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THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS345 E. 47 St., New York, N.Y. 10017

C The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections or printed in its publications.M Discussion is printed only if the paper is published in an ASME Journal. Papers are available]^^ from ASME for fifteen months after the meeting.

Printed in USA.

91-GT-143

Computation of a Wall Boundary Layerwith Discrete Jet Injections

P. KULISACharge de Recherches, C. N. R. S.

F. LEBOEUFProfessor

G. PERRINPh.D. Student

Ecole Centrale de LyonLaboratoire de Mecanique des Fluides et d'Acoustique

Ura Cnrs 26369131 Ecully Cedex-France

ABSTRACT Superscriptsspace average along z

Cooling of turbine blades is often achieved with cold discrete jets space fluctuation along zintroduced at the wall. In this paper, a new method for computation turbulent fluctuationof a wall boundary layer with discrete jet interactions is " crossflowpresented. The jets are assumed to be arranged in rows and the A jet partflow is assumed locally periodic in the row direction. n- turbulent average

The conservation equations are spatially averaged betweentwo jet orifices. The resulting equations look like two-dimensionalboundary layer equations, but with three-dimensional jet sourceterms.

The numerical method solves the boundary layer equationswith a Keller box method. A strong interaction with inviscid flowis also introduced in order to avoid numerical difficulty in the jetregion. Three-dimensional jet conservation equations are solvedwith an integral method, under the boundary layer influence. Acoupling of the two methods is performed.

Comparisons with low speed experimental data arepresented, particularly near the jet orifices. It is shown that theagreement between the results of computation and the experimentsdepends on the jet behaviour very near to the jet exit.

NOMENCLATURE

Ht stagnation enthalpyK curvatureP static pressureST jet source term(u, v, w) velocity components along (x, y, z) respectivelyV velocity modulus(s, n, b) curvilinear coordinates linked to the jet trajectoryx crossflow streamwise coordinatey normal to the wall coordinatez in the wall, lateral coordinatet turbulent eddy viscosity8q mass flow defect0 jet diameter at the orificep static density

Subscriptse inviscid flowmax maximum value in the jeto condition at the orifice

INTRODUCTION

The high temperatures reached in modern high pressureturbines require the use of cooling techniques in order to protectthe walls. As an example, cold flow may be introduced throughholes drilled at the wall, thereby creating a film of discrete jets,which coalesce after some streamwise distance (fig 1). Thedifficulty of predicting this type of flow is connected to the greatnumber of interacting phenomena. The flow in a turbine isstrongly accelerated. As a consequence, the transition is spread outas the turbulence is partly inhibated ; similarly, heat transfer atthe wall is reduced, owing to the low level of turbulence. However,complications occur in real engines ; for example, the flow may belocally decelerated, and thus separated, near the leading edge atoff-design conditions, under the influence of shock-waveimpingments or jet injections.

The behaviour of discrete jets, introduced in a crossflow,has been the subject of many experiments. Although the variouspublications are not always fully detailed, they produce a rationalpicture of the jet and crossflow interactions, provided the jet tocrossflow velocity ratio is sufficiently high. At first, for smalllength to diameter ratio of the injection holes, flow separation maybe induced near the entrance of the tube. As a consequence in the jetnear the orifice, the turbulence is high and the jet velocity profileis also very uniform, as shown by Pietrzyck, Bogard and Crawford(1988). This may be found even for low values, of the blowingparameter, as low as 0.25. Just after the exit, the jet behaves likea flexible body whose structure becomes strongly three-

Presented at the International Gas Turbine and Aeroengine Congress and ExpositionOrlando, FL June 3-6, 1991

This paper has been accepted for publication in the Transactions of the ASMEDiscussion of it will be accepted at ASME Headquarters until September 30, 1991

Copyright 1991 by ASME

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dimensional. On the upstream jet side, the outer flow decelerateslike in front of a solid body. On the downstream side, a low value ofthe static pressure may be measured (Moussa, Trischka, Eskinazi,1977). As a consequence, a pressure force exists, which isdirected towards the normal to the local jet trajectory at theorifice and which induces the jet to bend.

Figure 1 : Film Cooling.

A set of vortices appears in the vicinity of the jet. Some ofthem have a low influence on the jet global behaviour or shape, ie.the Karman vortices in the wake (Moussa et al., 1977) or thehorseshoe vortices which wrap around the upstream jet side(Andreopoulos and Rodi, 1984). However, the particular shape ofthe jet cross-section is probably a direct consequence of thelongitudinal counter-rotating vortex pair induced by theinteraction of the strong shear on the jet boundary. According tothe computation performed by Sykes, Lewellen and Parker(1986), this longitudinal vorticity exists immediately after theorifice, and is probably maximum in the immediate vicinity. Asthe jet bends downstream, dissipation of the vorticity is theprevailing phenomenon ; positive production of the vorticityoccurs according to the jet trajectory curvature, while a negativeproduction appears when the jet velocity decreases. Carotte andStevens (1988) also mention that the viscous crossflow, wrappingaround the jet, induces a similar set of counter-rotating vorticeswhich seems to dominate the bound vortex system after somedistance downstream in the case of multiple jets injection.

The neighbouring jets, in the case of jets row injection,have a strong influence on the behaviour of a particular jet in therow ; Bario, Leboeuf, Onvani and Seddini (1989) have shown thatthe jet in a row is pushed closer towards the wall, in comparisonwith the single jet case. Similar behaviour is also computed byHuang (1989), taking the velocity field induced by the counter-rotating vortex system into account ; however, only the casewithout mixing between jets was treated. In the multiple rowconfiguration, the upstream row sees a thinner upstream viscouswall layer than the downstream rows ; as a consequence, thedownstream jets will be less deviated towards the wall (Sinha,Bogard, Crawford, 1990). Similar behaviour is observed for jetsemerging from tubes introduced in a uniform flow. For that case,Huang (1989) has shown that this could be reproduced by takingthe velocity field induced by the counter-rotating vorticity intoaccount.

Introduction of film cooling in a turbine also needs to takethe high curvature of the walls into account. For low velocity ratioand owing to the pressure gradient normal to the wall , the lateralmixing of the jets is increased on concave walls (Schwarz and

Goldstein, 1988). For the same reason, the low velocity jet ispushed towards the convex walls, (Schwarz, Goldstein, Eckert,1990).

Detailed velocity and shear stress measurements,performed by Pietrzyk, Bogard, and Crawford (1988), indicatecoincident peaks in the mean velocity gradient and in the turbulentquantities. This may point to the possible use of a turbulentviscosity model. However, computations realized with this modeldo not prove to be very efficient (Dibelius, Pitt and Wen,1990).The preceding description of the film cooling behaviour emphasizesthe need for sophisticated computational method in order to take allthe flow details into account. Beside analytical approachs which areoften used for heat transfer study, many numerical methods havebeen developed. Dibelius, Pitt, Wen, (1990) use three-dimensional equations in an elliptic method of solution, whichallows access to the flow inside the jet orifice. All of the thermaleffects inside the blade have to be taken into account to defineproperly the heat transfer at the blade wall (Camci, 1988). Inorder to avoid excessive computational time in the case of multiplejet injection, a two-dimensional boundary layer approach is alsoused (Herring, 1975, Miller and Crawford, 1984, Tafti andYavuzkurt, 1989, Sch6nung and Rodi, 1987).

In this paper, we present a new method of computation for awall boundary layer, with discrete jets injection. Starting withthe Navier-Stokes equations, we first apply a spatial average inthe direction of the row of jets, thereby forcing periodicity of theflow between two neighbouring jets in the same row. A similartechnique has already been used by Herring (1975). We thusobtain a set of two-dimensional equations with three-dimensionalsource terms. Boundary layer approximation is then applied inorder to reduce the computational effort and to allow a betterdescription of the flow near to the wall. In order to allow thetreatment of local flow separation, a strong interaction model hasbeen added between the outer inviscid flow and the wall viscouslayer (Kulisa, Leboeuf, Klinger, Bernard, 1990). Compared toprevious two-dimensional methods, we avoid the use of analyticallaws for the description of the jet source terms. By contrast, wesolve three-dimensional jet equations written in a curvilinearframe of reference, based on the local jet trajectory. An integralmethod is used for the solution of these equations. Closures arenecessary for the description of the interactions between the jetand the outer flow. In the following, we shall describe the flowmodel and the interactions between the jet and the viscous walllayer. Then, the jet computation will be described. The completecomputational method will there be described. Comparison withexperimental results will be presented for a row of jets,introduced inside a low speed boundary layer.

THE JET AND CROSSFLOW COMPUTATION

Basic Idea of the ModellingAs mentioned in the introduction, turbine film cooling is a

complex phenomenon. Our objective is to build a method whichallows most of these phenomena to be taken into account, ifpossible at a fine closure turbulent level, but however with areasonable computation effort ; this last requirement implies thatthe use of Navier-Stokes equations for the computation of thewhole flow should be avoided.

Our flow model is based on an iterative exchange between aviscous wall layer and a three-dimensional jet computations. In


the basic flow configuration, the discrete jets are assumed to beintroduced through rows of orifices, located perpendicularly to theouter crossflow, along the z direction. Periodicity in the rowdirection is then assumed, over the distance G between twoneighbouring jets. As a consequence of this hypothesis, every flowquantity is separated in two terms :

4=4+4 (1)

where q is the spatial average of q, defined as:

G

1 (2)q (x,y) = G q(x,yz) dz

0

Therefore, q is a two-dimensional quantity which contains

implicitly the jet effects. q" is the spatial fluctuation of q ; it is athree-dimensional quantity which expresses the lateral flow non-homogeneity.

When this spatial averaging is applied to the flowconservation equations, a two-dimensional averaged problemresults, whose unknowns are the space-averaged quantities.Nevertheless, additional terms also appear in the resultingequations, associated with their non-linearity with respect to thebasic flow variables (V, Ht). This is similar, but not equivalent, tothe appearance of Reynolds stresses in the momentum equationswhen turbulent averaging is applied. In our case, thesesupplementary terms will be considered as source terms for thetwo-dimensional averaged problem, and will come from the three-dimensional jet computation. The jet model will be presented in thenext paragraph.

In brief, the whole film cooling flow calculation leads to aninformation exchange between a thin wall layer computation,which includes a viscous-inviscid interaction as well as additionalthree-dimensional jet source terms, and a jet computation, whichproduces these source terms. From the numerical point of view,the thin wall layer computation uses a space marching procedurewhich is suited to the parabolic character of this approximatedflow. The viscous-inviscid interaction reintroduces an ellipticeffect, which is allowed for by repeating the space marchingiteratively. After each upstream to downstream sweep, a jetcomputation is performed in order to update the three-dimensionalsource terms ; these terms are computed on a three-dimensionalmesh, and then averaged in the transverse Z direction. Theiterative procedure is described on the figure 2.

For convenience, we define the following notation : 6 willdefine a crossflow variable ; Ao = o -o corresponds to the excess ofo-variable in the jet, by reference to the crossflow. Note that Ao isnot necessary small, compared to o. The evolutions of AVs and ATtalong the n and b directions, are described by analytical laws ; theAVs and ATt maxima are localized on the s-line ; on the jetboundary, the jet excess quantities AVs and ATt tend towards zero.AVn and AVb are neglected for the results presented in this paper ;however, a complete formulation may be found in Leboeuf and

SPACE-AVERAGED VISCOUSWALL LAYER COMPUTATION (2D)

WITH VISCOUS-INVISCID INTERACTION

JET SOURCE TERMS

SOURCE TERMS SPACE-AVERAGEDQUANTITIES q

JET COMPUTATION (3D)

Figure 2 : Iterative procedure for the film cooling flowcomputation

The Jet ComputationThe behaviour of the jet is defined in a curvilinear

orthogonal coordinate system (s, n, b), the direction s beingassociated with the jet trajectory (fig. 3), the third direction bbeing identical to z, and n is orthogonal to s and b ; an is the anglebetween the direction s and the axial direction x.

Jet computation hypothesisVarious hypotheses allow the simplification of the problem.

The s direction is located in the (x, y) plane. The transverse jetsurface (n, b) is assumed to be flat. Along directions n and b, thes-line radius of curvature is assumed great, compared to acharacteristic jet width. The characteristic lengths of the jet inthe transverse directions are small compared to crossflow lengthscales. The consequence of this is that the crossflow conditions areassumed constant over the jet transverse surface a. The diffusionphenomenon along the direction s is neglected. Moreover, the staticpressure gradient along s is governed by the crossflow. The lasttwo hypotheses are typical of parabolized flow problems. Inaddition, the static jet density is also constant over a and iscomputed from the crossflow static pressure and the averagetemperature in the section of the jet.


Huang (1990). Similarly, heat fluxes and shear stress work areneglected on that boundary, in the jet computation.

A potential core is assumed to exist at the jet orifice,according to the observations made by Pietrzyck, Bogard andCrawford (1988) and Snel (1971). In this area, tVs,max andeTt,max are constant along s, although oV s and ATt are allowed tovary as the potential core disappears progressively. At thepotential zone extremity, the jet cross-section tends towards anellipse.

Figure 3 : Jet coordinate system.

Intearal formulationFor a steady, single phase flow, we consider the

conservation equations for mass, momentum and energy. They arewritten in a conservative form, in a fixed frame of reference.These equations are expressed with respect to the curvilinearorthogonal coordinate system (s, n, b). Using the Leibnitz limitrule, the equations are integrated over the jet cross-section(Huang, 1989). The integral equations are given below:

Defining the entrainment coefficient as

A da fE pVsds p V

nKsndap(OVn n+AVb b )dC (3)

C

the mass conservation equation givesn

dV

dsJJ Pvsd ass(4)

where C is the contour of the transverse surface a. K sn andK sb are the curvatures of the jet s line. The equation (3) showsthat the entrainment is a consequence of three effects : the growthof surface a, a curvature effect, and the transverse jet inducedvelocity component.

For the momentum conservation equation, a deficitformulation with respect to the crossflow is used. According to theprevious hypothesis, it allows to eliminate the static pressureterm in the s component of the momentum equation. The s and n,components are given below :

J J A(PV2 )da- 1 A(pVs )dC_+2 JJA((pV sVn )Km }daJC aIi -9-3 1

+ A((pVsVn ) n }dC - Ats dC=0

C C (5)

do- JJ A(PVsVn)da-C+ (PVsVn)dC + J J A((PVn)Ksn)da

a C

2''

JJ 2 r ^ t f

+ C ((p '/))n))

dC A(pVs) Ksn da + C P Ti . dC - ( C is dC = 0a ) J

(6)

The viscous and turbulent forces appear in the ets and Atnterms ; the contributions of the shear stresses have been neglectedin the curvature and in the da/ds term.

The variation of the momentum excess is associated withvarious effects : the growth of the surface da/ds, a curvatureeffect, the momentum flux induced by the jet itself, pressureforces, viscous and turbulent stresses.

For the conservative equation of energy, neglecting the heatflux in the s direction and the curvature terms, gives the followingintegral form as :

ds

d l i J

pVSHtda=ii pVSHtdC JJ P^nHtKsn daC

p t H t Vn n dC C(qn n +qb b) dC +O i s V s dCC

JC(7)

On the jet boundary, the shear work is dominated by thecontribution of the longitudinal velocity component Vs. The

enthalpy transport in the' direction is created by the variation ofa, a curvature effect, the heat flux through the jet boundary q n , qband the friction power t sV s , associated with the viscous forces. Thelast two terms are usually neglected in the present computations,except for the condition on the wall.


The integral formulation has two consequences. It changesthe parabolic nature of the previous equations to an hyperbolictype ; this allows a marching process to be used from the jet exitto the downstream location. But, the averaging process must beassociated with particular closure relationships, especially at thejet boundary.

Closure relationshipsAccording to the experimental results, the transverse jet

section has a very particular shape which may be observed fardownstream of the exit ; most of the preceding studies approximatethis shape by a simple geometry. It seems that an ellipse is a goodcompromise. In the present model, the ratio of the major andminor axes (av/bv) varies according to a linear relationav

=-3K+C (8)

where K allows to introduce a potential core : K=1 for acircle at the exit and K=0 when the end of the potential region isreached ; C varies usually between 2 and 4. Depending onparticular injection conditions, a potential region may exist nearthe orifice. By hypothesis, we shall assume that, for a three-dimensional jet, the longitudinal velocity component is constant inthe potential core and is equal to the injection velocity Vsmax,o Inthe potential region, the maximum jet velocity is then known, butthe extent of the potential core is unknown. The previous equationsystem may then be used in order to calculate the length of thepotential region.

According to the model, entrainment has two origins(El+E2) : the diffusion effect El and the lee side effect E2 . Theturbulent diffusion effect is modelized as in a one-dimensional jetaccording to Herring (1975), but including the potential core :

n Z

El = K1 PP Vsmax `^ O' 1 K (9)

with K1 = 0.14

For the E2 part, we ignore the vorticity which appears inthe jet wake, and take only the two counter-rotating vortices onthe lee side of the jet into account. Because these vortices do notinduce any mass flow, we add a sink singularity on this lee side,according to Le Grives's method (1977). Computing the velocityfield induced by this sink singularity downstream of an assumedcircular jet cross-section, we get, in the frame of potential flow,the drag on the jet boundary AFp with

4FP = (C OP dC = EZV. (1 0 )

jc

where Vn is the crossflow velocity component in the cross-section surface. Introducing a drag coefficient CD by

n n

4 FP 2 CDPVn (1 1 )

A relationship is postulated between the lee sideentrainment E2 and the drag force

n n

E2 =2 CDPVn (1 2 )

In practice, the jet cross-section is not circular but has a"bean" shape ; however, this relation is assumed to be valid even

when the cross-section of the jet does not remain circular. As aconsequence, the preceding formulas will be used with CD=2.0.

The momentum equations include a viscous force that wechoose to compute in terms of the jet velocity excess aspostulated

-- 4 4

4 n n n n

ec=-CT PtV-Vf V-V fa (13)The negative sign in the above expression allows the

momentum to be dissipated. A value CT=0.3-0.6 has been used forthe present results.

Analytical laws have been used in order to describe thelongitudinal velocity component and temperature distributionsalong the transverse directions. For the velocity distribution, weget

5

( b2 n2 l2(14)4Vs=sma, 1 2 - 2 I

a. bVIn the potential region, a constant value tVsmax,o is

assumed in a core of similar shape as the jet cross-section, butwhose dimensions are reduced by the factor K (eq. 8) ; a similarexpression as (14) is retained for oV s between the core and thejet boundary.

For the temperature distribution, we assume a gaussianlaw :

b2 2nP bV (15)

AT, = 4T tmax exP

The diffusion coefficient j is defined so thatoT/ATmax=1/10000 at the ellipse boundary.

The jet equation solution is obtained by using a Gauss-Newton method. The previous equations (4 to 7) are used tocompute four main unknowns, which may be defined arbitrarily.We choose to compute a characteristic jet maximum velocityAVsmax, the transverse surface area c, the angle a n , and a jetmaximum temperature LTtmax In the potential core region, as themaximum jet velocity is constant, we use the ratio of the potentialcore to the jet transverse area K, as one of the main variable ; as aconsequence, the potential core length comes out from thecomputation.

Jet computationsThe full jet model has been extensively tested for various

jet injection conditions, including uniform as well as boundarylayer crossflows (Huang,1989, Leboeuf and Huang, 1990). Wepresent here two examples of results obtained with theapproximate model. Figures 4 and 5 give the evolution along thetrajectory S of the ratio ATtmax/ATtmax,o, for a jet introducedperpendicularly in an uniform crossflow ; comparisons are shownwith the experimental results of Kamotani and Greber (1972) fortwo values of the jet and crossflow velocity ratio. Agreementbetween the results of computation and experiments is good when apotential core is introduced in the model.

The figure 6 gives the evolution of the jet in a crossflowviscous wall layer (Bousgarbies, Brizzi, Deniboire, Gdron,1991). The jet is introduced with an angle an=45, a jet to


crossflow velocity ratio of 2/3. In that particular case, thecrossflow velocity gradients near the orifice are very important,and our hypothesis of constant crossflow conditions over the jetcross-section is no longer justified. For that purpose, a slightlymodified form of the jet equations has been used, eliminating thatparticular hypothesis. The results show that the jet model is ableto reproduce the growth of the jet cross-section, and thetrajectory of the jet, as defined from the maximum values of thejet velocity and temperature. Note a slight reduction of the jetcrosswidth in the n-direction, very near to the orifice ; this is infact connected to the bigger growth of the jet in the b-direction, asa consequence of eq (8).

1.00 0

0.80 ------Calculation without potential coreCalculation with potential core D Experiments

0.60

0.40

0.20

o 0.00 S/D

0.00 10.00 20.00

Figure 4 : Thermal evolution along the jet trajectory. Velocityratio 9.7 ; an =90.

- Calculation without potential core1.00 _- Calculation with potential core

0 0 Experiments

0.80

0.40

020

S/D

0.00 1000 20.00

Figure 5 : Thermal evolution along the jet trajectory. Velocityratio 5.9 ; an=90.

Calculation

Y/D 0000 Experiments Maximum velocity positionz00 00000 Experiments Maximum temperature position

00000 Experiments : Thermal jet expansion

0

Potentialcore 0

0 oa

Walldet orifice

00-zo -^.o ao is zo ^.o as 5a

X/D

Figure 6: Jet evolution - experiments (Bousgarbibs,1991).Velocity ratio 2/3 ; an =45.

Space Averaged Flow Computationation

Basic equationsWe start from the Navier-Stokes equations. The

conservation equation for the stagnation enthalpy is added to takethe thermal effects into account.

In fact, as far as the average level of the film coolingproblem is concerned, a dominant convection direction may oftenbe found ; however, the local flow has a strong three-dimensionalbehaviour, which is influenced by the jet itself. As a consequence,the spatial averaging between neighbouring jets must be appliedbefore an introduction of a parabolization hypothesis, which willlead to the boundary layer model.

Applying first the spatial averaging, then we get for the x-momentum equation

a a7 a a pu ax + pv a +a pu u)+a pv u)+2Kxpu v +2Kxpu v =

Y Y

ap kxx(16)

aTxY ax + ax -

+2xXcaxxY

The stresses t i j correspond both to laminar and turbulent

contributions ; K x is the wall curvature, assumed constant in the zdirection. Similar equations are obtained for the mass and energyconservation equations. These new equat ions present a simil arform with and without jets ; however, the ( pu"u" ) and ( pu"v" )terms explicitly take the jet effects into account. It is interestingto remark that, following this averaging technique, firstly the Z-derivatives have disappeared from these equations, secondly onlythe space averaged stresse s remain, and finally the momentum Z-component is reduced to D w /Dt =0 ; as a consequence, if the

6


upstream flow is two-dimensional and the jet trajectory is kept inthe X-Y plane, then w is always zero.

The thin layer approximation may now be used in order toeliminate the diffusion terms in the X-direction. We obtain :

based on the use of the real flow mass defect Sq with regard to theinviscid flow, defined as

(^ 0

s9=J (Peue - pu) dy (20)

0

a a pu+pv=0 (17)ax ay

au au -- aP xy a a (18)puax+pva + 2KXPuv= - aX+a -a (Pu)u)-a (Pv)u)

Y Y Y

pU- -T+Pvy _ ^TXy -qy - (P^)- ((Pu) Hi)-^(pv) H)i (1 9)ax a y ay ax ay

qy is the heat flux component in the Y direction.

The unknowns are ( u , v , Ht) ; in the present

computation, the density is computed according to the perfect gaslaw. The static pressure has been assumed constant in the Y-direction. Although this a usual hypothesis in classical boundarylayer approach, it is probably wrong when high v velocitycomponent exists under jet influence. As a consequence, thepresent computation must be seen as a first step only.

Although the static pressure P is identified with theinviscid value Pe , outside the film cooled layer, this inviscid staticpressure is not prescribed as in direct mode computation. On thecontrary, it is a supplementary unknown of our problem. The Pevalue is obtained from an equation which is a part of the strongcoupling between the inviscid outer flow and the averaged filmcooled layer. The main features of this coupling are brieflydescribed in the following paragraphs.

Viscous-inviscid interaction and numericalmethodThe coupling method between the inviscid and viscous wall

flows has been described in detail by Kulisa, Leboeuf, Klinger,Bernard (1990). It is well known that the classical direct modeboundary layer approach fails near the separation area or when astrong perturbation occurs near the wall, because of jet injectionsfor example. In order to stabilize the boundary layercomputations, it is necessary to reintroduce an elliptic effect nearthe wall, which has been lost with the parabolic hypothesis. Forthat purpose, we solve a supplementary single equation whichsimulates the inviscid response to the wall layer evolution. Thistechnique has a few advantages. Firstly, it allows us to solvesimultaneously the inviscid equation with the viscous wall layersystem, thereby allowing excellent stabilization of the computationwithout any relaxation requirement. Secondly, it avoids excessivecomputation as only that inviscid part, which is needed, issimulated.

We have developed an overlapping coupling method, in thesense that only one calculation domain is considered for thesolution of viscous and inviscid equations. As a consequence, theinviscid flow close to the wall is a fictitious one. The two flowsexchange information at the coupling boundary, which is located atthe wall. This maintains the subcritical character of the walllayer, which behaves like a subsonic flow whatever the outerinviscid Mach number may be. The viscous-inviscid interaction is

These defects Sq are introduced in the inviscid flow at thewall, and induced curvatures of the streamlines. The correction ofthe inviscid axial velocity component may be derived then from thePoincare 's formula (Kulisa, 1990) ; after some computation, weget :

1 1 ax(s9)+(pv)y=0 (2 1)Sue(M) _ n (x x ) dxLp M

(21) is called the "interaction system", and relates thevelocity correction at a point M on the wall to the mass flow defectdistribution, according to the required elliptic character. It takesthe real flow injection into account through the (pv) y= 0 term.

The space averaged flow equations (17) to (19) arediscretized according to the Keller box method which is a secondorder implicit scheme. The main feature of this scheme is to allowthe treatment of first order space derivatives only. As aconsequence of the parabolized hypothesis, a space marchingprocess is used. After discretization of the equation (21), weextract that line of the resulting matrix which corresponds to thecurrent computation station, and solved it with the system (17) to(19). The space marching procedure must be iteratively repeatedin order to update downstream defect mass flow values Sq.

Turbulence modelFor the present computations, the turbulence model is

based on the Boussinesq hypothesis and uses the standard two-layer model of Cebeci-Smith (1974) with some modifications inorder to take the jet injections into account. Mixing length modelfor film cooling application have already been developed. Herring(1975) multiplies the two-dimensional turbulent viscosity by afactor which depends on the jet effects ; however as the inner lawof t tends towards zero at the wall, this model is unable to takethe high level of turbulence encountered at the jet orifice intoaccount. Yavuzkurt, Moffat, Kays (1980) has developed a mixinglength model which is only suitable for computation in the mixingregion of the jets, far downstream the orifices.

In our model, the turbulent field is separated in a boundarylayer part without injection tBL and a jet part tjet, so that

t=tBL+tiet and :

21 (22)tjet = P 1 ^'SmaxY ^ 1 = DV d6

V S.a

y is the classical Van Driest function which allows todistribute tjet on the jet section. Note that tjet is a three-dimensional quantity which is space averaged in the z direction.


RESULTS OF COMPUTATION

First, the code has been tested on various flows withoutinjections ; our purpose was to demonstrate the performance of theviscous-inviscid interaction method and to verify the stability ofthe code (Kulisa, Leboeuf, Klinger, Bernard, 1990).

Now we pay attention to the description of the resultsobtained for the film cooled flow.

Experimental ConfigurationWe present here some results for an experimental

configuration which has been studied by Bousgarbies, Brizzi,Deniboire and Geron, (1991). A flat plate is used, with a row offive holes. The injection holes have a diameter of 0.005m, with apitch (distance between two neighbouring orifices) of 0.015m.The boundary layer on that plate is turbulent, with a thickness of0.010m. The jets are introduced in that flow with an injection rateof 2/3 and an angle of 45 0 with respect to the wall ; the inviscidexternal velocity is 30m/s. The jets are heated to a temperature of340K while the downstream flow has a temperature of 301K. Thetest wall is warmed internally ; a constant temperature of 310K isfixed at y=0, although variations have been observedexperimentally, especially near the orifices of the jets.

Aerodynamic AspectFirst, the aerodynamic results are presented. The evolution

of velocity vectors is shown in figure 7, for the spanwise positionz=0 ; the jet center line is located in that (x, y) plan. The velocityvector field is presented here at convergence of the numericalprocess ; it results from the superposition of the pitchwise spatialaverages and jet fluctuations.

The computational mesh was constructed with 40 points inthe Y direction ; three stations of computation are located in the jetorifice, thus the jet exit may be clearly seen from the strongdeviation of the velocity vectors at the wall. Downstream the jetorifice, a no-slip condition is recovered on the wall. Moreover, thejet influence is indicated by a "hump" in the velocity profiles atsome distance from the wall. This moves off and reducesprogressively at it is displaced downstream. This is a directconsequence of the mixing of the jets and the crossflow ; thismixing is expressed in the numerical method by the decrease of thesource terms towards zero. In practice, this is obtained beyond 5to 6 diameters. Moreover, the influence of the injection may benoticed upstream of the jet orifice. This is a typical elliptic effect,which is preserved by the inviscid interaction method.Qualitatively, the computation reproduces evolutions according tothe experimental observations.

A more detailed comparison is presented in figure 8,concerning the streamwise component of the velocity profile. Theresults are given for the symmetry plane z=0, and for severalstreamwise stations, x=0.54, x=14, x=24, x=44, measured fromthe center of the hole. Note that the stations x=0.54 and x=14, infigure 8, are located in the hole. The calculated velocity at the wallis underpredicted in comparison with the expected injectionvelocity. The reason is that the velocity profile at the jet exit doesnot have a constant value as assumed in the potential core region.This is probably connected to deviation of the jet trajectory in theinjection tube, before the orifice is reached, (Dibelius, Pitt, Wen,1990 ; Charbonnier, J.M., Leblanc, R., 1990). On the whole, thevelocity peak, which is associated with the jets, is calculated toofar from the wall in the viscous layer. However, the width of this

Figure 7 : Velocity vectors evolution; Z = 0 -CEAT experiments

U (m/s) 000 0

30 p

0 0 0 0 020 30

0 0 0 ^

p o

10 I 20 1 30 00 10 20 30 p O

O

O

0O

00 00 00000 X = O's 0 O X = 1 .0

0 10 20 AL AAA X = 2.000000 X = 4.0

0 10 z = 0

Y (mm)

6 8 10 12 14 16 1

Figure 8 : Streamwise velocity profile -CEAT experiments

velocity peak seems to be well predicted. At the outer edge of theviscous layer, the computation shows a mass excess which does notappear in the experiments. This effect must be connected with theviscous-inviscid interaction equation, which introduces anincrease of the inviscid outer velocity Ue under the effect of theinjections. This increase is reproduced on the complete layerthickness, according to the constant static pressure hypothesis inthe Y direction. In practice, a static pressure gradient exists, andsimilarly an inviscid velocity gradient also, in the directionnormal to the wall. Taking into account the previous remarks, weconsider that the experimental evolutions are qualitatively wellreproduced.

The spatial locations x=24 and x=40 indicate the mixing ofthe jets and of the viscous layer. The jet influence is clearlyreduced. Now, the computed maximum velocity agrees with theexperimental one. At these downstream locations, the jets arealmost completely deflected and in line with the crossflow. The jeteffects being less important, the influence of the interactionequation on the inviscid velocity Ue decreases and thus the outeredge velocity agrees with the experiments. However, theexperimental results near the wall show a mass defect which is notreproduced by the computation. This is probably connected with anoverestimation of the eddy viscosity.

Y


Thermal AspectNow, we consider the thermal aspect. As for the

aerodynamic field, the thermal quantities may be determined overthe whole domain, by reconstructing the complete three-dimensional field. The three-dimensional computed enthalpyevolution is shown figure 9. Because of the periodicity of the flowin the z direction, the visualisation around one jet only ispresented. In the x=0 station, the thermal effect, induced by thewarm jet exit, may be seen. Downstream, figure 9 shows themixing and the thermal diffusion.of the jets.

In figure 10, the spanwise planes (y, z) only are given, inorder to illustrate all the phenomena in the z-direction. Sixequidistant surfaces are located from the orifice (x=0) up to 7diameters downstream. We may note the elliptic shape of theenthalpy lines, which reproduce, in fact, the shape of the jetcross-section. As before, the decrease of these elliptic regions is aconsequence of the thermal diffusion.

Complementary views are presented in figure 11. Differentareas (x, z), parallel to the wall, are located from y=0.00025mup to y=0.003m. Downstream diffusion may be observed for theclosest plan to the wall. At y=0.003m, the downstream extensionis a consequence of the jet bending. Although no experimentalvalues were available at that time for this test case, it may benoted that the computed evolutions follow the general trends,observed by other authors.

3.3e+05

3e+05

X = 8D

Figure 9 : Computed enthalpy distribution

3.3e+05

p.

3e+05

X = 0 / // /

X = 7D


3.3e+05

3e+0E

Y = 0.25 mm


Finally, the position of temperature maximum is givenfigure 12, in the symmetrical plan of a jet z=0. The computedresults agree well with the experimental one. The "thermaltrajectory" of the jet is well predicted.

Figure 13 presen ts the evolutio n of the space averaged heattransfer coefficient h = b / ( T y =O - Te ), where b denotes

the averaged heat flux density at y=0. Up to x/4=-1, h decreasesas the consequence of the growth of the wall layer which is forcedto flow over the downstream jets. Between x/4=-0.7 and 0.7,negative value of h occur ; these stations are located inside the jetorifice. Downstream the orifice, the jet is in contact with the wall.

YID

1.0

09

0.8

0.7

0.6

05

0.4

0.3 Calculation

00000 Experiments0.2

0.1 X/D

0.00 1 2 3 4 5 6 7

Figure 12 Temperature maximum position. Z = 0 -CEAT experiments

n

9


Up to x/4=3.5, the jet temperature is higher than at the wall(figure 14) ; as a consequence, negative h values will appearlocally at the jet level ; as we show here, the space averaged valueh in figure 13, negative values only occur in the vicinity only of

the orifice; such behaviour has already been observed by Camciand Arts (1985) and Dorignac (1990). The jet influence seemsparticularly important up to x/$=4 ; downstream, the h evolutionis mostly dominated by the wall itself.

Note that the stability of the numerical method, reinforcedby the viscous-inviscid interaction equation, has been verified. Inthis case, 5 iterations between the viscous layer and the jetcomputations are necessary to obtain convergence of the completeprocedure.

h (W/m 2/K)

300

200

100

0

X/D-100

-5 0 5 10 15 20

Figure 13 : Space average heat transfer evolution

T t ( C)

70

60

50X = g 54

40 Z - 0

30

20Y (mm)

100 1 2 3 4 5 6 7 8 9 10

CONCLUSIONS

A method of modelling viscous flows on cooled turbineblades has been developed. Our approach is based on iterativeinformation exchange between a viscous wall layer computationand a jet computation.

The viscous wall layer modelling uses a space-averagetechnique which allows the problem to be kept twodimensional.However, source terms appears in the equations. The parabolicnature of these viscous layer equations allows the use of a spacemarching method. In order to ensure the stability of the method, astrong viscous-inviscid interaction method was introduced. Theelliptic effect, which is thus introduced, requires the repetition ofthis space marching procedure.

The jet computation is a three-dimensional computation,using an integral method. The jet equations are solved on the jettrajectory, and closure relations are used. The jet computationresults produce information necessary to determine the sourceterms, which are passed to the space-average viscous layercalculation.

Tests were performed on a film cooled flat plate. Theaerothermal physical evolutions are well reproduced. Moreover,the numerical stability of the method was verified.

In order to apply this method to the computation of flow incooled turbine, more work are needed. Firstly, the assumption ofconstant static pressure in the direction normal to the wall, has tobe eliminated ; improvements in the jet computations are needed inparticular to include the effects of the counter-rotating vortices ;the turbulence needs also to be modify in order to include specificjet effects. Secondly, it is important to have access to good qualityexperimental data, including both aerodynamic and thermalinformations.

Adknowledgements :

We would thank SNECMA for the financial support of thisstudy. The authors are also grateful to the staff and students ofCEAT of POITIERS which gives us access to experimental data, andespecially Mr Leblanc, BousgarbiOs and Vuillerme.

REFERENCES

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Bario, F., Leboeuf, F., Onvani, A., Seddini, A., 1990,"Aerodynamics of Cooling Jets introduced in the Secondary Flow ofa low Speed Turbine Cascade", ASME Journal of Turbomachinery,Vol. 112, n3, Toronto.

Bousgarbibs, J.L., Brizzi ,L., Deniboire, Geron ,M. ,1991,"Interaction of Jets with a Boundary Layer Crossflow" ,submittedto 1991, ASME Conference, Orlando.

Camci C., Arts T., 1985, "Short Duration Measurementsand Numerical Simulation of Heat Transfer along the Suction Sideof a Film cooled Gas Turbine Blade", ASME paper 85-GT-111,Houston.

Figure 14 : Temperature profile

10


Camci C. ,1988, "An Experimental and NumericalInvestigation of Near Cooling Hole Heat Fluxes on a Film CoolesTurbine Blade", ASME paper No 88-GT-9.

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Dorignac, E. ,1990, " Contribution A I'Etude de laConvection Forcee sur une Plaque Plane en presence de JetsParietaux dans un Ecoulement Subsonique ", PhD Thesis,University Poitiers.

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