Optimal Disturbances in Compressible Boundary Layers Complete - - PowerPoint PPT Presentation

Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis

Simone Zuccher & Anatoli Tumin

University of Arizona, Tucson, AZ, 85721, USA

Eli Reshotko

Case Western Reserve University, Cleveland, OH, 44106, USA

4th AIAA Theoretical Fluid Mechanics Meeting, 6–9 June, 2005, Westin Harbour Castle, Toronto, Ontario, Canada.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 1

A classical transition mechanism

Tollmien-Schlicting (TS) waves first experimentally detected by Schubauer and Skramstad (1947), “Laminar boundary-layer oscillations and transition on a flat plate”, J. Res. Nat. Bur. Stand 38:251–92, originally issued as NACA-ACR, 1943.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 2

Are TS waves the only mechanism?

If the disturbances are not really infinitesimal (real world!)... ...streaks (instead of waves) can develop where the flow is stable according to the classical neutral stability curve. Alternative mechanism to TS waves: Transient growth.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 3

Modelling

• ✁

Acoustic wave Boundary layer profile Wall roughness Wall vibration Vorticity wave Output Input Box Black

=

A boundary layer, and its governing equations, can be thought in an input/output fashion.

• Inputs. Initial conditions and boundary conditions.
• Outputs. Flow field, which can be measured by a norm.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 4

Optimal perturbations

Question. What is the most disrupting, steady ini- tial condition, which maximizes the energy growth for a given initial energy

• f

the perturbation?

• ✁

y Boundary layer edge z Flat plate x Optimal perturbation

In this sense the perturbations are optimal.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 5

Goals/Tools

Goals Efficient and robust numerical determination of optimal perturbations in compressible flows. Formulation of the optimization problem in the discrete framework. Coupling conditions automatically recovered from the constrained optimization. Effect of energy norm choice at the outlet. Tools Lagrange Multipliers technique. Iterative algorithm for the determination of optimal initial condition.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 6

Problem formulation

• Geometry. Flat plate and sphere.
• Regimes. Compressible, sub/supersonic. Possibly

reducing to incompressible regime for M → 0.

• Equations. Linearized, steady Navier-Stokes equations.
• ✁

θ

Bow shock

φ z x y r

M >> 1

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 7

Scaling (1/2)

Lref is a typical scale of the geometry (L for flat plate, R

for sphere, etc.)

Href =

• νrefLref/Uref is a typical boundary-layer scale in

the wall-normal direction Flat plate. Href = l =

• ν∞L/U∞;

∞ = freestream.

• Sphere. Href =
• νrefR/Uref; ref = edge-conditions

at xref.

ǫ = Href/Lref is a small parameter.

Flat plate. ǫ = Re−1/2

L

, ReL = U∞L/ν∞.

• Sphere. ǫ = Re−1/2

ref

, Reref = UrefR/νref.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 8

Scaling (2/2)

From previous works, disturbance expected as streamwise

• vortices. The natural scaling is therefore

x normalized with Lref, y and z scaled with ǫLref. u is scaled with Uref, v and w with ǫUref. T with Tref and p with ǫ2ρrefU2

• ref. ρ eliminated through the

state equation. Due to the scaling, (·)xx << 1. The equations are parabolic! By assuming perturbations in the form q(x, y) exp(iβz) (flat plate – β spanwise wavenumber) and q(x, y) exp(imφ) (sphere – m azimuthal index)...

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 9

Governing equations

(Af)x = (Dfy)x + B0f + B1fy + B2fyy f = [u, v, w, T, p]T; A, B0, B1, B2, D 5 × 5 real matrices.

Boundary conditions

y = 0 : u = 0; v = 0; w = 0; T = 0 y → ∞ : u → 0; w → 0; p → 0; T → 0

More compactly

(H1f)x + H2f = 0

with H1 = A − D(·)y; H2 = −B0 − B1(·)y − B2(·)yy

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 10

Objective function (1/2)

Caveat! Results depend on the choice of the objective function. Physics dominated by streamwise vortices. Common choices of the energy norms.

• Inlet. vin = 0 and win = 0 (uin = Tin = 0).
• Outlet. vout = 0 and wout = 0 (uin = 0; Tin = 0).

Blunt body. Largest transient growth close to the stagnation point. Due to short x-interval, a streaks-dominated flow field might not be completely established. Contribution of vout and wout could be non negligible. Outlet norm. FEN vs. PEN

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 11

Objective function (2/2)

Mack’s energy norm (derived for flat plate and temporal problem), after scaling and using state equation,

Eout = ∞

• ρsout(u2
• ut + v2
• ut + w2
• ut) +

psoutT 2

• ut

(γ − 1)Ts2

• utM2
• dy
• r in matrix form as Eout =

∞

• fT
• ut

M outfout

• dy, with
• M out = diag
• ρsout, ρsout, ρsout,

psout (γ − 1)Ts2

• utM2, 0
• .

Initial energy of the perturbation

Ein = ∞

• ρsin(v2

in + w2 in)

• dy ⇒ Ein =

∞

• fT

in

M infin

• dy

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 12

Constrained optimization (1/3)

Our constraints are the governing equations, boundary conditions and the normalization condition Ein = E0. After discretization (M 0 ⇔

M in and M N ⇔ M out),

• bjective function J = fT

NM NfN

constraint Ein = E0 ⇒ fT

0 M 0f0 = E0

governing equations (BC included) Cn+1fn+1 = Bnfn The augmented functional L is

L(f0, . . . , fN) = f T

NMNfN + λ0[f T 0 M0f0 − E0] + N−1

• n=0
• pT

n (Cn+1fn+1 − Bnfn)

• with λ0 and (vector) pn Lagrangian multipliers.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 13

Constrained optimization (2/3)

By adding and subtracting pT

n+1Bn+1fn+1 in the summation,

N−1

• n=0
• pT

n (Cn+1fn+1 − Bnfn)

• =

N−1

• n=0
• pT

nCn+1fn+1 − pT n+1Bn+1fn+1

• +

N−1

• n=0
• pT

n+1Bn+1fn+1 − pT nBnfn

• =

N−1

• n=0
• pT

nCn+1fn+1 − pT n+1Bn+1fn+1

• +

pT

NBNfN − pT 0 B0f0,

L(f0, . . . , fN) = f T

NMNfN + N−1

• n=0
• pT

nCn+1fn+1 − pT n+1Bn+1fn+1

• +

pT

NBNfN − pT 0 B0f0 + λ0[f T 0 M0f0 − E0].

Stationary condition

δL = 0 ⇒ δL δf0 δf0 +

N−2

• n=0
• δL

δfn+1 δfn+1

• + δL

δfN δfN = 0

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 14

Constrained optimization (3/3)

δL δf0 = −pT

0 B0 + 2λ0f T 0 M0 = 0

δL δfn+1 = pT

nCn+1 − pT n+1Bn+1 = 0,

n = 0, . . . , N − 2 δL δfN = 2f T

NMN + pT NBN = 0

Inlet conditions : f0j =          (pT

0 B0)j

2λ0M0jj

if

M0jj = 0

if

M0jj = 0 “Adjoint” equations : pT

nCn+1 − pT n+1Bn+1 = 0

Oulet conditions : BT

NpN = −2MT NfN

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 15

An optimization algorithm

• 1. guessed initial condition f(0)

in

• 2. solution of forward problem with the IC f (n)

in

• 3. evaluation of objective function J (n) = E(n)
• ut. If

|J (n)/J (n−1) − 1| < ǫt optimization converged

• 4. if |J (n)/J (n−1) − 1| > ǫt outlet conditions provide the

“initial” conditions for the backward problem at x = xout

• 5. backward solution of the “adjoint” problem from x = xout

to x = xin

• 6. from the inlet conditions, update of the initial condition

for the forward problem f(n+1)

in

• 7. repeat from step 2 on

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 16

Results

Discretization. 2nd-order backward finite differences in x and 4th-order finite differences in y. Uneven grids in both x and y. Code verified against results by Tumin & Reshotko (2003, 2004) obtained with spectral collocation method. Inlet norm includes vin and win only. Outlet norm. Partial Energy Norm (PEN) uout and Tout only. Full Energy Norm (FEN) uout, vout, wout, Tout. FEN depends on Re, PEN is Re-independent.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 17

Results – Flat plate

Re = 103 Re = 104 Re → ∞ β G/Re 1 0.8 0.6 0.4 0.2 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002

Objective function G/Re: effect of Re and β for M = 3, Tw/Tad = 1, xin = 0 xout = 1.0, FEN. ⇒ Reynolds number effects only for Re < 104.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 18

Results – Flat plate

β G/Re 1 0.8 0.6 0.4 0.2 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002

Objective function G/Re: effect of β, Tw/Tad and norm choice (PEN vs. FEN) for M = 0.5, Re = 103, xin = 0 xout = 1.0. ✷, Tw/Tad = 1.00; ◦, Tw/Tad = 0.50; △, Tw/Tad = 0.25. ⇒ No remarkable norm effects; cold wall destabilizing factor.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 19

Results – Flat plate

β G/Re 1 0.8 0.6 0.4 0.2 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005

Objective function G/Re: effect of β, Tw/Tad and norm choice (PEN vs. FEN) for M = 1.5, Re = 103, xin = 0 xout = 1.0. ✷, Tw/Tad = 1.00; ◦, Tw/Tad = 0.50; △, Tw/Tad = 0.25. ⇒ Shift of the curves maximum, enhanced difference between norms (Tw/Tad = 1.00).

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 20

Results – Flat plate

β G/Re 1 0.8 0.6 0.4 0.2 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002

Objective function G/Re: effect of β, Tw/Tad and norm choice (PEN vs. FEN) for M = 3, Re = 103, xin = 0 xout = 1.0. ✷, Tw/Tad = 1.00; ◦, Tw/Tad = 0.50; △, Tw/Tad = 0.25. ⇒ Up to 17% difference for low values of β.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 21

Results – Flat plate

β G/Re 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0025 0.002 0.0015 0.001 0.0005

Objective function G/Re: effect of xin and β and norm choice (PEN vs. FEN) for M = 3, Tw/Tad = 1, xout = 1.0. ✷, xin = 0.0; ◦, xin = 0.2; △, xin = 0.4. ⇒ Up to 60% difference for xin = 0.4 and β = 0.1.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 22

Results – Flat plate

FEN PEN w v y vin(y), win(y) 20 15 10 5 0.4 0.3 0.2 0.1

• 0.1
• 0.2
• 0.3
• 0.4

FEN PEN w v y vout(y), wout(y) 20 15 10 5 0.2 0.15 0.1 0.05

• 0.05
• 0.1
• 0.15
• 0.2
• 0.25

Inlet and outlet profiles: effect of norm choice (PEN vs. FEN) for M = 3.0, Re = 103, xin = 0.4, xout = 1.0 and β = 0.1. ⇒ No significant changes in vin, some discrepancies in win; larger effects on vout, rather than on wout. No significant effects on uout and Tout.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 23

Results – Sphere

θ ∈ [25; 30] θ ∈ [15; 20] θ ∈ [10; 15] θ ∈ [5; 10] θ ∈ [3; 5] θ ∈ [2; 5] Tumin & Reshotko (2004)

• m

Gǫ2 1 0.8 0.6 0.4 0.2 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002

Objective function Gǫ2: effect of interval location and m = mǫ for θref = 30.0 deg, Tw/Tad = 0.5, ǫ = 10−3. PEN. ⇒ Largest gain for small θout − θin; strongest transient growth close to the stagnation point.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 24

Results – Sphere

• m

Gǫ2 0.1 0.08 0.06 0.04 0.02 0.0012 0.0011 0.001 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004

Objective function Gǫ2: effect of ǫ, energy norm (PEN vs. FEN) and m = mǫ for θin = 2.0 deg, θout = 5.0 deg, θref = 30.0 deg, Tw/Tad = 0.5. ✷, ǫ = 1 · 10−3; ◦, ǫ = 2 · 10−3; △, ǫ = 3 · 10−3. ⇒ Maximum appreciable difference within 1%. Effect increases with ǫ.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 25

Conclusions

√ Efficient and robust numerical method for computing

compressible optimal perturbations on flat plate and sphere.

√ Adjoint-based optimization technique in the discrete

framework and automatic in/out-let conditions.

√ Analysis including full energy norm at the outlet. √ Flat plate. For Re = 103, significant difference in G/Re

(up to 62%) between PEN and FEN. Effect of M and

• xin. No effect in subsonic basic flow. If Re > 104, vout

and wout do not play significant role.

√ Sphere. Largest Gǫ2 close to the stagnation point and

for small range of θ. No significant role played by vout and wout in the interesting range of parameters.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 26

The End!

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 27

Are we missing something?

At non-infinitesimal level of disturbance streaks are

• bserved on a flat plate, instead of

Tollmien–Schlichting waves. Linear Stability Theory (classical modal approach) fails even for the simplest geometries (Hagen-Poiseuille pipe flow, predicted stability vs. Recrex ≈ 2300)! Certain transitional phenomena have no explanation yet, e.g. the “blunt body paradox” on spherical fore-bodies at super/hypersonic speeds. There must exist another mechanism, not related to the eigenvalue analysis: transient growth.

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 28

Alternative paths of BL transition

• M. V. Morkovin, E. Reshotko, and T. Herbert,

(1994),“Transition in open flow systems – A re- assessment”, Bull. Am. Phys. Soc. 39, 1882.

“At the present time, no mathematical model exists that can predict the transition Reynolds number on a flat plate”! Saric et al., Annu. Rev. Fluid

• Mech. 2002. 34:291–319

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 29

Transient growth

M1 ϕ M2 M1 M2 ϕ M1 ϕ M2 M1 ϕ M2 |M1 + M2| |M2| |M1| Angle between modes ϕ = 150 [deg] Normalized time or space Normalized amplitude 14 12 10 8 6 4 2 1 0.8 0.6 0.4 0.2

Non-normality of the oper-

• ator. For most flows the lin-

ear stability equations are not self-adjoint (the eigen- functions are not orthogo- nal)

Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers – Complete Energy Norm Analysis, Paper AIAA-2005-5314 – p. 30