1

High-Fidelity Multidisciplinary DesignUsing an Integrated Design Environment

Antony Jamesonand

Juan J. AlonsoDepartment of Aeronautics and Astronautics

Stanford University, Stanford CA

AFOSR Joint Contractors Meeting for Applied Analysis andComputational Mathematics Programs

14-15 June 2004

2

Outline

• Aero-Structural Wing Shape and PlanformOptimization

• Fast Time Integration Methods for Unsteady Problems :application to shape optimization of a pitching airfoil

• Filtering the Navier Stokes Equations with an InvertibleFilter: implication for LES subgrid modeling

3

Aero-Structural Wing Shape andPlanform Optimization

Boeing 747 -Planform Optimization

Baseline profile (red)Redesigned profile (blue)

Baseline planform (green)Redesigned planform (blue)

WeakenedShock

Shape Optimization

Unstructured Mesh Optimization

4

Background: Levels of CFD

Flow Prediction

Automatic Design

Interactive Calculation

• Integrate the predictive capability into an automatic design method that incorporates computer optimization.

• Attainable when flow calculation can be performed fast enough• But does NOT provide any guidance on how to change the shapeif performance is unsatisfactory.

• Predict the flow past an airplane or its important components indifferent flight regimes such as take-off or cruise and off-designconditions such as flutter.• Substantial progress has been made during the last decade.

HIGHEST

LOWEST

5

Optimization and Design using SensitivitiesCalculated by the Finite Difference Method

†

The simplest approach is to define the geometry as

f (x) = a ibi(x)

where a i = weight, bi(x) = set of shape functions Then using the finite difference method, a cost function I = I(w,a) (such as CD at constant CL )

has sensitivities ∂I∂a i

ªI(a i + da i) - I(a i)

da i

If the shape changes is a n +1 = a n - l∂I

∂a i

(with small positive l)

The resulting improvements is I + dI = I -∂IT

∂ada = I - l

∂IT

∂a∂I∂a

< I

More sophisticated search may be used, such as quasi - Newton.

f(x)

6

Disadvantage of the Finite Difference Method

The need for a number of flow calculations proportionalto the number of design variables

Using 4224 mesh points on the wing as design variables

Boeing 747

4225 flow calculations~ 30 minutes each (RANS)

Too Expensive

7

Application of Control Theory

Drag Minimization Optimal Control of Flow Equationssubject to Shape(wing) Variations

†

≡

†

Define the cost function I = I(w,F)

and a change in F results in a change

dI =∂I∂w

È

Î Í ˘

˚ ˙

T

dw +∂I∂F

È

Î Í ˘

˚ ˙

T

dF

Suppose that the governing equation R which expressesthe dependencd of w and F as

R(w,F) = 0and

dR =∂R∂w

È

Î Í ˘

˚ ˙ dw +∂R∂F

È

Î Í ˘

˚ ˙ dF = 0

GOAL : Drastic Reduction of the Computational Costs

(for example CD at fixed CL)

8

Application of Control Theory

†

Since the variation dR is zero, it can be multiplied by a Lagrange Multiplier yand subtracted from the variation dI without changing the result.

dI =∂IT

∂wdw +

∂IT

∂FdF -yT ∂R

∂wÈ

Î Í ˘

˚ ˙ dw +∂R∂F

È

Î Í ˘

˚ ˙ dFÊ

Ë Á

ˆ

¯ ˜

=∂IT

∂w-yT ∂R

∂wÈ

Î Í ˘

˚ ˙ Ï Ì Ó

¸ ˝ ˛ dw +

∂IT

∂F-yT ∂R

∂FÈ

Î Í ˘

˚ ˙ Ï Ì Ó

¸ ˝ ˛ dF

Choosing y to satisfy the adjoint equation

∂R∂w

È

Î Í ˘

˚ ˙

T

y =∂I∂w

the first term is eliminated, and we find that dI = GTdF

where

GT =∂I∂F

T

-yT ∂R∂F

È

Î Í ˘

˚ ˙ Ï Ì Ó

¸ ˝ ˛

One Flow Solution + One Adjoint Solution

(Adjoint Equation)

(Gradient)

4224 design variables

9

Advantages of the Adjoint Method:• Gradient for N design variables with cost equivalent to two flow solutions

• Minimal memory requirement in comparison with automatic differentiation

• Enables shapes to be designed as free surface• No need for user defined shape function• No restriction on the design space

4224 design variables

10

Outline of the Design Process

Flow solution

Adjoint solution

Gradient calculation

Sobolev gradient

Shape & Grid Modification

Rep

eate

d un

til C

onve

rgen

ce

to O

ptim

um S

hape

11

Discrete versus ContinuousAdjoint Methods

• The discrete adjoint method evaluates the adjoint and gradient equations algebraically from the discretized flow equations.

• The continuous adjoint method evaluates the costate solution from the partial differential adjoint equation.

• The continuous adjoint method leads to no inconsistency as long as it is combined with a compatible search method

• In the limit of grid convergence the two approaches yield identical gradients.

• Numerical tests of a model problem verify slightly superior accuracy with the continuous formulation ( Jameson and Vassberg 2000)

12

Summary of the ContinuousFlow and Adjoint Equations

†

With computational coordinates x i Euler equations for the flow :

(1) ∂ ∂x i

Sij f j (w) = 0

where Sij are metrices, f j (w) the fluxes.Adjoint equation

(2) Ci∂y ∂x i

= 0, Ci = Sij∂f j ∂w

Boundary condition for the Inverse problem

(3) I =12

(p - pt )2 dsÚ y2nx +y3ny +y3nz = p - pt

Gradient

(4) dI = -∂yT

∂x i

dSij f jdDÚ D - dS21y2 + dS22y3 + dS23y4( )pdx1b wÚ dx3Ú

13

Point-wise Gradient and ShapeParameterizations

†

If the shape is parameterized as

f (x) = akbk (x)k

Â

then dI = g(x)df (x)dxÚ

= da kk

 g(x)bk(x)dxÚ

= Gkda kk

Â

where Gk = g(x)bk(x)dxÚ

14

Sobolev Gradient

Continuous descent path†

Define the gradient with respect to the Sobolev inner product

dI = < g,df > = gdf + eg'df '( )dxÚSet

df = - lg, dI = - l < g,g >

This approximates a continuous descent process dfdt

= -g

The Sobolev gradient g is obtained from the simple gradient g by the smoothing equation

g -∂∂x

e∂g∂x

= g.

Key issue for successful implementation of the Continuous adjoint method.

15

Computational Costs with N Design Variables

(Note: K is independent of N)O(K )Sobolev GradientO(N )Quasi-NewtonO(N2)Steepest Descent

Cost of Search Algorithm

O(N )Adjoint Gradient+ Quasi-Newton Search

(Note: K is independent of N)

O(K )Adjoint Gradient+ Sobolev Gradient

O(N2)Finite Difference Gradients+ Quasi-Newton Search or Response surface

O(N3)Finite Difference Gradients+ Steepest Descent

Total Computational Cost of Design

- N~2000- Big Savings- Enables Calculations on a Laptop

16

Trailing Edge CrossoverLeading term in gradient for drag reduction may lead totrailing edge crossover

This corresponds to a sink in a free stream and hencenegative drag.

This is prevented by Sobolev gradient or by shapeparameterizations which don’t allow crossover.

crossover

17

Example: Viscous RAE Drag Minimization

Initial Shape andNon-Smoothed Gradient

Initial Shape andSmoothed Gradient

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Example: Viscous RAE Drag Minimization

Final Shape andNon-Smoothed Gradient

Final Shape andSmoothed Gradient

19

Good Shape Parameterization of an Airfoilvia Conformal Mapping

†

Define the mapping by

log dzds

=Ck

s kÂOn C

log dsdq

+ i(a -q -p2

) = Ckk= 0

n

 e- ikq

= ak cos(kq) + bk sin(kq)( )k= 0

n

 + i bk cos(kq) + ak sin(kq)( )k= 0

n

Â

Fourier coefficients define the mapping Spectral convergence with n Trialing edge gap is 2piC1, so prevent crossover by setting C1 = 0.

a(s)z

C

s

20

Sobolev Gradient for Partial RedesignOne may wish to freeze some fo the profile, e.g. thestructure box

In this case shape changes needed to be smoothly blendedinto the frozen geometry. This is accomplished by addingmore derivatives to the Sobolev gradient

Then the corresponding smoothed gradient satisfies

Hence a patch can be smoothly blended.

Fixed (Structure box)

†

< u,v >= uv + e2 ¢ u ¢ v + e4 ¢ ¢ u ¢ ¢ v ( )Ú dx

†

g - ∂∂x

e2∂g ∂x

Ê

Ë Á

ˆ

¯ ˜ +

∂ 2

∂x 2 e4∂ 2g ∂x 2

Ê

Ë Á

ˆ

¯ ˜ = g,

g = g x = 0 at both end points.

21

Potential Modification of F8 BearcatRacer to win the RENO Air Racer

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Bearcat “Shock Free” ResultInitial Cp

Final Cp

23

Redesign of the Boeing 747 Wingat its Cruise Mach Number

Constraints : Fixed CL = 0.42: Fixed span-load distribution: Fixed thickness 10% wing drag saving

(3 hrs cpu time - 16proc.)

~5% aircraft drag saving

baseline

redesign

RANS Calculations

24

Redesign of the Boeing 747 Wing at Mach 0.9“Sonic Cruiser”

Constraints : Fixed CL = 0.42: Fixed span-load distribution: Fixed thickness Same CD @Cruise

We can fly faster at the same drag.

RANS Calculations

25

Redesign of the Boeing 747: Drag Rise( Three-Point Design )

Improved Wing L/D

Improved MDD

Lower drag at the same MachNumber

Fly faster with the same drag

benefit

benefit

Constraints : Fixed CL = 0.42: Fixed span-load distribution: Fixed thickness

RANS Calculations

26

Planform and Aero-StructuralOptimization

• Design tradeoffs suggest an multi-disciplinary design and optimization

†

Range =VLD

1sfc

logWo + W f

Wo

Maximize Minimize

Planform variations can further maximize VL/D but affects WO

27

Aerodynamic Design Tradeoffs

†

The drag coefficient can be split into

CD = CDO +CL

2

peAR

†

LD

is maximized if the two terms are equal.

Induced drag is half of the total drag.

If we want to have large drag reduction, we shouldtarget the induced drag.

†

Di =2L2

perV 2b2

Design dilemmaIncrease b

Di decreases

WO increases

Change span by changing planform

28

Break Down of Drag

270Total___

27015Other25520Nacelles23520Tail21550Fuselage16545Wing friction

(15 shock, 105 induced)120 counts120 countsWing Pressure

CumulativeCD

CDItem

Boeing 747 at CL ~ .47 (including fuselage lift ~ 15%)

Induced Drag is the largest component

29

Wing Planform Optimization

†

I = a1CD + a212

(p - pd )2 dSÚ + a3CW

where

CW =Structural Weight

q•Sref

Simplified Planform Model

Wing planform modification can yield largerimprovements BUT affects structural weight.

Can be thoughtof as constraints

30

Additional Features Needed• Structural Weight Estimation• Large scale gradient : span, sweep, etc…• Adjoint gradient formulation for dCw/dx• Choice of a1, a2, and a3

Use fully-stressed wing box to estimate the structural weight.

Large scale gradient

• Use summation of mappedgradients to be large scalegradient

31

Choice of Weighting Constants

†

Breguet range equation

R =VLD

1sfc

logWO + W f

WO

With fixed V, L, sfc, and (WO + W f ≡ WTO ), the variation of R can be stated as

dRR

= -dCD

CD

+1

logWTO

WO

dWO

WO

Ê

Ë

Á Á Á Á

ˆ

¯

˜ ˜ ˜ ˜

= -dCD

CD

+1

logCWTO

CWO

dCWO

CWO

Ê

Ë

Á Á Á Á

ˆ

¯

˜ ˜ ˜ ˜

Minimizing

†

I = CD +a3

a1

CW

†

≡ using

†

a3

a1

=CD

CWOlog

CWTO

CW0

MaximizingRange

32

Planform Optimization of Boeing 747

Baseline

Redesign

Constraints : Fixed CL=0.42

45087Optimize both section and planform45594Optimize Section at Fixed planform455108BaselineCWCD

1) Longer span reduces the induced drag2) Less sweep and thicker wing sectionsreduces structure weight3) Section modification keepsshock drag minimum

Overall: Drag and Weight Savings

33

Planform Optimization of MD11

Baseline

Redesign

Constraints : Fixed CL=0.45

344 138Optimize both section and planform346145Optimize Section at Fixed planform345159BaselineCWCD

34

Pareto Front: “Expanding the Range of Designs”Use multiple a3/a1 ==> Multiple Optimal Shapes

Boundary of realizable designs

Pareto front of Boeing 747

35

Super 747

• Design a new wing for the Boeing 747• Strategy

– Use the same fuselage of Boeing 747– Use a new planform (from the Planform Optimization

result)– Use new airfoil sections (AJ airfoils)– Optimized for fixed lift coefficient

36

Baseline Boeing 747

37

Super 747

38

Comparison

427(70,620 lbs)

121.0(83.8 pressure, 37.2 viscous)

.451Super 747

499(82,550 lbs)

146.2(112 pressure, 34.3 viscous)

.452Boeing 747

CW

countsCD

countsCL

• Save ~10% of airplane drag

• Save ~ 7% of airplane structural weight

39

Automatic design for the Complete AircraftGeometry on an Unstructured Mesh

(SYNPLANE)• Key step: reduce the gradient to a surface integral

independent of the mesh perturbation(Jameson, A., and Kim, S., "Reduction of the Adjoint Gradient Formula inthe Continuous Limit", 41 st AIAA Aerospace Sciences Meeting & Exhibit,AIAA Paper 2003-0040, Reno, NV, January 6-9, 2003. )

†

dI = yT dS2 j f j + C2dw*( )dx1b wÚ dx3Ú - dS21y2 + dS22y3 + dS23y4( )pdx1b w

Ú dx3Ú

†

dI = -∂yT

∂x i

dSij f jdDÚ D - dS21y2 + dS22y3 + dS23y4( )pdx1bwÚ dx3Ú

Compared to the previous formulation

This field integral is converted to boundary integral

40

Question :

If the entire profile or wing is translated as a rigid body,the flow is unchanged. Therefore if one calculates thepoint-wise gradients for movement in the xi direction,they should satisfy the relation

†

gxi

†

gxiÚ ds = 0

Is this true for the computed gradients?

41

Actually it can be verified for some implementations of thefield integral formula for the gradient.

But the boundary integral formula presents a difficulty in theneighborhood of the front and rear stagnation points where

is not well defined.

†

—w

42

• RAE 2822 airfoil,Grid 128x32, Euler calculation• Translate the mesh as a rigid body (Far-field boundary is not changed.)• Expect summed gradient around theairfoil to be zero.

Validation of the Reduced Adjoint

Initial Grid

Translated Grid(1 chord length in

x direction)

x

43

Validation of the Reduced Adjoint Gradient

Comparison of Original Adjoint,Reduced Adjoint, and Complex-Stepmethods

Comparison of Original Adjointand Reduced Adjoint methods

44

Gradient ComparisonPoint-wise gradient Integral of the point-wise

gradient over the airfoil surface

Theoretically there is no changein the flow, the integral of thegradient should become zero.

FD 1E-6

Grad 1E-7

45

Deformation of Unstructured Meshes

Method 1) Spring method: treat edges as springs.

Computationally inexpensive but doesn’t guarantee to prevent crossovers.

2) Treat the mesh as a pseudo elastic body.

3) Traction method : treat the mesh as an elastic body, but subject to force inputs proportional to the gradient, instead of displacements - alternative to the Sobolev-gradient.

Movement of anindividual surface pointinfluences entire mesh

†

Solve∂

∂x j

s ij = 0, s ij =∂ui

∂x j

+∂u j

∂xi

s = stress, u = displacementfor a given boundary displacement.

46

Redesign of FalconComplete aircraft calculation on Unstructured Mesh

Shock

CD = 234 counts

47

Redesign of FalconUsing SYNPLANE

Drag reduction 18 counts at fixed CL = 0.4

Weakened Shock

CD = 216 counts

48

Fast Time Integration Methodsfor

Unsteady Problems

49

Potential Applications

• Flutter Analysis,

• Flow past Helicopter blades,

• Rotor-Stator Combinations in Turbomachinery,

• Zero-Mass Synthetic Jets for Flow Control

50

Dual Time Stepping BDF

The non-linear BDF is solved by inner iterations which advance in pseudo-time t*The second-order BDF solves

†

dwdt* +

3w - 4wn + wn-1

2Dt+ R(w)

È

Î Í

˘

˚ ˙ = 0

Implementation via Dt*

†

Dt =1Dt

1qq=1

k

 (D-)q where D-wn +1 = wn +1 - wn

The kth-order accurate backward difference formula (BDF) is of the form

• RK “dual time stepping” scheme with variable local (RK-BDF)• Nonlinear SGS “dual time stepping” scheme (SGS-BDF) with Multigrid

51

Pressure Contours at Various Time Instances (AGARD 702) Results of SGS-BDF Scheme(36 time steps per pitching cycle,3 iterations per time step )

12.36million

ReynoldsNumber

0.202Reduced Freq.

+/- 1.01deg.Pitchingamplitude

0.796Mach NumberTest Case: NACA64A010pitching airfoil (CT6 Case)

Cycling to limit cycle

52

Payoff of Dual-time Stepping BDF Schemes

• Accurate simulations with an order ofmagnitude reduction in time steps. order

• For the pitching airfoil:from ~ 1000 to 36 time steps per pitchingcycle with three sub-iterations in each step.

53

Frequency Domain and Global Space-TimeMultigrid Spectral Methods

Application : Time-periodic flows

Using a Fourier representation in time, the time period T is divided into N steps.

Then, the discretization operator is given by

†

ˆ w k =1N

wn

n= 0

N-1

 e-iknDt

†

Dtwn = ik ˆ w ke

iknDt

k=-N2

N2

-1

Â

54

Method 1 (McMullen et.al.) : Transform the equations intofrequency domain and solve them in pseudo-time t*

Method 2 (Gopinath et.al.) : Solve the equations in the time-domain. The space-time spectral discretization operator is

This is a central difference operator connecting all time levels,yielding an integrated space-time formulation which requiressimultaneous solution of the equations at all time levels.

†

d ˆ w kdt* + ik ˆ w k + ˆ R k = 0

†

Dtwn = dmwn +m

m=-N2

+1

N2

-1

 , dm =12

(-1)m +1 cot(pmN

),m ≠ 0

55

Comparison with Experimental Data -CL vs. a

RANS Time-Spectral Solution with 4 and 8 intervals perpitching cycle

Computed Results

ExperimentalData

- Time spectral 4 time intervals- Time spectral 8 time intervals* AGARD-702:Davis

56

• Engineering accuracy with very small number of timeintervals and same rate of convergence as the BDF.

• Spectral accuracy for sufficiently smooth solutions.

• Periodic solutions directly without the need to evolvethrough 5-10 cycles, yielding an order of magnitudereduction in computing cost beyond the reduction alreadyachieved with the BDF, for a total of two orders ofmagnitude.

Payoff of Time Spectral Schemes

57

Example of Shape Optimization inPeriodic Unsteady Flow

(Nadarajah, McMullen, and Jameson “AIAA-2003-3875)

“Find the (fixed) shape which minimizesthe averaged drag coefficient of a pitchingairfoil with the constraint that the averagelift coefficient is maintained.”

58

Optimization Results

Comparison of the Final Airfoil Geometrybetween the Time Accurate and NLFDmethod with Lift Constraints.

Comparison of the Initial and Final AirfoilGeometry using the NLFD method withLift Constraints.

Initial Shape Final ShapeNon-Linear Frequency Domain resultMatched Time Accurate result

59

Pressure Distributions

Comparison of the Final PressureDistribution between the TimeAccurate and NLFD methodwith Lift Constraints.

Comparison of the Initial andFinal Pressure Distributionusing the NLFD method withLift Constraints.

Initial ShapeFinal Shape

60

Convergence HistoryTime Accurate Vs. Non-Linear Frequency Domain method

61

Flow Past Helicopter Blades

62

Challenges

• Blade runs into its own wake.• Resolve blade-vortex interaction• Simulate fully articulated hub that is capable of

lead-lag, flapping motions coupled withaeroelastic solver

• Reduce simulation time through convergenceacceleration techniques such as space-timemultigrid spectral methods

63

Test Case: Non-Lifting Hover

• 2-bladed NACA0012• Mesh size: 128¥48¥32• 0.52 Tip Mach number• 0º Collective pitch• Only one section of the

blade needs to becalculated because ofperiodicity

• Euler Calculation usingBDF scheme

64

Preliminary Results using theBackward Different Formula

• Plots of -Cp atdifferent span stations

• Results lookreasonable but needsrefinement

• Subtle modification tothe mesh should yieldbetter agreement withexperiments

65

Future Work

• Apply space-time multigrid spectral methods• This will enable solving Navier-Stokes calculations in

reasonable time• Couple the flow solution with an aeroelastic model• Simulation of forward flight that will allow blade flapping,

lagging and change of cyclic pitch• Eventually, introduce aero-structural shape optimization to

improve rotor performance

66

Filtering the Navier Stokes Equationswith

an Invertible Filter

67

Consider the incompressible Navier--Stokes equations

where

In large eddy simulation (LES) the solution is filtered to remove the small scales.Typically one sets

where the kernel G is concentrated in a band defined by the filter width. Then thefiltered equations contain the extra virtual stress

because the filtered value of a product is not equal to the product of the filteredvalues. This stress has to be modeled.

†

r∂ui

∂t+ ru j

∂ui

∂x j

+ r∂p∂xi

= m∂ 2ui

∂xi∂x j

(1)

†

∂ui

∂xi

= 0

†

u i (x) = G(x - ¢ x )u( ¢ x )dÚ ¢ x (2)

†

t ij = uiu j - ui u j (3)

68

A filter which completely cuts off the small scales or the high frequencycomponents is not invertible. The use, on the other hand, of an invertiblefilter would allow equation (1) to be directly expressed in terms of thefiltered quantities. Thus one can identify desirable properties of a filter as

1. Attenuation of small scales2. Commutativity with the differential operator3. Invertibility

†

Suppose the filter has the form ui = Pui (4)which can be inverted as Qui = ui (5)where Q = P-1. Moreover Q should be coercieve, so that Qu > c u (6)for some positive constant c.

69

†

Note that if Q commutes with ∂∂xi

then so does Q-1, since for any quantity f which is

sufficiently differentiable ∂∂xi

(Q-1 f ) = Q-1Q ∂∂xi

(Q-1 f )

= Q-1 ∂∂xi

(QQ-1 f )

= Q-1 ∂∂xi

( f )

Also since Q commutes with ∂∂xi

,

∂u∂xi

= 0 (7)

As an example P can be the inverse Helmholtz operator, so that one can write

Qui = 1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ ui = ui (8)

where a is a length scale proportional to the largest scales to be retained. One may also introduce a filtered pressure p, satisfying the equation

Qp = 1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ p = p (9)

70

†

Now one can substitute equation (8) and(9) for ui and p in equation (1) to get

r∂∂t

1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ ui + r 1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ u j

∂∂x j

1-a 2 ∂ 2

∂xl∂xl

Ê

Ë Á

ˆ

¯ ˜ ui +

∂∂xi

1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ p

= m∂ 2

∂x j∂x j

1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ ui

Because the order of the differentiations can be interchanged and the Helmholtz operator satisfies condition(6), it can be removed. The product term can be written as

r ∂∂x j

1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ ui 1-a 2 ∂ 2

∂xl∂xl

Ê

Ë Á

ˆ

¯ ˜ u j

Ï Ì Ó

¸ ˝ ˛

= r∂

∂x j

ui u j -a 2 ui∂ 2 u j

∂xk∂xk

-a 2 u j∂ 2 ui

∂xk∂xk

+ a 4 ∂ 2 ui

∂xk∂xk

∂ 2 u j

∂xl∂xl

Ï Ì Ó

¸ ˝ ˛

= r∂

∂x j

ui u j -a 2 ∂ 2

∂xk∂xk

ui u j( ) + 2a 2 ∂ui

∂xk

∂u j

∂xk

+ a 4 ∂ 2 ui

∂xk∂xk

∂ 2 u j

∂xl∂xl

Ï Ì Ó

¸ ˝ ˛

= rQ ∂∂x j

ui u j + a 2Q-1 2∂ui

∂xk

∂u j

∂xk

+ a 2 ∂ 2 ui

∂xk∂xk

∂ 2 u j

∂xl∂xl

Ê

Ë Á

ˆ

¯ ˜

Ï Ì Ó

¸ ˝ ˛

According to condition (6), if Qf = 0 for any sufficiently differentiable quantity f , then f = 0.

71†

Thus the filtered equation finally reduces to

r ∂ui

∂t+ r

∂∂x j

ui u j( ) +∂ p∂xi

= m∂ 2 ui

∂xk∂xk

- r∂

∂x j

t ij (10)

with the virtual stress

t ij = a 2Q-1 2∂ui

∂xk

∂u j

∂xk

+ a 2 ∂ 2 ui

∂xk∂xk

∂ 2 u j

∂xl∂xl

Ê

Ë Á

ˆ

¯ ˜ (11)

The virtual stress may be calculated by solving

1-a 2 ∂ 2

∂xk∂xk

Ê

Ë Á

ˆ

¯ ˜ t ij = a 2 2∂ui

∂xk

∂u j

∂xk

+ a 2 ∂ 2 ui

∂xk∂xk

∂ 2 u j

∂xl∂xl

Ê

Ë Á

ˆ

¯ ˜ (12)

Taking the divergence of equation (10), it also follows that p satisfies the Poisson equation

∂ 2 p∂xi∂xi

+ r∂

∂xi

∂∂x j

ui u j( ) + r∂ 2

∂xi∂x j

t ij = 0 (13)

72

†

In a discrete solution scales smaller than the mesh width would not be resolved, amountingto an implicit cut off. There is the possibility of introducing an explicit cut off off in t ij. Also one could use equation (8) to restore an estimate of the unfiltered velocity.

In order to avoid solving the Helmholtz equation (12), the inverse Helmholtz operator could be expanded formally as

1- a2D( )-1= 1+a2D +a4D2 + ...

where D denotes the Laplacian ∂ 2

∂xk∂xk

.Now retaining terms up to the fourth power of a,

the approximate virtual stress tensor assumes the form

t ij = 2a2 ∂ ui

∂xk

∂ uj

∂xk

+a4 2D∂ ui

∂xk

∂ uj

∂xk

Ê

Ë Á

ˆ

¯ ˜ + DuiDuj

È

Î Í Í

˘

˚ ˙ ˙ (14)

One ay regard the forms (11) or (14) as prototypes for subgrid scale (SGS) models.

73†

The inverse Helmholtz operator cuts off the smaller scales quite gradually. One could design filters with a sharper cut off by shaping their frequency response. Denote the Fourier transform of f as

ˆ f = Ffwhere (in one space dimension)

ˆ f (k) =12p

f (x)e- ikxdx-•

•

Ú

f (k) =12p

ˆ f (k)e- ikxdk-•

•

Ú

Then the general form of an invertible filter is

F PfŸ

= S(k) ˆ f (k)

F QfŸ

=1

S(k)ˆ f (k)

where S(k) should decrease rapidly beyond a cut off wave number inversely proportional toa length scale a .

74

†

In the case of a general filter with inverse Q, the virtual stress follows from the relation Quiu j = uiu j = Qu iQu j

Then t ij = uiu j - u iu j = Q-1(Qu iQu j - Q(u iu j ))

This formula provides the form for a family of subgrid-scale models.

75

Quotation From John Vassberg

“You are better off with a design method which optimizes 5000 design variables with 10 function evaluationsthan one

which requires 5000 function evaluationsto optimize 10 design variables.”