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• 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Declarations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction 1

2 Smoothed Particle Hydrodynamics 4 2.1 The SPH discrete approximation . . . . . . . . . . . . . . . . . . 6

2.1.1 The approximation function . . . . . . . . . . . . . . . . . 6 2.1.2 The first derivative . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 The second derivative . . . . . . . . . . . . . . . . . . . . 10 2.1.4 The kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 SPH applied to fluid dynamics . . . . . . . . . . . . . . . . . . . 13 2.2.1 The continuity equation . . . . . . . . . . . . . . . . . . . 13 2.2.2 The momentum equation . . . . . . . . . . . . . . . . . . 15 2.2.3 The energy equations . . . . . . . . . . . . . . . . . . . . 17 2.2.4 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.5 Equation of state . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.6 Integrals of motion . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.1 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 Variable Resolution Implementations . . . . . . . . . . . . 25 2.3.3 Neighbouring particle list . . . . . . . . . . . . . . . . . . 27 2.3.4 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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

3 The α-SPH turbulence model 30 3.1 LANS-α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 α-SPH: equations of motion . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 The filtered velocity . . . . . . . . . . . . . . . . . . . . . 33 3.2.2 The momentum equation . . . . . . . . . . . . . . . . . . 35

3.3 Integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.1 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.2 Iteration for filtered velocity . . . . . . . . . . . . . . . . . 43

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

i

• 4 One-Dimensional Tests 44 4.1 Burgers’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.1 Colliding shocks . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.2 The steepening shock front . . . . . . . . . . . . . . . . . 49

4.2 One-dimensional Navier-Stokes . . . . . . . . . . . . . . . . . . . 54 4.2.1 The Euler system . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.2 Forced Navier-Stokes simulations . . . . . . . . . . . . . . 72

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

5 The Kelvin-Helmholtz instability 82 5.1 Constant velocity fluids in relative motion . . . . . . . . . . . . . 83

5.1.1 Linear results . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1.2 Computational configuration . . . . . . . . . . . . . . . . 84 5.1.3 Determination of mode growth rates . . . . . . . . . . . . 86 5.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 The hyperbolic tangent velocity profile . . . . . . . . . . . . . . . 92 5.2.1 Linear results . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2.2 Computational configuration . . . . . . . . . . . . . . . . 96 5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

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

6 Two-Dimensional Turbulence 103 6.1 Computational configuration . . . . . . . . . . . . . . . . . . . . 107

6.1.1 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.1.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.1.3 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.1.4 Equation of state . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Intermediate scale forcing . . . . . . . . . . . . . . . . . . . . . . 113 6.3 Large scale forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3.1 Quasi-steady solutions . . . . . . . . . . . . . . . . . . . . 121 6.3.2 Steady solutions . . . . . . . . . . . . . . . . . . . . . . . 123 6.3.3 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 145 C.1 Non-linear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 C.2 Iteration for filtered velocity . . . . . . . . . . . . . . . . . . . . . 147 C.3 Timestepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 C.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 any other degree or diploma in any university or other institution. To the best of my knowledge, this thesis contains no material previously published or written by another person, except where due reference is made within the text of the thesis.

iv

• Summary

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

In Chapter 2 an extensive review of the SPH method is undertaken. First the details of SPH as a general numerical method are considered, followed by specifics of the SPH application to fluid dynamics, including derivation of equa- tions of motion within a Lagrangian framework. Likewise in Chapter 3, the particulars 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, highly accurate 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 to correct non-linear energy cascades and the influence of secondary SPH pressure gradient terms. Simulations of α-SPH demonstrate it to be successful in induc- ing closure of Euler dynamics, though results for small values of the turbulence parameter are found to be unsatisfactory.

Linear regime studies of the Kelvin-Helmholtz simulations are presented in Chapter 5. Expected growth rates from linear stability theory are recovered for SPH simulations where sufficient resolution is utilised. For poorly resolved Kelvin-Helmholtz perturbations, incorrect growth rates are shown to be directly related to deficiencies of the SPH pressure gradient. We demonstrate that the α-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 these do not appear to significantly influence large scales. Results for α-SPH method are given, though simulations are restricted to small values for the turbulence parameter, and findings are inconclusive.

v

• Chapter 1

Introduction

The advent of digital computers in the twentieth century heralded an era of strong proliferation in the mathematical sciences. A new avenue of investigation was opened, and previously intractable problems were now able to be tackled using the brute numerical force afforded by the digital calculator. Indeed the potential application of computational techniques has increased in line with the ever increasing processing power of the computer. Despite recent pessimism, technological developments continue to yield advancements in line with Moore’s law, which states that computational speed doubles approximately every two years. Even so, for many classes of problems, solution via pure numerical force is 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 fluid turbulence. The solution to problems of turbulence defies numerical calculation by 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 magnitude greater than what is currently available. It is perhaps ironic then that most flows encountered in problems of engineering interest are turbulent in nature.

While the straightforward approach of resolving all relevant dynamics is usually not a possibility, numerous techniques have been developed w