Stability Analysis of a Plane Poisseuille Flow. Course Project: Water Waves and Hydrodynamic Stability 2016 Matthias Steinhausen

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1 Stability Analysis of a Plane Poisseuille Flow Course Project: Water Waves and Hydrodynamic Stability 2016 Matthias Steinhausen

2 y Plane Poiseuille Flow (PPF): Characteristics y Velocity profile for PPF Pressure-driven flow between two resting plates Scaled velocity profile (only in y): U y = U U Cl = 1 y 2 x No slip boundary conditions at wall: U y = ±1 = Ux Stability properties from literature: Re cr,energy = 49.6 Re cr,linear =

3 PPF: Governing Equations 1/3 Start with incompressible Navier-Stokes equation: u i t = u j u i x j p x i + 1 Re 2 u i x j x j Introduce the disturbances: u i = U b,i + u i, p = P b + p Rearrange and linearize: u i t + U b,j u i x j + u j U b,i x j = p x i + 1 Re 2 u i x j x j

4 PPF: Governing Equations 2/3 Assume parallel flow: U b,i = U y δ 1i u t = U u b x U b v p x u Re x j x j v t = U v b x p y v Re x j x j w t = U w b x p z w Re x j x j u x + v y + w z = 0 Eliminate the pressure divergence of momentum equation: 2 p = 2U v x i x b i x

5 PPF: Governing Equations 3/3 Orr-Sommerfeld equation: New BCs at walls: v = v = η = 0 v t = U b x 2 + U b x + 1 Re 4 v Introducing η = u w z x leads to the squire equation: η t = U b x + 1 Re 2 η U b v z In matrix form: t v η = L OS 0 v C L S η external forcing + Bf

6 Eigenvalue Stability Analysis Assume wavelike solutions: v x, y, z, t = v y ei αx+βz ωt iω k2 D 2 0 C 1 v η + L OS 0 v iβu b L S η = 0 with L OS = iαu k 2 D 2 + iαu b + 1 Re k2 D 2 2 ; L SQ = iαu + 1 Re (k2 D 2 ) what leads to the EV problem: (L iωm) q = 0 with the solution q = q 0 exp(tl)

7 Long-time Stability: Maximum Eigenvalues

8 Long-time Stability: 2D-Waves Stable α cr = 1.02 Neutral line Unstable Re cr,linear =

9 Eigenvalue Spectra: α=1.02, β=0 A P S Eigenvalue responsible for instability

10 Eigenvalue Spectra: Re=

11 Eigenvalues: Summary

12 Transient Growth: G(t) t max ~ Re 1 G max ~ Re

13 Numerical Abscissa: α=0, β=

14 Numerical Abscissa: α=1.02, β=

15 Transient Growth: Re=2000, α=

16 Transient Growth: Summary

17 Resolvent Norm: Re=2000 d dt q = Lq + f with f = fexp(iωt) R ω = max መ f q E 2 2 መf E 2 = iωi L 1 E

18 Pseudospectra

19 Optimal Disturbance: Re=2000, α=0, β=

20 Conclusion Eigenvalues: Onset of linear instability at α = 1.02 and Re = 5772 Instable eigenvalue in the A-branch P-,S-branch stays stable Different eigenvalue distribution for α=0, no instability Eigenvalue responsible for instability relatively stable against disturbances Transient Growth: Onset of transient growth at Re=49.6 at β=2 and α=0 Maximum amplification for β=2 and α=0 Forcing and Optimal Response: For α=0 only one peak in resolvent Two peaks for α 0 due to eigenvalue distribution distance Streaks: Energy transfer from v to η

21 Thank you for your attention!

22 Instability analysis Long time Dynamics: Eigenvalue distribution: L = SΛS 1 Only maximum Eigenvalue Determines maximum growth for infinite time Short time Dynamics: Numerical Range: Lq,q q,q Combines both: Matrix exponential: exp tl E

23 Transient Growth: α=0, β=2 t ~ Re 1 G ~ Re

24 Transient Growth over α and Re

25 Resolvent Norm: Re=

26 Optimal Disturbance: Re=2000, α=1, β=0 Energy transfer Eta!!

SG2221 Wave Motion and Hydrodinamic Stability. MATLAB Project on 2D Poiseuille Flow Alessandro Ceci

SG2221 Wave Motion and Hydrodinamic Stability MATLAB Project on 2D Poiseuille Flow Alessandro Ceci Base Flow 2D steady Incompressible Flow Flow driven by a constant pressure gradient Fully developed flow