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Inviscid instability of a unidirectional flow sheared in two
transverse directions
Kengo Deguchi, Monash University
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Shear flow stability
•Navier-Stokes equations: nonlinear PDEs having a parameter called Reynolds number
•In stability analysis consider base flow + small perturbation
•Linearised NS can be solved by using normal mode (eigenvalue problem)
•Given wavenumber alpha, the complex wave speed c is obtained as eigenvalue
•Imaginary part of c is the growth rate of the perturbation (Im c positive is unstable case, i.e. the perturbation grows exponentially)
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Inviscid stability of shear flows
(delta=1/R)
Picture taken from Maslowe (2009)
•Rayleigh’s equation (viscosity is ignored) •Singular at the critical level: viscosity is needed in the critical layer•Matched asymptotic expansion must be used to analyse the critical layer
Classical case: U(y)
U=c: wave speed
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Inviscid stability of shear flows
•However, most physically relevant unidirectional flows vary in two transverse directions, so more general base flow U(y,z) must be considered!
•E.g. stability of flows over corrugated walls, or through non-circular pipe
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Streaks
Streaks in a boundary layer flowover flat plate
x
•Streaks can be visualized as thread-like structures
•Streamwise velocity naturally creates inhomogeneity in transverse direction
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VWI / SSP
Vortex-wave interaction(Hall & Smith 1991,Hall & Sherwin 2010)
Self-sustaining process (Waleffe 1997,Wang, Gibson & Waleffe 2007)
Nonlinear theory for shear flows
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Derivation of the generalized problem
Hocking (1968), Goldstein (1976), Benney (1984), Henningson (1987), Hall & Horseman (1991)
Neglecting the viscous terms,
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Classical stability problem for U(y)
The method of Frobenius can be used to show that the local expansion of the solution contains the term like
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Classical stability problem for U(y)
Inner analysis shows that the outer solution must be written in the formr
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Three 2nd order ODEs: Boundary conditions?
(Hereafter we set c=0)
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Generalised problem for U(y,z)
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Streak-like model flow profile
Wall
Wall
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Line: NS result, R=10000Blue triangle: Rayleigh, usual methodRed circle: Rayleigh, new method
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Red: eta constantGreen: zeta constant
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Complicated functions of U!
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Full NS
Rayleigh
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Remark 1: We only need singular basis function near the critical layer
Remark 2: Computationally much cheaper than solving NS
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Remark 3: Necessary condition for existence of a neutral mode
For the classical case
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Remark 3: Necessary condition for existence of a neutral mode
So if the classical problem has a neutral mode, =0 somewhere in the flow.
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Remark 3: Necessary condition for existence of a neutral mode
So if the classical problem has a neutral mode, =0 somewhere in the flow.
If the generalised problem has a neutral mode,
somewhere in the flow.
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Conclusion
•In order to solve the generalized inviscid stability problem (a singular PDE) the method of Frobenius is used in curved coordinates to construct appropriate basis functions
•The new Rayleigh solver is more efficient than the full NS solver
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End