report vt09 09 new

Institutionen f

or informationsteknologi

Project 9

Evaluation of Turbulence Models for Prediction

of Flow Separation at a Smooth Surface

Eric Furbo, Janne Harju & Henric Nilsson

Report in Scientic Computing Advanced Course

June 2009

PROJECTREPORT

Abstract

This project was provided by the Swedish Defence Research Agency, FOI. The task is totest several RANS (Reynolds-averaged Navier-Stokes) models on a generic case, constructedfor flow separation at a smooth surface. This is of very high importance for applicationsin the auto industry, ship construction and for aeronautics. The models tested are imple-mented in the open source CFD (Computational Fluid Dynamics)-program, OpenFOAM.OpenFOAM uses the finite volume method and the simple algorithm as solution procedure.The main flow features evaluated in this project is the shape, position and size of the flowseparation. Most of the models that we have tested have problems describing the complexdynamics of flow separation in this particular case.

1

Contents

1 Introduction 41.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theory 72.1 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Numerical solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 The Finite Volume Method (FVM) . . . . . . . . . . . . . . . . . . . . . . 82.2.2 The solution algorithm SIMPLE . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Direct numerical simulation (DNS) . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Large Eddy Simulation (LES) . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Reynolds-averaged Navier-Stokes models (RANS) . . . . . . . . . . . . . . 9

3 Simulation Parameters 123.1 OpenFOAM implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Input data for a computational case . . . . . . . . . . . . . . . . . . . . . 123.1.2 Our case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Boundary Conditions for the turbulent quantities . . . . . . . . . . . . . . . . . . 13

4 Results 154.1 Short description of the physical experiment . . . . . . . . . . . . . . . . . . . . . 154.2 Results from our simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Global behaviour of the models tested . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3.1 The k model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.2 The Nonlinear k Shih model . . . . . . . . . . . . . . . . . . . . . . . 184.3.3 The RNG k model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.4 The k SST model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3.5 The realizable k model . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.6 The Spalart Allmaras model . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Comparison to experimental and LES data . . . . . . . . . . . . . . . . . . . . . 23

5 Conclusions 28

A Examples of OpenFOAM case-files 29

B Technical specification of the OS cluster 34

2

List of Figures

1 Computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Inlet velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 The computational grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Case directory in OpenFOAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 One type of inlet boundary conditions for k and . . . . . . . . . . . . . . . . . . 146 Results from the k model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Results from the k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Results from the Nonlinear k Shih model. . . . . . . . . . . . . . . . . . . . . 189 Results from the Nonlinear k Shih model. . . . . . . . . . . . . . . . . . . . . 1810 Results from the RNG k model. . . . . . . . . . . . . . . . . . . . . . . . . . 1911 Results from the RNG k model. . . . . . . . . . . . . . . . . . . . . . . . . . 1912 Results from the k SST model. . . . . . . . . . . . . . . . . . . . . . . . . . . 2013 Results from the k SST model. . . . . . . . . . . . . . . . . . . . . . . . . . . 2014 Results from the realizable k model. . . . . . . . . . . . . . . . . . . . . . . . 2115 Results from the realizable k model. . . . . . . . . . . . . . . . . . . . . . . . 2116 Results from the Spalart Allmaras model. . . . . . . . . . . . . . . . . . . . . . . 2217 Results from the Spalart Allmaras model. . . . . . . . . . . . . . . . . . . . . . . 2218 Comparison with LDV experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 2319 Comparison with LDV experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 2420 Comparison with LDV experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 2521 Comparison with LDV experiment and LES data. . . . . . . . . . . . . . . . . . . 2622 Comparison with LDV experiment and LES data. . . . . . . . . . . . . . . . . . . 27

List of Tables

1 Inlet conditions for k and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Table of working turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . 153 Table of turbulence models that encountered problems. . . . . . . . . . . . . . . . 164 Table of separation and re-attachment points. . . . . . . . . . . . . . . . . . . . . 23

3

1 Introduction

The computational prediction of flow separation of a turbulent boundary layer from a smoothedsurface is a process of primary concern. The fluid flow becomes detached from the surface andinstead takes the form of eddies and vortices. Despite the simple shape (see figure 1) of the hillof which the flow separation occurs, it is a challenging computational problem and experimentsindicate that the flow in the recirculation zone is complex and strongly time-dependent.

In the simulations described in this report we apply turbulence models based on RANS(Reynolds-average Navier-Stokes) equations. The domain that the flow have been simulated inis constructed to be very smooth without edges and has no real life application. However, thephysical phenomena that arise is a subject of interest for many engineering components andsystems. Streamlined car bodies, low-pressure turbine blades and highly loaded aircraft wingsare some examples of where flow separation can have significant influence of the ability of thedevice in question to perform effectively.

The main objective is to evaluate the tested RANS-models prediction of the flow over thehill. We mainly evaluate the results with respect to the location and shape of the separationzone and the velocity distribution in the near wake of the wall. Our results will be compared toresults from experiments made by Byun and Simpson [9] and LES simulations made by Perssonet al. [7]

1.1 Problem Formulation

In order to simulate the turbulent flow and separation we have used the open source CFD-program OpenFOAM [3]. We have only used RANS turbulence models and OpenFOAM hasa variety of such models implemented. Since all of the 14 RANS turbulence models have beentested, no extensive work have been done to models that in some way encountered problems toconverge.

The geometry is a three-dimensional axisymmetric hill placed on the floor of a wind tunnel.The computational domain is given in figure 1.

Figure 1: Computational domain seen from the inlet boundary.

4

The shape of the hill is defined by

y(r)H

= 16.04844

[J0()I0

(r

a

) I0()J0

(r

a

)](1)

where = 3.1926, a = 2H. H = 0.078 m, is the height of the hill,a is the radius of the circularbase of the hill and r2 = x2 + z2. J0 and I0 are the Bessel function of the first kind and themodified Bessel function of the first kind, respectively. A Cartesian coordinate system is used forthe simulations, and in the domain coordinates (x, y, z) in this report also denoted by (x1, x2, x3)the velocity field is U = (ux, uy, uz) also denoted by U = (u1, u2, u3). The x-axis is parallel tothe centerline of the domain and pointing downstream. The y-axis is normal to the floor, pointingupward. The origin is fixed at the center of the bump, with y = 0 corresponding to the windtunnel floor.

The top and bottom patch seen in figure 1 are set to be wall boundaries. This implies theuse of no slip conditions, i.e. U = 0 at these boundaries. The left, right and outlet patches areused to limit the computational domain. Therefore we will use homogeneous Neumann boundaryconditions for these boundaries, hence Un = 0.

The inlet velocity has a maximum flow velocity of 27.5 m/s. The inlet velocity is not uniformlydistributed, it is instead a profile with lower velocities near the top and bottom wall and higherin the middle of the inlet patch corresponding to turbulent boundary layers of approximatethickness of H/2. The inlet velocity has only one non-zero component, that is in the x-direction.The complete inlet profile can be seen in figure 2.

Figure 2: Velocity profile for the inlet patch of the computational domain

The Reynolds number is a dimensionless quantity, defined as

Re =UL

,

and it represents the relation between inertial and viscous forces. Here U is a characteristicvelocity, L is a characteristic length and is the kinematic viscosity. The Reynolds number for

5

our case, based on the maximum inlet velocity of 27.5 m/s, the height of the hill, H = 0.078 mand the kinematic viscosity of air = 1.65 105 m2/s is Re = 1.3 105.

As for computational mesh we where given a mesh with about one million computational cellsfrom FOI [7]. In the simulations made by FOI, a mesh with about 4 million cells was primaryused. The mesh used in our simulations have the same distribution of cells as that one. Themesh is a structured mesh that have been refined near the wall boundaries to better resolve theboundary layers and the flow near this regions. The number of cells in x, y and z-direction is69 119 119 respectively. The y+ value is related to the distance between a boundary and thenearest computational point. The y+ value for our mesh is in the range between 1.44 and 2.25,depending on which patch is considered. A wireframe plot of the mesh can be seen in figure 3.

On this computational domain the incompressible Reynolds-averaged Navier-Stokes (RANS)equations have been solved together with the different turbulence model. The Navier-Stokesequations will be briefly discussed in section 2.1, the RANS equations and turbulence modelswill be described in section 2.3.3. All the computations have been carried out in OpenFOAM.OpenFOAM uses the Finite Volume Method and we have chosen the SIMPLE algorithm assolution procedure. This will be discussed in section 2.2.1 and 2.2.2 respectively.

(a) (b)

(c)

Figure 3: (a) The computational grid showing the cells on the surface of the hill and in the planez = 0. (b) Zoomed in at the hill, notice the refined mesh near the hill boundary. (c) Mesh seenfrom the side.

6

1.2 Previous Research

This particular case has been extensively investigated by a number of computational researchgroups such as Garcia-Villalba et al. [4], Krajnovic [2] and Person et al. [7]. Since most ofthe previous research and the references therein for this case have relied on other turbulencemodels than RANS, such as DES (Detached Eddy Simulation), LES (Large Eddy Simulation) andhybrid LES-RANS, our objective is to compare how the different implemented RANS models inOpenFOAM performs compared to the other more computational costly models. In comparisonwe also have LDV (Laser doppler velocity) measurements for this case conducted by Byun andSimpson [9].

2 Theory

2.1 The Navier-Stokes equations

The fundamental basis for fluid dynamics are the Navier-Stokes equations. The fluid is assumedto be described as a continuum and the governing equations of the physics of fluid dynamicsare that mass, momentum and energy are conserved. Derivation begins with application ofthe conservation of momentum, using control volumes and Reynolds transport theorem. Theincompressible form of N-S equations and the conservation of mass is described as

(dUdt

+U U) = p+ 2U (2)

U = 0 (3)where U is the flow velocity, p is the pressure, viscosity and is the density. See Pope [8] fora complete derivation of the Navier-Stokes equations.

7

2.2 Numerical solution method

2.2.1 The Finite Volume Method (FVM)

The solution domain is subdivided into a finite number of small control volumes (polyhedra) andthe conservation equations are applied to each control volume. FVM uses the integral form of ageneral convection-diffusion equation for a quantity as its starting point which is described as

d

dt

V

dV +S

u ndS =S

() ndS +V

qdV (4)

where V is a control volume, S its bounding surface, n unit normal, diffusion coefficient andq some external term. The i:th component of the momentum equation is of the same form asequation 4 with = ui.

The centroid of each control volume is assigned to be the computational node at which thevariable values are to be calculated. Interpolation is used to express variable values at the controlvolume surface in terms of the centroid nodal values. Surface and volume integrals are approxi-mated using suitable discretization methods. As a result, one obtains an algebraic equation foreach control volume, in which a number of neighbour nodal values appear. One advantage withFVM is that it can handle complex geometries, however one disadvantage compared to othercomputational methods is that methods of higher order than 2nd, are more difficult to developin 3D. A more detailed description of the Finite Volume Method can be found in chapter 4 of [1].

2.2.2 The solution algorithm SIMPLE

When the RANS approach for turbulence is used, a stationary problem arises. RANS and manyother methods for steady problems in computational fluid dynamics can be regarded as unsteadyproblems until a steady state is reached. If an implicit method is used in time, the discretizedmomentum equations at the new time step are non-linear. Due to this and that the underlyingdifferential equations are coupled, the equations system resulting from discretization cannot besolved directly. Iterative solution methods are the only choice.

The momentum equations are usually solved sequentially for each component. The pressureused in each iteration is obtained from the previous time step and therefore the computed ve-locities normally do not satisfy the discrete continuity equation. In order for the velocities tofulfill this equation one have to modify the pressure field. This can be done by solving a discretePoisson equation for the pressure.

After solving this new equation for the pressure the final velocity field at the new iterationis calculated. This new velocity field satisfies the continuity equation, but the velocity andpressure fields do not satisfy the momentum equations. Therefore, the procedure describedabove is iterated until a velocity field is obtained that satisfy both the momentum and continuityequations.

Methods of this kind which first construct velocity fields that do not satisfy the continuityequation and then correct them are known as projection methods. The SIMPLE algorithm issuch a method, and it is the solving procedure used in OpenFOAM for our computations.

2.3 Turbulent Flows

Laminar flow changes into turbulent flow when the Reynolds Number becomes larger than acritical value. This value is dependent of the flow, thus the geometry of the domain. As mentionedin section 1.1 the Reynolds number for our case is Re = 1.3 105. Since this indicate thatturbulence will occur we have to account for the turbulence in some way.

8

Turbulent flows are highly unsteady. The direction of the velocity field fluctuates rapidly inall three spatial dimensions. These flows (are also vortical) and fluctuate on a broad range oftime and length scales. Therefore it can be very hard to correctly simulate these types of flows.

2.3.1 Direct numerical simulation (DNS)

The most natural and straight forward approach to turbulence simulations is to solve the Navier-Stokes equations without any approximation of the turbulence other than numerical discretiza-tions. These type of simulations are called Direct Numerical Simulation (DNS). In such simula-tions all the motions contained in the flows are resolved. In order to capture the structures ofthe turbulence one have to use extremely fine grids and time steps for these type of simulations.Due to this DNS is very costly from a computational point of view and this method is onlyuseful for flows with low Reynolds numbers. The applications of DNS is therefore limited toturbulence research and results from DNS simulations can be important to verify results fromother turbulence models.

2.3.2 Large Eddy Simulation (LES)

A simulation that is less computationally costly is Large Eddy Simulation (LES). This type ofsimulation uses the fact that the large scale motions are generally more energetic than the smallscale ones. These large eddies are more effective transporters of the conserved properties andtherefore LES threats the large eddies more exactly than the small scale ones. LES is preferredover DNS when the Reynolds number is too high or the computational domain is too complicated.

2.3.3 Reynolds-averaged Navier-Stokes models (RANS)

Both DNS and LES-modelling give a detailed description of turbulent flows. If one is not inter-ested in that much details of the turbulence, a Reynolds-Averaged Navier-Stokes model (RANS-model) can be a good choice. This type of model is much less costly compared to both DNS andLES. In the RANS approach to turbulence, all of the unsteadiness in the flow is averaged outand regarded as part of the turbulence. When averaging, the non-linearity of the Navier-Stokesequations gives rise to terms that must be modeled. These terms can be modeled differentlyleading to various types of RANS models, each behaving differently depending on the Reynoldsnumber of the flow, the computational domain etc.

As mentioned above, the unsteadiness of the flow is averaged out. The flow variable, in thisexample it is one component of the velocity, is divided into two terms, one time-averaged valueand the fluctuation about that value:

ui(xi, t) = ui(xi) + ui(xi, t)

where

ui(xi) = limT

1T

T0

ui(xi, t)dt

xi are the spatial coordinates, also known as x, y and z. T is the averaging interval and mustbe large compared to the typical time scale of the fluctuations.

If the flow is unsteady, time averaging cannot be used and it has to be replaced with ensembleaveraging. The concept of this is to imagine a set of flows in which all the variables that can be

9

controlled (energy, boundary conditions etc.) are identical but the initial conditions are gener-ated randomly. This will give flows that differ considerably from the others. An average over alarge set of such flows is an ensemble average. In mathematical form written as

ui(xi, t) =1N

Nn=1

uni(xi, t) (5)

where N is the number of members of the ensemble. The term Reynolds averaging refers toany of the processes above and applying it to all the variables in the Navier-Stokes equationsyields the Reynolds-averaged Navier-Stokes (RANS) equations which are written as

(ui)t

+

xj

(uiuj + uiu

j

)= p

xi+

xj

(uixj

+ujxi

)(6)

uixi

= 0 (7)

In the equations above, is the density of the fluid and is the dynamic viscosity of the fluid.The equation for the mean of a scalar quantity, temperature for example, can be written as

()t

+

xj

(uj+ uj

)=

xj

(

xj

). (8)

As mentioned before the RANS equations give rise to a stationary problem. The time derivativesin equation 6 and 8 are used to advance the equations in an imaginary time to reach steadystate.

The RANS equations contain terms such as uiuj called the Reynolds stresses and u

j which

is the turbulent scalar flux. The presence of these new quantities in the conservation equationsmeans that they contain more variables than there are equations, The system of equations isnot closed. Therefore one have to introduce approximations for the Reynolds stresses and theturbulent scalar flux. The approximations introduced are called turbulence models. To modelthe Reynolds stress the eddy-viscosity model is used, which is written as

uiuj = t

(uixj

+ujxi

)+23ijk (9)

where the eddy-diffusion model is used for the turbulent scalar flux

uj = t

xj. (10)

The turbulent kinetic energy k is described by

k =12(uxux + uyuy + uzuz

)(11)

and t is the eddy viscosity and will be defined later.

The exact equation for the turbulent kinetic energy k, is given by

(k)t

+(ujk)xj

=

xj

(k

xj

) xj

(2uju

iui + pu

j

) uiuj

uixj

ui

xk

uixk

. (12)

10

The terms on the left-hand side of this equation and the first term on the right-hand sideneed no modelling. The second term on the right-hand side represents the turbulent diffusion ofkinetic energy and is modelled according to

2uju

iui

tk

k

xj(13)

where k is the turbulent Prandtl number whose value is approximately 1. The third term inequation 12 represents the rate of production of turbulent kinetic energy by the mean flow. Usingthe eddy-viscosity model for the Reynolds stress, this term is modelled to

uiujuixj

t(uixj

+ujxi

)uixj

. (14)

The equation for the dissipation of the kinetic energy, , in it most commonly used form, iswritten as

()t

+(uj)xj

= C1Pk

k C2

2

k+

xj

(t

xj

). (15)

In this model, the eddy viscosity is expressed as

t = Ck2

. (16)

Equation 15 for epsilon together with equation 12 is called the kmodel and is a commonly usedRANS-model for turbulent flows. This model contains five parameters and the most commonlyused values are

C = 0.09 C1 = 1.44 C2 = 1.92 k = 1.0 = 1.3 (17)

The most important difference in using a turbulence model like the k model compared tojust using the ordinary laminar equations is that two new equations have to be solved. One alsohave to replace the ordinary Navier-Stokes equations with the RANS-equations derived earlier.

One other thing that should be mentioned is that because the time scales associated withturbulence are much shorter than those connected to the mean flow. Therefore the equationsconnected to any turbulence model are much stiffer than the laminar equations. This has to betaken in to consideration in the numerical solution procedure.

Near the physical walls of a computational domain the profiles of the turbulent kinetic energyand its dissipation show a more significant peak than the mean velocity profile. One way to resolvethe turbulent quantities near the wall is to use a finer grid. This will of course generate morecomputational points and therefore be more computational costly. One can also use so calledwall functions for the velocity profile near walls. When these wall functions are used for thevelocity, the normal derivative of k at the walls is assumed to be zero, i.e.

k

n= 0. (18)

At inflow boundaries it is quite hard to know what values to set for k and . One common wayfor k at inlets are to set some small enough value times the square of the mean inlet velocity ofthe flow. To set the proper condition for one can use an approximation, written as

k3/2

L(19)

11

where L is the characteristic length off the flow. There are several other two-equation modelsthat try to describe the turbulence. Many of them are based on the k model described above.Such models are the k model, the RNG k model and the Nonlinear k Shih model.They are in general more complicated and contain more parameters than the k model.

There are also one-equation models like the Spalart Allmaras model. This model solves atransport equation for a viscosity-like variable .

The last family of models that can be used to model the turbulence and are used togetherwith the RANS equations are the Reynolds stress models. The assumption about the eddy-viscosity according to equation (9) is often a valid approximation for two dimensional problems.For three-dimensional flows this approximation is not always accurate enough, since the eddy-viscosity may become a tensor quantity. One way to model this is to have a dynamic equationfor the Reynolds stress tensor itself. These equations can be derived from the Navier-Stokesequations.

3 Simulation Parameters

3.1 OpenFOAM implementation

Open Field Operation and Manipulation (OpenFOAM) [6] Computational fluidmechanics tool-box is an open source program that can simulate a variety of physics, mainly fluid flows. Theprogram is based on finite volume methods and is an object oriented C++ program consistingof modules that can be used to create a desired setup for a solver. The program also comes withtools for both pre- and post-processing of the solution.

3.1.1 Input data for a computational case

To use OpenFOAM a case must be set up, the case is basically a file structure that contains all thedata that the program needs to process the problem. The picture below shows the file structureof an OpenFOAM case. To begin with, the case must contain mesh-files that the program canprocess to obtain a domain for the solution. Further more, the case must also contain files thatspecifies boundary conditions, solvers, time to start, stop and step, models to use and some morecase specific data. When building the case the user has great freedom of choice regarding whatsolvers, preconditioners etc to use.

Figure 4: Case directory in OpenFOAM.

The three directories that the case consists of are: system, constant and time directories.The system directory is for setting solution procedure parameters and must contain at least

12

three files. The controlDict, fvScheme and fvSolution. The constant directory contains the meshinformation and specified physical properties of the domain. The time directories contain theinitial values and further the solution over the domain.(for all the time steps that the user hasspecified to be written to file.) [5]

3.1.2 Our case

The parameters, preconditioners and solvers used in the simulations are specified in some filesthat are shortly be presented file by file below. Detailed examples of the structure of these filescan be found in appendix A.

controlDict: In controlDict the solver is specified and in all of the simulations presented, sim-pleFoam is used as the solver. SimpleFoam is a steady state solver for incompressible, turbulentflow of non-Newtonian fluids, and it uses the solution algorithm SIMPLE described earlier insection 2.2.2.

fvSolution: To begin with, in the fvSolution file all solvers for the different parts of the equationhave been specified. For solving the pressure, the preconditioned conjugate gradient methodis used with diagonal incomplete Cholesky preconditioner. For the velocity and turbulenceterms the preconditioned biconjugate gradient method is used, where diagonal incomplete LU-factorization is the preconditioner. Relaxation factors determine the convergence rate and are setfor all quantities. The solution from the previous time iteration is multiplied with the relaxationfactor before being employed in the current iteration. Lower relaxation factors are more likelyto grant convergence but lead to longer computational time. Higher factor could speed up theprocess, but might also lead to divergence.

fvSchemes: In the fvSchemes files the schemes for discretizing different types of terms arespecified, and for the ddt terms steady state solver is used. For gradient terms Gauss linearis being used, divergence terms are being discretized using Gauss linear or Gauss upwind andlaplacian terms are being discretized with Gauss linear corrected and interpolation linear scheme.

RASproperties: RASproperties file is for specification of the turbulence model and the pa-rameters included in the specified model. Also wall function coefficients are specified in thisfile.

3.2 Boundary Conditions for the turbulent quantities

All of the turbulence models that we have used except two (Spalart Allmaras and k SST), usethe turbulent kinetic energy, k, and its dissipation, , to describe the turbulence. The boundaryconditions for k and are set to a constant value on the inlet patch. For k the value is 0.756m2/s2 where we have used 27.5 m/s as reference velocity and computed the boundary conditionaccording to section 2.3.3. The value for is set to 1.08 m2/s3 at the inlet, which is computedfrom the k inlet value according to equation 19. We used one tenth of the width of the domainas the characteristic length to compute the epsilon inlet value. This setup is the type 1 setupfor the inlet conditions. For all the other boundary patches, Neumann boundary conditions wereused.

To investigate the sensitivity of the flow to inlet turbulence, we also tried this model withfour times higher and four times lower values of k, and the value of according to equation 19

13

Types of inlet conditions for k and Turbulent variable type 1 type 2 type 3 type 4

k [m2/s2] 0.756 3.024 0.189 non uniform profile,see figure 5 a

[m2/s3] 1.08 8.64 0.135 non uniform profile,see figure 5 b

Table 1: Table of the four types of inlet condition that have been used.

resulting from this new k value. These are the type 2 and type 3 setups and were only usedfor the k model.

A fourth setup for the inlet values of k and that was not constant and hopefully morephysical when modelling the turbulent boundary layer upstream of the hill, was also tried. Toachieve this we ran a k simulation in the channel flow environment without the hill. Afterconvergence to steady state we used a plane parallel to the inlet, but inside the domain, withcell values of k and epsilon. The distribution of k and for this plane can be seen in figure 5.These inlet conditions were used for the k model, the Non linear k Shih model and therealizable k model. This setup is known as the type 4 inlet condition. For an overview ofthe types of inlet conditions, see table 1.

As for the k SST model it uses identical boundary conditions for k as the k model.The inlet boundary condition for is simply = k .

The Spalart Allmaras model, which is a one equation model as mentioned in section 2.3.3,has only one turbulent quantity . The boundary condition for at the inlet boundary was setto 3 105.

0 0.05 0.1 0.15 0.2 0.250

1

2

3

4

5

6

7

y

k [m2/s2]

(a)

0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

10000

12000

14000

16000

18000

y

epsilon [m2/s3]

(b)

Figure 5: Inlet boundary conditions according to type 4. (a) Distribution of k at inlet. (b)Distribution of at inlet.

14

4 Results

4.1 Short description of the physical experiment

Results from the LDV measurements by Byun and Simpson [9] states that there is no separationin front of the bump but that the flow decelerates there and then accelerate until the top of thebump. The mean flow on the lee side is closly symmetric around the centerline and complexvortical separation occurs downstream from the top and merge into large-scale turbulent eddieswith two large streamwise vorticies. The flow along the streamwise centerline at x/H = 3.63 is adownwashing reattachment flow. The LDV experiment shows, with resulting velocity vectors inthe plane of z/H = 0, that the mean location of separation is at x/H = 0.96. A more detaileddescription of the results from the LDV experiment and comparison with our results is discussedin 4.4.

4.2 Results from our simulations

All simulations of the different turbulence models have been conducted on the UPPMAX (Upp-sala Multidisciplinary Center for Advanced Computational Science) cluster called Os. A detailedlist of the technical specifications of the Os cluster can be seen in appendix B. The wall clocktime for performing a simulation on 4 cores for 10 time-steps was 3 min and 44 seconds.

The models that converge to a stationary solution are shown in table 2. Notice that even thesimplest k model needed almost 3000 time-steps ( about 18 hours 40 minutes of wall-clocktime using 4 cores ) to converge. Keep in mind that the size of the time step is of no importancesince a steady state solver was used.

Turbulence Model Description CommentskEpsilon Standard k model with wall

functionsConvergence reached after ap-proximately 3000 time-steps.

RNGkEpsilon RNG k model with wall func-tions

Convergence reached after ap-proximately 3000 time-steps.

NonlinearKEShih Non-linear Shih k model withwall functions

Convergence reached after ap-proximately 11 000 time-steps.

SpalartAllmaras Spalart-Allmaras 1-eqn mixing-length model for external flows


kOmegaSST k SST two-equation eddy-viscosity model with wall func-tions


realizablekEps Realizable k model with wallfunctions

convergence reached after ap-proximately 17 000 time-steps.

Table 2: Table of working turbulence models

15

Turbulence Model Description CommentsLienCubicKE Lien cubic k model with wall

functionsUnable to reach convergence

QZeta q model without wall func-tions

Solution did converge though theresult was inconsistent with thephysical solution.

LaunderSharmaKE Launder-Sharma low-Re k model without wall functions

Unable to reach convergence af-ter 30 000 time-steps (no differ-ence in residuals for last 10 000time-steps)

LamBremhorstKE Lam-Bremhorst low-Re k model without wall functions

Solution starts to diverge afterapproximately 1000 time-steps

LienLeschzinerLowRE Lien-Leschziner low-Re k model without wall functions

Solution start to diverge after ap-proximately 800 time-steps

LRR Launder-Reece-Rodi Reynoldsstress model with wall functions


LaunderGibsonRSTM Launder-Gibson Reynolds stressmodel with wall-reflection termsand wall functions


Table 3: Table of turbulence models for which the simulation encountered problems, preventingit to produce a converged, steady solution.

The models that in any way failed to give accurate results are shown in table 3. Recall thatthe main objective was to try as many RANS turbulence models as possible for this particularcase. The time frame and the priorities of the project excluded the possibility to more closelyinvestigate models for which problems were encountered. It is quite possible that with an extraeffort, the turbulence models in table 3 may work for this case as well.

The k types of models that failed to converge were tested with more suitable underrelaxation factors and different types of inlet boundary conditions without success. The Reynoldsstress models that diverged early in the simulations were tested numerous ways. We tried runninga k method first, start the simlutation after 1000 time-steps with the Reynolds stress modelswithout success. We also tried suitable relaxation factors which also did not make any differenceeither.

4.3 Global behaviour of the models tested

All of the models that converged capture the magnitude of the velocity quite reasonably comparedto the physical solution. Separation of the flow is found in all models except the k model.Generally the separation region is too large compared to the physical solution, the start ofthe separation is however reasonably consistent with the physical solution. In all of the casesthe motion of the turbulent eddies is incorrectly resolved, such as the flow direction and themagnitude of this flow. Near the centerline two large eddies appear which should not be there.A more detailed description of the solutions computed with the models in table 2 is listed belowand comparisons to the LDV measurements and LES data are found in section 4.4.

16

4.3.1 The k modelThe first model that we tried for our domain was the basic k model. It is often used as firstmodel to test with. The result from this model was not accurate, the simulated flow was notseparated which is shown in figure 6 and 7 , and no eddies or vortices were captured.The result from the simulation that used the inlet setup 2 and setup 3 did not differ substantiallyfrom the result with inlet conditions according to the first setup described in 3.2. When we usedsetup 4 for the inlet conditions we saw only very minor differences in the solution compared tothe two other inlet setups for this model. Since the difference was none or very small with theseconditions, no results in form of graphs will be presented.

(a) (b)

Figure 6: (a) The magnitude of the velocity at z = 0. (b) The velocity field in the wake on theleeward side of the hill. Observe that there is no recirculation or separation.

(a) (b)

Figure 7: (a) The turbulent kinetic energy k, at z = 0. Compared to other models the k model produces a much more spread out profile of this quantity. (b) Streamlines of the velocityin x-direction through the line from [0.125 0.02 0.15] to [0.125 0.02 -0.15].

17

4.3.2 The Nonlinear k Shih modelThe results of this model differ from the results from other models. It was tested both withinlet conditions accordning to setup 1, the same as the rest of the models, and also with setup 4.As seen in figure 9, both cases gave unrealistic results, with different asymmetric solutions andoccurrence of only one large vortex either to the right or to the left of the center line downstreamof the hill. The results in both cases can be considered bad compared to the physical solution andthe LES-simulation. It is however an interesting result that this turbulence model supportsan asymmetric solution (and its mirror on the centre plane, of course).

(a) (b)

Figure 8: (a) The magnitude of the velocity at z = 0. (b) The velocity field in the wake on theleeward side of the hill.

(a) (b)

Figure 9: The streamlines from this model are completely different from other models that weretested. (a) Streamlines of the velocity in x-direction through the line from [0.125 0.02 0.15] to[0.125 0.02 -0.15]. (b) Streamlines of the velocity in x-direction when using inlet setup 4. throughthe line from [0.125 0.02 0.15] to [0.125 0.02 -0.15].

18

4.3.3 The RNG k modelThis model was used with inlet conditions according to setup 1 and setup 4 but converged onlyfor setup 1! As seen in figure 10 (a), the flow layer near the bottom of the domain deceleratesbefore the hill and then accelerates on the top to finally decelerate again on the leeward side ofthe hill. The flow separates around x/H 0.96 and is re-attached around x/H 2.12. Theseparation and re-attachment can be seen in figure 10 (b). Two small and two bigger vorticesdevelop downhill of the hill in the positive x-direction. This is shown in figure 11 (a). Theturbulent kinetic energy is shown in figure 11 (b). As expected more turbulent kinetic energyappears in the separation zone where the eddies are as greatest.

(a) (b)

Figure 10: (a) The magnitude of the velocity at z = 0. (b) The velocity field in the wake on theleeward side of the hill.

(a) (b)

Figure 11: (a) Streamlines of the velocity in x-direction through the line from [0.125 0.02 0.15]to [0.125 0.02 -0.15]. (b) The turbulent kinetic energy k, at z=0.

19

4.3.4 The k SST modelThis model was used with inlet conditions according to setup 1 and setup 4 but converged onlyfor setup 1! The model managed to clearly capture the flow separation. The flow separationoccurs around x/H 0.71 and is re-attached around x/H 2.24. This can be seen in figure 12(a).Two vortices on the leeward side of the hill occur and they are almost symmetrically placedaround the plane z = 0. This is shown in figure 13 (a).

(a) (b)

Figure 12: (a) The magnitude of the velocity at z=0. (b) The velocity field in the wake on theleeward side of the hill.

(a) (b)


20

4.3.5 The realizable k modelThis model was used with inlet conditions according to setup 1 and setup 4 and converged forboth setups. The results from the two inlet setups were almost identical and only the resultsfrom setup 1 will be presented in the graphs. Similar to the k SST model the realizable k model also managed to capture the flow separation. The flow separation occurs at x/H 0.64and is re-attached around x/H 2.24. This can be seen in figure 14 (a).Two vortices on theleeward side of the hill turn up and they are almost symmetrically placed around the plane z = 0.This is shown in figure 15 (a).

(a) (b)

Figure 14: (a) The magnitude of the velocity at z=0. (b) The velocity field in the wake on theleeward side of the hill.

(a) (b)


21

4.3.6 The Spalart Allmaras model

This model was only used with inlet setup 1. The result of this model is a bit different comparedto the other turbulence models that managed to capture the flow separation. In the plane z =0, no separation is appearing and the wake is divided in two symmetrically placed separationbubbles. This is shown in figure 16. In difference to all other models this model generates fourvortices, two at each side of the centerplane, at the leeward side of the hill.

(a) (b)

Figure 16: (a) The velocity field in the wake on the lee side of the hill. Notice that no separationof the flow is found at this cut. (b) z=0.02

(a) (b)

Figure 17: (a) Streamlines of the velocity in x-direction through the line from [0.125 0.02 0.15]to [0.125 0.02 -0.15]. Notice that this model captures 4 vortices at the lee side of the hill. (b)The magnitude of the velocity at z=0.

22

4.4 Comparison to experimental and LES data

Results of the LDV investigations of the flow around the hill where reported by Byun andSimpson 2005 [9]. Their result for the velocity field lee ward of the hill in the z/H = 0 plane isshown in figure 18. The flow separates at x/H = 0.96. Note that the k model did not haveany flow separation and the Non linear k Shih model showed obvious strange behaviour andtherefore these models have not been included in the comparison section. The Spalart Allmarasmodel shows no separation at all for the plane z/H = 0, however separation is found in the planez/H = 0.02 at x/H = 0.90.

Model Separation point Re-attachment pointLDV, figure 18 (a) x/H = 0.96 x/H 2.0RNG k , figure 10 (b) x/H = 0.96 x/H = 2.12Realizable k , figure 14 (b) x/H = 0.64 x/H = 2.24k-Omega SST, figure 12 (b) x/H = 0.71 x/H = 2.24

Table 4: Table of separation and re-attachment points for different models and LDV experiment.Note that no obvious re-attachment point is found in the LDV-experiment.

As we can see from the comparison in table 4 our results in the same plane show that theRNG k model have similar separation region. The realizable k and k model separatethe flow to early at x/H = 0.64 and x/H = 0.71 respectively.

(a) (b)

Figure 18: (a) Velocity field from LDV measurements in the plane z = 0. (b) Contours of themagnitude of the velocity U , normalized with reference velocity Uref = 27.5 m/s and the vectorfield of Uy and Uz in the plane x/H = 3.63 for the LDV measurements. Note that the view isfrom behind the hill. Plots made by Byun and Simpson [9]

If we look at the results from the LDV-experiment in the y, z-plane with x/H = 3.63, thereare two vortices centred at z/H 1.35. This can be seen in figure 18 b. All of our mostsuccessful models fail to predict this result, however some similar results can be found from acouple of our RANS models.

The Spalart Allmaras model produces two large vortices centred at z/H 0.45, and twovery small ones at z/H 1.92. This is shown in figure 20 a.

23

(a)

(b)

Figure 19: (a) Surface plot of the magnitude of velocity U and the transverse vector field(in they, z-plane), for the RNG k model. (b) k SST model. Note that the view is from behindthe hill.

The k SST model manage to produce two vortices at z/H 1.41 which are quite similar tothe LDV experiment. However, this model behaves strangely around z/H 0 with two smallvortices that collide. This is shown in figure 19 b.

The realizable k model produces a result that is quite similar to the k SST model.There are two vortices at z/H 1.54 and they are a bit smaller than the ones produced bythe k SST model at around the same place. This model, similarly to the k SST model,fail by producing two vortices that collide around z/H 0. These vortices can be seen in figure20 b.The RNG k model produces two vortices at z/H 1.40, which is consistent to the LDVexperiment. In difference with the LDV experiment some inaccurate behaviour occurs aroundz/H = 0.38. These vortices can be seen in figure 19 a.

24

Figure 20: Surface plot of the magnitude of velocity U and the vector field of Uy and Uz. Notethat the view is from behind the hill. (a)The Spalart Allmaras model (b) The realizable k model.

The line plots in figure 21 and figure 22 show the ux and uz respectively at x/H = 3.63for different values of z/H. Note that these plots show the velocity components behind theseparation region. Therefore ux is positive for all y/H-values in each plot. However, there arestill vortices in this region, but they appear in the y, z-plane according to figure 19 and figure20.

The results for the ux velocity from our RANS turbulence models are quite similar withLDV-experiment except some larger deviations for the Spalart Allmaras and the Non linear k Shih models. The latter of these two models produces for almost all z/H points inconsistentresults compared to the reference data.

If we look at the uz velocity almost all of the RANS turbulence models produce differentresults compared to the LDV-experiment. As one can see in figure 22 for z/H = 0.8143 andz/H = 0.3257, the uz velocity points in opposite direction for small values of y/H. The flowdirection of the eddies are opposite compared to the LDV and LES data. If we look in thecenterplane, hence z/H = 0, we can see in figure 22 that both the realizable k model and thek SST model produce incorrect results, which have been mentioned before.

25

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Ux/U0

y/H

RANSmodels compared with LDV measurements and LES data, z/H = 0.8143

LDV dataLES datakEpsilonSpalartAllmarasNonlinearKEShihRNGkEpsilonkOmegaSSTrealizablekEps

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Ux/U0

y/H



0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Ux/U0

y/H

RANSmodels compared with LDV measurements and LES data, z/H = 0


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Ux/U0

y/H



0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Ux/U0

y/H



0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Ux/U0

y/H



Figure 21: Comparison of RANS models with LDV experiment and LES data. Ux for RANSand LDV is measured at x/H = 3.63 and at x/H = 3.69 for LES. Data is for six different valuesof z/H.

26

0.1 0.05 0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

1.2

1.4

Uz/U0

y/H



0.2 0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

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y/H



0.25 0.2 0.15 0.1 0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

1

1.2

1.4

Uz/U0

y/H

RANSmodels compared with LDV measurements and LES data, z/H = 0


0.4 0.3 0.2 0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

Uz/U0

y/H



0.4 0.3 0.2 0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

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y/H



0.15 0.1 0.05 0 0.050

0.2

0.4

0.6

0.8

1

1.2

1.4

Uz/U0

y/H



Figure 22: Comparison of RANS models with LDV experiment and LES data. Ux for RANSand LDV is measured at x/H = 3.63 and at x/H = 3.69 for LES. Data is for six different valuesof z/H.

27

5 Conclusions

The results from the RANS models implemented in OpenFOAM of the flow over a three-dimensional axisymmetric hill have been reported, analysed and compared to experiment- andLES-data. Many other research groups like Wang et al. [10] and Person et al. [7] have testedRANS turbulence models on the same geometry. Their results are confirmed by our study, theRANS turbulence models tested for this geometry do not give accurate results.

Even though none of the tested models give good enough result some models do capture theseparation region fairly well. Our conclusion is that the best performing model in these tests isthe RNG k model. The wake in the centerplane, z/H = 0 is most similar in size and positioncompared to the results from LDV experiment. In addition to the RNG k model two moremodels give results that are acceptable. These are the realizable k and the k SST models.

All of the tested RANS models, even the three that produces acceptable results, produceseddies in the plane x/H = 3.63 near the centerline that do not exists in the experimental data.

Improvements may be possible if the meshes are refined, especially in the turbulent boundarylayer near the hill. Another possible way to improve the results is to try other types of inletboundary conditions for the turbulent quantities. Finally one could also further develop betteradapted wall functions to handle the turbulent boundary layer close to the walls.

References

[1] Milovan Peric Joel H. Ferziger. Computational Methods for Fluid Dynamics. Springer, 3rdedition, 2002.

[2] Sinisa Krajnovic. Large eddy simulation of the flow over a three-dimensional hill. FlowTurbulence Combust, 81:189204, December 2008.

[3] OpenCFD LTD. http://www.opencfd.co.uk/openfoam/.

[4] W. Rodi M. Garcia-Villalba, N. Li and M. A. Leschziner. Large-eddy simulation of separatedflow over a three-dimensional axisymmetric hill. J. Fluid Mech., 000:142, 2009.

[5] OpenCFD Limited. OpenFOAM - Programmers Guide, 1.5 edition, July 2008.

[6] OpenCFD Limited. OpenFOAM - User Guide, 1.5 edition, July 2008.

[7] Benson R. E. Persson T., Liefvendahl M. and Fureby C. Numerical investigation of the flowover an axisymmetric hill using les, des and rans. Journal of Turbulence, 7(4):117, 2006.

[8] Stephen B Pope. Turbulent Flows. Cambridge University Press, 2000.

[9] G.Byun Roger L. Simpson. Structures of three-dimensional separated flow on an axisym-metric bump. 43rd AIAA Aerospace Sciences Meeting and Exhibit, 23, 2005. Paper No.2005-0113.

[10] Jang Y. J. Wang C. and Leschziner M. A. Modelling two- and three-dimensional separationfrom curved surfaces with anisotropy-resolving turbulence closure. International Journal ofHeat and Fluid Flow, 25(3):499512, 2004.

28

A Examples of OpenFOAM case-filesFile: controlDict/*---------------------------------------------------------------------------*\| ========= | || \\ / F ield | OpenFOAM: The Open Source CFD Toolbox || \\ / O peration | Version: 1.5 || \\ / A nd | Web: http://www.openfoam.org || \\/ M anipulation | |\*---------------------------------------------------------------------------*/

FoamFile{

version 2.0;format ascii;class dictionary;object controlDict;

}

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

application simpleFoam;

startFrom startTime;

startTime 0;

stopAt endTime;

endTime 1.0;

deltaT 0.0001;

writeControl timeStep;

writeInterval 1000;

purgeWrite 0;

writeFormat ascii;

writePrecision 6;

writeCompression uncompressed;

timeFormat general;

timePrecision 6;

runTimeModifiable yes;

// ************************************************************************* //

29

File: fvSchemes/*--------------------------------*- C++ -*----------------------------------*\| ========= | || \\ / F ield | OpenFOAM: The Open Source CFD Toolbox || \\ / O peration | Version: 1.5 || \\ / A nd | Web: http://www.OpenFOAM.org || \\/ M anipulation | |\*---------------------------------------------------------------------------*/FoamFile{

version 2.0;format ascii;class dictionary;object fvSchemes;

}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

ddtSchemes{

default steadyState;}

gradSchemes{

default Gauss linear;grad(p) Gauss linear;grad(U) Gauss linear;

}

divSchemes{

default none;div(phi,U) Gauss upwind;div(phi,k) Gauss upwind;div(phi,omega) Gauss upwind;div(phi,R) Gauss upwind;div(R) Gauss linear;div(phi,nuTilda) Gauss upwind;div((nuEff*dev(grad(U).T()))) Gauss linear;

}

laplacianSchemes{

default none;laplacian(nuEff,U) Gauss linear corrected;laplacian((1|A(U)),p) Gauss linear corrected;laplacian(DkEff,k) Gauss linear corrected;laplacian(DomegaEff,omega) Gauss linear corrected;laplacian(DREff,R) Gauss linear corrected;laplacian(DnuTildaEff,nuTilda) Gauss linear corrected;

}

interpolationSchemes{

default linear;interpolate(U) linear;

}

snGradSchemes{

default corrected;}

fluxRequired{

default yes;p;

}

// ************************************************************************* //

30

File: fvSolution/*--------------------------------*- C++ -*----------------------------------*\| ========= | || \\ / F ield | OpenFOAM: The Open Source CFD Toolbox || \\ / O peration | Version: 1.5 || \\ / A nd | Web: http://www.OpenFOAM.org || \\/ M anipulation | |\*---------------------------------------------------------------------------*/FoamFile{

version 2.0;format ascii;class dictionary;object fvSolution;

}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

solvers{

p PCG //Preconditioned Conjugate gradient{

preconditioner DIC; //Diagonal Incomplete Choleskytolerance 1e-06;relTol 0.01;

};U PBiCG //Preconditioned Biconjugate gradient{

preconditioner DILU; //Diagonal Incomplete LUtolerance 1e-05;relTol 0.1;

};k PBiCG{

preconditioner DILU;tolerance 1e-05;relTol 0.1;

};omega PBiCG{

preconditioner DILU;tolerance 1e-05;relTol 0.1;

};}

SIMPLE{

nNonOrthogonalCorrectors 3;}

relaxationFactors{

p 0.3;U 0.7;k 0.5;omega 0.5;R 0.7;nuTilda 0.7;

}

// ************************************************************************* //

31

File: RASProperties/*--------------------------------*- C++ -*----------------------------------*\| ========= | || \\ / F ield | OpenFOAM: The Open Source CFD Toolbox || \\ / O peration | Version: 1.5 || \\ / A nd | Web: http://www.OpenFOAM.org || \\/ M anipulation | |\*---------------------------------------------------------------------------*/FoamFile{

version 2.0;format ascii;class dictionary;object RASProperties;

}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

RASModel kOmegaSST;

turbulence on;

printCoeffs on;

laminarCoeffs{}

kEpsilonCoeffs{

Cmu 0.09;C1 1.44;C2 1.92;alphaEps 0.76923;

}

RNGkEpsilonCoeffs{

Cmu 0.0845;C1 1.42;C2 1.68;alphak 1.39;alphaEps 1.39;eta0 4.38;beta 0.012;

}

realizableKECoeffs{

Cmu 0.09;A0 4.0;C2 1.9;alphak 1;alphaEps 0.833333;

}

kOmegaSSTCoeffs{

alphaK1 0.85034;alphaK2 1.0;alphaOmega1 0.5;alphaOmega2 0.85616;gamma1 0.5532;gamma2 0.4403;beta1 0.0750;beta2 0.0828;betaStar 0.09;a1 0.31;c1 10;

Cmu 0.09;}

NonlinearKEShihCoeffs

32

{Cmu 0.09;C1 1.44;C2 1.92;alphak 1;alphaEps 0.76932;A1 1.25;A2 1000;Ctau1 -4;Ctau2 13;Ctau3 -2;alphaKsi 0.9;

}

SpalartAllmarasCoeffs{

alphaNut 1.5;Cb1 0.1355;Cb2 0.622;Cw2 0.3;Cw3 2;Cv1 7.1;Cv2 5.0;

}

wallFunctionCoeffs{

kappa 0.4187;E 9;

}

// ************************************************************************* //

33

B Technical specification of the OS clusterHardware:

* IBM X3455 dual Opteron 2220SE, 2.8GHz nodes. 4*10=40 cores* 8GB RAM,* Gigabit interconnect

Software:

* Scientific Linux SL release 5.2* SGE version 6.2, Grid Engine queue system

o "man sge_intro" for an overview.* GCC version 4.3 c,c++,fortran and java

o module load gcc* PGI version 7.1, 64 bit compilers (C,C++,F77,F95,HPF)

o module load pgi* Intel version 10.1, 64 bit compilers (C,C++,Fortran)

o module load intel* MPI, OpenMPI libraries

o module load pgi openmpi_pgio module load gcc openmpi_gcco module load intel openmpi_intelo mpicc or mpif90 to compileo mpirun to execute.

* qsub -pe dmp8 16, to submit a 16 slot mpi job* openMP is available in pgi, intel and gcc

34

report vt09 09 new

Documents

shih model

sst model

model2116 results

model2115 results

and146 results

k178 results

spalart allmaras model

computational case