parameter sensitivity analysis for hypersonic viscous flow...

Parameter Sensitivity Analysis for Hypersonic ViscousFlow using a Discrete Adjoint Approach

Brian A. Lockwood and Dimitri J. Mavriplis

Mechanical EngineeringUniversity of Wyoming

January 4, 2010

B.A. Lockwood, D.J. Mavriplis (U. of WY.) Hypersonic Adjoint Sensitivity Jan. 4, 2010 1 / 32

Outline

Outline

Introduction

Overview of Flow Solver and Physical Models

Solution Scheme Details

Sensitivity Formulation

Flow and Sensitivity Results for Perfect Gas Model

Flow and Sensitivity Results for Real Gas Model

Conclusion


Introduction

Simulation of complexproblems rely heavily onempirical relations

Relations can requirehundreds of parameters todefine

Sensitivity and uncertaintycan enhance analysis anddesign capability

Sampling currently used dueto nonlinear nature of flows

Thousands of flowsolutions requiredComputationallyexpensive


Introduction

Localized sensitivity calculated usingflow adjoint

Sensitivity to large number of inputswith cost approximately equal to flowsolve

Possibilities for uncertaintyquantification, adaptation andsimulation optimization.

Should be possible to augmentweaknesses of sampling with an adjointbased approach


Flow Solver Details

Navier Stokes Equations:

∂U

∂t+∇ · ~F (U) = ∇ · ~Fv (U) + S(U) (1)

Two dimensional, cell-centered finite volume solver using unstructuredtriangles and/or quadrilaterals

Solver uses a fully implicit, pseudo-time stepping method

Perfect gas and Real gas models examined


Physical Model

Perfect gas variables

U =

ρρ~uρet

~F =

ρ~uρ~u ⊗ ~u + P

ρ~uht

~Fv =

0τ

τ · ~u − ~q

Sutherland’s Law used for viscosity

5 Parameters required to define model:

Ratio of Specific Heats γReynolds NumberPrandtl NumberTwo constants within Sutherland’s Law


Physical Model

5 species, Two Temperature Real Gas Model

U =

ρsρ~uρetρev

~F =

ρs~uρ~u ⊗ ~u + P

ρ~uhtρ~uhv

S =

ωs00∑

s ωsD̂v ,s + QT−V

~Fv =

−ρs Ṽsτ

τ · ~u − ~q − ~qv −∑

s ht,sρs Ṽs−∑

s hv ,sρs Ṽs − ~qv

Dunn-Kang chemical kinetics model used

Transport quantities calculated using curve fits from Blottner et al

Approximately 250 constants required to define physical model


Solver Description

Solution marched to steady state using implicit pseudo-time stepping

Method of Lines:dU

dt+ R(U) = 0 (2)

BDF1 discretization used for pseudo-time derivative

Unsteady residual given by:

J(Un,Un−1) =Un −Un−1

∆t+ R(Un) (3)

Nonlinear equation J(Un,Un−1) = 0 solved approximately at eachtime step


Solution Scheme

Fixed number of inexact Newton iterations performed per time step(∼ 10)

δUk = − [P]−1 J(Uk ,Un−1) (4)Uk+1 = Uk + λδUk (5)

[P] chosen to approximate ∂J∂Uk

, 1st order Van-Leer-Hänel

Preconditioner matrix and transport quantities calculated once pertime step and frozen.

Global time stepping used for start-up, local time stepping for fullconvergence

λ used to keep updates sensible


Spatial Discretization

Gradient reconstruction of primitives

Green-Gauss contour integration used to calculate gradients

Smooth Van Albada Limiter with Pressure Switch used:

Ψk = max(0, 1− Kνk)1

∆−(∆+

2+ ε2)∆− + 2∆−

2∆+

∆+2 + 2∆− + ∆−∆+ + ε2(6)

νi =

∑k |PR − PL|∑k PR + PL

(7)

Face based Gradients calculated using averaging and correction term:

∇Vk = ∇̃V +VR − VL − ∇̃V ·∆~T

|∆~T |∆~T

|∆~T |(8)

Inviscid Flux Calculated Using AUSM+UP flux function with FrozenSpeed of Sound


Sensitivity Derivation

Let the objective and constraint have following functional dependence

L = L(D,U(D)) (9)

R = R(D,U(D)) = 0 (10)

Objective and Constraint may be differentiated using the Chain rule

dL

dD=∂L

∂D+∂L

∂U

∂U

∂D(11)

dR

dD=∂R

∂D+∂R

∂U

∂U

∂D= 0 (12)

Solve Constraint Equation for ∂U∂D


Sensitivity Derivation

Forward Sensitivity Equation Given by:

dL

dD=∂L

∂D− ∂L∂U

∂R

∂U

−1 ∂R

∂D(13)

Transpose Equation

dL

dD

T

=∂L

∂D

T

− ∂R∂D

T ∂R

∂U

−T ∂L

∂U

T

(14)

Define Flow Adjoint with Equation:

∂R

∂U

T

Λ = − ∂L∂U

T

(15)

Sensitivity Calculated by:

dL

dD

T

=∂L

∂D

T

+∂R

∂D

T

Λ (16)


Parameter Sensitivity

For Model Parameters, D dependence enters through field variables µ

L = L(U(D),µ(D,U(D))) (17)

R = R(U(D),µ(D,U(D))) (18)

Sutherland’s Law Example

µ

µref=

C1T3/2

T + S(19)

Forward Sensitivity Equation for Model Parameters

dL

dD=∂L

∂µ

∂µ

∂D+

(∂L

∂U+∂L

∂µ

∂µ

∂U

)∂U

∂D(20)



Adjoint Sensitivity Equation for Model Parameters

∂L

∂D

T

=∂µ

∂D

T(∂L

∂µ

T

+∂R

∂µ

T

Λ

)(21)

Flow Adjoint Found by Solving Equation:[∂R

∂U+∂R

∂µ

∂µ

∂U

]TΛ = −

(∂L

∂U

T

+∂µ

∂U

T ∂L

∂µ

T)

(22)

Adjoint Equation solved with Defect Correction:

[P]T δΛk = − ∂L∂U

T

−

[∂µ

∂U

T ∂R

∂µ

T

+∂R

∂U

T]

Λk (23)

Λk+1 = Λk + δΛk (24)


Field Variable Sensitivity

Set ∂µ∂D = 1 and∂µ∂U = 0:

∂L

∂µ

T

=∂L

∂µ

T

+∂R

∂µ

T

Λµ (25)

Formally, Adjoint Solution with Frozen µ required:[∂R

∂U

)µ

]TΛµ = −

∂L

∂U

)Tµ

(26)

Field Variables defined throughout domain (at cell centers or facecenters)

May be used for Uncertainty Propagation or Model Adaptation

δL2 =∑

i

∂L

∂µi

2

δµ2i (Ui ) (27)


Perfect Gas Results

5 km/s cylinder test case

Fixed Wall temperature

Results compared againstLAURA and FUN2D

Table: Benchmark Flow Conditions

V∞ = 5 km/sρ∞ = 0.001 kg/m

3

T∞ = 200 KTwall = 500 KM∞ = 17.605Re∞ = 753,860Pr∞ = 0.72


Perfect Gas Results

Temperature Contour for 5 km/s case


Perfect Gas Results

Temperature along Stagnation Streamline Temperature along Stagnation Streamline(Boundary Layer)


Perfect Gas Results

Surface Pressure Distribution Surface Heating Distribution


Perfect Gas Sensitivity

Geometric and Parameter Sensitivities for 5 km/s Benchmark CaseSensitivities computed for Integrated Surface heating Objective:

L =

∫A−k∇T · ~ndA (28)

Sensitivities Validated using Finite DifferenceAdjoint Solution for Velocity


Geometric Sensitivity

Sensitivity of Surface Heating to surface geometry calculated

Normal displacement of surface grid points used as design variable

31 total design variables

-50 0 50Angle (degrees)

-2e+06

0

2e+06

4e+06

Sen

siti

vity

Adjoint SensitivityFinite Difference Result

Heat Flux Sensitivity to Surface Point Displacements



Sensitivity to Perfect Gas Parameter γ

Fixed Freestream Velocity and Fixed Freestream Mach NumberConsidered

Table: Sensitivity of Surface Heating to γ

Fixed Velocity Fixed Mach Number

Objective Value 1.628× 10−2 1.628× 10−2Finite Difference −3.111× 10−2 −1.198× 10−2

Adjoint −3.109× 10−2 −1.206× 10−2Relative Error 7.279× 10−4 7.186× 10−4


Real Gas Results

5 km/s cylinder test case

Fixed Wall temperature

Super-catalytic Wall

Results compared withLAURA

Table: Benchmark Flow Conditions

V∞ = 5 km/sρ∞ = 0.001 kg/m

3

T∞ = 200 KTwall = 500 KM∞ = 17.605Re∞ = 753,860Pr∞ = 0.72


Real Gas Results

Temperature along Stagnation Streamline Temperature along Stagnation Streamline (Log Scale)


Real Gas Results

Mass Fraction along Stagnation Streamline for 5 km/s case


Real Gas Results

Skin Friction Distribution Surface Heating Distribution


Real Gas Sensitivity

Sensitivities computed for Integrated Surface heating Objective:

L = −∫

Ak∇T · ~n + kv∇Tv · ~ndA (29)

Sensitivities Validated using Finite Difference

Geometric Sensitivity, 61 Design Variables



Sensitivity of Surface Heating to Arrhenius Coefficients:

Kf = Cf Tηfa e−

Ea,fkTa (30)

Kb = CbTηba e−

Ea,bkTa (31)

Sensitivities Expressed as fractional changes (i.e. dL/LdD/D )



Sensitivity of Surface Heating to Species Viscosity Curve FitParameters:

µs = 0.1e(As ln(T )+Bs)ln(T )+Cs (32)

Significantly Higher Sensitivity than Reaction Parameters



Sensitivity of Surface Heating to Face-centered Viscosity throughoutDomain7800 Faces within Computational GridExact and Approximate Adjoints used to Compute Sensitivity



Viscosity Sensitivity along Stagnation Streamline Viscosity Sensitivity at 45 degrees


Conclusions

Conclusions:

Sensitivity to large number of parameters with minimal effort possiblewith adjoint

Parameter sensitivity can be used to determine most important modelcomponents

Field variable sensitivity possible; identifying regions/ranges ofgreatest importance

Future Work:

Extend valid range of sensitivities by including higher order terms

Investigate simulation adaptation using adjoint

Explore hybrid sensitivity/sampling approaches to uncertaintyquantification


parameter sensitivity analysis for hypersonic viscous flow...

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