Transient growth on boundary layer streaks. Luca Brandt, Dan Henningson Department of Mechanics, KTH, Sweden

Transcription

1 Transient growth on boundary layer streaks Jérôme Hœpffner Luca Brandt, Dan Henningson Department of Mechanics, KTH, Sweden

2 Primary instability: TS vs TG

3 Secondary instability of the streaks Movie of streaks breakdown (Luca Brandt) Onset for streaks instability is at amplitude 26% for sinuous disturbances, and even larger for varicose.

4 Unstable sinuous eigenmode (From PhD thesis of Luca Brandt) Temporal growth for unstable sinuous disturbance: amplitude A=0.28, 0.32, 0.34, 0.36.

5 Secondary transient growth Maybe transient growth plays a role in secondary instability? Primary growth: streaks Secondary growth Free stream turbulence Conjectured by Schoppa&Hussain(2002) and Lundell(2004).

6 Base flow for stability Inflow condition from optimal vortices on Blasius flow Streaks are generated and grow downstream Extract Base flow +streaks at downstream location of maximum amplitude Freeze this flow and assume invariance in x U(y, z) Use U(y, z) as base flow for linear stability analysis. floquet formulation for the spanwise periodic base flow U(y, z) From Anderson et al (2001) Eigenvalues tell about asymtoptic stability Singular values tell about transient behaviour

7 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

8 Floquet analysis Base flow is spanwise-periodic Fundamental mode and its harmonics are coupled Detuned modes are decoupled to each other We look at fundamental modes in this study

9 Dynamic system with initial condition: Transient growth q = Lq, q(0) = q 0 Input-output operator H τ : q(t) = H τ (q 0 ) 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

10 Computation by power iterations Power Iteration: Consider initial guess q 0 (0) March forward in time with dynamic equation : q 0 (τ) = H τ q 0 (0) March backward in time with adjoint equation: q 1 (0) = H τ + q 0 (τ) Renormalize energy Each of these power iteration magnifies the component of the initial guess on the optimal initial condition. Convergence in less than 20 iterations well separated eigenvalues

11 Model limitations Infinitely elongated and frozen streak: Disturbance wavelength small compared to streak evolution in x α > 0 Disturbance should be quick to reach high energy Look at amplification for short times and streamwise dependent perturbations

12 Results

13 Introduction to energy envelope Energy envelope for several α (a) 1000 G eplacements 500 Decreasing α α =0.01, 0.1, 0.2,..., 0.6 The envelope shows for each time the greatest reachable energy τ

14 Sinuous energy evolution For 4 different streak amplitudes: energy envelope for several α (a) 2000 (b) lacements lacements τ G G PSfrag replacements Decreasing α τ τ G Decreasing α (c) PSfrag replacements reasing α Decreasing α τ G α =0.01, 0.1, 0.2,..., 0.6 (a),(b),(c),(d): Amplitudes: A=0.14, 0.2, 0.25, (d)

15 Sinous: onset of instability for large streak amplitude (d) eplacements τ G ecreasing α wavenumber α = 0.2 is linearly unstable, but can reach quickly high energy due to transient growth.

16 lacements lacements τ G reasing α G Varicose energy evolution For 4 different streak amplitudes: energy envelope for several α. Decreasing α (a) PSfrag replacements τ α = 0.25 τ G Decreasing α α = 0.25 (c) PSfrag replacements (b) α = 0.25 α = τ G Decreasing α α =0.01, 0.1, 0.2,..., 0.6 (a),(b),(c),(d): Amplitudes: A=0.14, 0.2, 0.25, (d)

17 Varicose: TS-like instability for low amplitude eplacements G Decreasing α (a) 200 b τ TS-like instability progressively disapear for streaky base flow... (cf Cossu&Brandt 2003)

18 Amplitude and Reynolds number lacements Gmax Increasing A Re 160 (a) 140 PSfrag replacements G max Re τmax varicose sinuous (b) Re Amplitudes: A=0.14, 0.2, 0.25, Maximum growth increases with increasing streak amplitude and Reynolds number Time for maximum increases for increasing amplitude and Reynolds number But no obvious scaling (we will see later)

19 Flow structures: Sinuous Optimal disturbance: Optimal response (Isosurface of constant velocity=20% of maximum)

20 Flow structures: Varicose Optimal disturbance: Optimal response (Isosurface of constant velocity=20% of maximum)

21 Unstable sinuous eigenmode (From PhD thesis of Luca Brandt) Similarity in structure with our optimal response!

22 where does the energy come from? Analysis

23 K t = Energy balance ( 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

24 lacements Production Dissipation Production and dissipation PSfrag replacements Production Dissipation Sinuous T T K D t t t Spanwise shear allways contributes to energy growth y z t t K D T z y t Varicose Wall-normal shear gives then takes: Orr mechanism related to structure tilting Disturbance can gain energy by interaction with both mean shear

25 Conclusion Possibility of energy growth of O(1000) before onset of instability Time to reach peak is small (compared to streak evolution time scale) Optimal response resembles the unstable mode Two production mechanisms acting together: lift up +Orr Observed transition from streaks may be a TG mechanism Remaining: how likely are those initial excitation in a boundary layer?

26 Submitted