Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis

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, 4416, USA 4th AIAA Theoretical Fluid Mechanics Meeting, 6 9 June, 25, Westin Harbour Castle, Toronto, Ontario, Canada. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 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-25-5314 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-25-5314 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-25-5314 p. 4

Optimal perturbations Question. What is the most disrupting, steady initial condition, which maximizes the energy growth for a given initial energy of the perturbation? z Flat plate Boundary layer edge y 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-25-5314 p. 5

Goals/Tools Goals Tools 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. 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-25-5314 p. 6

Problem formulation Geometry. Flat plate and sphere. Regimes. Compressible, sub/supersonic. Possibly reducing to incompressible regime for M. 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-25-5314 p. 7

Scaling (1/2) L ref is a typical scale of the geometry (L for flat plate, R for sphere, etc.) H ref = ν ref L ref /U ref is a typical boundary-layer scale in the wall-normal direction Flat plate. H ref = l = ν L/U ; = freestream. Sphere. H ref = ν ref R/U ref ; ref at x ref. ɛ = H ref /L ref is a small parameter. Flat plate. ɛ = Re 1/2 L, Re L = U L/ν. Sphere. ɛ = Re 1/2 ref, Re ref = U ref R/ν ref. = edge-conditions Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 8

Scaling (2/2) From previous works, disturbance expected as streamwise vortices. The natural scaling is therefore x normalized with L ref, y and z scaled with ɛl ref. u is scaled with U ref, v and w with ɛu ref. T with T ref and p with ɛ 2 ρ ref Uref 2. 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-25-5314 p. 9

Governing equations (Af) x = (Df y ) x + B f + B 1 f y + B 2 f yy f = [u, v, w, T, p] T ; A, B, B 1, B 2, D 5 5 real matrices. Boundary conditions y = : u = ; v = ; w = ; T = y : u ; w ; p ; T More compactly (H 1 f) x + H 2 f = with H 1 = A D( ) y ; H 2 = B B 1 ( ) y B 2 ( ) yy Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 1

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. v in and w in (u in = T in = ). Outlet. v out = and w out = (u in ; T in ). 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 v out and w out 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-25-5314 p. 11

Objective function (2/2) Mack s energy norm (derived for flat plate and temporal problem), after scaling and using state equation, E out = [ ρ s out(u 2 out + v 2 out + w 2 out) + or in matrix form as E out = ( M out = diag ρ s out, ρ s out, ρ s out, p s outtout 2 ] 2 dy (γ 1)T sout M 2 (fout T M ) out f out dy, with ) p s out 2 (γ 1)T sout M,. 2 Initial energy of the perturbation E in = [ ρs in(v 2 in + w 2 in) ] dy E in = (f T in M in f in ) dy Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 12

Constrained optimization (1/3) Our constraints are the governing equations, boundary conditions and the normalization condition E in = E. After discretization (M M in and M N M out ), objective function J = f T N M Nf N constraint E in = E f T M f = E governing equations (BC included) C n+1 f n+1 = B n f n The augmented functional L is L(f,..., f N ) = f T N M N f N + λ [f T M f E ] + N 1 n= [ p T n (C n+1 f n+1 B n f n ) ] with λ and (vector) p n Lagrangian multipliers. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 13

Constrained optimization (2/3) By adding and subtracting p T n+1 B n+1f n+1 in the summation, N 1 n= [ ] p T n (C n+1 f n+1 B n f n ) = N 1 n= N 1 ] [p T n C n+1 f n+1 p T n+1b n+1 f n+1 + [p T n+1b n+1 f n+1 p T n B n f n ] = n= N 1 n= ] [p T n C n+1 f n+1 p T n+1b n+1 f n+1 + p T N B N f N p T B f, L(f,..., f N ) = f T N M N f N + Stationary condition N 1 n= ] [p T n C n+1f n+1 p T n+1 B n+1f n+1 + p T N B N f N p T B f + λ [f T M f E ]. δl = δl N 2 [ ] δl δf + δf n+1 + δl δf N = δf δf n= n+1 δf N Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 14

Constrained optimization (3/3) δl δf = p T B + 2λ f T M = δl δf n+1 = p T n C n+1 p T n+1 B n+1 =, n =,..., N 2 δl δf N = 2f T N M N + p T N B N = Inlet conditions : f j = (p T B ) j 2λ M jj if M jj if M jj = Adjoint equations : p T n C n+1 p T n+1 B n+1 = Oulet conditions : B T N p N = 2M T N f N Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 15

An optimization algorithm 1. guessed initial condition f () in 2. solution of forward problem with the IC f (n) in 3. evaluation of objective function J (n) = E (n) out. If J (n) /J (n 1) 1 < ɛ t optimization converged 4. if J (n) /J (n 1) 1 > ɛ t outlet conditions provide the initial conditions for the backward problem at x = x out 5. backward solution of the adjoint problem from x = x out to x = x in 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-25-5314 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 (23, 24) obtained with spectral collocation method. Inlet norm includes v in and w in only. Outlet norm. Partial Energy Norm (PEN) u out and T out only. Full Energy Norm (FEN) u out, v out, w out, T out. 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-25-5314 p. 17

Results Flat plate.18.16.14 Re Re = 1 4 Re = 1 3.12 G/Re.1.8.6.4.2.2.4 β.6.8 1 Objective function G/Re: effect of Re and β for M = 3, T w /T ad = 1, x in = x out = 1., FEN. Reynolds number effects only for Re < 1 4. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 18

Results Flat plate G/Re.2.18.16.14.12.1.8.6.4.2.2.4.6 β.8 1 Objective function G/Re: effect of β, T w /T ad and norm choice (PEN vs. FEN) for M =.5, Re = 1 3, x in = x out = 1.., T w /T ad = 1.;, T w /T ad =.5;, T w /T ad =.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-25-5314 p. 19

Results Flat plate.35.3.25 G/Re.2.15.1.5.2.4.6 β.8 1 Objective function G/Re: effect of β, T w /T ad and norm choice (PEN vs. FEN) for M = 1.5, Re = 1 3, x in = x out = 1.., T w /T ad = 1.;, T w /T ad =.5;, T w /T ad =.25. Shift of the curves maximum, enhanced difference between norms (T w /T ad = 1.). Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 2

Results Flat plate.18.16.14.12 G/Re.1.8.6.4.2.2.4 β.6.8 1 Objective function G/Re: effect of β, T w /T ad and norm choice (PEN vs. FEN) for M = 3, Re = 1 3, x in = x out = 1.., T w /T ad = 1.;, T w /T ad =.5;, T w /T ad =.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-25-5314 p. 21

Results Flat plate.25.2 G/Re.15.1.5.1.2.3.4 β.5.6.7.8 Objective function G/Re: effect of x in and β and norm choice (PEN vs. FEN) for M = 3, T w /T ad = 1, x out = 1.., x in =.;, x in =.2;, x in =.4. Up to 6% difference for x in =.4 and β =.1. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 22

Results Flat plate vin(y), win(y).4.3.2.1 -.1 -.2 -.3 -.4 v w 5 1 y PEN FEN 15 2 vout(y), wout(y).2.15.1.5 -.5 -.1 -.15 -.2 -.25 v w 5 1 y PEN FEN 15 2 Inlet and outlet profiles: effect of norm choice (PEN vs. FEN) for M = 3., Re = 1 3, x in =.4, x out = 1. and β =.1. No significant changes in v in, some discrepancies in w in ; larger effects on v out, rather than on w out. No significant effects on u out and T out. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 23

Results Sphere Gɛ 2.14.12.1.8.6.4.2 Tumin & Reshotko (24) θ [2; 5] θ [3; 5] θ [5; 1] θ [1; 15] θ [15; 2] θ [25; 3].2.4.6 m Objective function Gɛ 2 : effect of interval location and m = mɛ for θ ref = 3. deg, T w /T ad =.5, ɛ = 1 3. PEN. Largest gain for small θ out θ in ; strongest transient growth close to the stagnation point..8 1 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 24

Results Sphere Gɛ 2.12.11.1.9.8.7.6.5.4.2.4.6 m.8 Objective function Gɛ 2 : effect of ɛ, energy norm (PEN vs. FEN) and m = mɛ for θ in = 2. deg, θ out = 5. deg, θ ref = 3. deg, T w /T ad =.5., ɛ = 1 1 3 ;, ɛ = 2 1 3 ;, ɛ = 3 1 3. Maximum appreciable difference within 1%. Effect increases with ɛ..1 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 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 = 1 3, significant difference in G/Re (up to 62%) between PEN and FEN. Effect of M and x in. No effect in subsonic basic flow. If Re > 1 4, v out and w out do not play significant role. Sphere. Largest Gɛ 2 close to the stagnation point and for small range of θ. No significant role played by v out and w out in the interesting range of parameters. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 26

The End! Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 27

Are we missing something? At non-infinitesimal level of disturbance streaks are observed 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. Re cr ex 23)! 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-25-5314 p. 28

Alternative paths of BL transition 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. 22. 34:291 319 M. V. Morkovin, E. Reshotko, and T. Herbert, (1994), Transition in open flow systems A reassessment, Bull. Am. Phys. Soc. 39, 1882. Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 29

Transient growth M2 ϕ M1 M2 ϕ M1 M1 ϕ M1 M2 M2 ϕ Angle between modes ϕ = 15 [deg] Non-normality of the operator. For most flows the linear stability equations are not self-adjoint (the eigenfunctions are not orthogonal) Normalized amplitude 1.8.6.4.2 M1 + M2 M1 M2 2 4 6 8 1 12 Normalized time or space 14 Zuccher, S., Tumin, A., Reshotko, E., Optimal Disturbances in Compressible Boundary Layers Complete Energy Norm Analysis, Paper AIAA-25-5314 p. 3