CFD: Modified Robin-type wall functions for turbulence industries ∗

Adi Susila G.†, Utyuzhnikov S.V.‡

University of Manchester, Manchester M60 1QD, UK

Dec , 2010

Abstract

CFD is the systematic analysis of computer based simulation to determine dynamic fluid flow, heattransfer and other fluid properties. Airbus researchers have found that commercial airliners commonlyencounter physical problems with friction drag, 40% of which are caused by a turbulent boundary layer,which is a thin layer of air located just above the skin of a wing/airfoil and body of an aircraft. Draghabitually happens in various instances of fluid flow. It is sometime necessary; however, the disturbancecaused by this friction should be optimized for the use of industrial requirement. This has resulted inconstant challenge to find appropriate solutions to reduce and ultimately eliminate this effect altogether.

This challenge promotes the need for extra emphasize on the importance of further boundaries totreat the near the wall areas in fluid mechanics. Studies on wall functions (mathematical profile) forturbulence modelling has been carried out and improved. Channel flow test case has been tested forturbulent flow model. To accomplish this, the finite difference method along with the computationalcode-simulation was implemented. For a fully developed turbulent channel flow, Cabot Moin’s turbulentviscosity was used, characterized by Low & High Reynolds numbers, Ret = 395 up to 10950, respectively.

Keywords: CFD, Finite Difference Method, Robin type Wall Function, Turbulent Viscosity.

1 Introduction

Computational Fluid Dynamics (CFD) is one of major strides for turbulent fluid currently under inves-tigation. It is a very powerful technique encompassing a wide range of industrial and non-industrialareas of application. In the engineering area, for example, it covers aerodynamics of aircraft and vehi-cles, hydrodynamics of ships, power plant and turbo-machinery, electrical/electronic application, chemicalprocess, biomedical, external/internal environment building, marine and environment, including hydrol-ogy/oceanography, meteorology as shown in figure 1 and many other fluid flow field.

Turbulence occurs in various aspects provided there is a flow of energy distribution, such as in the turbulentlayer of an aircraft wing, combustion processes in jet streams, chemical reactions within gas or liquid mixed,etc [1]. Richard E. Klabune of cardiovascular physiology has pointed out that in the operation of humanbody, turbulence can be seen in both large and narrow (stenotic) arteries at branch points, in the arteriesdisease. Reduced arteries flow area due to the disease will leads unbalance flow pressure along the arteriescouses unstable blood pressure.

For years studies have been conducted on the friction drag by a boundary wall. Related approaches have alsobeen developed to suit each unique physical surface, while several analytical solutions have been generatedto eliminate such drag on the surface/skin. However, turbulence model problems still exist. The mainhitches are the flow problem near by the wall, i.e the thin viscous layer and the thin near-wall viscosityaffected by the sub-layer which is predominantly due to affected by molecular diffusion. Figures 23 areillustrated sample cases for pressure distribution (load coefficient).

CFD consists of pre-processor, solver and post-processor which are mostly the numerical algorithms. Theaccuracy of the solutions will depend on the design of number of cells in the grid to simulate flow problemssuch as velocity, pressure, temperature, etc. The numerical simulations used to review turbulent flow arethen listed in three ways through which the phenomenon is predicted. They are:

∗An preliminary study of CFD for Large Edy Simulation†(PhD program) - Univesity of Udayana, Bali, Indonesia,‡(Former supervisor) - Mechanical. Aerospace & Civil Engineering School, Univesity of Manchester, UK,

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Figure 1: Schematic fluid flow of the 3D model of aircraft wing, efflux smoke (chimney) and wind pressurearound the building.[10]

1. RANS (Reynolds-averaged Navier-Stokes): Under RANS, average or even small scales only aremodelled. This approach is averaged to all the unsteadiness of the non-linear Navier-Stokes equations.This gave rise to the Reynolds stress term within the equations which is the lowest in term of costand time consumed.

2. LES (Large Eddy Simulation) Under this approach the largest scale of motions is explicitly presentedwhile the small scales are modelled. This is regarded as the middle ground between the DNS andRANS in term of cost and time.

3. DNS (Direct Numerical Simulation) This approach employs techniques in which all scales of turbulentflow motions are computed from large contained energy (integral scales) to the dissipative scales(viscous or Kormogorof scales) [3].

Figure 2: Turbulent flow (LES) resulted pressure coefficient (Cp) on Cooling tower model and DynamicPressure distribution on sphere model using CFD (Fluent) [9]

The result of LES was compared with those turbulent models based on RANS and experimental work shownin figure 23

The problem near by the wall is still highlighted as the thin viscous layer close to the wall is cruciallyimportant, which often causes the turbulent layer located away from it to be significantly affected. Numericalstudies were revisited to investigate the pattern of this thin layer using wall function (mathematical profile).

Previous wall functions [12] were modified using Cabot Moins turbulent viscosity profile in order to find morerobust solutions close to the wall phenomenon. This kind of a logarithmic profile is needed to change thepiecewise profile to match with the benchmark profile, i.e Reichardt profile. The finite differential methodwas also used to simulate the mathematical problems along with the computational code as the way CFDworks. Channel flow cases were then carried out to check if the modified wall function solution is workingproperly or not.

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Figure 3: (Top) Plan view: pressure coefficients for y/d = hemisphere [8] (1966), (Left) Plan view ofpressure coefficient contour y/d=h/D=1/2 [9] & (Right) Mean pressure coefficient LES result on a sphere[9]

2 Problem Description

There are two broad strategies to resolve those problems, namely:

1. Employing smooth/fine numerical meshes where the viscous influences are involved,

2. Employing wall function by taking into account the overall resistance of the sub-layer (momentumand heat transfer).

The RANS is particularly chosen to avoid wasting time during the numerical process of turbulent model. Themethod assumes a solution under which the vicinity of the wall is replaced by some appropriate boundaryconditions.

Such wall functions are widely used in industrial applications already. The turbulence modelling used wasthe eddy/turbulent viscosity on the layer as the major study which tackles the problems described. In thisoccasion, turbulent modelling would be prioritized to modify the Robin-type wall function. This type ofboundary method has been modified properly by involving Cabot Moin profile turbulent viscosity to takeinto account the near-wall turbulence problem.

2.1 Wall Functions - (WF)

Professor Launder (1960); in Chieng and Launder (1980) undertook the first effort to incorporate dissipationin the viscous sub-layer into a treatment for wall-function. This earlier attempt refined the conventional ap-proach of wall function and has ever since become the basic procedure for average generation and dissipationrates of k over the near wall-cell.

The underlying turbulence energy is used as the developed procedure for averaged generation rate in whicha cell near wall extending to the height of the wall yn, at the following [7]:

P =1

yn

∫ yn

0

uvdU

dydy (1)

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In [6], the WF are analytically obtained with constant assumptions on all variables apart from diffusivity.This analysis has adopted integration of boundary-layer-type equations in the vicinity of a wall. The cellnearest to the wall can be rebuilt by the near-wall solution using analytical profiles of an effective viscosity.

2.2 Standard WF

In most cases, the WF are semi-empirical and have very limited application [6]. First wall-function is basedon the log-law profile for velocity [4],[1]. The main disadvantage of this wall-function is strong dependencetowards the mesh point closest to the wall where the wall-function is taking place. This has particularlycreated a problem if the first mesh point is located in the viscous sub-layer. In order to avoid this problem,the scalable wall approach and pressure gradient must be taken into account in Gotjans, H. paper in 1998,[2]. To satisfy the turbulent approach of WF, there is a set of WF by[2] in the inertial sub-layer as follow:

U = uτ

[1

Kln(uτyv

)+B

](2)

or written as

U+ =1

Kln(Ey+

)κ =

u2τ√β∗

; ω =uτ

(β∗)12 κy

; vτ = uτκy (3)

Figure 4: The law of the wall [1] & result of Robin Type WF combined with Cabot Moin turbulent viscosityprofile at Re=5950 in log law representing high Reynolds number

These WF are regarded as the standard method. In the turbulent boundary layer, the strongest velocitygradient is found near by the wall. Based on these wall functions for a turbulence model in which coarsemesh utilized relatively, it is impossible to resolve these near wall gradients[4].

2.3 Transfer of the wall boundary condition (Robin Type WF)

The previous study requires separate WF for both low and high Reynolds number. However, [11] suggestedthat the new numerical wall function method can be used in either the high or the low Reynolds numbermodel directly. This Robin-type boundary condition in a differential form was introduced which transmittedthe boundary implied, i.e. from the intermediate range of boundary to the near wall area.

The WF can be used as an analytical easy-to-implement form which does not necessarily need to be placedat the first point of nearest location. The method of conveying boundary condition is described in [11]: atransporting boundary can be either approximate (analytically) or exact (numerically) which is influencedby the source in governing equations.

Analytical WF are evolved by integrating boundary layer-type equations under some simplicity assumption.The model equation is written below [12]:

(µuy)y + ynuy = C or Lu ≡ (µuy)y = Rh (y) ; 0 ≤ y ≤ y∗ (4)

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Defined in a domain Ω = [0, 1], where;

µt = µlKy+[1− e

−yA

]2; A = 19.0 ; K = 0.41 ; R 1 ; n > 0 (5)

equation 4 described; the first term simulates the dissipative in the Navier-Stokes equations, the second termis a model of the contribution of convective term and the right hand side represents pressure gradient termor source in the transport equations. The corresponds to the effective viscosity coefficient. The coefficient

is rapidly changed from a relatively small value (laminar viscosity) to µt = µlKy+[1− e

−yA

]2(turbulent

viscosity, [5]). 1st Integration of equation 4:

µdu

dy= Ry + C1 : µ

du

dy−Ry = τw ; C1 = τw (6)

and index w of mean value y=0 (at wall). The 2nd Integration;

uy = u0 + τw

∫ y

0

dξ

µ+Rh

∫ y

0

ξ

µdξ (7)

Considering equation 6 & 7

uy = u0 +

(µ

du

dy−Rh

)∫ y

0

dξ

µ+Rh

∫ y

0

ξ

µdξ ; 0 < y

Block in parenthesis is an intermediate term (y∗).

u∗y = u0 + F1du

dy(y∗)− Rh

µ(y∗)F2 (8)

F1 =

∫ y∗

0

µ(y∗)

µ(y)dy ; F2 =

∫ y∗

0

µ(y∗)

µ(y)(y∗ − y) dy (9)

Implementation of Robin-Type condition from: 1. Finite Difference and 2. Finite Volume. If Rh = Rh(y),then;

u∗y = u0 + F1du

dy(y∗)−

(∫ y∗

0

Rh(y)

)F2

y∗µ(y∗)

F1 =

∫ y∗

0

µ(y∗)

µ(y∗)dy ; F2 =

∫ y∗

0

µ(y∗)

µ(y∗)

(1−

∫ y∗0Rhdy∫ y∗

0Rhdy

)dy (10)

µ =

µw if 0 ≤ y ≤ yµw + (µ∗ − µw) y−yvy∗−y if yv ≤ y ≤ y∗

(11)

Intermediate boundary condition at y = y∗

du

dy(y∗) =

(τw +

∫ y∗0Rhdy

)µ∗

(12)

For the use of the high-Reynolds-number RANS model, the intermediate Robin boundary conditions areintroduced. They represent an approximate transfer of the boundary conditions from the wall to the inter-mediate boundary usually situated outside the viscous sub-layer. The previous Robin-type wall functions[12] have been modified to take into account the near-wall turbulent viscosity profile more fittingly in viscouslayer as shown in figure 6.

The aim is to change the piecewise linear profile shown into an exponential profile described in the in figure5.

The turbulent viscosity implementation: [5]; µt = µlKy+[1− e

−yA

]2followed by solution:

µeff = µl + (µ∗ − µl)y[1− e

−CyA

]2y∗[1− e−Cy

A

]2 if 0 ≤ y∗ , [10] (13)

Effective /efficient viscosity µeff are combined viscosity between turbulent and the laminar (µeff = µ∗t + µl).For this study, trade off has also been introduced in which effective viscosity is defined as (µeff = µ∗t + 0.2125µl).

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Figure 5: Hypothetic figure of Piecewise linear profile & Exponential Cabot-Moin’s profile

2.4 Turbulent Boundary Layer

To resolve the gradient near the wall, a law should be specified that correlates the outer flow (the velocityat first grid point) and the shear stress at the wall, when the grid is not fine enough. This allows us to placethe first node at y+ ≈ 30− 200 .

The law of the wall assumes that, for attached flow, the logarithmic portion of the boundary layer behavesaccording to:

u+ ≡ u∗U

= f(y+)

(14)

The non-dimensional characteristic wall coordinate, y+, is defined by:

y+ =yu∗ρwµw

(15)

Where y is the dimensional distance between the wall and the first grid point of the wall, µ is the absoluteviscosity obtained from Sutherlands law, and ρ is the density. The wall friction velocity, µ∗ , thus definedas:

u∗ =

√τwρw

; u∗ = uτ (16)

The turbulent boundary layer near the wall is characterized by the following quantities in a two dimensionalflow where the x direction is the predominant flow direction, with flow velocity u, and the wall is facing they direction. Here,τw is the wall shear stress and the quantity uτ or u∗ is called the friction velocity.

3 Finite Difference Methods: Boundary Value Problem Solver

The understanding of the numerical solution algorithms is crucially important in solving the mathematicalconcept. There are three main deciding notions in determining the success of such algorithms concepts, i.econvergence, consistency & stability. For these three, confirmations are needed in terms of: the property ofnumerical methods to produce a clear approach to the exact solution, produced system of algebraic equationwithin which the developed numerical scheme is demonstrated as equivalent to the original equation andassociated damping with error as numerical method process.

3.1 Two-equation Model

This model describes the transport of two scalars of kinetic energy (k) and the dissipation ε where theReynolds tensor-stress computed by using the variable into the velocity gradient and an eddy viscosity. Thevalue of (k) and come directly from the differential transport equations for turbulence kinetic and turbulencedissipation rate. The typical transport equation solved for kinetic energy k typically taken as one-equationmodel:

Dk

Dt= Pk − ε+

∂

∂xj

[(µ+

µtσk

)∂k

∂xj

](17)

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The generating rate is given by Pk = −uivj dUidxj, while dissipation rate ε = k

32

lεand turbulent viscosity

µt = cµk12 lµ . Both length scales lµ and lε are prescribed as increasing in linear with distance from the

wall. In two-equation model a second variable for ε = k32

lεis solved as the following:

Dε

Dt= Cε1

ε

kPk − Cε2

ε2

kε+

∂

∂xj

[(µ+

µtσε

)∂ε

∂xj

](18)

cµ =−u′v′∂u∂y

ε

k2; σk = 1.0 ; σε = 1.3 ; cµ = 0.09 ; Cε1 = 1.44 ; Cε2 = 1.92 (19)

Governing equations correspond to the Reynolds averaged Navier-Stokes equations (RANS) closed by theHRk−ε (High Reynolds number) model. There are diffusion parallel to the wall, the momentum, enthalpyand kinetic energy transport equations that can be written in the Cartesian coordinate system (x,y); Thisis easy to confirm by the equation 21

(µuy)y = Rh(y) or∂

∂y

(µ∂u

∂y

)= Rh(y) (20)

∂

∂y

[(µl + µt)

∂U

∂y

]= ρU

∂U

∂x+ ρV

∂U

∂y+∂P

∂x;

∂

∂y

[(µl + µt)

∂V

∂y

]= ρU

∂V

∂x+ ρV

∂V

∂y+∂P

∂y

∂

∂y

[(µl + µt)

∂U

∂y

]= ρU

∂U

∂x+ ρV

∂U

∂y+∂P

∂x;

∂

∂y

[(µlPr

+µtPrt

)∂U

∂y

]= ρU

∂U

∂x+ ρV

∂U

∂y+∂P

∂x

∂

∂y

[(µl + µt)

∂U

∂y

]= ρU

∂T

∂x+ ρV

∂T

∂y;

∂

∂y

[(µl +

µtPrk

)∂k

∂y

]= ρU

∂k

∂x+ ρV

∂k

∂y− Pk + ρε (21)

µl and µt are the laminar and turbulent viscosities; Pr ,Prt and Prk are Prandtl number; U and V are velocitycomponents in the (x, y) coordinate system; ρ is density and P is the pressure; T is the temperature; k isthe turbulent kinetic energy; Pk is its production;ε is the dissipation of k.

Upon substitution of U, V, T or k instead of u in equation 21, we obtained Robin type WF for this function.Rh , from to be evaluated at y = y∗ by the right-hand side as follows [12]:

Rh = Rhu ≡ ρ(U∂U

∂x(y∗) + V

∂U

∂y(y∗)

)+∂P

∂x(y∗) ; Rh = Rhv ≡ ρ

(U∂V

∂x(y∗) + V

∂V

∂y(y∗)

)+∂P

∂x(y∗)

Rh = Rht ≡ ρ(U∂T

∂x(y∗) + V

∂T

∂y(y∗)

); Rh = Rhk ≡ ρ

(U∂k

∂x(y∗) + V

∂k

∂y(y∗)

)+ ρε− µt

(∂U

∂y

)2

(22)

By the Convective term evaluation, the following expression for RHS (right hand side) of Rh obtained:

(µl + µt)∂U

∂y=

[(µl + µ∗t )U(y∗) + F2Rhu]

F1+ (y − y∗)Rhu (23)

µt =

0 if 0 ≤ y ≤ yvµ∗t

y−yvy∗−yv if yv ≤ y ≤ y∗

(24)

ε(y) =

(k∗)

32

C1ydif y < yd

µ∗ty−yvy∗−yv if else

(25)

The equation 23, 24 is replaced by introducing equation 13 as the result of replacement the piece-wiselinear with the Cabot Moin’s turbulent viscosity (exponential profile). This approach is effort to improveprediction turbulent near the wall by using WF combined with the Cabot Moin’s viscosity profile.

3.2 Turbulent Channel Flow

As in the case of the channel flow, the uniform pressure or velocity profile of fluid would come into thechannel, figure 6. As the fluid flow enters the channel, the wall will retract the velocity and development ofa boundary layer will occur. The velocity will turn to zero close/near to the wall due to the appearance ofviscosity resistance along the wall.

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Figure 6: Test case of Channel Flow with Wall Function implementation (reproduced figure),[2] & Viscouslayer grid illustration near wall

4 Result and Conclusion

The near wall regions of the flow are resolved by coarse mesh and fine mesh spacing with varied timescale, using numerical methods. In such case, piecewise linear profile (eddy-viscosity) has been changed byintroducing the exponential (Cabot Moin’s - turbulent viscosity) profile.

Figure 7: Comparison of turbulent viscosity profile between piece-wise linear and exponential profile as theresult of simulation [10]

The eddy-viscosity has been modified which a wall damping function is included on Cabot Moin’s profile.The exponential term was damped turbulent viscosity near the wall region. The result is considerableimproved then the previous profile. Fully developed turbulent motion occurs beyond a distance sufficientlyremote from the wall that a very smooth eddy is not damped by the vicinity to the wall. The distancethat is very close to the wall e.g. y+ = 1, gives over prediction of a velocity profile. It is due to turbulentviscosity profile that cannot fully dumped beyond the viscous sub-layer distance y+ < 5. This case is verysensitive as it means that the turbulent viscosity is not sensitive enough to predict areas very close to thewall which is fully laminar flow region.

The laminar viscosity is dominantly referred to the 1Re times the distance y+ < 5. Automatically, the tur-

bulent viscosity profile which is includes damping factor[1− e

−yA

]will predict turbulent viscosity relatively

close to laminar viscosity. Then, it will change rapidly when the distance away from the wall y+ > 50. Theturbulent viscosity will be much greater. It will occurs, however, due to the damping function has takeninto account, the velocity profile can be predicted closer to the benchmark (LR). The result of the velocityprofile can be sheen in figures 8 and 9

All the results in the case were generated on Fortran-code to approach stream wise velocity, Reynoldsstress, eddy viscosity, k-epsilon, flow rate, etc. It is expected to result in clear difference between theprevious turbulent viscosity (linear profile) and Cabot Moin’s profile.

The form of the latter profile was relatively more sophisticated since the involvement of the damping factor

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Figure 8: Velocity Profile Re = 395 and 3950 with Cabot Moin profile [10]

Figure 9: Velocity Profile Re = 5950 and 10950 with Cabot Moin profile [10]

Figure 10: Velocity Profile Re = 395 (exponential ”Cabot Moin’s” profile-updated uτ ) and the trade offprofile model) [10]

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[1− e

−yA

]which control is much closer to the log law profile as well as control prediction with regard to

mean velocity fluctuation. The curve produced by the exponential profile is bending at the initial stageslightly bigger than the linear profile more than y

δ=0.15-0.2 (distance of the total region). This was expectedsince the assumption to change the linear into the exponential profile was confirmed between figure 5 andfigure7.

Figure 11: Eddy-viscosity distribution in pipe flow according to the measurements of Laufer (circle), Nunner(box) and turbulent viscosity distribution using the exponential profile [10]

It has concluded that modified Robin-type WF by introducing the Cabot Moin’s/exponential profile hasprovided turbulent modelling prediction in considerably with better solution and agreement with the bench-mark profile (LR) for the velocity profile. The modification has been observed by providing more robustsolution due to the fact that coarse mesh and time scale sensitivity were the main concern for computationalstrategy (avoid time consuming). Result for flow rate and strain rate in which the fluid is flew along thelength of the channel flow can be seen in the appendix. Relevant future works: Fluid-structure interaction(low-rise structure)

Acknowledgments

The author conveys his sincere gratitude to Dr. Sergei Utyuzhnikov as the former supervisor of this paper forhis insightful ideas. I wish to express tremendous appreciation to National Education Council of Republic ofIndonesia and Udayana University Bali, for their financial assistance under the Batch 2 Scholarship DIKTI,without which this paper would not have been possible.

References

[1] The First Course in Turbulence. MIT Press, 1972.

[2] Turbulence Modeling for CFD. DWC Industries, Inc., 1994.

[3] Simulation and Modeling of Turbulent Flows. Oxford University Press, 1996.

[4] J. Bredberg. On the wall boundary condition for turbulence models. Technical report, ChalmersUniversity of Technology, Goteborg Sweden, 2000.

[5] . M. P. Cabot, W. Approximate wall boundary conditions in the large-eddy simulation of high reynoldsnumber flow. Flow Turbulence Combustion, 2000.

[6] A. V. G. e. a. Craft, T. J. Progress in the generalization of wall-function treatments. InternationalJournal of Heat and Fluid Flow, 2002.

[7] S. E. G. e. a. Craft, T. J. Development and application of wall-function treatments for turbulent forcedand mixed convection flows. Fluid Dynamics Research, 2006.

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[8] F. J. Maher. Wind loads on dome-cylinders and dome-cone shapes. Journal of Structural, 1966.

[9] I. G. A. Susila. Predicting pressure distribution on surface of arbitrary geometry from cfd sructuralengineering. Master’s thesis, Civil Engineering Department , Newcastle Upon Tyne, 2001.

[10] I. G. A. Susila. Wall functions for large eddy simulation (les). Technical report, University of Manchester, UK, 2009.

[11] S. V. Utyuzhnikov. Generalized wall functions and their application for simulation of turbulent flows.International Journal for Numerical Methods in Fluids, 2005.

[12] S. V. Utyuzhnikov. Robin-type wall functions and their numerical implementation. Applied NumericalMathematics, 2008.

Appendix

Figure 12: Eddy-viscosity distribution in pipe flow according to the measurements of Laufer (circle), Nunner(box) and turbulent viscosity distribution using the exponential profile