PhD%20Preliminary%20Oral%20Exam%20CHARACTERIZATION%20AND%20PREDICTION%20OF%20CFD%20SIMULATION%20UNCERTAINITIES

1
PhD Preliminary Oral ExamCHARACTERIZATION AND
PREDICTION OF CFD SIMULATION UNCERTAINITIES

• by
• Serhat Hosder
• Chair Dr. Bernard Grossman
• Committee Members
• Dr. Raphael T. Haftka Dr. William H. Mason
• Dr. Reece Neel Dr. Rimon Arieli
• Department of Aerospace and Ocean Engineering
• Virginia Tech.
• Blacksburg, VA

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Outline of the Presentation

• Introduction
• Classification of CFD Simulation Uncertainties
• Objective of the Present Work
• Previous Studies
• Transonic Diffuser Case
• Results, findings and discussion about different
  sources of uncertainty
• Conclusions

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Introduction (1)

• The 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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Introduction (2)
Drag Polar Results for DLR F-4 Wing at M0.75,
Rec3x106 (taken from 1st AIAA Drag Prediction
Workshop (DPW), Ref. 1)  
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Classification of CFD Simulation Uncertainties

• 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.)
• Uncertainty due to Discretization Error
• Numerical replacement of PDEs and continuum B.C
  with algebraic equations
• Consistency and Stability of PDEs
• Spatial (grid) and temporal resolution
• Uncertainty due to Iterative Convergence Error

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Definition of Uncertainty and Error

• Oberkampf and Trucano (Ref. 2) defined
• Uncertainty as a potential deficiency in any
  phase or activity of modeling process that is due
  to the lack of knowledge (uncertainty of
  turbulence models, geometric dimensions,
  thermo-physical parameters, etc.)
• Error as a recognizable deficiency in any phase
  or activity of modeling and simulation
• Discretization errors can be estimated with
  certain methods by providing certain conditions
• In this work, well refer the inaccuracy in the
  CFD simulations due different sources as
  uncertainty

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Objective of the Present Work

• Characterize different sources of CFD simulation
  uncertainties
• Consider different test cases
• Apply different grids, solution
  schemes/parameters, and physical models
• Try to quantify/predict the magnitude and the
  relative importance of each uncertainty
• Compare the magnitudes of CFD simulation
  uncertainties with other sources of uncertainty
  (geometric uncertainty, uncertainty in flow
  parameters, etc.)

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Previous Studies

• Previous CFD related studies mainly focused on
  discretization and iterative convergence error
  estimations
• Grid Convergence Index (GCI) by Roache (Ref. 3)
• Discretization Error of Mixed-Order Schemes by C.
  D. Roy (Ref. 4)
• Trucano and Hill (Ref. 5) proposed statistical
  based validation metrics for Engineering and
  Scientific Models
• Hemsch (Ref. 6) performed statistical analysis of
  CFD solutions from 1st AIAA DPW.
• Kim (Ref. 7) made statistical modeling of
  simulation errors (from poorly converged
  optimization runs) and their reduction via
  response surface techniques

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Description of Transonic Diffuser Test Case (1)

• Known as Sajben Transonic Diffuser case in CFD
  Validation studies
• Top wall described by an analytical equation
• Although geometry is simple, the flow-field is
  complex.
• The Shock strength and the location determined
  by exit-pressure-to-inlet-total pressure ratio
  Pe/P0i
• Pe/P0i0.72 (Strong shock case), Pe/P0i0.82
  (Weak shock case),

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Description of Transonic Diffuser Test Case (2)
Mach contours for the weak shock case
Mach contours for the strong shock case
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Simulation Code, Solution Parameters, and Grids
(1)

• General Aerodynamic Simulation Program (GASP)
• 3-D, structured, multi-block, finite-volume, RANS
  code
• Inviscid fluxes calculated by upwind-biased 3rd
  (nominal) order spatially accurate Roe-flux
  scheme
• All viscous terms were modeled (full N-S)
• Implicit time integration to reach steady-state
  solution with Gauss-Seidel algorithm

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Simulation Code, Solution Parameters, and Grids
(2)

• Flux-Limiters
• Van Albadas limiter
• Min-Mod limiter

• Turbulence Models
• Spalart-Allmaras (Sp-Al)
• k-w (Wilcox, 1998 version)

• Grids Generated by an algebraic mesh generator
  
• Grid 1 (g1) 41x26x2
• Grid 2 (g2) 81x51x2
• Grid 3 (g3) 161x101x2
• Grid 4 (g4) 321x201x2
• Grid 5 (g5) 641x401x2 (Used only for Sp-Al,
  Min-Mod, strong shock case)
• y 0.53 (for g2) and y 0.26 (for g3) at the
  bottom wall

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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 Total number of experimental points
(Nexp36)   di Orthogonal distance from the ith
experimental data point to Pc(x) curve
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Uncertainty due to iterative convergence error (1)

• Normalized L2 Norm Residual of the energy
  equation for the case with Sp-Al turbulence
  model, Van-Albada and Min-Mod limiters at the
  strong shock case.

• Same convergence behavior with respect to the
  limiters observed for the k-w case.

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Uncertainty due to iterative convergence error (2)
Poor L2 norm convergence does not seem to effect
the convergence of the neff results at different
grid levels
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Uncertainty due to iterative convergence error (3)
Roy and Blottner (Ref. 8) proposed a method to
estimate, the iterative convergence error at
time level (cycle) n
Assuming exponential decrease for
Need three time levels in the exponential region
where
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Uncertainty due to discretization error (1)
For each case with a different turbulence model,
grid level (resolution) and the flux-limiter
affect the magnitude of the discretization error

• The effect of the limiter observed at grid
  levels g1 and g2
• At grid levels g3 and g4, the effect is much
  smaller

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Uncertainty due to discretization error (2)

• Richardsons extrapolation method

h a measure of grid spacing p The order of the
method.

• Assumptions needed to use Richardsons method
• Grid resolution is in the asymptotic region
• The order of the spatial accuracy, p should be
  known. Usually observed order of spatial accuracy
  is different than the nominal value. The observed
  order should be determined.
• Monotonic grid convergence. Mixed-order schemes
  can cause non-monotonic convergence. Roy (Ref. 4)
  proposed a method for for the discretization
  error estimate of mixed-order schemes.

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Uncertainty due to discretization error (3)
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Uncertainty due to discretization error (4)

• p values are dependent on the grid levels used
• However the difference between the (neff)exact
  values are small compared to overall uncertainty

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Uncertainty due to discretization error (5)

• The uncertainty due to discretization error is
  bigger for the cases with strong shock compared
  to the weak shock results at each grid level. The
  flow structure has significant effect on the
  discretization error.
• For the monotonic cases, largest errors occur for
  the Sp-Al, Min-Mod, strong shock case and the
  smallest errors are obtained for the k-w ,
  Van-Albada, weak shock case
• Non-monotonic convergence behavior for the cases
  with k-w and the strong shock as the mesh is
  refined

23
Uncertainty due to discretization error (6)
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Uncertainty due to discretization error (7)

• Noise due to discretization error observed at
  grid levels 1 and 2.
• Noise error small compared to the systematic
  discretization error between each grid level.
  However, this can be important for gradient-based
  optimization.
• Kim (Ref. 7) successfully modeled the the noise
  error due to poor convergence of the optimization
  runs by fitting a probability distribution
  (Weibull) to the error.
• The noise error can be reduced via response
  surface modeling.

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Uncertainty due to turbulence models (1)

• Uncertainty due to turbulence modeling (in
  general physical modeling) should be investigated
  after estimation of the discretization and
  iterative convergence error.
• Difficult to totally separate physical modeling
  errors from discretization errors
• Validation of the Engineering and Scientific
  Models deals with accuracy of the physical model
• Need high-quality experimental data

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Uncertainty due to turbulence models (2)

• Orthogonal distance error, En is used for
  comparison of CFD results with the experiment

En for each case is scaled with the maximum
value obtained for k-w , Min-Mod, strong shock
case
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Uncertainty due to turbulence models (3)
For each case (strong shock or weak shock), best
match with the experiment is obtained with
different turbulence models at different grid
levels
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Uncertainty due to turbulence models (4)

• Experimental uncertainty should be considered

• With the experimental geometry, a perfect match
  with CFD and experiment can be observed upstream
  of the shock
• Upstream of the shock, discrepancy between CFD
  simulations and experiment is most likely due to
  the experimental uncertainty

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Uncertainty due to turbulence models (5)

• A better way of using En for this example would
  be to evaluate it only downstream of the shock
• The discretization and iterative convergence
  error should be estimated for En in a similar
  way used for the nozzle efficiency
• An estimate of exact value of (En ) can be used
  for approximating the uncertainty due to
  turbulence models
• The relative uncertainty due to the selection of
  turbulence models can also be investigated by
  using (neff)exact values obtained by Richardsons
  extrapolation

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Uncertainty due to turbulence models (6)

• Hills and Trucano (Ref. 5) proposed a Maximum
  Likelihood based model validation metric to test
  the accuracy of the model predictions
• Uncertainty in the experimental measurements and
  the model parameters are considered
• Model parameters
• Material properties
• Geometry
• Boundary or Initial Conditions
• This method requires prior knowledge about the
  measurement and the model parameter uncertainty
  (modeling with probabilistic distributions)
• Looks for statistically significant evidence that
  the model validations are not consistent with the
  experimental measurements

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Uncertainty due to turbulence models (7)

• PDF(d) PDF of measurement vector occurrence
• PDF(p) PDF of model parameter vector
  occurrence
• PDF(d, p) PDF(d) x PDF(p)
• Find the maximum likely values for the mode of
  the measurements d and the model parameters p
• Find the maximum value of Joint PDF via
  optimization
• Evaluate the probability of obtaining a smaller
  PDF assuming that the model is correct
• If this value is bigger than the level of
  significance that we assumed for rejecting a good
  model, than the model predictions are consistent
  with the experiment

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Uncertainty due to turbulence models (8)

• Possible application to test the accuracy of the
  turbulence models
• Takes into account the experimental uncertainty
• Requires prior knowledge of uncertainty in the
  measurements and the model parameters
• Selection of model parameters
• No simple relationship with the model parameters
  and the output quantities. Using response surface
  techniques may be needed to find a functional
  form.

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Additional Test Cases

• Need more cases to generalize the results
  obtained in Transonic Diffuser Case
• Next possible case Steady, turbulent, flow
  around an airfoil (RAE2822 or NACA0012)
• Consider transonic and subsonic cases
• Consider a range of AOA
• Output quantities to monitor Cl, Cd, Cp
  distributions
• Orthogonal distance error may be used for
  characterizing Cp distributions
• Consider a case with a more complex geometry

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Conclusions (1)

• Different sources of uncertainty in CFD
  simulations should be investigated separately.
• Discretization and iterative convergence errors
  can be estimated by certain methods in certain
  conditions
• Limiters affect the iterative convergence and the
  discretization error.
• L2 norm convergence affected by the use of
  different limiters
• Poor L2 norm convergence do not seem to affect
  the neff results
• Asymptotic Grid convergence hard to obtain
• Flow structure has a strong effect on the
  magnitude of the discretization error.
• Iterative convergence error small compared to the
  discretization error
• Uncertainty due to turbulence model should be
  investigated after the estimation of
  discretization and iterative convergence error.

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Conclusions (2)

• Comparison with the experiment is needed to
  determine the accuracy of the turbulence models
• Experimental uncertainty should be considered
  possibly by using a statistical method
• More cases need to be analyzed to generalize the
  results

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References

1. Levy, D. W., Zickuhr, T., Vassberg, J., Agrawal
   S., Wahls. R. A., Pirzadeh, S., Hemsch, M. J.,
   Summary of Data from the First AIAA CFD Drag
   Prediction Workshop, AIAA Paper 2002-0841,
   January 2002
2. Oberkampf, W. L. and Trucano, T. G., Validation
   Methodology in Computational Fluid Dynamics.
   AIAA Paper 2000-2549, June 2000
3. Roache, P. J. Verication and Validation in
   Computational Science and Engineering.Hermosa
   Publishers, Albuquerque, New Mexico, 1998.
4. Roy, C. J., Grid Convergence Error Analysis for
   Mixed-Order Numerical Schemes, AIAA Paper
   2001-2606, June 2001
5. Hills, R. G. and Trucano, T. G., Statistical
   Validation of Engineering and Scientific Models
   A Maximum Likelihood Based Metric, Sandia
   National Loboratories, SAND2001-1783
6. Hemsch, M. J., Statistical Analysis of CFD
   Solutions from the Drag Prediction Workshop, AIAA
   Paper 2002-0842, January 2002
7. Kim, H., Statistical Modeling of Simulation
   Errors and Their Reduction Via Response Surface
   Techniques, PhD dissertation, VPISU, June 2001
8. Roy, C. J. and Blottner F. G., Assesment of
   One-and Two-Equation Turbulence Models for
   Hypersonic Transitional Flows, Journal of
   Spacecraft and Rockets, Vol.38, No. 5,
   September-October 2001