[American Institute of Aeronautics and Astronautics 29th AIAA, Fluid Dynamics Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Fluid Dynamics Conference

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

JIM A AA98-32817

AIAA 98-2554Some Improvements in Menter'sk-co SST Turbulence ModelA. HellstenHelsinki University of TechnologyLaboratory of AerodynamicsEspoo, Finland

29th AIAA Fluid Dynamics ConferenceJune 15-18, 1998/Albuquerque, NM

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AIAA-98-2554

SOME IMPROVEMENTS IN MENTER'S k - u> SSTTURBULENCE MODEL

Antti Hellsten*Helsinki University of Technology, Laboratory of Aerodynamics

P.O. Box 4400, FIN-02015 HUT, FINLAND

Some improvements in the well-known k—u SST turbulence model are introduced in this paper. Its sensitizationfor the effects of system rotation and streamline curvature is the most important of the present modifications.Prior to this, a minor modification was introduced in order to make the model rotationally invariant, and thusapplicable for flows in rotating frames of reference. Furthermore, an improved method for specifying the solid-wall boundary condition for w is presented. Also, a trick is introduced, which slightly improves the numericalbehaviour of the model.

NomenclatureCDku positive portion of the cross-diffusion in w-

transport equationC-rc constant coefficient in the rotation and

curvature sensitizationFI , FI , F3, F4 auxiliary functions in turbulence modelH channel half-widthL reference lengthP production of the kinetic energy of turbu-

lenceRe Reynolds numberPi Richardson numberRo rotation number 2H£l/VmSij strain-rate tensor, (dui/dxj + duj/dxi)/2\Sij\ scalar measure of the strain-rate tensor,

/OC.. C.V "y^'J

SR nondimensional function in the rough-wallmethod

Vm bulk velocity in a channel flowVQO free-stream velocityQI Bradshaw's structural parameterCf local skin friction coefficientd distance to the nearest surface pointk kinetic energy of turbulence, u"u"/2ks sand-roughness height/ turbulent length scalep pressurer radius of curvature

s coordinate in the wall-tangential directiont timeUi velocity components (u, v, w) in Cartesian

directions: x,y,zu+ nondimensional velocity, u/urUfef defect velocity, (up — U)/UTUp potential velocityur friction velocity, \frw/pXi Cartesian coordinates: x, y, zy+ nondimensional distance from the surface,

uTd/vF, F2 auxiliary variables in turbulence modelA Clauser thickness, J0 u^fdrjAc/i thickness of the first computational cell ad-

jacent to a solid wallO angular velocity of the coordinate system

vorticity tensor, (dui/dxj - duj/dxi)/2scalar measure of the vorticity tensor,\fl

'Research Scientist, Member AIAAe-mail [email protected] Copyright ©1998 by A. Hellsten. Pub-lished by the American Institute of Aeronautics and Astronautics, Inc.,with permission.

ij

/?, /3* , 7 turbulence-model coefficients6ij Kronecker's deltae dissipation of kinetic energy of turbulence77 coordinate in the wall-normal directionK von Karman constantH dynamic viscosityUT turbulent viscosityv kinematic viscosity, p / pp ___ density—pu"u" Reynolds-stress tensorOk , ffu turbulence-model coefficientsTW surface shear-stressu> specific dissipation of turbulence kinetic

energy

1American Institute of Aeronautics and Astronautics


AIAA-98-2554

1 IntroductionMenter's k — u> SST turbulence model1'2 has gained pop-ularity in the aeronautical community due to its good per-formance in typical aerodynamic flows. It has been espe-cially designed for predicting adverse pressure gradientflows where most of the two-equation models, and evenhigher-order models, are known to fail severely.

It is well known that turbulence is sensitive to stream-line curvature and to system rotation, which are, in prin-ciple, similar phenomena. In a boundary layer on a con-cave wall, or on the pressure side of a rotating duct, tur-bulence is enhanced. The opposite occurs in a bound-ary layer on a convex surface, or on the suction side ofa rotating duct. The Boussinesq eddy-viscosity models,such as the k - w SST model, do not capture these ef-fects properly. The primary aim of this work is to finda unified sensitization for rotation and curvature effectsapplicable to general three-dimensional flows, and to in-corporate such a modification into the SST model.

The original SST model cannot be applied to flows inrotating frames of reference since it is not rotationallyinvariant. First, a modification is introduced in order tomake the model applicable to flows in rotating coordinatesystems. Next, a sensitization for rotation and curvatureeffects is designed and implemented into the model, andcalibrated for a two-dimensional fully developed channelflow with spanwise rotation. The rotation and curvatureadaptation is further tested by computing flow over aconvex-curved surface.

Also, two minor improvements to the original modelare introduced. The first one improves the numerical be-haviour of the blending function, which plays an import-ant role in the SST model. The second one concernsthe solid-wall boundary condition of w. An improvedmethod for specifying the wall-value for u> has been de-veloped. This method allows for the simulation of bothsmooth and rough surfaces, and on the other hand, it sig-nificantly decreases the sensitivity of the solution to thesublayer grid-spacing.

2 Turbulence ModelIn Menter's model1'2 the k - u> equations are solved onlyinside the boundary layers and the standard k — e modelis utilized elsewhere. This is because the pure k — uimodel3'4 is known to be harmfully sensitive to the free-stream turbulence, whereas the k - e model does notshare this defect with the k — w model. Menter trans-formed the standard k — e model into the k - u> form, anddeveloped a blending function FI that is equal to one inthe inner region and goes gradually towards zero near theedge of the boundary layer. The resulting equations for

k and u are as follows:dk dk = P- /3*pku (D

dk

(2)

d«—OXj

MT \ $^J

-Fi dk du>+

The function F4 does not belong to the original model.Instead, it is the rotation and curvature sensitization func-tion, which is the main focus of the present study. It willbe discussed in Section 2.1.

The model constants in Eqs. (1) and (2) are evaluatedfrom

= Fl - F,) (ak(3)

where subscripts 1 and 2 refer to constants in Wilcox'smodel and the transformed k— e model, respectively. Themodel constants of the original k — u model have thefollowing values:

akl = 2.0 = 2.0 fa = 0.075(4)

and the respective values of the transformed standard k —e model are

= 1-0 = 1.168 #2 = 0.0828(5)

The coefficients K and {3* have constant values of 0.41and 0.09, respectively, and 7 is calculated from

7 (6)

In Eqs. (1) and (2) P is the production of the kin-etic energy of turbulence and is modelled using theBoussinesq approximation as follows

CD

The last term in the w-equation originates from the trans-formed e-equation and it is called the cross-diffusionterm. This term makes the model insensitive to the free-stream u. The blending function is given by

=tanh(r4)

where

F = min I max

(8)

(9)

American Institute of Aeronautics and Astronautics


AIAA-98-2554

Here, the first term is the turbulent length scale dividedby the wall-distance. This ratio is about 2.5 in the log-layer and tends towards zero in the defect layer. Thesecond term exceeds unity only in the sublayer. The thirdterm is designed to ensure correct behaviour of FI incases of very low free-stream w. CD^w stands for thepositive portion of the cross-diffusion term

CDkw = max 1p dk (10)

Menter suggests using a value of 10~20 for CDkum\n-However, according to our tests, this choice may lead tobad numerical behaviour of the last term in Eq. (9) andinduce an undesired rise of FI back to a value of 1 aroundthe outer edge of the boundary layer. A better behaviouris obtained when CDkw min is made proportional to themaximum value of the cross-diffusion term detected inthe flow field. For the sake of simplicity, the maximumvalue of CDku in each grid-block is used in this study,instead of the maximum value of the whole flow field.This value is multiplied by a factor of 10~8 to obtain aproper value for CDku mm-

The k — (jj model in the form presented above (withoutthe modifications introduced by the author of this paper)is known as the k - u BSL model. The reader shouldnote that this model does not need any other informa-tion about solid walls except the distance to the nearestwall-point. This distance depends only on the geometryand can be calculated while generating the grid. Thismeans that the model is effectively pointwise, which isan invaluable feature when applying the model to com-plex geometries.

The SST model is further developed from the BSLmodel by Menter.1'2 His idea is to introduce an upperlimit for the principal turbulent shear stress in bound-ary layers in order to avoid excessive shear-stress levelstypically predicted with Boussinesq eddy-viscosity mod-els. In the original SST model, the eddy viscosity [IT isdefined as

max

where = 0.31 and

(11)

a scalarmeasure of the vorticity tensor. The vorticity tensor isdefined by

1Xj dx

The lower limit of the denominator of Eq. (11) isbased on Bradshaw's assumption, that in boundary-layerflows, the principal turbulent shear stress, denoted hereas \pu"v"\ (although, in general, it may be anyone of theshear-stress components) depends on k as follows

The conventional Kolmogorov-Prandtl expressionpk/uj is used as far as it does not exceed the value

\pu"v"\ a\pk (14)

This is called the SST limitation for HT- The SST lim-itation significantly improves the model performance foradverse pressure-gradient boundary layers since the con-ventional formulation clearly overestimates /J>T in thepresence of adverse pressure gradients.1'2 The same val-ues for the model constants apply to both the BSL andthe SST models, except that the Schmidt number Uki hasthe value of 1.176 in the SST model. The SST limitationis suitable only for wall-bounded flows. The purpose ofthe function F2 in Eq. (11) is to prevent the activationof the SST limitation in free shear flows. Otherwise F2behaves like FI except that it remains at unity further outin the defect layer and it is given by

F2 = taring)

where

F2 = max/ 2\/fc SOOiA

/3*u}d'

(15)

(16)

In this study, the formula for eddy viscosity is slightlymodified from its original form in Eq. (11)

Mr =

where |5»j| =strain-rate tensor

(17)

a scalar measure of the

r> «i2 \oxj ~r (18)

The corresponding scalar measure of the vorticity tensor\fiij\ in the original model has been replaced by \Sij\ inorder to achieve rotational invariance. It is considered tobe a sounder scalar measure of fluid distortion also in aninertial frame of reference than the absolute value of thevorticity, since the Boussinesq approximation is basedon the strain-rate instead of the vorticity. For simpleboundary-layer flows both quantities are equivalent toeach other.

The function FS is designed to prevent the SST limita-tion from being activated in the roughness layer in rough-wall flows, i.e. the layer very close to the rough wall.This is necessary because Bradshaw's assumption is notvalid there. The function FS is an extension to the ori-ginal model developed by Hellsten5 in co operation withMenter.6 FS is given by

F3 = 1 - tanh\pu"v"\ = a\pk (13)

/150z/\4

(19)



AIAA-98-2554

2.1 Rotation and Curvature Adaptation

The transport equations for the Reynolds stresses, writ-ten in a rotating coordinate system, contain terms de-pendent on the angular velocity of the frame of reference.The effects of the Coriolis forces enter into the Reynolds-stress tensor via these terms, sometimes referred to asrotation production terms (see, e.g. Launder, Tselepida-kis and Younis7). However, these terms cancel out eachother when the equations for the diagonal stresses aresummed together to obtain the transport equation for k.Therefore, the two-equation models, like the k — e andthe k — uj models, are primarily insensitive to the systemrotation. The same applies for the effects of streamlinecurvature, because both phenomena are basically gov-erned by additional production terms of stress compon-ents proportional to secondary strain components.

The primary aim of this work is to construct a uni-fied sensitization for rotation and curvature effects ap-plicable to general three-dimensional flows, and to in-corporate such a sensitization into the SST model. Thesensitized model is calibrated for a rotating channel flow(see Section 4). Following this, is also a survey of howthe resulting new model version behaves for a flow over aconvex-curved surface studied experimentally by So andMellor8'9 (see Section 5). This survey showed that thenew model version produced very accurate results forthis flow. Also, the flow past the ONERA-A aerofoilin a high-lift situation was computed, although this isnot presented in this paper. In this case, the new modelversion did not give any further improvement in the ac-curacy compared with the original SST model, whichalready produces a much more accurate result in thiscase than the k — e model or the algebraic ones. Thisis because in this flow, the effects of the adverse pressuregradient significantly dominate the effects of the stream^line curvature. On the other hand, this means that thenew model version effectively approaches the originalSST model, as it is designed, in such cases where therotation or curvature effects are not present, or they arenegligibly small.

In future, the new model version should be tested for afew other "building-block" flows where the side-strainsplay an important role. At least, it should be checkedfor a flow over a concave-curved surface and an axialflow around a spinning cylinder. Furthermore, its per-formance should be studied for a few complex three-dimensional flows, for instance, for flow around a delta-wing with vortex breakdown and flow past a ship hullwith highly curved stern geometry.

There are several proposals in the literature for modi-fications to the Boussinesq eddy-viscosity turbulencemodels to sensitize them for the effects of system rota-tion as well as for streamline curvature, e.g. Refs.10"14

The vast majority of them are based on the modificationof the turbulent length scale as10-13

or = (20)

Because the eddy viscosity HT is directly proportionalto the turbulent length scale /, the eddy viscosity can bedirectly multiplied by one of the above functions. In thecase of two-equation turbulence models, however, it ismore attractive to modify the coefficients of one or moreof the source terms in the transport equations for the tur-bulence variables. Most authors have selected to modifythe decay term in the e-equation of the k — e model. Thischoice is mostly based on the work of Launder, Prid-din, and Sharma.12'13 Following this approach, the de-cay term in the w-equation is chosen to be modified inthe present study, see F4 in Eq. (2).

There are also several definitions for the Richardsonnumber M. Most of them are designed specifically for acertain type of flow and coordinate system only. For in-stance, Bradshaw's definition for the Richardson numberin a flow over curved surfaces reads10'12

2u d(ur) t

r2 dr 'dudrTT- +

9(u/r)dr

(21)

Launder et al. replaced the mean-flow time scale in Eq.(21) by the turbulent time scale k/e obtaining a newdefinition

k\ 2u<9(ur)dr

(22)

They also extended this definition for axial flows aroundspinning cones with the cone angle a by multiplying thenumerator of Eq. (22) by cos a. Bradshaw also presen-ted a definition for the Richardson number in a two-dimensional channel flow with spanwise rotation10' "• 13

= -20 - (23)

A simpler alternative for Eq. (21) is found in Ref.

2

13

m = -207 (24)

Following the idea of Launder et al., the mean-flow timescale l/(du/dy) in Eq. (23) can be replaced by k/e toobtain

~dy

(25)

Wilcox has proposed a rotation correction for the k —u model.13 Unlike in the above examples, he modifiedthe transport equation for k. An additional source term



AIAA-98-2554

is added in the transport equation for kin Wilcox's approach.

All of the above definitions for the Richardson numberas well as Wilcox's correction share one serious defect.They are bound to a certain type of flow and a certaincoordinate system, i.e. they are not generalizable for ar-bitrary three-dimensional flows. Considering the modernCFD methods, we must abandon the above formulas andtry to find a more general approach.

Recently, Spalart and Shur15 have studied general andcoordinate-invariant methods for sensitizing the eddy-viscosity models for the effects of rotation and curvature.However, their proposed technique is highly complic-ated and involves, e.g. second-order velocity derivatives.Therefore, their approach is not followed as such in thiswork. Furthermore, Khodak and Hirch14 have general-ized the definition of the Richardson number for arbitrarythree-dimensional flows as

(26)

When applied to a rotating channel flow in a coordinatesystem fixed to the channel, Eq. (26) reduces to

the Richardson number can be written as

» _ 1%

dudy dy

(27)This looks quite similar to Eq. (25). In fact, it givesexactly the same values as Eq. (25) everywhere exceptfor 0 < (du/dy)/fl < 2, as Khodak and Hirsch haveshown.14

In this work, Eq. (26) has been tested in conjunctionwith the k — u} model using the former one of Eqs. (20)as ^4 in Eq. (2). Reasonably good results were obtainedfor the two-dimensional fully developed rotating channelflow with modest rotation numbers fto = 2Htt/Vm <0.1. However, when applied to a complex developingthree-dimensional flow through a radial compressor im-peller passage, it produced unphysically high values fork in certain parts of the flowfield. The same occurredwith the k — c model applied to a flow through a re-turn channel with a deswirl cascade.16 The reason forthis is that the turbulent time scale may in some condi-tions become very high or very low compared with \flij \or (\Sij\ — |fiij|). This is the reason why Khodak andHirsch14 had to introduce some damping in Eq. (20). Inprinciple, the factor fc/e has been introduced in Eqs. (23)and (26) as a scaling factor. However, for flows whereturbulence is not near its equilibrium state, the turbu-lent time scale is not a good choice as a scaling factorof K, which is a measure of the mean-flow deforma-tion. Therefore, in this work the turbulent time scale inEq. (26) has been replaced by a mean-flow time scalel/|Sj_/| = l/-\/2SijSij. The resulting new definition for

(28)

Eq. (28) is a parabola with the global minimum valueof -1/4 when |ny|/|Sy| is equal to 1/2. The formeralternative of Eqs. (20) is selected for the rotation andcurvature correction function F&, appearing in Eq. (2) as

F± = l+1c R (29)

Thus, the present formulation of K ensures that F4 re-mains always bounded, provided that the constant Cre <4. On the other hand, |5y has to be limited from be-low to a small number in order to avoid division by zero.When \Sij —> 0, K may become high, if |f2y re-mains non-zero. This means, that turbulence is highlydamped. This is not a problem, since in such regionswhere \Sij —> 0, the turbulent stresses anyway tend to-ward zero, according to the Boussinesq approximation.

Cn has been calibrated for the fully developed two-dimensional channel flow with spanwise rotation11 withlow and modest rotation numbers (see Section 4). Avalue of 3.6 was found to be suitable for Cre.

With the present modifications introduced in the k — u>SST turbulence model, it will be called the k - u RC-SST model. The abbreviation RC stands for rotation andcurvature. The computational expense due to the evalu-ation of Pi and ^4 is only marginal since all the velocityderivatives needed in Eq. (28) have to be evaluated any-way in order to compute the production term of k, i.e.Eq. (7).

2.2 Boundary Conditions on Solid WallsThe value of pk on solid no-slip surfaces is zero simplybecause the no-slip condition forces the velocity fluctu-ations to vanish on the wall. The boundary conditionof puj is not that straightforward. Theoretically, when anideally smooth solid surface is approached, u> —¥ oo as3'4

w = (30)

where d is the distance to the nearest wall-point, and thesubscript w refers to a wall-value. In practice, Eq. (30)cannot be used for the evaluation of the boundary valueof (jj due to its infinite value on the wall. On the otherhand, on a rough surface, w has a finite value of

u>,,, = (31)

where UT is the friction velocity ^/rw/p, and SR is anondimensional function defined as

(32)for k+ < 25100//C+ for k+ > 25



AIAA-98-2554

Here A;+ is the nondimensional sand-grain height

1.+ _ ,. J. /.. /TT\Ks — UrKs/V \i~>)

Surface value ww is evaluated using Eqs. (31), (32), and(33). This rough-wall procedure is also used for the sim-ulation of flows over smooth surfaces. This is possible,since the surface roughness does not affect the boundarylayer if fc+ has a value lower than four or five. The simu-lation of flows over ideally smooth surfaces is facilitatedby limiting fc+ from below to a value of k^m-ln. Thus Eq.(32) becomes

_ / (50/(max(A;+;fc+rnin)))2 forfc+ < 25for k+ > 25

(34)In this work, a value of 1 was first chosen for k~^m-m. Lateron, it was found that in the computations the skin frictionstill depends somewhat on the k*min althought its value isless than four. It was also found that the lower the valueis chosen for A;^,,, the more grid-sensitive the solutionbecomes. The specification of the smooth-wall value ofw has been extensively studied in Ref.,17 and as a resultof that study, the following new formula for k*m-in waschosen for fct, ;T,

,,+0.85 (35)

Here, yjf" = uT&d-\_/v denotes the nondimensionalthickness of the first computational cell adjacent to thewall. Using this approach, the high grid-sensitivity ofthe skin friction can be avoided. The use of t/+ does notprevent the algorithm from being pointwise, since it doesnot have to be evaluated for the internal computationalpoints but merely on the solid-wall boundaries.

2.3 Free-Stream Boundary ConditionsMenter1'2 suggests the following formulas for the free-stream values of k and u;

— constant x V^/L= 10~3"TOO = ~ z /

fcoo = "Too 00

(36)(37)(38)

where a value, ranging from 1 to 10, may be selectedfor the constant, and L stands for the approximate lengthof the computational domain. The author of this paper,however, uses the characteristic length of the flow prob-lem in question as L. The characteristic length is thelength scale used in the definition of the Reynolds num-ber of the flow. It is specified by the user and is thussomewhat arbitrary. This may introduce some variationof the value of uj^, from its original definition. How-ever, it is thought that the characteristic length is usually

chosen in a way that it really characterizes the flow, andthus gives a consistent free-stream uj for each flow. Forinstance, in the case of flow past a wing, the mean aero-dynamic chord is an obvious choice for the characteristiclength. Furthermore, it is thought that usually the charac-teristic length is shorter than the dimensions of the wholecomputational domain. Therefore, the value of 1 is ad-opted for the constant instead of the value of 10.

In order to make sure that the turbulence variables re-main always positive, they can be limited from belowto a small positive value. For k, its free-stream value isused as a lower limit. For u, the value of 10~3Woo ischosen as a lower limit. This choice has been made bear-ing the safety reasons in mind. It is considered for suchsituations that the characteristic length has been chosenin some peculiar way. If an extraordinarily high valueis used as the lower limit of u, there is a risk that thesolution of possible free shear-layers, i.e. wakes, jets,etc. may become corrupted. This may occur when toohigh a lower limit for w is used since u may be quitelow through a free shear-layer compared with the valuesfound in a near-wall region of a boundary layer.

3 The Flow-Solution MethodFor solving the rotating channel flow, a simple one-dimensional solver, called FULFLO, for fully developedincompressible channel or pipe-flow was constructed. Abrief description of the FULFLO code can be found inRef.'7

For the curved boundary layer computations, a Navier-Stokes solver called FINFLO18'20 is utilized. The turbu-lence model is coupled with the compressible-flow equa-tions, i.e. conservation equations for mass, momentumand energy. The FINFLO code is developed at HelsinkiUniversity of Technology in the Laboratory of Aerody-namics. It is based on the finite-volume method and util-izes structured multi-block grids. The solution methodis an implicit time integration and a multigrid methodis used to accelerate convergence. The inviscid fluxesare evaluated using Roe's method with MUSCL-type dis-cretization, while the central-differencing scheme is usedin the calculation of the viscous fluxes.

4 Channel Flow with SpanwiseRotation

The rotating channel flow studied experimentally byJohnston, Halleen, and Lezius11 is selected for the basiccalibration and test case of the k - u RCSST model. Theflow is assumed to be fully developed and incompress-ible. With these assumptions, the flow is computationally



AIAA-98-2554

1.2-

1.1-

1.0 H

0.9-

0.8

0.7-

0.6-

n ^ -,

%^\&r

In"~—-UJ. f^H

V Re^ZOOOOOO /?e=35000n /?e=11500A /?e=60000 /?e=2950

LJ3 '

— R-— _J

tn E

lo <

^ ^?~o- - - • — C<

! (> '

L -- - -j (

0.00 0.05 0.10 0.15

Ro0.20 0.25

Fig. 1: The skin-friction asymmetry illustrated by UT/UTO in therotating channel flow at various fle-numbers. The solid curve ap-proximates the experimental data.11

very inexpensive to solve. The FULFLO code is utilizedfor this purpose. For a fully developed two-dimensionalrotating channel flow in a coordinate system fixed to thechannel, Eq. (28) reduces to

\du/dy-2fl\du/dy \du/dy\ -1 (39)

One case, where Bo = 0.068 and Re = 35000, isalso solved with the FINFLO code in order to checkthe correct implementation of the rotation and curvaturesensitization, i.e. Eqs. (28) and (29), in the generalthree-dimensional method. In the FINFLO computa-tion, the Mach-number was set to 0.2. There are someslight differences between the velocity profiles com-puted with FINFLO and FULFLO (see Fig. 2). Thereason is probably that, in the flow solved with FINFLO,there are slight variations of density due to the non-zeroMach-number required by the compressible flow solu-tion method of FINFLO. In the FULFLO solution, dens-ity is fixed to a constant value. Inspite of the slight differ-ences, it is concluded that the general implementation ofEqs. (28) and (29), in the FINFLO code, works correctly.

The friction velocities UT on the pressure and suc-tion sides of the channel normalized by the friction ve-locity UTQ of the corresponding non-rotating channel arepresented in Fig. 1 computed at various Jfe-numbers.The upper branch of the curves represent the pressureside, where the skin friction is increased due to the en-hanced turbulence, and the lower branch represent thesuction side, respectively. The computed results are com-pared with the curves approximating the experimentaldata." In fact, there is a considerable scatter in the ex-perimental data and the curves are merely selected to

approximate the average behaviour of the actual exper-imental data. Most of the experimental data is from .fib-numbers less than 0.1. There are only few data pointsfrom Ro > 0.1.

It was observed in an early phase of testing that thepresent technique gives too distorted velocity profiles athigh Jib-numbers (see Fig. 2). Several alternative formsfor rotation correction as well as for fS-number weretested in this phase but none of them removed this defect.Even the sensitized model is still a Boussinesq model,which means that the predicted zero point of the shearstress and the point of velocity maximum will always bealigned. At high .fib-numbers the offset between the zeroshear-stress point and the maximum velocity point maybecome quite wide. In such conditions, a Boussinesqmodel produces too distorted velocity profiles, i.e. themaximum velocity point is predicted too near the suctionside-wall. This is clearly visible in cases with Bo — 0.21and 0.42 in Fig. 2.

It is probably so that eddy-viscosity models in gen-eral cannot be sensitized for arbitrary .fib-numbers, un-less perhaps by recalibrating the model separately forhigh fib-flows. Such an approach is not practical, thuswe simply accept that the new model version gives ac-curate results only in a limited .fib-range.

Because of the above observation, and because the ex-perimental data may be unreliable above .fib-numbers ofabout 0.1, it was decided to calibrate the present modelfor flows with 0 < Ro < 0.1. Fig. 1 shows that for smallJib-numbers the computed results with (7re = 3.6 agreewell with the average experimental curves regardless ofthe Jfe-number. At Jib « 0.1 the computed pressure sidefriction begins to deviate from the average experimentalcurve predicting somewhat too high a friction.

A computational grid of 128 cells through the channelis utilized in all of the rotating channel-flow computa-tions. The sublayer grid-spacing is designed so that forall cases the first cell thickness on the pressure-side is nomore than one in y+.

5 Boundary Layer overConvex-Curved Surface

As the model is now calibrated to give reasonably ac-curate results for the rotating channel flow with modestJib-numbers, it is interesting to see how accurately it pre-dicts a curved boundary-layer flow. This is surveyed bycomputing the convex-curved boundary layer studied ex-perimentally by So and Mellor.8'9 This case is computedalso with the original SST model, the low-Jfe-numberk — € model of Chien, and the algebraic Baldwin-Lomaxmodel. Here, the SST model is, in fact, not completely inits original form. Instead, the modifications concerning



AIAA-98-2554

Ro-0.069, Re=11500RCSST/FULFLO

• Experiment

0.00 0.25 0.50

Ro=0.068, Re=35000RCSST/FULFLO

......RCSSJ/FINFLO• Experiment

0.00 0.25 0.50

y/D

Ro=0.21, Re-77500RCSST/FULFLO

Ro=0.42, Re=35000RCSST/FULFLO

1.00

0.75 1.00 1.00

Fig. 2: Velocity profiles in the rotating channel flow.

the blending function Fj (see Section 2) and the solid-wall boundary condition for ui (see Subsection 2.3) areincluded. The FINFLO code is utilized in this case.

In order to isolate the effects of streamline curvature,the pressure gradient has been minimized in the area,where the boundary layer is studied. This has been ac-complished by suitable contouring of the opposite wall,see Fig. 3. In this case the curvature effects are quitestrong the value of A/r being on the average equal to0.33. Here r is the local radius of curvature and A is theClauser thickness defined as

(40)

with Ufa/ being the defect velocity (up - U)/UT, and upis the potential velocity. Here and when presenting theresults, we employ a curvilinear s - r?-coordinate sys-tem, where s is aligned with the local tangential direc-tion of the surface, and TJ with the local surface-normaldirection, respectively. However, all the computationsare performed in the Cartesian coordinate system.

The computational domain consists of a straight duct,where the boundary layer initially develops, the curved

section, and a straight outlet section. In the computa-tions, the length of the straight inlet duct has been chosenso that the boundary layer displacement and momentumthicknesses and the skin friction coefficient match the ex-perimental data at s = 24 in. The convex wall geometryobeys the geometry of the curved-flow wind tunnel,8'9while the opposite concave wall is contoured to obtaina pressure distribution as close to the measured one aspossible. As can be seen in Fig. 3, there are some slightdifferences in the range: 50 in < s < 60 in, whereas inthe area: 60 in < s < 80 in, the computed and meas-ured pressures agree well. The opposite wall geometryis optimized only once instead of doing it separately foreach turbulence model. This is why there are some dif-ferences between the pressure distributions obtained withdifferent turbulence models.

The computational grid of the curved section consistsof 240 cells in the streamwise direction and 128 cells inthe cross-stream direction. The resolution in the viscoussublayer is considered to be sufficient, since the thick-ness of the first cell adjacent to the convex wall variesbetween 0.4 and 0.6 in y+.

The skin friction coefficient along the wall is plotted in



AIAA-98-2554

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where pw is the static pressure on the wall and p0e isthe total pressure outside the boundary layer. The wall-potential velocity is used as a reference velocity also forthe velocity and the Reynolds-stress profiles presented inFig. 5. The profiles are plotted in three sections: s = 59,67, and 71 inches. The velocity profiles predicted withthe RCSST model are in a good agreement with the ex-perimental data. The same applies for the u'V-profiles.It should be noted that the measured value on the wallin Fig. 5 represents the total wall shear-stress, instead ofu"v".

The direct influence of the rotation and curvature sens-itization occurs in the defect layer, since in the inner por-tion of a boundary layer M -^ Q. Therefore, it is import-ant to inspect how accurately the model can predict theshear stress in the outer layer of the boundary layer. Atleast, in this case it does it very well. However, furthertesting is needed with the boundary layer over a concave-curved wall. Another suitable case for further testing isan axial flow past a spinning cylinder.

6 ConclusionsThis paper proposes a few improvements to the well-known Menter's k — u SST turbulence model. First,the model is modified to be rotationally invariant, hence

being applicable in rotating frames of reference. Forthe second modification, the numerical behaviour of theblending function, which plays an important role in theSST model, has been slightly improved. An improvedmethod for specifying the solid-wall boundary conditionfora; is also presented. This method reduces remarkablythe sensitivity of the predicted skin friction to the sub-layer grid-spacing. Furthermore, it allows one to simu-late also rough surfaces.

The most important modification is the design and in-corporation of a generalized rotation and curvature sens-itization. The new model version, including these modi-fications, is referred to as the k - w RCSST model. It hasbeen tested for a channel flow with span wise rotation andfor a boundary layer over a convex-curved surface. Thenew model gives accurate results for the rotating chan-nel flow with rotation numbers below 0.1. For the flowover a convex-curved wall, the results predicted with theRCSST model are in better agreement with the exper-imental data than the results obtained with the originalSST model, or with the k - e model of Chien or the al-gebraic Baldwin-Lomax model. However, more testingis needed. In future, the RCSST model will be tested atleast for a flow over a concave-curved wall and an axialflow around a spinning cylinder.

7 AcknowledgementsThis work was supported by the Finnish Technology De-velopment Centre under the National CFD TechnologyProgramme. Their support for this research is gratefullyacknowledged.



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AIAA-98-2554

References1 Menter, E, "Zonal Two Equation k - u> Turbu-

lence Models for Aerodynamic Flows," in 24th FluidDynamics Conference, (Orlando), AIAA paper-93-2906, July 1993.

2 Menter, E, "Two-Equation Eddy-Viscosity Turbu-lence Models for Engineering Applications," AIAAJournal, Vol. 32, August 1994, pp. 1598-1605.

3 Wilcox, D., Turbulence Modeling for CFD.Canada, California: DCW Industries Inc., 1993.

La

4 Wilcox, D., "Reassesment of the Scale-DeterminingEquation for Advanced Turbulence Models," AIAAJournal, Vol. 26, November 1988, pp. 1299-1310.

5 Hellsten, A. and Laine, S., "Extension of the k — u-SST Turbulence Model for Flows over Rough Sur-faces," in AIAA Atmospheric Flight Mechanics Con-ference, (New Orleans, LA, USA), 1997.

6 Menter, F. Private communication.7 Launder, B. E., Tselepidakis, D., and Younis, B.,

"A second-moment closure study of rotating channelflow," Journal of Fluid Mechanics, Vol. 183, 1987,pp. 63-75.

8 So, R. and Mellor, G., "An Experimental Investiga-tion of Turbulent Boundary Layers along Curved Sur-faces," NASA CR-1940, 1972.

9 So, R. and Mellor, G., "Experiment on con-vex curvature effects in turbulent boundary layers,"Journal of Fluids Engineering, Vol. 60,1973, pp. 43-62.

10 Bradshaw, P., "The analogy between streamlinecurvature and buoyancy in turbulent shear flow,"Journal of Fluid Mechanics, Vol. 36, 1969, pp. 177-191.

11 Johnston, J., Halleen, R., and Lezius, D., "Effects ofspan wise rotation on the structure of two-dimensionalfully developed turbulent channel flow," Journal ofFluid Mechanics, Vol. 56, December 1972, pp. 533-557.

12 Launder, B. E., Priddin, C., and Sharma, B., "TheCalculation of Turbulent Boundary Layers on Spin-ning and Curved Surfaces," Journal of Fluids Engin-eering, 1977, pp. 231-239.

13 Howard, J., Patankar, S., and Bordynuik, R.,"Flow Prediction in Rotating Ducts Using Coriolis-Modified Turbulence Models," Journal of Fluids En-gineering, Vol. 102, December 1980, pp. 456-461.

14 Khodak, A. and Hirsch, C., "Second Order Non-Linear k — e Models with Explicit Effect of Curvatureand Rotation," in Proceedings of the Third ECCO-MAS Computational Fluid Dynamics Conference,(Paris, France), September 1996.

15 Spalart, P. and Shur, M., "On the Sensitizationof Turbulence Models to Rotation and Curvature,"Aerospace Science and Technology, No. 5, 1997,pp. 297-302.

16 Rautaheimo, P. and Ojala, J., "Description of theNumerical Methodology for the ERCOFTAC TestCase F3," Helsinki University of Technology, Labor-atory of Applied Thermodynamics, MemorandumCFD/TERMO-21-97,1998. Unpublished.

17 Hellsten, A., "On the Solid-Wall Boundary Conditionof u> in the k - w-Type Turbulence Models," HelsinkiUniversity of Technology, Laboratory of Aerodynam-ics, Report B-50, Series B, 1998.

18 Siikonen, T, Kaurinkoski, P., and Laine, S., "Tran-sonic Flow over a Delta Wing Using a k — c Tur-bulence Model," in 19th Congress of the Interna-tional Council of the Aeronautical Sciences, (Ana-heim, California, USA), 1994.

19 Kaurinkoski, P. and Hellsten, A., "FINFLO: the Par-allel Multi-Block Flow Solver," Helsinki Universityof Technology, Laboratory of Aerodynamics, ReportA-17, Series A, 1998.

20 Hellsten, A., "Implementation of a One-EquationTurbulence Model in the FINFLO Flow Solver," Hel-sinki University of Technology, Laboratory of Aero-dynamics, Report B-49, Series B, 1996.


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[American Institute of Aeronautics and Astronautics 29th AIAA, Fluid Dynamics Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Fluid Dynamics Conference