AIAA 2002-5531 OBSERVATIONS ON CFD SIMULATION UNCERTAINITIES

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AIAA 2002-5531OBSERVATIONS ON CFD SIMULATION
UNCERTAINITIES

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

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Introduction

• Computational fluid dynamics (CFD) as an
  aero/hydrodynamic analysis and design tool
• Increasingly being used in multidisciplinary
  design and optimization (MDO) problems
• Different levels of fidelity (from linear
  potential solvers to RANS codes)
• CFD results have a certain level of uncertainty
  originating from different sources
• Sources and magnitudes of the uncertainty
  important to assess the accuracy of the results

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Objective of the Paper

• To illustrate different sources of uncertainty in
  CFD simulations, by a careful study of
• 2-D, turbulent, transonic flow
• In a converging-diverging channel (primary case)
• Around a transonic airfoil
• To compare the magnitude and importance of each
  source of uncertainty
• Use different turbulence models, grid densities
  and flux-limiters
• Use modified geometries and boundary conditions

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Uncertainty Sources

• Physical Modeling Uncertainty
• PDEs describing the flow (Euler, Thin-Layer N-S,
  Full N-S, etc.)
• Boundary and initial conditions (B.C and I.C)
• Auxiliary physical models (turbulence models,
  thermodynamic models, etc.)
• Discretization Error
• Originates from the Numerical replacement of PDEs
  and continuum B.C with algebraic equations
• Consistency and Stability
• Spatial (grid) and temporal resolution
• Iterative Convergence Error
• Programming Errors

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Transonic Diffuser Problem (primary case)
strong shock
weak shock
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Transonic Airfoil Problem

• RAE 2822 Airfoil
• Test case Rec6.2 x 106,
• Mach0.75, ?3.19?
• (AGARD case 10)
• Test case Rec6.2 x 106,
• Mach0.30, ?0.0?
• Test case Inviscid,
• Mach0.30, ?0.0?

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Computational Modeling

• General Aerodynamic Simulation Program (GASP)
• Reynolds-averaged, 3-D, finite volume
  Navier-Stokes (N-S) code
• Inviscid fluxes calculated by upwind-biased 3rd
  (nominal) order spatially accurate Roe-flux
  scheme
• Flux-limiters Min-Mod and Van Albada
• In viscous runs, full N-S equations are solved
• Turbulence models
• Spalart-Allmaras (Sp-Al)
• k-? (Wilcox, 1998 version) with Sarkars
  compressibility correction
• Implicit time integration to reach steady-state
  solution with Gauss-Seidel algorithm

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Grids Used in the Computations
Transonic diffuser (original geometry)
RAE 2822 Airfoil
Grid level Mesh Size (number of cells)
1 40 x 25
2 80 x 50
3 160 x 100
4 320 x 200
5 640 x 400
Grid level Mesh Size (number of cells)
1 92 x 16
2 184 x 32
3 368 x 64
4 736 x 128

• A single solution on grid 5 requires
  approximately 1170 hours of total node CPU time
  on a SGI Origin2000 with six processors (10000
  cycles)
• Typical grid levels used in CFD applications
• For transonic diffuser case Grid level 2
• For RAE 2822 case Grid level 3

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Output Variables (1)
Nozzle efficiency, neff H0i Total enthalpy
at the inlet  He Enthalpy at the exit  Hes
Exit enthalpy at the state that would be reached
by isentropic expansion to the actual pressure
at the exit
Throat height
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Output Variables (2)

• Orthogonal Distance Error, En
• A measure of error in wall pressures between the
  experiment and the curve representing the CFD
  results

Pc Wall pressure obtained from CFD
calculations   Pexp Experimental Wall Pressure
Value   Nexp Number of experimental data points
  di Orthogonal distance from the ith
experimental data point to Pc(x) curve
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Uncertainty Sources Studied

• In transonic diffuser case, uncertainty in CFD
  simulations has been studied in terms of five
  contributions
• Iterative convergence error
• Discretization error
• Error in geometry representation
• Turbulence model
• Changing the downstream boundary
• condition

Numerical uncertainty
Physical modeling uncertainty
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Discretization Error
(Richardsons Extrapolation)
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Discretization Error
The approximations to the exact value of nozzle
efficiency and p depend on the grid levels
used in the estimations.
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Discretization Error
Noise error small compared to the systematic
discretization error between each grid level.
However, this can be important in a
gradient-based optimization.
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Discretization Error

• Complexity level of the flow structure affects
  the grid convergence
• RAE case, Mach 0.3, ? 0.0 deg, Re6.2x106
  Attached flow
• RAE case, Mach 0.75, ? 3.19 deg, Re6.2x106
  Shock-induced
  
  
  separation region

Case Grid level CL CD (drag counts)
Mach 0.3, ? 0.0 deg, Re6.2x106 1 0.15940 191
Mach 0.3, ? 0.0 deg, Re6.2x106 2 0.19694 104
Mach 0.3, ? 0.0 deg, Re6.2x106 3 0.20546 85
Mach 0.3, ? 0.0 deg, Re6.2x106 4 0.20550 83
Mach 0.75, ? 3.19 deg, Re6.2x106 1 0.68992 353
Mach 0.75, ? 3.19 deg, Re6.2x106 2 0.75094 298
Mach 0.75, ? 3.19 deg, Re6.2x106 3 0.77889 295
Mach 0.75, ? 3.19 deg, Re6.2x106 4 0.79341 302
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Discretization Error
3.8 difference in CL between the cases with and
without the limiter at grid level 2 (RAE 2822,
inviscid, Mach0.3, and ?0.0 deg.)
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Discretization Error

• Major observations on the discretization errors
• For transonic diffuser cases and the RAE 2822
  case with flow separation, grid convergence is
  not achieved with grid levels that have moderate
  mesh sizes.
• Shock-induced flow separation has significant
  effect on the grid convergence
• Discretization error magnitudes are different
  for the cases with different turbulence models.
  The magnitude of numerical errors are affected by
  the physical models used.

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Error in Geometry Representation

• Upstream of the shock, discrepancy between the
  CFD results of original geometry and the
  experiment is due to the error in geometry
  representation.
• Downstream of the shock, wall pressure
  distributions are the same regardless of the
  geometry used.

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Turbulence Models

• Compare orthogonal distance error calculated
  downstream of the shock at grid level 4 for each
  case
• Difficult to isolate the numerical errors from
  the physical uncertainties
• For each flow condition, highest accuracy
  obtained with a different turbulence model
• In some cases, physical modeling uncertainties
  may cancel each other, and the closest result to
  the experiment can be obtained at intermediate
  grid levels

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Turbulence Models
Effect of the Sarkars compressibility correction
on the nozzle efficiency
Strong shock
Weak Shock
Turbulence model Grid level nozzle efficiency
k-? w/ Sarkars Comp. Correct. 1 0.8113
k-? w/ Sarkars Comp. Correct. 2 0.79362
k-? w/ Sarkars Comp. Correct. 3 0.78543
k-? w/o Sarkars Comp. Correct 1 0.78117
k-? w/o Sarkars Comp. Correct 2 0.75434
k-? w/o Sarkars Comp. Correct 3 0.74271
Sp-Al 1 0.81827
Sp-Al 2 0.76452
Sp-Al 3 0.73535
Turbulence model Grid level nozzle efficiency
k-? w/ Sarkars Comp. Correct. 1 0.86563
k-? w/ Sarkars Comp. Correct. 2 0.84093
k-? w/ Sarkars Comp. Correct. 3 0.83271
k-? w/o Sarkars Comp. Correct 1 0.86494
k-? w/o Sarkars Comp. Correct 2 0.83561
k-? w/o Sarkars Comp. Correct 3 0.82465
Sp-Al 1 0.87577
Sp-Al 2 0.83956
Sp-Al 3 0.82048
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Turbulence Models
Effect of the Sarkars compressibility correction
on the wall pressure
Strong shock
Weak Shock
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Downstream Boundary Condition

• Extending the geometry or changing the exit
  pressure ratio affect
• location and strength of the shock
• size of the separation bubble

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Uncertainty on Nozzle Efficiency

• Nozzle efficiency as a global indicator of CFD
  results
• Cloud of the results that a reasonably informed
  user may obtain from CFD calculations

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Uncertainty on Nozzle Efficiency

• Major observations on the uncertainty in nozzle
  efficiency for the strong shock case
• The maximum variation is about 10 (original
  geometry)
• Magnitude of the discretization error is larger
  than that of the weak shock case. This error can
  be up to 6 at grid level 2.
• Depending on the grid level used, relative
  uncertainty due to the selection of turbulence
  model can be larger than the discretization error
  (can be as large as 9 at grid level 4)
• Contribution of the error in geometry
  representation to the overall uncertainty
  negligible compared to the other sources of
  uncertainty

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Uncertainty on Nozzle Efficiency

• Major observations on the uncertainty in nozzle
  efficiency for the weak shock case
• The maximum variation is about 4 (original
  geometry)
• The maximum value of the discretization error is
  3.5
• The maximum value of the relative uncertainty
  due to the selection of turbulence model is 2
• Nozzle efficiency values more sensitive to the
  exit boundary conditions. The difference between
  the results of the original geometry and the
  extended geometry can be as large as 7 depending
  on the exit pressure ratio used.
• Contribution of the error in geometry
  representation to the overall uncertainty can be
  up to 1.5

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Conclusions

• For attached flows without shocks (or with weak
  shocks), informed CFD users can obtain
  reasonably accurate results
• More likely to get large errors for the cases
  with strong shocks and substantial separation
• For transonic diffuser cases and the RAE 2822
  case with flow separation, grid convergence is
  not achieved with grid levels that have moderate
  mesh sizes.
• The shock induced flow separation has
  significant effect on the grid convergence
• The magnitudes of numerical errors are
  influenced by the physical models (turbulence
  models) used.
• Difficult to isolate physical modeling
  uncertainties from numerical errors

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Conclusions

• Depending on the flow structure, highest
  accuracy is obtained with a different turbulence
  model
• In some cases, physical modeling uncertainties
  may cancel each other, and the closest result to
  the experiment can be obtained at intermediate
  grid levels
• In nozzle efficiency results,
• range of variation for the strong shock is much
  larger than the one observed in the weak shock
  case ( 10 vs. 4)
• discretization error can be up to 6 at grid
  level 2 (strong shock)
• relative uncertainty due to the selection of the
  turbulence model can be as large as 9 (strong
  shock)
• changing the boundary condition can give 7
  difference (weak shock)