AIAA Observations on CFD Simulation Uncertainties

Transcription

1 AIAA Observations on CFD Simulation Uncertainties Serhat Hosder, Bernard Grossman, Layne T. Watson William H. Mason Virginia Polytechnic Institute State University, Blacksburg, VA Raphael T. Haftka University of Florida, Gainesville, FL 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis Optimization September 4-6, 2002 /Atlanta, GA

2 OBSERVATIONS ON CFD SIMULATION UNCERTAINTIES Serhat Hosder, Bernard Grossman, Raphael T. Haftka, William H. Mason, Layne T. Watson Multidisciplinary Analysis Design (MAD) Center for Advanced Vehicles Virginia Polytechnic Institute State University Blacksburg, VA Abstract Different sources of uncertainty in CFD simulations are illustrated by a detailed study of 2-D, turbulent, transonic flow in a converging-diverging channel. Runs were performed with commercial CFD code GASP using different turbulence models, grid levels, flux-limiters to see effect of each on CFD simulation uncertainties. Two flow conditions were studied by changing exit pressure ratio: first is a complex case with a strong shock a separated flow region, second is weak shock case with no separation. The uncertainty in CFD simulations has been studied in terms of five contributions: (1) iterative convergence error, (2) discretization error, (3) error in geometry representation, (4) turbulence model, (5) downstream boundary condition. In this paper we show that for a weak shock case without separation, informed CFD users can obtain reasonably accurate results, whereas y are more likely to get large errors for strong shock case with substantial flow separation. We demonstrate difficulty in separating discretization errors from physical modeling uncertainties originating from use of different turbulence models in CFD problems that have strong shocks shock-induced separation. For such problems, interaction between different sources of uncertainty is strong, highly refined grids, which would not be used in general applications are required for spatial convergence. This study provides observations on CFD simulation uncertainties that may help development of sophisticated methods required for characterization quantification of uncertainties associated with numerical simulation of complex turbulent separated flows. Graduate student, Department of Aerospace Ocean Engineering, Student Member AIAA Professor, Department of Aerospace Ocean Engineering, Fellow AIAA Distinguished Professor, Department of Aerospace Engineering, Mechanics Engineering Science, University of Florida, Gainesville, FL, Fellow AIAA Professor, Department of Aerospace Ocean Engineering, Associate Fellow AIAA Professor, Departments of Computer Science Mamatics Copyright c 2002 by authors. Published by AIAA, Inc. with permission. 1. Introduction Computational fluid dynamics (CFD) has become an important aero/hydrodynamic analysis design tool in recent years. CFD simulations with different levels of fidelity, ranging from linear potential flow solvers to full Navier-Stokes codes, are widely used in multidisciplinary design optimization (MDO) of advanced aerospace ocean vehicles. 1 Although low-fidelity CFD tools have low computational cost are easily used, full viscous equations are needed for simulation of complex turbulent separated flows, which occur in many practical cases such as high-angle-of attack aircraft, high-lift devices, maneuvering submarines missiles. 2 Even for cases when re is no flow separation, use of highfidelity CFD simulations is desirable for obtaining higher accuracy. Due to modeling, discretization computation errors, results obtained from CFD simulations have a certain level of uncertainty. It is important to underst sources of CFD simulation errors ir magnitudes to be able to assess magnitude of uncertainty in results. Recent results presented in First AIAA CFD Drag Prediction Workshop 3 4 also illustrate importance of understing uncertainty its sources in CFD simulations. Many of performance quantities of interest for DLR-F4 wingbody configuration workshop test case, such as lift curve slope, drag polar, or drag rise Mach number, obtained from CFD solutions of 18 different participants using different codes, grid types, turbulence models showed a large variation, which revealed general issue of accuracy credibility in CFD simulations. The objective of this paper is to illustrate different sources of uncertainty in CFD simulations, by a careful study of a typical, but complex fluid dynamics problem. We will try to compare magnitude importance of each source of uncertainty. The problem studied in this paper is a two-dimensional, turbulent, transonic flow in a converging-diverging channel. CFD calculations are done with General Aerodynamic Simulation Program (GASP). 5 Runs were performed with different turbulence models, grid densities, flux-limiters to see effect of each on CFD simulation uncer- 1

3 tainties. In addition to se, contribution of error in geometry representation to CFD simulation uncertainties is studied through use of a modified geometry, based on measured geometric data. The exit station of diffuser exit pressure ratio are varied to determine effects of changes of downstream boundary conditions on results. The results of this study provide detailed information about sources magnitudes of uncertainties associated with numerical simulation of flow fields that have strong shocks shockinduced separated flows. 2. Uncertainty Sources To better underst accuracy of CFD simulations, main sources of errors uncertainties should be identified. Oberkampf Blottner 6 classified CFD error sources. In ir classification, error sources are grouped under four main categories: (1) physical modeling errors, (2) discretization solution errors, (3) programming errors, (4) computer round-off errors. Physical modeling errors originate from inaccuracies in mamatical models of physics. The errors in partial differential equations (PDEs) describing flow, auxiliary (closure) physical models boundary conditions for all PDEs are included in this category. Turbulence models used in viscous calculations are considered as one of auxiliary physical models, usually most important one. They are used for modeling additional terms that originate as result of Reynolds averaging, which in itself is a physical model. Oberkampf Blottner 6 define discretization errors as errors caused by numerical replacement of PDEs, auxiliary physical models continuum boundary conditions by algebraic equations. Consistency stability of discretized PDEs, spatial (grid) temporal resolution, errors originating from discretization of continuum boundary conditions are listed under this category. The difference between exact solution to discrete equations approximate (or computer) solution is defined as solution error of discrete equations. Iterative convergence error of steadystate or transient flow simulations is included in this category. A similar description of discretization errors can also be found in Roache. 7 8 Since terms error uncertainty are commonly used interchangeably in many CFD studies, it will be useful to give a definition for each. Uncertainty, itself, can be defined in many forms depending on application field as listed in DeLaurentis Mavris. 9 For computational simulations, Oberkampf et al described uncertainty as a potential deficiency in any phase or activity of modeling process that is due to lack of knowledge, whereas error is defined as a recognizable deficiency in any phase or activity of modeling simulation. Considering se definitions, any deficiency in physical modeling of CFD activities can be regarded as uncertainty (such as uncertainty in accuracy of turbulence models, uncertainty in geometry, uncertainty in rmophysical parameters etc.), whereas deficiency associated with discretization process can be classified as error. 11 Discretization errors can be quantified by using methods like Richardson s extrapolation or gridconvergence index (GCI), a method developed by Roache 8 for uniform reporting of grid-convergence studies. However, se methods require fine grid resolution in asymptotic range, which may be hard to achieve in simulation of flow fields around complex geometries. Also, non-monotonic grid convergence, which may be observed in many flow simulations, prohibits or reduces applicability of such methods. That is, it is often difficult to estimate errors in order to separate m from uncertainties. Therefore, for rest of paper, term uncertainty will be used to describe inaccuracy in CFD solution variables originating from discretization, solution, or physical modeling errors. 3. Simulation Case 3.1. Description of physical problem The test case presented in this paper is simulation of a 2-D, turbulent, transonic flow in a converging-diverging channel, known as Sajben Transonic Diffuser in CFD validation studies. 12 Figure 1 shows a schematic of two versions of geometry used in computations. The flow is from left to right, in positive x-direction. The y-direction is normal to bottom wall. All dimensions are scaled by throat height, h t. The throat section, which is minimum cross-sectional area of channel, is located at = 0.0. Both geometries have inlet stations located at = The exit station is at = 8.65 for geometry shown at top part of Figure 1. This is original geometry used in computations a large portion of results with different solution physical modeling parameters are obtained with this version. The exit station is located at = for or geometry shown in Figure 1. This extended geometry is used 2

4 y/h t grid 2 (80x50 cells) y/h t grid 2 ext (90x50 cells) Figure 1: Original Original geometry, geometry, Grid Grid 2 (top), (top), extended extended geometry, geometry, Grid Grid 2 ext (bottom) used in computations. ext (bottom) used in computations. is to located study ateffect of = varying 0.0. Both geometries downstreamhave boundary location stationsonlocated CFD at x/h simulation t = results. TheFor exitboth sta- inlet tion geometries, is at x/h t bottom = 8.65wall for of geometry channel is shown flat at converging-diverging top part of Figure 1. section This isof original top wall geometrscribed usedby inan analytical computations function of a large x/h portion of t defined in is de- Bogar results et al. with 13 Indifferent additionsolution to se twophysical geometries, modeling a third parameters version of are same obtained diffuser with ( this modified-wall version. The exit geometry) station has is located been developed at = for this for research or geometry has been used shown in in our Figure calculations. 1. This extended This version geometry has same is used inlet to study exit locations effect of as varying original downstream geometry, but boundary upper location wall is described on CFD by natural simulation cubicsplines results. For fitted both to geometries, geometric data bottom points wall that of were channel measured is in flat experimental converging-diverging studies. Having section observed of top wall fact is that described upper by an wall analytical contour obtained function of by x/h t analytical defined in equation Bogar et al. 13 In contour addition described to se two by experimental geometries, data a third points version are slightly of same different, diffuser ( modified-wall modified-wall geometry geometry) is used has to been find developed effects for of this geometric research uncertainty has been on used numerical in our calculations. results. This version has same inlet exit locations as Despite original geometry, relatively but simple upper geometry, wall is described flow by has natural a complex cubic-splines structure. fitted The to exit geometric pressure ratio data points P e that sets were strength measured in location experimental of a shock studies. that appears Having observed downstream of fact that throat. In upper our studies, for wall contour obtained original by analytical modified-wall equation geometries, we contour described define P e /P by 0i experimental = 0.72 as data strong points shock are slightly case different, P e = 0.82 modified-wall as weak geometry shock case. is used A to separated determine flow region effects exists of geometric just afteruncertainty shock on at P e /Pnumerical 0i = results. Although a nominal exit station was defined at = 8.65 for diffuser used in experiments, Despite relatively physicalsimple exit station geometry, is located flow at has x/h a complex structure. The exit pressure ratio t = In experiments, P e was mea- e as sets strength for location strongof a shock Psured that weakappears shock cases downstream respectively of at throat. physical In our studies, location. for Table original 1 gives a summary modified-wall of different geome- exit tries, versions weof term transonic P e = diffuser 0.72 asgeometry strong shock exit case pressure ratios P e used = 0.82 in ascomputations. weak shock case. A separated A large set flowofregion experimental exists just dataafter for a range shock of exit at pressure P e = ratios are Although available. a nominal 13 In ourexit study, station top was bottom defined at wall x/h pressure t = 8.65 values for were diffuser used for used incom- parison experiments, of CFD results physical with exit station experiment. is located Note at that = diffuser In geometry experiments, used in P e /Pexperiments 0i was measured suction as slots placed at x/h for strong t = 9.8 bottom has weak shock side cases walls respectively to limit at growth physical of boundary exit location. layer. Table The 1 existence gives a summary of se of slots can different affect versions of accuracy of transonic quantitative diffuser comparison geometry between P e /P 0i experiment ratios used in computations. computation at A large downstream set of exlocations. Table 1: Different versions of transonic diffuser geometry P e ratios used in computations Geometry at exit station P e 0i original original modified-wall modified-wall , extended , extended , , perimental Computational data for a range modeling of P e values is available. CFD 13 In calculations our study, are topperformed bottomwith wallgasp, pressure a Reynolds-averaged, values were used forthree-dimensional, comparison offinite-volume, CFD results Navier-Stokes with experiment. code, It which shouldisbecapable noted that of solving diffuser geometry (time usedasymptotic) in experiments time-dependent has suction steady-state problems. slots placedfor at his t problem, = 9.8 on bottom inviscid fluxes were side calculated walls to limit by an upwind-biased growth of boundary third-order layer. spatially The accurate existenceroe of se fluxslots scheme. can affect The minimum accuracy modulus of (Min-Mod) quantitative comparison Van Albada s between flux limiters experiment were used to prevent computation non-physical at downstream oscillations locations. in solution. 3 American American Institute Institute of of Aeronautics Aeronautics Astronautics Astronautics

5 Table 2: Mesh size nomenclature. Grid at exit station mesh size g g g g g g1 ext g2 ext g3 ext g4 ext g1 mw g2 mw g3 mw All viscous terms were included in solution two turbulence models, Spalart-Allmaras 14 (Sp- Al) k-ω 15 (Wilcox, 1998 version), were used for modeling viscous terms. The iterative convergence of each solution is examined by monitoring overall residual, which is sum (over all cells in computational domain) of L 2 norm of all governing equations solved in each cell. In addition to this overall residual information, individual residual of each equation some of output quantities are also monitored. In simulations, five different grids were used for original geometry: Grid 1 (g1), Grid 2 (g2), Grid 3 (g3), Grid 4 (g4), Grid 5 (g5). The finest mesh is Grid 5 or grids are obtained by reducing number of divisions by a factor of 2 in both x y-directions at each consecutive level (grid halving). Grid 5 is used only for case with Sp-Al turbulence model, Min-Mod limiter, P e = Four grid levels were used for extended geometry: Grid 1 ext (g1 ext ), Grid 2 ext, (g2 ext ), Grid 3 ext (g3 ext ), Grid 4 ext (g4 ext ). The grids of extended geometry grids generated for original geometry are essentially same between inlet station = For modifiedwall geometry, three grid levels were used: Grid 1 mw (g1 mw ), Grid 2 mw (g2 mw ), Grid 3 mw (g3 mw ). All grids have same mesh distribution in y-direction. The size of grids used in computations are given in Table 2. Grid 2 (top) Grid 2 ext (bottom) are shown in Figure 1. To resolve flow gradients due to viscosity, grid points were clustered in y-direction near top bottom walls. In wall bounded turbulent flows, it is important to have a sufficient number of grid points in wall region, especially in laminar sublayer, for resolution of near wall velocity profile, when turbulence models without wall-functions are used. A measure of grid spacing near wall can be obtained by examining y + values defined as y + = y τ w /ρ, (1) ν where y is distance from wall, τ w wall shear stress, ρ density of fluid, ν kinematic viscosity. In turbulent boundary layers, a y + value between 7 10 is considered as edge of laminar sublayer. General CFD practice has been to have several mesh points in laminar sublayer with first mesh point at y + = O(1). In our study, maximum value of y + values for Grid 2 Grid 3 at first cell center locations from bottom wall were found to be respectively. The grid points were also stretched in x-direction to increase grid resolution in vicinity of shock wave. The center of clustering in x-direction was located at = At each grid level, except first one, initial solution estimates were obtained by interpolating primitive variable values of previous grid solution to new cell locations. This method, known as grid sequencing, was used to reduce number of iterations required to converge to a steady state solution at finer mesh levels. It should be noted that grid levels such as g5, g4, g4 ext are highly refined, than those normally used for typical two-dimensional problems well beyond what could be used in a three-dimensional flow simulation. A single solution on Grid 5 required approximately 1170 hours of total node CPU time on a SGI Origin2000 with six processors, when cycles were run with this grid. If we consider a threedimensional case, with addition of anor dimension to problem, Grid 2 would usually be regarded as a fine grid, whereas Grid 3, 4, 5 would generally not be used. 4. Results Discussion For transonic flow in converging-diverging channel, uncertainty of CFD simulations is investigated by examining nozzle efficiency (n eff ) as a global output quantity obtained at different P e ratios with different grids, flux limiters (Min- Mod Van Albada), turbulence models (Sp-Al k-ω). The nozzle efficiency is defined as n eff = H 0i H e H 0i H es, (2) 4

6 where H 0i is total enthalpy at inlet, H e enthalpy at exit, H es exit enthalpy at state that would be reached by isentropic expansion to actual pressure at exit. Since enthalpy distribution at exit was not uniform, H e H es were obtained by integrating cell-averaged enthalpy values across exit plane. Besides n eff, wall pressure values from CFD simulations are compared with experimental data. In addition to visual assessment of graphs, comparison with experiment is also performed quantitatively by introducing a measure of error between experiment curve representing CFD results, orthogonal distance error where d i = E n = 1 N exp d i, (3) N exp i=1 [ min (x xi ) 2 + x inlet x x exit (P c (x) P exp (x i )) 2] 1/2 (4) In equations (3) (4), d i represents orthogonal distance between i th experimental point P c (x) curve ( wall pressure obtained from CFD calculations), P exp is experimental wall pressure value, N exp is number of experimental data points used. Pressure values are scaled by P 0i x values are scaled by length of channel. In transonic diffuser study, uncertainty in CFD simulation results has been studied in terms of five contributions: (1) iterative convergence error, (2) discretization error, (3) error in geometry representation, (4) turbulence model, (5) changing downstream boundary condition. In particular, (1) (2) contribute to numerical uncertainty, which is subject of verification process; (3), (4), (5) contribute to physical modeling uncertainty, which is concern of validation process. In our study, we have seen that contribution of iterative convergence error to overall uncertainty is negligible. A detailed analysis of iterative convergence error in transonic diffuser case is given in Appendix A The discretization error In order to investigate contribution of discretization error to uncertainty in CFD simulation results, we study Sp-Al k-ω cases separately. Grid level flux-limiter affect magnitude of discretization error. Grid level determines spatial resolution, limiter is part of discretization scheme, which reduces spatial accuracy of method to first order in vicinity of shock waves. A qualitative assessment of discretization error in nozzle efficiency results obtained with original geometry can be made by examining Figure 2. The largest value of difference between strong shock results of Grid 2 Grid 4 is observed for case with Sp-Al model Min-Mod limiter. For weak shock case, difference between each grid level is not as large as that of strong shock case when results obtained with Sp-Al turbulence model are compared. Weak shock results in Figure 2 also show that k-ω turbulence model is slightly better than Sp-Al in terms of discretization error for this pressure ratio. Non-monotonic behavior of k-ω results can be seen for strong shock case as mesh is refined, whereas same turbulence model shows monotonic convergence for weak shock cases. The Sp-Al turbulence model exhibits monotonic convergence in both shock conditions. Richardson s extrapolation technique has been used to estimate magnitude of discretization error at each grid level for cases that show monotonic convergence. This method is based on assumption that f k, a local or global output variable obtained at grid level k, can be represented by f k = f exact + αh p + O(h p+1 ), (5) where h is a measure of grid spacing, p order of method, α p th -order error coefficient. Note that Equation 5 will be valid when f is smooth in asymptotic grid convergence range. In most cases, observed order of spatial accuracy is different than nominal (oretical) order of numerical method due to factors such as existence of discontinuities in solution domain, boundary condition implementation, flux-limiters, etc. Therefore, observed value of p should be determined used in calculations required for approximating f exact discretization error. Calculation of approximate value of observed order of accuracy ( p) needs solutions from three grid levels, estimate of f exact value requires two grid levels. The details of calculations are given in Appendix B. Table 3 summarizes discretization error in n eff results obtained with original geometry. The cases presented in this table exhibit monotonic convergence with refinement of mesh size. For each case with a different turbulence model, limiter, exit pressure ratio, approximation to exact value of n eff is denoted 5

7 n eff grid level: A P e A B original geometry modified-wall geometry extended geometry extended geometry Min-Mod Van Alb. Min-Mod Van Alb. Min-Mod Van Alb. Min-Mod Van Alb. k-ω k-ω Sp-Al Sp-Al k-ω k-ω Sp-Al Sp-Al B Figure 2: Nozzle efficiencies obtained with different grid levels, turbulence models, limiters, geometries, boundary conditions for strong shock case (A) weak shock case (B). by (ñ eff ) exact discretization error at a grid level k is calculated by error(%) = (n eff ) k (ñ eff ) exact 100 (ñ eff ) exact. (6) When results at grid level g2 are compared, Sp-Al, Min-Mod, P e = 0.72 case has highest discretization error (6.97%), while smallest error (1.45%) is obtained with k-ω turbulence model at P e = The finest grid level, g5 was used only for Sp-Al, Min-Mod, strong shock case obtained with original geometry. Table 7 in Appendix B gives discretization error values of this case, which are less than 1% at grid level g5. In Table 3, observed order of accuracy p, is smaller than nominal order of scheme its value is different for each case with a different turbulence model, limiter, shock condition. The values of both (ñ eff ) exact p also depend on grid levels used in ir approximations. For example, p value was calculated as for Sp-Al, Min-Mod, strong shock case with different grid levels (See Appendix B, Table 7). This may add more uncertainty to approximation of discretization error at each grid level by Richardson s extrapolation. The difference in n eff values due to choice of limiter can be seen in results of Grid 1 Grid 2 for strong shock case Grid 1 for weak shock case. The maximum difference between Min-Mod limiter Van Albada limiter occurs on Grid 1 with Sp-Al model. The relative uncertainty due to choice of limiter is more significant for strong shock case. For both pressure ratios, solutions obtained with different limiters give approximately same n eff values as mesh is refined. n eff grid 1, Sp-Al, Min-Mod grid 2, Sp-Al, Min-Mod grid 1, k-ω, Min-Mod grid 2, k-ω, Min-Mod grid 3, Sp-Al, Min-Mod P e grid 3, k-ω, Min-Mod Figure 3: Nozzle efficiency vs. exit pressure ratio for different grids obtained with original geometry, Sp-Al k-ω turbulence models, Min-Mod limiter. Figure 3 shows significance of discretization uncertainty between each grid level. In this fig- 6

8 Table 3: Discretization error results obtained with original geometry turbulence limiter P e p (ñ eff ) exact grid discretization model level error (%) Sp-Al Van Albada Sp-Al Min-Mod Sp-Al Van Albada Sp-Al Min-Mod k-ω Van Albada k-ω Min-Mod g g g g g g g g g g g g g g g g g g g g g g g g ure, noisy behavior of n eff results obtained with Grid 1 can be seen for both turbulence models. The order of noise error is much smaller than discretization error between each grid level, however this can be a significant source of uncertainty if results of Grid 1 are used in a gradient-based optimization. When we look at Mach number values at two points in original geometry; one, upstream of shock ( = 1.5) or, downstream of shock ( = 8.65, exit plane), both of which are located at mid point of local channel heights (Figure 4), we see convergence of Mach number upstream of shock for all cases. However, for strong shock case, lack of convergence downstream of shock at all grid levels with k-ω model can be observed. For Sp-Al case, we see convergence only at grid levels g3 g4. For weak shock case, downstream of shock, convergence at all grid levels with k-ω model is also seen. At this pressure ratio, Sp-Al model results do not seem to converge, although difference between each grid level is small. These results may again indicate effect of complex flow structure downstream of shock, especially separated flow region seen in strong shock case, on grid convergence. Major observations on discretization errors: 1. Grid convergence is not achieved with grid levels that have moderate mesh sizes. For strong shock with flow separation, highly refined grids, which are beyond grid levels we use in this study, are needed for spatial convergence. Even with finest mesh level we can afford, achieving asymptotic convergence is not certain. 2. At each grid level, discretization errors of strong shock case are larger than that of weak shock case. The shock induced flow separation observed in strong shock case has a significant effect on grid convergence. 3. The discretization error magnitudes are different for cases with different turbulence models, when nozzle efficiency results with same limiter grid level are compared at each shock condition. This indicates effect of turbulence model on grid convergence implies that magnitudes of numerical errors are influenced by physical models. 7

9 Mach =8.65, y/h t =0.75, P e =0.72 =-1.5, y/h t =0.5558, P e =0.72 & 0.82 =8.65, y/h t =0.75, P e =0.82 %error in y/h t g1 g2 g3 g4 g1 g2 g3 g4 g1 g2 g3 g4 g1 g2 g3 g4 Min-Mod k-ω Van Albada k-ω Min-Mod Sp-Al Van Albada Sp-Al Figure 4: Mach number values at upstream of shock ( = 1.5), downstream of shock ( = 8.65, exit plane) for different grids obtained with original geometry, Sp-Al k-ω turbulence models, Min-Mod Van Albada limiters. The values of y/h t correspond to mid points of local channel heights Error in geometry representation The contribution of error in geometry representation to CFD simulation uncertainties is studied by comparing results of modified-wall original geometry obtained with same turbulence model, limiter, grid level. Figure 5 gives % error distribution in y/h t (difference from analytical value) for upper wall of modified-wall geometry at data points measured in experiments. Natural cubic-splines are fit to se data points to obtain upper wall contour. The maximum error is approximately 7% observed upstream of throat, at = Starting from = 1.2, error is approximately constant with an average value of 0.9%. The difference between upper wall contours of original modifiedwall geometry in vicinity of throat location is shown in Figure 6. The flow becomes supersonic just after throat is very sensitive to geometric irregularities for both P e = From top wall pressure distributions shown in Figures 7 8, a local expansion/compression region can be seen around = 0.5 with modified-wall geometry. This is due to local bumps created by two experimental data points, third fifth one from throat (Figure 6). Since neir wall pressure results obtained with original geometry nor experimental values have this local expansion/compression, values of se problematic points may contain some measurement error. The locations of se two points were modified by moving m in negative y-direction halfway between Figure 5: Error distribution in y/h t for upper wall of modified-wall geometry at data points measured in experiments. y/h t upper wall contour of modified-wall geometry (cubic-spline fit to modified data points) upper wall contour of modified-wall geometry (cubic-spline fit to data points) modified experimental data points 0.99 upper wall contour obtained with analytical equation Figure 6: Upper wall contours of original modified-wall geometry in vicinity of throat location. ir original value analytical equation value obtained at corresponding locations. These modified locations are shown with black circles in Figure 6. The wall pressure results of geometry with modified experimental points (Figures 7 8) show that local expansion/compression region seems to be smood, although not totally removed. One important observation that can be made from same figures is improvement of match between CFD results experiment upstream of throat when modified-wall geometry is used. 8

10 Sp-Al, Min-Mod, grid 2, wall contour from equattion P e =0.72 Top Wall Sp-Al, Min-Mod, grid 2, wall contour from equation P e =0.82 Top Wall Sp-Al, Min-Mod, grid 2 mw, wall contour from modified experimental data 0.8 Sp-Al, Min-Mod, grid 2 mw, wall contour from modified experimental data P 0.6 P 0.6 experiment experiment Sp-Al, Min-Mod, grid 2 mw, wall contour from experimental data Figure 7: Top wall pressure distributions obtained with original modified-wall geometry for strong shock case (The results of Sp-Al model, Min-Mod limiter, Grids g2 g2 mw are shown). 0.3 Sp-Al, Min-Mod, grid 2 mw, wall contour from experimental data Figure 8: Top wall pressure distributions obtained with original modified-wall geometry for weak shock case (The results of Sp-Al model, Min- Mod limiter, Grids g2 g2 mw are shown). Major observations associated with uncertainty in geometry representation: 1. The main source of discrepancy between CFD results of original geometry experiment upstream of shock is error in geometry representation. Since viscous effects are important only in a very thin boundary layer near wall region where re is no flow separation, contribution of Sp-Al or k-ω turbulence models to overall uncertainty is very small upstream of shock for both P e = Downstream of shock, wall pressure results obtained with same turbulence model limiter are approximately same regardless of geometry used. This may imply that difference between experiment CFD results downstream of shock is more likely due to turbulence models when finest grid levels are used to minimize contribution of discretization error Evaluation with orthogonal distance error The quantitative comparison of CFD simulation results with experiment can be done considering different measures of error. In transonic diffuser case, we use orthogonal distance error, E n to approximate difference between wall pressure values obtained from numerical simulations experimental data. The error E n was evaluated separately in two regions: upstream of experimental shock location (UESL) downstream of experimental shock location (DESL). The calculations were made by using equations 3 4. The parameters used in se equations for UESL DESL are given in Appendix B. Table 4 lists top wall scaled error En ˆ values obtained for UESL with original geometry, different grids, turbulence models, flux-limiters. Table 5 gives DESL results. In se tables, scaled error values, En ˆ were obtained by scaling E n as Eˆ E n n = 100, (7) (E n ) max where (E n ) max is maximum E n value calculated DESL at strong shock case with Grid 4, Min-Mod limiter, k-ω turbulence model. It can be seen from Table 4 that results obtained with Sp-Al k-ω turbulence models are very close, especially for weak shock case, when values at grid level g4 are compared. For each P e, small difference between results of each turbulence model at finest mesh level originate from difference in shock locations obtained from CFD calculations. This again shows that a large fraction of uncertainty observed upstream of shock (UESL) in wall pressure values originate from uncertainty in geometry representation. The difference in E ˆ n between each grid level for each turbulence model P e is very small indicating that wall pressure distributions upstream of shock obtained at each grid level are approximately same. In or words, grid convergence is achieved upstream of 9

11 Table 4: Top wall orthogonal distance error E ˆ n calculated upstream of experimental shock location (UESL) for each case obtained with original geometry. P e Grid Sp-Al, Min-Mod Sp-Al, Van Albada k-ω, Min-Mod k-ω, Van Albada 0.72 g g g g g g g g Table 5: Top wall orthogonal distance error E ˆ n calculated downstream of experimental shock location (DESL) for each case obtained with original geometry. P e Grid Sp-Al, Min-Mod Sp-Al, Van Albada k-ω, Min-Mod k-ω, Van Albada 0.72 g g g g g g g g shock discretization error in wall pressure values at each grid level is very small. Recall that experimental data also contains uncertainty originating from many factors such as geometric irregularities, difference between actual P e its intended value, measurement errors, heat transfer to fluid, etc. We have discussed error due to geometric irregularities in previous section. In a way, this error in geometry representation can also be regarded as a part of uncertainty in experimental data. By evaluating orthogonal distance error in two separate regions, DESL UESL, we tried to approximate contribution of geometric uncertainty to CFD results obtained with original geometry. However, experimental wall pressure values may still have a certain level of uncertainty associated with remaining factors Turbulence models To approximate contribution of turbulence models to CFD simulation uncertainties in transonic diffuser case, En ˆ values calculated for top wall pressure distributions downstream of shock (DESL) (Table 5) at grid level g4 are examined. By considering results of finest mesh level, contribution of discretization error should be minimized, although it is difficult to isolate numerical errors completely from physical modeling uncertainties, especially for strong shock case. The Sp-Al turbulence model is more accurate than k-ω model for strong shock case. In fact, difference is significant, with k-ω giving highest error of all cases, which is larger than Sp-Al error by a factor of 3.6. With Sp-Al model, orthogonal distance error gets smaller as mesh is refined, while k-ω model gives largest error value at grid level g4. When compared to error values presented in Table 4, for strong shock, uncertainty of k-ω turbulence model is 3.7 times larger than error due to geometric uncertainty. On or h, uncertainty of Sp-Al model has approximately same magnitude as geometric uncertainty. As opposed to strong shock case, k-ω turbulence model gives more accurate wall pressure distributions than Sp-Al model when weak shock results of grid g4 are compared (Table 5). The orthogonal distance error of Sp-Al is twice as big as that of k-ω model. The minimum error for Sp-Al model is obtained at grid level g2, while wall pres- 10

12 sure distributions of k-ω model get closer to experimental distribution as mesh is refined. The results of Sp-Al model show that most accurate results are not always obtained at finest mesh level. The error due to geometric uncertainty is bigger than uncertainty of k-ω model by a factor of 2.6 in weak shock case. The uncertainty of Sp-Al model is slightly smaller than geometric uncertainty for same shock condition. Major observations on turbulence model uncertainties: 1. The strong weak shock results show that for each flow condition, highest accuracy in terms of wall pressure distributions are obtained with a different turbulence model, although Sp-Al model gives reasonable results for both shock conditions. 2. Uncertainties associated with turbulence models interact strongly with discretization errors. In some cases, numerical errors physical modeling uncertainties may cancel each or, closest results to experiment can be obtained at intermediate grid levels. Figure 9: Streamline patterns of separated flow region obtained with different versions of diffuser geometry exit pressure ratios for strong shock case Downstream boundary condition The effect of downstream boundary location variation on CFD simulation results of transonic diffuser case has been investigated by using extended geometry, which has physical exit station at same location as geometry used in actual experiments. For strong shock case, runs were performed with Sp-Al model two P e ratios, The second pressure ratio is same value measured at physical exit station of geometry used in experiments for strong shock case. The results obtained with extended geometry were compared to results of original geometry. Figure 9 shows streamline patterns of separated flow region obtained with different geometries P e ratios in strong shock case. The comparison of separation bubble size is given in Figure 10. The separation bubble obtained with extended geometry P e = 0.72 is bigger extends farr in downstream direction compared to or two cases. The separation bubbles obtained with original geometry, P e = 0.72; extended geometry, P e = are approximately same in size. These results are also consistent with top wall pressure distributions given in Figure 11. With extended geometry P e = 0.72, flow accelerates more under separation bubble, pressure is lower compared to or Figure 10: Comparison of separation bubbles obtained with different versions of diffuser geometry exit pressure ratios for strong shock case. P experiment Sp-Al, Van Albada, grid 3 ext, P e = Top Wall Sp-Al, Van Albada, grid 3 ext, P e =0.72 Sp-Al, Van Albada, grid 3 Figure 11: Top wall pressure distributions obtained with different versions of diffuser geometry exit pressure ratios for strong shock case (The results of Sp-Al model, Van Albada limiter, grids g3 g3 ext are shown). cases where separation bubbles have smaller thickness. Moving exit location furr downstream increases strength of shock 11

13 size of separation region. As shock gets stronger, its location is shifted downstream. On or h, increasing P e reduces strength of shock, moves shock location upstream. As can be seen from Figure 9, separated flow region in original geometry is close to downstream boundary. This may be thought as one of factors that affect grid convergence in strong shock case. However, discretization error analysis of n eff values obtained with extended geometry do not show any improvement in terms of grid convergence (Appendix D) Discussion of uncertainty on nozzle efficiency We use nozzle efficiency as a global indicator of CFD results scatter in computed values of this quantity originates from use of different grid levels, limiters, turbulence models, geometries, boundary conditions for each shock strength case. A graphical representation of this variation is given in Figure 2. This figure shows a cloud of results that a reasonably informed user may obtain from CFD calculations. The numerical value of each point is presented in Table 6. We will analyze scatter in nozzle efficiency results starting from grid level 2, since coarse Grid 1 will not be used by those that have significant experience in performing CFD simulations. On or h, grid levels 3 4 would generally not be used in practical CFD applications, particularly in three dimensions, due ir computational expense. For purpose of determining variation in nozzle efficiency in terms of a % value, we use g4, Sp-Al, Van-Albada result as comparator. When we consider cases obtained with original geometry, maximum variation for strong shock condition is 9.9% observed between results of g2, k-ω, Min-Mod g4, Sp-Al, Van Albada. Maximum difference in weak shock results is 3.8% obtained between results of g2, k-ω, Van Albada g4, Sp-Al, Min-Mod. For each case with a different turbulence model limiter, variation between results of g2 g4 may be used to get an estimate of uncertainty due to discretization error. The maximum variation for strong shock is 5.7% obtained with Sp-Al model Min-Mod limiter. For weak shock case, maximum difference is 3.5% obtained with same turbulence model limiter. We can approximate relative uncertainty originating from selection of different turbulence models by comparing nozzle efficiency values obtained with same limiter grid level. At grid level 4, maximum difference between strong shock results of Sp-Al k-ω model is 9.2% obtained with Min-Mod limiter. For weak shock case, maximum difference at grid level 2 is 2.2%, obtained with same limiter. It should be noted that, at each grid level, relative uncertainty due to turbulence models is different resulting from interaction of physical modeling uncertainties with numerical errors. For strong shock case, at each grid level, difference between nozzle efficiency values of original geometry results of modified-wall geometry is much smaller than variations originating from or sources of uncertainty regardless of turbulence model limiter used. On or h, this difference is notable for weak shock case varies between 0.9% 1.4%. Nozzle efficiency values of extended geometry show considerable deviation from results of original geometry at certain grid levels, when are used as exit pressure ratios for strong weak shock cases respectively. For exit pressure ratio of , maximum difference is 1.8% obtained with grid level 3. The maximum difference for exit pressure ratio of is 6.9% observed at grid level 4. The difference between results of original extended geometry is smaller when exit pressure ratios of are used. For exit pressure ratio of 0.72, maximum difference is 0.8% observed at grid level 3. A maximum difference of 1.1% is obtained at grid level 2 for exit pressure ratio of Major observations on uncertainty in nozzle efficiencies for strong shock case: 1. The range of variation in nozzle efficiency results is much larger than one observed in weak shock case. The maximum variation is about 10% for strong shock case, 4% for weak shock case, when results of original geometry are compared. 2. Magnitude of discretization errors is larger than that of weak shock case. The discretization errors at grid level 2 can be up to 6% for strong shock case. 3. Relative uncertainty due to selection of turbulence model can be larger than discretization errors depending upon grid level used. This uncertainty can be as large as 9% at grid level The contribution of error in geometry representation to overall uncertainty is negligible compared to or sources of uncertainty. 12

14 Table 6: Nozzle efficiency values obtained with different grid levels, limiters, turbulence models, geometries boundary conditions. turbulence model k-ω k-ω Sp-Al Sp-Al limiter Min-mod Van Albada Min-mod Van Albada strong shock grid extended modifiedwalwall modified- level original geometry original geometry geometry P e P e geometry geometry weak shock extended geometry P e P e Major observations on uncertainty in nozzle efficiencies for weak shock case: 1. Discretization error is dominant source of uncertainty. The maximum value of discretization error is 3.5%, whereas maximum value of turbulence model uncertainty is about 2%. 2. The nozzle efficiency values are more sensitive to exit boundary conditions associated error magnitudes can be larger than size of or sources. The difference between results of original geometry extended geometry can be as large as 7% when exit pressure ratio of is used. 3. The contribution of error in geometry representation to overall uncertainty can be up to 1.5%. 5. Conclusions Different sources of uncertainty in CFD simulations are illustrated by examining a 2-D, turbulent, transonic flow in a converging-diverging channel at various P e ratios by using commercial CFD code GASP. Runs were performed with different turbulence models (Sp-Al k-ω), grid levels, fluxlimiters (Min-Mod Van Albada). Two flow conditions were studied by changing exit pressure ratio: first one was a complex case with a strong shock a separated flow region; second was a weak shock case with attached flow throughout entire channel. The uncertainty in CFD simulation results was studied in terms of five contributions: (1) iterative convergence error, (2) discretization error, (3) error in geometry representation, (4) turbulence model, (5) downstream boundary condition. In addition to original geometry used in calculations, contribution of error in geometry representation to CFD simulation uncertainties was studied through use of a modified geometry, based on measured geometric data. Also an extended version of transonic diffuser was used to determine effect of change of downstream boundary location on results. Overall, this paper demonstrated that for a weak shock case without separation, informed CFD users can obtain reasonably accurate results, whereas y are more likely to get large errors for strong shock case with substantial separation. In particular, following conclusions can be made based on results obtained in this study: 1. Grid convergence was not achieved with grid levels that have moderate mesh sizes. For strong shock with flow separation, highly refined grids, which are beyond grid levels we use in this study, are needed for spatial convergence. Even with finest mesh level we can afford, achieving asymp- 13