

where we use equation (2.67) to determine the smoothing length derivative.Putting all this together, we arrive at our momentum equation for variable-hSPH:

dvadt

= −∑b

mb

(Pa

Ωaρ2a

∇aWab(ha) +Pb

Ωbρ2b

∇aWab(hb)). (2.75)

As for the constant-h version, the above conserves angular and linear momen-tum, and total energy. An equivalent formulation has been given by Springeland Hernquist (2002), though derived in a slightly different manner. The earliermomentum equation (2.41) is recovered for constant smoothing length, as wewould expect. It should be noted that additional terms now present in momen-tum equation (2.75) result from a new density specification. Such terms areoften dubbed the ∇h terms, and while not incorrect, this view is misleading inthat it implies an intrinsic reliance of dynamics on the smoothing length h. Amore natural perspective is found by forgetting smoothing length, noting thatall kernels may instead be written as functions of density:

W (ra − rb, f(ρa), f(ρb)).

The new algorithm still only requires a single numerical parameter (the relatedparameter σ). Where particle distributions are anisotropic, problems may beencountered due to a lack of interacting particles in certain directions. In thiscase, perhaps radial kernels are not ideal and isotropic kernels may be a betterchoice (see section 2.1.4). Another option may be to use remeshing strategies.

2.3.3 Neighbouring particle list

Where global kernels, such as the Gaussian, are used, summations involve con-tributions from all particles (though small for distant particles). As such, com-puations scale as N2 for a SPH population of N particles; highly expensivecomputations follow. This motivates choice of kernels with compact support,


which restrict the number of particle-particle interactions. Though this leadsto substantial savings, there are some minor extra costs incurred. Namedly, wemust construct list of particles such that our algorithms know which particlesare required for summations.

For constant smoothing length (ie constant kernel support size), the proce-dure is straightforward. For an interaction radius Ri (for the cubic spline, wehave Ri = 2h), we divide the domain into cells of width Ri. For each cell, wethen create a list of particles contained within that cell. For two-dimensionalsimulations, summations for any said particle in cell Ci,j , then only considerparticles within the same cell (Ci,j), as well as the eight cells surrounding cells(Ci−1,j+1, Ci,j+1, Ci+1,j+1, Ci−1,j , Ci+1,j , Ci−1,j−1, Ci,j−1, Ci+1,j−1). Particle-particle interaction symmetries are also exploited to improve efficiency.

We may use this method for variable smoothing length implementations,where the largest particle interaction radius must be used to construct cells.However efficiency declines as our variation in h increases, and some other neigh-bour finding algorithm such as rank listing may be preferable.

2.3.4 Boundaries

For fluid dynamics problems, we usually require some form of boundary condi-tion. Boundaries may serve the simple purpose of confining the fluid within aprescribed domain, or perhaps may form a more integral part of dynamics, suchas flow past a cylinder. Various types of boundary conditions may be required,such as free slip, no slip, periodic or open. Such conditions may be included inSPH computations through a number of methods.

Perhaps the most straightforward implementation for impenetrable bound-ary conditions is to simply use fixed SPH fluid particles. We calculate densityfor these boundary particles in the same way as for fluid particles, with respec-tive pressure forces yielding the desired condition. For a free slip requirement,we simply exclude boundary particles from viscosity calculations, else we havea no-slip boundary. There are a number of advantages to using this approach.As symmetry is not altered, we retain conservation of momentum (where forceson boundary are not disregarded). Initial settling periods are also not required(on account of boundaries), as boundaries apply equivalent forces to fluid par-ticles. The disadvantages to this approach are as follows. Firstly, additionalcosts are incurred as we require numerous boundary particle layers to preventparticle penetration. The number of layers will depend on the kernel support,with boundary particles being required to extend beyond fluid particle interac-tion radii. Secondly, such boundaries present non-constant constituitive forces,as felt by particles moving along the boundary. This may cause difficulty insimulations which are sensitive to such noise, such as Poiseuille flow. Boundaryforce noise issues are increasingly problematic for insufficiently resolved curvedboundaries.

An alternate boundary methodology instead gives boundary particles someexplicitly defined repulsive force to prevent particle penetration. This force actsalong the line of centre for any fluid/boundary particle interaction. A typicalchoice is the Lennard-Jones force, though many possibilities exist (Monaghan,1994). The noise issues discussed above may now be addressed by interpolat-ing boundary forces (Monaghan et al., 2003), and in a similar fashion we mayconstruct smooth curved boundaries (Monaghan, 1995). We only require one


line of particles for boundaries, though we must now use the time evolutionforms for continuity, as summations will lead to errors near the boundaries.Bonet and Rodriguez-Paz (2005) propose a renormalisation correction for sum-mations near boundaries, which they also include in their variationally derivedmomentum equation.

Another option is the use of mirror particles, where particles are reflectedthrough the boundary plane using temporary ghost particles. This yields a lownoise boundary condition, though is only really applicable for flat boundaries.

We may alternatively wish to implement periodic boundary conditions. Forour simulations, the linked list data structures fascilitate periodicity. For ex-ample, particles near the right boundary, ie those in the rightmost linked listcell, directly interact with particles in the leftmost linked list cell, though therighthand particles percieve a transposed version of the lefthand particles. Anequivalent (if slightly more costly) situation sees the lefthand particles copiedtemporarilty beyond the right boundary. We may also implement other sym-metries in a similar fashion, such as point symmetry about the origin.

2.4 Summary

An outline of the SPH numerical method has been given. Initial attention wasdirected towards analytic considerations of the SPH approximation. In section2.1.1 we found that where symmetric kernels are used, the smoothing functionyields O(h2) approximations, and for equispaced data, the discrete smoothingfunction also tends towards O(h2) accuracy. Next the first and second deriva-tives were considered (sections 2.1.2 and 2.1.3). It was shown for the firstderivative that improved accuracy is found where we write summations in sym-metric form, while second derivatives are found via an integral approximation.Kernel requirements were discussed in section 2.1.4.

Next the application of SPH to Navier-Stokes equations were considered.Various forms of the continuity equation were given, their appropriateness tovarious simulations discussed (section 2.2.1). The various ensuing forms for mo-mentum were determined via a variational framework (section 2.2.2), along withcorresponding energy terms (section 2.2.3). Viscosity formulations are given insection 2.2.4. The equation of state, and coefficients required to determine soundspeed and compressibility, are discussed in section 2.2.5. We finally establishthe conservation properties for our governing discrete equations (section 2.2.6).

Issues related to computational time integration of forementioned equationsare then discussed. Various qualities of the Verlet timestepping algorithm arediscussed, along with the Courant timestep requirement (section 2.3.1). Section2.3.4 deals with the different implementations for boundary conditions. A self-consistent variable resolution formulation is discussed in section 2.3.2, alongwith the shortcomings of fixed smoothing length algorithms.

Chapter 3

The α-SPH turbulencemodel

Turbulent flow regimes are characterised by irregular fluid motion acting overa very broad spectrum of length scales in a seemingly chaotic nature. Themathematical realisation of these flows contain many degrees of freedom, whichcomputationally translate to expensive simulations.

We may characterise our flow by a Reynolds number, which we define asRe = UL/ν, for length and velocity scales L and U and kinematic viscosityν. Typically, turbulent regimes occur for medium to high Reynolds numbers(Re ≥ 103). Such systems will tend to exhibit energy cascade phenomona,where we have large eddies breaking into smaller eddies, which themselves fol-low on to seed even smaller eddies, and so forth. Cascades also act in theopposite direction, with small eddies combining to spawn larger eddies, as isthe predominant behavior in two-dimensional turbulence. This phenomena is adefining characteristic of turbulent flows, and continues down to a scale whereviscosity becomes dominant. This length scale is known as the Kolmogorovscale ηK , and it can be shown that in three dimensions it varies with Reynoldsnumbers as

ηKL

= Re−3/4 (3.1)

(Kolmogorov, 1941). For a three-dimensional Navier-Stokes flow, the numberof degrees of freedom then scales as Re9/4 and so becomes large for turbulentReynolds numbers. Indeed doubling the Reynolds number will result in fivetimes the number of degrees of freedom, and for a spectral code will increasethe computational work by an order of magnitude. Techniques such as directnumerical simulation (DNS), where all energy containing modes are resolved, re-main completely intractable using today’s processing technology. Furthermore,at the rate at which computational power is increasing, a full DNS of mostturbulent flows will remain beyond reach for the foreseeable future.

Generally, numerical turbulence methodologies use a model to account forlength scales beyond resolution limit, while directly simulating bulk dynamics.One such method, known as Reynolds averaged Navier-Stokes (RANS), involvesaveraging out fluctuation from our variables, with the flow split into a mean andfluctuating component. Differential equations for the mean components are thenderived from the Navier-Stokes equations, and a model is used to account for

30

31 The α-SPH turbulence model

the effect of fluctuating components on the mean. Given that all fluctuationis removed from direct computation, the outcome relies heavily on the modelused. Experiments are generally required to calibrate the subgrid model, andfinal results are accepted as approximations. A method which uses modelingless aggressively is large eddy simulation (LES). Here the flow is resolved downto computationally realistic length scales, and a model used to account forscales beyond this point. To this end, velocity is filtered to remove fluctuationsbelow the required length scale, with the required differential equations beingdetermined through filtration of the Navier-Stokes equations. As with RANS,the filtered equations posses a dependence on subgrid terms for which a modelis required. There are many different models used for the subgrid dependence,such as the Smargorinsky model and the scale similarity model.

An alternate method involves averaging at the variational principle level.Fluid elements are moved with a local filtered velocity, and a new Lagrangian isdefined from which the so called Lagrangian-averaged Navier-Stokes equations(LANS) are obtained. This derivation naturally yields subgrid stress terms, andthe resulting modified acceleration equation ensures that circulation is conservedfor non viscous flows. Furthermore, a quadratic invariant related to the kineticenergy is conserved.

Analysis due to Monaghan (2002) led to an SPH equivalent of LANS knownas α-SPH, where particles are moved with a filtered velocity under action of amodified acceleration equation. Originally, a version of SPH known as XSPHwas devised to help prevent particle interpenetration (Monaghan, 1999). In sim-ilarity to α-SPH, XSPH moved particles with a local averaged velocity, thoughenergy conservation was compromised. Conservation is recovered by derivingour SPH equations from a discrete averaged Lagrangian, yielding the α-SPHformulation. The derivation of α-SPH is given here, along with discussion ofit’s conserved quantities and time integration.

3.1 LANS-α

A relatively recent technique devised to simulate turbulent fluid dynamics isthe Lagrangian Averaged Navier-Stokes alpha (LANS-α) method. The key dis-tinguishing feature of this method is that an averaging is performed at theLagrangian level, and following this averaging we derive the required equations.This contrasts methods such as large eddy simulation, and Reynolds averageNavier-Stokes, where an averaging (or filtering) procedure is performed on theactual equations of motion.

LANS-α originates from the one-dimensional Camassa-Holm equations (Ca-massa and Holm, 1993), which deals with non-linear shallow water wave dynam-ics. Generalisation of the Camassa-Holm equations to three dimensions yieldsthe Lagrangian averaged Euler alpha equations (LAE-α) (Holm et al., 1998a,b).These equations provide a closed realisation of Euler dynamics, and we beginto see hints of the methods potential for modeling turbulent fluid regimes. Theinclusion of viscous dynamics by Chen et al. (1998,1999a) brings us to the fullLANS-α scheme, with application to turbulence given impetus by favourableresults for turbulent pipe flow. Here analytic solutions for steady mean velocityprofiles reproduced those found in experiments at moderate to high Reynoldsnumbers. Direct numerical simulations of LANS-α were performed for three-

3.1 LANS-α 32

dimensional isotropic turbulence in Chen et al. (1999b). Key results indicatedthat large scale Navier-Stokes dynamics were reproducible using LANS-α, whilethe latter only requires a reduced bandwidth for complete spectral representa-tion.

The LANS-α equations can be written (Holm, 1999)

∂v

∂t+ u · ∇v +∇uT · v +∇P = ν∆v (3.2)

withv = (1− α2∆)u (3.3)

and∇ · u = 0. (3.4)

We note that two different velocities are required, v and u. They are relatedby equation (3.3). The parameter α gives a length scale which will be discussedshortly.

The vector u represents the velocity with which the fluid is transported. Itis determined via inversion of equation (3.3), and as such takes values whichare weighted averages of velocity v, with averaging performed over the lengthscale α. Velocity u is therefore a smoother realisation of velocity v. Where theabove equations are derived of an averaged Lagrangian, the velocity v can beinterpreted as the momentum per unit mass of the Lagrangian averaged motion(Holm, 1999).

A new length scale α is introduced to dynamics in the LANS-α system.Technically α is defined as the distance of typical deviations of a Lagrangian pathfrom its time-averaged trajectory (Holm et al., 2005). It may also be interpretedas the scale below which fluctuations are passively advected by the flow, anddo not participate in cascades, nor affect their own advection. Typically it isdesirable to make maximum utilisation of numerical resources, and so α maythen be chosen to be some fixed multiple of the numerical resolution length.In this way we maximise the bandwidth of Navier-Stokes like dynamics, i.e.,those which occur at lengths scales above α. Care must be taken however toensure that the LANS-α dynamics are fully resolved, or that numerical schemesbehaves ‘agreeably’ for marginally resolved dynamics.

Comparative studies of LANS-α with other closure methods such as LEShave been performed in Domaradzki and Holm (2001) and Geurts and Holm(2002), and it is found that α modeling techniques generally perform at leastas well the best LES methods.. While similarities exist between such methods,LANS-α methods modify the nonlinear advection of the Navier-Stokes system,where LES methods introduce energy diffusion or dispersion. Indeed LES canoften be overly dissipative when compared to LANS-α techniques which conservecirculation (for inviscid flow regions) and as such are able to produce greatervariability. It should be mentioned that this increased degree of small scaledynamics may not always be desirable however, with Geurts and Holm (2002)reporting overly large growth rates, and instability.

Of course the purpose of all subgrid schemes is to reduce the resolutionrequirements of turbulence calculations, and savings naturally depend on howaggressively modeling is used. For LANS-α, dimensional arguments indicatethat energy scales as k−3 for length scales below α (Foias et al., 2001; Monaghan,2002). It follows that resolution requirements for fully resolved LANS-α scale as


Re2, which presents a considerable saving compared with Re3 requirements forfull Navier-Stokes dynamics, though overheads involved in calculating additionalquantities must also be considered.

Further information on the LANS-α model can be found in Foias et al. (2001)and references therein.

3.2 α-SPH: equations of motion

We seek an SPH equivalent to the continuum LANS-α equations. The derivationgiven in Monaghan (2002) is outlined. We start with the discrete Lagrangian

L =∑b

mb

[12vb · vb − u(ρb, sb)

], (3.5)

where in addition to the previously defined quantities, we have a new velocityvb. This velocity represents an averaged or smoothed realisation of the velocityvb. This is the velocity with which particles are advected. We consider it’sparticulars now.

3.2.1 The filtered velocity

The precise specification of vb may take a number of forms. For the calculationsfound herein, we use an implicit definition given by

va = va + ε∑b

mb

ρab(vb − va)W ab, (3.6)

where we have the parameter ε which modulates the degree of smoothing. Thekernel W ab is an averaged kernel, as given in equation (2.70). Choice of kernelis an open question, though the cubic spline has been used here (see 2.1.4) forsimplicity. Furthermore, the basis size of our kernel (with respect to particlespacing) will affect the filtering qualities of (3.6). An alternate specification forthe filtered velocity is

va = va + ε∑b

mb

ρab(vb − va)W ab. (3.7)

This is the form used for XSPH implementations. The implicit version, equation(3.6), is preferred for our current simulations for a number of reasons, which wewill come to shortly. To gain further understanding of the behavior of equation(3.6), we considering the limiting situation as smoothing length goes to zero.Taylor expansions reveal

vj = vj +A2

ρ∇ ·(ρ∇vj

)+

12(∇ρ · ∇vj

) ∂h∂ρ

∂A2

∂h(3.8)

withA2 =

ε

2

∫(x′ − x)2W (r′ − r, h(r)) dr′. (3.9)

Where we assume variations in density (and therefore smoothing length) can beneglected, equation (3.8) reduces to

vj = vj +A2∇2vj , (3.10)

3.2 α-SPH: equations of motion 34

which may be compared with the LANS-α equivalent (3.3). In the above, super-script j denotes dimensional component. We find that our smoothed velocity isequivalent to that of LANS-α under constant density conditions, where we nowwrite

α2 = A2 =ε

2

∫(x′ − x)2W (r′ − r, h(r)) dr′. (3.11)

In one dimension, for the cubic spline kernel with smoothing length h, we have

α2 = 16h

2ε. (3.12)

So, at least for the limiting constant density one-dimensional situation, we canalter ε or smoothing length to modulate α, and therefore the degree of smooth-ing. The effect of doubling smoothing length will be identical to that of qua-drupling ε.

The action of (3.10) in filtering components of v is best understood in spec-tral space. For simplicity, we restrict dynamics to one dimension, and expandour velocity fields in trigonometric functions:

v =∑m

vm exp(ikmx) (3.13)

v =∑m

¯vm exp(ikmx), (3.14)

for imaginary number i and wavenumber km = 2πm/L. Inserting (3.13) and(3.14) into equation (3.10) brings us to

¯vm =1

1 + (kmα)2vm = Fi(km, α) vm (3.15)

The factor Fi(km, α) acts to attenuate mode amplitudes as we push into higherwavenumbers. Using (3.12), and setting g = h/λ for wavelength λ, we recastFi:

Fi =1

1 + (kmα)2=

11 + ε(2πg)2

. (3.16)

This attenuation factor is displayed in Figure 3.1 for the values of ε = 0.1, 0.25,0.5 and 1.0. We see that the smoothing operation acts as a low pass filter, withthe expected result of greater mode damping for larger values of ε.

We turn our attention briefly to the explicit smooth velocity (3.7). For theconstant density situation, we find the following continuum equivalent

vj = vj + α2∇2vj , (3.17)

which is equivalent to (3.10) but with the Laplacian operating on the standardvelocity v. Mode coefficients are determined by the following:

¯vm =(1− (kmα)2

)vm = Fe(km, α) vm. (3.18)

We plot Fe in Figure 3.2. It is found that this form of smoothing may leadto mode phase changes, and more critically in amplification of certain modes.These issues present where values of ε > 0.5 are taken. Application of this formof smoothing is therefore limited, as we may not be able to alter frequencies at


0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

Fi

g = h / λ = hk/2

ε = 0.10ε = 0.25ε = 0.50ε = 1.00

Figure 3.1: Implicit filtered velocity attenuation profile at various values of ε.

the lower part of the spectrum without causing instabilities for high frequen-cies. In practise, these issues are not found to be as severe as Figure 3.2 maysuggest. This is possibly owing to the fact that modes where stability problemsare expected are marginally represent on the SPH grid, and as such may benumerically damped.

Other possibilities for filtered velocity include using higher order Laplacianoperators (or their SPH equivalents), leaving a larger portion of the spectrumunchanged below some cut off frequency, while strongly damping modes beyond.Resulting dynamics have not been investigated.

3.2.2 The momentum equation

Equip with filtered velocity (3.6), we return to the Lagrangian. Our Euler-Lagrange equations for this system read:

ddt

(∂L

∂via

)− ∂L

∂ria= 0, (3.19)

and we note that the advection velocity v is used for our generalised velocitycoordinate. Canonical momentum is considered first:

∂L

∂via=

∂

∂via

∑b

mb

[12vb · vb + u(ρb, sb)

].


-1

-0.5

0

0.5

1

0 0.05 0.1 0.15 0.2 0.25

Fe

g = h / λ = hk/2

ε = 0.10ε = 0.25ε = 0.50ε = 1.00

Figure 3.2: Explicit filtered velocity attenuation profile at various values ofε. Note that large values of ε may cause excitation of high order wavenumbers,leading to instability.

The thermal energy term in the above disappears. Using (3.6) we rewrite thekinetic energy term:

12

∑b

mbvb · vb =12

∑b

mbvb · vb −ε

2

∑b

∑c

mbmc

ρbcvb · (vc − vb)W bc

=12

∑b

mbvb · vb +ε

4

∑b

∑c

mbmc

ρbc(vc − vb)

2W bc. (3.20)

The canonical momentum now becomes

∂L

∂via=

∂

∂via

12

∑b

mbvb · vb

=∂

∂via

12

∑b

mbvb · vb +ε

4

∑b

∑c

mbmc

ρbc(vc − vb)

2W bc

= mav

ia + ε

∑b

mamb

ρab

(vib − via

)W ab

= mavia. (3.21)

So we find that the canonical momenta for the current system reduces to thestandard momentum for the Navier-Stokes regime. We turn to the second term


of equation (3.19):

∂L

∂ria=

∂

∂ria

∑b

mb


]. (3.22)

Thermal energy terms in the above yield pressure terms in the momentumequation, as per standard SPH (see Section 2.2.2):

∂

∂ria

∑b

mbu(ρb, sb)

= −ma

∑b

mb

(Paρ2a

+Pbρ2b

)∇aWab. (3.23)

For clarity and simplicity, we neglect terms which may arise owing to non-constant smoothing lengths. Letting v2

bc = (vc − vb)2, We consider the velocityterms in (3.22):

12∂

∂ria

∑b

mbvb · vb

=∂

∂ria

ε

4

∑b

∑c

mbmc

ρbcv2bcWbc

=ε

4

∑b

∑c

mbmcv2bc

∂

∂ria

1ρbc

Wbc

=ε

4

∑b

∑c

mbmcv2bc

Wbc

∂

∂ria

1ρbc

+

ε

4

∑b

∑c

mbmcv2bc

1ρbc

∂Wbc

∂ria

(3.24)

Now working with the first term:

ε

4

∑b

∑c

mbmcv2bcWbc

∂

∂ria

1ρbc

=ε

4

∑b

∑c

mbmcv2bcWbc

∂

∂ria

12

(1ρb

+1ρc

),


symmetry then allows us to write

=ε

4

∑b

∑c

mbmcv2bcWbc

∂

∂ria

1ρb

= − ε4

∑b

∑c

mbmcv2bcWbc

1ρ2b

∂ρb∂ria

= − ε4

∑b

∑c

mbmcv2bcWbc

1ρ2b

∂

∂ria

∑d

mdWbd

= − ε4

∑b

∑c

mbmcv2bcWbc

1ρ2b

∑d

md∂Wbd

∂ria

= − ε4

∑b

∑c

mbmcv2bcWbc

1ρ2b

∑d

md∂Wbd

∂rib(δab − δad)

= − ε4ma

∑c

∑d

mcmdv2acWac

1ρ2a

∂Wad

∂ria− ε

4ma

∑b

∑c

mbmcv2bcWbc

1ρ2b

∂Wab

∂ria

= − ε4ma

∑d

md1ρ2a

∑c

mcv2acWac

∂Wad

∂ria− ε

4ma

∑b

mb1ρ2b

∑c

mcv2bcWbc

∂Wab

∂ria

= − ε4ma

∑d

mdζaρ2a

∂Wad

∂ria− ε

4ma

∑b

mbζbρ2b

∂Wab

∂ria

= − ε4ma

∑b

mb

(ζaρ2a

+ζbρ2b

)∂Wab

∂ria, (3.25)

where we designateζa =

∑c

mcv2acWac. (3.26)

Now for the second term of (3.24):

ε

4

∑b

∑c

mbmc1ρbc

v2bc

∂Wbc

∂ria

=ε

4

∑b

∑c

mbmc1ρbc

v2bc

∂Wbc

∂rib(δab − δac)

=ε

4

∑c

mamc1ρac

v2ac

∂Wac

∂ria+ε

4

∑b

mamb1ρab

v2ab

∂Wab

∂ria

=ε

2ma

∑b

mb1ρab

v2ab

∂Wab

∂ria(3.27)

We put all this together (equations (3.21), (3.23), (3.25) and (3.27)) to arriveat our α-SPH momentum equation:

dvadt

= −∑b

mb

Paρ2a

+Pbρ2b

− ε

2

(v2ab

ρab− ζa

2ρ2a

− ζb2ρ2b

)∇aWab. (3.28)

We note that the co-moving derivative is defined using the advection velocity v:

ddt

=∂

∂t+ v · ∇. (3.29)


Insight to the behavior of the α-SPH momentum equation is gained by consid-ering it’s continuum limit. Letting particle spacing tend to zero and performingTaylor series expansions, we find (see Appendix B):

dv

dt=−∇Pρ

+α2

ρ

[∇vl

(∇ · (ρ∇vl)

)+ρ

2∇(∇vl · ∇vl

)](3.30)

with α (defined by (3.11)) held constant as smoothing length goes to zero. Su-perscripts define spatial components and summation convention is used. Wherewe assume an incompressible flow field, (3.30) reduces to

∂vi

∂t+ v · ∇vi + vj

∂vj

∂xi+

∂

∂xi

(P

ρ− 1

2vj vj − α2

2∂vj

∂xk∂vj

∂xk

)= 0

together with velocity filter (3.10). This is equivalent to LANS-α momentumequation given in Holm (1999), and so our system can be interpreted as a particlediscretisation of LANS-α.

Where a variable smoothing length has been considered, the required equa-tion is (see Appendix A):

dvadt

= −∑b

mb

[Pa

Ωaρ2a

− ε

4

(v2ab

ρab+

1Ωa

∂ha∂ρa

νa −2ζaΩa

)∇aWab(ha)

+

PbΩbρ2

b

− ε

4

(v2ab

ρab+

1Ωb

∂hb∂ρb

νb −2ζbΩb

)∇aWab(hb)

],

(3.31)

with

νk =∑c

mc

ρkcv2kc

∂Wkc

∂hk(hk)

ζk =∑c

Akcmc

ρ2kc

v2kcW kc

Akl =∂ρkl∂ρk

Ωk = 1− ∂hk∂ρk

∑c

mc∂Wkc

∂hk(hk),

and where smoothing length h may be determined according to (2.67).Viscous dynamics may now be included in the same manner as undertaken

for standard SPH (see Section 2.2.4). We introduce viscosity term Πab (2.52)to our momentum equation, though it now depends on smooth velocity v:

Πab = − cabαρab

(vab · rab|rab|

). (3.32)

The use of the smoothed velocity is desirable to ensure that the addition ofviscosity leads to an increase in thermal energy (see Monaghan (2002) for further

3.3 Integrals of motion 40

details). We may now write our momentum equation

dvadt

= −∑b

mb

[Pa

Ωaρ2a

− ε

4

(v2ab

ρab+

1Ωa

∂ha∂ρa

νa −2ζaΩa

)+

12

Πab

∇aWab(ha)

+

PbΩbρ2

b

− ε

4

(v2ab

ρab+

1Ωb

∂hb∂ρb

νb −2ζbΩb

)+

12

Πab

∇aWab(hb)

].

(3.33)

3.3 Integrals of motion

A key quality of α-SPH/LANS-α modeling methodologies is their conservationof certain integrals of motion. The derivation of these systems through varia-tional methods provides a natural pathway to such conservation, whereas otherclosure techniques usually do not provide robust assurances.

As discussed in Section 2.2.6 for standard SPH, conservation may be de-termined through inspection of the Lagrangian. Conservation of momentumfollows from homogeneity of space, which requires linear translation invarianceof the Lagrangian (3.5). The invariance of thermal energy term u(ρ, s) has beenestablished for standard SPH in Section 2.2.6 and is unchanged for the currentsystem. Where we take canonical variables r and v, we now have a kineticenergy term which is a function of variable r. We consider equation (3.20). Thefirst term is not dependent on r, while the second term is translation invariantowing to the kernel’s invariance (see Section 2.2.6), and so we have establishedmomentum conservation for the current system. The conserved momentum isthe same as for the standard Navier-Stokes system (see equation (3.21)).

Conservation of angular momentum is determined in a similar fashion, fol-lowing from the Lagrangian’s invariance to rotation. We note from equation(3.20) that only velocity magnitudes enter the Lagrangian, and these are un-changed by rotations. Density terms also remain constant for rotations of thecoordinate system, completing the kinetic energy’s invariance, while also assur-ing invariance for the thermal energy. We thus have conservation of the angularmomentum:

ddt

∑b

mbrb × va = 0.

As Lagrangian (3.5) has no explicit dependence on time, we expect to con-serve a final additive integral of motion, that of energy. Manipulation of theEuler-Lagrange equations reveal:

E =∑b

vb ·∂L

∂vb− L

=∑b

mb


]=∑b

mb

[12vb · vb +

ε

4

∑c

mc

ρbc(vc − vb)

2W bc + u(ρb, sb)

],

where we have used (3.20). The SPH summation leads to the following contin-


uum equivalent where density variations have been neglected:∑c

mc

ρbc(vc − vb)

2W bc =

12α∇vib · ∇vib.

Where parameter α is set to zero, we recover conservation of the same energy asthat of the Navier-Stokes system. Furthermore, where high order velocity modesare zero or small, the conserved energy is expected to be close to that of theNavier-Stokes system. So in summary, we have a conserved energy, quadratic invelocities, which constrains the flow in a similar fashion to the Navier-Stokes en-ergy, and only diverges from Navier-Stokes conservation where energy occupiestypically under-resolved velocity modes.

A further quantity conserved for the α-SPH system is a discrete equivalentof circulation. Monaghan (2002) gives an outline of this invariant using a so-called necklace transformation, which is followed here. We imagine a closedpath within the domain defined by a necklace of particles. Now each particle inthe loop is translated to the location of its neighbour (with all particles movingin the same sense), and given it’s neighbour’s velocity v. Where we assume allparticles have identical mass and entropy, we can write:

δL =∑j

(∂L

∂rijδrij +

∂L

∂vijδvij

)= 0,

with subscript j giving consecutive loop particle labels, δrij = rij+1 − rij andδvij = vij+1 − vij . We apply the Euler-Lagrange equation to the above:

0 =∑j

(∂L

∂rijδrij +

∂L

∂vijδvij

)

=∑j

(ddt

(∂L

∂vij

)δrij +

∂L

∂vijδvij

)

=∑j

ddt

(∂L

∂vijδrij

)

=ddt

∑j

∂L

∂vijδrij ,

and using (3.21),

0 =ddt

∑j

vijδrij .

Our turbulence closure therefore does not impinge upon the conservation of fluidcirculation. This owes to the fact that α models in effect enslaves the circulationpresent at short length scales to the dynamics of the larger scales, whereas otherturbulence closures result in a diffusion of short scale circulation (Holm et al.,2005).

3.4 Implementation

A number of implementation issues arise resulting from the above modifica-tions to standard SPH. Firstly we have an acceleration equation which explic-


itly depends on particle velocities, and so scheme (2.59) requires modification.Secondly, advection velocity v is determined as an implicit solution of equa-tion (3.6), and therefore an iterative scheme is required. We investigate thesematters here.

3.4.1 Timestepping

For the current system, the set of ordinary differential equations for which weseek solution are:

dradt

= va (3.34a)

dvadt

= ga(r1, . . . , rn, v1, . . . , vn), (3.34b)

along with

va = fa(r1, . . . , rn,va, v1, . . . , vn) (3.34c)

for functions fa and ga determined respectively by (3.6) and (3.33), with fasolved interactively. The Verlet scheme may now be written

r1/2a = r0

a + 12∆t v0

a (3.35a)

v1/2a = v0

a + 12∆t g0

a (3.35b)

v1/2a = v0

a + 12∆t g1/2

a (3.35c)

v1a = v1/2

a + 12∆t g1/2

a (3.35d)

r1a = r1/2

a + 12∆t v1

a, (3.35e)

For the above system, iteration is required at a number of points. Most sig-nificant is the requirement of iteration at equation (3.35c), which requires thecalculation of summation (3.33), along with solution to (3.6), which itself re-quires iteration. In practise, convergence is usually attained with only a fewiterations, though this will depend on the value of parameter α, with largervalues requiring larger iteration counts. As particles coordinates do not changeduring this iteration, it is possible during the first pass to create lists whichdetermine exact particle interactions. These lists may be used at future iteratesfor improved efficiency.

The construction of a smoothed velocity at the final time point (v1) alsopresents difficulty, as particles are advected to their final position (r1) usingthis velocity, though the final particle positions are required in its construction.So we again require point iteration, now about v1 and r1. In practise, we ofteninstead construct the filtered velocity using the approximation v1 = 2v1/2− v0.This approximation suffices where we have smoothing parameter ε less thanunity. For larger values, it is found that an instability develops in the filteredvelocity field where such approximations are used, and instead we are requiredto perform iteration about the final step.

Nested iterations lead to substantially increased computational cost. Ex-plicit timestepping schemes may be constructed with equivalent order of ac-curacy, though such schemes will not respect geometric properties of our La-grangian system. As such, the additional cost of the Verlet scheme may prove


worthwhile for the maintenance of reversibility, and where timestep is held con-stant, symplecticity. Also, though the variational properties of our system areviolated where viscosity is introduced, use of the geometric scheme may stillbe desirable given that many particle-particle interactions occur with minimalviscous forcing. Such issues warrant further investigation.

3.4.2 Iteration for filtered velocity

We require solution to implicit equation (3.6). The simplest approach is to usepoint iteration until convergence is found, with unfiltered velocity v used as aninitial guess. Faster convergence may be found through use of a Jacobi iterativetype scheme. We write:

va = va + ε∑b

mb

ρab(vb − va)W ab

= va + ε∑b

mb

ρabvbW ab − εva

∑b

mb

ρabW ab

=(va + ε

∑b

mb

ρabvbW ab

)(1 + ε

∑b

mb

ρabW ab

)−1

.

Further improvement may be found by utilising the latest iterated velocity valuesas they are determined in a Gauss-Seidel approach. Again, as particles do notmove within iterations, we may construct particle interaction lists to improvecomputational speed.

3.5 Summary

A new methodology due to Monaghan (2002), the SPH-α model, has beenoutlined. This new scheme is constructed for the calculation of turbulent fluidregimes, and it has been shown that it is analogous to the LANS-α turbulencemodel. Integrals of motion have been discussed, with the equivalence of particleversions established. Issues relating to implementation have also been covered.

Chapter 4

One-Dimensional Tests

We start our investigation of SPH and α-SPH by considering one-dimensionaltest problems for which non-linear advection plays a central role. Restrictingdynamics to one-dimension allows for a thorough exploration of parameter spacewhile still retaining the key physics of higher dimensional systems. Furthermore,a greater degree of analytic tractability is often found, as is the case for Burgers’equation which is consider first. Following this, we turn to simulations of theone-dimensional Euler system, and a forced compressible Navier-Stokes system.

4.1 Burgers’ equation

The difficulties of simulating and comprehending the full Navier-Stokes equa-tions arise due to their complex geometric nature and nonlinearity, which giverise to turbulent energy cascades and shock formation. Burgers’ equation pro-vides a simplified framework for studying such phenomenon. It is the simplestequation which leads to the competition between convection and diffusion. Assuch, Burgers’ equation is often first considered when investigating new modelsfor hydrodynamics. It is most commonly written

∂u

∂t+ u

∂u

∂x= ν

∂2u

∂x2, (4.1)

which first appeared in a paper due to Bateman (1915). Burgers (1948) in-vestigated a recast version of this equation as a toy model to Navier-Stokesturbulence, demonstrating the cascade of energy through different length scalesconsequent of non-linearity, along with the arrest of this cascade due to diffusion.This early work led to equation (4.1) coming to be known as the Burgers’ equa-tion. Analytic solutions to Burgers’ equation are available due to Cole (1951),who derived a transformation which leads to a general solution for known initialand simple boundary conditions. Application has been found to a wide range ofdisciplines, and so it is perhaps no surprise that many distinct and significantsolutions have appeared in the literature. A collation in tabular form of manysuch solutions may be found in Benton and Platzman (1972).

Another aspect of Burgers’ dynamics which make it attractive as a numericaltesting ground is the potential for shock formation. Many natural phenomenaare subject to shocks, and the ability to model them accurately, or at the least

44

45 One-Dimensional Tests

prevent corruption due to their presence, is often of prime concern. A commonapproach is to add a von Neumann-Richtmyer type artificial viscosity to helpprevent the steepening of shock fronts, though such methodologies may proveoverly dissipative, both at the shock and throughout the domain. We investi-gate here an alternate route to regularised shocks, where dispersion is utilised tomobilise energy away from short length-scales, as opposed to removing this en-ergy via viscous diffusion. To this end, averaged Lagrangian techniques providethe required framework.

We consider standard SPH, the α-SPH model, and a spectral method equiv-alent (see Appendix C), with influence alpha modifications impart on the dy-namics being of concern. Two initial conditions are utilised: firstly, that whichresults in two approaching shock fronts which eventually merge; secondly, theclassical sine wave initial condition.

4.1.1 Colliding shocks

In this section we consider the situation where initially two shock fronts movetowards each other, and eventually merging. Here the analytic solution to Burg-ers’ equation is given by

u(x, t) = −2βνsinh(βx)

coshβx+ exp (−β2νt), (4.2)

(Benton and Platzman, 1972), where β is a parameter which determines ourlength scale and ν is a kinematic viscosity. This solution may be found in Figure4.1 for various times. Our velocity scale, as determined by equation (4.2), isU ∼ βν, while we define a length-scale of L ∼ 1/β. Using these definitionsto determine a Reynolds number Re = UL/ν, results in a constant Reynoldsnumber, and so non-linear terms scale with viscous terms. While this limitsthe potential insight to be gained, solution (4.2) still provides a worthwhilepreliminary to more complex dynamics.

Parameters are determined by first selecting a value for SPH viscosity param-eter α, then using (4.4) to determine ν. We fix Umax = 1, from which β follows.Initial time was chosen to give a clear initial separation of the approaching wavefronts. To meet inflow boundary conditions, particles were added as required atthe boundaries (±Lmax) and given initial velocities of u(±Lmax, t) = ∓Umax,Lmax being chosen sufficiently large that u(±Lmax, t) ' u(±∞, t). For the pre-sented integrations we set α = 1, and initially 126 particles have been used tospan the domain, though due to the inflow condition, this grows to 260 particlesby the end of the simulation. Most of this resolution is in effect not utilised,and many particles could be removed if efficiency were a concern.

Acceleration equation (3.31) is used where pressure terms are removed toproduce the required dynamic. We use a version of the SPH viscosity termwritten

Πab = − αρb

(vab · rab|rab|

). (4.3)

with corresponding kinematic viscosity given by

α =715ν

h. (4.4)

4.1 Burgers’ equation 46

vv

-1

-0.5

0

0.5

1

time = -0.500 time = -0.100

-1

-0.5

0

0.5

1

-0.4 -0.2 0 0.2 0.4

time = 0.000

-0.4 -0.2 0 0.2 0.4

time = 0.100

Figure 4.1: Solution where a compressible SPH viscosity term has been used.Density gradients result in increased viscosity at fronts. Solid curve correspondsto analytic solution (4.2).

This yields dissipation terms of the form found in (4.1). Where an averagedensity is used in the denominator of (4.3), we yield viscous terms involvingdensity gradients (see Section 2.2.4). This results in departure from the Burg-ers’ dynamics of equation (4.1), and also therefore solution (4.2). We compareFigures 4.1 and 4.2 where respectively a compressible and incompressible vis-cosity term have been used. The solution flow field represented in Figure 4.2clearly reproduces the analytic solution with greater accuracy. While these aresimply questions of consistency, they are worth noting and often overlookedby SPH practitioners. Subtle change to the SPH equations can yield signifi-cant changes in dynamics. Often a symmetric form for the SPH equations isdeemed the highest priority due to the conservation properties which follow. Asdemonstrated here however, it may be worth sacrificing conservation for strictconsistency. This will naturally depended on the relative significance of thedifferent dynamical terms being modelled.

Returning to Figure 4.2, it is noted that in the final frame, minor discrepan-cies develop at the front crests, with numerical solutions tending to overshootthe correct result, leading to a steeper shock front than required. An expansionon this region is given in Figure 4.3, where we also display profiles resultingfrom variations to our algorithm, which we now discuss.

Thus far we have utilised an SPH implementation for which smoothing lengthh is held constant. We introduce a variable smoothing length together with selfconsistent density evaluation (see Section 2.3.2 for details). Increasing densi-


vv

-1

-0.5

0

0.5

1

time = -0.500 time = -0.100

-1

-0.5

0

0.5

1

-0.4 -0.2 0 0.2 0.4

time = 0.000

-0.4 -0.2 0 0.2 0.4

time = 0.100

Figure 4.2: Solution where incompressible viscosity term (4.3) has been used. Asexpected, numerical solutions correspond with greater accuracy to exact results.

ties, owing to particle accumulation at shock front centres, leads to diminishingsmoothing length, and correspondingly small timesteps. We therefore limit theminimum size for smoothing length by applying the following rule for it’s deter-mination:

ha = σma

β

(β − 1ρa

+ 1), (4.5)

where for the cubic spline we use σ = 1.3, and the parameter β determines aminimum limit for for smoothing length. For results presented, the value β = 10has been taken, which allows the smoothing length to take a minimum valueof one tenth it’s initial setting. We must now also modify our viscosity term,as kinematic viscosity will vary with smoothing length h, according to equation(4.4). Instead of (4.3) we write

Πab = − αρb

(vab · rab|rab|

)h0

ha. (4.6)

In comparing Figures 4.3a and 4.3b we see that the largest differences occursnear the origin. Here the fixed smoothing length routines result in dissipationacting over a larger area than expected, owing to large particle number densi-ties, and corresponding increased contributions to SPH viscosity summations.Furthermore, density summations tend to give underestimates in such regions,and particle contributions may then be overestimated due to poor quadrature.We recognise that smoothing length defines an implicit resolution limit for sim-ulations. Allowing for a variable smoothing length addresses these issues, and


(a) (b)

(c) (d)

(e) (f)

vv,v

v,v

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-0.09 -0.07 -0.05 -0.03 -0.01 0.01 -0.09 -0.07 -0.05 -0.03 -0.01 0.01

Figure 4.3: Comparison of velocity fields for algorithm variations: (a) Stan-dard SPH with a constant smoothing length; (b) variable smoothing length; (c)constant smoothing length and filtered velocity; (d) full α-SPH algorithm againusing constant smoothing length; (e) equivalent to (c) but using incompressiblefiltered velocity (4.7); (f) equivalent to (d) but using momentum (4.8) in additionto filtered velocity (4.7). We have symbol (+) for standard velocity, and (×) forfiltered velocity.

hence Figure 4.3b exhibits reduced spurious diffusion.

We next introduce the filtered velocity to investigate it’s influence on dy-namics, again using the fixed smoothing length algorithm. Firstly the filtered(3.6) is used along with the standard acceleration equation (2.41). A smoothingparameter of ε = 1 is used for these calculations, which results in a frequencycutoff parameter given by α2 = 1

6h2. The two velocity fields v and v are given

in Figure 4.3c. The filtered field v produces the desired result of a smoother ve-locity field. The standard velocity v appears to have suffered less diffusion thatthat of standard SPH (Figure 4.3a), owing to reduced energy in short lengthscales and therefore reduced viscosity (note that the filtered velocity v is usedfor the viscosity calculation). Figure 4.3d shows the outcome of using the fullα-SPH algorithm. Results correspond well with analytic solution, and we notethat velocity v is not pulled down with smooth velocity v, as in Figure 4.3c.This behavior may possibly be attributed to the energy conservation found forthe full scheme.

Filtered velocities have been given by equation (3.6), however such summa-tions lead to dependencies on density gradients, viz equation (3.8). We may


instead construct filtered velocities using the following:

va = va + ε∑b

mb

ρb(vb − va)Wab, (4.7)

which results in a continuum equivalent given by equation (3.10). We also mod-ify our additional acceleration equation terms to suit, writing equation (3.28)as

dvadt

=∑b

mb

ε

2

(v2ab

ρb− ζa

2ρa− ζb

2ρb

)∇aWab. (4.8)

together withζa =

∑c

mc

ρcv2acWac.

It is perhaps not clear which form of filtered velocity and acceleration is prefer-able. On one hand, our Burgers’ equation (4.1) has no dependence on densityand perhaps we should expect our discretisation to follow suit, with density onlyacting as a numerical tool to determine quadrature. However, the compressibleversion is derived of a variational framework, and as such exhibits better sym-metry properties which may be desirable, though we note that the addition ofviscosity perhaps voids any benefits. For the colliding waves problem, there islittle to separate the formulations, as a comparison of Figures 4.3c, 4.3d, 4.3eand 4.3f reveals.

In summary, there is little to separate the methods considered under thisconfiguration, unless we expand upon regions where difficulties tend to present,though the significance of differences is questionable. Largest improvement ispossibly attributed to allowing for a variable smoothing length, though thiscomes at greatly increased cost due reduced timestep. We now consider theclassic Burgers’ realisation of a flow field initiated with a single sinusoidal mode.

4.1.2 The steepening shock front

Our Burgers’ simulation is now initiated using

u0(x) = sin(2πx), (4.9)

which is defined in the periodic domain 0 ≤ x < 1. The parameter whichdetermines behavior of the system is the Reynolds number which we now defineas

Re =UL

ν(4.10)

where U is a velocity scale, L is a length scale and ν is the kinematic viscosity.For the sine wave initial perturbation, we take L ∼ 1 and U ∼ 1. The value thekinematic viscosity coefficient takes will depend on the SPH viscosity coefficientα and the form of the SPH viscosity term used.

For the lower Reynolds number regimes, energy largely resides in largelengths scales, being diffused away before nonlinear advection takes it into theshort scales. Velocity fields are correspondingly smooth, and tend to be repre-sented with greater easy via discrete representations. For larger Reynolds num-bers, viscosity is less able to arrest energy propagation to short length scales.For any discrete method then, there exists a regime where energy is carried


time = 0.1 time = 0.2

time = 0.3 time = 0.4

vv

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9

Figure 4.4: Multivalued solutions resulting from insufficient resolution of viscouslength scales. This simulation utilised a fixed smoothing length with one-hundredparticles, Re = 102.

beyond resolution allowances, and ensuing evolution will depend on the detailsof the discretisation.

While exact analytic solutions exist for the current configuration (Cole,1951), where Reynolds numbers are high (Re > 100), evaluation becomes im-practical due to slow convergence of series solutions. We rely on comparisonwith other numerical results, though there appears to be some inconsistenciesin the literature. For simulations with Re ∼ 105, Zhang et al. (1997) have com-pared simulations with those of Kakuda and Tosaka (1990). They found signifi-cant differences in early time evolution, where velocity fields were still relativelysmooth, though late time velocity profiles appeared to agree well. They havealso compared results with Varoglu and Finn (1980), with good correspondenceat all points in the domain with the exception of near the discontinuity, wherespecific characteristics of the particular numerical method may be of influence.More recently Wei and Gu (2002) have given a method by which they are ableto determine numerical solutions up to the inviscid regime, while Xu and Duan(2001) have also calculated results using Reynolds numbers of up to Re = 105,though some artifacts are apparent. SPH simulations will be compared withthese author’s results.

For large Reynolds numbers, we correspondingly have small dissipative lengthscales which must be resolved by the numerical method. Where insufficient res-olution is used, SPH simulations yield solutions which have multivalued velocityfields (see Figure 4.4), though Burgers’ equation (4.1) cannot admit such solu-


time = 0.1 time = 0.3

time = 0.5 time = 1.0

vv

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9

Figure 4.5: Burgers solution at various times. Re = 102, 100 particles, β = 10.

tions. In this situations, the SPH realisation of viscosity is in effect ‘watereddown’ due summations over large number of particles with few contributing sig-nificant viscous effect. We also note that the assumption of local smoothnessused in the deriving continuum viscosity equivalents (see Section 2.2.4) is nolonger valid. To correct this, it is desirable to reduce smoothing length. Wemay increase the particle population while using a constant-h implementation,though as we only require increased resolution at the shock, it is more efficientto simply allow smoothing length to vary. Indeed SPH lends itself very well tosuch simulations, concentrating resolutions where most required.

We require the use of limited smoothing length rule (4.5) to maintain atimestep which allows for reasonable time evolution. All calculations are per-formed using one-hundred particles, though we allow smoothing length parame-ter β to vary such that higher Reynolds number regimes are effectively calculatedat higher resolution. Regimes considered are Re = 102, 103 and 105, with re-spectively β = 10, 50 and 600 (Figures 4.5 - 4.7). All particle cross-streaminghas been prevented through use of increased resolution. We also note that therepresentations given do not convey the near shock velocity profile well. Forinstance, the Re = 100 simulation, Figure 4.5, at time t = 1.0, shows a velocityprofile which appears to have discontinuous derivative near the shock, where inactual fact particles follow a smooth path past the peak, and into the centreof the domain. Comparison with results of Varoglu and Finn (1980) (Figure4.7) reveal only minor discrepancy in Re = 105 simulations, lending support totheir solutions over those of Kakuda and Tosaka (1990). Results of Wei and Gu(2002) are also shown in Figure 4.7 and are in excellent correspondence with


time = 0.1 time = 0.3

time = 0.5 time = 1.0

vv

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9

Figure 4.6: Burgers solution at various times. Re = 103, 100 particles, β =50. The solid curve represents solutions found via a spectral method using 256trigonometric modes.

SPH results.Solutions have also been calculated using a spectral method (Re = 103), and

excellent agreement with SPH solutions is found (Figure 4.6). Energy spectrumscorresponding to given times are to be found in Figure 4.8. Specific definitionsfor energy spectrum e(k) are to be found in Appendix C, with only kineticenergy components being considered here. The energy cascade process canbe clearly identified, with all energy (originally residing in the fundamentalmode) eventually propagating through the spectrum down to the dissipativelength scale. Once the ‘turbulence’ has developed, an inertial subrange alsobecomes evident. For Burgers’ equation, we expect this range to exhibit a k−2

scaling (Burgers, 1948), which is indeed what is found. Burgers also predictedan exponential decay within the dissipation range which can be seen. As timeprogresses energy can be seen to decay under viscous forces, with the dissipativerange growing as peak velocity falls. Finally, we note the minor artifact at thespectrum tail which results from energy reaching our highest wavenumber mode.The cascade process must finish at this point, and so energy accumulates untilviscous dissipation is sufficiently activated.

For the spectral simulation, the Re = 103 regime may be considered largelyresolved, so energy accumulation at short scales does not present an issue,with viscosity dissipating energy before amplitudes become significant. Hada larger Reynold’s regime been simulated however, this accumulation wouldbecome large, and spurious oscillations throughout the physical domain would


time = 0.1 time = 0.3

time = 0.5 time = 1.0

vv

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Figure 4.7: Burgers solution at various times. Re = 105, 100 particles, β = 600.Blue curve corresponds to Varoglu and Finn (1980), while green curve is solutiondue to Wei and Gu (2002).

ensue. We note that the behavior of solutions under such under-resolved cir-cumstances will rest upon the details of the particular numerical scheme, withspectral methods yielding unphysical high order oscillations throughout the do-main, while SPH techniques result in the related issue of multivalued solutions.

Of course, these issues may be addressed by improving resolution. In SPH wemay increase particle numbers or allow smoothing length to vary, or both. Forspectral techniques we introduce further modes. While this achieves the desiredresult, largely increased computational cost is incurred due to reduced timestep,which may be compounded by increased operation count per step where moremodes or particles are required. In analogy to computational Navier-Stokes tur-bulence, we may wish to instead provide closure to the Burgers’ regime at somescale larger than that required of viscous closure. It seems that the α-SPH modelmay provide means of this end, though unfortunately it is found that the modelproves to be unstable under the current configuration. Though a formal stabil-ity analysis has not been performed, the instability occurs and similar in naturefor both SPH and spectral simulations. From this we may possibly concludethat the instability is intrinsic to the differential dynamics and not a productof the particular discretisation taken. We also note that the instability is onlyapparent where turbulence terms are included in our momentum equation, anddoes not present where only the filtered advection velocity is used. Geurts andHolm (2002) also makes mention of this Burgers’ instability, attributing it toexcessive energy back-scatter resulting from antidiffusive turbulence terms. We

4.2 One-dimensional Navier-Stokes 54

time = 0.1 time = 0.3

time = 0.5 time = 1.0

E(k

)E

(k)

k/π k/π

10-9

10-7

10-5

10-3

10-9

10-7

10-5

10-3

10 100 10 100

Figure 4.8: Energy spectrum at various times for the Re = 103 spectral solutiongiven in Figure 4.6. The dashed line gives a reference for the k−2 energy spectrumscaling expected for the Burgers’ inertial range.

finally note that the instability occurs regardless of which momentum equationis utilised, whether it be the compressible form given by momentum equation(3.31) and velocity filter (3.6), or the incompressible version defined by momen-tum equation (4.8) and velocity (4.7). Some examples of unstable solutions maybe found in Figure 4.9.

4.2 One-dimensional Navier-Stokes

As we are unable to perform simulations of Burgers’ regime incorporating al-pha turbulence terms, we instead wish to consider an alternate system whichmost importantly presents energy cascade phenomena by which we may inves-tigate the action of both standard SPH and α-SPH. Various one-dimensionalmodels have been proposed by numerous authors which are designed to mimicbehavior of multidimensional turbulent flows, or rather to recreate various en-ergy scalings found in higher dimensional systems. Bartello and Warn (1988)have given a model where in effect severe mode truncation is performed in allbut one dimension, which has proven successful in reproducing two-dimensionalenergy scalings of Kraichnan (1967) and Batchelor (1969), but not as successfulfor the three-dimensional Kolmogorov (1941) scaling (Bartello, 1992). An alter-native model due to Qian (1984) modifies the one-dimensional advection termand introduces an artificial pressure term such that Kolmogorov’s k−5/3 energyscaling is reproduced. Furthermore, a model which is based on a postulated


time = 0.10 time = 0.15

time = 0.20 time = 0.25

vv

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9

Figure 4.9: Solution demonstrating instability of alpha model to the steepeningshock Burgers’ configuration. Solution computed using incompressible versionof alpha model. SPH solution represented with + symbols, Re = 5 × 103, 100particles, β = 10, ε0 = 1, εa = ε0h

20/h

2a. Solid curve solution is computed using

spectral methods with 1024 modes.

stochastic advection is given by Kerstein (1999). However the behavior of alphaturbulence terms under such models are not necessarily expected to mimic thoseof higher dimensional systems, and contruction of such models within the SPHframework presents difficulty.

We instead move on to consider a related system to that of Burgers equa-tion which simply incorporates a pressure gradient term, hence yielding theone-dimensional compressible Navier-Stokes system. This provides the neces-sary ingredients of conservation of total energy, along with the non-linear termsrequired for energy cascades. Naturally we still do not expect dynamics im-parted by turbulence terms to necessarily represent those found in higher dimen-sions, owing to the sacrificed physics in restricting dynamics to one-dimension.We approach results with the view of gaining perspective to the mathematicsof cascade processes realised in standard and α-SPH simulations. Our one-dimensional system is then (see Appendix B for details):

∂v

∂t+ v

∂v

∂x= −1

ρ

∂P

∂x+α2

ρ

((∂v

∂x

)2∂ρ

∂x+ 2ρ

∂v

∂x

∂2v

∂x2

)+ν

ρ

∂

∂x

(ρ∂v

∂x

)(4.11)

v = v +α2

ρ

∂

∂x

(ρ∂v

∂x

), (4.12)


with continuity equation

∂ρ

∂t+ v

∂ρ

∂x= −ρ∂v

∂x, (4.13)

and equation of state

P = ργ . (4.14)

Pressure perturbations then move at the sound speed Cs, where C2s = ∂P/∂ρ.

We initiate all flows with velocity perturbation

v0 = v(x, 0) = 0.05Cs sin(2πx), (4.15)

where the mach number has been taken to ensure density perturbations arewithin five percent of the base density. Integrations are performed over a peri-odic domain 0 ≤ x < 1, and so we find that the fundamental mode perturbationwill oscillate with period T = 1/Cs. We define the nondimensional time t = t∗/Tfor dimensional time t∗.

4.2.1 The Euler system

First considered is the dynamics of the Euler system, and a modified equivalentwhere alpha terms have been included. In the standard guise, the Euler equa-tions present a open system in the sense that infinite spectral components willbecome active given sufficient time. The multidimensional physical realisationof this is an infinite inertial energy cascade range, where large eddies spawnsmaller eddies, which themselves spawn further eddies onto infinity. In one di-mension we have a related situation where as with the Burgers’ simulation wefind a shockfront which steepens with time, though now the shockfront oscil-lates. This contrasts the full Navier-Stokes regime where eventually energy willbe acted upon by viscosity, halting any further cascade. Numerically, turbulentEuler simulations only represent continuum dynamics up until the point whereresolution limitations prevent or slow further cascades.

The modified Euler algorithm introduces the length scale α beyond which itis expected that energy cascades will be inhibited, thus providing an alternateroute to closure. We present here simulations for the modified algorithm, alongwith it’s standard counterpart. Both SPH and spectral techniques are utilisedand compared.

Standard Euler

First considered is the integration of Euler’s equations using the spectral al-gorithm. Figure 4.10 gives the velocity and density profiles for a simulationutilising 256 trigonometric modes, with Figure 4.11 showing closer views at cer-tain time points. The initial sinusoidal perturbation evolves to forms two shockfronts which oscillate back and forth within the domain. Eventually this steep-ening process cannot continue as the spectral decomposition is unable to repre-sent shorter length scales, with resulting dynamics given in the latter frames ofFigure 4.10.


Figure 4.10: Velocity and density profiles for compressible Euler simulationusing spectral algorithm. Velocity and density are represented with red and graycurves respectively. Frames are given in 0.4T increments. Density scale showsvalues within five percent of static density.

Another perspective is found in considering the energy spectrum of the flow,with energy modes defined in Appendix C. We observe in Figure 4.12 the cas-cade of energy down from the initial fundamental mode perturbation, with whatappears to be a tendency towards k−2 energy scaling. Naturally no developedturbulent state can be reached given the infinite spectral modes required. Wesee that once energy reaches the final mode, it is unable to be passed downany further, and we have so-called spectral blocking where energy is continuallydumped into this final mode. Preceding modes soon also struggle to pass downenergy, owing to excited state of the final mode, so they also begin to grow.Eventually the solution is dominated by this behavior. For this calculation,total system energy is conserved to within less than one percent of the initialperturbed energy up to time t = 10.

We now turn to SPH simulations of the compressible Euler configuration.First attempted are calculations using a fixed smoothing length implementation.Unfortunately, these are not met with success. While qualitatively we havesimilar behavior to that found using our spectral algorithm, there are a numberof significant departures.


time = 2.3 time = 5.4

time = 5.9 time = 10.0

vv

ρρ

-1

0

1

0.95

1

1.05

-1

0

1

0.95

1

1.05

Figure 4.11: Velocity and density profiles for compressible Euler simulationusing spectral algorithm. Accumulation of energy at the spectral limit becomesevident at later time points, physically realised as oscillations throughout thedomain. Velocity is represent with red curve, while density is given by greycurve.

time = 2.3 time = 5.4

time = 5.9 time = 10.0

E(k

)E

(k)

k/2π k/2π

10-13

10-11

10-9

10-7

10-5

10-3

10-13

10-11

10-9

10-7

10-5

10-3

1 10 100 1 10 100

Figure 4.12: Energy spectrum for flow configurations found in Figure 4.11. Thedashed line gives a reference for k−2 energy spectrum scaling.


E(k

)

k/2π

spectral

h = 1.3 ∆x

h = 2.0 ∆x

h = 2.5 ∆x10-16

10-14

10-12

10-10

10-8

10-6

10-4

1 10 100

Figure 4.13: Energy spectrum for different implementations at time t = 2.5.

We first perform simulations using one-thousand particles, with smoothinglength set at h = 1.3∆x for initial particle spacing ∆x. It is found that pertur-bations do no not move at the expected velocity, with results lagging behind thespectral method equivalent. Further to this, it appears the nature of the non-linearity is modified, with energy cascades processes taking longer to reach anequivalent state. We note that even if we rescale the integration time, our sim-ulations still do not coincide, with cascade processes being slowed to a greaterextent than sound speed. Simulations appear convergent, with increased parti-cle numbers and shortened timestep leaving results unchanged.

Increasing smoothing length to h = 2 ∆x leads to more favourable results.Pressure perturbations now in effect travel with only slightly larger than expectspeeds, and energy cascades proceed at a comparable rate to those of spectralcalculations. If we now increase smoothing length further still to h = 2.5 ∆x,sound speed is again found to fall short of the expected figure. Now however,the energy cascade is enhanced and energy propagates through the spectrumfaster than found in the spectral integration. These results are demonstrated inFigure 4.14 where velocity profiles in time are given, and Figure 4.13 which showsenergy spectrums. Details for the determination of spectral mode coefficientscorresponding to SPH particle simulations are to be found in Appendix D.


Fig

ure

4.1

4:

Vel

oci

typro

file

sin

tim

ein

crem

ents

of

0.1T

.T

he

bla

ckdash

edline

repre

sents

resu

lts

for

the

spec

tral

alg

ori

thm

;th

egre

en,

blu

eand

magen

tacu

rves

corr

esp

ond

tosi

mula

tions

usi

ngh

=1.3

∆x

,h

=2.0

∆x

andh

=2.5

∆x

resp

ecti

vel

y.


Figure 4.15: Velocity profiles for variable-h SPH simulation (blue) alongsideresults obtained using spectral algorithm (red). Frames are given in 0.4T incre-ments.

While best results appear to correspond to h = 2.0 ∆x, we are left unsure asto whether this choice of smoothing length parameter will best suit all configu-rations, or even all regimes within a simulation (such as transition to turbulence,or developed turbulence). We instead seek solutions using a variable-h imple-mentation, which are found to correspond closer to spectral solutions. Figure4.15 gives ensuing velocity profiles, where we find that results are nearly iden-tical up until the point where resolution limits are met.

In spectral space (Figure 4.16), a few differences become apparent. Firstlywe notice that mode growth rates fall slightly short of those found for thespectral method, though the difference is relatively minor. More significant isthe attenuation of high order modes at later times. A divergence first occurswhen energy reaches modes of order one-hundred, at approximately time t = 3.From this time on a second spectral range forms with a steeper energy scaling,as can be seen in the third frame of Figure 4.16. We have now in effect reached apoint where solutions will begin to diverge from strict Euler dynamics, with thehigher order details of the SPH technique becoming significant. Energy howeverstill attempts to filter down the spectrum from above, with Euler dynamicstending towards a k−2 energy scaling. This results in accumulation of energyat particular modes, with more energy entering than is able to leave, and thespectrum then tending to ‘buckle’ forming kinks at certain modes (final frameFigure 4.16). This is not dissimilar to the spectral blocking observed for thespectral algorithm. Here however, overly excited modes are eventually ableto pass energy down, with kinks then moving through the spectrum. SPHthus provides a less abrupt cascade cutoff, though this occurs at a much larger


time = 2.0 time = 4.0

time = 5.3 time = 6.0

E(k

)E

(k)

k/2π k/2π

10-13

10-11

10-9

10-7

10-5

10-3

10-13

10-11

10-9

10-7

10-5

10-3

1 10 100 1 10 100

Figure 4.16: Energy spectrum for variable smoothing length SPH implemen-tation (blue), alongside spectrum for spectral algorithm (red). The dashed linegives a reference for k−2 energy spectrum scaling.

wavelength than found for the spectral technique.The variable smoothing length is defined according to equation (2.67), for

which the parameter σ determines the number of particles which fall over thekernel’s support domain, and therefore modifies the smoothing length. Someexamples demonstrating the influence of varying σ (and therefore smoothinglength) are to be found in Figure 4.17. We can clearly see modulation of thecascade cutoff point as smoothing length is increased, with cutoff point movingto larger scales for increased smoothing lengths. Similarly, we may instead varyparticle numbers, maintaining a smoothing length parameter σ = 1.3 (Figure4.18). We are still here in effect varying smoothing length, and so it is not sur-prising that we find similar results to those of Figure 4.17, with the consequenceagain being an alteration of cutoff frequency. A similar change in cutoff pointis observed for equivalent changes in smoothing length (i.e. n = 2000 → 1000or σ = 1.3→ 2.6).

To test the convergence of the SPH summation interpolant (2.5), we wish toincrease σ while keeping the absolute base smoothing length fixed, so thereforemust increase particle numbers (Figure 4.19). What is found is a slight increasein cutoff point, indicating that σ = 1.3 is not sufficient for true convergenceof the summation interpolant. We note that the difference is relatively minor,and increased costs incurred may perhaps be better utilised simply increasingparticle numbers while using σ = 1.3. Finally, we consider the influence ofthe omega terms (2.74) found in the momentum equation (2.75). These terms


E(k

)

k/2π

σ = 1.3

σ = 2.0

σ = 2.6

spectral10-16

10-14

10-12

10-10

10-8

10-6

10 100

Figure 4.17: Energy spectrum for different smoothing lengths with fixed parti-cles numbers. Results are taken at time t = 5.0.

E(k

)

k/2π

n = 500

n = 1000

n = 2000

n = 4000

spectral10-16

10-14

10-12

10-10

10-8

10-6

10 100

Figure 4.18: Energy spectrum for different particle populations using smoothinglength parameter σ = 1.3. Results are taken at time t = 5.0.


E(k

)

k/2π

n = 1000

n = 2000

Ω = 1

spectral10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

100

Figure 4.19: Energy spectrum keeping base smoothing length fixed at h0 =1.3 ∆x0 with ∆x0 = 0.001. Results are taken at time t = 5.0.

are related to the gradient of the smoothing length, and are required for avariationally consistent SPH derivation. We can see in Figure 4.19 that insetting Ω = 1 we find medium-scale modes suffer significantly less damping.However, the sacrificed variational consistency appears to result in phase speedinaccuracies, and in physical space results are found to step away from thespectral solution, though not as severely as found where smoothing length iskept fixed. So despite the increased mode damping, we choose to retain ourcalculation of Ω according to equation (2.74), though the differences found insetting Ω = 1 are worth noting.

In conclusion, we can say that while the SPH technique is able to producethe bulk dynamics of Euler flow, mode attenuation becomes significant fromlength scales of order ten times larger than smoothing length. This imposes asignificant restriction on the simulation of medium to short scale Euler dynamicswhich can only be resolved through use of very large particle populations. Whilein one-dimension this is perhaps on option, even stepping up to two-dimensionswe will find computer resources quickly overwhelmed.

On the other hand, we may interpret this as an implicit turbulence typebehavior, with large scales being calculated as desired and short scales beinginhibited. For a simulation such as Navier-Stokes flow, where dissipation willprovide eventual closure, this may indeed prove useful in preventing spuriousshort-scale behavior corrupting bulk dynamics. Further investigation of thisimplicit turbulence modelling behavior is certainly warranted, though for nowwe turn to the explicit turbulence modelling scheme outlined in Chapter 3.


time = 2.3 time = 4.4

time = 5.9 time = 10.0

E(k

)E

(k)

k/2π k/2π

10-25

10-20

10-15

10-10

10-5

10-25

10-20

10-15

10-10

10-5

1 10 100 1 10 100

Figure 4.20: Energy spectrum for standard Euler (red) and modified Euler(blue) simulations. 256 modes are utilised with kα ' 596. Dashed line gives areference for k−2 energy spectrum scaling.

Modified Euler

Our modified Euler systems is defined as the one-dimensional version of thealpha continuum model, given by equations (3.30) and (3.8):

∂v

∂t+ v

∂v

∂x= −1

ρ

∂P

∂x+α2

ρ

[(∂v

∂x

)2∂ρ

∂x+ 2ρ

∂v

∂x

∂2v

∂x2

]

v = v +α2

ρ

∂

∂x

(ρ∂v

∂x

),


∂ρ

∂t+ v

∂ρ

∂x= −ρ∂v

∂x.

It is expected that the additional terms will act to slow and then halt energycascade processes for modes of order kα ∼ 1, therefore closing the Euler systemat some wavenumber where kα 1. We consider results for the spectral algo-rithm first. Figure 4.20 gives the progression of energy modes for the modifiedscheme alongside the results found above for standard Euler. For this calcula-tion 256 modes are used, and the effective alpha cutoff wavenumber is kα ' 596,where we define kα = 1/α. This choice is taken to coincide with values used forSPH simulations.


Figure 4.21: Velocity and density profiles for compressible Euler simulationusing spectral algorithm with alpha parameter kα ' 596. Velocity and densityare represented with red and gray curves respectively. Frames are given in 0.4Tincrements. Density scale shows values within five percent of static density.

We observe that both simulations proceed identically (first frame Figure4.20) until the point where modes begin to feel the affects of alpha turbulenceterms, which under the current parameters occurs from mode m = 30. The en-ergy spectrum is then turned down with energy propagation inhibited (secondframe Figure 4.20). As seen previously in SPH simulations, we then find kinksdeveloping in the energy spectrum as energy mode net fluxes become non-zero(third frame Figure 4.20), and a train of such kinks eventually develop fur-ther along the energy spectrum (fourth frame Figure 4.20). The spectrum hasnow reached a largely statistically steady state, and so we may conclude thatclosure has been attained. We note that for large length scales, the spectrumappears able to maintain the expected k−2 energy scaling, whereas where turbu-lence terms have not been included eventually noise overwhelms all wavenumbermodes. The simulation in physical space is shown in Figure 4.21 for density andvelocity, and may be compared with results without turbulence terms foundin Figure 4.10. The most obvious change is the reduction in short scale noise,which for standard Euler simulations appears to overwhelm the solution. Thisis especially evident in the final frame of 4.20. It may be said that solutions are


E(k

)

k/2π

kα =∞kα = 5958

kα = 2665

kα = 1884

kα = 843

kα = 596

kα = 37710-25

10-20

10-15

10-10

10-5

10 100

Figure 4.22: Spectral algorithm energy spectrum for different alpha turbulenceparameters at time t = 5.

regularised by the addition of alpha turbulence terms.

We find in Figures 4.22 and 4.23 results for different values of parameterkα. At the earlier time (Figure 4.22), we see that all simulations inhibit energypropagation with strength relative to alpha cutoff parameter kα, as expected.For higher wavenumber cutoff parameters, energy is still able to reach the high-est integrated wavenumber (n = 256), though at this simulation time minimalenergy accumulation appears to be occurring. Turning to the later time results(4.23) it is found that simulations with higher alpha cutoff wavenumbers (such askα ' 2000) are able to produce improved large scale dynamics despite requiringgreater resolution for true closure. Lower wavenumber cutoff parameters, whilesignificantly reducing noise at short scales, tend to result in large scales whichdeviate from Euler dynamics. Instead, an overabundance of energy is found dueto inhibited propagation, with incorrect scaling caused by nearby energy kinks.In general it is desirable to resolve most if not all of the alpha dispersive energysubrange, though computational cost may rule this out. Bearing these points inmind, for closest approximation to Euler dynamics the alpha cutoff wavenumbershould be set at some value short of the maximum wavenumber which allowsfor reasonable integration of the dispersive subrange. Returning to Figure 4.23,we find that parameter kα = 843 meets these requirements, perhaps giving thebest compromise between short scale noise and large scale accuracy.

In turning to α-SPH simulations, a few points need to be considered. Firstly,where we use variable resolution, our filtered velocity cutoff length scale willdepend on smoothing length according to equation (3.11). However, we insteadwish to have a fixed cutoff frequency for comparison with the spectral algorithm.


E(k

)

k/2π

kα =∞kα = 5958

kα = 2665

kα = 1884

kα = 843

kα = 596

kα = 37710-25

10-20

10-15

10-10

10-5

10 100

Figure 4.23: Energy spectrum for different alpha turbulence parameters at timet = 10.

This may be achieved by allowing parameter ε to be vary with smoothing length:

εa =h2

0

h2a

ε0, (4.16)

for initial values ho and ε0. In application, the best approach (fixed or variablecutoff frequency) is unclear. Using a variable cutoff allows us to slow energycascading as they approach our resolution scale, which practically may be de-sirable. For the current simulation, density variations are small and so we donot expect a large difference between approaches though for consistency withthe spectral algorithm we make use of equation (4.16).

Secondly, as observed above for the SPH Euler simulations, there exists asecondary dynamic which acts to disperse energy away from short length scales.Taylor series expansions reveal the following for the SPH pressure gradient term:

∑b

mb

Paρ2a

+Pbρ2b

∇aWab ' ρ−1P ′ + β2

(ρ−2ρ′′′P

+Pf ′′′ + 3P ′f ′′ + 3P ′′f ′ + P ′′′f)

where primes denote derivatives with respect to coordinate (example P ′ = ∂P∂x ),

we set f = 1/ρ, and in analogy to the definition of α (equation (3.11)) we definethe following:

β2 =12

∫(x∗ − x)2W (r∗ − r, h(r)) dr∗.


Where we use equation of state P = ργ , the above becomes

∑b

mb

Paρ2a

+Pbρ2b

∇aWab ' γργ−2ρ′

+ β2(γργ−2ρ′′′ + 3ργ−3ρ′ρ′′(1− γ)(2− γ)

− ργ−4(ρ′)3(1− γ)(2− γ)(3− γ)).

For the cubic spline kernel, the parameter β takes the value β2 = h2/6. We pos-tulate that these terms are responsible for the dispersive behavior observed forthe earlier SPH Euler simulations. If we wish to investigate the effects of alphaterms introduced in α-SPH, we must set α β to minimise the influence ofbeta terms. Unfortunately this leads to expensive computations, as we requiredε 1 which necessitates many iterations for filtered velocity convergence.

Bearing the above in mind, we proceed to consider the α-SPH simulations,and in Figure 4.24 we find results where one-thousand particles are utilised anddifferent values of ε are taken. Interestingly, it is found that for small values ofε, the additional turbulence terms tend to allow increased propagation of en-ergy into short scales. The exact mechanisms by which this is able to occur isunclear, though it appears that turbulence terms perhaps reduce the strengthof the secondary terms discussed above. In comparing with the equivalent spec-tral algorithm results found in Figure 4.22, we find that reasonable agreementis found for values ε greater than unity, with improved correspondence for in-creasing values of ε, as predicted above. Results at a later time are to be foundin Figure 4.25, and can be compared with Figure 4.23. As for results at theearlier time, best correspondence is found for largest values of ε.

Simulations are performed to investigate dynamics where the relative strengthof α and β terms are varied (Figure 4.26). To this end, a fixed value of kα = 1884is taken, with the value of ε is varied with values of smoothing length chosen tosuit. For the value ε = 1, results appear to be in good agreement on averagewith the spectral results, though stepping up to ε = 2 we see that short scaleenergy is bolstered. Simulations for ε = 5 and ε = 10 closely correspond tothose found via spectral methods. We also note that for all simulations modesbeyond n = 300 are inactive, with exception of results for ε = 2 where modesbeyond n = 380 are inactive. This indicates that improved results are notdirectly consequent of increased SPH grid resolution, but rather due to the di-minished significance of the SPH modified differential equation terms (i.e. thebeta terms).

To investigate this further, simulations for fixed absolute smoothing lengthand fixed parameter ε are performed while the particle spacing is decreased(or equivalently particle populations increased). Therefore, summations involvelarger number of particles (so smoothing length parameter σ is increased), andit is expected that such summations will form better approximations to the inte-gral approximant. Turning to Figure 4.27, we indeed find significant differencesat all but the largest scales. While results where σ = 1.3 on average replicatethe spectral result, the spectrum form appears to be generally incorrect. Con-vergence appears to be attained for h0 ≥ 2.0∆x, though considerably increasedcomputational cost is incurred. These results echo those found earlier for stan-dard SPH (Figure 4.19). Parameter configurations will in general require some


E(k

)

k/2π

kα =∞,kα = 5958,

kα = 2665,

kα = 1884,

kα = 843,

kα = 596,

kα = 377,

ε = 0

ε = 0.1

ε = 0.5

ε = 1.0

ε = 5.0

ε = 10

ε = 2510-25

10-20

10-15

10-10

10-5

10 100

Figure 4.24: Energy spectrum for different alpha turbulence parameters attime t = 5. Results for α-SPH simulations, using one-thousand particles. To becompared with the equivalent spectral results found in Figure 4.22.

E(k

)

k/2π

kα =∞,kα = 5958,

kα = 2665,

kα = 1884,

kα = 843,

kα = 596,

kα = 377,

ε = 0

ε = 0.1

ε = 0.5

ε = 1.0

ε = 5.0

ε = 10

ε = 2510-25

10-20

10-15

10-10

10-5

10 100

Figure 4.25: Energy spectrum for different alpha turbulence parameters. Re-sults for α-SPH simulations, using one-thousand particles. To be compared withthe equivalent spectral results found in Figure 4.23. Time t = 10.


E(k

)

k/2π

spectral

ε = 1

ε = 2

ε = 5

ε = 1010-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

100

Figure 4.26: Energy spectrum where ε is varied with fixed parameter kα = 1884.Particle populations for ε = 1, 2, 5 and 10 are respectively n = 1000, 1414, 2236and 3162. Time t = 5.

E(k

)

k/2π

spectral

n = 1000, σ = 1.3

n = 1308, σ = 1.7

n = 1424, σ = 1.85

n = 1500, σ = 1.95

n = 1538, σ = 2.0

n = 2308, σ = 3.010-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

100

Figure 4.27: Energy spectrum for varied particle densities, with base smoothinglength kept at the same value of h = 0.0013 for all simulations. For all simulationswe use ε = 1. Results are at time t = 5 with kα = 1884.


Energy

time

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0 5 10 15 20

Figure 4.28: Time progression of energy in modes and total energy. Uppermostcurve gives total energy, with curves below showing energy in the first, second,third and fourth mode respectively. Spectral algorithm utilising 256 modes, withRe = 1000.

degree of tuning for an optimal balance of accuracy and efficiency, though itappears that values σ ' 2 are desirable.

4.2.2 Forced Navier-Stokes simulations

Integrations are performed for the full one-dimensional Navier-Stokes equa-tions. The inclusion of viscosity introduces a length scale where energy dis-sipation will eventually outweight non-linear cascades, and hence bring aboutclosure. For real multi-dimensional turbulent flows, this process usually occursat scales which fall well beyond the limits of computational resolution. For one-dimensional simulations, full direct numerical simulations of governing equa-tions is a realistic prospect however, which allows for useful comparison withthe modified dynamics of LANS, despite the incurred simplification of physics.

The forcing regime essentially maintains the initial energy of the fundamen-tal mode. In effect, we force the fundamental mode velocity amplitude suchthat it always accelerates towards the inviscid linear solution (where total fun-damental energy remains constant in time). Density perturbations follow suit.Specifically, for the spectral algorithm we apply the follow for velocity forcing

∂v1

∂t= fAB1 + C(vL(t)− v1) (4.17)

with fAB being the forcing from our Adam-Bashforth scheme, and vL(t) beingthe linear solution. The constant C adjusts the aggression with which forcing


time = 2 time = 5

time = 6 time = 8

E(k

)E

(k)

k/2π k/2π

10-13

10-11

10-9

10-7

10-5

10-3

10-13

10-11

10-9

10-7

10-5

10-3

1 10 100 1 10 100

Figure 4.29: Energy spectrum for Re = 1000 simulation, with 256 modes. Thedashed line gives a reference for k−2 energy spectrum scaling.

is applied, and typically is set such that forcing acceleration is one tenth thestrength of maximum linear mode acceleration.

Total energy is found to increase until the point where modes of short scalebecome sufficiently active for viscous dissipation to become appreciable. Even-tually a balance is found where the forced energy input is equal to the viscousenergy dissipation (Figure 4.28). We also observe a relatively small degree ofenergy oscillation which may be attributed to the oscillation of the fundamentalmode, consequent of the forcing scheme. Alternatively we may have chosen toexplicitly drive the fundamental velocity and density modes according to thelinear regime, though an equivalent SPH simulation would have been difficult.

Simulations are performed using the spectral scheme with 256 modes. Wedefine a Reynolds number as previously:

Re =UL

ν(4.18)

for length scale L, velocity scale U and kinematic viscosity ν. The length andvelocity scales are respectively defined as the domain size and initial velocity.The energy spectrum at various times is displayed in Figure 4.29. Since viscositypredominantly acts upon short scales, only once these scales become sufficientlyactive do we have appreciable deviation from the inviscid simulation. Energyin short scales is dissipated under the action of viscosity, and therefore tapersoff within the dissipative subrange. By the final time frame, a statisticallysteady state has been achieved, and we expect the energy spectrum to remainstationary. This may be confirmed with reference to the energy progressionof Figure 4.28, and also by inspection of the solution in the physical domain


Figure 4.30: Velocity and density profiles for Navier-Stokes simulation usingspectral algorithm. Velocity and density are represented with red and gray curvesrespectively. Frames are given in 0.4T increments. Density scale shows valueswithin five percent of static density.

(Figure 4.30). The physical space solution is observed to evolved from thesinusoidal initial condition to eventually form two shock fronts which oscillateback and forth. We note that the viscous dissipation range appears to not becompletely resolved, with a small degree of spectral blocking to be found at thetail of the spectrum. Only the slightest hint of this is found in the physical spacesolution, with very slight oscillation at shock fronts. Resolution is sufficient suchthat viscosity ensures that energy does not accumulate. The inertial subrangescaling of k−2 may be clearly seen.

Forced spectral simulation is performed for various viscosity levels, withsteady state results found in Figure 4.31. Larger Reynolds numbers push thedissipation range to higher and higher wavenumbers, with the result being thatlarger amounts of energy reach the numerical limit. Naturally, a larger degreeof energy accumulation occurs at these short scales, with the physical spacerealisation being short scale noise overlayed on solutions. Indeed, since a smallerportion of the viscosity range is simulated, the short scales must increase inenergy to effect equivalent viscous dissipation. It is not until we step up toRe = 106 that viscosity is insufficient to ensure stability, and the entire spectral


E(k

)

k/2π

Re = 250

Re = 500

Re = 1000

Re = 2000

Re = 4000

Re = 1000010-14

10-12

10-10

10-8

10-6

10-4

10-2

1 10 100

Figure 4.31: Energy spectrum at various Reynolds numbers. All results aretaken at time t = 10, where a steady state has been reached.

range becomes corrupted, with results similar to those of Figure 4.12. In thiscase forcing leads to a continual increase in total energy which we expect toeventually cause simulation failure.

Forced Navier-Stokes simulations are also performed using the SPH algo-rithm. The forcing scheme is equivalent to that of the spectral algorithm. Weuse the following equation for particle velocity forcing:

∂va∂t

= fSPHa + C(vL(t)− v1) sin(2πxa). (4.19)

Here fSPHa is the standard SPH summation force, va is the particle velocity,and xa is the particle position. Function vL(t) is as prior, the linear mode solu-tion. The value v1 is the fundamental mode coefficient, and is determined usingmethods outlined in Appendix D. Figure 4.32 shows total energy progression,and is in excellent agreement with the equivalent spectral result (Figure 4.28),though a slight increase in total energy oscillation is observed.

The SPH energy spectrum is given in Figure 4.33. As insufficient resolutionis utilised, neither the SPH nor spectral methods give correct behavior at shortscales. The SPH solution however appears to suffer to a larger extent owing tobeta terms which slow energy cascades, resulting in energy accumulation pointsin the spectrum. SPH solutions also exhibit a degree of oscillation in the highwavenumber section of the spectrum, though statistically solutions are steady.

Steady state solutions are compared for both the spectral and SPH codeat different resolutions in Figure 4.34. The spectral method’s superiority isevident, with the coarser simulation replicating the higher resolution result forall but it’s highest wavenumbers. It appears that beta terms implicit in SPH


Energy

time

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0 5 10 15 20

Figure 4.32: Time progression of energy in modes and total energy for SPHsimulation using 1000 with Re = 1000. Uppermost curve gives total energy,with curves below showing energy in the first, second, third and fourth moderespectively.

integrations, and the resulting impedance of energy cascades, leads to unphysicalenergy accumulation in the coarse SPH simulation (at n ' 40). For the higherresolution simulation, the viscosity range is sufficiently resolved to prevent suchaccumulation, though we still do not find the expected exponential decay. Thissuggests that even where two-thousand particles are used, SPH simulations arestill under-resolved for Re = 1000. Results for SPH simulations at variousReynolds numbers are given in Figure 4.35. Results appear to be sufficientlyresolved for the Re = 100 and Re = 250 simulations. The Re = 500 simulationsbegins to show evidence that the poorly simulated high wavenumbers are havinginfluence, while results for larger Reynolds number are clearly subject to theseinadequacies.

We proceed to perform calculations for the alpha modified forced Navier-Stokes regime. Solutions at varying alpha parameter are carried out for aReynolds number of Re = 10000. Results are presented in Figure 4.36. Sim-ilarities are to be found with previous modified Euler (and SPH) simulationswith the formation of kinks in the spectrum where energy propagation is slowed.All simulations (with the exception of that with smallest alpha parameter) aresuccessful in reducing energy at the shortest scales, though spectrums in the in-termediate range now certainly deviate largely from the Navier-Stokes solution.We note that for one-dimensional simulations, viscous dissipation ensures solu-tions remain regular even for very large Reynolds numbers (as demonstrated forRe = 10000), despite spectral blocking. It appears there is no practical appli-cation for the alpha turbulence methodologies in one-dimension, which despite


time = 2 time = 5

time = 6 time = 8

E(k

)E

(k)

k/2π k/2π

10-13

10-11

10-9

10-7

10-5

10-3

10-13

10-11

10-9

10-7

10-5

10-3

1 10 100 1 10 100

Figure 4.33: Energy spectrum for Re = 1000 simulation. SPH simulationsgiven by blue points, while spectral results are given in red. The dashed linegives a reference for k−2 energy spectrum scaling.

reducing short scale energy, yield large deviates at all but the smallest wavenum-bers. Regardless, the alpha turbulence dynamics are worth investigating in theirown right.

Though we do not expect to produced any ‘improvement’ to solutions, thealpha modifications to our SPH algorithm are also explored. A Reynolds numberof Re = 2500 is taken while the parameter ε (or conversely α) is varied (Figure4.37). Qualitatively results are again very similar to those found previous, bothwhere alpha terms have been included, and for standard SPH. Similarities are ofcourse due to the slowing of energy cascades at high wavenumbers with energyat larger scales trying to flow down at a higher rate. Results echo those foundfor the modified Euler SPH simulations, where low values of ε result in a boostin energy at short scales. As we reach higher values of ε however, increasedimpedance of cascade processes results, leading to reduced energy propagationinto high wavenumbers.

We note that for both the spectral and SPH simulations, the decrease inenergy at short scales reduces the ability of viscosity to effect energy dissipation.This results in a increased transition period to reach a steady state, and also anincreased system total energy.


E(k

)

k/2π

Spectral

Spectral

SPH

SPH

256 modes

512 modes

1000 particles

2000 particles10-14

10-12

10-10

10-8

10-6

10-4

10-2

1 10 100

Figure 4.34: Energy spectrum for SPH and spectral codes at different resolu-tions. All results are taken at time t = 10, where a steady state has been reachedwith Re = 1000.

E(k

)

k/2π

Re = 100

Re = 250

Re = 500

Re = 1000

Re = 200010-14

10-12

10-10

10-8

10-6

10-4

1 10 100

Figure 4.35: Energy spectrum at various Reynolds numbers using SPH method.All results are taken at time t = 10, where a steady state has been reached.


E(k

)

k/2π

kα =∞kα = 10053

kα = 5027

kα = 2513

kα = 1257

kα = 62810-14

10-12

10-10

10-8

10-6

10-4

10-2

1 10 100

Figure 4.36: Energy spectrum for different values of parameter α. All resultsgiven are for the steady state solution using the spectral algorithm with 256modes and at Re = 10000.

E(k

)

k/2π

kα =∞kα = 1826

kα = 1289

kα = 576

kα = 408

kα = 258

ε = 0

ε = 0.5

ε = 1

ε = 5

ε = 10

ε = 2510-14

10-12

10-10

10-8

10-6

10-4

10-2

1 10 100

Figure 4.37: Energy spectrum for different values of parameter α using SPHalgorithm with 1000 particles and h0 = 1.9∆x. Results given are for steady statesolutions at Re = 2500.

4.3 Summary and Conclusions 80

4.3 Summary and Conclusions

Numerous one-dimensional results have been presented, with comparison beingmade between SPH and spectral algorithms. The SPH technique’s behaviorin the limit of marginally resolved dynamics is examined, with the influenceof simulation parameters and methodologies considered. Further to this, theinclusion of alpha terms and dynamics therein have been investigated.

Initially simulations of Burgers’ equation were tested. Where a variable res-olution implementation has been used, standard SPH proved very capable ofproducing highly accurate results, assisted by the natural concentration of reso-lution where most required. For moderate Reynolds regimes (Re = 103) , resultshave been verified through comparison with spectral integrations, for which ex-cellent correspondence was found. For higher Reynolds numbers (Re = 106),we have compared SPH simulations with the Burgers results of Wei and Gu(2002), among others. Again excellent agreement was found. Less fortune wasfound however in investigating the effects of alpha terms in Burgers’ regime sim-ulations, with an instability resulting, possibly owing to excessive back-scatter(Geurts and Holm, 2002).

Next the SPH algorithm was recast to perform simulations of Euler dynam-ics. Here we integrate an open system, and it is expected that eventually reso-lution limits will be encountered. The behavior of our schemes in approachingand reaching this limit has been considered. The spectral technique providedbenchmark results by which the accuracy of SPH could be quantified. Naturallythe spectral method also has resolution limits which eventually lead to solutioncorruption. Up to this point accuracy is generally very good, and comparisonwith larger mode number simulations show that only a small portion at the highend of the spectrum are effected by resolution shortcomings.

Turning to the SPH simulations of Euler dynamics some interesting resultsare observed. Firsty, it is found that where a constant SPH smoothing lengthis used, pressure perturbations do not travel at the speed determined by thelinearised system. While we may tune smoothing length parameter such thatcorrect sound speeds are obtained (Figure 4.14), the perhaps more significantissue of incorrect cascade rates persists (Figure 4.13). The use of a variablesmoothing length corrects both of these issues, though we now encounter sec-ondary effects which become dominant at length scales of order 10h for smooth-ing length h. These effects tend to slow energy cascades into shorter scales(Figure 4.16). We relate this to beta terms which appears in the SPH resultingmodified differential equation. Also worth note is the non-dissipative nature ofthese terms which may be attributed to the variational derivation of the SPHscheme.

The possible function of beta terms as an implicit turbulence modellingmethodology perhaps undermines the usefulness of SPH alpha turbulence terms.Spectral simulations incorporating alpha terms indeed reveal behavior which isat least qualitatively very similar to that encountered for standard SPH (com-pare Figures 4.16 and 4.20). With regards to the spectral results, we note thatwhere a large enough alpha parameter has been taken, turbulence terms areindeed able to bring about closure, with the largest scales still producing Eulerenergy scaling and energy conservation maintained. As such, eventual solutioncorruption due to resolution limitations is averted, so solutions may be saidto be regularised. Where alpha terms have been included in SPH simulations,


some interesting results are found. Large values of parameter epsilon lead tothe expected result of greater energy inhibition. For small values of the epsilon,we have the counter-intuitive result of decreased inhibition with greater energyreaching high wavenumber energy modes. Beta and alpha terms are expected tobe of similar order in this limit, so this behavior is perhaps owing to an interplaybetween terms. We may fix parameter alpha and take larger values of epsilonvia modulation of smoothing length, hence increasing the prominence of alphaterms with respect to beta terms. Then for large values of epsilon, the α-SPHsimulations produce results in line with those obtained using the spectral algo-rithm. Resulting SPH simulations are however computationally expensive withlarge iteration counts required for convergence.

The inclusion of viscosity yields the full Navier-Stokes equation. A forc-ing regime has been implemented which ensures constant energy in the funda-mental mode, with integrations eventually yielding a statistically steady state.Spectral simulations for underresolved conditions yield the expected energy ac-cumulation at the highest mode. While higher Reynolds regimes exacerbatethis shortcoming, it is not until we go above Re = 105 that this leads to en-tire solution corruption. Increase mode number simulations show that at leastfor Re = 1000, solution error is restricted to the final twenty percent of modes(Figure 4.34). SPH simulations reveal similar behavior to that found earlier andattributed to the beta terms of the modified differential equation. So where vis-cosity is insufficient to prevent significant energy propagation to scales shorterthan approximately 10h, we have a slowing of further cascades and character-istic formation of kinks in the spectrum. Again we find similar behavior forspectral results inclusive of alpha turbulence terms, with the slowing of energycascades. We perhaps cannot say that an improvement in accuracy is realisedin the use of these terms for the underresolved simulations, though as with Eu-ler simulations, they may prevent noise at high wavenumber from permeatingthrough the spectral domain. SPH results also reflect those found for Eulersimulations, with a boost in short scale energy for small values of parameter εand similar behavior to spectral results for large values of ε where alpha termsdominate.

With regards to the SPH numerical method, all results indicate that a vari-able smoothing length should always be chosen over a fixed smoothing lengthimplementation. For Burgers’ equation, this requirement is born of the needto implement viscosity summations terms over short scales. The reasons forthe vastly improved accuracy realised in Navier-Stokes simulations is less clear.Results also indicate that where we define a smoothing length according toequation (2.67), best summation accuracy is obtained for σ ≥ 2, though theadditional cost incurred may warrant use of smaller values.

In conclusion we emphasise that results in one-dimension will not necessaryreflect those to be found in higher dimensions, with very different physics tobe encountered in two and three dimensions. Perhaps most significant is theappearance of transverse waves, which are likely to influence the action of bothalpha and beta terms. At least for one-dimensional simulations we can say thatthere appears to be no benefit in using alpha terms where instead we may varythe significance of beta terms through modulation of parameter σ.

Chapter 5

The Kelvin-Helmholtzinstability

We consider the linear instabilities which may present within a fluid containinglayers in relative motion, the so-called Kelvin-Helmholtz instability. While orig-inal mention of this instability is generally attributed to Helmholtz (Helmholtz,1868), it is Lord Kelvin who first presented a thorough mathematical investiga-tion of it’s properties under various configurations (Kelvin, 1871). A minimumrequirement for the instability of inviscid unidirectional two-dimensional flowis the presence of an inflection point in the lateral velocity profile (Rayleigh,1880). Meeting this requirement is the configuration whereby a fluid containingtwo regions of different but constant velocity is separated by a surface of disconti-nuity (see Figure 5.1). The instability of this flow was established by Helmholtz(1868), and it is found that sinusoidal perturbations of all wavelengths growexponentially in time with growth rate proportional to wavenumber.

Instability may also occurs for flows where the lateral profile of the meanstreamwise velocity is given by a hyperbolic tangent function (see Figure 5.1).Such profiles correspond to free boundary layers that form where two fluids ofdifferent velocity but same direction meet, as can be found for instance at theedge of jets and wakes, and at the trailing edge of an asymmetric aerofoil. Sim-ilarly, splitter plate experiments produce profiles approximated very well by thehyperbolic tangent function (Dimotakis and Brown, 1976; Slessor et al., 1998).Unlike the constant shearing configuration discussed above, the splitter plateregime only exhibits instability over a finite bandwidth of sinusoidal modes. Assuch, there exists a particular wavelength of fastest growth, and an initial whitenoise perturbation will eventually be dominated by growth of this mode. Theensuing non-linear evolution yields coherent vorticies and the phenomenologyof mixing layers.

In this chapter the above two regimes are considered, with growth rates foundfor SPH simulations compared with values from stability theory. A number oflimitations for this comparison must be noted. Firstly, the linear theory assumesan incompressible fluid, while SPH integrations approximate the compressibleNavier-Stokes system. The convective Mach number (Bogdanoff, 1983), definedas M = (U1 − U2)/(c1 + c2), for mean stream velocities U1 and U2 and cor-responding soundspeeds U1 and U2, is the most appropriate nondimensional

82

83 The Kelvin-Helmholtz instability

U1

U2 U0 tanh(z)

Figure 5.1: Mean velocity profiles.

number to parametrise mixing layer compressible dynamics (Papamoschou andRoshko, 1988; Ragab and Wu, 1989). To simulate a nearly compressible fluid, weseek a Mach number M ' 0.05, and so for U = U1 = −U2 and Cs = C1 = C2, weset soundspeed as Cs = 20U . This yields Mach number M = 0.05, from whichit may be shown that δρ/ρ ∼ M2 = 0.0025. Density variations throughoutsimulations should therefore be minimal. Furthermore, numerous compressiblecalculations (Ragab and Wu, 1989; Sandham and Reynolds, 1991; Sauvage andKourta, 1999) have shown that dynamics are very similar to incompressible sim-ulations for Mach numbers M < 0.5, with growth rates reduced by less thanfive percent at M = 0.2 and approximately twenty percent at M = 0.5 (Vremanet al., 1996).

Another potential source of deviation from linear theory occurs due to thefinite domain within which SPH simulations are performed. Linear solutioneigenfunctions decay exponential however, so we can expect velocity perpendic-ular to boundaries to be negligible for sufficiently large computational domainsize. A clearer definition of sufficiently large will follow, though for now we sim-ply state that a domain size some multiple of the perturbed mode wavelengthshould suffice.

For the presented linear regime results, viscosity is not included. As inte-gration are only performed over a long enough time to capture the exponentialmode growth, it is not expected that any significant excitation of overly short(with respect to grid resolution) modes will occur, and so Euler simulations arevalid. We also note that while linear results for the constant shearing velocityconfiguration gives mode growth proportional to wavenumber, SPH simulationstend to attenuate growth as wavelengths approach resolution length scale, pre-venting these modes outgrowing perturbed modes. Any addition of viscosityhowever will have a stabilising effect on all wavelength modes, with reducedgrowth rates (Ragab and Wu, 1989).

5.1 Constant velocity fluids in relative motion

We consider the Kelvin-Helmholtz instabilities that may arise at the interfacebetween two fluid bodies of constant velocity in relative motion.

5.1 Constant velocity fluids in relative motion 84

5.1.1 Linear results

The onset of instabilities may be precipitated with perturbations of the form

u(x, z, t) = u(z) exp(ikx+ nkt) (5.1)w(x, z, t) = w(z) exp(ikx+ nkt) (5.2)

where k is a wavenumber, n is a growth exponent and i =√−1. These pertur-

bations are superimposed over the counter-streaming flow given by

u(z) =

+U z > 0−U z < 0. (5.3)

for a domain of infinite extent. The linearised inviscid Navier-Stokes equationsthen yields (see Chandrasekhar (1961) for details),

u(z) =B( nk − ikU) exp(+kz) z < 0B(−nk − ikU) exp(−kz) z > 0 (5.4)

and from continuity we have

w(z) =B(−ink + kU) exp(+kz) z < 0B(−ink − kU) exp(−kz) z > 0. (5.5)

The characteristic equation (Chandrasekhar, 1961) then requires that

nk = ±kU (5.6)

and for mode growth we take the positive exponent. Clearly mode growth isdirectly proportional to shearing velocity U , and also directly proportional towavenumber.

5.1.2 Computational configuration

SPH simulations are performed as follows. Particles are initially arranged on aregular square grid with separation ∆x, within a domain of width Lx and heightLz (see Figure 5.2). The domain is periodic in the horizontal, and enclosed fromabove and below with four layers of boundary particles (see Section 2.3.4 for anoutline of boundary implementations). Particles (including boundaries) intiallymove with the mean velocities (5.3). To this mean velocity, the perturbationsdefined by (5.1), (5.2), (5.4) and (5.5) are added, with perturbation strengthgiven as some fraction of mean velocity U .

For the initial field of equispaced particles, we have density set to unity, withparticle mass calculated accordingly. The equation of state used for calculationsis

P = P0

(ρ

ρ0

)γ− 1

(5.7)

with γ = 7 taken to yield strong variations in pressure for small changes indensity 1. The constant P0 will determine the soundspeed and degree of com-pressibility. As stated above, we set P0 such that Mach number M = 0.05 is

1Other values for gamma have been tested and results do not appear to be sensitive to theparticular choice (within an appropriate range). For tested perturbations, values 1.4 ≤ γ ≤ 7yielding identical growth exponents.


Lz

Lx

Figure 5.2: Initial SPH particle configuration. Red and blue particles representboundary particles and fluid particles respectively.

expected. In Figure 5.3 results for three simulations are given where the Machnumbers M = 0.1, 0.05 and 0.025 are taken. The expected trend of reducedmode growth with increased Mach number is observed, with calculated growthexponents of nk = 11.4, 11.8 and 11.9 for M = 0.1, 0.05 and 0.025 respectively(linear result give nk = 4π). So we choose M = 0.05 as further reduced Machnumber increases computational cost while not yielding significant change ingrowth rate. Details of growth rate measurement are to be found below.

For all calculations, a domain of half-width Lx = 1 is taken. To determinean appropriate domain height, we may first consider the eigenfunctions (5.4)and (5.5). It is easily shown then that for a wavenumber of wavelength λk, theperturbation drops to one percent of it’s maximum value (found at the interface)at a height of approximately z = 0.8λk. For a domain of height Lz = λk, wedo not then expect significant perturbation velocity at the boundary. However,this alone cannot assure us that boundary effects will not be significant, andthe large soundspeeds used for calculations ensures that boundaries are knownthroughout the domain. It does appear however that interface perturbationgrowth is relatively insensitive to vertical domain size, with fundamental modetests using Lz = 0.25λk, 0.5λk and λk showing minimal difference. In all threecases, mode saturation occurs at almost the same time and largely identicalgrowth rates are observed. The main requirement is that interface perturbationsare allowed room to grow, and so a minimal domain height of Lz = 20∆x isused for all simulations. Given the above concerns, we define domain height asLz = max(20∆x, 2λk).

Simulations in this section utilise both the self-consistent summation density(2.69) and a fixed smoothing length, along with momentum equation (2.75) andtimestepping regime (3.35). It is noted that some acoustic activity is evidentin these simulations (often of greater amplitude than the perturbation), thoughmode growth appears unaffected.


A(t

)

time

M = 0.1

M = 0.05

M = 0.025

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0 0.1 0.2 0.3 0.4 0.5

Figure 5.3: Perturbed mode displacement amplitude growth A(t), k = 4π. Wenote the reduction in growth rate for increasing Mach number.

5.1.3 Determination of mode growth rates

To determine mode growth exponents, we do not directly measure the velocitygrowth of the perturbation. Instead, the interface between the two fluids isconsidered, with the progression of disturbances observed and measured. Tounderstand the expected behavior of the interface, we use (5.2) to write anequation for the interface:

wI(x, t) = wI(x) exp(nkt),

where wI represents the vertical velocity at some position x along the interfaceat time t. Integration with respect to time yields

I(x, t) =1nkwI(x) exp(nkt) +D(x),

for interface displacement I(x, t) with respect to original position, and arbitraryfunction D(x). The interface is initially unperturbed, so I(x, 0) = 0, and wehave

D(x) = − 1nkwI(x)

and may then write

I(x, t) =1nkwI(x) (exp(nkt)− 1) .

Where wI(x) is given by a trigonometric function, such as in (5.2), the term(exp(nkt)− 1) acts to modulate the amplitude of perturbations. For sufficientlylarge values of time, the above becomes

I(x, t) ' 1nkwI(x) exp(nkt) (5.8)

=1nkwI(x)A(t). (5.9)


A(t

)

time

k = π

k = 10π10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Figure 5.4: Perturbed mode displacement amplitude A(t) for modes k = π andk = 10π. Measurements of growth rates are possible at earlier times for shorterwavelength mode.

The value of A(t) may be determined at each timestep of the simulations, andwe may then measure the value of nk by first taking the logarithm of A(t):

log(A(t)) = nkt, (5.10)

from which it is clear that the gradient of the curve of log(A(t)) plotted againsttime should be determined by the mode growth exponent nk.

To get an indication of what may be considered a sufficiently large timefor (5.8) (and therefore (5.10)) to be valid, we may insist that exp(nkt) ≥20, in which case we have t ' 3/nk. For the current regime, with expectedgrowth rates given by equation (5.6), modes of shorter wavelength fulfill thisrequirement much quicker than those of long wavelength. Considering Figure5.4, it can be seen that a curve gradient may be extracted confidently at earlytimes (0.2 < t < 0.5) for the k = 10π mode, while for k = π, a later time range(0.8 < t < 1.3) must be utilised. For all simulations, growth curves such asthose of Figure 5.4 are visually inspected to determine the best time range toconsider, with a least squares routine then used to give a line of best fit fromwhich a gradient (and thus growth rate) is found.

Initial velocity perturbations (equations (5.1), (5.2), (5.4) and (5.5)) takeamplitudes of strength B = 0.00001U . Such small perturbation velocities allowsufficient integration time for growth rate extraction to be performed confidently.Larger perturbations result in earlier transition to non-linear dynamics, andfor long wavelength modes this may not allow sufficient integration time fora definite growth rate calculation. For high order modes, larger perturbationstrengths may be used, though growth rates are not changed.

The final required detail is the calculation of interface perturbation ampli-tude A(t). The technique used consists of tracking the interface, and performinga Fourier analysis to determine interface perturbation modes and amplitudes.The details of this procedure are now outlined, and should be considered withreference to Figure 5.5. Particles are initially configured such that rows occurs


Figure 5.5: Interface tracking for the determination of mode amplitude. Boldgreen circles represent the SPH particles which are tagged as interface particles,with linear interpolation between these being used to determine data at equis-paced points.

at z = ±0.5∆x. The particles at z = +0.5∆x are flagged as the interface par-ticles, though the z = −0.5∆x particles (or any row actually) could have beenused. Linear interpolation is then used between interface particles to constructa continuous curve from which interface heights at regular intervals may be de-termined. Fourier analysis is then applied to the evenly spaced interface heightsto determine mode amplitudes. To avoid aliasing issues, evenly space points aretaken at four times the frequency of SPH particles.

5.1.4 Results

Simulations are performed for various SPH parameters to develop an under-standing of the influence of SPH resolution on perturbation growth regimes,both with respect to particle populations and smoothing length. A summary ofsimulation parameters is given in Table 5.1.

The SPH development of instabilities for a typical simulation are given inFigure 5.6, with the corresponding interface amplitude given in Figure 5.7. Theexpected behavior of exponential growth is observed. Owing to the small initialperturbation, much of the early exponential growth is not distinguishable inFigure 5.6, though can be clearly seen for the logarithmic scale amplitude givenin Figure 5.7. Exponential growth continues until the perturbation reachessaturation (just before time t = 1), after whichpoint the wave ‘breaks’ and theinterface ceases to be one to one (final frame 5.6). For the current Euler regimesimulations, dynamics beyond this point are largely chaotic and non-physical.Where viscosity is utilised, the classic Kelvin-Helmholtz vorticities emerge.

Resulting growth rates for simulations using single mode perturbations areto be found in Figure 5.8. All simulations result in slowed growth rates asperturbation wavelengths approach the resolution length scale. Certainly asless particles per wavelength are available, we expect integration accuracy tosuffer. The damping of poorly resolved modes is in ways a desirable artifact,


Table 5.1: Parameters for Kelvin-Helmholtz simulations

Parameter Value Description∆x 0.025, 0.0125 Initial particle separationh0 1.3∆x, 1.9∆x Initial SPH smoothing lengthα 0 Viscosity parameterU 1 Shearing velocityB 0.00001U Perturbation strengthM 0.05 Mach numberLx 1 Horizontal domain extentLz Lz = max(20∆x, 2λk) Vertical domain extentε 0.0, 0.1, 0.5, 1.0 Turbulence cutoff parameter

minimising their influence.In Figure 5.8 we also note the difference in behavior where smoothing lengths

of h = 1.3∆x and h = 1.9∆x are taken. The larger smoothing length exhibitsmore accurate growth rates in nearly all cases, but perhaps more importantlyis the consistency of results, and predictable change in growth rate as perturba-tion length scales approach smoothing length. Where h = 1.3∆x is used, growthwithin the linear regime for short wavelength perturbations do not track an ex-ponential curve with the accuracy found for h = 1.9∆x simulations (see Figure5.8). Hence the erratic calculated growth rates found in Figure 5.8 are a re-flection of both the inaccurate exponential growth, and the ensuing difficulty indetermining growth rate exponents. We conclude that a smoothing length ofh = 1.3∆ does not appear to be sufficient for accurate discrete representationof the approximation integral 2.1. It is noted however that larger smoothinglengths result in increased neighbouring particles withing respective interac-tion radii, and so computational costs per timestep are increased significantly.For the current simulations, this increase is of order fifty percent, though largertimesteps may be taken which negate additional cost. Regardless, for the currentKelvin-Helmholtz simulations, it appears additional cost are warranted. Indeed,the h = 1.9∆x simulations give excellent clear exponential growth regimes forall perturbations up to k = 35π (approximately five particles per perturbationwavelength).

The lower resolution simulations of Figure 5.8 exhibit a similar trend to thosefound at higher particle numbers, with results scaling according to smoothinglength. Figure 5.10 gives numerical growth rates nk scaled against theoreticalgrowth rates nk as a function of particles per perturbation wavelength. Re-sults scaled very well for h = 1.9∆x, with at least ten particles per wavelengthrequired for growth rates within twenty percent of predicted values. Wheresmoothing length h = 1.3∆x is used, the inconsistency again becomes evident,although short lengthscale growth rates are closer to the theoretical result. Inter-estingly, for both high resolution simulations, a fundamental mode perturbationyields growth rates up to twenty percent higher than expected.

Results presented above were almost identical for both variable and con-stant smoothing length algorithms. However, only the linear regime has beenconsidered, with one-dimensional simulations indicating that variable smoothinglengths give significant improvements where non-linearity is of importance.


Figure 5.6: Development of Kelvin-Helmholtz instability. Bold points markout particles used to determine interface growth. For this particular simulation ak = 4π perturbation has been used, with parameters ∆x = 0.025 and h = 1.9∆x.Corresponding interface amplitude growth is to be found in Figure 5.7.

A(t

)

time

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.2 0.4 0.6 0.8 110-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Figure 5.7: Interface perturbations amplitudes for solution displayed in Figure5.6. Broken line gives solution over a logarithmic scale, while solid line showslinear scale solution.

91 The Kelvin-Helmholtz instabilitynk

k/π

∆x = 0.0125, h = 1.9∆x

∆x = 0.0125, h = 1.3∆x

∆x = 0.025 , h = 1.9∆x

∆x = 0.025 , h = 1.3∆x

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

Figure 5.8: Perturbation growth rates for SPH simulations at different fixedsmoothing lengths and particle separations of ∆x = 0.0125 and ∆x = 0.025. Thebold line gives growth rates predicted by linear theory.

A(t

)

time

h = 1.9∆x

h = 1.3∆x

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 5.9: Comparison of exponential growth for different smoothing lengths.Perturbation amplitudes are shown on a logarithmic scale to highlight differences.Shown results correspond to a k = 20π perturbation with ∆x = 0.125.

5.2 The hyperbolic tangent velocity profile 92

nk

h = 1.9∆x h = 1.3∆x 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

10 100 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

10 100

Figure 5.10: Scaled growth rates nk = nk/nk against particles per perturbationwavelength. Filled circles and crosses give results for ∆x = 0.025 and ∆x =0.0125 respectively.

Simulations have been performed incorporating the α-SPH turbulence terms,with results using h0 = 1.3∆x given in Figure 5.11. The general observed trendover all simulations is a damping of mode growth, with damping increasingfor larger wavenumbers, and for larger values of parameter ε. Results for thesimulations with ε = 0.1 and ε = 0.5 are almost indistinguishable. For ε = 1.0,mode damping is observed for all modes greater than k = 4π. Here modesgreater than k = 17π do not produce sufficiently clear growth for an accuratedetermination of exponents. Indeed, as with standard SPH at h0 = 1.3∆x,there is some uncertainty in exponent calculation (see Figure 5.9) for all testedmodes, with the degree of uncertainty increasing with wavenumber. For thisreason there is some scatter in the data presented in Figure 5.11.

For standard SPH, as discussed above, a clearer exponential growth range isencountered where smoothing length parameter h0 = 1.9∆x is utilised. Unfortu-nately α-SPH simulations with h0 = 1.9∆x under the current Kelvin-Helmholtzconfiguration results in an interface instability which appears to be related to thediscontinuous velocity profile. A constant vertical velocity is observed to growacross the entire interface, and in the absence of viscosity this leads to horizontallayers of particles below the interface being transported upwards, and vice-versa,with layers shearing past each other. Therefore Kelvin-Helmholtz mode growthdoes not occur. We leave this configuration and instead consider the continuousinterface provided by the hyperbolic tangent mean velocity profile.

5.2 The hyperbolic tangent velocity profile

We now turn to the instabilities which may arises where the mean velocityprofile takes the hyperbolic tangent form (5.1). Such profiles are found wheretwo fluids of different velocity but coincident direction meet, with a boundarylayer then resulting in a continuous velocity across the interface.

5.2.1 Linear results

The linear stability results for hyperbolic tangent mean flow are given by Michalke(1964). These results have been used to provide the initial velocity disturbances

93 The Kelvin-Helmholtz instabilitynk

k/π

ε = 0.0

ε = 0.1

ε = 0.5

ε = 1.0

0

10

20

30

40

50

60

70

2 4 6 8 10 12 14 16 18 20

Figure 5.11: Perturbation growth rates for α-SPH simulations using h = 1.3∆x.

from which exponential growth follows, and we outline the required methodhere. The flow is decomposed into mean and perturbed quantities according to

u(x, z, t) = u(x, z, t) + U(z) (5.11a)v(x, z, t) = v(x, z, t) (5.11b)

with mean velocityU(z) = tanh(z), (5.12)

for which perturbations defined by

u(x, z, t) = B exp(kct) φ′r(z) cos(kx)− φ′i(z) sin(kx) (5.13a)v(x, z, t) = kB exp(kct) φi(z) cos(kx) + φr(z) sin(kx) (5.13b)

are appropriate (Lin, 1955), with primes denoting differentiation with respectto z. These perturbations are derived of a stream function which guarantees thedivergence-free condition. The function φ(z) = φr(z)+iφi(z) and it’s derivativesdefine the perturbation lateral profile and from here dependence on z will notbe explicated. Disturbance growth is dictated by the value n ≡ kc, with phasespeed for all wavelength disturbances being zero as a result of chosen meanvelocity (5.12) (Michalke, 1964). Perturbation amplitude may be determinedby the free parameter B. Inserting equations (5.11) and (5.13) into the Eulerequations yields the Rayleigh stability equation

[U − c][φ′′ − k2φ

]− U ′′φ = 0, (5.14)


n

k

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.12: Perturbation growth rates n variation with wavenumber k forhyperbolic tangent mean velocity configuration.

where we have assumed u U and v U and terms second order in the pertur-bations are discarded. For given wavenumber k, we then require correspondingeigenvalue c (and hence growth exponent n = kc), along with eigenfunctionφ. Solution to equation (5.14) is sought over a domain periodic in x (which isfulfilled by our perturbations (5.13) and mean velocity (5.12)) and infinite inthe vertical direction. We require that perturbations go to zero as the verticalextent becomes infinite.

To determine eigenvalues, we simplify equation (5.14) by setting

φ(z) = exp(∫ z

0

Φ(z∗)dz∗)

from which the Riccati equation is obtained:

Φ′ = k2 − Φ2 +U ′′

U − c. (5.15)

To reduce our domain to a finite interval, the transformation y = tanh(z) isintroduced. The following equations are then obtained for Φ(y) = Φr(y) +iΦi(y):

dΦrdy

=k2 − Φ2

r + Φ2i

1− y2− 2y2

y2 + c2(5.16)

dΦidy

= −2ΦrΦi1− y2

− 2cyy2 + c2

. (5.17)

The above equations have been simultaneously solved using a fourth-orderRunge-Kutta scheme coupled with a shooting method to home in on the requiredeigenvalue for chosen wavenumber k. The procedure is as follows. The symmetryof equations (5.16) and (5.17) is exploited, with integrations performed fromy = −1 to y = 0. Boundary conditions follow from the requirement that


φr

φi

z

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Figure 5.13: Solution eigenfunctions. From top to bottom, functions correspondto wavenumbers k = 0.1, 0.3, 0.5, 0.7 and 0.9. The dashed curve corresponds tou = tanh(x) mean velocity profile, which has been included to illustrate the scaleof eigenfunctions.

eigenfunctions vanish at infinity:

Φr(−1) = k

Φi(−1) = 0dΦrdy

(−1) =−2

(1 + c2) (k + 1)dΦidy

(−1) =2c

(1 + c2) (k + 1).

Integration is then performed for different eigenvalues (holding wavenumberconstant), with the correct value being found where the following conditions aremet:

Φr(0) = 0dΦidy

(0) = 0.

Initial bounding values of c = 0 and c = 0.5 are taken, with Newton’smethod being utilised to locate the correct eigenvalues. Growth rate results aredisplayed in Figure 5.12. Maximal mode growth is found for wavenumbers k '0.44, with growth tending to zero as perturbation wavelength become infinite.


Table 5.2: Parameters for hyperbolic tangent mean velocity simulations

Parameter Value Description∆x λk/Nx Initial particle separationNx 20, 40, 60 Number of particles per perturbationh0 1.9∆x Initial SPH smoothing lengthα 0 Viscosity parameterU0 1 Hyperbolic tangent amplitudeB 0.0001U0 Perturbation strengthM 0.05 Mach numberLx λk Horizontal domain extentLz From perturbation Vertical domain extentε 0.0, 0.5, 1.0, 5.0 Turbulence cutoff parameter

As wavelengths approach sizes of order of the boundary layer thickness (asdetermined by mean profile (5.12)), mode growth also goes to zero.

Having determined eigenvalues, we may return to the Rayleigh stabilityequation to find corresponding eigenfunctions. Equation (5.14) is integratednumerically again using the fourth-order Runge-Kutta method. Symmetry of φfollows from symmetry of Φ, so we are only required to integrate for z ≥ 0. Theconditions at z = 0 are:

φr(0) = 1, φi(0) = 0φ′r(0) = 0, φ′i(0) = Φi(0).

Resulting eigenfunctions may be found in Figure 5.13. These functions,together with their derivatives (also obtained during eigenfunction integration),complete the perturbation specification given by equation (5.13).

While results given in this section assume an infinite domain, solutions fora finite domain (IJzerman, 2000) are almost indistinguishable provided that asufficient domain size has been taken. We define a sufficient domain with respectto any given eigenfunctions, and require that the eigenfunction is of negligiblemagnitude at the domain extent. Taking for example wavenumber k = 0.7perturbations (Figure 5.13), a domain of size Lz = 10 would result in negligibledifference between finite and infinite domain linear solutions.

Obtained eigenfunctions and eigenvalues compare favourably with those ofMichalke (1964) and IJzerman (2000).

5.2.2 Computational configuration

The numerical configuration used for the hyperbolic tangent simulations is muchthe same as used earlier for the discontinuous velocity profile simulations. Thekey difference here is that for all simulations within a series, the number of par-ticles per disturbance wavelength is held constant, so we write ∆x = λk/Nx,for perturbations of wavelength λk, and Nx particles spanning the domain hor-izontally. Simulation domain is then defined by the perturbation wavelengthand eigenfunctions, taking Lx = λk and Lz chosen to sufficiently represent therequired eigenfunction of Figure 5.13.


n

k

Nx = 60Nx = 40Nx = 20

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.14: Measured growth rates for SPH simulations at different resolution.

Perturbation strength takes the value of B = 0.0001U0. Tests performedshow negligible difference for perturbation amplitudes in the range 0.00001U0 ≤B ≤ 0.01U0 (see Section 5.2.3 for further details). All other configuration detailsare identical to those used earlier. A summary of parameters is given in Table5.2.

5.2.3 Results

We perform all tests over the spectrum of wavenumbers for which mode growthis expect. The particular wavenumbers simulated are those from k = 0.1 tok = 0.9 in k = 0.1 increments. First considered is the change in mode growthwhere different particle numbers are used to represent each perturbation wave-length. Tests are performed for Nx = 20, 40 and 60 with results given in Figure5.14. While modes are simulated using equivalent number of particles (for eachseries of simulations), we note that this gives different resolutions in the lat-eral direction with respect to the mean velocity profile. This however does notappear to be a limitation, with all simulations producing the required growthrates very well over the entire spectrum. It is also noted that in the limit of di-minishing wavenumber, growth rate results reduce to those of the discontinuousvelocity profile (equation (5.6)), in which case the coarsely represented verticalprofile is not a limitation. For the simulations in Figure 5.14, best results areobtained for Nx = 40 and Nx = 60, with each series showing improved ac-curacy at different wavenumbers, but neither clearly superior. The Nx = 20simulations, while still producing the required mode growth trend well, doesexhibit a greater degree of scatter across the spectrum.

The above simulations have been performed for relatively weak perturba-tion amplitude of B = 0.0001U0. While for any sufficiently small perturba-tion the linear stability results presented above are valid, simulations havebeen performed to determine any sensitivity to perturbation strength. Fig-


n

k

B = 10−5U0B = 10−4U0B = 10−3U0B = 10−2U0B = 10−1U0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.15: Growth rates observed for different perturbation strengths. Forthese simulations, parameter Nx = 40 has been used.

ure 5.15 gives measured growth rates for perturbation strengths ranging rang-ing from B = 0.00001U0 to B = 0.1U0. Results are visually identical for0.00001U0 ≤ B ≤ 0.001U0. For B = 0.01U0 an almost indistinguishable re-duction in mode growth is found at high wavenumbers, and for B = 0.1U0 afurther reduction is observed for medium to high wavenumbers. Given that onlyminor differences are observed over the range of perturbation strengths tested,we are reassured of the validity of using small perturbaton strengths.

We turn to Kelvin-Helmholtz simulations which utilise the α-SPH algorithm.Simulations are performed for turbulence parameter ε = 0.5, 1.0 and 5.0, withresulting growth rates given in Figure 5.16 and growth rate change (with respectto standard SPH) given in Figure 5.17. We see that for simulations using valuesof ε ≤ 1.0, growth rates are almost identical to simulations without turbulenceterms. No clear trend is apparent in the small deviations for these results,though generally a slight increase in growth rates is observed. For the ε =5.0 simulations, all modes exhibit a reduction in growth rates. The changein growth rates brought about by the introduction of turbulence term, ∆n =nα − nsph, is given in Figure 5.17, where we define nα and nsph as respectivelythe growth rates for α-SPH and standard SPH. It is found that the reduction ingrowth exponent is approximately constant for the spectrum of modes tested.This is an expected result, as the effective turbulence model cutoff lengthscaleis proportional to the smoothing length, and so is held constant in relationto perturbed wavenumbers for given simulations. With reference to equation(3.11), we may write in two dimensions:

α2 =31196

h2ε. (5.18)

So for ε = [0.5, 1.0, 5.0] we have α/λk = [0.013, 0.019, 0.042]. Clearly the cutofflengthscale α is much smaller than the perturbed wavelength λk, though in light


n

k

ε = 0.0ε = 0.5ε = 1.0ε = 5.0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.16: Perturbation growth rates for α-SPH simulations.

∆n

k

ε = 0.5ε = 1.0ε = 5.0

-0.08

-0.06

-0.04

-0.02

0

0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.17: Change in growth rates for α-SPH simulations, ∆n = nα − nsph.

of the low order Helmholtz velocity filtering, we still expect a degree of modeattenuation (see Figure 3.1) at all tested turbulence parameters.

The results for the α-SPH simulations may be contrasted with findings whereonly the filtered velocity has been used to advect particles, with turbulenceterms removed from the acceleration equation. Growth rates for simulationsusing equivalent turbulence parameters to those of Figure 5.16 are given in Fig-ure 5.18, with change in growth rate ∆n displayed in Figure 5.19. We observethat for these simulations growth rates appear to be slowed proportionally to


n

k

ε = 0.0ε = 0.5ε = 1.0ε = 5.0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.18: Perturbation growth rates for standard SPH simulations withparticles advected using filtered velocity.

∆n

k

ε = 0.5ε = 1.0ε = 5.0

-0.08

-0.06

-0.04

-0.02

0

0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.19: Change in growth rates for filtered velocity SPH simulations,∆n = nα − nsph.


parameter ε. Comparison of Figures 5.17 and 5.19 reveals that mode attenua-tion is significantly larger where turbulence terms have not been included. Wemay perhaps conclude that the additional constrains of energy and circulationconservation yields correct integration of coarse scale (with respect to α cut-off parameter) dynamics despite the damping effect of the filtered velocity atthese length scales. However where a sufficiently large turbulence parameteris utilised, it is certainly expected that even α-SPH simulations will give re-duced growth rates as the perturbaton lengthscale approaches α. Sign of thisis observed for the ε = 5.0 simulation where reduced growth is observed forα-SPH, though to a lesser extent than what is found for filtered velocity advec-tion alone. Indeed the value of turbulence parameter ε (or α) determines whichlengthscales are be considered as coarse scales, and which are to be consider assub-grid scales. Choice of appropriate turbulence parameter will be dependenton computational resources, and the required minimum wavenumber for whichdynamics will be expected to be governed by Navier-Stokes dynamics. This willbe determined by the problem being considered, and by the behavior of thestandard SPH algorithm where resolution is limiting.

With a filtered velocity alone, results indicate that mode growth attenuationacts over a broader scale than encountered for α-SPH. Furthermore, we specu-late that violation of energy conservation will lead to more striking differencesfor non-driven simulations2, such as perhaps a travelling wave simulation, or de-caying turbulence, where the filtered velocity will act to dissipate any wave-likemotion.

5.3 Conclusion

The results presented here indicate that SPH is able to correctly reproduceKelvin-Helmholtz growth rates where sufficient particles are utilised to representperturbations. The growth rates are observed to be attenuated as perturbationlengthscale approaches SPH smoothing length scale. This slowing of growthrates may be linked to a ‘weakness’ of the SPH pressure gradient operator inthe limit as perturbation wavelengths approach the SPH smoothing length. Fur-thermore, the noise observed for simulations where we set h0 = 1.3∆x may alsobe traced to innacuracies in the pressure gradient calculation for this parameter.To quantify the performance of the pressure gradient, we define the quantityQP (k):

Q2P (k) =

[∇sphP

∗·∇sphP

]k[

∇P∗ ·∇P

]k

. (5.19)

Full details for the calculation of equation (5.19) may be found in Section 6.1.2.For now suffice to say that it gives a measure in spectral space of the SPHpressure gradient (as determined by (2.41), and represented in the numerator of(5.19)), relative to an analytically calculated pressure gradient (represented inthe denominator of (5.19)). Where the SPH pressure gradient produces resultsidentical to the analytic pressure gradient, we expect the value QP = 1, andif the SPH pressure gradient is weaker than the analytic equivalent, we expectthe values 0 ≤ QP ≤ 1. The function QP (k) is displayed in Figure 5.20,

2We consider the Kelvin-Helmholtz simulations driven in the sense that the mean flowprovides a large kinetic energy potential from which perturbation growth is driven.

5.3 Conclusion 102

nkQP (k)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 100

Figure 5.20: Scaled growth rates nk = nk/nk and pressure gradient qualityfactor against particles per wavelength.

alongside the scaled growth rates of Figure 5.10. Clearly there is a very strongcorrelation between the weakness of the SPH gradient and the slowed growthrates of perturbation where insufficient particles per wavelength are used. Notethat the data used for calculation of QP (k) is from the periodic turbulencesimulations of Chapter 6, though we expect the profile of QP (k) will largelyidentical for the Kelvin-Helmholtz simulations.

The second mean velocity profile considered was the hyperbolic tangent,with the eigenvectors and eigenvalues determined as outlined in Section 5.2.1.Theoretical growth rates were again found to be reproduced accurately by SPHsimulations. Investigation of α-SPH indicate that it is successful in reducinggrowth rates at short scales, though not as aggressively as found for simulationswhere only the filtered velocity is utilised. The difference between the simu-lations lies in the additional acceleration terms of α-SPH which act to restorethe energy conservation violated by a filtered velocity, and appear to counterslowing of growth rates due to the velocity filtering.

These findings give indication that α-SPH may perform successfully as aturbulence model, though linear regime simulations are certainly not challengingenough to draw strong conclusions about the potential of the model. We nowtherefore consider full non-linear simulations of turbulent two-dimensional flow.

Chapter 6

Two-DimensionalTurbulence

It is often contended that turbulence cannot truly exist in two dimensions. Ofcourse all physically realised flows must contain some degree of three dimension-ality, but at times motion may be largely constrained in a particular direction.This may be due to domain limitations, or other limiting forces, such as thosewhich can arise due to rotation. These flows may then be categorised as two-dimensional turbulence in the sense that turbulent dynamics (i.e. displayingsignificant variability and irregularity) only occurs in two dimensions, or in thesense that they are approximated well mathematically by a two-dimensionaltruncation of the Navier-Stokes equations. This reduced system of equationsin some ways presents a simplification of governing dynamics, with for instancevorticity stretching being eliminated.

It is perhaps ironic then that the two-dimensional Navier-Stokes system givesrise to seemingly counter-intuitive dynamics which differ significantly from thethree-dimensional counterpart. In particular, it is observed that contrastingthe three-dimensional situation, energy in two-dimensional turbulence tends tocascade towards smaller wavenumbers (larger scales). The physical space reali-sation of the inverse cascade processes is the coalescence of similar sized vorticies(Frisch and Sulem, 1984). This process of self organisation culminates in thecreation of large coherent structures which persist over long times, travelling un-der the action of almost inviscid advection. The most obvious example of thisbehavior in nature may be found within the atmosphere1 where we find that in-verse cascade processes yield the phenomena of cyclones. Similarly, some aspectsof ocean dynamics are approximated well by two-dimensional turbulence. In thelaboratory, a number of approaches have been used to study two-dimensionalturbulence. Paret and Tabeling (1997) have performed experiments using twothin layers of stably stratified fluid to create a quasi two-dimensional config-uration, the upper layer in effect moving inviscidly. With turbulence drivenby magnetic fields, the formation of an inverse energy cascade was successfullyobserved. Another common approach uses thin soap films as a fluid base uponwhich excitations may be imposed (see for instance Martin et al. (1998)). A

1Note that the vertical extent of the atmosphere is orders of magnitude smaller than thehorizontal scale.

103

104

comprehensive review of related experimental results is given in Kellay andGoldburg (2002).

The theory of two-dimensional turbulence has been developed in the pio-neering works of Kraichnan (1967) and Batchelor (1969). A number of keyconjectures were given in these publications:

1. An inverse energy cascades exists, with kinetic energy moving from smallto large scales.

2. A direct cascade of enstrophy (mean squared vorticity) exists, enstrophybeing transferred to shorter and shorter scales until it is acted upon byviscosity.

3. Under an appropriate scaling, the energy spectrum is self similar.

This applies for a turbulent two-dimensional homogeneous isotropic fluid ofsufficiently high Reynolds number such that an inertial range may form. Fol-lowing Batchelor, the arguments leading to the above conjectures begin withthe evolution equation for vorticity in two dimensions. Taking the curl of theNavier-Stokes equations yields the required condition:

dωdt

= ν∇2ω, (6.1)

where in two-dimensions we have ω = (0, 0, ω). It can be seen that in theinviscid limit, vorticity is then advected as a passive scalar. Most importantly,the mechanisms of vortex stretching are absent in equation (6.1). This keyomission results in the conservation of kinetic energy in the limit as viscositydiminishes, the most significant departure from three-dimensional turbulencewhere energy dissipation instead approaches a constant as viscosity disappears.Vortex stretching dynamics are integral to this finite dissipation in the three-dimensional case, with any reduction in viscosity resulting in an intensification ofvortex stretching, followed by amplification of small scale vorticity, and hencecompensation of dissipation. In two dimensions, the kinetic energy evolutionequation may be written

12

ddt

v · v = −ν ω2 (6.2)

where spatial homogeneity has been assumed and overbars denote averages.Further to this, the equation for the evolution of enstrophy is written

12

ddtω2 = −ν∇ω · ∇ω, (6.3)

from which it can be seen that enstrophy must decline monotonically. Equa-tions (6.2) and (6.3) illustrates the previous contention that kinetic energy isapproximately conserved for two-dimensional turbulence in the limit of vanish-ing viscosity. We argue that since enstrophy is bound from above via equation(6.3), energy dissipation therefore must vanish as viscosity goes to zero. As such,any significant cascade of energy from large to small scales (which would even-tually lead to dissipation) must be precluded where viscosity is small, hintingat the possibility of an inverse cascade.

In contrast to kinetic energy, as viscosity falls the dissipation of enstrophyneed not vanish. We again note that vorticity will be advected as a passive

105 Two-Dimensional Turbulence

ε2/3k−5/3

β2/3k−3

E(k)

kf k

Figure 6.1: Log-log plot of the predicted energy spectrum E(k) where turbu-lence is forced at wavenumber kf .

scalar in the non-viscous limit (according to equation 6.1), and we expect vor-ticity within any closed subsection of the domain to be continually twisted andwound up, with layers of differing vorticity eventually being brought closer to-gether, therefore driving vorticity gradients to ever higher values. Analogous tothe three-dimensional result of energy dissipation tending to a finite value fordiminishing viscosity, it may be shown that smaller viscous forcings are compen-sated by increased vorticity gradients, hence giving rise to constant dissipation ofenstrophy as viscosity vanishes. This process gives an indication that we mightexpect a forward enstrophy cascade, facilitating finite enstrophy dissipation.

Kraichnan (1967) considers the situation where an infinite two-dimensionalfluid is excited by a small bandwidth forcing centered on some wavenumber kf .We define the total energy per wavenumber E(k) such that for total kineticenergy Ekin and mass m we have:

Ekin =12

∫R

ρv · v dx = m

∫ ∞0

E(k) dk (6.4)

for the domain R. For the inertial section of the spectrum E(k), two similarityranges are then identified. Where it is assumed that the spectrum dependsonly on wavenumber k and net energy transfer per unit mass ε, a scaling lawequivalent to Kolmogorov’s three-dimensional result is found:

E(k) = Cε2/3k−5/3. (6.5)

Here C is some constant which is expected to be different to that of three-dimensional turbulence. As eluded to above, we also expect the sign of thetransfer rate ε to be reversed for two-dimensional turbulence. We may instead

106

assume that spectrum E(k) depends only on k and the net enstrophy dissipationrate β, from which we find the scaling

E(k) = Dβ2/3k−3, (6.6)

for a constant D. Kraichnan gives weight to the consistency of relations (6.5)and (6.6) through use of Fourier mode triad interactions, showing that where(6.5) is valid, energy transfer ε is independent of k, and β is identically zero.Likewise, where (6.6) is valid, β is independent of k and the energy transferrate ε is zero. So where energy is supplied at wavenumber kf , we may expecttwo inertial ranges to form. For wavenumbers k < kf , a range of the formgiven by (6.5) is expected, with energy being carried to lower wavenumbers bythe inverse energy range. For a finite periodic domain simulation, the inversecascade continues until modes of wavelength similar to the domain size areexcited. Energy will be continually injected at these large scales until suchmodes are sufficiently energetic for viscosity to act (Lesieur, 1990; Tran andBowman, 2004). This process has been likened to Bose-Einstein condensationby Kraichnan.

For wavenumbers k > kf , an enstrophy cascade range may be expected,with spectrum given by (6.6), and enstrophy carried to higher wavenumbers.Kraichnan also introduced a logarithmic correction to equation (6.6) in light ofnon-localness of interactions in spectral space, though there is little evidence tosupport this correction. For the enstrophy cascade, viscosity must eventuallybecome significant, halting further cascades.

The earliest simulations of forced two-dimensional turbulence dates to workof Lilly (1972) who used a finite difference simulation forced at kf ' 8 alongwith a large scale friction and was able to produce a k−5/3 inverse energy cas-cade range, along with the k−3 enstrophy cascade spectrum. Indeed the inverseenergy cascade scaling appears to be robust with many authors observing thepredicted k−5/3 energy scaling in numerical simulations (Frisch and Sulem, 1984;Maltrud and Vallis, 1991; Boffetta et al., 1999; Tran and Bowman, 2004). Ex-perimental evidence also suggests the existence of the k−5/3 range, includingthe electromagnetically driven shallow flows of Paret and Tabeling (1997) anddriven soap films of Rivera and Wu (2002). The enstrophy cascade range how-ever appears to be a much more elusive phenomena, with most reporting energyscaling as k−α with exponent α ≥ 3. While some have found the exponent αto be within the range 3 ≤ α ≤ 4 (for instance Maltrud and Vallis (1991) andLindborg and Alvelius (2000)), others report values as large as α = 6 (Dahlburget al., 1990; Basdevant et al., 1981). The experimental results of Martin et al.(1998) suggests an exponent α ' 3.3. Interestingly, the k−3 energy scaling hasbeen observed in the atmosphere, but on the infrared side of the spectrum,with the inverse energy cascade k−5/3 instead found at shorter scales (Lind-borg, 1999). A possible explanation is found in the energy spectrums of longtime integrations by Tran and Bowman (2004). These show a striking similar-ity to the spectrum obtained via wind data in Lindborg (1999), with a largescale k−3 spectrum eventually appearing as a result of long time Bose-Einsteincondensation.

In this chapter we consider both SPH and α-SPH simulations of forced two-dimensional turbulence. Forced simulations have been chosen to be investigatedas they provide a basis for clear evaluation of SPH over a broad spectrum of


lengthscales, yielding stationary spectrums which allow for direct comparisonsbetween different simulations. In this respect, the perhaps fleeting dynamicsof decaying turbulence are in ways more difficult to qualify in terms of theBatchelor theory, with spectrums which are transient in nature and thereforedo not make for easy comparison between simulations.

The main purpose of performing these simulations is to evaluate the abil-ity of the SPH and α-SPH algorithms to yield results in line with theoreticalexpectations and with data presented in the literature. Given the geometricsimplicity of the simulations, spectral methods are a better choice for an inves-tigation of two-dimensional turbulence theory, and results here instead largelyserve to illuminate difficulties the SPH numericist may encounter. The chapteris organised as follows. In Section 6.1 details of the computation configurationare considered. This is followed by simulation results and discussion for simu-lations utilising short scale forcing in Section 6.2, followed by results for largescale forcing in Section 6.3 where simulations including the α-SPH model arealso presented. Finally concluding comments are made in Section 6.4.

6.1 Computational configuration

SPH simulations are carried out within a square domain periodic in both thehorizontal and vertical direction. Periodic boundaries are set at Lx = ±0.5 andLy = ±0.5. As with previous simulations, density is initialised at ρ = 1, withparticle masses determined accordingly. Particles are initialised on a regulargrid of particle seperation ∆x, with forcing then applied until an irregular con-figuration is achieved. Subsequently, damping and viscosity are used to bringparticles to rest. The initial SPH smoothing length is set at h = 1.9 ∆x, andthe variable smoothing length implementation is used.

6.1.1 Forcing

To initiate and drive turbulent hydrodynamics in the sense of Kraichnan (1967),a random forcing regime is required to inject energy at wavenumbers fallingwithin a thin annulus in wavenumber space. Many methods exist for imple-menting such a random forcing. Nadiga and Shkoller (2001) use a forcing whichensures a constant amplitude for all modes within a certain bandwith. A varia-tion to this instead ensures that total energy within any particular waveband ismaintained at some predetermined level. A perhaps more physically realistic al-ternative proposed by Alvelius (1999) instead applies a forcing of constant powerover a wavespace annulus, with the energy input expected to be eventually bal-anced by dissipation. These methods all apply trigonometric mode forcing, as isconvenient where spectral techniques have been employed. Alternatively, forcingwithin physical space may be employed. Boffetta et al. (1999) have used a Gaus-sian forcing f(r, t) with correlation 〈f(r, t)f(0, t′)〉 = F0 exp(−(r/lf )2)δ(t− t′),such that forcing should rapidly decline for r lf . We note that this forcingcannot be used to investigate the direct enstrophy cascade of Batchelor (1969),and as most results presented in the literature use spectral forcing, it is appro-priate for us to use similar techniques so that clear comparisons may be made.We consider the method of Alvelius in further detail.

6.1 Computational configuration 108

For the verlet timestepping scheme (2.59), the velocity timestep is written

v1 = v0 + ∆t (g1/2 + f1/2) (6.7)

for some predetermined random forcing function f = f(x, t). Dominant termscontributing to kinetic energy power input averaged over a timestep are givenby

P =∆t2

f1/2 · f1/2 + f1/2 · v0 = P1 + P2. (6.8)

Here overbars represent volume averages over the entire spatial domain. Werequire that the force-force correlation function P1 makes the most significantcontribution so that power input may be controlled and constant. To remove thetimestep size dependence of P1, forcing will be written as a function of timestep(f ∼ (∆t)−1/2). The secondary contribution P2 should be much smaller, thoughwhere a small number of forcing modes are utilised and the timestep is small,the forcing function will be large and P2 may correspondingly become signifi-cant for any given step. Alvelius considers forced three-dimensional simulationswhere forcing is implemented at large scales only, and involves a relatively smallnumber of modes. In such situations and where precise energy input is required,measures to ensure the term P2 is zero may be necessary. For the current sim-ulations where we force at intermediate wavenumbers, typically over a hundredcontributing modes are utilised, so we may be confident that P2 averages to zeroover short times (relative to total simulation time). Simulations below for largescale forcing do exhibit significant variability in the energy input, and wherethis is of concern we instead force to achieve constant total energy within theforcing wavespace annulus. Further details for this are given in Section 6.3.

Additional terms also arise due to the Navier-Stokes forces and are neglectedin equation (6.8). These terms are either zero on average, or tend to zero withshortening timestep (Alvelius, 1999). Where P2 = 0, we write (dropping thetimestep index)

P =∆t2

f · f

=∆t2

∑kx

∑ky

fx(kx, ky)f∗x(kx, ky) + fy(kx, ky)f∗y (kx, ky)

(6.9)

for complex spectral forcing component f(kx, ky) where

f(x, y) =∑kx

∑ky

f(kx, ky) exp(i(kxx+ kyy)).

We define functions F (k) and G(k) which give respectively the total and averagepower input due to all components of (6.9) falling within a wavespace annulusof radius k =

√k2x + k2

y. We may then write equation (6.9) as

P =∆t2

∫ ∞0

F (k)dk =∆t2

∫ ∞0

k

2πG(k)dk. (6.10)

To ensure that forcing does not introduce compressible moments to the velocityfield, we impose the constraint

k · f(k) = 0


which must be met for any forcing mode f(k). This may be satisfied by writing

f(k) =1k

(kyAran ,−kxAran

)with complex random number Aran. It is required that the power input for amode of this form is given by the function G(k):

G(k) = AranA∗ran =

2πkF (k),

which is satisfied by

Aran = exp(i θ)

√2πkF (k)

with random number θ = [0, 2π]. This yields a forcing where all modes withinany particular shell have equal amplitude, but random phase. It is noted thatthis forcing is not truly isotropic, as there is some dependence on the principalaxes with which wavenumbers are defined. A further random rotation couldpossibly be applied to forcing component f(k), though this would not be com-patible with the square periodic domain (which in itself presents a degree ofanisotropy). New random numbers are taken at each timestep.

It remains to define the function F (k) which will determine the shape of theforcing spectrum. A common choice is the Gaussian profile, centered on somedominant forcing wavenumber kf :

F (k) = A exp(−(k − kfc

)2 )The constant A is determined such that the required power input is P . Withreference to equation (6.10) we have:∫ ∞

0

A exp(−(k − kfc

)2 )dk =

2P∆t

.

For kf sufficiently large, and c sufficiently small, we may write

A =2P

c√π∆t

.

For the SPH algorithm, we define the particle forcing fb for particle b as

fb =nm∑i=1

fi exp(iki · xb), (6.11)

where we have a list of contributing forcing coefficients fi with wavenumber ki,and nm total contributing modes. This equation is added to the left hand sideof the SPH momentum equation.

6.1.2 Viscosity

Viscosity in numerical simulations of two-dimensional turbulence is often rele-gated to a secondary role. The more important Euler dynamics usually takes


centerplace with viscosity considered a tool to simply remove energy that ap-proaches the spectral limit in a quasi-physical manner. Indeed the Newtonianviscosity is most often replaced with so-called hyperviscous terms, which areconstructed of higher order Laplacian operators:

fν = (−1)n+1νn∆nv. (6.12)

Here the parameter choice n = 1 yields the standard Newtonian viscosity. Hy-perviscosity is obtained where higher values of n are taken, with values rangingup to n = 8 often found in the literature (see for instance Maltrud and Vallis(1991)). Large values of n effectively yield higher order velocity filtration, withmodes less than some cutoff wavenumber largely unchanged, and modes beyondstrongly attenuated. Hyperviscosity in effect acts over a shorter bandwidth,with a larger portion of the available spectrum then to be considered inviscid.

The SPH viscosity operator may take a number of forms, as outlined inSection 2.2.4. Generally, it attempts to replicate a Newtonian viscosity. Forsimulations presented in this section we take a viscosity forcing akin to that ofMorris et al. (1997):

fsphν (ra) = −2ν∑b

mb

ρbvab

(1rab

∂Wab

∂ra

). (6.13)

Taylor series expansions of the above give to leading order fν with n = 1, sowe expect the correct behavior for sufficiently smooth velocity fields, thoughthis leads to questions of what may be considered sufficiently smooth. Wewish to understand the performance of (6.13) for short length-scale velocityfluctuations, which are expected to be of significant for turbulent flows. Spectralrecompositions of SPH particle fields provide a powerful means by which thebehavior of fsphν and fν may be compared quantitatively.

The SPH particle velocities va and viscous dissipation fsphν (ra) are firstrecomposed in trigonometric functions with component amplitudes v and fsphν

respectively (Appendix D gives details of the required technique). We may thenconsider the total viscous dissipation rates

∫R

v · fsphν dx and∫R

v · fνdx. Theterm fν is first evaluated analytically:

fν(x) = ν∆∑

k

v(k) exp(ik · x) = −ν∑

k

k2v(k) exp(ik · x), (6.14)

where∑

k =∑kx

∑ky

. For any two vector functions a and b we may write∫R

a · b dx =∫R

∑ka

∑kb

a(ka) · b(kb) exp(i(ka + kb) · x) dx

=∑ka

∑kb

a(ka) · b(kb)∫R

exp(i(ka + kb) · x) dx,

and for a square domain of size length L we have

= L2∑ka

∑kb

a(ka) · b(kb) δka,−kb


h = 1.3∆x

h = 1.7∆x

h = 1.9∆x

Qν(k)

k/2π

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

20 40 60 80 100 120 140 160

Figure 6.2: The function Qν(k) (equation (6.15)) quantifies the integrity of theSPH Laplacian operator. The above results are obtained for a 640000 particlesimulation, where identical velocity fields are used to obtain the required quan-tities. The particle configuration and velocity field is obtained from a typicalturbulence simulation, as given in the results section of this chapter.

where δ is the Kronecker delta function. Therefore we have∫R

a · b dx = L2∑kx

∑ky

(a · b∗

)k

= L2∑k

[a · b∗

]k

with the term[a · b∗

]k

representing the sum of all terms a · b∗ within an annulusk − 1

2 ≤ |k| ≤ k + 12 in wavenumber space. We consider the function

Qν(k) =

[v∗ · fsphν

]k[

v∗ · fν]k

. (6.15)

This function effectively gives a measure of the quality of the SPH Laplacianoperator. It is given in Figure 6.2 with different values of smoothing length usedfor the kernel in (6.13). We first note the for all simulations, the SPH derivativegives values short of expectations as we move to higher wavenumbers. Indeedwhere approximately five particles are utilised per wavelength (k/2π = 160), theSPH viscous dissipation falls approximately thirty percent short of the expectedresult. For smaller smoothing lengths, there is less fall in dissipation strengthwith wavenumber, though results now exhibit a large degree of noise through-out the spectrum. For a smoothing length of h = 1.7∆x a good compromise isperhaps found, with only a slight increase in noise over the h = 1.9∆x result,


though only minor improvements in dissipation strength is obtained. Simula-tions presented in this chapter use h = 1.9∆x, though the potential use of ashorter smoothing length for viscosity certainly warrants further investigation.Noise at the small wavenumber limit is simply a result of the denominator inequation (6.15) vanishing, with relative errors actually quiet small. Importantly,the sign of SPH viscous forcing modes is everywhere correct, and where smooth-ing length is sufficient, viscosity diminishes in strength regularly. Note that allcalulations in this chapter utilise a variable smoothing length, with smoothinglength values given here representing the initial values.

These results highlight the deficiencies of the SPH viscosity. On one hand,we would like to minimise the viscosity such than an inertial range may form.However given the weak dissipation at short scales, we must select a sufficientlylarge kinematic viscosity to prevent energy accumulation at the spectral limit.This proves to be a significant limitation, with viscosity then required to actover a much broader spectrum than even standard Newtonian viscosity, and cer-tainly much broader than the hyperviscous dissipation often utilised in spectraltechniques.

6.1.3 Scales

Integration times are normalised using an eddy-turnover time defined as Z−1/2

where Z is the total enstrophy which is written

Z = ω2 =∫ ∞

0

k2E(k)dk. (6.16)

For simulations presented where only a quasi-steady state is reached, the inverseenergy cascade results in energy accumulation at small wavenumbers, and sothe spectrum in this region is not stationary. For time scaling, integrations areperformed until the spectrum is steady down to some wavenumber kl kf ,at which time the integral (6.16) is calculated and label Z0. We note howeverthat equation (6.16) converges to value Z0 very early in the simulation. Unlessotherwise stated, times are given in units of the eddy-turnover time Z−1/2

0 .An appropriate Reynolds number to characterise the flow may be written

Re =Lf√Ek

ν(6.17)

where Lf is the forcing length scale. For an assumed enstrophy cascade rangescaling of E(k) = β2/3k−3, it may then be shown (Lesieur, 1990) that resolutionrequirements will be dictated by

kdkf

=√Re (6.18)

where kd is the viscous dissipation wavenumber. Therefore a doubling of Reynoldsnumbers in two-dimensional turbulence leads to an increase in computationalwork of order 2

√2, a more favourable result than the order 23 increase encounter

in three-dimensional turbulence.


Table 6.1: Parameters for forced turbulence simulation

Parameter DescriptionA B Run

∆x 0.00125 0.00125 Initial particle separationν 1.88× 10−6 1.88× 10−6 Kinematic viscosityP 4.81× 10−5 9.61× 10−5 Forcing power inputkf/2π 30 30 Forcing wavenumberc 2.5 2.5 Forcing bandwidthM 0.05 0.05 Maximum run Mach numberRe 266 380 Maximum run Reynolds numberEk 2.26× 10−4 4.52× 10−4 Maximum run kinetic energyZ 4.56 7.88 Maximum run enstrophytime 110 115 Total run time

6.1.4 Equation of state

Results presented in this section use a similar equation of state to that of earliercomputations:

P = ργ − 1. (6.19)

The value γ = 1.4 is used, though use of other values (1 ≤ γ ≤ 7) does not affectresults significantly. Removal of the offset term on the right hand side of (6.19)does however cause a large change in dynamics with simulations exhibiting atendency towards certain particle configurations which appears to overwhelmresults.

We wish to simulate flows which may be considered largely incompressible,and so require that Mach numbers are kept to a small value. This is achievedsimply by use of sufficiently small forcing amplitudes, with ensuing Mach num-bers monitored to ensure they remain small throughout the simulation. Re-sults for weakly compressible Kraichnan forced turbulence have been reportedby Dahlburg et al. (1990), with Mach numbers up to M = 0.3 utilised. Im-portantly, kinetic energy scalings were found to be almost identical for bothcompressible and incompressible calculations.

6.2 Intermediate scale forcing

The growth in time of kinetic energy and enstrophy for the run A parameters ofTable 6.1 are to be found in Figure 6.3, with the corresponding kinetic energyspectrum at various times found in Figure 6.4. Kinetic energy is observed toinitially grow at a rapid rate until a time of approximately t ' 10 at whichpoint the viscous dissipation range becomes sufficiently active to slow the energygrowth. From here on, total energy continues to grow at a slowed rate, andcorresponds to the inverse energy cascade carrying some percentage of the energyinput to large scales. This is apparent in Figure 6.4 where the short scale regionof the spectrum is largely stationary for times greater than t ' 20, while largescales continue to become more energetic. Energy continues to rise until thesimulation is stopped, though growth rate is gradually diminishing with viscousdissipation eventually expected to balance energy input. The total enstrophy

6.2 Intermediate scale forcing 114

in Figure 6.3, as calculated using 6.16, rapidly rises to a value of Z ' 4.5 atapproximately t = 10. As total enstrophy is largely dictated by the short scalewhich are stationary beyond this point in time, enstrophy correspondingly doesnot vary significantly despite the growth in energy at large scales.

Turning to Figure 6.4, a number of observations are to be made. We first notethat a peak forms at the forcing wavenumbers, with energy within the forcingrange approximately of four times greater magnitude than of the surroundingwavenumbers. This is a result of the strong viscosity required to prevent anexcessive accumulation of energy at short scales. Indeed even with a relativelystrong viscous dissipation, a degree of energy accumulation at the resolutionlimit2 occurs, as is evident in the upward turned tail of the energy spectrum inFigure 6.4. With respect to energy spectrum scaling, the inverse energy rangeappears to trend with the expected Kolmogorov k−5/3 law, though the spectrumis still not fully developed at the conclusion of the simulation. For this reason wealso do not observed the accumulation of energy in the fundamental mode whichleads to the development of domain scale coherent structures. The predictedenstrophy cascade section of the spectrum (k > kf ) deviates greatly from theKraichnan k−3 law, and instead energy scales as k−6 in this region. Reasons forthis deviation will be explored shortly.

Figure 6.5 gives the vorticity in physical space. Early times are clearlydominated by the random forcing function. By the third frame of Figure 6.5 (t =23.5), we see the emergence of coherent eddies which appear to have lengthscalestypically of order of the forcing lengthscale. For greater times very little changeis observed in the characteristics of the vorticity field, despite the growth inkinetic energy. This may be understood with reference to the correspondingenstrophy spectrums found in Figure 6.6. As the inverse energy spectrum followsa k−5/3 spectrum, the peak in enstrophy is expected to be found at the forcingwavenumber. This is only true while significant energy has not accumulatedat the domain scale, which would turn the spectrum up at the large scalespotentially causing a peak in the enstrophy spectrum. We also note the tails ofthe enstrophy spectrum turn at the shortest scales, an artifact of the marginalresolution use for simulations.

Due to the excessively slow growth of energy at the large scales, the simu-lation energy input was increase, with all other parameters left unchanged (seeparameters for run B in Table 6.1), resulting in a larger simulation Reynoldsnumber. The late-time kinetic energy spectrum is given in Figure 6.7. A moredeveloped inverse energy range is obtained, though energy at the largest scalesis still growing slowly at the time the simulation is stopped. As with the lowerpower simulations, a Kolmogorov scaling is observed in this range. The en-strophy cascade section of the spectrum is largely unchanged in form from theprevious run, with a scaling of approximately k−6 found within this range.Naturally energy across the spectrum is increased from the previous simulation,though notably short scales are significantly more energetic as a result of theinsufficient resolution (or viscosity) utilised.

Results given above demonstrate the it is possible to produce Kraichnan like

2Note that the most appropriate choice for a resolution maximum wavenumber is definedby the SPH smoothing length, though there is no clear spectral cutoff. For the currentsimulations, the smoothing length implies a maximum wavenumber of approximately k/2π =210. Consideration of the kinetic energy as determined using (6.4) however suggests theslightly smaller value of k/2π = 200.


Ek × 104 Z

time

0

0.5

1

1.5

2

2.5

0 20 40 60 80 100 0

1

2

3

4

5

Figure 6.3: Evolution of total kinetic energy Ek (solid green curve) and totalenstrophy Z (broken green curve) in time for simulation A.

dynamics within the SPH framework. An inverse energy cascade is observedwith the correct spectrum scaling, with kinetic energy observed to be growingdespite a stationary energy spectrum for k > kf . Also observed is the appear-ance of coherent structures which are of order of the forcing lengthscale, andsurvive for numerous eddy turnover times. These vorticies are commonly en-countered in both forced and decaying two-dimensional turbulence (for instanceLegras et al. (1988) and McWilliams (1990)), and their significant to the inertialrange theory is still a subject of debate.

The enstrophy cascade for the simulations presented has an approximatek−6 spectrum. While this is far from the predicted Kraichnan k−3 spectrum,it is not out of line with the literature, many authors reporting spectrums k−α

with 3 ≤ α ≤ 6. The presence of coherent vortices is often cited as the causefor enstrophy range spectra with exponents α > 3. These long lived vorti-cies introduce a spatial and temporal intermittency which inhibits nonlineartransfers leading to a steepening of the enstrophy spectra (Benzi et al., 1986).Basdevant et al. (1981) considered the spectral effects of this intermittency, con-cluding that it resulted in localness of spectral space interaction which tended tosteepen spectra. However simulations have been performed in Maltrud and Val-lis (1991) which exhibited no coherent structures yet still resulted in enstrophyrange spectrum steeper than k−3. In these test various means were introducedto inhibit the formation of vorticies.

Tran and Bowman (2004) offer an alternative explanation for the steep en-strophy spectra based on global conservation of energy and enstrophy, alongwith the inclusion of viscosity. Their arguments are based on defining a quan-tity r which determines the strength of the inverse cascade, with the cascadeconsidered weak unless 1 − r 1. It is conjectured that unless the inversecascade is strong, the enstrophy range spectra must be steeper that k−5. Fur-thermore, the inverse cascade is shown to be extremely robust, persisting insimulations where approximately eighty percent of energy is dissipated by vis-cosity. For our simulations, we consider the quantity ε0 = dEk/dt which is the


E(k)

k/2π

t = 2.3 t = 12.9

t = 23.5 t = 44.5

t = 65.8 t = 108.0

Figure 6.4: Log-log plot of kinetic energy spectrum for run A. The peak in thespectrum corresponds to the forcing wavenumbers kf/2π = 30. The green brokenline gives a reference for the Kolmogorov k−5/3 scaling, while the blue broken linecorresponds to an enstrophy cascade k−3 scaling.


t = 2.3 t = 12.9

t = 23.5 t = 44.5

t = 65.8 t = 108.0

Figure 6.5: Fluid vorticity given at times corresponding to energy spectrumsfound in Figure 6.4.


k2E(k)

k/2π

t = 2.3 t = 12.9

t = 23.5 t = 44.5

t = 65.8 t = 108.0

Figure 6.6: Log-log plot of enstrophy spectrum for run A. The peak in thespectrum corresponds to the forcing wavenumbers kf/2π = 30.


E(k)

k/2π

Figure 6.7: Log-log plot of kinetic energy spectrum for run B (red) alongsiderun A (blue). The peak in the spectrum corresponds to the forcing wavenum-bers kf/2π = 30. The green broken line gives a reference for the Kolmogorovk−5/3 scaling, while the blue broken line corresponds to an enstrophy cascadek−3 scaling.

εν × 105 ε0/ε

time

0

1

2

3

4

5

0 20 40 60 80 100-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Figure 6.8: Energy dissipation rate εν (green) and inverse cascade factor ε0/ε(blue) in time. The upper graph corresponds to simulation parameters A, whilethe lower curve is for simulation parameters B. Energy input due to forcing isdenoted by the red line.


ˆv∗·fν

˜k

εν

k/2π

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

10 100

Figure 6.9: Viscous dissipation spectrum normalised by total viscous dissipa-tion. SPH viscosity is given by red plus symbols, while analytic dissipation (asgiven by equation (6.14)) is represented by green cross symbols. Data is forsimulation A parameters at time t = 105.

rate of energy injected into larger scales due to inverse cascades. In effect, thisquantity is the gradient of the kinetic energy given in Figure 6.3 from the pointwhere total enstrophy has stabilised. Alternatively we may consider the totalSPH viscous dissipation:

εν =∑b

mv · fsphν (6.20)

for a sum of all SPH particles b in the domain and SPH viscous forcing fsphν . Weassuming a constant energy input rate ε. This is a reasonable assumption giventhat we use a random white forcing of constant amplitude, and as discussedin Section 6.1.1 the velocity-forcing correlation P2 can be expected to be smallsince approximately five-hundred discrete modes are forced in the presentedsimulations. Therefore once the flow has evolved to a stationary enstrophy rangespectrum, we can write ε0 = ε−εν , and hence define an inverse cascade strengthr = ε0/ε. The evolution of viscous dissipation rate εν and cascade factor ε0/εare found in Figure 6.8. We consider times beyond t = 20 afterwhich the shortscale spectrum region is stationary. It can be seen for both simulations that atearly times the inverse cascade factor is approximately r = 0.06 and thereforeonly a very weak inverse cascade is present. As the inverse cascade graduallyfeeds long scale modes, these become more energetic and viscous dissipationcorrespondingly increases at these scales. Hence we see the inverse cascade factordrop to approximately r = 0.01 by the end of the run, which is reflected in thekinetic energy gradient falling in Figure 6.3. At this point the inverse cascade islargely exhausted with energy input being balanced by viscous dissipation. Wetherefore do not expect the k−5/3 range to extend to the domain scale, and sothe appearance of large scale coherent vorticies which follow from Bose-Einsteincondensation are also not expected.

The viscous dissipation spectrum (Figure 6.9) reveals that significant dis-sipation is to be found across the enstrophy cascade range. Approximately


Table 6.2: Parameters for quasi-steady turbulence simulation

Parameter DescriptionC Run

∆x 0.002 Initial particle separationν 1.88× 10−6 Kinematic viscosityP 1.11× 10−5 Forcing power inputkf/2π 10 Forcing wavenumberc 0.5 Forcing bandwidthM 0.06 Maximum run Mach numberRe 1064 Maximum run Reynolds numberEk 4.04× 10−4 Maximum run kinetic energyZ 0.79 Maximum run enstrophytime 116 Total run time

twenty percent is dissipated within the forcing band, and fifty percent of totaldissipation occurs within the first fifty modes. As such, requirements of theKraichnan theory are not satisfied. However the broad spectrum nature of theSPH viscosity (as discussed in Section 6.1.2) makes it difficult to reduce viscos-ity sufficiently such that minimal dissipation occurs at long and intermediatescales. Indeed particle populations used for the above simulations appears to bemarginally sufficient, and any further reduction in viscosity would result in ex-cessive unphysical energy accumulation at short scales. An analytic viscosity, asper equation (6.14), is also show. While for wavenumbers less than k/2π = 100,the SPH viscosity approximates a Newtonian viscosity with good accuracy, atshort scales the weakness of the SPH viscosity is evident. So for equivalenttotal viscous dissipation, the SPH viscosity must also be significantly strongerat intermediate scales.

A quality spectrum may be determined for the SPH pressure gradient terms,as has been done for the viscosity term in Section 6.1.2. Here the power spec-trum

[∇P ∗ · ∇P

]k

is constructed. A similar trend as that of Figure 6.2 isrecovered (see Figure 5.20), and so the SPH pressure gradient term also suffersa similar weakness to the viscosity term. The significance of this with respectto the current simulations has not been investigated, though it is expected tocontribute to short scale deficiencies.

Shorter tests were performed at higher resolutions (one million particles)which exhibited reduced spurious energy at short scales (as compared with Fig-ure 6.7 for instance), though results appeared otherwise unchanged.

6.3 Large scale forcing

6.3.1 Quasi-steady solutions

We turn to simulations utilising a larger scale forcing. Given that the forcingwaveband will be subjected to less viscous dissipation than the intermediateband forcing used above, a stronger inverse cascade should result. Relevantparameters are given in Table 6.2. Here the wavespace annulus kf/2π = 10 ±0.5 is subjected to a continual random forcing as discussed in Section 6.1.1.

6.3 Large scale forcing 122

Ek × 104 Z

time

0

1

2

3

4

0 20 40 60 80 100 120 0

0.2

0.4

0.6

0.8

Figure 6.10: Large scale forcing simulation. Evolution of total kinetic energyEk (solid green curve) and total enstrophy Z (broken green curve) in time forsimulation A.

While on average over a large enough time the dominant energy contributionis going to be due to the force-force correlation P1, because here only thirtydiscrete modes are forced, we cannot expect the force-velocity correlation P2 tobe negligible, though it’s long time contribution should still be small. For thecurrent simulation this is not of concern as we simply wish to present an SPHsimulation using large scale forcing, with repeatability not required.

The progression of total energy and enstrophy are given in Figure 6.10. Theearly time energy and enstrophy variability results from the random fluctuationsin forcing power due to P2, and is effectively determine only by the energywithin the forcing band. By a time of t = 40, the total enstrophy has reach alargely steady value, signaling the convergence of the short scales (k > kf ) to asteady state spectrum. From this point forward, the total energy and enstrophyexhibits a robustness to variations in forcing owing to an active dissipationrange.

Unlike the previous simulations, with the inverse cascade eventually haltedby viscosity, we find here a strong inverse cascade which presists throughout therun. This is evident in Figure 6.10, where the kinetic energy is increasing at asteady rate up until where the simulation is halted. At this point, the inversecascade strength paramater is the relatively large r = 0.14, and approximatelyhalf all kinetic energy resides within modes k ≤ 3.5.

The energy and enstrophy spectrums are displayed in Figure 6.11, along withthe corresponding velocity and vorticity fields in Figure 6.12. We first note thatthe small energy spike at the forcing wavenumber is absent in this simulationdue to reduced viscosity in the forcing range.. Again the inverse cascade rangetrends according to a Kolmogorov k−5/3 scaling. The enstrophy range for thissimulation follows approximately k−5, and so is shallower than the k−6 scalingencountered for intermediate scale forcing. This is most probably on account ofthe larger inverse cascade strength, in line with the theory of Tran and Bowman(2004). The horizontal velocity field (Figure 6.12) exhibits predominatly largescale features, as dictated by the energy spectrum. Similarly, the structures


of the vorticity field appear to be of order of the forcing length scale, whichcorresponds to the peak in the enstrophy spectrum. Also evident are linearstructures, or sheets, of constant vorticity, giving weight to the earlier qualitativearguement for an enstrophy cascade. Dark regions for both the vorticity andenergy field correspond to negative values, and we note that a reversed colourimage is qualitatively identical. For domain scale vorticity structures, we expectthat the enstrophy spectrum must peak at the smallest wavenumbers, so we arerequired to continue the simulation until sufficient energy has accumulated inlong wavelength modes. Time constraints did not permit the simulation to becontinued unfortunately.

6.3.2 Steady solutions

A series of simulations are performed to determine the behavior of SPH asresolution is reduced. We also consider some preliminary simulations utilisingthe α-SPH algorithm, as well as simulations where the filtered velocity is usedwith a standard acceleration equation. For clarity of comparison, we maketwo modifications to the turbulence regime. Firstly, forcing is implementedto achieve some predetermined total energy Emf within the forcing bandwidth.This is implement by applying forcing where Ef (t) < Emf , and no forcing forEf (t) > Emf , where total energy within the forcing band is Ef (t). Secondly, weapply large scale dissipation to shut down the inverse energy cascade for modesk < 3.5, allowing the simulation to eventually achieve a statistically steadystate. Large scale dissipation is used extensively in spectral two-dimensionalturbulence simulations (for instance Gotoh (1998) and Lindborg and Alvelius(2000)), and may be implemented natively within the spectral framework.

A form for an analogous SPH large scale dissipation term, perhaps akinto the standard SPH viscosity, is not obvious. Instead, large scale dissipationis effected by explicitly removing the required velocity components from thevelocity field. This is applied using the velocity stepping,

v1a = v0

a −∆t∑|k|<3.5

νL|k|

v1/2 exp(ik · x1/2a ) (6.21)

with large scale dissipation parameter νL. The coefficients v1/2 are determinedusing the methods found in Appendix D.

Simulations are executed until the kinetic energy reaches a statisticallysteady state. Parameters may be found in Table 6.3. Changes in the steady-state spectrum as resolution is varyed are displayed in Figure 6.14. For thisfigure, energy spectrum data is averaged over approximately ten eddy turnovertimes to compose the presented spectra. A small peak occurs at the forcingwavenumber on account of the lower Reynolds numbers encountered. Shortscales exhibit increasing energy accumulation as resolution is reduced. Thisresult is expected given the larger velocity amplitudes required to effect anequivalent total viscous dissipation. Also worth noting is the reduced effective-ness of viscous dissipation as mode wavelengths approach the SPH smoothinglength. This SPH shortcoming will be more pronounced for low resolution sim-ulations, resulting in a further increase in short scale energy to compensatefor deficiencies of the viscosity operator. The dimensional arguements in Sec-tion 6.1.3 indicate a dissipative wavenumber of kd/2π ∼ 200. The maximum


E(k) k2E(k)

k/2π

(a) (b)

Figure 6.11: Log-log plot of kinetic energy (a) and enstrophy (b) spectrum forrun C. The peak in the spectrum corresponds to the forcing wavenumbers kf/2π =10. The green broken line gives a reference for the Kolmogorov k−5/3 scaling,while the blue broken line corresponds to an enstrophy cascade k−3 scaling. Datais taken at t = 85.

(a) (b)

Figure 6.12: Horizontal velocity (a) and fluid vorticity (b) corresponding tospectrums found in Figure 6.11.


Table 6.3: Parameters for steady-state forced turbulence simulation

Parameter DescriptionD Run

∆x 0.002, 0.0027, 0.004 Initial particle separationν 1.88× 10−6 Kinematic viscosityνL 0.625 Large scale dissipation parameterP 1.11× 10−5 Forcing power inputEmf 2× 10−5 Target forcing band energykf/2π 10 Forcing wavenumberc 0.5 Forcing bandwidthM 0.035 Maximum run Mach numberRe 500 Maximum run Reynolds number

wavenumber of the highest resolution simulation is k/2π ' 130 (based on theSPH smoothing length), which is somewhat shy of the dissipative wavenumber.While the validity of dissipative length scale definition 6.18 must be questionedfor the current simulations given the absence of a k−3 enstrophy range scaling,the high resolution simulation does indeed appear to be insufficiently resolvedin Figure 6.14. A true dissipative lengthscale for the current regime is unclear,though energy at small wavenumbers seems to be largely convergent for the twohighest resolution simulations, which correspond well at all but the shortestscales. We conclude that the N = 250000 simulation is sufficiently resolved togive accurate large scale dynamics.

The lowest resolution spectrum deviates significantly from the higher res-olution counterparts however, with energy at medium and large scales greatlyreduced. This appears to be due to insufficient particles being available to re-solve coherent structures. Instead, the vorticity field is largely dominated by therandom forcing, with only weak vorticies presenting (see Figure 6.13). However,the coalescence of coherent structures is believed to be integral to the inverseenergy cascade (Frisch and Sulem, 1984), and so we postulate that the absenceof strong vorticies weakens the SPH inverse cascade. With a total particle pop-ulation of 62500, there are approximately 625 particles per vorticy (assumingstructures are of order of the forcing lengthscale), which we conclude is insuffi-cient to correctly produce the behavior of interacting two-dimensional vorticies.For the range k > kf , we note with reference to Figure 6.4 that the enstrophycascade range amplitude is not determined by the forcing mode amplitude, butrather by the amplitude of the k < kf range. This is due to non-localnessof interactions in spectral space, and may not be the case where a sufficientlywide forcing range is applied. Interestingly, Figure 6.13 indicates significantdifferences in the vorticity field between the N = 137000 and N = 250000 sim-ulation, which is perhaps not obvious in the energy spectra of Figure 6.14. ForN = 250000, the vorticity field exhibits a certain ‘robustness’, with seeminglystronger features such as sheets of constant vorticity.

A further simulation artifact becomes evident where the squared densityspectrum is considered (Figure 6.15). Details for the determination of spectrum[ρ2]f are similar to those used in Section 6.1.2. While we note that maximumdensity variations are within one percent of the mean value, it is apparant that


N = 62500 N = 137000 N = 250000

Figure 6.13: Vorticity field for steady-state simulations at different resolutions.

E(k)

62500 particles137000 particles

250000 particles

10 100

Figure 6.14: Log-log plot of kinetic energy for simulation series D. Presenteddata corresponds to averages taken over ten eddy turnover times.


most of this variation is realised as very short scale noise. Furthermore, this noisebecomes more pronounced as the degree of non-resolution is increased, with anorder of magnitude increase for the ∆x = 0.04 simulation over the ∆x = 0.02simulation at the SPH smoothing parameter lengthscale. A similar increase isfound in the kinetic energy spectrum, and we conclude that this short scale noiseis a product of energy accumlation at short scales, together with the inherentweakness of the SPH pressure gradient. This pressure gradient weakness occursat short scales (see Figure 5.20), and results in the SPH pressure force beingunable to counter short-scale density variations. The signficance of this shortscale noise has not been investigated, though we note that the accuracy of theSPH density calculation is fundamental to the calculation of all SPH quantaties.There is no evidence in Figure 6.15 of the squared density scalings observed byDahlburg et al. (1990), though the Mach numbers achieved here (M = 0.035)are much lower than those of Dahlburg et al. (M ∼ 0.3).

As with the earlier one-dimensional results, simulations appear to tend regu-larly to the correct result, and do not suffer catastrophic failure (as would oftenbe found for a spectral algorithm) where largely under-resolved. It is notedthat this robustness may not always be desired, with simulation failure per-haps signaling the operator to the insufficient resolution utilised. Scaling withinthe enstrophy cascade range is largely consistent between different resolutionintegrations (approximately k−5 for all simulations). For the reasons outlinedabove, the inverse cascade range differs for the lowest resolution simulation,though is largely consistent for the other two (note that noise at large scales forthe highest resolution simulation is due to an insufficient time range used foraveraging).

6.3.3 Steady solutions incorporating α-SPH

We present preliminary results for simulations which incorporate the α-SPHmethodologies. As outlined previously, in α-SPH particles are advected by afiltered velocity v (equation 3.6), with the momentum equation then modifiedto restore energy conservation (equation 3.31). We note that for this system, theconserved quantity of energy does not contain the standard kinetic energy, withthe averaged Euler-Lagrange equation dictating that instead the appropriatedconserved quantity is

E =∑b

mb


](6.22)

with thermal energy u(ρb, sb) (see Chapter 3 for further details). Given the lowMach numbers encountered in these simulations, we expect thermal energy toremain small. The kinetic energy

∫v · v dx is important in the sense that in the

incompressible limit, this is the mathematically conserved quantity. However, inthe sense that we wish to model a physical system, the standard kinetic

∫v ·v dx

is also of significance. We consider the spectrums for both quantities, notingthat at large scales we expect them to coincide. Furthermore, a spectrum forthe advection velocity

∫v · v dx is of interest. Parameters for these simulations

are given in Table 6.4.Energy spectrums for the ε = 1.0 simulation may be found in Figure 6.16.

It is first noted that while a statistically steady total kinetic energy has been


[ρ2]k

62500 particles137000 particles

250000 particles

10 100

Figure 6.15: Log-log plot of squared density spectrum for simulation series D.

Table 6.4: Parameters for α-SPH steady-state forced turbulence simulations

Parameter DescriptionE Run

∆x 0.0027 Initial particle separationν 1.88× 10−6 Kinematic viscosityνL 0.625 Large scale dissipation parameterP 1.11× 10−5 Forcing power inputEmf 2× 10−5 Target forcing band energykf/2π 10 Forcing wavenumberc 0.5 Forcing bandwidthM 0.035 Maximum run Mach numberRe 500 Maximum run Reynolds numberε 0.5, 1.0 α-SPH turbulence parameter


attained, integrations have not been performed for sufficient time to average outall spectral variations, most notably at the largest scales. While we observe thatthe velocity filtering is successfuly in reducing short scale energy amplitudes forfiltered quantities, all energy spectra now exhibit increased energy levels overstandard SPH, significantly so for unfiltered spectrum. For wavenumbers lessthan k/2π = 30, all spectrums approximately coincide. For the enstrophy range(k > kf ), the energy spectrum are largely identical to those encountered forstandard SPH. It appears that a slightly shallower scaling of E(k) ∼ k−4.8

is appropriate (as compared to E(k) ∼ k−5 for standard SPH), though giventhe uncertainty owing to insufficient integration times, it is premature to makeconclusions about energy scaling. While the filtered velocity energy spectrum([ 1

2 v · v]k) for the α-SPH simulation appears to coincides with the standard SPHenergy spectrum, this appears to be largely coincidence, and we note that thisis not observed for ε = 0.5 simulations (Figure 6.17). Interestingly, the lowerturbulence parameter simulations yield increased mode attenuation at shortscales. These results are remarkable remiscent of the earlier one-dimensionalα-SPH findings, where it was found that a turbulence parameter ε & 5 wasrequired to consistently produce energy spectrums with reduced short scaleenergy. Indeed for the parameters ε = 0.5 and ε = 1.0, short scale energy wasfound to be slightly increased, and more so for the latter parameter, echoingwhat is observed here (see Figures 4.24 and 4.37). Though the exact mechanismsby which this occurs are unclear, it appears that at these small values of ε, SPHpressure gradient effects may dominate. The results of Geurts and Holm (2002)also suggest a minimum turbulence parameter requirement for finite volumesimulations, where this minimum is defined with respect to simulation resolution(as is ours effectively through parameter ε.). A specific minimum value will mostlikely be dependent on the particular discritisation used.

Simulation using only the filtered advection (therefore removing turbulenceterms from α-SPH momentum equations) result in minor changes from the fullα-SPH scheme for the parameter ε = 1 (Figure 6.18). Only a slight reductionin short scale energy is observed in comparison with α-SPH, which is consis-tent with earlier findings for Kelvin-Helmholtz instability growth rates. Energyspectrums constructed of the filtered velocity exhibit similar trends. We notethat for the standard velocity spectrum ([ 1

2v ·v]k), energy at short scales is stillsignificantly greater than for standard SPH. While the addition of momentumturbulence terms appears to have minimal influence on the energy spectrum(at least for ε = 1), we still cannot dismiss their importance in maintainingthe variationally consistent framework. Indeed, for our one-dimensional simu-lations, phase velocities are produced more accurately for Lagrangian derivedequations of motion, and we must remember that information such as phasevelocity is invisible on an energy spectrum.

Similar simulations to those presented here are given in Nadiga and Shkoller(2001), where spectral methods have been used for solution to the LANS equa-tions. A steepened short scale energy spectrum is reported, along with anapparent enhancement of the inverse energy cascade. No evidence of a steependenergy spectrum or an inverse energy enhancement is observed in our simula-tions, though we note that Nadiga and Shkoller performed simulations with aminimum turbulence parameter equivalent of ε ∼ 100. While for spectral sim-ulations, the velocity filtering operation is applied analytically at effectively nocost, the SPH equivalent requires an iterative process, with iteration count in-


E(k)

[ 12v · v]k

[ 12v · v]k

[ 12 v · v]k

10 100

Figure 6.16: Log-log plot of kinetic energy for α-SPH simulation with parameterε = 1.0. The solid curve corresponds to the equivalent standard SPH simulationenergy spectrum found in Figure 6.14. Presented data is averaged over one eddyturnover time.

E(k)

[ 12v · v]k

[ 12v · v]k

[ 12 v · v]k

10 100

Figure 6.17: Log-log plot of kinetic energy for α-SPH simulation with parameterε = 0.5. The solid curve corresponds to the equivalent standard SPH simulationenergy spectrum found in Figure 6.14. Presented data is averaged over one eddyturnover time.


E(k)

α-SPHFiltered advection

10 100

Figure 6.18: Log-log plot of kinetic energy ([ 12v ·v]k) for α-SPH simulation and

SPH simulation with particles advected by filtered velocity. Both simulationsuse turbulence parameter ε = 1.0. The solid curve corresponds to the equivalentstandard SPH simulation energy spectrum found in Figure 6.14. Presented datais averaged over one eddy turnover time.

creasing with ε parameter. Currently a Gauss-Seidel algorithm is utilised for theimplicit velocity, yielding an iteration count which rises approximately linearlywith turbulence cutoff parameter α = 31/196h

√ε. Unfortunately, this rapidly

overwhelms current computational resources, and time constraints do not allowfor simulations to be performed with larger turbulence parameters.

6.4 Conclusion

A two-dimensional forced turbulence regime has been implemented and inves-tigated in what constitutes the first such simulations using the SPH particletechnique. It is established that Kraichnan-like dynamics may be reproducedwithin the SPH framework at large scales, though at short scales the shortcom-ings of SPH operators becomes significant. However, the impact of these SPHdeficiences on the important large scales appears to be relatively minor wheresimulation parameters within an appropriate range are used.

For simulations forced at intermediate length scales (kf/2π = 30), a weakinverse cascade was observed, leading to an energy cascade scaling in good agree-ment with the Kolmogorov k−5/3 law. The weakness of the energy cascade wasdue largely to the excessive viscosity the forcing band was subjected to, result-ing in a large portion of the input energy dissipated within or in close spectralproximity to the forcing wavenumber. While the choice of an intermediate forc-ing wavenumber naturally makes the input energy more accessible to viscousdissipation, it also highlights the broad spectrum nature of the SPH viscosity(see Figures 6.2 and 6.20). This is in effect a weakness of the SPH Laplacianoperator at large wavenumbers, and for an equivalent total viscous dissipationto a Newtonian viscosity, the SPH viscous operator must act over a larger spec-

6.4 Conclusion 132

trum. For our simulations, this broad viscosity brings the inverse cascade to apremature halt. Spectral numericists often implement a hyperviscosity whichhas the exact opposite effect, reducing the viscous dissipation bandwidth. Anequivalent SPH hyperviscosity implementation is unclear, and while hypervis-cosity is certainly a numerical convenience (and widely utilised), some questionit’s validity (Bartello and Warn, 1996; Gotoh, 1998). Regardless, measures torestore the ‘strength’ of the SPH viscosity at short scales are certainly desirableand would increase the application of SPH.

While the criterion of Kraichnan (1967) turbulence are therefore not trulymet, we note the persistance of the k−5/3 range. This supports the theory ofTran and Bowman (2004) who have also found inverse energy cascades evenwhere signficant viscosity is applied across the energy range. Tran and Bowman(2004) also gives theoretical basis for our observed enstrophy range scaling ofk−6, in light of the extremely weak inverse energy cascade.

Forcing at a smaller wavenumber of kf/2π = 10 presents less numerical dif-ficulty due to reduced resolution requirements. The forcing band is now ableto be removed to a greater extent from the effects of viscosity. This is reflectedin the increased inverse energy cascade strength observed in these simulations,which is approximately three times larger than found for intermediate wavebandforcing. This larger inverse energy cascade results in a shallower enstrophy cas-cade scaling of k−5, in accordance with Tran and Bowman (2004). The k−5/3

scaling is again observed for the inverse energy range, though we note that asthe domain scale is approached, the assumption of homogeneity and isotropyare less strictly adhered to, and may lead to deviation from Kraichnan turbu-lence (Lowe and Davidson, 2003). For this reason, and to bring about a trulystatistically steady state, a large scale forcing has been applied. In comparingequivalent simulations at varying resolution, a number of observations are made.Importantly, the increase in short scale kinetic energy with reduced resolution isnoted (Figure 6.14), along with significantly reduced coherent structures for thecoarsest simulation, possibly owing to insufficient particle populations. How-ever, another potential error source may result from the increase in short scalesquared density amplitudes (Figure 6.15), which is consequence of the SPH pres-sure gradient deficiencies. It is noted that accuracy of SPH density summationis paramount to correct numerical quadrature. Further investigation of thesephenomona are warranted.

Findings for α-SPH simulations are inconclusive. The intent of the LANSand α-SPH frameworks are to bring about closure within a smaller spectralbandwidth than what would be required for equivalent turbulent Navier-Stokessimulations, therefore reducing computational requirements. However, at thesmall values of turbulence parameter ε tested, an increase in kinetic energy atshort scales is instead observed, effectively increasing resolution requirements.A similar increase in energy has been observed in the earlier one-dimensionalsimulations, with consistent short scale energy reductions only found for ε & 5.However, α-SPH two-dimensional simulations using parameter ε = 1 lead to overthree times the computational cost required of standard SPH, largely owingto the iterative scheme required for velocity filtering. Further increases in εare currently not possible given limited resources, so we are unable to drawsolid conclusions about the potential application of α-SPH. However unless amore efficient velocity filtering operation is constructed, along with a fastertimestepping routine, it seems that resources will be better directed simply


increasing particle numbers.

Chapter 7

Conclusion

In this thesis, a comprehensive study of the SPH particle discretisation has beenundertaken, with emphasis on the behavior where length scales approach theSPH smoothing length. A new modification of SPH known as α-SPH (Mon-aghan, 2002) has also been considered and investigated. The intention of thisnew SPH implementation is to provide a mathematically rigorous pathway toclosure of turbulence regimes, the simulation of which would otherwise be wellbeyond computational means. Throughout the thesis, we often return to thetheme of spectral recompositions of the SPH solutions using the novel tech-nique outlined in Appendix D. This has allowed quantitative appraisal of thedifferent scales of motion present in SPH simulations, allowing for the spectralexamination of dynamical solutions. For the one-dimensional problems, a di-rect comparison has been made between an SPH solution and a highly accuratespectral solution which has been used as a benchmark. Important insights weremade by observing the SPH solutions in the spectral domain, such as the signif-icance of a variable smoothing length to non-linear cascade rates, and the finescale behavior of the α-SPH scheme as we vary the turbulence parameter. Thesedetails would remain largely invisible in physical space. In two dimensions, spec-tral recompositions again proved invaluable in the analysis of forced turbulence.Importantly, it was determined that the Kraichnan (1967) theory and energyscaling laws, commonly observed in spectral methodologies, were largely able tobe produced within the SPH framework. Deviations from Kraichnan turbulencewere given theoretical basis. Similarly, reconstruction and Fourier analysis ofa fluid-fluid interface has allowed for accurate determination of SPH Kelvin-Helmholtz instability growth rates. We have thus shown that where sufficientresolution is utilised, growth rates correspond very well to analytic expectations.

While there are many variations possible to the standard SPH algorithm,perhaps the most important changes are related to the SPH smoothing length,which effectively determines the interaction radius for SPH particles. It wasconsistently observed that the often used smoothing length parameter σ = 1.3(for smoothing length h = σ∆x and kernel support radius 2h) is insufficient.It is postulated that his is due to poor convergence of the summation approx-imant (2.5) to the integral approximant (2.1). For the one-dimensional SPHand α-SPH simulations, energy spectra are found to be convergent where val-ues σ & 1.9 are utilised. Similarly, for the Kelvin-Helmholtz simulations, usingσ = 1.3 resulted in interface progression which wavered from strict exponential

134

135 Conclusion

growth, and SPH growth rate attenuation was not found to scale consistentlywith smoothing length. In contrast, simulations with σ = 1.9 exhibited pertur-bation growth which tracked the expected exponential very closely. Resultinggrowth rates were also largely consistent and predictable. Further evidence ofthe σ = 1.9 requirement may be found in our analysis of the two-dimensionalSPH viscosity operator, where it was shown that although the larger smooth-ing lengths reduced viscosity strength at short scales, results exhibited signifi-cantly reduced noise. While additional costs are incurred in choosing a largersmoothing length parameter, and some compromise in accuracy is often deemedworthwhile, it is a false economy to chose overly small values of σ. Any per-ceived resolution gains are certainly questionable in light of evidence outlinedabove, and additional summation costs for increased smoothing lengths may benegated by a larger timestep.

The superiority of the variable smoothing length definition (2.67) togetherwith the self-consistent density (2.69) is established in Chapter 4. Whereas theconstant smoothing length implementation required tuning of the σ parame-ter to produce the required soundspeeds and cascade rates, these were natu-rally obtained for the variable smoothing length calculations. Here parameterσ had less influence, though values σ > 1.9 were shown to be still desirable.Similarly, simulations of Burgers’ equation demonstrate the importance of avariable smoothing length to the SPH viscosity operator. For two-dimensionalKelvin-Helmholtz simulations however, no major changes were found in usingthe variable smoothing length, though we note that these studies were concernedwith the linear growth regime, while most significant changes in one dimensionwere observed in nonlinearities. We also touched upon the significance of theomega terms which results from the variationally consistent SPH derivation.One-dimensional simulations indicate that the inclusion of these terms yieldssolutions which exhibit excellent phase speed properties, though at a penalty ofincreased artificial mode attenuation for medium length scales. Further investi-gation in higher dimensions is certainly warranted.

The SPH pressure gradient has been implicit in shortcomings observed forboth one and two-dimensional applications. In one-dimensional Euler simula-tions, we observed that energy spectrum cascades where inhibited at intermedi-ate scales (relative to smoothing length) resulting in premature spectral closureof dynamics. We note that this behavior was not observed in the Burgers’simulations where pressure terms were not present. The wavenumber at whichdeviations occurred was shown to be directly related to the smoothing length,with wavenumber decreasing for larger smoothing lengths. Taylor series expan-sion of the SPH pressure gradient revealed the modified differential equationhigher order terms, which were postulated to be responsible for these secondaryeffects. Interestingly, the one-dimensional behavior was remarkably reminiscentof spectral simulations where alpha turbulence methodologies had been incorpo-rated. It was speculated that perhaps these secondary effects may equivalentlyprovide an implicit means of turbulent closure. Though for our two dimensionsimulations, the nature of this behavior appears to differ. This is perhaps owingto low simulation Mach numbers and the appearance of transverse waves, to-gether which result in reduced density gradients. Mode attenuation is certainlystill observed for Kelvin-Helmholtz simulations however. While highly accurateinstability growth rates were found for perturbations significantly larger thanthe SPH smoothing length scale, perturbations approaching this scale grew at

136

reducing rates. This growth rate reduction was shown to be directly relatedto the increasing ‘weakness’ of the SPH pressure gradient for pressure modesapproaching the smoothing length scale. For the non-linear two-dimensionalturbulence simulations, this pressure gradient weakness was also found to al-low significant short scale density variations to persist, the significance of whichrequires further consideration.

A similar weakness has been observed for the SPH viscosity, with viscous dis-sipation strength falling short of analytic expectations at short scales. Therefore,for an equivalent total dissipation to a true Newtonian viscosity, the SPH viscos-ity must have influence over a larger bandwidth. This broad dissipation rangeresults in increased difficulty for turbulence forced at intermediate length-scales,with the inviscid requirements of Kraichnan (1967) turbulence not strictly met.While importantly an inverse energy cascade was observed with the requiredenergy scaling of k−5/3, it was found to be extremely weak owing to viscosity,resulting in an enstrophy range scaling of k−6, deviating for the k−3 Kraich-nan prediction. This steep energy scaling is given a theoretical basis throughthe work of Tran and Bowman (2004). Larger scale forcing yields a strongerinverse cascade for which an enstrophy range scaling of k−5 is observed, a resultalso consistent with Tran and Bowman (2004). While at large and intermediatescales SPH algorithms produce dynamics which are in line with current theory,a number of short scale deficiencies are apparent. Most evident is the accumu-lation of kinetic energy at short scales which results from the weak viscosityand insufficient resolution. This contrast the one-dimensional SPH simulationswhere viscosity and secondary SPH pressure gradient effects prevented energypropagating to short scales, even for very large Reynolds numbers.

The α-SPH methodologies present a possibly means by which this short scaleenergy accumulation may be addressed, potentially providing mathematical clo-sure at computable lengthscales. For the two-dimensional forced turbulencesimulations, only small values of the turbulence parameter (ε ≤ 1) were ableto be tested due to computational restrictions. Unfortunately we have consis-tently observed that values ε ≤ 2 are insufficient to produce mode attenuationat short scales. Indeed, no significant change is found for Kelvin-Helmholtzsimulations until values ε > 1 are utilised, and the opposite effect is observedfor one-dimensional simulations, with α-SPH resulting in increased short scalemode amplitudes for values up to ε = 5. Likewise for the two-dimensional tur-bulent simulations, increased short scale energy resulted for the tested values ofε = 0.5 and 1.0, and more so for the latter. Other authors (Geurts and Holm,2002) have similarly observed an equivalent minimum turbulence parameter re-quirement. The expected behavior of mode attenuation is observed for valuesε > 5 for one-dimensional simulations, and is consistent with the spectral algo-rithm equivalent. Using a large turbulence parameter for the two-dimensionalturbulence simulation unfortunately is prohibitively expensive. This is mainlydue to the iterative requirement for the implicit filtered velocity.

So the question of the need for a turbulence implementation for SPH shouldbe asked. Clearly for one-dimensional simulations there is no true requirement,with closure obtained due to secondary pressure effects and/or viscosity, thoughresults for α-SPH are of interest in their own right. In two dimensions, whileenergy accumulation is observed at short scales, it does not appear to have asignificant effect on large scale dynamics for the relatively wide range of pa-rameters tested. Certainly the robust nature of SPH ensures no catastrophic

137 Conclusion

failure occurs even where simulations appear to be largely under-resolved. Thisrobustness may in part be due to the weakness of the SPH pressure gradient,with reduced Kelvin-Helmholtz instabilities potentially slowing energy cascaderates. As such, though a weakend viscous dissipation occurs at short scales, itis only required to dissipate energy from a similarly weakened energy cascade.So in ways, the shortcomings of SPH are also it’s strengths. While a physicallymotivated turbulence closure, such as that offered by averaged Lagrangian tech-niques, is certainly desirable, it must foremost be numerically efficient. Howeverfor the current algorithm, the iterative velocity filtration, which itself is nestedwithin an iterative timestep, yields significantly increased costs over standardSPH (a factor of three increase at ε = 1). For the α-SPH turbulence model tobe of true utility, an alternative method of obtaining a filtered velocity is clearlyrequired. Currently computational resources are better directed towards simplyincreasing SPH particle populations.

Appendix A

α-SPH: variable-h terms

We consider the derivation of our acceleration equation where the variation ofour kernel with smoothing length is not neglected. The Euler-Lagrange equa-tions (3.19) are evaluated with Lagrangian (3.5). Our canonical momentum isas found in Section 3.2.2, equation (3.21). Terms deriving from thermal energyare equivalent to those for regular variable-h SPH (see Section 2.2.2):

∂

∂ria

∑b

mbu(ρb, sb)

= ma

∑b

mb

(Pa

Ωaρ2a

∂Wab;a

∂ria+

PbΩbρ2

b

∂Wab;b

∂ria

), (A.1)

with notation Wab;a = W (|ra − rb|, ha). We consider the velocity terms in(3.22):

12∂

∂ria

∑b

mbvb · vb

=∂

∂ria

ε

4

∑b

∑c

mbmc

ρbcv2bcW bc

=ε

4

∑b

∑c

mbmcv2bc

∂

∂ria

1ρbc

W bc

=ε

4

∑b

∑c

mbmcv2bc

W bc

∂

∂ria

1ρbc

+ε

4

∑b

∑c

mbmcv2bc

1ρbc

∂W bc

∂ria

. (A.2)

Considering the first term:

ε

4

∑b

∑c

mbmcv2bcW bc

∂

∂ria

1ρbc

= − ε4

∑b

∑c

mbmc

ρ2bc

v2bcW bc

∂ρbc∂ρb

∂ρb∂ria

+∂ρbc∂ρc

∂ρc∂ria

,

138

139 α-SPH: variable-h terms

we use symmetry to simplify, and letting Abc = ∂ρbc∂ρb

,

= − ε2

∑b

∑c

mbmc

ρ2bc

v2bcW bc

Abc

∂ρb∂ria

.

Now armed with density derivative (2.73) along with definition (2.74), we write

= − ε2

∑b

∑c

mbmc

ρ2bc

v2bcW bc

AbcΩb

∑d

md (δab − δad)∂Wbd;b

∂rib

= − ε2

∑c

mamc

ρ2ac

v2acW ac

AacΩa

∑d

md∂Wad;a

∂ria

− ε

2

∑b

∑c

mbmc

ρ2bc

v2bcW bc

AbcΩb

ma∂Wab;b

∂ria

= − ε2ma

∑c

∑d

mcmd

ρ2ac

v2acW ac

AacΩa

∂Wad;a

∂ria

+∑b

∑c

mbmc

ρ2bc

v2bcW bc

AbcΩb

∂Wab;b

∂ria

= − ε2ma

∑b

mb

1

Ωa

(∑c

Aacmc

ρ2ac

v2acW ac

)∂Wab;a

∂ria

+1

Ωb

(∑c

Abcmc

ρ2bc

v2bcW bc

)∂Wab;b

∂ria

= − ε2ma

∑b

mb

ζaΩa

∂Wab;a

∂ria+ζbΩb

∂Wab;b

∂ria

, (A.3)

where

ζk =∑c

Akcmc

ρ2kc

v2kcW kc.

We note that kernel derivatives given above are taken holding smoothing lengthconstant. Turning to the second term in (A.2):

ε

4

∑b

∑c

mbmc

ρbcv2bc

∂W bc

∂ria

=ε

8

∑b

∑c

mbmc

ρbcv2bc (δab − δac)

∂Wbc;b

∂rib:= A

+ε

8

∑b

∑c

mbmc


∂Wbc;c

∂rib:= B

+ε

8

∑b

∑c

mbmc

ρbcv2bc

∂Wbc;b

∂hb

∂hb∂ρb

∂ρb∂ria

:= C

+ε

8

∑b

∑c

mbmc

ρbcv2bc

∂Wbc;c

∂hc

∂hc∂ρc

∂ρc∂ria

:= D

140

We again note that kernel derivatives given above are taken holding smoothinglength constant, with the exception of W bc. We consider term A:

ε

8

∑b

∑c

mbmc


∂Wbc;b

∂rib

=ε

8

∑c

mamc

ρacv2ac

∂Wac;a

∂ria+ε

8

∑b

mbma

ρbav2ba

∂Wab;b

∂ria.

Likewise for term B:ε

8

∑b

∑c

mbmc


∂Wbc;c

∂rib

=ε

8

∑c

mamc

ρacv2ac

∂Wac;c

∂ria+ε

8

∑b

mbma

ρbav2ba

∂Wab;a

∂ria.

We group these together to find

A+B =ε

4ma

∑b

mb

v2ab

ρab

∂Wab;a

∂ria+v2ab

ρab

∂Wab;b

∂ria

.

Now working with term C:

ε

8

∑b

∑c

mbmc

ρbcv2bc

∂Wbc;b

∂hb

∂hb∂ρb

∂ρb∂ria

=ε

8

∑b

∑c

mbmc

ρbcv2bc

∂Wbc;b

∂hb

∂hb∂ρb

(1

Ωb

∑d

md∂Wbd;b

∂rib(δab − δad)

)

=ε

8

∑c

mamc

ρacv2ac

∂Wac;a

∂ha

∂ha∂ρa

(1

Ωa

∑d

md∂Wad;a

∂ria

)

+ε

8

∑b

∑c

mbmc

ρbcv2bc

∂Wbc;b

∂hb

∂hb∂ρb

(1

Ωbma

∂Wba;b

∂ria

)

=ε

8ma

∑d

md1

Ωa∂ha∂ρa

(∑c

mc

ρacv2ac

∂Wac;a

∂ha

)∂Wad;a

∂ria

+ε

8ma

∑b

mb1

Ωb∂hb∂ρb

(∑c

mc

ρbcv2bc

∂Wbc;b

∂hb

)∂Wab;b

∂ria

=ε

8ma

∑b

mb

1

Ωa∂ha∂ρa

νa∂Wab;a

∂ria+

1Ωb

∂hb∂ρb

νb∂Wab;b

∂ria

,

with

νk =∑c

mc

ρkcv2kc

∂Wkc;k

∂hk.

Symmetry yields the same result for term D:

ε

8

∑b

∑c

mbmc

ρbcv2bc

∂Wbc;c

∂hc

∂hc∂ρc

∂ρc∂ria

=ε

8ma

∑b

mb

1

Ωa∂ha∂ρa

νa∂Wab;a

∂ria+

1Ωb

∂hb∂ρb

νb∂Wab;b

∂ria

,

141 α-SPH: variable-h terms

and bringing together all contributions from terms A, B, C and D:

ε

4

∑b

∑c

mbmc

ρbcv2bc

∂W bc

∂ria

=ε

4ma

∑b

mb

[v2ab

ρab+

1Ωa

∂ha∂ρa

νa

∂Wab;a

∂ria+v2ab

ρab+

1Ωb

∂hb∂ρb

νb

∂Wab;b

∂ria

].

Finally, we combine this with the earlier results, equations (A.1) and (A.3):

dvadt

= −∑b

mb

[Pa

Ωaρ2a

− ε

4

(v2ab

ρab+

1Ωa

∂ha∂ρa

νa −2ζaΩa

)∇aWab;a

+

PbΩbρ2

b

− ε

4

(v2ab

ρab+

1Ωb

∂hb∂ρb

νb −2ζbΩb

)∇aWab;b

]. (A.4)

Appendix B

α-SPH: Resultingdifferential equations

We wish to determine the form α-SPH momentum equations take in the con-tinuum limit. The discrete equation is written:

dvadt

= −∑b

mb

Paρ2a

+Pbρ2b

− ε

2

(v2ab

ρab− ζa

2ρ2a

− ζb2ρ2b

)∇aWab. (B.1)

First consider the following term from (B.1)

ε

2

∑b

mbv2ab

ρab∇aWab (B.2)

=ε

41ρa

∑b

mb(va − vb)2∇aWab +ε

4

∑b

mb

ρb(va − vb)2∇aWab

In the limit as particle numbers becomes infinite, the first term on the left-handside becomes

ε

41ρa

∑b

mb(va − vb)2∇aWab

−→ ε

41ρ

∫R

ρ(r′) [v(r)− v(r′)]2∇W (r − r′, h)dr′,

for a domain R, where we assume boundaries are not within the kernel support.We consider the x component of the above, and for simplicity restrict calcu-lations to two-dimension. Equations generalise to three dimensions trivially.The following notation is adopted: non-numerical superscripts refer to the com-ponent in consideration (v = (vx, vy, vz)); subscripts indicate derivatives withrespect to the given subscript (vx = ∂v

∂x , vxx = ∂2v∂x2 ). We expand v(r′) and ρ(r′)

about r to second order assuming functions are sufficiently smooth in expansion

142

143 α-SPH: Resulting differential equations

range:

ε

41ρ

∫R

ρ+ ∆xρx + ∆yρy

×[−∆xvxx −∆yvxy − 1

2∆xvxxx −∆x∆yvxxy − 12∆y2vxyy

]2+[vx −→ vy

]2∂W∂x

∂x′∂y′

where ∆x and ∆y denote (x − x′) and (y − y′) respectively. Furthermore,where the notation [vx −→ vy] is used, we require that a term equivalent tothat previous should be included but with vy substituted for vx. The termsmultiplied by ρ (leaving out terms which disappear due to symmetries) are

ε

4

∫R

[∆x3vxx v

xxx + ∆x∆y2(vxx v

xyy + 2vxy v

xxy)]

+[vx −→ vy

]∂W∂x

∂x′∂y′.

Now using integration by parts and the definition of α2 we have

α2

2([

3vxx vxxx + 2vxy v

xxy + vxx v

xyy

]+[vx −→ vy

]).

We now combine terms to yield

α2

2(vlx∇2vl + (∇vl · ∇vl)x

)with summation on repeated indices. The terms multiplied by ρx and ρy,leaving out those which equate to zero due to symmetry, gives

ε

41ρ

∫R

[∆x3(vxx)2ρx + ∆x∆y2

((vxy )2ρx + 2vxx v

xyρy) ]

+[vx −→ vy

]∂W∂x

∂x′∂y′,

or

α2

2ρ([

3(vxx)2ρx + (vxy )2ρx + 2vxx vxyρy]

+[vx −→ vy

]).

Combining terms and using the summation convention this becomes

α2

2ρ(ρx(∇vl · ∇vl

)+ 2vlx

(∇ρ · ∇vl

)).

Thus, in the limit as particle numbers become infinite, we have to second order

ε

41ρa

∑b

mb(va − vb)2 ∂Wab

∂xa

−→ α2

2

[vlx∇2vl + (∇vl · ∇vl)x +

ρxρ

(∇vl · ∇vl

)+

2vlxρ

(∇ρ · ∇vl

) ].

(B.3)

The derivation of the continuum equivalent of the second term on the left-handside of (B.2)) is performed as above to yield

ε

4

∑b

mb

ρb(va − vb)2 ∂Wab

∂xa−→ α2

2

[vlx∇2vl + (∇vl · ∇vl)x

]

144

and this combined with (B.3) gives a contribution of

ε

2

∑b

mbv2ab

ρab

∂Wab

∂xa

−→ α2[vlx∇2vl + (∇vl · ∇vl)x +

12ρρx(∇vl · ∇vl

)+vlxρ

(∇ρ · ∇vl

) ].

We now consider the following terms from (B.1):

− ε4

∑b

mb

(ζaρ2a

+ζbρ2b

)∇aWab.

In the limit we have

ε

4

∑b

mb

(ζaρ2a

+ζbρ2b

)∇aWab −→

ε

4∇ζρ.

Also, the continuum equivalent of ζ becomes

ζa =∑k

mkv2akWab −→

2ρα2

ε(∇vl · ∇vl),

so we then have

ε

4∇ζρ

=α2

2ρ∇(ρ(∇vl · ∇vl)

).

Therefore, combining the continuum limit of all terms in (B.1), we find

dv

dt=−∇Pρ

+α2

ρ

[∇vl

(∇ · (ρ∇vl)

)+ρ

2∇(∇vl · ∇vl

)],

where we have let particle separation go to zero holding parameter α constant.

Appendix C

The spectral method

A spectral code has been constructed for the integration of one-dimensionalsystems found in Chapter 4. Solutions obtained are compared with equivalentresults found via α-SPH methodologies. Details of the spectral algorithm aregiven here.

We write a one-dimensional version of the governing equations (3.30) and(3.8):

∂v

∂t+ v

∂v

∂x= −1

ρ

∂P

∂x+α2

ρ

[(∂v

∂x

)2∂ρ

∂x+ 2ρ

∂v

∂x

∂2v

∂x2

](C.1)

v = v +α2

ρ

∂

∂x

(ρ∂v

∂x

), (C.2)


∂ρ

∂t+ v

∂ρ

∂x= −ρ∂v

∂x, (C.3)

where we note that the filtered velocity is used. These equations, together withequation of state

P = ργ , (C.4)

form our one-dimensional system. We introduce truncated Fourier expansions:

vN (x, t) =N∑

m=−Nvm(t) exp(ikmx)

vN (x, t) =N∑

m=−N

¯vm(t) exp(ikmx)

ρN (x, t) =N∑

m=−Nρm(t) exp(ikmx).

where km = 2πm/L, L is the domain size, N is the mode beyond which spectralcomponents are truncated, and we have the complex number i =

√−1. Spec-

tral coefficients are denoted with an overbar, and take complex values. Inserting

145

C.1 Non-linear terms 146

these approximate forms in place of variables v, v and ρ in our one-dimensionalsystem, we arrive at a set of equations defining the evolution of spectral coeffi-cients:

∂vm∂t

+[vN

∂vN

∂x

]m

= −[

1ρN

∂PN

∂x

]m

+ α2

[1ρ

(∂vN

∂x

)2∂ρN

∂x+ 2

∂vN

∂x

∂2vN

∂x2

]m

¯vm = vm + α2

[1ρN

∂

∂x

(ρN

∂vN

∂x

)]m

∂ρm∂t

+[vN

∂ρN

∂x

]m

= −[ρN

∂vN

∂x

]m

where

gm = [g]m =1L

∫ L

0

g(x) exp(−ikmx)dx.

Subscripts in m denote the mth Fourier component.

C.1 Non-linear terms

The non-linear terms in the above equations lead to the coupling between modesresponsible for cascade processes. We may evaluate such terms through a num-ber of methods.

Quadratic non-linear terms may be evaluated directly. For any two functionsaN and bN we have

[aNbN ]m =1L

∫ L

0

aNbN exp(−ikmx)dx

=1L

∫ L

0

N∑o=−N

N∑p=−N

aobp exp(ikox) exp(ikpx) exp(−ikmx)dx

=1L

N∑o=−N

N∑p=−N

aobp

∫ L

0

exp(2πix(o+ p−m)/L)dx

=∑

o+p=m−N≤o,p≤N

aobp.

However, this approach leads to an expensive operation count of N2. Also,where non-linearities can not be constructed of quadratic terms, the above is notapplicable. An alternative is to utilise so called pseudospectral approximations.Here terms are multiplied in physical space, then returned to spectral space:

[aNbN ]m =1L

∫ L

0

aN (x)bN (x) exp(−ikmx)dx

' ∆xL

N∑j=−N

aN (xj)bN (xj) exp(−ikmxj),

with xj = jL/N . Where fast Fourier transforms are used, this results in costsof order N logN . The difference in direct evaluation and the pseudospectral

147 The spectral method

approximation gives the aliasing errors incurred in using the latter. Where aquadratic term is considered, per mode the error is written

Err(m) =∑

o+p=m±N−N/2≤o,p≤N/2

aobp.

Techniques exist to remove this error while still retaining the N logN operationcount, such as the 3/2 or phase shift rules (Canuto et al., 1988). As aliasingerrors are most evident in high order modes, where these are expected to be ofimportance, additional costs of dealiasing (or direct evaluation) are warranted.Given that we consider turbulent dynamics which by nature involve integra-tion of marginally resolved modes, we wish to calculate these modes as bestwe can. Therefore, where possible, non-linear terms have been evaluated usingthe pseudospectral approximation dealiased via the 3/2 rule. This method ef-fectively involves evaluating each Fourier series at fifty percent more physicalpoints, before multiplication and conversion back to spectral space. Naturallythis incurs greater costs, though fast Fourier techniques still result in hugelyimproved efficiency over direct evaluation.

C.2 Iteration for filtered velocity

An iterative method is required to determine v. We rewrite the equation for vas

vm = ¯vm − α2

[1ρN

∂

∂x

(ρN

∂vN

∂x

)]m

.

Having used pseudospectral methods to handle non-linearities, the above can beinterpreted as the mth equation of a linear system of equations of form F = LU .We wish to solve for U . Letting V 0 be our initial approximation to U , we usethe Richardson method to give the iterative scheme

V i+1 = V i + ω(F − LV i),

where ω is our relaxation parameter. This is currently chosen on a trial anderror basis, taking values between zero and one. We take our initial guess to bethe exact result where ρ is constant

V 0m =

vm(kmα)2 + 1

.

C.3 Timestepping

The equations of motion are integrated using a second order Adams-Bashforthtimestepping algorithm. For equations written in the form

∂v

∂t= F1

∂ρ

∂t= F2

C.3 Timestepping 148

the Adams-Bashforth algorithm is given as

vn+1 = vn +∆t2[3Fn1 − Fn−1

1

](C.5a)

ρn+1 = ρn +∆t2[3Fn2 − Fn−1

2

]. (C.5b)

This technique results in slow exponential error growth which may be madesufficiently small by choosing appropriate timestep. We consider a linearisedversion of our spectral evolution equations:

F1 =∂v

∂t= −γργ−2

0

∂ρ′

∂x

F2 =∂ρ′

∂t= −ρ0

∂v

∂x− ρ0α

2 ∂3v

∂x3

v = v + α2 ∂2v

∂x2,

where we have set ρ = ρ0 +ρ′, for mean density ρ0 and pertubed value ρ′. Onlyterms to order O(α2) have been retained. The equations for each spectral modeat timestep n are then:

Fn1 (m) ≡ −iγργ−20 kmρ

nm

Fn2 (m) ≡ −iρ0km(1− α2k2m)vnm

where km = 2πm/L. We have not included overbars for notational simplicity.Using Taylor’s expansions, the values at previous timestep can be written

Fn−11 (m) = ∆tγργ−1

0 k2m(1− α2k2

m)vnk − iγργ−20 kmρ

nm

Fn−12 (m) = −iρ0km(1− α2k2

m)vnm + ∆tγργ−10 k2

m(1− α2k2m)ρnm.

These values are inserted into (C.5):[vn+1m

ρn+1m

]=[1− 1

2∆t2γργ−10 k2

m(1− α2k2m) −i∆tγργ−2

0 km−i∆tρ0km(1− α2k2

m) 1− 12∆t2γργ−1

0 k2m(1− α2k2

m)

] [vnkρnk

].

We find the eigenvalues of the 2× 2 matrix in the above equation

det[1− 1

2∆t2γργ−10 k2

m(1− α2k2m)− λm −i∆tγργ−2

0 km−i∆tρ0km(1− α2k2

m) 1− 12∆t2γργ−1

0 k2m(1− α2k2

m)− λm

]= 0(

1− 12

∆t2γργ−10 k2

m(1− α2k2m)− λm

)2

+ ∆t2γργ−10 k2

m(1− α2k2m) = 0

(1− 1

2∆t2γργ−1

0 k2m(1− α2k2

m)− λm)

= ±i∆tkm√γργ−1

0 (1− α2k2m).

We let Gm =√γργ−1

0 (1− α2k2m) = Cs

√1− α2k2

m, where Cs is the soundspeed. (

1− 12

∆t2γργ−10 k2

m(1− α2k2m)− λm

)= ±i∆tkmGm

λm =(

1− 12

∆t2γργ−10 k2

m(1− α2k2m))± i (∆tkmGm)

149 The spectral method

|λm| = 1 +14

(∆tkmGm)4. (C.6)

We have eigenvalues which are always slightly greater than unity resulting in aweak instability. This instability has an O(∆t)4 dependence. We require thatthe eigenvalues are kept as close to unity as possible, and to this end we chooseour timestep such that

∆t 1Gmkm

=L

2πmCs√

1− α2k2m

The instability results in a slow numerical growth, whereas the true solutionhas no growth. For a sufficiently small value of ∆t, this numerical growth isbounded by

|λm|t/∆t ≤[1 + (∆tkmGm)4

]t/∆t(C.7)

≤ exp(∆t3k4mG

4m)t.

We can see from (C.7) that the alpha smoothing has the effect of reducing theerror bounds. This reduction is greater for larger order modes, which is desirableas it is these modes which exhibit the greatest error growth.

C.4 Normal modes of linearised energy

The total energy is written

ET =

L∫0

12ρvv +

ργ

γ − 1

dx. (C.8)

The kinetic term can be calculated exactly, while the thermal term may be cal-culated utilising the pseudospectral approximation. We also wish to determinethe energy distribution across the spectrum of modes, and to this end we lin-earise and expand (C.8). Setting ρ = ρ0 +δρ and assuming velocity componentshave zero mean, (C.8) becomes

ET =

L∫0

12

(ρ0 + δρ)vv

dx+

L∫0

(ρ0 + δρ)γ

γ − 1

dx = Ek + Eth.

Expanding the thermal energy and only retaining terms up to second order gives

Eth =

L∫0

ργ0 + γργ−1

0 (ρ− ρ0) + 12γ(γ − 1)ργ−2

0 (ρ− ρ0)2

γ − 1

dx

=

L∫0

ργ0 + 1

2γ(γ − 1)ργ−20 (ρ2 − 2ρρ0 + ρ2

0)γ − 1

dx

=Lργ0γ − 1

− 12Lγργ0 +

12γργ−2

0

N∑l,m=−N

ρlρm

L∫0

exp(i(kl + km)x) dx

=Lργ0γ − 1

+ Lγργ−20

N∑l=1

ρlρ−l.

C.4 Normal modes of linearised energy 150

Now turning to kinetic energy terms, we use equation (C.2) and retain up tosecond order terms:

Ek =

L∫0

12ρ0vv dx

=12ρ0

L∫0

vv − α2 v

ρ

∂

∂x

(ρ∂v

∂x

)dx

' 12ρ0

L∫0

vv − α2v

∂2v

∂x2

dx

=12ρ0

N∑l,m=−N

(¯vl ¯vm + α2k2m

¯vl ¯vm) L∫

0

exp(i(kl + km)x)dx

= Lρo

N∑l=0

¯vl ¯v−l(1 + α2k2

l

)We can then write

E0 =Lργ0γ − 1

El = Lρo ¯vl ¯v−l(1 + α2k2

l

)+ Lγργ−2

0 ρlρ−l l = 1, . . . , N.

The total energy from (C.8) is also calculated and is compared with the to-tal energy determined by summing modal components. Total energy (C.8) iscalculated using pseudospectral techniques.

Appendix D

Fourier mode constructionusing particle data

We wish to expand SPH particle data in trigonometric modes. Coefficients aredetermined as follows. For a function f(x) periodic on the domain −L ≤ x < L,we write a Fourier series:

f(x) =12a0 +

∞∑n=1

[an cos(knx) + bn sin(knx)] ,

withkn =

2πnL

.

Coefficients are then determined according to

an =2L

∫ L

−Lf(x) cos(knx)dx

bn =2L

∫ L

−Lf(x) sin(knx)dx.

For particle data fb = f(xb) taking general configuration, we make the approx-imation

an '2L

N∑b=1

fb cos(knxb)∆xb (D.1)

bn '2L

N∑b=1

fb sin(knxb)∆xb. (D.2)

for N particles spanning the domain. The volume element ∆xb is determineusing SPH density summations, such that we write ∆xb = mb/ρb for a particleof mass mb and density ρb.

Tests are performed to establish confidence limits in the above approxima-tion. Particle density and coordinate data obtained for SPH Euler flow simula-tions (see Section 4.2.1) are utilised, with various test spectrums then overlayedon particles. Test spectrums include a white noise spectrum, a linear decline

151

152

(a) (b)

(c) (d)

ρρ

0.95

1

1.05

0.95

1

1.05

Figure D.1: Density profiles at various times for a compressible Euler SPHsimulation. These particle and density configurations are used for spectral tests,with various functions overlayed at particle locations.

spectrum, a physically motivated k−2 falloff spectrum, and a random noise spec-trum. One-thousand particles are used to span the domain, and test spectrumsinclude up to five-hundred cosine modes. We give mode amplitudes determinedvia equation (D.1) where particle configurations correspond to the SPH densityprofiles of Figure D.1.

First considered is the white noise test spectrum, with values fb = f(xb)defined by

fb =500∑n=0

aTn cos(knxb)

with

aTn = 1.

Errors in the determined mode coefficients along with the white noise test spec-trum are given in Figure D.2. Errors are define according to

errn =

∣∣an − aTn ∣∣|aTn |

.

153 Fourier mode construction using particle data

(a)

(b)

(c)

(d)

aTn

errn

n

0.95

1

1.05

10-14

10-13

0.01

0.1

1

0.01

0.1

1

0.01

0.1

1

0 100 200 300 400 500

Figure D.2: White noise spectrum: Lower four graphs giving errors errn cor-responding to density profiles of Figure D.1.

Figure D.2 gives the errors corresponding to density profiles of Figure D.1.We find excellent correspondence with test mode coefficients up to mode n '430. This is perhaps not surprising, given that smoothing length should possiblybe considered the actual resolution limit for simulations. For a maximum densityperturbation of five percent, smoothing length considerations then lead to aNyquist mode of approximately n = 420. This is supported by consideration ofFigure D.2, where errors are observed to be small for modes up to n ' 440, withthe exception of frame (a), where density values are initialised at their exactvalues.

Next is the linear spectrum test,

aTn = (500− n)/500,

with errors given in Figure D.3. Results are similar to those found for whitenoise, though errors as a percentage of expect value are at times much largerfor modes over n = 450. This is possibly due to aliasing of neighbouring modes

154

(a)

(b)

(c)

(d)

aTn

errn

n

0

0.2

0.4

0.6

0.8

1

10-14

10-13

0.01

0.1

1

10

0.01

0.1

1

10

0.01

0.1

1

10

0 100 200 300 400 500

Figure D.3: Linear spectrum: Lower four graphs giving errors errn correspond-ing to density profiles of Figure D.1.

which may have larger amplitudes. Importantly, modes less n = 450 are recov-ered with minimal error.

A test spectrum which reflects typically what may be found for developedone-dimensional compressible Navier-Stokes is given by aTn = k−2

n , with aT0 = 0.Results are again similar to those present previously, with slightly increase noiseat the initial instance reflecting the smaller gap between mode amplitudes andmachine precision. More interesting is the increased error in low order modesin the final frame, though still accurate to within three percent.

The final test involves a random noise spectrum with zero mean and variancedefined by σn = (500 − n)/n. Excellent correspondence with expect valuesis found again up to approximately n = 440. The final slide shows slightlyincreased errors, though certainly within acceptable limits. In the final frame,a few outlying points have larger error (less than three percent), though thesemodes also have small expected amplitudes.

We also note that for all performed tests, sine mode coefficients given byequation (D.2) yields values |bn| ≤ 10−11 for all modes. In conclusion, results

155 Fourier mode construction using particle data

(a)

(b)

(c)

(d)

aTn

errn

n

10-610-510-410-310-210-1100

10-12

10-10

0.01

0.1

1

0.01

0.1

1

0.01

0.1

1

0 100 200 300 400 500

Figure D.4: Physical spectrum: Lower four graphs giving errors errn corre-sponding to density profiles of Figure D.1.

from performed tests indicate that for an SPH simulation using one-thousandparticles with a self-consistent smoothing length, we may have confidence in aspectral analysis using equations (D.1) and (D.2) up to mode n = 420, which ap-proximately corresponds to the resolution limit defined by a minimum smooth-ing length.

The above technique generalises to higher dimensions trivially. Similar testsin two dimensions recover amplitudes (for wavespace annulus totals) within onepercent of the test function value for modes of wavelength down to approx-imately 1.3h for smoothing length h. Tests utilised particle data from two-dimensional forced turbulence simulations.

156

(a)

(b)

(c)

(d)

aTn

errn

n

-3-2-10123

1e-14

1e-12

0.01

0.1

1

10

0.01

0.1

1

10

0.01

0.1

1

10

0 100 200 300 400 500

Figure D.5: Random spectrum: Lower four graphs giving errors errn corre-sponding to density profiles of Figure D.1.

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Download - SPH and -SPH: Applications and Analysis · Both SPH and -SPH are applied to test problems, with solutions analysed within physical and spectral space. In Chapter 2 an extensive review

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