Stochastic excitation of streaky boundary layers. Luca Brandt, Dan Henningson Department of Mechanics, KTH, Sweden

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1 Stochastic excitation of streaky boundary layers Jérôme Hœpffner Luca Brandt, Dan Henningson Department of Mechanics, KTH, Sweden

2 Boundary layer excited by free-stream turbulence Fully turbulent inflow and flat plate Turbulent free-stream receptivity streaks streak instability turbulent spots turbulent boundary layer (Image by Philipp Schlatter, LES of bypass transition, Tuesday, D31 )

3 Boundary layer stability TS waves: Large Reynolds 2D waves Exponential growth Streaks: Subcritical Reynolds Large external disturbances Transient growth

4 Secondary instability of streaks Contours of streamwise velocity, yz plane: Streaks instability is at amplitude 26% of the free stream velocity inflectional profile Inviscid instability Maybe an other mechanism at lower amplitude? (Luca Brandt, PhD thesis)

5 Secondary transient growth of the streaks Free stream turbulence Primary growth: streaks Secondary growth Assume the boundary layer is excited by the FST at all downstream position streaks is generated, then streaks is disturbed Possibility of transient growth on top of the streaks: Schoppa & Hussain, JFM(22).

6 Energy growth: sinuous perturbations 25 2 Transient growth: energy envelopes α=.1 α=.1 (c) Energy 15 1 α= time Streak amplitude: 25% of free-stream velocity

7 Question How likely are those initial excitation in a FST boundary layer? Will this growth mechanism be active with realistic disturbances?

8 Statistical description Farrell& Ioannou, POF(1993):Stochastic forcing of the linearized Navier Stokes equations. FST is erratic, instationnary, we describe it by its statistics two point correlation Average, or expectation operator E. Two-point correlation: P u (x, x ) = E[u(x)u(x )] of u at x and x. For 3D flow: P(x, x, y, y, z, z, t, t ).

9 p x1 (x ) Covariance matrix Correlation is covariance normalized to unit variance. x x 1 x x x 1 p x2 (x ) x 2 p x3 (x ) x x 2 x 3 x P(x, x ) x 3 Two point correlation, varying in x Correlation matrix

10 Two-point correlation of FST Covariance in y: Two points covariance alpha=.5, beta=1 uu vv ww vw uv uw distance alpha=.5, beta= Energy spectra: E(k) = 2 3 kl a(kl) 4 (b+(kl) 2 ) (17/6) Energy L= wave number Covariance matrix: L=1 L=1 Two points covariance distance Isotropic turbulence: Von-Karman spectrum

11 Zero disturbances inside the boundary layer, Threshold at y = 3δ 7 6 Base flow FST is only in the free-stream 5 FST amplitude 4 y 3 2 Streak Threshold u Covariance matrix with ramping

12 POD modes from the covariance matrix Two-point correlation data has many dimensions show coherent structures Energy content of the flow structures tells about flow coherence POD modes are the eigenmodes of the covariance matrix

13 POD modes of the FST Faster oscillation more damped, many spatial frequencies are present 5 6

14 Lyapunov equation Explicit state solution: P(t, t) t q = Aq + w, q() = q, q(t) = e At q + E[q(t)q(t) H ] }{{} P(t,t) =e At P {}}{ E[q q H ]e AHt + =e At P e AHt + State covariance: e A(t τ) e A(t τ) We AH (t τ) dτ τ= Wδ(τ τ ) Differentiating this convolution integral: e A(t τ) w(τ)dτ {}}{ E[w(τ)w(τ ) H ]e AH (t τ ) dτdτ = A eat P e AHt + e A(t τ) We AH (t τ) dτ + eat P e AHt + e A(t τ) We AH (t τ) dτ AH +e A We AH }{{}}{{}}{{} W P(t,t) P(t,t)

15 Lyapunov equation P t = AP + PAH + W, P() = P Initial condition problem No stochastic forcing: W = Stochastic initial condition Forced problem With stochastic forcing: W Long time after initial condition P t = AP + PA H, P(, ) = P = AP + PA H + W Covariance varies in time Reached statistical steady state

16 Streaky flow excited by FST: Stochastic initial value Time evolution of u rms 12 1 α=.1 problem rms profile: 6 5 w rms Time evolution, α= u rms u rms 6 y 3 4 α= α=.1 1 α= time The FST is the flow initial condition rms

17 POD modes, α =.5 POD modes energy mode 1 Kinetic energy mode 2 mode time Flow structure with maximum energy

18 Comparison FST/optimal Worst case initial condition: u,v,w FST initial condition: u,v,w

19 Flow structures at time of max energy

20 Comparison of flow structures: Streamwise velocity (LES of bypass transition : Philipp Schlatter)

21 Comparison of flow structures: Streamwise shear (LES of bypass transition : Philipp Schlatter)

22 Conclusions 1. Possibility of energy growth of O(1) for subcritical streak 2. FST description using two-point correlation 3. Computation of state two-points correlation using Lyapunov equation 4. Response to FST involves transient growth mechanism 5. Secondary transient growth explains observed streak structure Bypass transition involves TG mechanism twice Does this explain streak breakdown?

23 Thank you!

24 Computation of the transient growth Power Iteration: Consider initial guess q () March forward in time with dynamic equation : q (τ) = H τ q () March backward in time with adjoint equation: q 1 () = H τ + q (τ) Renormalize energy Each of these power iteration magnifies the component of the initial guess on the optimal initial condition. Convergence in less than 2 iterations well separated eigenvalues

25 Numerical solution of Lyapunov equation Solve: AX + XA H + W = 1. Schur decomposition A = UA U H, A upper diagonal, U orthogonal. 2. Resulting equation A X {}}{ U H XU + X {}}{ U H XU A H + W {}}{ U H WU = 3. Use Kronecker product A B a 11 B a 12 B... a 1m B a 21 B a 22 B... a 2m B a n1 B a n2 B... a nm B (A X + X A H + W ) = = (I A + A I) (X }{{} ) + (W ) F 4. Solve by backward substitution F has upper diagonal structure

26 Energy balance in streak secondary transient growth K t = ( uv U y }{{} T y uw U z }{{} T z ω ω/re) dy dz dx, }{{} D K t : time variation of kinetic energy T y : production due to interaction with wall normal mean shear T z : production due to interaction with spanwise mean shear D: dissipation due to viscosity

27 Disturbance can gain energy Production and dissipation in streak secondary transient growth Varicose Production Dissipation Sinuous t T T y z K D t Spanwise shear always contributes to energy growth Wall-normal shear gives then takes: Orr mechanism related to structure tilting

28 Streaky flow excited by FST: Forced problem The flow is constantly excited by FST 1 2 Energy Mean energy in POD modes POD number

29 Unstable sinuous eigenmode.1 Unstable eigenmode λ i.1 For large streak amplitude α =.3 eigenvalues: λ r Corresponding eigenfunction:

30 Transient growth analysis To build the dynamical operator: Use PSE to optimize disturbance Input forcing in DNS Extract fully saturated streamwise velocity profile U(y, z) Apply linear stability analysis, using Floquet Build matrix A eigenvalues for asymptotic stability singular values for transient growth

31 Stability equations Perturbation (v, η) on the base flow U(y, z) Wavelike behaviour in the streamwise direction: [v, η] = [ v(y, z, t), η(y, z, t)] e iαx + c.c. Derivation similar to the Orr Sommerfeld/Squire equation: v t + U v x + U zz v x + 2U z v xz U yy v x 2U z w xy 2U yz w x = 1 Re v, η t + Uη x U z v y + U yz v + U y v z + U zz w = 1 Re η. (with w xx + w zz = η x v yz ) + Floquet analysis: base flow and disturbance are periodic in spanwise direction. Look only at fundamental modes Chebyshev discretization in wall-normal direction

32 Computation of the transient growth Dynamic system with initial condition: q = Aq, q() = q Input-output operator H τ : q(t) = H τ (q ) Maximum possible growth: G(τ) = max q H τ q E q E = max q (H τ q, H τ q) (q, q) max q (q, H τ + H τ q), (q, q) with adjoint operator: (Hq 1, q 2 ) = (q 1, H + q 2 ), q 1, q 2 Max G(τ) is the largest eigenvalue of operator H + τ H τ

33 Varicose energy evolution For 4 different streak amplitudes: energy envelope for several α. G Decreasing α (a) τ α =.25 (c) (b) (d) α =.1,.1,.2,...,.6 (a),(b),(c),(d): Amplitudes: A=.14,.2,.25,.29.

34 Sinuous: Flow structures for streak transient growth Varicose: Optimal disturbance: Optimal disturbance: Optimal response Optimal response

35 Sinuous energy evolution For 4 different streak amplitudes: energy envelope for several α. 15 (a) 2 (b) 25 2 G 1 5 Decreasing α τ (c) (d)