Adjoint-based Trailing-Edge Noise Minimization via Porous ... EuroAd Workshop - Beckett Zhou -...

Adjoint-based Trailing-Edge Noise Minimization viaPorous Material

Beckett Y. Zhou1,2, Nicolas R. Gauger1,

Seong R. Koh3, Matthias Meinke3, Wolfgang Schroder3

1Chair for Scientific Computing, TU Kaiserslautern, Germany2Aachen Institute for Advanced Study in Computational Engineering Science (AICES),

RWTH Aachen University, Germany3Institute of Aerodynamics (AIA), RWTH Aachen University, Germany

19th Euro AD Workshop, Kaiserslautern

April 7, 2016

Motivation

Why do we need to perform aeroacoustic optimization?

FAA forecast in 2012: Demand for air travel to DOUBLE over the next20 years

This situation is highly unsustainable as most major airports around theworld are already saturated

Long-term exposure to air traffic noise extremely hazardous to groundpopulation near airports

EU ‘FLIGHTPATH 2050’ goal: reduce perceived aircraft noise by 65%(from the 2000 level) by the year 2050

Aeroacoustic considerations must be included in the initial design phaseof an aircraft to meet noise-specific challenges

Beckett Y. Zhou et al. Adjoint-based Noise Minimization via Porous Material 2/ 24

Aircraft NoiseJet Noise

© Boeing

Airframe Noise

Flap

Slat

Landing Gear

Trailing-edge noise


Trailing-Edge Noise Generation and Propagation

Computational Challenges:

Fine grid & small time-steps required to resolve noise generating 3-Dturbulent structure

For low-speed flows, turbulence develops slowly and many thousands oftime steps required to simulate one cycle (of shedding)

Important ramifications on optimization

Run-timeMemory/storage overheadAccuracy and stability of adjoint solver


Trailing-Edge Noise ReductionThree Predominant Approaches

Shape Optimization

Adjoint-based [Rumpfkeil & Zingg, 2010; Economon & Alonso, 2012; Nilsen et al., 2015]

Active Flow Control

Injection of gas mixtures [Koh, Schroder & Meinke, 2011]

Trailing-edge blowing for fan tonal noise [Enghardt et al., 2015]

Suction and blowing for blunt airfoil trailing-edge [Ramirez & Wolf, 2015]

Surface modification with porous material

Experimental work: Geyer et al. 2010; Herr et al. 2014

Airfoil with porous trailing-edge [Fassmann et al., 2015]

Optimization of porous trailing-edge on a flat plate [Schulze & Sesterhenn, 2013]

Challenges: i) No clear design guidelines exist for ideal placement of porous media –

an uninformed choice may amplify noise

ii) Noise reduction accompanied by a marked loss of lift (Herr et al.)

Research Goal: Systematically search for optimal distribution of porousmedia via discrete adjoint optimization


Flow through Porous Media

Navier-Stokes equations for flow through porous media:[Liu & Vasilyev, 2007]

∂ρ

∂t= −

∂ρuj

∂xj−(

1

ε− 1)χ

∂ρuj

∂xj

∂ρui

∂t= −

∂

∂xj(ρuiuj ) −

∂p

∂xi+

1

Rea

∂τij

∂xj−χ

Kv(ui − Uoi )

∂e

∂t= −

∂[(e + p)uj ]

∂xj+

1

Rea

∂uiτij

∂xj+

1

ReaPr(γ − 1)

∂

∂xj(µ∂T

∂xj)−

χ

Kt(T − To)

Allows to model flow inside permeable media and external flowmonolithically (χ = 1: porous; χ = 0: external)

Porosity ε = (fluid volume)/(total material volume) (ε→ 0: solid)

Viscous Kv and thermal Kt permeabilities: flow conductance at a givenporosity (Kv ,Kt → 0: impermeable solid)

Parameters can be transformed to pore sizes


Optimization Framework

Objective Function: J =√

1NtNp

∑Nt

j

∑Np

i J 2(tj ,~xi ),

J : instantaneous measure of turbulence intensity/noiseNp: user-defined observation points.Nt : number of time steps.

Design Variables: spatially varying ε, Kv and Kt

Optimization Algorithm: Gradient-based Quasi-Newton BFGS algorithm withline search and box constraints for design variablesto prevent singularity

Design Gradient: Computed using algorithmic differentiation (AD)


AD-based Discrete Adjoint Framework

Algorithmic Differentiation (AD)

View a complex solver as sequence of elementary operations→ Successive application of chain-rule for derivatives

Forward (tangent) mode: exact but one evaluation per gradientcomponent

Reverse (adjoint) mode: exact AND entire gradient vector in onestroke. High memory requirement tackled by checkpointing

Techniques: Operator overloadingSource-code transformation

Advantages of AD-based Adjoint

Algorithmically differentiates through entire solver (LES filtering,turbulence models, flux limiter, dynamic grid movement, etc.)

Adjoint solver inherits convergence properties of the primal solver

Flexibility in defining objective function w/o special interface treatment


Minimization of Trailing Edge Pressure Fluctuation

Configuration:

c

h

d

z

x

y

s

εi

, Ki

v, K

i

t

Flat plate with a finite thickness(c = 33h)

Noise characterized by a strongtonal component due to periodicvortex shedding (δ ≈ 2.5h)

Spanwise porous strips in last12% chord (d = 0.12c)

M∞ = 0.20, Rec = 135000

1.5 million elements in x-y plane,solved with LES with RK-5time-marching scheme(1)

(1). Implementation details on the LES solver can be found in: S. Koh, M. Meinke and W. Schroder,

Impact of Wall Permeability on Trailing-Edge Noise at High Reynolds Number , AIAA Paper 2015-2368



x/c

y/c

0.8 0.85 0.9 0.95 1 1.050.05

0

0.05

0.1

0.15

5h

h

0.06c 0.06c

Observer Location 1 Observer Location 2 Observer Location 3

εi, K

i

v, K

i

t

d = 0.12c

Reduce pressure fluctuation (p′) nearthe T.E. directly: J = f (p′)

p′ measured at 3 observer locationssituated 5h above porous T.E. alongplate centerline

Design variables: ε, Kv & Kt in eachsegment ⇒ 30 DV’s

Constraints: ε ∈ [0.3, 0.5]Kt ∈ [0.0005, 0.1]Kv ∈ [0.005, 0.1]

Baseline: εi = 0.40, K iv = 0.001,

K it = 0.05 for all segments

(‘hardplate’)



Pressure History at Observer Location 2

0 10 20 30 40 50 600.675

0.68

0.685

0.69

0.695

tUo/h

p/ρoa2 o

Optimization Window

Minimize root-mean-squareof the pressure fluctuation atall 3 observer locations

J =√

1NtNp

∑Nt

j

∑Np

i [(p′)ji ]2,

where (p′)ji = pji − pi

Optimized over 3 periods ofshedding

Computation re-started fromfreestream at each designstage until optimizationwindow → J and Gre-evaluated

5 design updates performed


Optimal Porosity & Permeability Distributions

1 2 3 4 5 6 7 8 9 100.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

Streamwise Strip Number

Porosity (ε)

Baseline

OptimizedConstraint

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12


Viscous Permeability (Kv)

Baseline

OptimizedConstraint

1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06


Thermal Permeability (Kt)

Baseline

OptimizedConstraint

1 2 3 4 5 6 7 8 9 1010

−7

10−6

10−5

10−4

10−3

10−2

10−1


Magnitude of Gradient Component

εKv

Kt


Pressure Fluctuation

30 40 50 60 70tUo/h

p′/ρoa2 o

Pressure Fluctuation at 3 Observer Locations

BaselineOptimized

2 × 10−3

Optimization Window

Obs. Loc. #1

Obs. Loc. #2

Obs. Loc. #3

x/c

y/c

0.8 0.85 0.9 0.95 1 1.050.05

0

0.05

0.1

0.15

5h

h

0.06c 0.06c

1 2 3

d = 0.12c

Observer NoiseLocation Reduction (dB)

1 17.12 15.83 12.4

RMS of p′ reducedby 82% over 3observer locations


Vorticity Magnitude

Baseline Optimized


Pressure Fluctuation Field

Baseline Optimized



Overall Sound Pressure Level (OASPL)

θ

OASPL (dB)

0

30

60

90

120

150

180

210

240

270

300

330

Baseline

Optimized

5dB

12dB reduction in normal direction(where the observer locations aresituated)

Noise reduction achieved in alldirections. Up to 18dB in upstreamdirection



Overall Sound Pressure Level (OASPL)

θ

OASPL (dB)

0

30

60

90

120

150

180

210

240

270

300

330

Baseline

Optimized

High Kv & K

t

5dB

12dB reduction in normal direction(where the observer locations aresituated)

Noise reduction achieved in alldirections. Up to 18dB in upstreamdirection

Uniformly high permeability doesnot lead to further noise reduction

Advantage of adjoint-basedoptimization: achieves significantnoise reduction without resorting tounnecessarily porous/permeablesurfaces.

Critical when considering practicalaerodynamic constraints such as lift


Transitioning to a Hybrid Noise Prediction Strategy

Observer

x

r

ΓpΩ

1 (CFD)

Ω2 (CAA)

To obtain the same accuracy of acoustic signals ata given offbody position, the grid resolution for adirect DNS/LES noise prediction is M−3

∞ timeshigher than that is necessary to resolve theacoustic field (λ

lt∝ M−1

∞ ).

A hybrid two-step approach more favourable

Allows for more efficient noise predictions due tothe separation of hydrodynamic and acousticcomputations

Many hybrid CFD-CAA frameworks exist(URANS-FWH, LES-FWH, URANS-Kirchoff...)

Near-body turbulent flow field resolved using LES

Noise source propagated to near-far field usingAcoustic Perturbation Equations (APE)(Ewert & Schroder, 2003)


Acoustic Perturbation Equations (APE-4)

Re-arranging Navier-Stokes equations, dropping viscous and nonlinearfluctuating terms:

∂p′

∂t+ c2

o∇ ·(ρo~u

′ + ~uop′

c2o

)=

c2oρocp

Ds ′

Dt

∂~u ′

∂t+∇

(~uo · ~u ′)+∇

(p′

ρo

)= −(~ω × ~u)′ + T ′∇s − s ′∇T

An ‘acoustic analogy’ in which sound is generated by vorticity andentropy inhomogeneities

Dominant vortex sound source (fluctuating Lamb vector): ~L′

= (~ω × ~u)′

Spatial discretization: 6-th order FD scheme using summation-by-partsoperator

Temporal discretization: alternating 5-6 stage low-dispersion andlow-dissipation Runge-Kutta scheme

Mean flow/source quantities ( · )o based on an offline sampling stage

0

0 0


Coupled CFD-CAA Framework

~x =

εKv

Kt

LES APE

Mean Flow Field~Qo = [ρo , ~uo , po ]T

Mean Source Term~Lo = (~ω × ~u)o

J =√

(p′)2ρ, ~u, p ρ′, ~u ′, p′

6

Full chain algorithmically differentiated for dJd~x

Mean source term (~Lo) and flow field ( ~Qo) recomputed from free-streamafter each design update


Acoustic Pressure in Near-Far Field

35 40 45 50 55 60tUo/h

p′/ρoa2 o

Acoustic Pressure at 3 Observer Locations

BaselineOptimized

2 × 10−3

Optimization Window

Obs. Loc. #1

Obs. Loc. #2

Obs. Loc. #3

x/c

y/c

0.8 0.85 0.9 0.95 1 1.050.05

0

0.05

0.1

0.15

c=33h

h

0.06c 0.06c

1 2 3

d = 0.12c

Observer NoiseLocation Reduction (dB)

1 15.52 17.53 14.2

RMS of p′ reducedby 87% over 3observer locations


Conclusion and Future Work

ConclusionFirst work in noise-minimization to couple discrete adjoint with ahigh-fidelity LES and LES-APE solver

Significant noise reduction achieved by minimizing the pressurefluctuation at off-body observation points – 12dB in the normaldirection and up to 18dB in the upstream directions.

Preliminary result on coupled LES-APE solver also shows effectivesuppression of acoustic pressure fluctuation in the near far-field observerpoints via optimal distribution of porous material in the trailing edge

Take-away MessagesAdjoint-based method allows for exploration of large design spacesNon-intuitive designs possible without unnecessarily penalizing otherperformance metricsAlgorithmic differentiation leads to accurate & stable adjoint informationover long integration times – particularly well-suited for design problemsin aeroacoustics


Conclusion and Future Work

Future WorkPerform optimizations on a full 3D turbulent case at increased spanwiseresolution

Allow variations of porosity/permeability parameters in the spanwisedirection(Stay tuned and join us at the 22nd AIAA Aeroacoustics Conference in Lyon,

France: Session AA-14, May 30)

Apply methodology to aerodynamic shapes – airfoil or wing with poroustrailing edge

Incorporate practical design constraints – lift-constrained noiseminimization


Acknowledgements

Financial support from German Research Foundation (DFG) andCanadian Postgraduate Scholarship (NSERC-PGS-D)

Computing resources provided by the “Alliance of High PerformanceComputing Rheinland-Pfalz” (AHRP), via the “Elwetritsch” Cluster atthe TU Kaiserslautern

Thank you for your attention


Optimal Porosity & Permeability Distributions (LES)

1 2 3 4 5 6 7 8 9 100.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44


Porosity (ε)

Baseline

OptimizedConstraint

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12



Baseline

OptimizedConstraint

1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06



Baseline

OptimizedConstraint


Optimal Porosity & Permeability Distributions (LES-APE)

1 2 3 4 5 6 7 8 9 100.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54


Porosity (ε)

Baseline

OptimizedConstraint

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12



Baseline

OptimizedConstraint

1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06



Baseline

OptimizedConstraint


Validation of AD Gradients (LES Case)

i Finite Difference Forward AD Reverse AD

1 1.167141014946083 1.167141550726590 1.1671415507266312 0.979045005067292 0.979047994868520 0.9790479948685393 -0.037741010316950 -0.037741196023650 -0.0377411960236574 -0.172801009057366 -0.172800017556677 -0.1728000175566765 2.172569004699199 2.172574472509500 2.172574472509508

Table: Comparison between the gradients computed using 2nd order finitedifference (δ = 10−6), forward-mode and reverse-mode of AD, over 100 time steps

*generated using AD tool TAPENADE


Validation of AD Gradients (LES-APE Case)

i Finite Difference Forward AD Reverse AD

1 1.169212892736943E-6 1.169212543283378E-6 1.169212543283365E-62 -2.727110638006680E-6 -2.727110359586795E-6 -2.727110359586793E-63 -2.627478041996451E-6 -2.627478879823261E-6 -2.627478879823263E-64 -4.622594407949210E-6 -4.622594173893573E-6 -4.622594173893574E-65 -9.390040828227697E-7 -9.390036763006530E-7 -9.390036763006709E-7


Pressure Fluctuation Field

Baseline Optimized


AD-based Unsteady Discrete Adjoint Framework

Consider a system of semi-discretized PDEs as follows:

dU

dt+ R(U) = 0

U: spatially discretized state vectorR(U): is the discrete spatial residual vector.

Second-order backward difference is used for time discretization:

R∗(Un) =3

2∆tUn + R(Un)− 2

∆tUn−1 +

1

2∆tUn−2 = 0, n = 1, . . . ,N

Dual-time stepping method converges R∗(Un) to a steady state solution at eachtime level n through a pseudo time τ :

dUn

dτ+ R∗(Un) = 0

Implicit Euler method is used to time march the above equation to steady state:

Unp+1 − Un

p + ∆τR∗(Unp+1) = 0, p = 1, . . . ,M



The resultant nonlinear system can be linearized around Unp to solve for the state

Unp+1:

Unp+1 − Un

p + ∆τ

[R∗(Un

p ) +∂R∗

∂U

∣∣∣∣np

(Unp+1 − Un

p )

]= 0, p = 1, . . . ,M

This can be written in the form of a fixed-point iteration:

Unp+1 = G n(Un

p ,Un−1,Un−2), p = 1, . . . ,M, n = 1, . . . ,N

G n: an iteration of the pseudo time steppingUn−1: converged state vector at time level n − 1Un−2: converged state vectors at time level n − 2

The fixed point iteration converges to the numerical solution Un:

Un = G n(Un,Un−1,Un−2), n = 1, . . . ,N



The discretized unsteady optimization problem over N time levels:

minβ

J =1

N

N∑n=1

J(Un, β)

subject to Un = G n(Un,Un−1,Un−2, β), n = 1, . . . ,N

β: vector of design variables. One can express the Lagrangian associated with theabove constrained optimization problem as follows:

L =1

N

N∑n=1

J(Un, β)−N∑

n=1

[(Un)T (Un − G n(Un,Un−1,Un−2, β)

)]Un: adjoint state vector at time level n.

∂L

∂Un= 0, n = 1, . . . ,N (State equations)

KKT :∂L

∂Un= 0, n = 1, . . . ,N (Adjoint equations)

∂L

∂β= 0, (Control equation)



The unsteady discrete adjoint equations can be derived in the fixed point form as:

Uni+1 =

(∂G n

∂Un

)T

Uni +

(∂G n+1

∂Un

)T

Un+1 +

(∂G n+2

∂Un

)T

Un+2 +1

N

(∂Jn

∂Un

)T

, n = N, . . . , 1

Un+1: converged adjoint state vector at time level n + 1Un+2: converged adjoint state vector at time level n + 2

The unsteady adjoint equations above are solved backward in time.The sensitivity gradient can be computed from the adjoint solutions:

dL

dβ=

N∑n=1

(1

N

∂Jn

∂β+ (Un)T

∂G n

∂β

)High-lighted terms computed using AD in reverse mode

Reverse accumulation used at each time level to ‘tape’ the computationalgraph for AD

Adjoint iterator inherits the same convergence properties as primal iterator

G includes: turbulence model, grid movement, limiters, etc


Run-time and Memory Usage

3.5 million cellsPrimal solution: 10GB, constant at all time stepsReverse-mode AD: 42GB per time step, scales with number of time stepsSlow-down factor: ∼15 (primal vs. black-box reverse mode AD)


Minimization of Trailing Edge Turbulence Intensity

Strip Viscous Permeabilities

1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Strip Number

Vis

co

us P

erm

eab

ilit

y (

Kv)

baseline

optimized

Optimizer makes the laststrip almost an impermeablesolid (constraint active)

Unclear a priori why thispermeability distribution isoptimal

Highlights the power ofcombining high-fidelitysimulation with numericaloptimization – opportunity toexplore non-intuitive andunconventional designs


Adjoint-based Trailing-Edge Noise Minimization via Porous ... EuroAd Workshop - Beckett Zhou -...

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Transcript of Adjoint-based Trailing-Edge Noise Minimization via Porous ... EuroAd Workshop - Beckett Zhou -...