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.- 4 . NASA Technical Memorandum 100878 Numerical Analysis of Three-Dimensional Viscous Internal Flows :NASA-TM-130878) N U B E E I C A L ANALYSIS i ~ 2 N88-211 16 TIIR EE- DIMENS1 O N A L VISCOUS I N T E R N A L EL 3 US (NASA) 10 p CSCL U l b IJnclas G3/C 1 C 1363 23 RodrickV. Chima NASA Lewis Research Center Cleveland, Ohio and Jeffrey W. Yokota Sverdrup Technology, Inc. (Lewis Research Center Group) NASA Lewis Research Center Cleveland, Ohio Prepared for the First National Fluid Dynamics Congress cosponsored by the AIAA, ASME, ASCE, SIAM, and APS Cincinnati, Ohio, July 24-28, 1988 NUMERICAL ANALYSIS O F THREE-DIMENSIONAL VISCOUS INTERNAL FLOWS Rodrick V. Chima NASA Lewis Research Center Cleveland, OH 44135 and Jeffrey W. Yokota Sverdrup Technology, Inc. (Lewis Research Center Group) NASA Lewis Research Center . ABSTRACT Cleveland, OH 44135 A three-dimensional Navier-Stokes code has been de- centrifugal impeller. Each of these two-dimensional vis- veloped for analysis of turbomachinery blade rows and cous effects can be expected to lead t o secondary flows in other internal flows. The Navier-Stokes equations are three dimensions. Complex geometries, endwall bound- written in a Cartesian coordinate system rotating about ary layers, tip clearance effects, etc. also lead to three- the z-axis, and then mapped t o a general body-fitted dimensional flows in turbomachinery. It is the goal of coordinate system. Streamwise viscous terms are ne- the present work t o begin to predict some of these three- glected using the thin-layer assumption, and turbulence dimensional viscous effects. effects are modelled using the Baldwin-Lomax turbulence Several steady three-dimensional analyses for tur- model. The equations are discretized using finite differ- bomachinery have been published lately. Among them ences on stacked C-type grids and are solved using a mul- are the work of Dawes3 and Subramanian and Bozzola4. tistage Runge-Kutta algorithm with a spatially-varying Both used Runge-Kutta schemes implemented on sheared time step and implicit residual smoothing. H-type grids. H-grids are particularly easy t o generate Calculations have been made of a horseshoe vortex and implement for turbomachinery calculations, but suf- formed in front of a flat plate with a round leading edge fer from poor leading-edge resolution. Rai5 has published standing in a turbulent endwall boundary layer. Compar- a notable analysis of unsteady threedimensional rotor- isons are made with experimental data taken by Eckerle stator interaction in an axial turbine. He used a third- and Langston for a circular cylinder under similar con- order accurate upwind implicit scheme and a system of ditions. Computed and measured results are compared patched and overlaid 0- and 11-type grids for good reso- in terms of endwall flow visualization pictures and total lution of viscous phenomena. His analysis has not been pressure loss contours and vector plots on the symme- used for steady flows, however. try plane. Calculated details of the primary vortex show In this paper we describe a numerical method for excellent agreement with the experimental data. The cal- analyzing three-dimensional viscous flows in isolated tur- culations also show a small secondary vortex that was not bomachinery blade passages. The underlying Cartesian seen experimentally. The calculations required about 1.6 formulation allows the method t o be applied easily to million words of storage and 1.3 hours of CPU time on a both Cartesian and cylindrical geometries. Stacked C- Cray X-MP computer. type grids give good resolution of critical leading-edge Calculations have also been made of an annular tur- regions. bine stator that has been tested experimentally a t NASA A multistage Runge-Kutta scheme i used t o solve s Lewis. The Mach number ranged from from about 0.21 a t the finite-difference form of the thin-layer Navier-Stokes the inlet to 0.67 a t the exit, and the axial chord Reynolds equations with a Baldwin-Lomax turbulence model. A number was about 1.7 x lo5. Computed surface pressure spatially-varying time step and implicit residual smooth- distributions compare well with measured values at three ing are used to accelerate convergence of the scheme to a span-wise locations. The endwall boundary layers pro- steady state. Two calculations are presented to validate duce horseshoe vortices a t the leading edge of the blade. the analysis. The first calculation shows the formation Computed wake profiles resemble the measured profiles, of a horseshoe vortex at the leading edge of a flat plate but computed efficiencies are lower than measured values with a round leading edge that stands in a turbulent end- by 3 factor of two. wall boundary layer. Comparisons made between calcu- lated results and experimental data for a circular cylin- der under similar conditions show excellent agreement INTRODUCTION between static pressure distributions and flow visualiea- tion pictures on the endwall, and between static and total Much of our work in the past few years has in- pressure contours and velocity vector plots made on the volved the analysis of two-dimensional or quasi-three- symmetry plane. The second calculation is of the flow dimensional blade-to-blade flows in turbomachinery'*2. through an annular turbine stator. Comparisons made In these two references, both Euler and Navier-Stokes re- between calculated and measured static pressure distri- sults were presented for each blade row considered, and in butions compare well at three span-wise locations. each case a significant viscous effect was observed. These effects included such things as a pressure-surface or 'cove' separation on an axial turbine blade, a reduction in the GOVERNING EQUATIONS choking mass flow and a change in shock location for an axial compressor blade, and a reduction in the peak Mach The Navier-Stokes equations are written in a Carte- number and shock strength near the leading edge of a sian ( 2 ,y, z ) coordinate system rotating with angular ORIGINAL PAGE IS OF POOR QUALITY. velocity R about the axis. Th rotatic introduces and source termn in the y- and z- momentum equations. The Cartesian equations are mapped to a general body-fitted Cl = q:: + rl,' + Or2 (E, q ,<) coordinate system using standard techniques. q c,= 3 rlrafJu+ rluaov + Vratlw) The (-coordinate direction is assumed t o follow the flow c = rlrfz + rlvs, + 3 VSSS direction and the thin-layer approximation is used to drop all viscous derivatives in this direction. All v s i- c = $ ( < z a r U + < y a r V + <za,w) 4 cous t e r n in the cross-channel (q,s) plane are retained, (35 = tlraru + qvarv + qrarw with an option in the code to delete all crossderivatives (74 if desired. The resulting equations are as follows. 1 Terms multiplied by C and C2 lead to non-mixed second derivative viscous terms like u,, while terms multiplied ,,, by C 4 5 lead to mixed-derivative terms like uVr. The viscous flux vector 8 can be written similarly, with di- where: rections 1 and < everywhere interchanged. Metric terms are defined wing the following rela- [:; ;; :I tions. L rlr sr where =J 1 Yo% - Yr% ZqYr - Yr-? - Y V r z ~ z , , z,,z~ ~ € 2- Z ~ Z ( Z,,.ZE - ZrYo , zrY( - ZQYr YC% ZcY, - YvzC - ZCZ,, - ZvYt 1 (8) J= The velocities in 4 are absolute with respect to the co- ordinate system fixed to the blade. Relative velocities (Z(YrjZ, + ZrYtz, + ZoYrzC - ZtYrz, - ZnYtZr - Z < Y , ~ C ) - ~ (denoted by a prime] are given by: (9) The equations are nondimensionahed by arbitrary reference quantities (here the inlet total density pore/ u' = u and the total sonic velocity core! were used,) and the v'=v-rlnZ Reynolds number Re and Prandtl number P r are defined in terms of these quantities. The equations assume that w' = w + Ry the specific heats C, and C, and Prandtl number are (3) constant, that Stoke's hypothesis is valid, and that the and the relative contravariant velocitiy components are effective viscosity for turbulent flows may be written as given by: Pe/l = Plam + Pturb ( 10) U' = Ezu + EvY' + EZW' where the laminar viscosity is calculated using a power V' = q,u + qyvf + q,w' law function of temperature: W' = <=u + <,v' + {,W' (4) Note that although u' = u, U' # U. The energy and static pressure are given by: with n = 3 for air. e=p [CJ + (u2 + Y2 + w')/2] (5) p = (7- 1) [e - p(u' + v z + w2)/2] (6) TURBULENCE MODEL Using Stoke's hypothesis, A = - $pl the viscous flux The Baldwin-Lomax algebraic two-layer eddy viscos- Pv can be written as follows: ity model6 is applied on cross-channel (11, <) planes. Two I modifications to the standard model are made to account pv = J - ' P [ O , F2, 41F4, F5IT (7 4 for the endwall boundary layer and the blade boundary layer and wake, and their interaction in corners. First, the distance from the wall is calculated using the Buleev' length scale d: where s , and sr are normal distances from the walls in , the q- and <- directions respectively. This length scale 2 has the desirable property that d approaches the normal For subsonic outflow the exit static pressure is spec- distance from one wall a t large distances from the other ified and ( p , pu, pu, p w ) are extrapolated. For Cartesian wall. geometries the exit pressure is constant. For annular Secondly the turbulent viscosities are calculated geometries the hub pressure is specified and the radial across each boundary layer or wake separately, then the pressure distribution is found by integrating the axisym- total turbulent viscosity is taken as the vector sum of the metric radial momentum equation: components, Le., dP - PVe2 - -P v z - ( wy)2 dr r 13 Sidewalls and the trailing-edge cut are treated as This assumption has the desirable properties that outside periodic boundaries. of one viscous layer p t u r b takes on values calculated for On the blade surface V' = 0, and for viscous flows the other layer, that it goes to sero in the core flow, and U' = W' = 0. Blade surface pressures are found from the that near corners it accounts for both walls. normal momentum equation. On the hub (e = 1) and tip ( 5 = 5rnaz): COMPUTATIONAL GRID W z +5 v L +c * W € P + (fzrlz + evvv + Szrl*)a,P Two-dimensional body-fitted grids for this work were generated using the GRAPE code developed by Sorenson '. Threedimensional grids were formed by stacking the 2-D grids. Figure 1 shows a 3-D grid around . , a plate with a round leading edge. For annular geome- On the blades (r) = 1) the normal momentum equation tries the 2-D grids were stacked along a radial stacking can be found from (18) by replacing everywhere by r) line and stretched in the r)-direction so that the blade and V' by W'. shape remained constant and the angular pitch of the outer (periodic) boundary remained constant. MULTISTAGE RUNGE-KUTTA ALGORITHM BOUNDARY CONDITIONS The governing equations are discretined using a node-centered finite difference scheme. Second order cen- A t the inlet, total temperature T re/ is specified as o tral differences are used throughout. a constant. A cdistribution of total pressure (PolPo ref) The multistage Runge-Kutta scheme developed by is specified, c/2/2- as a constant or as appropriate for an Jameson, Schmidt, and Turkel' is used to advance the inlet boundary layer with given thickness and a power- flow equations in time from an initial guess to a steady law velocity profile. For Cartesian geometries the (2, y) state. If we rewrite (I) as and (2, ) flow angles are specified. For cylindrical ge- z ometries the (2, y) flow angle is replaced by the inlet whirl rug. For supersonic inlet flows, all flow variables are spec- where Rr is the inviscid residual including the source ified a t the inlet. For subsonic flows the inlet condi- term, RV is the viscous residual, an D is an artificial tions are updated each iteration by extrapolating the dissipation term described in the next section, then the upstream-running Riemann invariant R - based on the multistage Runge-Kutta algorithm can be written as fol- + absolute total velocity Q = du2+ v 2 w 2 to the inlet. lows: 90 = Qn 91 = Qo - aiJAt[Rr Qo - (Rv D ) QO] + The total velocity is then found from the TO using: For efficiency both the physicial and artificial dissipation terms are calculated only a t the first stage, then are held (15) constant for subsequent stages. Velocity components are found from Q and the specified angles or whirl. Within the endwall boundary layer, that is, where Po/Porcf < 1, the v and w velocity components are found by extrapolation from upstream. The density ARTIFICIAL DISSIPATION is found using: The dissipative term D in ( 2 0 ) is a nonconservative vereion of that used by Jameson et al.' It is given by: DQ = (4 o +D + D,)Q (224 3 where the (direction operator is given by Jameson". The technique involves replacing the residual calculated in (20) with a value that has been smoothed Dcq = c (v29cc - VrPcctc) (22b) by an implicit filter, Le., where 1 c=- (224 JAt where 6,c, 6,,,, and 6,, are standard second difference in a coefficient that cancella similar terma in (21). To operators and E € , e,, and E , are smoothing parameters. minimize the artificial dissipation in viscous regions we Linear stability analysis has shown that the Runge- reduce C linearly acroas several grid points t o cero at the Kutta scheme with implicit residual smoothing may be walls. made unconditionally stable if the E smoothing parame- ters are made sufficiently large. In one dimension , The terms V and V. are given by: € 1f [ ( $ ) 2 - l ] (25) gives unconditional stability if A' is the Courant limit where of the unsmoothed scheme, and X is a larger operating Courant number. In three dimensions different E'S may be used in each direction, and their magnitudes may be and often be reduced below the value given by Eq. (25.) P2 = O(1) P = O(&) . RESULTS (2211 In smooth regions of the flow the dissipative terms are of Two sets of computed results are presented for pre- third order and do not detract from the formal second- liminary validation of the code described above. The first order accuracy of the scheme. Near shocks vi,j is large set of results shows the structure of a horseshoe vortex and the dissipative terms become locally of first order. formed a t the base of a cylinder standing in a turbulent boundary layer. The second set of results is for turbu- lent flow through an annular turbine cascade. Computed THREE-DIMENSIONAL STABILITY LIMIT results are compared to experimental data in each case. When a boundary layer approaches a local obstruc- Applying a linear stability analysis to the inviscid tion such as the leading edge of a turbine blade, the low form of (20-21) gives the following expression for the time momentum fluid in the boundary layer often cannot over- step. come the local pressure gradient and the flow separates from the wall. In front of the obstacle the separation CFL creates a vortex which convects around the sides of the At 5 I,lul+ l u p l + l E p l+ c d g q T i j T = obstacle and forms a characteristic horseshoe-shaped flow region. (234 Eckerle and Langston" have made detailed mea- where surements of the horseshoe vortex in front of and around a cylinder of diameter D centered between the sidewalls L = I L I + lllzl+ lfIl of a wind tunnel. Test conditions included an inlet Mach 1, = I&lI+ l l l Y l + Ifvl number of 0.084, R C D= 5.5 x lo5,and an upstream tur- 1, = ILI + lllrl+ If*( bulent boundary layer thickness 6 = 0.lD. Detailed sur- (23b) face flow visualization and static pressure measurements The Courant limit for a particular multistage scheme de- were made, and static and total pressure measurements pends on the number of stages and the choice of coeffi- were taken using a five-hole probe. cients ui. See Ref. 9 for several examples. Figure 1 shows the grid used to compute Eckerle To accelerate convergence to a steady state we use and Langston's flow. The grid shown has been coars- the maximum permissible time step a t each grid point so ened for clarity. The actual grid had 65 x 49 x 25 points that the Courant number is constant everywhere. The with an initial spacing a t the walls Asi = 0.0010. To time step is calculated once based on the initial condi- avoid questions of trailing-edge vortex shedding, a tail 4 tions. It is stored and is not updated during the calcula- board waa added from the back half of the cylinder to tions. the exit boundary. The base grid is approximately 6D square and .5D high, t o match the dimensions of Eck- erle and Langston's wind tunnel test section. We used a IMPLICIT RESIDUAL SMOOTHING symmetry condition at mid span but computed the full symmetric flow side-to-side. Residual smoothing was introduced by Lerat (see The experimental inlet Mach number of 0.084 is too for example Ref. 10) for use with the Lax-Wendroff low for the compressible algorithm used here, s the cal- o scheme and was later applied to Runge-Kutta schemes by i culations were run with M , = 0.2. The peak Mach 4 DR,IGHNAI; PAGE IS OF POOR QUALITY number was 0.34 at the cylinder-plate junction, so the junction. There is excellent agreement between the com- flow was essentially incompressible. The Reynolds num- puted and experimental data. ber and inlet boundary layer thickness were matched to The velocity vectors show how the flow rolls up to the experimental data. form a horseshoe ahead of the cylinder. Experimental Even with Min = 0.2 the four-stage Runge-Kutta velocity vectors are missing in areas where the flow an- scheme seemed t o converge poorly, so we eventually ran gle exceeded the calibrated range of the five-hole probe. the calculations with a two-stage scheme with ai = From this data Eckerle and Langston12 concluded that (1.2, l.),CFL = 4., and implicit residual smoothing at “The reverse flow did not roll up t o form a vortex, each stage with e t = 2., e,, = cr = 4. The initial conver- however. The vectors clearly show that a closed vor- 8 gence rate was fast, as shown in Fig. 2 by the histories tex was not present in the plane of symmetry, though of the maximum and r m s residuals, but after 1000 iter- positive pitch angles in a portion of the reverse flow at ations the solution showed only a tiny horseshoe vortex R I D = 0.72 and Y/D= 0.02 may indicate the start of that did not match the data. Over the next 3000 iter- vortex formation. Rather than rolling up, flow passed ations the residuals changed little, but the vortex grew out of the plane and proceeded tangentially around the and moved upstream until it stabilized a t the position cylinder.” shown later. The total solution took 1.3 hours on a Cray The computed vectors clearly show a vortex in the X-MP computer. symmetry plane. A small counterrotating secondary vor- Figures 3 and 4 compare the calculated and experi- tex is also shown a t the cylinder-endwall junction. The mental data on the endwall. Figure 3 shows contours of dimension of the secondary vortex is about 1.5 times the constant static pressure coefficient,defined by: diameter of the five-hole probe, and would have been nearly impossible to detect experimentally. The second set of results is for an annular cascade of constant profile turbine stator vanes developed and tested a t NASA The annular ring has 36 Computed contours are on the bottom and measured vanes with a hub-tip radius ratio of 0.85 and a tip di- contours are on the top. The calculations show excel- ameter of 508 mm. The vanes themselves are 38.10 mm lent agreement with the data from the symmetry plane high and have an axial chord of 38.23 mm. Design flow to about 45 degrees around the cylinder, where the in- conditions are for a fully axial inflow with a hub-static to fluence of the tailboard becomes apparent. The lower inlet-total pressure ratio of 0.6705. These conditions cor- part of Figure 4 shows a flow visualization picture made respond to average inlet and exit Mach numbers of 0.211 with ink dots on the endwall. It clearly shows the sepa- and 0.665 respectively. The Reynolds number based on ration line and reverse flow region ahead of the cylinder. axial chord is 1.73 x lo6. The computed vector plot at the top of the figure shows A grid consisting of 97 x 31 x 33 points with an initial close agreement with the measured separation line loca- spacing a t the wall of 0.0002 of a blade chord was used tion and flow directions. The vectors are one point o f f for the flow calculations and is shown in Figure 7. the endwall, and are all drawn to the same length, so they show direction only. The calculation was run with a four-stage scheme Figure 5 compares the static pressure coefficient dis- with ai = (1/4, 1/3, 1/2, 1) and C F L = 5.5, using tributions on the endwall along the symmetry line ahead implicit residual smoothing after each stage with e t = of the cylinder. The experimental data (circles) show e,, = et = 0.75. Convergence histories for the annular a general pressure rise upstream due to the cylinder cascade calculation are shown in Fig. 8 where the log of blockage, but also a large dip in the pressure inside the the maximum and rms-averaged residuals have dropped separated region. Two vertical bars indicate substan- approximately 3.5 orders of magnitude in 1500 iterations. tial unsteadiness in the experimental data. Eckerle and The total CPU time was approximately 54 minutes for Langston included a 2-D potential solution (solid line) this calculation. for comparison. The computed solution (line with trian- Mach number contours a t mid span are shown in Fig. gles) shows good agreement with the data ahead of the 9 to illustrate the blade boundary layer and wake thick- saddle point, but falls short of predicting the magnitude nesses. There are approximately 12 grid points across of the pressure dip. The discrepancy may be due to the the pressure surface boundary layer. unsteadiness in the real flow or to lack of resolution in The inlet boundary later thicknesses were specified the computed solution. as 1.9 percent span on the hub and 7.1 percent span on Figure 6 compares computed (bottom) and experi- the tip, corresponding to the measured data in Ref. 13. mental (top) velocity vectors and total pressure loss co- In Fig. 10, velocity vectors with superimposed contours efficient contours on the symmetry plane upstream of the of total pressure show how these boundary layers roll up cylinder. The total pressure loss coefficient is defined by: into horseshoe vortices at the leading edge of the blade. The primary vortices are considerably smaller than the inlet boundary layers, and each primary vortex has an (27) even smaller counterrotating secondary vortex associated with it. These contours show nearly horizontal boundary layer- Figure 11 shows a comparison between the calcu- like flow upstream that rolls up into a vortex with a high lated surface static pressure distribution and data ob- loss core. The low loss fluid above the boundary layer tained by Goldman and S e a ~ h o l t s ’ ~ locations of 13.3, at curves down the face of the cylinder and carries high 50.0, and 86.7 percent span. The calculated results agree momentum fluid to the region near the cylinder-endwall very well with the experimental data a t all spanwise lo- 5 cations. Although the blade section is constant from hub It is felt that with additional work and experience to tip, the increased pitch at the tip increases the loading this analysis will prove to be a useful tool for investigat- of the tip section considerably. ing three-dimensional viscous flow phenomena in turbo- Figure 12 compares calculated (top) and measured machinery. (bottom, Ref. 14) efficiency contours on a cross-channel surface located 1/3 axial chord downstream of the trail- ing edge. The kinetic energy efficiency is theoretically REFERENCES independent of the axial position and is defined by the following: I. Chima, R. V., “Inviscid and Viscous Flows in Cas- v =QLal/QLeal cades with an Explicit Multiple-Grid Algorithm,” AIAA Journal, Vol. 23, No. 10, Oct. 1985, pp. where 1556-1563. 2. Chima, R. V., “Explicit Multigrid Algorithm for Q%,, = 2CpTo 1 - - ( T ;a) a: ‘ Quasi-Three-Dimensional Viscous Flows in Turbo- machinery,” J. Propulsion and Power, Vol. 3, No. and TOis taken as constant. 5, S e p t . 4 c t . 1987, pp. 3 9 7 4 0 5 . The efficiency contours clearly delineate the endwall 3. Dawes, W. N., “A Numerical Analysis of the Three- boundary layers and wake. The computed wake is thin- Dimensional Viscous Flow in a Transonic Com- ner than the measured wake and has higher losses (lower pressor Rotor and Comparison With Experiment,” efficiency) a t the center. Integrating the efficiencies over J. Turbomachinery, VoL 109, Jan. 1987, pp. 83-90. the entire area gives a total efficiency of 0.960 for the real 4. Subramanian, S. V., and Boccola, R., “Numeri- machine and 0.923 for the calculations. The discrepancy cal Simulation of Three-Dimensional Flow Fields could be due to inadequate resolution of the thick, round in Turbomachinery Blade Rows Using the Com- trailing edge, an inadequate turbulence model, or possi- pressible NavierStokes Equations,” AIAA Paper 87- bly to unsteady vortex shedding in the real flow. The 1314, June, 1987. high calculated losses seem not to be due to numerical 5. Rail M. M.,“Unsteady Three-Dimensional Navier- dissipation. Indeed, the computed wake appears to be Stokes Simulations of Turbine RotorStator Interac- less dissipative than the measured wake, and the com- tions,” AIAA Paper 87-2058, June, 1987. puted loss is remarkably insensitive to the artificial vis- 6. Baldwin, B. S., and Lomax, H., “Thin-Layer Ap- cosity coefficient. Considerable work is clearly needed in proximation and Algebraic Model for Separated Tur- the area of modelling the flows leaving the blunt trailing bulent Flows,” AIAA Paper 78-257, Jan. 1978. edges commonly found on turbomachinery blades. 7. Gessner, F. B., Po, J. K., ‘A Reynolds Stress Model for Turbulent Corner Flows - Part I1 Comparisons Between Theory and Experiment,” J. Fluids Enur., C 0NCLUDING REMARKS June, 1976, pp. 269-277. 8. Sorenson, R. L., “A Computer Program to Generate A numerical analysis has been developed for three- Two-Dimensional Grids About Airfoils and Other dimensional viscous internal flows. The analysis solves Shapes by the Use of Poisson’s Equation,” NASA the 3-D NavierStokes equations written in a general TM-81198, 1980. body-fitted coordinate system, including rotation about 9. Jameson, A., Schmidt, W., and Turkel, E., ”Numer- the z-axis. The thin-layer approximation is made in the ical Solutions of the Euler Equations by Finite Vol- streamwise direction but all viscous terms are included ume Methods Using Runge-Kutta Time-Stepping in the cross-planes. The Baldwin-Lomax eddy-viscosity Schemes,” AIAA Paper 81-1259, June 1981. model is used for turbulent flows. 10. Hollanders, H., Lerat, A., and Peyret, R., “Three- An explicit multistage Runge-Kutta scheme is used Dimensional Calculation of Transonic Viscous Flows to solve the finitedifference form of the flow equations. by an Implicit Method,” AIAA Journal, Vol. 23, No. A variable time step and implicit residual smoothing are 11, Nov. 1985, pp. 1670-1678. used to accelerate the convergence of the scheme. Con- 11. Jameson, A., and Baker, T. J., “Solution of the Eu- vergence rates are slow a t low Mach numbers but reason- ler Equations for Complex Configurations,” AIAA able for typical turbomachinery applications. We hope Paper 83-1929, July 1983. to improve convergence rates by adding multigrid to the 12. Eckerle, W. A., and Langston, L. S., “Measurements code. of a Turbulent Horseshoe Vortex Formed Around Results showing the development of a horseshoe vor- a Cylinder,’ NASA Contractor Report 3986, June, tex in a turbulent boundary layer ahead of a cylinder are 1986. presented to validate the analysis. Excellent agreement 13. Goldman, L. J., and Seasholtc, R. G., ‘Laser is found between the computed results and experimental Anemometer Measurements in an Annular Cascade data for this case. of Core Turbine Vanes and Comparison With The- Results for an annular turbine cascade also show ory,” NASA Technical Paper 2018, June, 1982. horseshoe vortex development. Very good agreement was 14. Goldman, L. J., and McLallin, K. L., “Cold-Air found between measured and computed surface pressure Annular-Cascade Investigation of Aerodynamic Per- distributions. Computed losses were high and show the formance of Core-Engine-Cooled Turbine Vanes. I: need for improved modelling of flows around blunt trail- Solid-Vane Performance and Facility Description,” ing edges commonly found in turbomachinery-. NASA T M X-3224, 1975. 6 ECXEPLE I L A # G S I O S S MOPSESHOE VOI7IE.V C I P f R I M E Y l MACH 0.196 RE 538257. ALPHA 0.00 IIEP WOO 'igure 1. Computational grid for the horseshoe vorte, problem. rcxrar I LAYCSIOYS no+?sfsmr voerrx E x p r e i m r N I I- 2 NlCH 0.196 PC 538261. 1iPHO 0.00 IICP to00 I4Cn 0.196 RE 538257. ALPXA 0.00 Iff# WOO CPS C D U I D U P Z "I* -2.000 rrx .0 I00 IUC 0.IOO Figure 3. Comparison of measured (top) and computed Figure 4. Comparison of endwall flow visualization (bot- (bottom) static pressure contours on the endwall. tom) and computed flow direction vectors (top) ahead of - the cylinder. I , 1 1 1 I I 1 1 - 1 I 1 I 05 06 01 08 09 10 11 12 I: RADIAL DISTANCE FROM CYLINDER CENTER R I D Figure 5. Comparison of measured and computed static pressure distributions on the endwall symmetry line. rcxrsir I LANGSIONS n o n s f w o r vosfrx rXPrPiwY7 I = 33 NACH 0.196 er 538257. ~ P N A 0.00 /ire woo c p r coxrooas "iu o.000 "AX 0.600 INC 0.100 Figure 6. Comparison of measured (top) and computed (bottom) velocity vectors and total pressure loss contours on the symmetry plane. 7 " 9 -5 -IO G O L O l l N S A U Y U L l P CASCAOEr 3-0 Figure 8. Convergence history for the annular turbine cascade. cascade problem. TIP - _ GOLONAYS I N Y U L A P c*scmr. 3-0 I - 18 SOLOCANS ANNULAR CASCAOC. 3 - 0 x ~ c n 0.21~ # I C X CONfOUPS PI "IN 173000. 0.000 ALpna N4X 0.00 0.850 IrEn IYC 1500 0.050 XACH po/poIy 0.212 courouas P I 173000. 0.960 1LPnA 0.00 0.998 ITER ruc 1500 o.aa2 1 Figure 9. Mach number contours a t mid-span. 0 . h . 0.50 SOL0""YS IACX L?: " Y Y U L I P C*SC*OE. 0.212 PC 3-0 l.11000. hLPnA 0.00 IIrP .llP6 X . 1500 It GOIOM1YS IYUULAP C4SCAOI, 3-0 XlCl 0.212 crrlctcycr PI #I# 173000. 0.020 ALPMl MAX 0.00 0.980 ITCP IYC IS00 o.oto Figure 11. Comparison of measured and computed pres- pigure 12. Comparison of measured (bottom) and com- sure distributions a t three span-wise locations. puted (top) efficiency contours 1/3 axial chord down- stream of the trailing edge. a .. j.2 2--.q--g3 - 1 . :?-u,uT-y y-7 I Nallonal Aearonaulcr and Space M m i n i m r a l m Report Documentation Page L 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. NASA TM-I 00878 4. Title and Subtitle 5. Report Date N u m e r i c a l A n a l y s i s o f Three-Dimensional V i s c o u s I n t e r n a l Flows 6. Performing Organization Code 7. Author@) R o d r i c k V. Chima and J e f f r e y W . Yokota 8. Performing Organization Report No. E-41 12 . 10. Work Unit No. 505-62-21 9. Performing Organization Name and Address 11. Contract or Grant No. N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n Lewis Research C e n t e r 1 C l e v e l a n d , O h i o 44135-3191 13. Type of Report and Period Covered 112. Sponsoring Agency Name and Address I T e c h n i c a l Memorandum N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n 14. Sponsoring Agency Code Washington, D.C. 20546-0001 I 15. Supplementary Notes P r e p a r e d f o r t h e F i r s t N a t i o n a l F l u i d Dynamics Congress cosponsored b y t h e A I A A , ASME, ASCE, S I A M , and APS, C i n c i n n a t i , Ohio, J u l y 24-28, 1988. R o d r i c k V. Chima, NASA Lewis Research C e n t e r ; J e f f r e y W . Yokota, S v e r d r u p Technology, I n c . , ( L e w i s Research C e n t e r Group), NASA Lewis Research C e n t e r , C l e v e l a n d , O h i o 44135. I 16. Abstract - A three-dimensional Navier-Stokes code has been developed f o r a n a l y s i s o f t u r b o m a c h i n e r y b l a d e rows and o t h e r i n t e r n a l f l o w s . The Navier-Stokes equati,ons a r e w r i t t e n i n a C a r t e s i a n c o o r d i n a t e system r o t a t i n g a b o u t t h e x - a x i s , and then mapped t o a g e n e r a l b o d y - f i t t e d c o o r d i n a t e system. Streamwise v i s c o u s terms a r e n e g l e c t e d u s i n g t h e t h i n - l a y e r assumption, and t u r b u l e n c e e f f e c t s a r e m o d e l l e d u s i n g t h e Baldwwin-Lomax t u r b u l e n c e model. The e q u a t i o n s a r e d i s c r e t i z e d u s i n g f i n i t e d i f f e r e n c e s on ' s t a c k e d C-type g r i d s and a r e s o l v e d u s i n g a m u l t i s t a g e Runge-Kutta a l g o r i t h m w i t h a s p a t i a l l y - v a r y i n g t i m e s t e p and i m p l i c i t r e s i d u a l smoothing. C a l c u l a t i o n s have been made o f a horseshoe v o r t e x formed i n f r o n t o f a f l a t p l a t e w i t h a round l e a d i n g edge s t a n d i n g i n a t u r b u l e n t e n d w a l l boundary l a y e r . Comparisons a r e made w i t h e x p e r i m e n t a l d a t a t a k e n by E c k e r l e and Langston f o r a c i r c u l a r c y l i n d e r under s i m i l a r c o n d i t i o n s . Computed and measured r e s u l t s a r e compared i n terms o f e n d w a l l f l o w v i s u a l i z a t i o n p i c t u r e s and t o t a l p r e s s u r e l o s s c o n t o u r s and v e c t o r p l o t s on t h e symmetry pJane. C a l c u l a t e d d e t a i l s o f t h e p r i m a r y v o r t e x show e x c e l l e n t agreement w i t h t h e e x p e r i m e n t a l d a t a . The c a l c u l a t i o n s a l s o show a s m a l l secondary v o r t e x t h a t was n o t seen e x p e r i m e n t a l l y . The c a l c u l a t i o n s r e q u i r e d a b o u t 1 . 6 m i l l i o n words o f s t o r a g e and 1.3 h o u r s o f CPU t i m e on a Cray X-MP computer. 1 C a l c u l a t i o n s have a l s o been made o f an a n n u l a r t u r b i n e s t a t o r t h a t has been t e s t e d e x p e r i m e n t a l l y a t NASA Lewis Research C e n t e r . The Mach number ranged rom a b o u t 0.21 a t t h e i n l e t t o 0.67 a t t h e e , i i t , 6 and t h e a x i a l c h o r d Reynolds number was a b o u t 1 . 7 ~ 1 0 . Computed s u r f a c e p r e s s u r e d i s t r i b u t i o n s compare w e l l w i t h measured v a l u e s a t t h r e e span-wise l o c a t i o n s . The e n d w a l l boundary l a y e r s produce horseshoe v o r t i c e s a t t h e l e a d i n g edge o f t h e b l a d e . Computed wake p r o f i l e s resemble t h e measured p r o f i l e s , b u t computed e f f i c i e n c i e s a r e l o w e r t h a n measured v a l u e s by a f a c t o r o f two. 17. Key Words (Suggested by Author@)) 18. Distribution Statement Navier-Stokes equations Unclassified - Unlimited T h r e e - d i m e n s i o n a l flow S u b j e c t C a t e g o r y 01 Turbomachinery 19. Security Classif. (of this report) 20. Security Classif. (of this page) Unclassified Unclassified 10 A02