turbulence modeling in an immersed-boundary rans method

Center for Turbulence ResearchAnnual Research Briefs 2002

415

Turbulence modeling in an immersed-boundaryRANS method

By Georgi Kalitzin AND Gianluca Iaccarino

1. Motivation and Background

Virtually all the applications of the Immersed Boundary (IB) technique have beenin the low-Reynolds number regime (up to 104) either as Direct Numerical Simulations(DNS) or Large Eddy Simulations (LES). For those applications simple off-boundaryconditions (usually based on linear interpolations) are used for the velocity fields toaccount for the effect of the immersed boundary on the flow field: see Fadlun et al.(2000) and Verzicco et al. (2000). The application of IB to industrially relevant turbulentflows at high Reynolds numbers requires its adaptation in the framework of the Reynolds-Averaged Navier-Stokes equations.

When IB is used in conjunction with RANS additional care must be devoted to theapplication of suitable turbulence models. In particular, classical approaches (e.g. modelsof k–ε type) are based on differential equations for turbulent scalars that characterizea velocity and a length scale which, in turn, define the eddy-viscosity. Those quantitiestypically exhibit large gradients and local extrema in the near vicinity of solid walls;for example the turbulent kinetic energy has a peak in the logarithmic layer and decaysquadratically towards solid walls. It is evident that the straightforward application ofan off-wall linear interpolation would introduce errors in the representation of such aquantity; on the other hand, even at high Reynolds numbers, provided that the gridresolution is sufficient, a linear interpolation for the velocity would still be consistentwith the linear behavior of the velocity in the viscous sublayer.

Previous investigations by Majumdar et al. (2001) of more sophisticated off-wall bound-ary conditions, mainly for flows at low Reynolds number, did not demonstrate a conclu-sive advantage over the simpler linear interpolation. Furthermore, a linear interpolationscheme can easily be recast in a fully-implicit form, thus introducing no time step limi-tation or additional stiffness for steady-state problems. The implicit treatment of moresophisticated interpolation schemes (quadratic, inverse-distance based, etc.) introducenon-linearities that may have an impact on the robustness and overall stability of thecomputational procedure.

In this paper we focus on the development of IB conditions for turbulence modelsand the initial validation of an IB RANS solver. The application consists of a turbineblade passage for which detailed DNS results are available (see Kalitzin et al. (2002)).The paper compares the IB technique with a standard body-fitted method to isolate theeffect of the off-wall boundary conditions on the results. A three-dimensional test case ofa turbine blade with a 10% tip gap is included to demonstrate the potentials of the IBtechnique for more complicated situations.

A major concern in applying the IB technique to high-Reynolds number flow is theresolution of thin boundary layers on curvilinear bodies using Cartesian grids. This res-olution issue will be addressed in future work with a grid-refinement technique. Thepresent paper considers coarse-grid solutions to demonstrate that smooth surface distri-

416 Kalitzin & Iaccarino

butions of pressure and friction can be obtained with the present IB technique. To modelturbulence, we first considered the model of Spalart and Allmaras (1992) and the k-gmodels, discussed by Kalitzin (1997) because of their simple wall boundary conditions.The k-ω model of Wilcox (1993) was implemented at a later stage. The test case con-sidered is not sensitive to the turbulence model, especially for the grid used. Thus, theobjective of considering several turbulence models is to show that these models can beused in conjunction with the IB technique.

2. Flow solver and turbulence models

The steady-state incompressible Reynolds-averaged Navier-Stokes equations are solvedusing a second-order, cell-centered finite-volume scheme on Cartesian non-uniform grids.The momentum equation is solved sequentially for each component of an intermediate ve-locity. The divergence of this velocity field provides the source term of a Poisson equationfor a pressure correction (SIMPLE procedure, Vandoormaal et al. (1984)) which enforcesa solenoidal velocity field. The eddy viscosity is obtained from an one-equation or two-equation turbulence model which is solved separately from the mean flow. The equationsare linearized and solved in a fully-implicit fashion, resulting in a linear algebraic systemof the form:

anpφn+1p +

∑

l=w,e,s,n,b,t

anl φn+1l = Sn

p (2.1)

where φ is either one of the velocity components, the pressure or a turbulence variable.While the usual boundary conditions (inflow, outflow, periodicity and solid walls) areapplied at the boundary of the computational domain, the curvilinear solid body is placedentirely or partly within the computational domain and immersed using the IB proceduredescribed in section 2.1. The system is then solved using a LU-type decomposition or aKrylov iterative solver; the computer program is based on Ferziger and Peric (2002).

2.1. Immersed Boundary Approach

The IB-methodology is based on the work of Fadlun et al. (2000) and Verzicco et al.(2000). It has been suitably modified to work in the current RANS environment. In apreliminary step, a given geometry which is described with a closed polygon in 2D andan STL file in 3D is overlaid on a Cartesian non-uniform mesh. The computational cellsin the fluid and in the solid body are tagged as internal and external cells, respectively.The cells containing both fluid and solid body are tagged as interface cells (figure 1).In this phase, care must be taken to ensure that the layer of interface cells completelysurrounds the immersed surface. Once all the cells are tagged, the distance from the wallis computed if needed for later use by the turbulence models.

The equations that is solved for the variables φ in the interface cells is:

αnb φ

n+1p +

∑

l=w,e,s,n,b,t

αnl φ

n+1l = 0 (2.2)

where the α’s are the interpolation weights. These weights are computed from the basecells (i.e. the internal cells) and the target cells (i.e. the internal cells surrounding eachbase cell). The base and target notation is commonly used for Chimera grids. The weightsare computed once, at the start of the calculation, and stored.

Equation (2.2) represents the discrete form of the direct-forcing approach proposed byFadlun et al. (2000). The interpolation is applied to the velocity components and to the

Turbulence models for immersed boundaries 417

SOLIDFLUID

INTERFACE

Figure 1. Result of the cell-tagging procedure for the Immersed Boundary Technique

turbulent scalars. The pressure equation is not modified and no boundary conditions areenforced at the immersed boundary. This has proven to be effective in producing smoothpressure distributions, even for thin airfoils such as the one presented in the Resultssection. Alternative treatments of the pressure have been attempted but further analysisis required to identify improvements to the current approach.

2.2. Spalart-Allmaras turbulence model

The model of Spalart and Allmaras (1992) exhibits good convergence properties and hasa remarkably accurate response to pressure gradient. It consists of one transport equationfor a modified eddy-viscosity, ν:

~u · grad ν =1

cb3

[

div ((ν + ν) grad ν) + cb2( grad ν · grad ν)]

+Q (2.3)

in which the source term Q is:

Q = cb1(1− ft2)Sν + (cb1κ2ft2 − cw1fw)

(

ν

d

)2

(2.4)

The eddy viscosity is:

µt = ρνfv1 (2.5)

The model damping functions, auxiliary relations and trip term are defined as:

fv1 =χ3

χ3 + cv13, fv2 = 1−

χ

1 + χfv1, χ =

ν

ν(2.6)

fw = g

[

1 + cw36

g6 + cw36

]

16

, g = r + cw2(r6 − r), r =

ν

Sκ2d2(2.7)

S = S +ν

κ2d2fv2, S =

√

2ΩijΩij , ft2 = ct3exp(−ct4χ2) (2.8)

in which d is the distance to the nearest wall, κ the von Karman constant and S thevorticity expressed in terms of the rotation tensor Ωij =

12( ∂ui

∂xj− ∂uj

∂xi). Finally, the model


closure coefficients are:

cb1 = 0.1355, cb2 = 0.622, cb3 = 2/3, cv1 = 7.1 (2.9)

cw1 =cb1κ2

+1 + cb2cb3

, cw2 = 0.3, cw3 = 2, ct3 = 1.2, ct4 = 0.5 (2.10)

The wall boundary condition is:

νt = 0. (2.11)

2.3. Wilcox k-ω model

The k-ω model of Wilcox (1993) is implemented in its original form. The 1998 version ofthe model (see 2nd ed.) contains certain correction terms which will be considered at alater stage. The transport equations for the turbulent kinetic energy k and the specificdissipation rate ω ∝ ε/k are:

∂(ρuik)

∂xi=

∂

∂xi

[

(µ+µtσω

)∂ω

∂xi

]

+ Pk − ρkω, (2.12)

∂(ρuiω)

∂xi=

∂

∂xi

[

(µ+µtσω

)∂ω

∂xi

]

+ αω

kPk − βρω2. (2.13)

The eddy viscosity is:

µt = ρk

ω(2.14)

The model coefficients are:

σω = 2.0, β∗ = 0.09, α = 5/9, β = 0.075

The wall boundary condition for k is:

k = 0. (2.15)

The specific dissipation rate tends asymptotically to infinity at the wall as ∼ 1/y2.Wilcox (1993) describes the numerical errors associated with the numerical integrationof ω up to the wall. Menter (1993) suggested to use the following boundary condition:

ω =60ν

β1y21

(2.16)

where y1 is the distance from the wall to the center of the first cell above the wall.

2.4. Near-wall behavior

This section focuses on the near-wall behavior of the turbulence models presented above,in particular on the treatment of the wall boundary condition in the framework of theimmersed boundary.

The modified eddy viscosity ν is zero at the wall and it has the property of varying ina nearly linear fashion from the wall throughout the law-of-the-wall layer thus decreasingthe sensitivity to grid resolution and wall clustering (Durbin and Pettersson Reif (2001)).The linearity of ν makes it straightforward to implement the Spalart-Allmaras model withthe immersed boundaries, using in the interface cells the same linear interpolation stencilused for the velocity components. Inside the body the modified eddy viscosity ν is set tozero.

The turbulent kinetic energy k and the specific dissipation rate ω in the k-ω model


vary in the viscous sublayer as ∼ y3.23 and ∼ 1/y2, respectively. At the current stage thelinear interpolation method is applied directly to k. This may, however, not be sufficientlyaccurate, and further investigation is required. ω is set in the interface cells to the valuecomputed with (2.16). Some of the cell centers may lie exactly on the wall, which wouldlead to undefined values in those cells. The minimal distance y in equation (2.16) islimited by an arbitrary small constant which could be related to the Reynolds number.This basically cuts off the ω value at the wall. In the cells inside the body, ω is set to thevalue defined by this arbitrary constant.

In an alternative approach, we investigate the transformation of the ω-equation to adifferent independent variable that is more suitable for use with immersed boundaries.Kalitzin (1997) suggests recasting the ω-equation in terms of the variable g = 1/

√β∗ω:

then

∂(ρuig)

∂xi=

∂

∂xi

[

(µ+µtσg

)∂g

∂xi

]

− αg

2kPk +

βρ

2gβ∗−(

µ+µtσg

)

3

g

∂g

∂xi

∂g

∂xi. (2.17)

In the viscous sublayer g varies linearly with the distance to the wall. At the wall g iszero, and the interpolation stencil used for the mean velocities can be applied in theinterface cells.

In the continuous space the above transformation leads to identical eddy viscosityvalues, whereas when the discretized equations are solved this is not the case. Numericaldissipation is in particular influential in regions of large gradients and poor resolution.g increases away from the wall as shown for the flat plate in figure 2b. However, theouter part of the boundary layer is usually less resolved than the near-wall region. Thediffusion of g from the free stream into the boundary layer is in contrast to ω diffusingfrom the boundary layer to the freestream. To investigate the effect of this, the k-ω andk-g models have been combined into a two-layer formulation (hybrid model) where thelatter is used only in the vicinity of solid walls.

2.5. Implementation of the hybrid model

At the start of the computation, each cell is marked with an integer array γ whichidentifies the equations to be solved. In the near-wall region, where the g equations is tobe solved, γ is set to 1, elsewhere it is set to 0. The distance to the wall is used to definethe near-wall region. The implicitly-discretized turbulence equations can be written as:

anpφn+1p +

∑

l=w,e,s,n,b,t

anl φn+1l = RHSn

p (2.18)

where the variable φ is g or ω depending on the array γ. The coefficients anl and theright hand side RHSn

p , which depend on the variable φn from the previous iteration n,are computed according to the g- or ω-equation, respectively. Thus, the source term Sn

p

which is included in RHSnp is computed as:

Snp = (−α

φnp2kPk +

βρ

2φnpβ∗)γp + (α

φnpkPk − βρ(φnp )

2)(1− γp) (2.19)

The convective and diffusion terms contain the dependent variable from two adjacentcells. The dependent variable in the cell l adjacent to cell p is computed from:

φnl = (Γgg + Γωω)φnl + Γgω(

1√

β∗φnl) + Γωg(

1

β∗(φnl )2) (2.20)


0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

10-2

10-1

100

101

102

103

Figure 2. (a) Turbulent kinetic energy and (b) length scale variable for flat plate ( k-ω,k-g, hybrid (every cell centered value plotted))

with Γgg = γpγl; Γgω = γp(1− γl); Γωg = (1− γp)γl; Γωω = (1− γp)(1− γl); Only oneof these indicators is non-zero and equal to one. In the case of an interface cell, Γgg and

Γωω are zero and φnl is converted into φnl .Implicitness at the interface is achieved by modifying the coefficients anl with anl =

anl φnl /φ

nl ; only at the interface anl is not equal anl . At the interface the coefficient anl is

anl = anl ωn/gn and anl = anl g

n/ωn for Γωg = 1 and Γgω = 1, respectively. If the interfaceis close to the wall, the algebraic system becomes unstable as ω tends to infinity and gto zero. Therefore ω is treated explicitly at the interface:

anpφn+1p +

∑

l=w,e,s,n,b,t

anl φn+1l (1− Γωg) = Sn

p − anl φnl Γωg (2.21)

Figures 2 and 3 show the results of a calculation of flow over a flat plate with the k-ω, k-gand the hybrid model. The turbulent kinetic energy in figure 2a and the velocity profilein 3a are almost identical although the shear stress at the wall is slightly smaller forthe k-ω model, resulting in a larger U+ in the defect layer. Figure 2b demonstrates thesecond scalar of the hybrid model following g near the wall and switching at a prescribedlocation to ω. The convergence is very similar for all three models as shown in figure 3b.

2.6. Wall models

High near-wall grid resolution is usually required to perform RANS simulation using thediscussed turbulence models; the accepted rules in meshing turbulent boundary layersare: (i) the y+ of the first cell center should not be greater than 1 and (ii) about 20cells should be located inside the boundary layers. A careful selection of turbulencemodel might reduce those restrictions. However, it is reasonable to expect that for highReynolds numbers Cartesian mesh will not fully resolve the boundary layers. Computingskin friction in the usual way on such grids may result in values incorrect by several orderof magnitude. Therefore, the velocity fields are post-processed using a wall model of theform:

u/uτ =1

κln(1 + κy+) + c(1− e−y+/d+

−y+

d+e−by+

) (2.22)


0 1 2 3 4 50

5

10

15

20

25

30

500 1000 1500 2000

10-5

10-4

10-3

10-2

10-1

"!$# &%#

Figure 3. (a) Velocity and (b) convergence of L2 norm of velocity residual for flat plate( k-ω, k-g, hybrid (selected values are plotted))

b =1

2(d+κ

c+

1

d+) (2.23)

c =1

κln(

E

κ) (2.24)

where κ, E and d+ assume the values of 0.4187, 9.793 and 11 respectively. This expression,due to Reichardt (1951), reproduces the logarithmic layer and the linear sublayer with acontinuous switch in the buffer region.

The formulation (2.22) is used to compute uτ using the (tangential) velocity u at acertain distance (δ) from the immersed surface (this de facto corresponds to the creationof a body-fitted grid line). The friction velocity uτ and the formulation (2.22) are used toestimate the tangential velocity in the interface cell (at a distance δi < δ). Note that, ifthe interface cell is very close to the surface, the above approach returns a linear velocityinterpolation consistent with the standard immersed boundary technique. The choice ofthe distance δ is obviously critical for the application of this procedure. Preliminaryresults have been obtained using δ equal to the largest distance from the wall in all theinterface cells.

3. Results and discussion

The IB/RANS solver is applied to turbulent flow inside the T106 turbine blade passage.This test case has been studied extensively using DNS for various inflow conditions(Kalitzin et al. (2002)). The focus of the present study is to investigate the feasibilityof the IB approach for this flow, and its accuracy in representing surface quantities likepressure and skin friction. The representation of the latter using the IB-approach hasnot been discussed previously in the literature. Comparison are made with body-fittedresults obtained with FLUENT and the DNS data for a turbulence-free inlet.


Figure 4. (a) Computational domain and (b) computational grid 186× 110

(a) (b) (c)

Figure 5. Comparison of the velocity magnitude in the blade passage. (a) DNS, (b) ImmersedBoundary / SA model and (c) Body-fitted / SA model.

(a) (b) (c)

Figure 6. Comparison of the pressure coefficient in the blade passage; (a) DNS, (b) ImmersedBoundary / SA model and (c) Body-fitted / SA model.


(a) (b) (c)

Figure 7. Turbulent kinetic energy in the blade passage; (a) DNS, (b) Immersed Boundary /k-ω model, (c) Immersed Boundary / k-g model.

x/c

Ski

nF

rict

ion

coef

fici

ent

0.25 0.5 0.75 10

0.01

0.02

0.03

x/c

Pre

ssur

eC

oeff

icie

nt

0 0.25 0.5 0.75 1-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Figure 8. Surface pressure distribution (left), skin friction distribution (right). DNS,Immersed Boundary / SA model - Body-fitted / SA model.

x/c

Ski

nF

rict

ion

coef

fici

ent

0.25 0.5 0.75 10

0.01

0.02

0.03

x/c

Pre

ssur

eC

oeff

icie

nt

0 0.25 0.5 0.75 1-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Figure 9. Surface pressure distribution (left), skin friction distribution (right). DNS,k-ω model, k-g model, hybrid model. RANS is with Immersed Boundaries.


Iteration

L2(X

-mom

entu

mR

esid

ual)

1000 2000 300010-5

10-4

10-3

10-2

10-1

100

101

102

Iteration

L2(V

eloc

ityD

iver

genc

e)

1000 2000 300010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Figure 10. Convergence history for IB method. Velocity (left), Velocity divergence (right).k-ω, k-g, hybrid, SA

Figure 11. Velocity distribution in the wake of the three-dimensional blade with 10% tip gapshowing traces of a vortex and trailing edge separation, IB method.

(a) (b)

Figure 12. Friction lines on the surface of the three-dimensional blade: (a) suction side (b)pressure side, IB method.


A sketch of the computational setup and the grid is shown in figure 4. The IB andbody-fitted calculations are carried out on a Cartesian mesh of (186 × 112) cells anda comparable unstructured mesh of about 20,000 elements, respectively. The Reynoldsnumber is Re = 148, 000. The T106 case is extremely challenging for the current IB-approach because of the large pressure difference between the suction and pressure sidesof the blade. The airfoil is extremely thin and only a few cells (ranging from 4 to 15) areinside it.

Contours of the velocity magnitude are shown in figure 5. The overall qualitative agree-ment among the DNS, the IB and the body-fitted RANS calculations is satisfactory, eventhough the latter two methods predict a thicker boundary layer towards the trailing edgeon the upper side of the airfoil. This is caused by the under-resolution of the boundarylayers. However, the agreement between the IB and body-fitted results is good. Figure6 demonstrates that the IB technique is able to predict accurate pressure distributionsin the passage. The region near the leading edge is particularly challenging for the IBmethod due to the sharp pressure gradient and strong surface curvature. The comparisonof the turbulent kinetic energy contours to the DNS, shown in figure 7, is satisfactoryalthough the DNS predicts a higher level of turbulent kinetic energy on the upper sideof the airfoil. Transition is missed by the RANS simulations.

A quantitative assessment of the IB method is shown in figure 8. The wall pressuredistribution agrees well among the DNS, the IB and the body fitted RANS calculations,the latter two using the Spalart-Allmaras model. Differences among the skin frictiondistributions are more substantial. The RANS simulations do not show the sharp increasein the final 10% portion of the blade, which is caused by transition. Interestingly, theIB calculation shows friction levels that are overall closer to the DNS, especially on thelower side of the airfoil. This might be due to differences in the implementation detailsof the Spalart-Allmaras model or to differences in the post-processing.

Figure 9 shows wall-pressure and skin-friction distributions for the k-ω variants. Thehybrid model has been run with the g-equation in the near-wall region wrapped aroundthe blade. This layer is 0.1Cx thick, where Cx is the axial chord of the blade. Only slightdiscrepancies are observable in the skin friction, on the upper wall near the trailing edge.The pressure is slightly better predicted than using the Spalart-Allmaras model.

The convergence of the IB method is plotted in Figure 10. The steady state is achievedin about 2,000 iterations. The IB interpolation has no negative impact on convergenceto the steady state.

The flow around a three-dimensional version of the blade geometry, embedded betweenan endwall and a tip gap of 10% of the blade chord, is shown in Figure 11. This flowwas computed to demonstrate the potential of the IB method for complex flows. The tipgap adds substantial complexity to the flow with the formation of a strong tip vortex.The skin-friction lines on the blade surface, shown in Figure 12, reveal that the three-dimensionality of the flow starts at 50% span. The flow separates at the trailing edgeand there is a strong tip leakage flow from the pressure side to the suction side.

The main objective of future work is the evaluation of the accuracy of the present IBapproach. A local grid refinement will be implemented to resolve of the boundary layers.Future work will also include an investigation of coarse grid behavior of turbulence modelsin conjunction with the linear interpolation procedure of the IB method.


REFERENCES

Durbin, P. A. & Pettersson Reif, B. A. 2001 Statistical Theory and Modeling forTurbulent Flows. Wiley, New York.

Fadlun, E. A., Verzicco, R., Orlandi, P. & Mohd–Yusof, J. 2000 Combinedimmersed-boundary/finite-difference methods for three-dimensional complex flowsimulations. J. Comp. Phys. 161, 35–60.

Ferziger, J. H. & Peric, M. 2002 Computational Methods for Fluid Dynamics. 3rded. Springer, Berlin.

Kalitzin, G. 1997 Validation and development of two-equation turbulence models. InValidation of CFD codes and assessment of turbulence models.(W. Haase et al., eds.).Notes on Numerical Fluid Mechanics Series Vol. 57. Vieweg, 1997.

Kalitzin, G., Wu, X. & Durbin, P. A. 2002 DNS of fully turbulent flow in a LPTpassage. Presented at 5th Int. Sympo. on Engg Turbulence Modelling and Meas.(ETMM5), Mallorca, Spain.

Majumdar, S., Iaccarino, G. & Durbin P. A. 2001 RANS solver with adaptivestructured boundary non-conforming grids. Annual Research Briefs, Center for Tur-bulence Research, NASA Ames/Stanford Univ., 353–366.

Menter, F. R. 1993 Zonal two equation k − ω turbulence model predictions. AIAApaper 93-2906.

Reichardt, H. 1951 Vollstandige Darstellung der turbulenten Geschwindigkeits-verteilung in glatten Leitungen. Z. Math. Mech. 31, 11.

Spalart, P. R. & Allmaras, S. R. 1992 A one-equation turbulence model for aero-dynamic flows. AIAA paper 92-439.

Vandoormaal, J. P., & Raithby, G. D. 1984 Enhancements of the SIMPLE methodfor predicting incompressible fluid flows. Num. Heat Transf 7, 147-163.

Verzicco, R., Mohd–Yusof, J., Orlandi, P. and Haworth, D. 2000 LES in com-plex geometries using boundary body forces. AIAA J. 38, 427–433.

Wilcox D.C. 1993 Turbulence modeling for CFD. 1st ed. DCW Industries, La Canada.

turbulence modeling in an immersed-boundary rans method

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