Development and Testing of a Robust Free-SurfaceFinite Volume Method

Marwan Darwish, Ph.D.

Faculty of Engineering and Architecture, American University of Beirut

July 2003

Summary

A robust numerical technique for the simulation of Free-Surface Flows wasdeveloped, implemented and tested. The technique is based on the Finite Volumemethod, uses unstructured grids for better geometric flexibility, a pressure-basedmulti-fluid algorithm and a high order discretization in both the spatial and timedomains. The accuracy of the method was deemed to be an essential feature, to thisend in the spatial domain state-of-the-art high resolution (HR) advection schemes areused in addition to a family of fifth order accurate very High resolutions (VHR)schemes. In the time domain, in addition to the standard Crank-Nicholson scheme,two new discretization schemes were developed: a compressive transient scheme anda second order bounded transient scheme based on the second order backward Eulerscheme. A series of test were used to validate the accuracy and robustness of thecode. Result indicate that use of the widely popular first order accurate schemeswhether in the transient (First order Euler scheme) or spatial (UPWIND scheme)domains is not an option in free-surface simulation as it leads to substantialunphysical distortion in the simulated free surface. The Crank-Nicholson and SMARTschemes were found to be a good combination for the transient and spatial domaindiscretizations, with respect to accuracy and robustness. Robustness was tested on avery tough dam break simulation involving a returning wave, and results were verysatisfactory.

Key-words: Free-Surface, Finite Volume, Volume of Fluid, Multiphase Flow, VeryHigh Resolution Scheme, Rhie-Chow Interpolation

1. INTRODUCTION

The last decades have witnessed a sustained development effort in the area ofComputational Fluid Dynamics (CFD) that have led to important advances in thefield: increased numerical accuracy through the development of High ResolutionSchemes [1,2,3,4], increased robustness through the development of velocity-pressurecoupling algorithms for the simulation of compressible and incompressible flows atall speeds (subsonic, transonic and supersonic) [5,6], increased model complexitythrough the development of multi-fluid algorithms [7,8] to simulate complex multi-fluid flows, increased efficiency though the development of better solvers and robustmultigrid acceleration techniques [9,10,11,12]. A major driver behind thesedevelopment have been the increasing needs of a number of industries, such as theautomotive, chemical processing, aeronautic, industries to name a few, for anumerical simulation tool to help engineers and developers as they tackle more andmore complex problems. The intrusion of Computation Fluid Dynamics as a tool inthe design of ships, boats, and injection moulds has put a renewed focus on thedevelopment of numerical techniques for the simulation of Free-Surface flows. Inthis project we endeavor to develop of a simulation tool for Free-Surface flows thatinvolve immiscible fluids separated by a well-defined interface, which are ofimportance in engineering and environmental applications. Examples include fillingof fluid in metal casting operations, polymer injection molding, flow of air pollutantsaround industrial plants, etc.. . In these flows there is generally no inter-phase masstransfer, and generally the fluids are distinct in their composition. The simulation ofsuch physical phenomena is important in many respects: for metal casting andpolymer injection molding the final quality of the product is highly dependent on thetype of flow circulation occurring within the cast of mould, changing the castingdesign to minimize turbulent flow for example can drastically minimize void defectsin the product. A number of CFD approaches have been developed for specific free-surface flow applications, such as the potential-flow/boundary-element methodswhich are used for wave dynamics and water-entry problems, or the shallow-waterequations, incorporating depth-averaging and a hydrostatic-pressure approximation,which are frequently used in tidal flows and to simulate non-breaking wavepropagation and run-up. While these methods are powerful tools in their own areasthey are restricted in their use within these areas because of the approximationembodied in their development, such as neglecting viscous transport, or hydrostaticapproximations that become untenable when vertical accelerations are comparablewith that of gravity; an important example is wave impact on structures. Surfacetension can also play an important role [13,14,15,16,17] however in this work itseffect are neglected

REPORT ORGANIZATION

This report will be divided into five sections. In the first section we review free-

surface flow numerical methods and define the general features of our method. In thesecond section the governing equations for free-surface flow are derived and the basisof the discretization method described. In the next two sections specific techniquesdeveloped as part of this project for improved free-surface simulations are presentednamely the development of accurate numerics for the discretization of the volumefraction, and improvements to the Rhie-Chow interpolation for free-surfacesimulation these represent the major contributions of this thesis. Finally in the last twosections a number of test problems are solved for validation purposes.

4

2. REVIEW OF NUMERICAL METHODS FOR

FREE-SURFACE FLOWS

Numerical methods for free-surface flows can generally be divided into two basiccategories: moving mesh methods and fixed mesh methods, within each a number ofapproaches have been developed over the years. In what follow we propose tobriefly review these methods in order to better appreciate their advantages anddisadvantages.

MOVING MESH METHODS

All Moving Mesh methods involve a moving mesh that follows the boundarybetween the simulated fluids [18,19,20]. Either the mesh boundary and the interfaceare the same, or an internal boundary within the mesh follows the interface, see figure2.1. This requires a continuous re-meshing of the domain or part of it at each timestep so as to follow the interface movement. This can lead to complex algorithms thatdepend on whether elements are added/subtracted or simply moved during the re-meshing operation. Additionally a special procedure needs to be implemented in orderto enforce volume conservation in the moving cells. All of this in addition to specialtopological restraints related to agglomeration and dispersion of the simulated flowsmake these methods rather limited in the types of applications for which they areuseful. However, it should be mentioned that, where applicable, these methods allowa precise control of the boundary conditions at the Free-Surface interface [21].

(a) (b)Figure 2.1: Moving mesh methods (a) internal fluid/fluid interface (b) external

interface

5

FIXED MESH METHODS

In Fixed Mesh methods the fluid interface does not conform to the mesh boundary orinternal elements rather it is described using an internal representation. Fixed meshmethods are generally more robust and more general in their applications as theusually complex re-meshing step is avoided. In the Fixed Mesh group the interface ismoving within a fixed computational mesh, whether structured or unstructured and nore-meshing is needed. Three different approaches have evolved within this category,namely: The Point or Marker Methods in which a set of discrete particles ormarkers, convected in a Lagrangian manner, is used to define and follow the fluidinterface. The Surface or Interface Tracking Methods in which some interfacefunction is used to explicitly define the fluid/fluid interface. And the Volume orInterface Capturing Methods, where the interface is not explicitly defined but isreconstructed when needed using available information regarding the respective fluidcontent in different mesh elements.

Point or Marker Based Methods

The first method capable of modeling gas-liquid flow, separated by a movinginterface, was the Marker and Cell method (MAC) of Harlow and Welch [22]. Thiswas in fact a combination of a Eulerian solution of the basic flow field, withLagrangian marker particles attached to the liquid to distinguish it from the gas (seefigure 2.2). While the staggered mesh layout and other features of MAC have becomea model for many other Eulerian codes, the marker particles proved to becomputationally too expensive and are now rarely used. However, some improvedversions have been presented, for example by Chen et al. [23].

Figure 2.2: Schematic representation of a marker and cell mesh layout, the particlesare tracked individually in a Lagrangian manner

Surface or Interface Tracking Methods

A feature common to many Surface Tracking methods [24,14,15] is that theinterface is represented by a set of points connected by polynomial curves (see figure2.3). These points move in time and are advected in a Lagrangian manner using thefixed grid flow field as the advecting medium. During this movement the particlesmight tend to move apart and the marker density along the surface might decrease,this leads to a lower resolution of the interface. This problem can be partially

6

alleviated by the addition and deletion of particles dynamically wherever needed so asto preserve a near uniform particle density throughout the interface. This re-initialization is performed in tandem with a renumbering of the particles position sothat the particles are kept in sequence along the interface. Another approach followedis to mark the interface position by using some special function; examples of thisapproach include the height function [25,26] and level-set function methods[27,28,29,30,31,32].

Generally the advantage of these methods is that the interface position is knownthroughout the flow field and remains sharp as it is convected across the mesh. Inaddition the interface curvature can be precisely computed if needed for the inclusionof surface tension forces. But again limitations arise when complex interfacetopologies develop during the simulation such as coalescence and rupture of theinterface surface [33,34].

(a) (b)Figure 2.3: Schematic of the Interface Tracking methods (a) simple topology (b)

complex topology.

Volume or Interface Capturing Methods

In Volume or Surface Capturing methods [35,36,37,38,39,40,41,42,43,44] the fluidson either side of the interface are marked by an indicator function (volume of fluidsmethods). The exact position of the interface is thus not known explicitly and specialtechniques need to be applied to capture or reconstruct (reconstruction algorithm) awell-defined interface as part of the solution algorithm. This is the main drawback ofthis technique.

The use of volume fraction is currently the most economical method, and can bereadily integrated with other techniques for the simulation of solidification, fluid-solidinteraction, combustion or even multi-component simulation. In this method a scalarindicator function, between zero and one, known as the volume fraction, is used todistinguish between the different fluids. A value of zero indicates the presence of onefluid and a value of unity indicates the second fluid. On a computational mesh,volume fraction values between these two limits indicate the presence of the interfaceand the value itself gives an indication of the relative proportions occupying the cellvolume.

7

Volume Of Fluid

Through the SOLA-VOF code of Hirt & Nichols [45], the VOF method becamepopular in the 80s though development did occur before [46,47,48]. The main featureof the method is the use of a volume-fraction field defined on each control volume inthe mesh and computed in a Eulerian manner (see figure 2.4). The volume fractionfield indicates for each computational cell the volume fraction r(k) of fluid (k), so ifr(k)=0 then the cell does not contain fluid (k), if r(k)=1 then the cell is completely filledwith fluid (k) and for 0< r(k)<1 then the cell is partially filled by fluid (k) (see figure2.4). Using the volume fraction field the interface between the different fluids can beconstructed. Since r(k) is a property of the fluid with which it moves a simpleadvection equation Dr(k)/Dt=0 holds. An determining element of VOF methods is inthe technique used for advecting r(k) as standard advection schemes methods sufferfrom numerical diffusion, which would ruin the solution of the purely advected fieldr(k). Application wise the VOF is quite general and has been used for the simulation ofrising bubbles of air in a liquid bed [49,50]), for the simulation of liquid dropletimpact [51,52,53], jet and droplet break-up [ 54,55], breaking waves and waves onbeaches [56,57,58] and dam-break problems [ 59] to cite a few applications. Severalcommercial code also use variations of the VOF method such as SURFER [60] orRIPPLE [61], CFX , StarCD, Comet, FLUENT, for more details on the VOF methodrefer to [62,63].

Figure 2.4: Schematic of the VOF Method

In light of the above review, the volume fraction approach is clearly the most generalin its applications. However, while much work has been performed in thedevelopment of volume fractions methods [64,65,66,67,68] it has mostly concentratedon the development of improved surface reconstruction methods. Improvements topressure-velocity coupling, also known as the Rhie-Chow Interpolation [6], as appliedto the volume fraction method has been largely neglected and the application ofgeneral Very High Resolution Schemes (VHR), of order of accuracy higher that 5th

order, schemes has not been pursued. The Pressure-velocity coupling plays anessential role in the convergence and robustness of any CFD method and as wasdemonstrated in the all speed flow algorithms [5,6] and general multi-fluid flowalgorithms [7,8] developed by the author, proper formulation of the Rhie-Chowinterpolation for the volume fraction method can form the basis of a more general androbust technique for the simulation of a wider range of free-surface flows.Specifically in the handling of body forces and pressure gradients in the momentumequations, which is critical when dealing with fluids of high-density ratio.

8

Furthermore the inherent deficiency of the VOF is addressed by the development andapplication of a Very High Resolution scheme for use in the discretization of thevolume fraction equation.

9

3. GOVERNING EQUATIONS

INTRODUCTION

Three conservation laws, namely conservation of mass, momentum and energy,govern fluid flows. These laws determine the physical behavior of the flow and areindependent of the nature of the fluid, which is defined by additional properties(constitutive relations) such as viscosity, heat conductivity, surface tension andcompressibility. For bulk phases the equations of conservation are well known [69]and can all be cast in the form of a general scalar conservation equation written as:

€

∂(ρφ)∂t

+∇.(ρvφ) =∇.(Γ∇φ) + Q (3.1)

Where v is the velocity vector, ρ is the density, and Γ differs according to thephysical property

€

φ . The four terms in the above equation describe respectivelytransient, convection (or advection), diffusion and generation/dissipation and othersources effects, see figure (2.1).

Figure 3.1: Conservation over a control volume

The governing equations for flows involving free surfaces are more difficult toformulate. For their formulation two approaches have been devised, both start withthe conservation equations governing the flow of a single fluid. In the first known asthe multifluid homogeneous model or multifluid formulation, the multifluid equationsare averaged over a Representative Elementary Volume such that all properties aregoverning by mixture laws, then assuming a homogeneous velocity and temperaturefields the equations governing free surface flow are derived. In the second known asthe interface tracking model or single-fluid formulation, no averaging is assumed totake place and the interface relations between the fluids have to be accounted forexplicitly. For free-Surface flows where surface tension is negligible both approachesyield the same set of equations.

10

THE REPRESENTATIVE ELEMENTARY VOLUME (REV)

The view of a Representative Elementary Volume (REV) rests on the idea of time andspace averaging and implies the following:

1. Any small volume can be regarded as containing a volume fraction αk of the kth

phase, so that, if there are n phases then

1n

1kk =∑

=

α (3.2)

2. The mass flow rate of any phase across any elementary surface within the domainat any time can be expressed in terms of the quantity

€

αkρkr v k where

€

ρk is thedensity of the kth phase and vk is the local value of the velocity vector of thatphase.

3. When the content of the finite volume and the flow rates across finite areas are tobe computed over finite time intervals, a suitable averaging over space and timewill be carried out.

Observance of these rules amounts to treating each phase as a continuum in the spaceunder consideration, which requires choosing a proper scale with regard to the controlvolume used, or a Representative Elementary Volume (REV). For a multi-phasesystem the equations are derived over a representative element volume within whichthe different phases are present. The scale of the modeled phases has to be smallerthan the REV volume. An extensive review of this averaging approach can be found,for example, in Hassanizadeh [70,71].

We start by presenting the governing equations for multi-fluid flows before describingthe simplification leading to the free-surface flow equations.

THE MULTIFLUID FORMULATION

In what follows the fluidic equations that describe the multifluid model are describedfor an n-fluid flow.

Conservation of mass

The general form of the continuity equations is

€

∂ r(k )ρ(k )( )∂t

+∇. r(k )ρ(k )u(k )( ) = ˙ M (k )(3.3)

In this equation rk is the volume fraction, which represent the volumetric spaceoccupied by the kth fluid, ρk the density and uk, the velocity field of the kth fluid.

Note that the volume fractions have to satisfy the geometry conservation equation

€

r(k )k∑ =1 (3.4)

where

11

€

r(k ) =V (k )

V (n )

~n∑

(3.5)

where Vn represents the volume of the nth phase. Finally the term rkρk present in thecontinuity equation, denote the effective density. For a steady state flow equation(3.3) becomes

€

∇. r(k)ρ(k )u(k)( ) = ˙ M (k )(3.6)

Considering that each fluid is incompressible, then

€

∂( )∂ρ(k )

= 0 (3.7)

and equation () becomes

€

∂ r(k )( )∂t

+∇. r(k )u(k )( ) = ˙ M (k )(3.8)

and if there is no phase change

€

∂ r(k )( )∂t

+∇. r(k )u(k )( ) = 0 (3.9)

Summation over all the phases leads to the overall mass-conservation equation:

€

∂ r(k)ρ(k )( )∂t

+∇. r(k)ρ(k )u(k)( )

~k∑ = 0 (3.10)

The conservation equations can also be though of as volume fraction equations. Inthis form they can be used to calculate the volume fraction occupied by the phase inthe control volume, the equation can be rewritten as:

€

∂ ρ(k )r(k )( )∂t

+∇. ρ(k )u(k )r(k )( ) = 0 (3.11)

Conservation of momentum

The phasic momentum equation for the kth phase can be written as:

€

∂ r(k )ρ(k)u(k )( )∂t

+∇. r(k)ρ(k )u(k )u(k )( )=

∇. r(k)µ(k )∇u(k)( ) + r(k ) −∇P + B(k )( ) + I(k) + ˙ M n(k ) u(n ) −u(k )( )

~n∑

(3.12)

Here P stands for the pressure, which is regarded as being shared between the phases,Bk is the body force per unit volume of phase k, Ik is the momentum transfer to phasek resulting from interaction with the other phases. The latter term vanish when theglobal momentum equation is formed thus:

12

€

∂ r(k)ρ(k )u(k)( )∂t

+∇. r(k )ρ(k)u(k )u(k)( ) =∇. r(k )µ(k)∇u(k )( ) −∇P + r(k)B(k )

~k∑ (3.13)

Conservation of energy

Let hk stand for the enthalpy of phase k per unit mass, then the first law of thermo-dynamics implies:

€

∂ rkρ(k )h(k )( )∂t

+∇. r(k )ρ(k )u(k )h(k )( ) =∇. r(k )k(k )∇T (k )( ) + r(k )H (k) + J(k ) (3.14)

where Hk represents the heat transfer to phase k per unit volume, Jk represents theeffects of interactions with other phases occupying the same control volume. Onceagain the overall energy equation loses the interaction terms:

€

∂ r(k)ρ(k )h(k)( )∂t

+∇. r(k )ρ(k)u(k )h(k)( ) =∇. r(k )µ(k)∇T (k )( ) + r(k )H (k )

k∑ (3.15)

THE HOMOGENEOUS MODEL

Free surface phenomena are characterized by the presence of a distinct surface thatseparates one fluid from another. The simulation of this class of fluids can besimplified if the velocities of the fluids are assumed to be similar within each controlvolume, i.e. the effect of inter-fluid slip is neglected. This condition is representedmathematically as :

€

u = u1 = u2 =L = un (3.16)

We can also define a bulk density and a bulk viscosity as:

€

ρ = rkρk

~k∑

µ = rkµk

~k∑

(3.17)

In this case the multi-phase flow can be regarded as a single-phase fluid having itsown peculiar thermo-dynamic and transport property relations (see Figure 3.2).Mixture theory is used to compute these properties depending on the mass fraction ofeach phase in the control volume. No species equation is needed since each of thetwo fluids is assumed to have a constant composition.

13

Figure 3.2: (a) the Control Volume for the Multi-Fluid Formulation (b) as viewed inthe Homogeneous Model

The free surface model can now be obtained by assuming a homogeneous velocityfields i.e.

€

u = u(m ) = u(1) = u(2)

ρ = ρ(m ) = r(1)ρ(1) + r(2)ρ(2)(3.18)

By using the following definitions where ()m indicate the mixture value theconservation equations become:

Conservation of mass

The two equations of conservation of mass for the two fluids are added together toyield the following equation which is used to form the pressure equation

€

∂ r(1)ρ(1) + r(2)ρ(2)( )∂t

+r ∇ . r(1)ρ(1)u(1) + r(2)ρ(2)u(2)( ) = 0 (3.19)

using the homogeneous velocity field the equation is written in a simplified form as:

€

∂ρ(m )

∂t+∇. ρ(m)u( ) = 0 (3.20)

Volume Fraction Equation

Because the volume fraction r is not known throughout the field, an equation isneeded for its computation. The phasic continuity equations can be used as a volumefraction equation to determine the fraction of the phase in the control volumes,provided that not all are used for the computation of the volume fraction this isnecessary as the bulk continuity equation would become redundant. Thus for the lastvolume fraction compatibility equation is used, i.e.:

€

rn =1− rkk~1..n−1∑ (3.21)

14

Conservation of Momentum

€

∂ r(1)ρ(1)u(1) + r(2)ρ(2)u(2)( )∂t

+r ∇ . r(1)ρ(1)u(1)u(1) + r(2)ρ(2)u(2)u(2)( )

=r ∇ . r(1)µ(1)

r ∇ u(1) + r(2)µ(2)

r ∇ u(2)( ) − r(1) + r(2)( )

r ∇ P

(3. 22)

or more concisely

€

∂ ρ(m )u( )∂t

+∇. ρ(m )uu( ) =∇. µ(m )∇u( ) −∇P (3.23)

where

€

µ(m ) = r(1)µ(1) + r(2)µ(2) (3.24)

Conservation of energy

€

∂ ρ(m )h( )∂t

+∇. ρ(m )uh( ) =∇. k∇T( ) + ... (3.25)

THE ONE-FLUID FORMULATION

The exact or microscopic instantaneous equations governing a free surface flowsystem can also be formally written in terms of the component indicator function ofthe volume fraction field r(k)(x,t) at time t and point x defined by r(k) = 1 for x in phasek and r(k)=0 otherwise. Since r(k) is a property moving with the flow, its materialderivative is obviously zero [72]:

€

Dr(k )

Dt=∂r(k )

∂t+ u ⋅ ∇r(k) = 0 (3.26)

This is known as the topological equation describing the motion of a surface markedwith rk, moving with velocity v. note in particular that this is a weak formulation ofthe problem since the discontinuity in rk (across interface) makes it a non-derivablefunction.

Figure 3.3: Control Volume for the one fluid formulation

15

In the absence of heat and mass transfer, the mass balance equation for non-miscible,contacting fluids contained within volume V as shown in figure 3.3 can be written as:

€

r(k ) ∂ρ(k )

∂t+∇ ⋅ ρ(k )u( )

= 0 (3.27)

where ρ denote the density of phase k. The above two equations can be reformulatedin a more convenient form:

€

∂ r(k )ρ(k )( )∂t

+∇ ⋅ r(k )ρ(k )u( ) = 0 (3.28)

Combining equation (3.28) over all of the n phases yields a single conservationequation but with a variable density r dictated by the interface location.

€

∂ ρ( )∂t

+∇ ⋅ ρu( ) = 0 (3.29)

with

€

ρ = r(k )ρ(k )~k∑ (3.30)

Using equation () and taking

€

∂ρ(k )

∂t= 0 (3.31)

to be zero for incompressible fluids simplifies equation (3.29) to

€

∂ρ∂t

+ u ⋅ ∇ρ + ρ∇ ⋅u =

∂ r(k )ρ(k)k=1..n∑

∂t+ u ⋅ ∇ r(k )ρ(k )

k=1..n∑

+ ρ∇ ⋅u

= ρ(k )∂r(k)

∂t

k=1..n∑ + r(k ) ∂ρ(k )

∂t

k=1..n∑ + r(k) u ⋅ ∇ρ(k )( )

k=1..n∑ + ρ(k ) u ⋅ ∇r(k )( )

k=1..n∑ + ρ∇ ⋅u

= ρ(k )∂r(k)

∂t+ u ⋅ ∇r(k )

k=1..n∑ + r(k) ∂ρ(k )

∂t+ u ⋅ ∇ρ(k)

k=1..n∑ + ρ∇ ⋅u

= 0(3.32)

Using equation (3.31) and noting that the fluids are incompressible, equation (3.32)reduces to:

€

∇ ⋅u = 0 (3.33)

This result is important as it represents a geometric constraint rather than a massconstraint

16

4. HIGH-RESOLUTION SCHEMES

Free-surface flows are defined as problems where the fluids are separated by sharpinterfaces, hence the use of advection schemes that can preserve the sharpness of fluidinterfaces interface is essential. While standard advection techniques can be used toadvect some type of material indicator function, these methods are usually toodiffusive and cannot guarantee the sharp resolution of the multi-fluid interfacesessential in free surface problems on stationary meshes. Two approaches can befollowed to achieve this goal: in the first approach the sharpness is preserved by usingVery High Resolution scheme, while in the second approach low-order Compressiveschemes are used. Most if not all of the schemes used in Free-Surface flows havebeen of the compressive type (C-schemes) [73,74,75,76,77,78]. In this project a newVHR scheme is evaluated in addition to a number of C schemes.

In all these schemes a convection boundedness criterion (CBC) has to be enforced ona base interpolation profile. The role of the CBC is to eliminate numerical dispersionproblems [79] without increasing numerical diffusion [ 80,81,82,83]. Striking aproper balance in the enforcing of the CBC is essential as numerical dispersion andnumerical diffusion are “self-destructing”, that is it is by adding numerical diffusionthat we eliminate numerical dispersion and vice-versa. The VBV framework used inthis work is based on the Normalized Variable Formulation of Leonard [84], isdescribed in the next section. One issue of to keep in mind is the extremadeterioration that occurs due to the indiscriminate enforcing the CBC in all flowregions. In this case a new and subtle error is introduced that leads to a significantreduction in accuracy at locations where extrema (maxima or minima) with steepprofiles are present The reduction in accuracy is mainly due to the attenuation of theextrema levels. More specifically, the local extrema in the profiles gradually decreasein value with distance as if in the presence of a diffusion-like phenomenon.

The Convection Boundedness Criterion

The NVF Convection Boundedness Criteria can be defined in terms of normalizedvariable (NV) proposed by Leonard [85]. Considering face f of a control volume (seefigure 1), defining φU, φD, φC and φf as the Upstream (U), downstream (D), centralnodal values (C), and face value (f) for each cell face the normalized variable isdefined as:

€

˜ φ =φ − φUφD −φU

(4.1)

17

Figure 4.1: Notation for CBC

Note that with this normalization, ˜ φ D =1 and ˜ φ U = 0 . The use of the normalizedvariable simplifies the definition of the functional relationships of HR schemes and ishelpful in defining the conditions that the functional relationships should satisfy inorder to be bounded and numerically stable.

NV in unstructured grids

For unstructured grids defining the U,D and C nodes is not as straightforward as forstructured grids as there is no logical Upwind node. In this case a virtual Upwindnode is constructed (see figure 4.2). Noting that values for φD and φC represent thevalues of the nodes straddling the interface and thus are readily available forunstructured grid. Therefore, the normalized variables would be computable if theterm involving φU could be replaced by a known term. In this case

€

φD −φU( ) =∇φC ⋅ rUD= 2∇φC ⋅ rCD( )

(4.2)

and the formulation of Normalized Variable becomes

€

˜ φ =φ − φUφD −φU

=φ − φU

2∇φC ⋅ rCD( )( ) ( ) ( )

122

rCD

C

CD

CDCf −

φ−φ

⋅φ∇=

φ−φ

φ−φ−⋅φ∇= CDCD rr

(4.3)

which can be easily computed for unstructured grids.

18

Figure 4.2: NV for unstructured grids

Returning to the significance of the CBC it is clear that for values of ˜ φ C < 0 or ˜ φ C >1we have an extremum at C,

19

Figure 4.3: NV for ˜ φ C <0< and ˜ φ C >1

While for ˜ φ C ≈0 and ˜ φ C ≈1 we have a gradient jump

20

Figure 4.4: NV for ˜ φ C ≈0 and ˜ φ C ≈1

and for values of ˜ φ C between 0 and 1, 0< ˜ φ C <1, we have a monotone profile.

The NVF Convective Boundedness Criteria (CBC)

Based on the normalized variable analysis, Gaskell and Lau [83] formulated aconvection boundedness criterion (CBC) for implicit steady state flow calculation.The CBC states that for a scheme to have the boundedness property its functionalrelationship should be continuous, should be bounded from below by ˜ φ C and fromabove by unity, and should pass through the points (0,0) and (1,1), in the monotonicrange (0< ˜ φ C <1), and for 1< ˜ φ C or ˜ φ C <0, the functional relationship f( ˜ φ C ) shouldequal ˜ φ C . The above conditions illustrated in the figure below, can be formulated as:

€

˜ φ f =

f ˜ φ C( ) continuous

f ˜ φ C( ) =1 if ˜ φ C =1˜ φ C < f ˜ φ C( ) <1 0 < ˜ φ C <1

f ˜ φ C( ) = 0 if ˜ φ C = 0

f ˜ φ C( ) = ˜ φ C if ˜ φ C < 0 or ˜ φ C >1

(4.4)

21

Figure 4.5: CBC region

The interpretation of the Convection Boundedness Criteria is quite intuitive. WhenφC is in a monotonic profile the interpolation profile at the cell surface should notyield any new extremum. Thus it is constrained between the cell face φ values, as thevalue of φC get closer to of φD while still in the monotonic regime the value of φ fwill also tend toward φD until φC becomes equal to φD when φ f also becomes equalto φD . When the value of φC is in the ˜ φ C >1, φ f is assigned the upwind value, i.e. φC ,this has the effect of yielding the largest outflow condition possible while fulfillingthe condition that φ f is bounded by the cell face straddling nodes. This behaviormeans that any undue oscillation will be damped since φC will tend to a lower valuebecause outflow is larger than inflow in these conditions. Thus if there is no externalphysical mechanism to yield the extrema (a source term for example) the extrema willdie out. A similar mechanism takes place when ˜ φ C <0. However as φC gets closer toφU coming from the non-monotonic region φ f will be equal to the upwind value i.e.φC until φC =φU thus the condition that ( ˜ φ f , ˜ φ C ) pass through the point (0,0).

Many known advection schemes can be represented on the NV diagram, where theirplot can yield important information as to their behavior.

22

Figure 4.6: High Order schemes on NV diagram

Different schemes can be defined using the normalized variables, for example, theQUICK scheme can be formulated using the normalized variables as:

Scheme Functional Relationship Functional Relationship

(NVF)

First order upwinding (T.1) φ f = φC ˜ φ f = ˜ φ C

Second order upwinding (T.2)(Extrapolation of linear fit through φC and φU )

φ f =32φC −

12φU

˜ φ f =32

˜ φ C

Lax-Wendroff method (T.3)(Interpolation of linear fit through φC and φD )

φ f =12φC +

12φD

˜ φ f =12

+12

˜ φ C

Fromm's method (T.4)

(arithmetic mean of (T.2) and (T.3))φ f = φC +

φD −φU4

˜ φ f =14

+ ˜ φ C

QUICK (T.5)

(Interpolation of quadratic fitthrough φU , φC and φD )

φ f =φC +φD2

−φD − 2φU + φU

8˜ φ f =

38

+34

˜ φ C

Table 4.1: NVF for various linear schemes

Constructing a High-Resolution scheme using the NVD diagram is a relatively simpleexercise, any base high order scheme can be bounded using an ad-hoc set of curves. Anumber of well-known High-Resolution schemes built in this manner are illustrated infigure 4.6.

23

Figure 4.7: High Resolution schemes on NV diagram

Compressive Schemes

Compressive schemes are especially suitable for use with the Volume Fraction (VF)equation as they prevent the smearing associated with numerical diffusion, theseschemes can be recognized in the NV diagram as their NV line will tend toward theDownwind scheme line. The most compressive scheme is the Downwind scheme.,however it is very unstable in that it leads to increasing oscillations when used in thepresence of the smallest of step profiles. The Hyper-C scheme try to address thisproblem by blending the Downwind scheme with the upwind scheme in such a way asto preserve it compressive characteristics. Unfortunately the Hyper-C scheme on itsown is not suitable for the modeling of interfacial flow because of its steepeningcharacteristics that tend to wrinkle the interface [86]. This is because downwindingtends to compress any gradient into a step profile, even if the orientation of theinterface is almost tangential to the flow direction. This artificial steepening of thevolume fraction gradients was shown by Leonard [87] for the advection of a one-dimensional semi-ellipse profile that becomes disfigured into a step profile by thescheme.

Two C-schemes that avoid the steepening problems are the HRIC scheme ofMuzaferija [67] and the CICSAM scheme of Ubink [77]. Both these schemes defined

24

within the NVF framework have been implemented in commercial codes, and bothrequire a local Courant number be lower than 1. These schemes blend the Hyper-Cand U-QUICKEST schemes with the upwind scheme, the derivation of these schemesis now described

Hyper-C Scheme

The hyper-C scheme is a bounded Downwind scheme, i.e. it is constructed byenforcing the transient CBC criterion (see figure 4.8) onto the Downwind scheme.The transient CBC is expressed mathematically as:

€

CBC

˜ φ C ≤ ˜ φ f < min˜ φ Cc f

,1

for 0 ≤ ˜ φ f ≤1

˜ φ f = ˜ φ C for ˜ φ C < 0 or ˜ φ C >1

(4.6)

Figure 4.8: the CBC for transient flows

The NVF for the Hyper-C compressive scheme is shown in figure 4.9:

25

Figure 4.9: Hyper-Scheme

ULTIMATE-QUICKEST Scheme

The ULTIMATE-QUICKEST schemes is a bounded version of the QUICKESTscheme of Leonard [85] which is based on an transient upwind biased cubicinterpolation the QUICKEST scheme can be written using the normalized variableformulation as:

€

˜ φ f = c f ˜ φ C{ } + 1− c f( ) 34

˜ φ C +38

1− c f( )

= c f ˜ φ f (UPWIND ){ } + 1− c f( ) ˜ φ f (QUICK ){ }(4.5)

It is clear that the QUICKEST scheme is basically a linear blending between theUPWIND and QUICK schemes, with the blending function depending on the localCourant Number. The NV for the QUICKEST scheme is shown in figure 4.10

26

Figure 4.10: the NV for the QUICKEST scheme

By enforcing the transient CBC on the quickest scheme we get a bounded versionknown as the ULTIMATE-QUICKEST scheme. The NV for the U-QUICKESTscheme is shown in Figure 4.11.

Figure 4.11: the ULTIMATE-QUICKEST scheme

Blending for Interface Capturing Schemes

As shown in the description of the HRIC and CICSAM schemes, a strategy is used to

27

blend between a Compressive scheme and a high resolution scheme or upwindscheme, with the blending depending on the angle that the volume fraction makeswith the cell faces. The reasoning followed in this choice is illustrated in figure 4.12:if the cell is almost empty and the interface is parallel to the cell face then only thefluid present at the downstream cell will be convected through the cell face. Howeverif the interface is parallel to the cell face then it is expected that the convected fluidwould be of the same composition as the upwind cell. In the case where the interfaceis parallel to the cell face but most of the cell is filled then either of the two optionscould be used. Now these cases represent two extremes and interfaces that form anangle between 0 and 90 with the cell face use a blend of the extreme cases advectionscheme. Another constraint is that any advected quantity through the cell face shouldnot drain more fluid than is available in the cell. One option that can be used toenforce this condition is by using a local Courant number that is less than unity.

(a)

(b)

(c)

(d)Figure 4.12: Blending Strategy for Interface Capturing Schemes

CICSAM Scheme

The CISCAM scheme of Ubbink [77] was formulated based on the idea of the donor-acceptor formulation, i.e. as a scheme that varies as a function of the interface-cellface angle.i.e. that the discretization depends on the interface velocity direction and

28

the angle it makes with the integration cell face. However rather than choosing asbase schemes the downwind and upwind scheme it between theULTIMATE_QUICKEST scheme of Leonard and the Hyper-C scheme, with theHyper-C scheme being used when the cell face is directed perpendicular to theinterface normal vector and the ULTIMATE_QUICKEST scheme used when the facenormal vector is aligned with the normal to the interface. Mathematically, theCICSAM scheme can be written as

€

φ f (CICSAM ) = γ fφ f (Hyper−C ) + 1− γ f( )φ f (UQ ) (4.7)

where the weighing factor γf is based on the angle θf between the volume fractioninterface normal and the cell face interface see figure (4.13). the equations for theangle and blending factor are:

€

θ f = arccos∇rf ⋅dPF∇rf dPF

(4.8)

and

€

γ f =mincos 2θ f( ) +1

2,1

(4.9)

Figure 4.13: Angle between interface cell face

For an angle θf=90˚, i.e. when the interface normal is perpendicular to the cell facenormal, γf is zero and the ULTIMATE-QUICKEST scheme is used, and for qf=0, i.e.when the flow interface is aligned with the face normal the Hyper-C scheme is used.The NV for the CICSAM scheme is shown in figure 4.14.

29

Figure 4.14: the CICSAM scheme

HRIC Scheme

The High Resolution Interface capturing Scheme (HRIC) is based on a blending ofthe Bounded Downwind and Upwind Schemes (UD), with the aim of combining thecompressive property of the DD scheme with the stability of the UD scheme. TheBounded downwind scheme (BD) is formulated as

€

˜ φ f (BD ) =

2 ˜ φ C 0 < ˜ φ C ≤ 0.51 0.5 < ˜ φ C ≤1˜ φ C otherwise

(4.10)

and can be viewed as a steady-state version of the Hyper-C scheme.

30

Figure 4.15: NVF for the Bounded Downwind Scheme

The scheme is again modified to account for the angle θ between the normal to theinterface (defined by the gradient of the volume fraction r) and the normal of the cellface [88].

For an interface aligned with the cell face (angle θ=0) the bounded downwind schemeis used, while for an interface perpendicular to the cell face the upwind scheme isused, the blending formula is:

€

˜ φ f = ˜ φ f (θ ) = ˜ φ f (BD ) cos θ( ) + ˜ φ f UPWIND( ) 1− cos θ( )( ) (4.11)

The blending of upwind and downwind schemes is dynamic and accounts for the localdistribution of the volume fraction. HRIC further modifies the above scheme byaccounting for the local Courant number Co defined as:

€

Cof =v f ⋅S fΔtVf

(4.12)

For Courant number below 0.3 the scheme is not modified, while for a courantnumber above 0.7 the upwind scheme is used, the Courant range of 0.3 to 0.7 a linearblending with the upwind scheme is formulated to yield:

€

˜ φ f = ˜ φ f (θ ) + ˜ φ f (θ ) − ˜ φ f (UPWIND )( )0.7 −Cof0.7 − 0.3

(4.13)

The NV for the HRIC scheme is shown in figure 4.16.

Figure 4.16: NV for the HRIC scheme

VERY HIGH RESOLUTION SCHEMES (VHR)

All of the compressive schemes are based on a combination of a base third or lowerorder interpolation profile and the downwind scheme, on top of which a CBC isenforced. The idea is to remove some of the numerical diffusion associated with thebase interpolation profile by blending it with a downwind scheme. An approach thatcan be followed is to use a base scheme that has minimal numerical diffusion andbound it using the CBC criterion. This represents the basic idea behind Very HighResolution Scheme (VFH), in this case a fifth order interpolation profile to represent

31

the solution at the cell faces is used, with a higher accuracy surface integrationprocedure that determines the order of the integration at the cell face, and finally aboundedness criteria to filter out any unphysical oscillation associated with thereconstruction procedure [89,90,91,92,93]. In an unstructured framework aninterpolation profile can be described by a Taylor series as

φ r( ) = φ rP( ) + ∇φ( )P ⋅ r − rP( ) + 12 r − rP( ) :: ∇∇φ( )P + .... (4.17)

In the case of a firs order or quadratic interpolation profile equation (6.1) becomes:

φ r( ) = φ rP( ) + ∇φ( )P ⋅ r − rP( ) + 12 r − rP( ) :: ∇∇φ( )P (4.18)

we note that for the Compressive scheme only the first two terms on the right handside are needed, i.e:

€

φ r( ) = φ rP( ) + ∇φ( )P ⋅ r − rP( ) (4.19)

where the gradient can be computed using neighboring cells which should number atleast 2 in two dimension. For the quadratic profile determination of the hessian usingcontrol volume cell direct neighbors (neighbor 1 cells) lead to an under determineproblem, hence the need to used an extended cell with neighbor 2 cells (see figure4.17). This leads to an over-determined problem where the number of equations forthe Hessian computation is more than four.

Generalized Least Square Reconstruction

One way to resolve this problem is by using a least square method for the Hessiancomputations in this case the equation are written as:

Δx1 Δy1 12 Δx1( )2 1

2 Δy1( )2 Δx1Δy1M M M M M

ΔxNB ΔyNB 12 ΔxNB( )2 1

2 ΔyNB( )2 ΔxNBΔyNB

A[ ]1 2 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4

∂φ∂x∂φ∂y

∂2φ∂x 2

∂2φ

∂x 2

∂2φ∂x 2

d{ }1 2 3

=

φ1 − φPM

φNB −φP

Δφ{ }1 2 4 3 4

(4.20)

32

Figure 4.17: Neighbor 2 cells used in VHR schemes

Solving the above equation yields both the gradient and hessian forthe control volume.

Green-Gauss Gradient Quadratic Reconstruction

At first sight the generalized least-square method seems to be the only way to performa quadratic reconstruction with an arbitrary stencil. However, a quadratic extensionto the Green-Gauss linear reconstruction can be developed. By using a Taylor seriesexpansion of φ around a node n, the truncation error TE can be expressed as

€

) ∇ φ( )P = ∇φ( )P +

12

12 r − rP( )2( )dS

∂ΩP

∫ :: (∇∇φ)P

O(h )1 2 4 4 4 4 4 3 4 4 4 4 4

+O(h2) (4.21)

Substituting into equation 4.18 yields

€

φ r( ) = φ rP( ) + ∇φ( )P +12

12 r − rP( )2( )dS

∂ΩP

∫ :: (∇∇φ)P

O(h )1 2 4 4 4 4 4 3 4 4 4 4 4

+O(h2)

⋅ r − rP( ) + 12 r − rP( ) :: ∇∇φ( )P

= φ rP( ) + ∇φ( )P ⋅ r − rP( ) + 12 r − rP( ) :: ∇∇φ( )P +

12

12 r − rP( )2( )dS

∂ΩP

∫ :: (∇∇φ)P

quadratic part1 2 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4

(4.22)

The above formula clearly chows that the present quadratic reconstruction can beviewed as a correction of the O(h) linear Green-Gauss reconstruction. Hence this

33

provides us with the flexibility of using least square method for computing theHessian while using the green-gauss for the gradient computation. However unlikethe generalized least square method, the size of the system is reduced, because thenumber of unknowns is only equal to 3 second order derivatives.

€

12 Δx1( )2 1

2 Δy1( )2 Δx1Δy1M M M

12 ΔxNB( )2 1

2 ΔyNB( )2 ΔxNBΔyNB

A[ ]1 2 4 4 4 4 4 4 3 4 4 4 4 4 4

∂2φ

∂x 2

∂2φ

∂x 2

∂2φ

∂x 2

=

φ1 −φPM

φNB −φP

Δφ{ }1 2 4 3 4

(4.23)

It is worth noting that for fixed mehses, the matrices involved in the reconstructioncan be preprocessed and stored because they only depend on the geometriccharacteristic features of the mesh. The same remark obviously applies to the linearand quadratic least square methods.

34

5. TRANSIENT SCHEMES

For transient simulations, the governing equations must be discretized in both spaceand time. The spatial discretization for the time-dependent equations is identical to thesteady-state case. Temporal discretization involves the integration of every term in thedifferential equations over a time step t. The integration of the transient terms can takemany forms yielding different accuracy, as shown below.

A general expression for the time evolution of a variable φ is given by:

€

∂ ρφ( )∂t

= F φ( ) (5.1)

where the function F incorporates any special discretization. If the equation isintegrated over time we have:

€

∂ ρφ( )∂t

dtt

t+Δ

∫ = F φ( )dtt

t+Δ

∫

= TemporalAverage F φ( )( )dt(5.2)

expanding equation (4.21) using a pseudo temporal control volume yields. Integratingequation (5.2) we get:

€

ρφ( ) n+1/ 2( ) − ρφ( ) n−1/ 2( ) = F φ n( )Δt (5.3)

Figure 5. 1: temporal control volume

35

Two choices determine the accuracy and stability of our discretization

1. The order of accuracy chosen when discretization the transient term

2. The assumptions used to average the other terms over the interval t→ t + Δt

These choices can be made independent of each other increasing the number ofavailable schemes. In what follows four schemes are presented in the context of atemporal control volume.

First Order Upwind Euler Schemes

Using a first-order transient “upwind” interpolation:

€

ρφ( ) n+1/ 2( ) → ρφ( ) n( )

ρφ( ) n−1/ 2( ) → ρφ( ) n−1( )(5.4)

subsutituting in equation () we get

€

ρφ( ) n( ) − ρφ( ) n−1( )

Δt= F φ n( )( ) (5.5)

which is the first order backward euler scheme

Second Order Upwind Euler Schemes

Using a second-order transient “upwind” interpolation:

€

ρφ( ) n+1/ 2( ) →32ρφ( ) n( ) −

12ρφ( ) n−1( )

ρφ( ) n−1/ 2( ) →32ρφ( ) n−1( ) −

12ρφ( ) n−2( )

(5.6)

Substituting in equation () we get

€

3 ρφ( ) n( ) − 4 ρφ( ) n−1( ) + 2 ρφ( ) n−2( )

2Δt= F φ n( )( ) (5.7)

Second Order Central Schemes (Crank-Nicholson)

Using a second-order transient “upwind” interpolation:

€

ρφ( ) n+1/ 2( ) →12ρφ( ) n( ) +

12ρφ( ) n−1( )

ρφ( ) n−1/ 2( ) →12ρφ( ) n−1( ) +

12ρφ( ) n−2( )

(5.8)

Substituting in equation () we get

€

ρφ( ) n( ) + ρφ( ) n−2( )

2Δt= F φ n( )( ) (5.9)

36

Note that the CN scheme can as the averaging of the upwind and downwind transientschemes

€

UPWIND→ρφ( ) n( ) − ρφ( ) n−1( )

Δt= F φ n( )( ) (5.10)

€

DOWNWIND→ρφ( ) n+1( ) − ρφ( ) n( )

Δt= F φ n( )( ) (5.11)

€

Crank − Nicholson→ρφ( ) n+1( ) − ρφ( ) n( )

Δt+

ρφ( ) n( ) − ρφ( ) n−1( )

Δt= F φ n( )( ) + F φ n( )( )

→ρφ( ) n+1( ) − ρφ( ) n−1( )

2Δt= F φ n( )( )

(5.12)

What is useful in this reformulation is that enables us to implement the CN schemewithin the same implicit schemes framework, i.e.

solve using UPWIND scheme for time t(n) then update t(n+1) as:

€

Crank − Nicholson :1→ ρφ( ) n( ) − ρφ( ) n−1( )

Δt= F φ n( )( ) (5.13)

€

Crank − Nicholson : 2→ ρφ( ) n+1( ) − ρφ( ) n( )

Δt= F φ n( )( ) =

ρφ( ) n( ) − ρφ( ) n−1( )

Δt: 2→ ρφ( ) n+1( ) = 2 ρφ( ) n( ) − ρφ( ) n−1( )

(5.14)

However in order to move by a ∆t timestep at each iteration the ∆t for the CN schemeis chosen to be half that specified by the user, thus ∆tCN = 0.5 ∆t.

Bounded Second Order Upwind Euler Schemes

The second order transient “upwind” interpolation, like its spatial counterpart is not abounded scheme in that it can yield values for f that are above or below the localmaxima and minima respectively. For the Volume fraction equation this would leadto values above 1 or below 0 which are not acceptable. To resolve this problem abounded versions of the scheme is used so as to for φ(n) to be in the range [0,1]. Theimplementation of the bounding is similar to that of the CBC and will not be detailedhere.

37

6. RHIE-CHOW INTERPOLATION

Pressure based methods [94], for solving the velocity pressure coupling in theNaviers-Stoke’s equations, are now being applied to the simulation of a wide range offluid phenomena and will be the basic algorithm in the Free-Surface code. Anunderstanding of the basic features of these algorithms is essential in order to improveon their robustness.The first multi-dimensional pressure based method was developed by Harlow andWelch in 1965 [95]. The procedure was explicit in type. An implicit pressure basedmethod was proposed by Patankar and Spalding [96] in 1972. The calculation of thepressure field using the continuity equation was the principal idea in both proposals.The method proposed by Patankar and Spalding became very popular under theacronym SIMPLE [97] for Semi-Implicit Method for Pressure-Linked Equations, andthe less known SIVA procedure[98]. Many improvements were later introduced,SIMPLE Consistent [99], SIMPLE Revised [ 100], PISO [ 101], SIMPLE eXtended[102], PRIME [ 103] , see Darwish and Moukalled [ 94] for a review. In all of thesealgorithms the interpolation of the velocity field in the discretization of the continuityequatio, known as the Rhie-Chow interpolation, n is a determining factor in thesuccess of the method, especially as we will be using a collocated variablearrangement. Two facets of the Rhie-Chow interpolation where found to requireimprovement for the simulation of free surface flows, these two facets are treated inthis section, after a brief review of the Rhie-Chow interpolation

THE CELL-FACE VELOCITY

The Rhie and Chow formulation tries to mimics the staggered grid by forming apseudo-momentum equation at the Cell-face. It is because of this behavioral imitationthat the Rhie-Chow interpolation is successful. So a guiding principle to anymodification to the Rhie-Chow interpolation should whether the modifiedformulation is more similar to the staggered grid formulation, this will be theyardstick of the modification.

Starting with the discretized u-momentum equations for cell N and F

€

aNuN +VN∂P∂x

N

= aNBuNB( )NB (N )∑ (6.1)

€

aFuF +VF∂P∂x

F

= aNBuNB( )NB (F )∑ (6.2)

where the coefficients contain contributions from the diffusion and convection termsof the discretized equation:

€

a = aC + aD (6.3)

38

From conservation principles of the control volume formulation the uf velocitycomponent at a point on the face between the cell nodes should also satisfy adiscretized momentum equation of the form

af uf +Vf∂P∂x

f

= aNBuNB( )NB( f )∑ (6.4)

Since the coefficients of this equation are difficult to they are approximated throughinterpolation of the coefficients of the neighboring nodes.

Figure 6.1: Linear vs Rhie-Chow interpolation

Assuming that the right hand side of equations (6.4) may be approximated using alinear interpolation of the corresponding terms yields

aNBuNB( )NB( f )∑ =

12

aNBuNB( )NB(N )∑ + aNBuNB( )

NB(F )∑

=12aNuN + VN

∂P∂x

N

+ aFuF + VF∂P∂x

F

= af uf + Vf∂P∂x

f

(6.5)

we get after substitution

af uf =12aNuN + aFuF( ) − Vf

∂P∂x

f

−12VN

∂P∂x

N

+ VF∂P∂x

F

= afuf − Vf∂P∂x

f−Vf

∂P∂x

f

(6.6)

if we take af = af , we can write

uf = uf − Df∂P∂x

f− Df

∂P∂x

f

(6.7)

39

where

D =Va

(6.8)

it can be shown that

aNBuNB( )NB( f )∑ =

12

aNBuNB( )NB(N )∑ + aNBuNB( )

NB(F )∑

+ 0 Δx

2( )

=12aNuN + VN

∂P∂x

N

+ aFuF + VF∂P∂x

F

= af uf + Vf∂P∂x

f

(6.9)

and

Df =12DN +DF( ) +O Δx2( )

= Df +O Δx 2( )(6.10)

A prominent feature of that equation is that the face velocities depend on the pressureof the adjacent cells closely resembling the staggered grid practice. In the standardRhie-Chow formulation on the pressure gradient is treated in this manner

In the MWIM formulation equation is written as

uf = ˆ u f − Df∂P∂x

f(6.11)

where now

ˆ u f = uf + Df∂P∂x

f

= aNBuNB( )NB ( f )∑

(6.12)

TREATMENT OF BODY FORCES (DRAG, BUOYANCY, GRAVITY,

…)

Body forces pose instability problem if they are not properly treated in the Rhie-Chowinterpolation procedure. The treatment of Gu [104] who recognized the convergenceproblems of the collocated grid arrangement in buoyancy driven flows was tointroduce a second order body force term into the Rhie-Chow interpolation stencil.The equation was written as

uf = uf − CDf∂P∂x

f−

∂P∂x

f

+Df bf − Df bf (6.13)

where the body force source term is derived as

40

bf =12DNbN + DFbF( )

=12DN bf +

Δx2

∂b∂x

f

+DF bf −

Δx2

∂b∂x

f

=DN + DF( )2

bf +DN − DF( )2

∂b∂x

f

(6.14)

that the cell face equation becomes

uf = uf − CDf∂P∂x

f−

∂P∂x

f

+

DN − DN( )2

Δx2

∂b∂x

f (6.15)

Thus the source term is not adequately accounted for in the cell face equation, so it isnot surprising that the scheme was not stable enough, so Gu proposed another formulaas:

uf = uf − CDf∂P∂x

f−

∂P∂x

f

− K

∂u∂x

f−

∂u∂x

f

(6.16)

where C was computed from

C =

1 if Df∂P∂x

f

−∂P∂x

f

≤ uf

0 if Df∂P∂x

f

−∂P∂x

f

> uf

(6.17)

and the extra term used to smooth the velocity directly, the factor K in the extra termis a velocity field oscillation detector and has the form:

K =

∂u∂x

N

uF + 2 uN + uU(6.18)

It is a normalized second difference of the velocity component. Its value increases inthe region with oscillating velocity. Gu’s extra term is similar to Jameson;s artificialdissipation term [105]. Gu used the later formula for a turbulent buoyancy driven flowof Ra = 1011, which could not be converged with the original.

Rahman et al. [106] also proposed a body force correction but their correction madethe cell face velocity stencil first order accurate

When dealing with the staggered grids, the pressure and the source terms are definedin the same location, thus in the equation

at + aS + aCD( ) f u f = aNB

CDuNB( )NB( f )∑ − Vf

∂P∂x

f

+ aft u f

o + Vf bf (6.19)

the stencil for bf and that of pressure in (∂P/∂x)f are the same. In the case of acollocated arrangement, the body force, velocity and momentum variables are at the

41

same location so in order to have a discretization of the body force that retain asimilar stencil as the pressure, a redistribution procedure, thus when discretizing themomentum equation we get

aNCDuN = aNB

CDuNB( )NB (N )∑ − VN

∂P∂x

N

+ VNbN (6.20)

where the double bar indicate that two average procedures are taking place one tocompute b at the face of the cell and then the second to get an average of these facevalues at the cell center

figure 6.2: Treatment of Body Forces

bN =bfacex S facex

faces(N )∑

Sfacex

faces(N)∑

(6.21)

using the RhiChow interpolation thus yields

afCDuf = aNB

CDuNB( )NB( f )∑ − Vf

∂P∂x

f

+ Vf bf (6.22)

the bar indicate that the value is an average of the two cell centervalues straddling the face. The Rhie Chow formula become

afCDuf = af

CDuf − Vf∂P∂x

−∂P∂x

f

+Vf bf − bf( ) (6.23)

it is the mean of the neighboring nodes, and for the pressure gradient it is the cellsneighboring the face in question. The reason why this arrangement is optimal, is thatit enable more assistants might be needed,

42

TREATMENT OF TRANSIENT AND RELAXATION TERMS

Transient Term

When solving a transient problem the discretized momentum equation is written as

aNt + aN

CD( )uN = aNBCDuNB( )

NB (N )∑ − VN

∂P∂x

N

+ aNt uN

o(6.24)

an equation of a similar form is sought for the face

aft + af

CD( )uf = aNBCDuNB( )

NB( f )∑ − Vf

∂P∂x

f

+ aft u f

o(6.25)

Following a similar procedure as for the standard Rhi and Chow we get

uf = uf −Vf

aft + afCD( )∂P∂x

−∂P∂x

f

+aft

a ft + afCD( )uf

o − ufo( ) (6.26)

note that the above equation could be rewritten in the spirit of the MWIM formulationas

uf = uf +Vf

aft + af

CD( )∂P∂x

−af

t

a ft + af

CD( )uf

o

−

Vf

aft + af

CD( )∂P∂x

f

+af

t

a ft + af

CD( )uf

o

= ˆ u f −Vf

aft + af

CD( )∂P∂x

f+

aft

a ft + a f

CD( )uf

o

=

aNBCDuNB

NB ( f )∑af

t + afCD( )

−Vf

aft + af

CD( )∂P∂x

f

+af

t

a ft + af

CD( )uf

o

(6.27)

Relaxation Term

Similarly when including under relaxation in the discretization equation themomentum equation could be written as

aNCD +

1− urfurf

aN

t

uN = aNB

CDuNB( )NB( N)∑ − VN

∂P∂x

N

+1− urfurf

aN

t uNn−1

(6.28)

the Rhie-Chow interpolation equation should have a similar form as

afCD +

1− urfurf

af

t

uf = aNB

CDuNB( )NB( f )∑ − Vf

∂P∂x

f

+1 − urfurf

af

t u fn−1

(6.29)

upon substitution we get

uf = uf −Vf

afCD +

1 −urfurf

af

CD

∂P∂x

−∂P∂x

f

+ 1− urf( ) ufn−1 − uf

n−1( ) (6.30)

This concludes our treatment of the algorithmic aspects of the Free-Surface numerics.In the next section a series of tests are run for evaluation and validation purposes.

43

6. TEST PROBLEMS

SHAPE ADVECTION TESTS

In shape advection tests [107,108] a number of hollow shapes are advected with anoblique unidirectional velocity field, v(2,1) for a certain time. The accuracy of thedifferent schemes is evaluated by comparing the initial and advected shapes. Threebase shapes are considered as shown in Figure 6.1. The computational domain is a1x1 m square discretized for the structured grid into a 200x200 mesh.

a. a hollow square aligned with the co-ordinate axes with the outer side length of 0.2m and a 0.1 m inside side length, which for the structured mesh represents 40 and20 cells respectively

b. a hollow square at an angle of 26.57˚ to the x-axis with same dimensions as theprevious hollow square

c. a hollow circle with an outer diameter of 0.2m and inner diameter of 0.1 m. (40and 20 cells respectively)

All the hollow shapes are initially centered are at (0.25,0.25)m. the final advectedcenters are at (0.875,0.5325). These tests were carried, in addition to the UPWINDscheme, with the HRIC, CICSAM, U-QUICKEST, and VHR schemes for spatialdiscretization, and with the First Order Backward Euler scheme, the Second OrderEuler, the Crank-Nicholson scheme, and two bounded transient schemes namely theBounded Second Order Euler scheme and the Compressive Transient scheme,

The computational domain had a size of (200x200) cells, with the rotated square having an angle of 26.27˚, and a velocity field v(2,1), the time step chosen for all theproblem was defined using the Courant Number with the test performed for a CourantNumber of 0.375,. The tests are shown in figure 6.1 with the results displayed infigures 6.2, 6.3 and 6.4 using a variety of spatial discretization schemes along with thecrank-nicholson transient scheme.

44

(a) (b)

(c)Figure 6.1: Schematic for shape advection problems

UPWIND GAUSS-GAUSS

45

SMART GAUSS-LEASTSQUARE

LINEAT LEAST-SQUARE

Figure 6.2: results for square problem

Results for the compressive scheme were found not to be satisfactory at high courant number and were dropped from the comparison. More work on improving the robustness of these schemes will be carried out in the near future.

Upwind Gauss-Gauss

SMART Gauss-Leas square

46

Linear Least Square

Figure 6.3 Translation of a Circle

UPWIND GAUSS-GAUSS

SMART GAUSS-LEASTSQUARE

LINEAR LEAST SQUARE

Figure 6.4 Translation of a rotated square

The translation of a circle was also used for a comparison of spatial schemes withwere used with the SMART scheme for spatial discretization. Results are shown infigure 6.5. Results indicate that while preserving a good sharpness the compressiveEuler scheme is not shape preserving while the second order upwind schememsufferes from its unboundedness. This is shown to be better resolved with thebounded SOU scheme, which yielded slightly more accurate results than the crank-nicholson.

47

Euler Cracnk-Nicholson

Second OrderUpwind

Bounded Second-Order Upwind

Compressive Euler

Figure 6.5: Comparison of Transient Schemes

ROTATION OF A SLOTTED CIRCLE

The solid-body rotation of an object poses a test problem with a trivial exact solution[107,109,110]. However it’s a very tough problem with regard to the advectionschemes. The test in question involves the rotation of a circle with a slot around anexternal point. The computational domain is a square of dimensions [0-4,0-4]mdiscretized into 200x200 cells. The circle of diameter 1 (50 cells) has its centre at(2,2.65) and is cut by a slot of width 0.12 (6 cells). The rotation of the slotted circle isdriven by a rotational velocity field with a centre of rotation (2,2). The Courantnumber is Co=0.512. The circle performs a full rotation after (1262) time steps andits initial position should be recovered. The schematic of the test is shown in figure6.2.

48

Figure 6.6: Schematic for rotation of a slotted circle

Results for the test are shown in figure 6.7. It is clear that because of the relativelyhigh courant number nearly all schemes yield results with substantial numericaldiffusion in the slotted area, results using the quadratic schemes (fifth order accurate)are slightly better that the SMART and Linear schemes, with the UPWIND yieldingclearly unacceptable results

UPWIND Gauss

SMART Gauss-LeastSquare

Linear Least-Square

Least-Square

Figure 6.7: Results for rotation of a slotted circle

From the series of tests performed it is clear that a proper combination of spatial andtransient discretization schemes is needed for the proper simulation of free-surfaceflows in complex situations. Though quadratic schemes were slightly more accurate

49

than the SMART schemes they are computationally expensive as they involve thecomputation of a Hessian at each control volume in addition to the a gradient. To thisend SMART/Crank-Nicholson schemes combination was chosen as a goodcompromise between accuracy and economy. These will be used in the next test ofthe “broken dam”.

THE BROKEN DAM TEST

The problem under consideration is a column of water, in hydrostatic equilibrium,being confined between two vertical walls. Gravitational force acting downwardcauses a liquid to seek the lowest possible level, therefore the water column, cannotstand up on a flat surface. At the beginning of the calculation, when the right wall(dam) is suddenly removed, the water will flow out along a dry horizontal floor andform an advancing wave. When the wave hits an obstacle (the vertical wall on theextreme right) the momentum will carry it along the wall, before falling down andforming a reversed waves.

Figure 6.8: Dam break physical setup

This is a tough problem because of the reverse wave involved, it involves simpleboundary conditions and has a simple initial configuration. The appearance of both avertical and horizontal free surface, however, provides a check on the capability of themathematical models to treat free surfaces which are not single valued with respect tox or y coordinates.

Figure 6.8 shows the predicted free surface profiles for the flow with initialrectangular section column. Results in relation to the pictures starting from the topleft are the volume of fluid free surface drawn at a value of 0.5 contour line, at timeintervals of 1/32 seconds.

50

51

Figure 6.9 Break of a Dam, from left to right and top to bottom in intervals of

The results displayed in figure 6.9 are part of a movie sequence obtained from theresults of the run. An advancing wave is shown to be forming in the first 11 slideswhich then hits the right hand vertical wall. The momentum carries part of the fluidalong the vertical wall up to the height of the square cavity. As shown in slides 12-25.A returning wave is then formed as the fluid momentum is lost and gravity effectstakes over forcing the fluid along the vertical right wall to move back down. Thisreturning wave is show to accelerate in slides 26-32.

The ability to simulate the returning wave is a clear proof of the robustness of thecode. Such a situation involving an accelerating returning wave forming from a peakwave height is very demanding algorithmically because of the changes involved in thetime schemes of the phenomena.

52

CONCLUSION

The aim of this project was the development of a finite volume based robust andaccurate method for the simulation of free surface flows. this method wasimplemented in the form of a computer code validated and tested for accuracy androbustness in a number of increasingly complex test cases. In the final test thesimulation of a broken dam involving advancing and returning wave was performed.The numerics involved were of second order in the transient domain and third order inthe spatial domain, specifically a (Crank-Nicholson/SMART) schemes combination.This combination was chosen after comparing more than 10 different schemes in aseries of test cases involving the transport of hollow circles, square and rotatedsquares in addition to the rotation of a slotted disk. These tests though simple tocomprehend present substantial numerical difficulties in relation to the numericaldiffusion present in both spatial and transient schemes. The performance of thecompressive schemes was found to be below that of the other schemes in terms ofrobustness at a high courant number, this issue will require more work in the future todetermine whether it can be resolved.

Finally I would like to thank the LNCSR for the support that was provided during thisprojects.

Marwan Darwish,Associate Professor,American University of Beirut,Riad El Solh,Beirut 1107 2020,P.O. Box: 11-0236,Email: darwish@aub.edu.lb,Phone: 350000 ext 3636.

53

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