Comparison of viscous flow field calculations applicable ...€¦ · Comparison of viscous flow field calculations applicable to turbomachinery ... KWU Group, Gas Turbine Technology,

Comparison of viscous flow field

calculations applicable to turbomachinery

design with experimental measurements

L.J. Lenke," A.W. Reichert,* H. Simon"

" Institute of Turbomachinery, University of Duisburg,

47048 Duisburg, Germany

* Siemens AG, KWU Group, Gas Turbine Technology,

45473 Miilheim, Germany

Abstract

The aerodynamic design of turbomachinery components is a field of economicimportance where Computational Fluid Dynamics methods are applied formany years. For the high Mach numbers and the high Reynolds numbersappearing in the components, the accuracy requirements concerning the pre-dicted data are very high. Data of interest are the flow field losses, the flowdirection and the shock configuration in supersonic regions. For selectedexamples from the wide field of turbomachinery design comparisons of nu-merical and experimental results are given for these data. The numericalresults are calculated by a State Difference Splitting code of high accuracyrecently developed at the University of Duisburg. The numerical results areshown to be grid independent by calculations on different fine grids. In thiscomparison the accuracy, which can be gained by the use of a modern NavierStokes code becomes visible. In general, very good agreement between calcu-lated and measured data is shown. Some differences are found to be causedby the turbulence modelling. In these cases, the results of different turbu-lence models (k-u and Low-Reynolds-number k-e model) are investigated,showing their influence. The choice of appropriate boundary conditions isreferenced.

Introduction

The requirement for reliable turbulence models has become increasingly im-portant in the prediction of complex flows. Among the turbulence modelsused today, two equation eddy viscosity models appear to be favoured. Espe-cially the k-c. model by Launder & Spalding [7] is probably the most popular.One major difficulty with the k-c model involves its application to near-wallturbulent flows. Therefore a number of alternative models or modificationsof the k-c. model have been developed. A most notable alternative model isthe k-u model suggested by Wilcox [19]. It has the advantage that it does

Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X

356 Computational Methods and Experimental Measurements

not require damping functions in the viscous sublayer and the equations aremathematically simpler especially with regard to numerical stability.

For the development of efficient turbulence models a comparison withreliable experiments is required. For this reason and because of the abovementioned advantages offered by the k-u model a comparison of a modifiedk-LJ model suggested by Menter [11] including a function which describesthe influence of streamline curvature with the Low-Reynolds-number modelsuggested by Lam & Bremhorst [6] including a new function for higher con-vergence rates is presented in this paper.

First the computational results are shown for a Mach 2 flow behind arearward-facing step to describe the principal differences between the twoturbulence models. Then the flow fields of the plane SE1050 profile cascadewith two different extreme values of the incidence angle and of the VKI-1turbine cascade have been calculated.

The Numerical Scheme

The Governing EquationsThe turbulent flow of a perfect gas in an arbitrary domain may be describedby the averaged conservation equations for the averaged physical quantitiesspecific momentum /rv, density p and specific total internal energy %*, allquantities related to volume. For simplicity these conservative variables aregathered to a vector u. One can find the conservation law for this vector ina fixed domain V with the surface S to be

— / udV + <{>¥*ds = I F^ds + S. (1)at Jv Js Js

In these equations the flux tensors F^, F^ and the vector S are in-troduced, describing the convective (F*), diffusive (F ) transport and thesource terms of the conservative variables (S). The flux tensors connect thethree-dimensional physical space with the five-dimensional vector space ofthe vector u, which contains the physical one as a subspace. The flux ten-sor of the inviscid flow F^ depends on the vector u only, while the tensorF^ mainly depends on gradu. These dependencies are described by thethree-dimensional Navier-Stokes equations

u =pv

p

v -f P

0_TvH

I0V

- q

(2)

(3)

(4)


Computational Methods and Experimental Measurements 357

p = (7 - 1) ut - vv - (5)

T = (fi + fit) (grad v + grad? v - - div v l) - \pk I,V o / o

(6)

(7)

In these equations v abbreviates the velocity vector, p the static pressure,T the static temperature, 7 the ratio of the specific heat capacities, // and //<the molecular and turbulent viscosity, A and A, the molecular and turbulentheat conductivity and I abbreviates the identity tensor and the discretizationmethod of the gradients based on offset control volumes. Incorporating aneddy viscosity formulation, T and q describe the effective stress tensor andthe effective heat flux vector. The operator o declares the dyadic product,while the scalar product is not declared especially.

Low-Reynolds-Number k-e Model (LB-model)The conservation law (1) for the Standard k-c equations suggested by Laun-der & Spalding [7] are given analogous to equations (2-4) by

pkFf = u< o v,

grade

pt

(8)

(9)

(10)

P =T 2 \

grad v + grad v -- div v I j grad vo /

--pklgradv,

A - rA, - — c,.

(11)

(12)

The Standard k-e model neglects the molecular diffusion compared toturbulent diffusion so that this high-Reynolds-number form of the k-e, modelis not valid in the vicinity of solid walls. In the low-Reynolds-number formof the k-e, model devised by Lam & Bremhorst [6], the viscous effects areaccounted for by replacing the original set of constants by functions of theturbulent Reynolds numbers RT and Ry

(13)

(14)



with

/„ = [!- exp(-0.0165#»)]*(l + 20.5/Ar),

/2 = max[0.95; 1 - tanh(max[0; (/„ - 1)]')],

^ t/\/fcIlT = /) , -Tty = /9

(15)

(16)

(17)

(18)

to integrate the k and e transport equations down to the wall.The new function ji limits the function c^ to a minimum value of 0.096

in regions in which //< becomes very small (or equivalent f greater than 2.8).This function avoids very small (or negative) values of k in these regions sothat higher Courant numbers (Courant et al. [3]) can be used. The function/2 has no significant influence on the numerical results but higher conver-gence rates can be obtained.

k-u> Turbulence ModelUnlike any other two-equation model, the k-w model does not involve damp-ing functions to integrate through the viscous sublayer and allows simpleDirichlet boundary conditions. In this investigation a modified k-u modelsuggested by Menter [11] is used, given analogous to the LB-model by

pkpu

where 0 is the absolute value of the vorticity and F% is given by

T o / 2\/t 500/2Fz = tanh max^ ; -^-

L . \ >

The five constants of this model are set to:

/o* p

0.55 0.075 0.09 1.176 2.0 0.91

(19)

(20)

(21)

(22)

(23)

The function /sc in equation (22) describes the influence of streamlinecurvature which reduces in some cases drastically the discrepancy between



Prandtl number = 0.7 Prescription: o, 7\, ptExtrapolation: pvy, pk, pt or pw

Periodic boundary conditions

Prescription: pV = pk = 0, pwExtrapolation: p, T, pe

Periodic boundary conditions

Prescription: pExtrapolation: pV, Tt, pk, pt or pw

Figure 1: Boundary conditions for the turbulent flow through SE1050 andVKI-1 turbine cascade.

computed and experimental results. The function fsc suggested by Leschziner& Rodi [9] for the k-c model and is used in this investigation with the k-tomega model. This function is given by

= max n 1 • I 10.1, 1 , (24)

with the radius of curvature R^ the velocity parallel to the streamlines Usand the constant KI = 0.267 (details described by Leschziner & Rodi [9]).

Boundary ConditionsThe boundary conditions which are used for the k and e transport equationsare given by

* = 0; g = 0 (25)

at solid surfaces, where dc/dy describes the gradient of e normal to the wall.The Dirichlet boundary conditions for the k and w equations at solid

surfaces are given by

6 = 0; (26)

where At/i is the distance to the next gridpoint away from the wall.The complete boundary conditions used for the different turbine cascades

are shown in figure 1 with the prescription of the isentropic exit Mach numberM is (SE1050: M^ = 0.905 for t = -67° and M^ = 1.012 for i = 30°; VKI-1: Ma,-. = 1.2).

The boundary conditions at the wall used for the rearward-facing stepare the same compared to figure 1 with an inlet Mach number MI = 2 and



the prescription of the velocity and pressure at the inlet taken from measure-ments.

The Discretisation MethodDividing the surface S of the control volume V into a finite number of planes,given by their normal-vectors s*, and using proper averages for the flux-tensor F in the planes and for the unknown vector u in the domain, equation(1) becomes

^ = -£S>'. F = F'-F". (27)

To implement equation (27) into a numerical scheme, an assumption has tobe made: Since there is only one set of equations to determine the averages ofthe unknown values (at the domain center) the mean value of the flux tensorat the surface has to be assumed by an interpolation procedure. To indicatethe assumption made, the flux tensor determined at the domain surface shallbe called numerical flux tensor F^.

Introducing an unit vector g* with the same direction as the plane normal-vector s*, one can define the product with the flux tensor

f*N _ pNg^ (28)

which is called flux vector or numerical flux function. Using the norm s* ofthe plane vector s*, equation (27) gets:

This is a general finite volume formulation of the Navier-Stokes equation (1).The accuracy of equation (29) depends on the evaluation method for the fluxvector. The authors have developed a new method, picking up elements fromRoe's [17] and Osher's [12] scheme, which shall be described briefly.

The numerical flux vector is calculated according to the State DifferenceSplitting technique, using an intermediate state u*:

fiN = F'(u*)g'' - FV. (30)

This technique is used by Osher and differs from the flux difference splittingtechnique, which is used in Roe's scheme, where an intermediate flux vectoris calculated.

The intermediate state u* is calculated from the two states u+ and u~which are interpolated from two different sets of gridpoints as depicted infigure 2. The interpolation is done using the van Albada (van Albada etal. [1]) limiter or a polinominal fitting. It is carried out in characteristicvariables u&, which are those components of the vector u, to which knownpropagation speeds \c(&) can be related. They are generated by a proper



Finite volume face

Left set of gridpoints

Right set of gridpoints

Figure 2: Definition of states used in the flux calculation

choice of base vectors g& and g*. These base vectors and the propagationspeeds AC (a) are evaluated using the density weighted average proposed byRoe.

The differences between the two states u+ and u~ are regarded as wavespropagating in different directions corresponding to their propagation speeds.The intermediate state u*, needed to calculate the flux vector at the cell faceis composed of those wave components, which propagate towards the cellface. The intermediate state may be found by following a path connectingthe two states u+ and u~ and looking for the change of sign of the actualpropagation speed. In general, there are six locations along the path wheresuch a change of sign is possible, as described by Osher.

Since the linear transformation to characteristic variables is only appli-cable to differences, the left cell values (L) are chosen as a reference state ateach cell face.

u* = Ui + £Auag*; (31)

forAc(6)>0forAJ&i <0

(32)

To prevent expansion shocks, the difference AAc(&) in the propagationspeed between the states u+ and u~ is calculated assuming a linear variationbetween both states:

U+ — U" (33)

The derivative is evaluated using the density weighted average. An expan-sion wave is indicated by a positive sign of the difference A AC (a) correspond-ing to one of the two characteristic pressure waves. Using the abbreviation



s = 1/2 — Ac(c*)/AAc(a), the complete algorithm becomes:

l, max(0, s))x(u+ - u")g& for AAc(d) > 0 . (34)

Equation (32) else

This is a highly accurate difference splitting method, preventing unphysicalsolutions as known from Roe's scheme (Hart en & Hyman [4]) but using itssimple algebra for the difference split. This transformation to characteristicvariables is not nessecary for the components of the u^- or u^-vector.

The components of the effective stress tensor T, needed to calculate thecomponents of the flux vector F^g\ are discretised using central differences.

The Iterative MethodIn the investigation only steady state solutions are considered. For high con-vergence rates, an implicit, Newton-Raphson-like iterative method is usedto solve first the averaged conservation equations and then the k-c or k-u-equations (Rai & Chakravarthy [13]). According to this method equation(29) is rewritten:

(35)

The split flux jacobians A* — grad^F^ are evaluated using "first order" Roeflux difference splitting. The subscripts L and R denote values from the left(L) and right (R) cells sharing the surface s*. The different treatment of thediscrete operators on the left and right side of equation (35) allows the use ofa very accurate solution method combined with a low effort implicit operator.On the other hand, the method results in a mild restraint of the time step size,which is locally calculated using a constant Courant number CFL. For higherconvergence rates a smaller Courant number (CFLk = 0.2 . . . 0.5 • CFL) isused to solve the k-c- or &-w-equations.

Furthermore after each time step very small values of A;, e and w are setto a minimum value with

',lQ-*kmax], (36)

;10-*emaj, (37)

w = max[w; lO tcWc]- (38)

At the boundaries either the prescription of a physical quantity is neededor the extrapolation of this quantity from the interior to the boundary. The



Figure 3: H-C-grid. a) Grid topology of an H-C-grid (reduced number ofgridpoints). b) Part of the grid topology used for the rearward-facing step(87x70 points).

physical quantity can be chosen from a large variety of quantities, includingcharacteristic components for nonreflecting boundary conditions. All bound-ary conditions have the form

flu = (39)

with a being the physical quantity or its extrapolation function. To keep theconvergence rate high, this equation is implemented to the iterative methodin a Newton-Raphson formulation.

= a,*r<,e, " <^ (40)

Grid GenerationThe CFD program has been coded for a structured grid. The benefit of thisgrid type can be found in the low effort needed for the gridpoint adminis-tration. Problems occur with these grids due to the calculation of the flowthrough turbine cascades, if usual grid topologies as H- or C-grids are used.Using these topologies one will get extremely distorted cells due to the pe-riodic boundary conditions. Therefore a grid topology especially for turbinevanes has been developed, that shall be called H-C-grid (Reichert & Simon[14]).

Figure 3a) shows this grid topology used for the SE1050 and VKI-1 tur-bine cascade. One can see the grid distortion in the sensitive region of the



nozzle exit being low. Furthermore the trailing edge is captured very wellby the grid. In this region a supersonic expansion may appear, followed bycompression shocks, thus a very dense grid is needed to capture this phe-nomenon. In contrast to the trailing edge small Mach numbers appear at theleading edge, which are accompanied by small variations of the flow quanti-ties. For these small variations the grid density may be lower at the leadingedge compared to the trailing edge.

The structured grid used for the rearward facing step is shown in figure3b). One can see the very dense grid in the region where the oblique com-pression shock appears.

Numerical Results

Rearward-Facing StepFigures 4b) - c) show the calculated contour plots of the pressure distributioncompared to measurements for the Mach 2 flow over the rearward-facingstep (figure 4a) taken from McDaniel [10]. In figure 4a) it is seen that theflow expands from the free stream pressure across isobars that coincide withexpansion waves emanating from the step. The flow is then recompressedby the oblique shock wave which is formed away from the wall near the flowreattachement line. One can seen that the LB-model calculates a very thinoblique shock over only one grid cell compared to the k-uj model.

The measured pressure profiles at x/H = 3.99 in figure 5b) show theformation of the oblique shock wave near the flow reattachment line. Bothturbulence models show already the formation of the oblique shock wave atx/H = 1.75 in figure 5a) so that the turbulence models calculate a lowerpressure behind the step. But in the vicinity of the wall the pressure profilesof both models lie close together and agree with the measurements.

The u-velocity components at x/H = 3.99 in figure 5e) are seen to agreewell and both, the measurements and the turbulence models show clearlythat the flow has reattached to the wall at this station. But the k-u model(with fsc = 1) calculates a greater recirculation region due to higher returncurrent at x/H = 1.75 in figure 5d) compared to the LB-model so that thek-u model agrees something better with the measurements.

At station x/H = 7.67 in figure 5c) the strong compression is seen acrossthe fully developed oblique shock wave at y/H = 1.5. The location of thisoblique shock calculated by the two turbulence models is very similar butthe width of this shock differs between the two models.

Without regard to the above mentioned differences the results of bothturbulence models are similar and agree well with the measurements.



H

Figure 4: Mach 2 flow over a rearward-facing step: a)-c) pressure contours(p/Poo); (a): data taken from McDaniel et al. [10]; (b): LB-model; (c): k-umodel with fsc = 1



6 -

JLH

2 -

a o

LB

0.2 0.6 1.0P/Poo

d) o

= 3.99

6

4 -

_ _H

2 -

b) 0

6 -

0.6 . 1.0P/Poo

e) o

= 7.67

1.0

6 -

1LH

2 -

c) o

6 -

JLH

2 -

0.6 . 1.0P/Poo

f) o-0.2

Figure 5: Mach 2 flow over a rearward-facing step: a)-c) pressure profiles(P/Poo); d)-f) velocity profiles (I//WQO) at various locations x/H (see fig. 4a).



2000. 4000.

iterationsFigure 6: Convergence history of the SE1050 profile cascade (i = -67°).

SE1050 Profile CascadeThe results from the wind tunnel measurements of the SE1050 profile cascadewith two different extreme values of the incidence angle have been used toverify the 2D calculations with the turbulence models.

Figures 7a) and b) show the simulated density distributions (incidenceangle i = —67°) compared with an interferometric picture taken from Stastny& Safarik [18]. Both calculations are very similar and show good agreementwith the measurements. The location of the compression shock agrees wellwith the interferometric picture. Differences between the turbulence modelsare shown by the exit flow angle fa. The k-u model calculates a greaterdeviation from the measurements compared to the LB-model which is shownin table 2.

Table 2: outlet angle fa

incidence anglez = -67° 2 = 30°

MeasurementLB-modelk-u model

30.3031

181

oo0

303030

669

00o



b)

Figure 7: Turbulent flow through SE1050 profile cascade (i — —67°). Densitydistribution with Lam-Brernhorst model (a) and k-u model (b) comparedwith interferometric picture c) taken from Stastny & Safarik [18] (k-cu modelwith /„ = 1).



a)

c)

Figure 8: Turbulent flow through SE1050 profile cascade (i — 30°). Densitydistribution with Lam-Bremhorst model (a) and k-u model (b) comparedwith interferometric picture c) taken from Stastny & Safarik [18] (k-u modelwith /,c = 1).



The main differences between the turbulence models are shown by the flowseparation at the pressure side. The extension of the recirculation region bythe k-w model is approx. 15% greater compared to the LB-model.

Furthermore, another difference between the two turbulence models istheir convergence history shown in figure 6. The k-w model needs less itera-tion steps compared to the LB-model due to the greater CFL- and CFLk~Numbers which can be used even at the beginning of a calculation. Butthis advantage of the k-u} model will be reduced by the function /2 in theLB-model.

The flow through the SE1050 cascade with an incidence angle i = 30°is shown as an interferometric picture in figure 8c). The comparison withthe calculated density distributions (figure 8a) and b) shows good agreementwith the interferometric picture. The location of the compression shock atthe trailing edge and the width of the recirculation region at the suctionsurface agree very well with the measurements. The main difference of thek-w model is the approx. 10% greater recirculation region compared to theLB-model. The calculated outflow angles lie close to the experimental valuecompared to the calculation with i = —67°.

In both cases the function fsc in the k-w model is set to one. The functionfac will lead to greater extensions of the recirculation regions which will notagree furthermore with the measurements.

VKI-1-Turbine CascadeFigure 9 shows the comparison of the simulated Mach number distribution(isentropic exit Mach number M is = 1.2) with a schlieren picture taken fromLehthaus [8]. In the inlet flow field there are no differences between the twoturbulence models. The differences just exist in the trailing edge region. TheLB-model leads to greater Mach numbers in this region. But the locationsof the two shocks are very similar.

The interaction of the oblique compression shock with the boundary layerat suction surface results in a separation bubble in the viscous sublayer rec-ognizable in the schlieren picture. The main differences between the twomodels are shown in figure 10 by this separation bubble at 72% of chordlength and in figure 11 by the distribution of the isentropic surface Machnumber. The k-w model is able to calculate this separation bubble and anoblique compression shock caused by the flow diversion at the beginning ofthe separation bubble only if the function fac is included in the k-w model.The extension of the separation bubble cannot be reflected correctly. Theextension calculated with the k-w model is approx. 30% greater comparedto the LB-model but always smaller compared to the measurements whichis shown in figure 11 by the to small calculated plateau at chord length 72%.Without the function fac the k-w model is not able to calculate this separa-tion bubble which is shown in figure 11 by the nonexisting plateau at chordlength 72%.



b)

Figure 9: Turbulent flow through VKI-1 turbine cascade (isentropic exitMach number = 1.2). a)-b) Mach number distribution with Lam-Bremhorstmodel (a) and fc-u; model including fsc (b) compared with a schlieren picture(c) taken from Lehthaus [8].



Figure 10: Separation bubble at the suction surface of the VKI-1 turbinecascade. Mach number distribution with Lam-Bremhorst model (a) andk-u model including fsc (b).

25 50 75

% Chord length

100

Figure 11: Isentropic surface Mach number distribution compared to mea-surement taken from Baines et al. [2]. ( Lam-Bremhorst model; k-u>model including fsc't &-w model).



Figure 12: Exit flow angle and loss coefficient taken from Lehthaus [8](x Lam-Bremhorst model; D k-u model including /^; o k-uj model).

The exit flow angles of the turbulence models are very similar and showgood agreement with the VKI-measurements in figure 12. The influenceof the streamline curvature on the loss coefficient for the k-u model canbe clearly seen in figure 12. The loss coefficients calculated with the LB-model and the k-w model including /^ lie close together (£LB = 0.084;£k-u>,ac = 0.085) but without the function /,<. the k-w model differs byA£ = 16.5% from the LB-model (&_„ = 0.071).

Conclusions

In this investigation numerical simulations with the presented two turbulencemodels for various transonic flow examples have shown their capability togenerate reliable numerical results compared to measurements. The Machnumber, the density distributions and the location of compression shocks canbe calculated very well compared to interferometric and schlieren pictures.Furthermore the calculated isentropic surface Mach number distributions andexit flow angles lie mainly close to the measured values.

The main differences between the two turbulence models are shown bythe calculation of recirculation regions. The k-w model calculates generallygreater recirculation regions compared to the Lam-Bremhorst model. For thecalculation of the interaction of compression shocks with boundary layers the



influence of streamline curvature has to be included in the k-uo model.Another difference between the two models is shown by the calculation

of strong oblique compression shock waves. The Lam-Bremhorst model cal-culates very thin oblique compression shocks extending over only one gridcell compared to the wide oblique compression shocks calculated by the k-u:model.

References

[1] vanAlbada, G. D., van Leer, B., Roberts, B.B., A Comparative Studyof Computational Methods in Cosmic Gas Dynamics, Astronomy andAstrophysics.,, 1982, 108, 76-84

[2] Baines, N. C. et al., A Comparison of Aerodynamic Measurements ofthe Transonic Flow Through a Plane Turbine Cascade in Four EuropeanWind Tunnels, 8th Symposium on Measuring Techniques for Transonicand Supersonic Flow in Cascades and Turbomachines, Genoa, Italy,1985, 24-25 October

[3] Courant, R., Friedrichs, K. 0., Lewy, H., Uber die partiellen D inferential -gleichungen der mathematischen Physik, Mathematische Annalen, 1928,100, 32-74

[4] Harten, A., Hyman, J.M., Self Adjusting Grid Methods for One-Dimensional Hyperbolic Conservation Laws, Journal of Computational

s, 1983, 50, 235-269

[5] Kiock, R., Lehthaus, F., Baines, N. C., Sieverding, C.H., 1985, TheTransonic Flow Through a Plane Turbine Cascade as Measured in FourEuropean Wind Tunnels, ASME Paper 85-GT-44, 1985

[6] Lam, C.K.G., Bremhorst, K., A Modified Form of the k-t Model forPredicting Wall Turbulence, Journal of Fluids Engineering, 1981, 103

[7] Launder, B.E., Spalding, D.B., Mathematical Models of Turbulence,Academic Press, 1972

[8] Lehthaus, F., Experimented Untersuchungen an einem Gasturbinen-Laufrad-Gitter mit dem Schaufelprofil VKI-1, DFVLR AerodynamischeVersuchsanstalt Gottingen, Germany, 1975

[9] Leschziner, M.A., Rodi, W., Calculation of Annular and Twin ParallelJets Using Various Discretization Schemes and Turbulence- Model Vari-ations, Journal of Fluids Engineering, 1981, 103, 352-360



[10] McDaniel, J., Fletcher, D., Hartfield Jr., R., Hollo, S., Staged Trans-verse Injection into Mach 2 Flow Behind a Rearward- Facing Step: A

3-D Compressible Test Case for Hypersonic Combustor Code Validation,AIAA-91-5Q71, 1991

[11] Menter, F.R., Two-Eqation Eddy- Viscosity Turbulence Models for En-gineering Applications, AIAA Journal, 32, 1598-1605

[12] Osher, S., Solomon, F., Upwind Schemes for Hyperbolic Systems ofConservation Laws, Mathematics of Computation, 1982, 38, 339-377

[13] Rai, M. M., Chakravarthy, S. R., An Implicit Form for the Osher UpwindScheme, /1//W JowrW, 1986, 24, 735-743

[14] Rei chert, A.W., Simon, H., Numerical Investigations to the OptimumDesign of Radial Inflow Turbine Guide Vanes. International Gas Turbineand Aeroengine Congress, den Haag, ASME-Paper 1994, GT-61

[15] Reichert, A.W., Simon, H., A Numerical Scheme Based on the Differ-ence Splitting Technique Solving the Euler Equations. To be publishedin Computers and Fluids, Pergamon Press, 1995, Great Britain,

[16] Reichert, A. W., Simon, H., Design and Flowfield Calculations for Tran-sonic and Supersonic Radial Inflow Turbine Guide Vanes. To be pub-lished at the ASME-Gas Turbine and Aeroengine Congress, 1995

[17] Roe, P. L, Approximate Riemann Solvers, Parameter Vectors and Dif-ference Schemes, Journal of Computational Physics, 1981, 43, 357-372

[18] Stastny, M., Safarik,P., Experimental Analysis Data on the TransonicFlow Past a Plane Turbine Cascade, ASME Paper 90-GT-313

[19] Wilcox, D.C., A Half Century Historical Review of the k-u Model,Paper , 1991, 615


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Comparison of viscous flow field calculations applicable ...€¦ · Comparison of viscous flow field calculations applicable to turbomachinery ... KWU Group, Gas Turbine Technology,