Application of a Non-Linear Frequency Domain Solver to the...

Application of a Non-Linear Frequency Domain Solverto the Euler and Navier-Stokes Equations

Matthew McMullen and Antony Jamesonand Juan J. Alonso

Dept. of Aeronautics & AstronauticsStanford University

40th AIAA Aerospace Sciences Meeting& ExhibitReno, NV

January 14-17, 2002

AIAA 2001-0120 1

Outline

• Motivation

• Governing Equations

• Solution Method

• Gradient Based Variable Time Period

• Results - Pitching Airfoil

• Results - Cylinder

• Conclusions

AIAA 2001-0120 2

Motivation

• The goal of ASCI is to calculate the unsteady flow of an aircraft gasturbine engine. This includes component simulations of the compressor,combustor, and turbine.

Component % Wheel Total CPU Hours

Turbine 16 2.0 millionCompressor 16 5.4 million

• The job size typically varies between 500-1,000 processors, but estimatesfor job length are now provided in months.

AIAA 2001-0120 3

Governing Equations

• Navier-Stokes Equations in integral form

∫∂W

∂tdV +

∮~F · ~Nds = 0

W =

ρρuρvρE

~Fi = f =

ρuρu2 + p− σxx

ρuv − σxy

ρuH − uσxx − vσxy + qx

~Fj = g =

ρvρuv − σxy

ρv2 + p− σyy

ρvH − uσxy − vσyy + qy

AIAA 2001-0120 4

Governing Equations (cont.)

• Closure is provided by the following equations:

p = (γ − 1)ρ[e− 12(u2 + v2)]

σxx = 2µux − 23µ(ux + vy)

σyy = 2µvy − 23µ(ux + vy)

σxy = σyx = µ(uy + vx)

qx = κ∂T

∂x= − γ

γ − 1µ

Pr

∂pρ

∂x

qy = κ∂T

∂y= − γ

γ − 1µ

Pr

∂pρ

∂y

AIAA 2001-0120 5


• Approximating the volumetric integral for a given cell with volume V

∫∂W

∂tdV ≈ V

∂W

∂t

• Using the steady state residual variable R(W ) we can approximate theboundary integral as a sum of fluxes over a finite number of cell walls

∮~F · ~Nds ≈ R(W ) =

n∑

j=1

~Fj · ~Sj

AIAA 2001-0120 6


• Using the notation introduced above the integral form of the Navier-Stokes equation simplifies to

V∂W

∂t+ R(W ) = 0

• Expanding both W and R(W ) with a Fourier series in time

W =

N2 −1∑

k=−N2

Wkeikt R(W ) =

N2−1∑

k=−N2

Rkeikt

AIAA 2001-0120 7


• It follows that a periodic steady-state equation can be written for eachindependent wave number. We add in a pseudo-time derivative andnumerically integrate the equations.

VdWk

dτ+ ikV Wk + Rk = 0

• Each iterative step in the solution process requires the following dataflow.

AIAA 2001-0120 8

Solution Method

• Since we are dealing with solving a steady system of equations, we applyestablished methods to accelerate the convergence.

Multi-stage RK scheme with local time steppingImplicit residual averagingMultigrid V or W Cycle

• In addition, we are dealing with real functions where the Fouriercoefficients for the positive wavenumbers are equal to the complexconjugates of the Fourier coefficients for the negative wavenumbers.This eliminates computation required to integrate the negative wavenumbers forward in pseudo-time.

AIAA 2001-0120 9

Gradient Based Variable Time Period (GBVTP)

• In the development of the nonlinear frequency domain method, weassume the time period of the fundamental harmonic.

• A class of problems exist where the exact frequency of the discretizedequations can not be known in advance. GBVTP is an method thatiteratively determines this parameter.

• The process of finding a solution to the unsteady flow equations isanalogous to an optimization problem where the magnitude of theunsteady residual is minimized.

AIAA 2001-0120 10


• For this optimization problem we choose to minimize the square of themagnitude of the unsteady residual In. Taking a derivative of thisquantity we form a gradient with respect to the time period.

• This modification is not costly because the quantity In is alreadycalculated while monitoring the convergence of the solution.

AIAA 2001-0120 11


• The wavenumber k is calculated by normalizing the sinusoidal period ofoscillation 2π by the time period of interest T .

k =2πn

T(1)

• The unsteady residual can be can then be written as a function of thetime period T .

In =i2πnV

TWn + Rn (2)

AIAA 2001-0120 12


12

∂∣∣∣In

∣∣∣2

∂T= Inr

∂Inr

∂T+ Ini

∂Ini

∂T(3)

∂Inr

∂T=

2πnV Wni

T 2(4)

∂Ini

∂T= −2πnV Wnr

T 2(5)

• The gradient can be simplified by employing cross product notation.

12

∂∣∣∣In

∣∣∣2

∂T=

2πnV

T 2| ~In × ~Wn| (6)

AIAA 2001-0120 13


• The time period can be updated using the gradient information byselecting a stable step ∆T .

Tn+1 = Tn −∆T∂

∣∣∣In

∣∣∣2

∂T(7)

• Typically one can start with an initial guess in the vicinity of the finalanswer for the time period.

• The gradient can be used to adjust the time period at each subsequentiteration in the solution process.

AIAA 2001-0120 14

Results

• Pitching Airfoil - Forced Frequency

1. Euler2. Viscous

Baldwin-Lomax Turbulence Model

• Laminar vortex shedding from a cylinder - Variable Time Period

AIAA 2001-0120 15

Results - Pitching Airfoil

Description Variable Value

AGARD CaseNumber CT6 DI 55Airfoil 64A010Mean Angleof Attack αm 0.00Angle of AttackVariation α0 ±1.01◦

Reynolds Number Re∞ 12.56x106

Mach Number M∞ 0.796Reduced Frequency kc 0.202

Table 1: Experiment Parameters - AGARD Report 702 by Sanford Davis

AIAA 2001-0120 16

Results - Pitching Airfoil - Euler

Topology Dimensions Mean Mean GridBoundary SpacingDistance at Wall(Chords) (Chords)

O-mesh 81x33 128 0.0096O-mesh 161x33 128 0.0097C-mesh 129x33 22 0.0087C-mesh 193x49 22 0.0087

Table 2: Euler Grid Configurations

AIAA 2001-0120 17

Results - Pitching Airfoil - Euler

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

81x33 O-mesh 128x33 C-meshConformal Mapping Hyperbolic Mesh Generation

AIAA 2001-0120 18

Results - Pitching Airfoil - Euler - Cl Cm

• For each grid we ran separate cases employing 1,2 and 3 temporal modes.

−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125

Angle of Attack (degrees)

Coe

ffici

ent o

f Lift

ExperimentEuler Numeric Solutions

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent

ExperimentEuler Numeric Solutions

All grid and mode permutations All C-mesh grid and mode permutations

AIAA 2001-0120 19

Results - Pitching Airfoil - Euler Convergence

• Convergence results based on 129x33 C-mesh with 1 mode.

−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125


Coe

ffici

ent o

f Lift

1 Cycle5 Cycle10 Cycle15 Cycle

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


AIAA 2001-0120 20

Results - Pitching Airfoil - Viscous

• The measured coordinates of the CT6 airfoil were significantly differentthan the theoretical 64A010 airfoil. Two sets of grids were generatedbased on the different geometries.

Topology Dimensions Boundary Mean GridDistance Spacing(Chords) at Wall

(y+)

C-mesh 129x33 15 11.6C-mesh 193x49 12 6.9C-mesh 257x65 12 3.8

Table 3: Viscous Grid Configurations

AIAA 2001-0120 21

Results - Pitching Airfoil - Viscous

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

129x33 C-mesh 257x65 C-mesh

AIAA 2001-0120 22

Results - Pitching Airfoil - Viscous - Cl

• Results are depicted for all grid/mode permutations.

64A010 Coordinates CT6 Coordinates

AIAA 2001-0120 23

Results - Pitching Airfoil - Viscous - Cm

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent

Experiment1 Mode2 Mode3 Mode

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


64A010 airfoil 64A010 airfoil129x33 grid at 3 separate modes 193x49 grid at 3 separate modes

AIAA 2001-0120 24

Results - Pitching Airfoil - Viscous - Cm

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


64A010 airfoil CT6 airfoil257x65 grid at 3 separate modes All grid and mode permutations

AIAA 2001-0120 25

Results - Pitching Airfoil - Viscous - Convergence

• Convergence results based on 193x49 C-mesh with 1 mode.

−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125


Coe

ffici

ent o

f Lift


−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


AIAA 2001-0120 26

Results - Execution Time

• Timing results based on 129x33 Euler mesh and 193x49 viscous mesh.

• Results compiled on a 1.4Ghz AMD Athlon using 64 bit floating pointmath.

Mesh 1 Mode 2 Modes 3 Modes(secs/cycle) (secs/cycle) (secs/cycle)

129x33 0.59 1.0 1.5193x49 1.9 2.9 4.5

Table 4: Execution Time - Euler and Viscous Meshes

AIAA 2001-0120 27

Results - Cylinder

−2 −1 0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3

4

• The grid size is 256 by 128 cells. The mesh boundary is 200 chords fromthe center of the cylinder. An exponential stretching function is used inthe radial direction with the smallest grid spacing of 3.54e − 03 chordsoccurring at the wall. At the top of the cylinder roughly 15 grid pointscaptured the boundary layer.

AIAA 2001-0120 28

Results - Cylinder - Fixed Time Period

Experiment −Cpb Cd St

Williamson and Roshko (1990) 0.83Roshko (1954) 0.185Wieselsberger (1922) 1.3Henderson (1995) 0.83 1.34

Table 5: Time Averaged Experimental Data

Temporal Modes −Cpb Cd

1 0.832 1.2573 0.895 1.3065 0.903 1.3117 0.903 1.311

Table 6: Previously Reported Time Averaged Data versus Temporal Modes

AIAA 2001-0120 29

Results - Cylinder - Strouhal

60 70 80 90 100 110 120 130 140 1500.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

Reynolds Number

Str

ouha

l Num

ber

1 Harmonic3 Harmonic5 Harmonic7 HarmonicWilliamson 1988

Figure 1: Strouhal Numbers versus Reynolds Number For Laminar VortexShedding

AIAA 2001-0120 30

Results - Cylinder - Cpb

60 70 80 90 100 110 120 130 140 1500.5

0.6

0.7

0.8

0.9

Reynolds Number

Neg

ativ

e B

ase

Pre

ssur

e C

oeffi

cien

t

1 Harmonic3 Harmonic5 Harmonic7 HarmonicHenderson 1995Williamson 1990

Figure 2: Mean Base Pressure Coefficient versus Reynolds Number ForLaminar Vortex Shedding

AIAA 2001-0120 31

Results - Cylinder - Cd

60 70 80 90 100 110 120 130 140 1500.2

0.4

0.6

0.8

1

1.2

1.4

Reynolds Number

Mea

n C

oeffi

cien

t of D

rag

Viscous Component of Cd

Pressure Component of Cd

Total Cd

1 Harmonic3 Harmonic5 Harmonic7 HarmonicHenderson 1995

Figure 3: Mean Coefficient of Drag versus Reynolds Number For LaminarVortex Shedding

AIAA 2001-0120 32

Conclusions

• For the cylinder shedding test case, relatively accurate global coefficientswere obtained using three temporal modes.

• For the forced pitching airfoil case only one temporal mode was neededto accurately predict the coefficient of lift. Experimental data for thecoefficient of moment is poorly predicted.

• The dominant natural frequency can be predicted by the gradient basedvariable time period (GBVTP) method.

AIAA 2001-0120 33

AIAA 2001–0152 40th AIAA Aerospace Sciences Meeting & Exhibit

Acknowledgments

• Department of Energy - Accelerated Strategic Computing Initiative(ASCI)

• Kenneth C. Hall - Associate Professor in the Mechanical Engineering andMaterials Science Department, Duke University

• Sanford Davis - NASA AMES Research Center

• Charles Williamson - Professor in the Mechanical and AerospaceEngineering Department, Cornell University

34

AIAA 2001–0152 40th AIAA Aerospace Sciences Meeting & Exhibit

Future Work

• Continue verification process to turbomachinery cases where experimentaldata exists.

• Modify TFLO code to implement the nonlinear frequency domainmethod.

35

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