2012

Validation Guide

2012 Next Limit Technologies

Contents

1 Theory 1

2 Lid-driven cavity flow 9

3 Natural convection in a cavity 13

4 NACA-0012 airfoil at Re = 500 19

5 S825 airfoil 27

6 Vortex cell 35

7 Automotive aerodynamics 39

8 Multi-phase flows 45

ii

1Theory

In the literature there are several particle-based numerical approaches to solve the

computational fluid dynamics. They can be classified in three main categories:

(i) algorithms modeling the behavior of the fluid at molecular level (e.g. Direct

Simulation Montecarlo); (ii) algorithms which solve the equations at a macroscopic

level, such as Smoothed Particle Hydrodynamics (SPH) or Vortex Particle Method

(VPM), and finally, (iii) methods based on a mesoscopic framework, such as the

Lattice Gas Automata (LGA) and Lattice Boltzmann Method (LBM).

The algorithms that work at molecular level have a limited application, and they

are used in theoretical analysis. The methods that solve macroscopic continuum

equations are employed most frequently, but they also present several problems.

SPH-like schemes are computationally expensive and in their less sophisticated

implementations show lack of consistency and have problems imposing accurate

boundary conditions. VPM schemes have also a high computational cost and

besides, they require additional solvers (e.g. schemes based on boundary element

method) to solve the pressure field, since they only model the rotational part of the

flow.

Finally, LGA and LBM schemes have been intensively studied in the last years

being their affinity to the computational calculation their main advantage. Their

main disadvantage is the complexity to analyze theoretically the emergent behavior

of the system at macroscopic level from the laws imposed at mesoscopic level.

XFlows approach to the fluid physics takes the main ideas behind these schemesand extends them to overcome most of the limitations present on these schemes.

1.1 Lattice Gas Automata

LGA schemes are simple models that allow to solve the behavior of gases. The

main idea is that the particles move discretely in a d-dimensional lattice in one of

2 Theory

Figure 1.1: HPP model.

the predetermined direction at discrete times t = 0, 1, 2, ... and with velocity ci,

i = 0, ..., b, also predetermined.

The simplest model is the HPP, introduced by Hardy, Pomeau and de Pazzis,

in which the particles move in a two-dimensional square grid and in four directions,

as shown in Figure 1.1. The state of an element of the lattice at instant t is given by

the occupation number ni(r, t), with i = 0, ..., b, being ni = 1 presence and ni = 0

absence of particles moving in direction i.

The equation that governs the evolution of the system is as follows:

ni(r + cit, t+ t) = ni(r, t) + i(n1, ..., nb) (1.1)

where i is the collision operator, which for each previous state (n1, ..., nb) computes

a post-collision state (nC1 , ..., nCb ) conserving the mass, linear momentum and energy;

r is a position in the lattice and ci a velocity.

From a statistical point of view, a system is constituted by a large number

of elements which are macroscopically equivalent to the system under study. The

macroscopic density and linear momentum are:

=1

b

bi=1

ni, v =1

b

bi=1

nici (1.2)

1.2 Boltzmanns transport equation

Boltzmanns transport equation is defined as follows:

fi(r + cit, t+ t) = fi(r, t) + Bi (f1, .., fb) (1.3)

where fi is the distribution function in the direction i and Bi the collision operator.

From this equation and by means of the Chapman-Enskog expansion, the

compressible Navier-Stokes equations can be recovered [1]. The Chapman-Enskog

1.3 Lattice Boltzmann Method 3

expansion shows that it is possible to design LGA schemes that recover the

hydrodynamic macroscopic behavior at low Mach numbers.

The main advantage of these methods is their great affinity with computers.

They are easily programmed and very efficient. Some schemes have isotropy

problems (do not satisfy Galilean invariance) and produce very noisy results.

The main contribution of LGA schemes is that they are precursor of the Lattice

Boltzmann method.

1.3 Lattice Boltzmann Method

The origins of the Lattice Boltzmann Method (LBM) ([2, 3, 4]) lie in the LGA

schemes. While the LGA schemes use discrete numbers to represent the state of the

molecules, the LBM method makes use of statistical distribution functions with real

variables, preserving by construction the conservation of mass, linear momentum

and energy. It can be shown that if the collision operator is simplified under the

Bhatnagar-Gross-Krook (BGK) approximation [5], the resulting scheme reproduces

the hydrodynamic regime also for low Mach numbers. This operator is defined as

follows:

BGKi =1

(feqi fi) (1.4)

where feqi is the local equilibrium function and is the relaxation characteristic time

(which is related to the macroscopic viscosity).

Usually, the equilibrium distribution function adopts the following expression:

feqi (r, t) = ti

(1 +

civc2s

+vv2c2s

(cicic2s )

)(1.5)

where cs is the sound speed, v the macroscopic velocity, the Kronecker delta, and

ti are built preserving the isotropy in space.

(a) D2Q7 (b) D2Q9

Figure 1.2: Most common LBM schemes in two-dimensions.

LBM schemes are classified as a function of the spatial dimensions d and the

number of distribution functions b, resulting the notation DdQb. The most common

4 Theory

(a) D3Q19 (b) D3Q27

Figure 1.3: Most common LBM schemes in three-dimensions.

schemes in two dimensions are the D2Q7 and D2Q9 represented in Figure 1.2,

while in three dimensions the most used schemes are the D3Q13, D3Q15, D3Q19

and D3Q27 plotted in Figure 1.3.

Finally, the multiscale Chapman-Enskog expansion gives us the relation between

the macroscopic viscosity and the relaxation parameter:

= c2s( 1

2) (1.6)

For a positive viscosity, the relaxation time must be greater than 0.5. The most

interesting aspect is that these schemes are able to model a wide range of viscosities

(0,) in an efficient way even using explicit formulations.General references are the review by Chen and Doolen (1998) [6] and the book

by Succi (2001) [7].

1.4 Turbulence modeling

Following the dimensional analysis proposed by Kolmogorov at high Reynolds

numbers, the flow tends to break in smaller eddies to transform the kinetic energy

into internal energy. This process is known as Kolmogorov cascade and it explains

the turbulence phenomenon.

The time necessary to break an eddy in the flow is in the order of

Tbreak Leddyeddy

(1.7)

and the time to dissipate the kinetic energy through viscosity is expressed as

Tdissip,visc L2eddy

(1.8)

1.4 Turbulence modeling 5

ProblemBlood flow Vehicle airflow

Vehicle airflow Plane airflowBioengineering at low speed

Re 103 - 104 105 - 106 107 - 108 > 109

Nelements 106 - 109 1011 - 1013 1015 - 1018 > 1020

Table 1.1: Number of elements required for DNS.

For large eddies and high Reynolds numbers, the break time is smaller than time

employed to dissipate the energy and this produces the Kolmogorov cascade.

The kinetic energy of a turbulent structure can be estimated by

Ec eddy V 2eddy

The specific kinetic energy dissipation ratio is as follows

break V 2eddyTbreak

=V 3eddyLeddy

The smallest eddies present in the flow (of size Lcritical) have a breaktime equal to time necessary to transform their kinetic energy to viscous energy

(Tbreak Tdissip,visc). Then the kinetic energy dissipation ratio can be estimated by

viscous V3critical

Lcritical

3

L4critical V

3eddy

Leddy break (1.9)

and thus,

LeddyLcritical

V3eddyL

3eddyL

3critical

3L3eddy=L3criticalL3eddy

Re3eddy (1.10)

Finally,LcriticalLeddy

Re3/4eddy (1.11)

Taking into account this relation, if we want to solve explicitly every eddy in a

three-dimensional flow, the number of elements are in the order of

Nelements (

LeddyLcritical

)3 Re9/4eddy (1.12)

Table 1.1 summarizes the number of elements necessary to solve in a direct way

the turbulence at different Reynolds numbers. Direct numerical simulation (DNS)

of turbulence is currently possible only for low Reynolds numbers, while for typical

industrial Reynolds numbers some modeling for the unresolved scales is required.

Reynolds-Averaged Navier-Stokes (RANS) approach models the turbulence in

a global way. This approach is nowadays the most widely adopted and calculates

6 Theory

Figure 1.4: Turbulence modeling.

values averaged in time, removing the time dependence of the solution. Although

calculating averaged results is computationally less expensive, new terms appear in

the Navier-Stokes equations that have to be modeled by new transport equations.

Moreover there are several RANS models, each one suitable for a specific problem,

and the parameters of each model need to be adjusted empirically.

Another approach to the turbulence problem is the Large Eddy Simulation

(LES). These schemes solve the turbulence in a local way, modeling only the smallest

scales, and are closer to the physics (see Figure 1.4). The turbulence at smallest

scales has been extensively studied and its behavior can be reproduced without using

arbitrary parameters. These are the type of schemes employed in XFlow.

The LES scheme adopted by default by XFlow is the Wall-Adapting Local Eddy-viscosity (WALE), which has good properties both near and far of the wall and with

laminar and turbulent flows. This model is formulated as follows:

effective = molecular + turbulent (1.13)

turbulent = 2filterscale

(GdGd)

3/2

(SS)5/2 + (GdG

d)

5/4(1.14)

S =1

2

(vr

+vr

)(1.15)

Gd =1

2(g2 + g

2)

1

3g

2 (1.16)

g =vr

(1.17)

REFERENCES 7

References

[1] Y.H. Qian, D. DHumieres, and P. Lallemand. Lattice BGK models for Navier-

Stokes equation. Europhysics Letters, 17:479, 1992.

[2] G. McNamara and G. Zanetti. Use of the Boltzmann equation to simulate lattice-

gas automata. Physical Review Letters, 61:23322335, 1988.

[3] F.J. Higuera and J. Jimenez. Boltzmann approach to lattice gas simulations.

Europhysics Letters, 9:663668, 1989.

[4] H. Chen, S. Chen, and W. Matthaeus. Recovery of the Navier-Stokes equations

using a lattice-gas Boltzmann method. Physical Review A, 45:5339, 1992.

[5] P.L. Bhatnagar, E.P. Gross, and M. Krook. A model for collision processes in

gases. Phys. Rev., 94:511, 1954.

[6] S. Chen and G. Doolen. Lattice Bolzmann method for fluid flows. Annual

Reviews of Fluid Mechanics, 30:32964, 1998.

[7] S. Succi. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.

Clarendon Press, 2001.

8 REFERENCES

2Lid-driven cavity flow

The lid-driven cavity is a classical benchmark problem for viscous incompressible

fluid flow. It consists in a cavity where the upper boundary moves to the right, and

causes a rotation in the cavity. Side and bottom walls of the cavity are considered

no-slip, while the velocity at the upper wall is imposed to vx = 1 m s1 (see Figure

2.1). Although there is a discontinuity of the boundary conditions at the two top

corners where the side wall meet the lid, this corner singularity plays a minor role

in the overall solution field.

Figure 2.1: Geometry and boundary conditions.

The solution field will depend on the Reynolds number

Re = vx L

(2.1)

This validation case computes the laminar incompressible flow for a 2D driven cavity

at Re = 1000 and compares XFlows results with the ones presented in [1] and [2].Fluid properties are = 1 kg m3 and = 0.001 Pa s, in a L = 1 m square cavitydiscretized with 128x128, 256x256 and 512x512 cells lattices.

10 Lid-driven cavity flow

y 128x128 256x256 512x512 Ref. [1] Ref. [2]

0.9688 0.58578 0.57217 0.57970 0.57492 0.58031

0.9531 0.476 0.46869 0.47213 0.46604 0.47239

0.7344 0.18963 0.18866 0.18908 0.18719 0.18861

0.5 -0.06255 -0.06218 -0.06216 -0.0608 -0.06205

0.2813 -0.28175 -0.28020 -0.28064 -0.27805 -0.2804

0.1016 -0.29968 -0.30286 -0.30078 -0.2973 -0.30029

0.0625 -0.20118 -0.20493 -0.20248 -0.20196 -0.20227

Table 2.1: Horizontal velocity vx along the vertical centerline. Comparison of

XFlows results for different resolutions with the reference solutions.

The solution computed with XFlow is plotted in Figure 2.2. The vx velocity

Figure 2.2: Velocity field.

distribution along vertical centerline is shown in Figure 2.3 and Table 2.1. Additional

quantitative comparisons of pressure and vorticity are presented in Tables 2.2 and

2.3.

For this problem it is essential to activate the high order boundary conditions

(Project Tree > Engine > Advanced Options > High order boundary

conditions). This imposes the velocity at the lid through a high order scheme.

While using high order BC the velocity relative error is O(103), with the defaultBC the relative error is O(102).

11

Figure 2.3: Horizontal velocity vx distribution along the vertical centerline.

y 128x128 256x256 512x512 Ref. [1]

0.9688 0.055238 0.054295 0.054077 0.051514

0.9531 0.053995 0.053059 0.052855 0.050329

0.7344 0.013019 0.012739 0.012738 0.012122

0.5 0.0 0.0 0.0 0.0

0.2813 0.043511 0.042518 0.042447 0.040377

0.1016 0.112102 0.110042 0.109613 0.104187

0.0625 0.117450 0.115454 0.114904 0.109200

Table 2.2: Pressure p along the vertical centerline for Re = 1000. Comparison of

XFlows results for different resolutions with a reference solution.

y 128x128 256x256 512x512 Ref. [2]

0.9688 9.51527 9.11829 9.43159 9.47810

0.9531 5.04854 4.63587 4.81708 4.86280

0.7344 2.10078 2.09089 2.09418 2.09090

0.5 2.07863 2.06865 2.07152 2.06690

0.2813 2.27638 2.26145 2.26792 2.26780

0.1016 1.64462 1.61598 1.63825 -1.63520

0.0625 2.32787 2.30427 2.31797 -2.31740

Table 2.3: Vorticity along the vertical centerline. Comparison of XFlows resultsfor different resolutions with a reference solution.

12 REFERENCES

References

[1] U. Ghia, K.N. Ghia, and C.T. Shin. High-Re solutions for incompressible

flow using the Navier-Stokes equations and a multigrid method. Journal of

Computational Physics, 48:387411, 1982.

[2] C.-H. Bruneau and M. Saad. The 2d lid-driven cavity problem revisited.

Computers & Fluids, 35:326348, 2006.

3Natural convection in a cavity

Buoyancy-driven flow in a square cavity with vertical sides which are differentially

heated allows to test XFlow in thermal problems where the buoyancy forces aremodeled using the Boussinesq approximation. The aim is to calculate the flow and

thermal field for Rayleigh numbers of 103 and 106, and compare the results with the

benchmark solutions published by De Vahl et al. [1].

Figure 3.1: Geometry and boundary conditions.

The boundary conditions for the problem (see Figure 3.1) involve two vertical

walls at different temperatures, leading to a thermal gradient across the domain.

This thermal gradient produces varying buoyancy forces between the walls that

drive the flow. Horizontal walls are considered adiabatic.

The Boussinesq approximation assumes the linear variation of the density as a

function of the temperature ,

= 0[1 ( 0)]

and that the thermodynamic properties of incompressible fluids are constant except

14 Natural convection in a cavity

when considering the body force g in the momentum equation:

v = 0 (3.1)0

dv

dt= + 0[1 ( 0)]g (3.2)

0cpd

dt= k () + (3.3)

where is the thermal expansion coefficient, cp the specific heat at constant pressure,

k the thermal diffusion coefficient, 0 a reference temperature, 0 = (0), and is

the viscous heat dissipation.

Benchmark solutions for the natural convection problem were published by [1].

These solutions are compared against the numerical results obtained with XFlow.The non-dimensional distances and velocities used in the results are

X =x

L, Y =

y

L, Vx =

vxL

, and Vy =

vyL

with =k

cpthe thermal diffusivity. The Rayleigh number is defined as

Ra = g L3

(3.4)

= |1 2| is the temperature difference between the two vertical walls.For the simulations presented here, the solution domain consists of a L = 1 m

square, using 20x20, 40x40, 80x80, and 160x160 cell lattices. The fluid properties

for the Ra = 103 case are:

0 = 1 kg m3,

g = 10 m s2, = 0.1 K1, = 1 K (e.g. 1 = 293.65 K, 2 = 292.65 K),

= 0.0266458 Pa s,

k = 0.0375293 kg m s3 K1,cp = 1 m

2 s2 K1

The last three values are chosen to match the Prantl number of air, Pr = cpk

=

0.71. For the Ra = 106 case, only change = 0.0008426 Pa s and k = 0.0011868

kg m s3 K1. The viscous heat dissipation has been neglected as in [1].

The following quantities will be analyzed at the steady state:

Nu0, average heat flux in the hot vertical wall

Nu0 =

10

xdyX=0

15

maxVx, maximum value of the non-dimensional horizontal velocity on the

vertical centerline and its location Ymax

maxVy, maximum value of the non-dimensional vertical velocity on the

horizontal centerline and its location Xmax

Tables 3.1 and 3.2 show the comparison of XFlows results with the benchmarksolution published in [1] for Ra = 103 and Ra = 106. There is a good agreement in

both cases although the accuracy decreases for the highest Rayleigh number. This

deterioration is consistent with the findings of most contributors reported in [1].

In this test, the maximum and minimum locations are cell centered, none of the

interpolation techniques suggested by [2] have been employed.

20x20 40x40 80x80 160x160 Ref.[1]

maxVx 3.5428 3.6325 3.6422 3.6468 3.649

at Ymax 0.825 0.8125 0.8187 0.8156 0.813

maxVy 3.6359 3.6941 3.7023 3.7012 3.697

at Xmax 0.175 0.1875 0.1812 0.1781 0.178

Nu0 1.0916 1.1072 1.1129 1.1155 1.117

Table 3.1: Case Ra = 103. Comparison of XFlows results for different resolutionswith the benchmark solution.

20x20 40x40 80x80 160x160 Ref.[1]

maxVx 54.448 57.750 63.805 64.902 64.630

at Ymax 0.875 0.863 0.856 0.847 0.850

maxVy 127.259 193.479 215.178 219.817 219.360

at Xmax 0.075 0.038 0.044 0.041 0.038

Nu0 5.924 7.975 8.688 8.823 8.817

Table 3.2: Case Ra = 106. Comparison of XFlows results for different resolutionswith the benchmark solution.

For this case it is essential to deactivate the viscous heat dissipation

(in Project Tree > Engine > Advanced Options > Enable viscous term in

Energy equation: Off), set a Courant 1, and do not use turbulence modelneither wall models (Off-resolved). Too large Courant values introduce an

excessive compressibility in the flow and the results do not correspond to the

Boussinesq incompressible reference solution.

16 Natural convection in a cavity

(a) (b)

(c) (d)

Figure 3.2: Result fields for Ra = 103: (a) temperature, (b) horizontal velocity, (c)

vertical velocity, and (d) vorticity.

17

(a) (b)

(c) (d)

Figure 3.3: Result fields for Ra = 106: (a) temperature, (b) horizontal velocity, (c)

vertical velocity, and (d) vorticity.

18 REFERENCES

References

[1] G. DeVahl Davis and I.P. Jones. Natural convection in a square cavity: a

comparison exercise. International Journal for Numerical Methods in Fluids,

3:227248, 1983.

[2] G. DeVahl Davis. Natural convection of air in a square cavity: a benchmark

numerical solution. International Journal for Numerical Methods in Fluids,

3:249264, 1983.

4NACA-0012 airfoil at Re = 500

The NACA-0012 airfoil is a widely-used wing section that has zero camber and

a maximum thickness to chord ratio of 12 percent. Its two-dimensional profile is

symmetric and has very smooth aerodynamic shape, as shows Figure 4.1. This

validation case presents the results for the flow past a NACA-0012 at zero angle-

of-attack and Reynolds number 500 using XFlow, and compares the results withreference data. Due to lack of experimental data for such a low Reynolds number,

the comparison will be based on the CFL3D code from the National Aeronautics

and Space Administration (NASA) [1, 2].

Figure 4.1: NACA-0012 profile.

This two-dimensional single phase external aerodynamics analysis has been run

using a virtual wind tunnel of dimensions 60 40 m and a NACA-0012 profile ofchord length L = 1 m. In order to reach a Reynolds number equal to 500 based on

the chord length, the simulation parameters have been set according to Table 4.1.

The spatial resolution chosen is 2.56 m for the far field, and 0.005 m around

the airfoil profile and within the wake area, as shown in Figure 4.2. The spatial

discretization has been defined through a region of refinement instead of using an

adaptive refinement in order to ensure that the symmetry of the NACA-0012 is

respected. The discretization ended up with 1.3 million elements in 9 levels of

refinement. Since the flow is laminar, no wall functions have been used to model the

boundary layer.

Due to the fact that XFlow solver is inherently transient, the analysis has been

20 NACA-0012 airfoil at Re = 500

Free-stream velocity vref 50 m s1

Density 1 kg m3

Dynamic viscosity 0.1 Pa s

Chord length L 1 m

Reynolds number Re 500

Angle-of-attack 0 degree

Table 4.1: Simulation conditions.

run until the aerodynamic coefficients stabilize in time. The time step used was

0.004 s, which corresponds to a Courant number of 1 with respect to the lattice size

and the free-stream velocity.

Figure 4.2: Spatial resolution in XFlow.

Profiles of normalized X and Y velocity components, as well as pressure

coefficients, have been analyzed at five vertical sections: x/L = 0, 0.25, 0.5, 0.75

and 1 (see Figure 4.3).

Figure 4.4 shows the X-component of the velocity field normalized by the

reference velocity vref = 50 m s1. The results of XFlow and CFL3D are perfectly

matching for the five sections. The profiles are as expected in both codes: they tend

to zero in the airfoil thickness and to one (or slightly more) on the sides where the

boundary layer is fully developed.

21

Figure 4.3: Data plot lines.

XFlow CFL3D ObjectiveCd 0.1705 0.1741 0.1741

Cl 1013 0.538 105 0

Table 4.2: Aerodynamic coefficients comparison.

For the Y-component of the velocity field (normalized by vref ), again

XFlow results are almost perfectly matching with those of CFL3D, as shown inFigure 4.5. Nevertheless, one can observe some differences close to the airfoil wall,

specially for section x/L = 0.5. This might be due to the size of the first element

within the boundary layer which is not fine enough. However, the differences between

the two codes are small.

The pressure coefficient Cp is defined as Cp =pstatic12 v

2ref

, where pstatic is the

gauge static pressure. Figure 4.6 shows the pressure coefficient distribution at the

five sections. More differences are now noticeable, especially at x/L = 0.5, 0.75 and

1.0. In general, the Cp tends to be slightly over-estimated.

The aerodynamic coefficients predicted by XFlow are quite similar to the onesfrom CFL3D, see Table 4.2. The drag coefficient has a relative error of -2.0678%

with respect to CFL3D results, whereas the lift coefficient is actually even more

accurate since it should be equal to zero due to the symmetry of the NACA profile

at zero angle-of-attack.

Aknowledgements: Validation data have been kindly provided by courtesy of

NASA Langley Research Center and David P. Lockard.

22 NACA-0012 airfoil at Re = 500

Figure 4.4: X-component of velocity u(x, y) at x/L = 0.0, 0.25, 0.50, 0.75 and 1.0.

23

Figure 4.5: Y-component of velocity v(x, y) at x/L = 0.0, 0.25, 0.50, 0.75 and 1.0.

24 NACA-0012 airfoil at Re = 500

Figure 4.6: Pressure coefficient at x/L = 0.0, 0.25, 0.5, 0.75 and 1.0.

REFERENCES 25

References

[1] C. Rumsey, R. Biedron, and J. Thomas. CFL3D: Its history and some recent

applications. Technical report, NASA TM-112861, 1997.

[2] D.P. Lockard, L.-S. Luo, S.D. Milder, and B.A. Singer. Evaluation of

PowerFLOW for Aerodynamic Applications. Journal of Statistical Physics,

107(1/2):423478, 2002.

26 REFERENCES

5S825 airfoil

The S825 airfoil has been designed for horizontal-axis wind turbine applications by

the National Renewable Energy Laboratory (Colorado, USA). The report of the

design and experimentation of the S825 airfoil [1] exposes the different objectives

and constraints set for the design, as well as the methodology of measurements

which have been conducted in the NASA Langley Low-Turbulence Pressure Tunnel

(LTPT) [2]. As explained in [1], the main objectives were, first, to reach a maximum

lift coefficient of at least 1.40 at a Reynolds number of 2106. Second, a low profile-drag coefficients should be obtained between 0.40 and 1.20 of the lift coefficient. Two

main constraints were to keep the zero-lift pitching-moment coefficient greater than

-0.15, and also to have an airfoil thickness equal to 17% of the chord. The final two-

dimensional design is as shown in Figure 5.1, with a chord length equal to 0.45715

m.

Figure 5.1: S825 airfoil shape.

Experiments have been conducted at different Reynolds numbers based on the

chord length, however this validation case will only treat the Reynolds number 2106since it has been used for most of the data provided by [1]. The Mach number is

0.1 and the experimentation has been done with transition free (smooth) and with

transition fixed by roughness at specific locations.

The objective of this case is to validate the pressure distribution and

aerodynamic forces predicted by XFlow at low Mach number and different angles ofattack (AoA).

The calculations have been performed with XFlow for a range of angles of attack

28 S825 airfoil

between -4 and 10 degrees every two degrees. All the calculations are transient due

to the nature of the XFlow solver and use the Wall-Adapting Local-Eddy turbulencemodel, which belongs to the Large Eddy Simulation (LES) approach. Wall function

models in XFlow assume that the boundary layer is fully turbulent, therefore it isnot possible to model transition or prescribe a transition location.

Two-dimensional single phase analyses have been performed using a virtual wind

tunnel of 60 m 40 m and a velocity at the inlet of 43.7493 m s1. The angle ofattack is varying by rotating the geometry instead of projecting the inlet velocity

vector, since XFlow allows easy manipulation of the geometry.

The fluid has a density of 1 kg m3 and a dynamic viscosity of 105 Pa saccordingly to the Reynolds number based on the airfoil chord length (Re = 2106).The simulations have been run for 1 second of physical time, with a time step of

0.002 s. The resolution scale at the far field is 1.28 m, using the refinement near

walls and dynamically adapting to the wake available in XFlow. The walls and thewake are resolved with a scale of 0.0025 m, as shown in Figure 5.2.

Figure 5.2: Resolution refinement near the airfoil and the wake.

The solution for the static pressure and velocity flow variables at final time for

zero angle of attack can be observed in Figure 5.3.

For each angle of attack, the curve of pressure coefficients (Cp) has been

extracted in XFlow using a cutting plane field distribution which projects the selectedfield on the upper and lower sides of the airfoil. The Cp has been computed as

following, being Vref equal to 43.7493 m s1:

Cp =pstatic

12 V

2ref

(5.1)

29

(a) Static pressure

(b) Velocity

Figure 5.3: Static pressure and velocity flow fields at final time for AoA = 0 degrees.

For the angles of attack -4, -2, 0, 2, 4, 6, 8 and 10 degrees, the pressure coefficient

distribution along the airfoil has been compared with the transition free experimental

data presented in [1]. The results for angle of attack between 0 and 6 degrees are

in good agreement with the experiments, as shows Figure 5.4. On the upper side

of the airfoil, the pressure coefficients are slightly under-estimated when the angle

increases but still match reasonably with the experimental data.

However, when angles are increased to 8 and 10 degrees of AoA then results

are getting less accurate, as shown in Figure 5.5. The pressure coefficient tends

to be more under-estimated near the leading edge of the upper part. This could

be explained by the lack of transition model or the LES model, which is not fully

consistent for 2D simulations.

Another last series of angles of attack have been studied, this time for negative

incidence. Again, XFlow predicts with accuracy the pressure coefficient distributionfor -2 and -4 degrees, as shown in Figure 5.6.

Finally, Figure 5.7 compares the angle of attack vs. lift coefficient for theoretical

[1], experimental [1] and XFlow results. For positive angles, the lift coefficient isslightly over-predicted by XFlow but in good agreement with the theoretical results.

30 S825 airfoil

(a) (b)

(c) (d)

Figure 5.4: Airfoil pressure coefficient distribution for different AoA: a) 0 degrees,

b) 2 degrees, c) 4 degrees, d) 6 degrees.

31

(a) (b)

Figure 5.5: Airfoil pressure coefficient distribution for different AoA: a) 8 degrees,

b) 10 degrees.

(a) (b)

Figure 5.6: Airfoil pressure coefficient distribution for different AoA: a) -2 degrees,

b) -4 degrees.

32 S825 airfoil

Figure 5.7: AoA () vs. lift coefficient (Cl) for theoretical, experimental and

XFlow results.

REFERENCES 33

References

[1] D. Somers. Design and experimental results for the s825 airfoil. Technical report,

National Renewable Energy Laboratory, 2005.

[2] R. J. McGhee, W. D. Beasley, and J. M. Foster. Recent modifications and

calibration of the langley low-turbulence pressure tunnel. Technical report,

NASA TP-2328, 1984.

34 REFERENCES

6Vortex cell

Trapping vortices is a technique that prevents vortex shedding in flows past bluff

bodies. Vortices forming near bluff bodies tend to be shed downstream but, if the

vortex is kept near the body at all times, it is called trapped.

This validation case compares XFlow results with experimental data for the flowinside a vortex trapping cavity (vortex cell). The geometry consists in a rectangular

channel of section 520 52 mm with a spherical vortex cell of 45 mm depth locatedat mid-length as shown in Figure 6.1. The boundary condition at the inlet is set

to a constant fluid velocity of UL = 36 m/s and the gauge pressure to 0 Pa at the

outlet. The fluid has been initialized to UL in the whole domain except inside the

vortex cell where it is 0 m/s in order to reach quicker the pseudo-steady state.

Figure 6.1: Vortex cell geometry.

The experimental data from [1, 2] provide the normalized X-component of the

velocity measured along a vertical line going from the bottom of the sphere up to

the upper wall of the channel (see Figure 6.2). The vertical coordinate along the

line is normalized by the line length L = 52 mm.

The two-dimensional studies led by [3] based on steady RANS turbulence models

show how sensitive are the numerical results for the vortex cell flow depending

on the turbulence model and the choice of the numerical scheme. XFlow uses

36 Vortex cell

Figure 6.2: Line for measurement.

Large Eddy Simulation turbulence models, which are inherently three-dimensional.

Furthermore, although the vortex cell seems a two-dimensional flow, turbulence

effects (important near and inside the cavity) need a three-dimensional analysis to

be accurately modeled.

Unfortunately three-dimensional analyses may involve a large number of

elements and long simulation times. The refinement algorithms available in

XFlow (near the walls and adaptive wake) allow to minimize the number of elements,but tend to introduce numerical dissipation when passing from one element size to

another and has been found to be inaccurate especially in the boundary layer that

detaches from the leading edge of the vortex cell.

The following results were obtained using a uniform resolution of 1 mm in the

whole domain and a time step of 106 s. This resolution leads to a total of 1.1million elements. The total simulation time solved is 0.9 s at a frequency of 500

Hz. Averaged results are required in order to analyze the pseudo-steady state of the

solution.

Figure 6.3 shows that XFlow 3D results are globally in good agreement withthe experimental data. The areas of less accuracy are at y/L around -0.7 and the

peak around 0.1. Nevertheless XFlow is able to predict the experimental velocityprofile at the cavity entry (0.4 < y/L < 0) better than the RANS calculations andcorrectly predicts the vortex speed at y/L = 0.8.

Finally, Figure 6.4 shows the averaged velocity field computed by XFlow. It ispossible to observe the creation of vortices at the leading edge of the cavity, what is

not evident in steady calculations.

37

Figure 6.3: Comparison of XFlow 3D results with experimental data and RANSresults from [3].

Figure 6.4: XFlow averaged velocity field.

38 REFERENCES

References

[1] P. A. Baranov, S. V. Guvernyuk, M. A. Zubin, and S. A. Isaev. Numerical and

physical modeling of the circulation in a vortex cell in the wall of a rectilinear

channel. Fluid Dynamics, 35:663673, 2000.

[2] S.A. Isaev, S.V. Guvernyuk, M.A. Zubin, and Y.S. Prigorodov. Numerical and

physical modeling of a low-velocity air flow in a channel with a circular vortex

cell. Journal of Engineering Physics and Thermophysics, 73:337344, 2000.

[3] R. Donelli, P. Iannelli, S. Chernyshenko, A. Iollo, and L. Zannetti. Flow models

for a vortex cell. AIAA Journal, 47:451467, 2009.

7Automotive aerodynamics

In automotive aerodynamics it is usual to use reference geometries to validate CFD

codes [1]. This section uses the well known ASMO model, which comprises a square-

back rear, smooth surfaces, boat tailing, underbody diffuser and no pressure induced

boundary layer separation. The geometry does not have a well defined separation

line and is characterized by a low drag shape. For this model, experimental data

from Daimler Benz and Volvo model scale wind tunnel are available.

Mesh is one of the major issues in classic CFD approaches for aerodynamical

problems. Mesh quality may be as important as mesh resolution when high accuracy

in the calculations is aimed for. The particle-based approach of XFlow avoids thecostly generation of a good mesh.

For this validation the 1/5 wind tunnel test model has been adopted. Vehicles

length, width and height are 0.81 m, 0.29 m and 0.27 m respectively, while wind

tunnel dimensions are 9 x 1.5 x 3 m (see Figure 7.1). This corresponds to a blockage

ratio of 1.38%. The wind tunnel domain type available in XFlow is used, with aninlet uniform velocity of 50 m/s. The fluid properties are density = 1 kg m3 anddynamic viscosity = 1.5 105 Pa s.

Figure 7.1: ASMO body and wind tunnel geometries.

40 Automotive aerodynamics

The Reynolds number for this case is 2.7 106, taking the length of the vehicleas reference. In this problem it is essential to resolve the turbulent wake properly.

Particle resolution in the far field is 0.1 m, while in the wake and on the model

surface scales up to 2.5 mm are resolved. Dynamic wake refinement is applied, so

that the specified particle resolution is automatically adopted in regions with high

turbulence, while less turbulent regions are treated with fewer particles. In addition,

XFlow uses a Large Eddy Simulation (LES) approach for modeling the turbulence,in particular the Wall-Adapting Local Eddy-viscosity (WALE) model (see Section

1).

The turbulent wake structure can be observed in Figures 7.2 and 7.3, together

with the instantaneous pressure field and the skin friction distribution in Figure 7.4.

Figure 7.2: Snapshot of isosurface of vorticity.

41

(a)

(b)

(c)

Figure 7.3: Instantaneous velocity field: (a) on the vehicle surface, (b) in the

Y = 0.07 m plane, and (c) in the symmetry plane.

42 Automotive aerodynamics

Figure 7.4: Instantaneous pressure field in the symmetry plane and skin friction

distribution.

Figures 7.5 to 7.8 show validation results of surface pressure measurements.

XFlow results are compared with experimental data obtained from Volvo andDaimler Benz. Data are available in the symmetry plane and are shown for roof,

underbody, front and base region of the vehicle. It can be seen that the comparison

with the measurements is good, although some deviations can be observed especially

in the base pressure, which is slightly underpredicted. However the proper level of

the base pressure is not known exactly, as there is a large difference between both

experiments.

Figure 7.5: Front pressure distribution along the symmetry plane.

43

Figure 7.6: Roof pressure distribution along the symmetry plane.

Figure 7.7: Base pressure distribution along the symmetry plane.

Figure 7.8: Underbody pressure distribution along the symmetry plane.

44 REFERENCES

Typically, drag stabilizes in a characteristic time of the order of the flow traveling

the vehicle length. In 0.1 seconds, the flow has traveled more than six times the

whole body. The time averaged drag between 0.05 and 0.1 seconds of physical

simulation is Cd = 0.151 (see Figure 7.9), in good agreement with the values

measured in the experiments shown in Table 7.1. [2] showed that using transient

CFD simulations, surface pressure values can be computed fairly accurate. The

overall drag coefficient however, is not predicted satisfactorily. RANS turbulence

models tend to overestimate the drag. LES transient calculations are becoming an

integral part of the aerodynamic development process and affordable using XFlow.

Figure 7.9: Overall drag history.

XFlow 0.151Experiments Volvo 0.158

Experiments Daimler Benz 0.153

Table 7.1: Drag values for ASMO model.

References

[1] G. Le Good and K. Garry. On the use of reference models in automotive

aerodynamics. SAE paper, 2004-01-1308.

[2] S. Perzon and L. Davidson. On transient modeling of the flow around vehicles

using the reynolds equations. In ACFD 2000 Beijing, pages 720727, 2000.

8Multi-phase flows

When heavy fluid lies above lighter, the equilibrium in unstable and a small

perturbation of the interface from the horizontal will grow with time, producing the

phenomenon known as Rayleigh-Taylor instability [1]. This instability is a prototype

problem for computational studies of multi-phase flows.

The problem consists of two layers of fluid initially at rest in the rectangular

domain = (d/2, d/2) (2d, 2d), see Figure 8.1. The flow is characterized bythe density difference between the two fluids and their effective viscosity.

Figure 8.1: Initial configuration and physical properties of the fluids.

The density difference is represented by the Atwood number At = (A B)/(A + B). The Reynolds number is defined as Re = A d

3/2 g1/2/, where

d is the reference length, g the gravity acceleration and the dynamic viscosity of

the fluids (assumed uniform).

The growth and evolution of RayleighTaylor instability has been investigated

among others by Tryggvason [2] for inviscid incompressible flows, and by Guermond

& Quartapelle [3] and Ding et al. [4] for viscous flows. None of these studies has

taken into account surface tension.

We compare the XFlow results with those of [3] and [4] at At = 0.5 and Re =3000. The initial position of the perturbed interface is y(x) = 0.1 d cos(2pix/d).

46 Multi-phase flows

Figure 8.2: Vertical position of spike and bubble vs. time. Comparison of

XFlow solution with reference ones.

Computations are carried out on a 200800 grid and the time step is automaticallyset to 0.000144 s. Free-slip condition is enforced at all walls. The tracking of the

interface is done using the marker-and-cell method.

Results on the vertical position of the tip of the falling and rising fluid (spike and

bubble, respectively) are shown in Figure 8.2. XFlow solution is in good agreementwith the reference results [3, 4].

The evolution of the instability is shown in Figure 8.3 at dimensionless times

t =0, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, where t = tg At. Around t = 1.5 the heavy

fluid begins to roll up into two counter-rotating vortices (see also Figure 8.4). Later,

around t = 2, these two vortices become unstable and a pair of secondary vortices

appear at the tails of the roll-ups. The roll-ups and vortices in the heavy fluid

spike are due to the KelvinHelmholtz instability. The shapes of the fluid interface

obtained with XFlow compare well with those of the reference results [3, 4].

47

t = 0 t = 1 t = 1.25 t = 1.5

t = 1.75 t = 2 t = 2.25 t = 2.5

Figure 8.3: Rayleigh-Taylor instability evolution.

48 REFERENCES

Figure 8.4: Velocity field at t = 1.5 s.

References

[1] D.H. Sharp. An overview of Rayleigh-Taylor instability. Physica D, 12:318,

1984.

[2] G. Tryggvason. Numerical simulations of the Rayleigh-Taylor instability. Journal

of Computational Physics, 75:253282, 1988.

[3] J.-L. Guermond and L. Quartapelle. A projection FEM for variable density

incompressible flows. Journal of Computational Physics, 165:167188, 2000.

[4] H. Ding, P. Spelt, and C. Shu. Diffuse interface model for incompressible

two-phase flows with large density ratios. Journal of Computational Physics,

226:20782095, 2007.

TheoryLid-driven cavity flowNatural convection in a cavityNACA-0012 airfoil at Re = 500S825 airfoilVortex cellAutomotive aerodynamicsMulti-phase flows