From non-parallel flows to global modes André V. G. Cavalieri Peter Jordan (Visiting researcher, Ciência Sem Fronteiras)

The parallel-flow hypothesis A necessary step to arrive at the Orr-Sommerfeld equation (an ODE)

The parallel-flow hypothesis Strictly true only for some wall-bounded flows (e.g. Poiseuille, Couette) Free-shear flows are non-parallel due to momentum diffusion by viscosity

Non-parallel flows The parallel-flow assumption - U(y) - leads to: homogeneity in x, z and t: coefficients depend only on y normal-mode Ansatz Resulting problem is an ODE v(y) is the eigenfunction Degrees of freedom = Ny Suppose we had U(x,y) instead of a parallel flow: homogeneity in z and t: coefficients depend on x and y normal-mode Ansatz Exercise: estimate the computational time to solve an eigenvalue problem of size: N = Ny = 200 N = Nx x Ny = 40000 Resulting problem is a PDE v(x,y) is the eigenfunction Degrees of freedom = Nx x Ny

A simpler approach: slowly-diverging flows For parallel base-flows U(y) the normal-mode Ansatz is Definition: Slowly-diverging base-flows U(x,y) x is the slow variable y is the fast variable dU/dx = ε dU/dy With some math (method of multiple scales: Bouthier 1972, Gaster 1974, Crighton & Gaster 1976) one shows that the solution takes the form slow x-dependency fast x-dependency

A simpler approach: slowly-diverging flows Definition: Slowly-diverging base-flows U(x,y) x is the slow variable y is the fast variable dU/dx = ε dU/dy With some math (method of multiple scales: Bouthier 1972, Gaster 1974, Crighton & Gaster 1976) one shows that the solution takes the form slow x-dependency fast x-dependency Expand linearised Navier-Stokes in powers of ε to get v(x,y) and α(x) Only ODEs (direct and adjoint Rayleigh/Orr-Sommerfeld)

Parabolised Stability Equations With some math (method of multiple scales: Bouthier 1972, Gaster 1974, Crighton & Gaster 1976) one shows that the solution takes the form slow x-dependency fast x-dependency We know that the solution for a slowly-diverging flow has this shape. Use the Ansatz above in the linearised Navier-Stokes equations! Herbert 1997

Parabolised Stability Equations Blasius boundary layer

Parabolised Stability Equations Linear PSE As Im(α) changes sign, amplification switches to decay unstable near nozzle stable downstream

Global synchronisation in shear flows: the impulse response Disturbance equation (parallel-flow assumption) Impulse response (Green’s function) is defined as Stable Convectively unstable Absolutely unstable

Global synchronisation in shear flows: the impulse response Stable Convectively unstable Absolutely unstable Upstream disturbances are amplified as they are advected NOISE AMPLIFIER (cold jets, mixing layers, boundary layers) Global synchronisation; intrinsic dynamics; insensitive to upstream disturbances OSCILLATOR (wakes, hot jets) Huerre & Monkewitz Ann. Rev. Fluid Mech. 1990

Absolute instability of heated jets Monkewitz & Sohn 1988 C A

Absolute instability of heated jets Monkewitz et al. JFM 1990 “Cold” jet S = 0.91 Convective instability (K-H) Hot jet S = 0.62 Absolute instability

Absolute instability of heated jets Monkewitz et al. JFM 1990 Self-excited, periodic structures Side jets!

Absolute instability of heated jets Monkewitz et al. JFM 1990

Feedback mechanisms in shear flows: the impulse response Disturbance equation (parallel-flow assumption) Impulse response (Green’s function) is defined as Recall that this is based on the parallel-flow hypothesis Current work: non-parallel flows, global modes

Non-parallel flows: global modes The parallel-flow assumption - U(y) - leads to: homogeneity in x, z and t: coefficients depend only on y normal-mode Ansatz Resulting problem is an ODE v(y) is the eigenfunction Degrees of freedom = Ny Suppose we had U(x,y) instead of a parallel flow: homogeneity in z and t: coefficients depend on x and y normal-mode Ansatz Resulting problem is a PDE v(x,y) is the eigenfunction Degrees of freedom = Nx x Ny v(x,y) is a global mode

Non-parallel flows: global modes Parallel base-flow Slowly-diverging base-flow 2D base-flow 2D EVP 2D global mode “BiGlobal” 3D EVP 3D global mode “TriGlobal” 3D base-flow

Global modes: numerical issues Parallel flow DoF = Nx x Ny x Nz DoF = Nx x Ny DoF = Ny Estimated matrix size Estimated matrix size Estimated matrix size Theofilis, Prog. Aerospace Sci. 2003 Simplify whenever possible Stability of non-parallel flows is a currently feasible calculation Direct solution of eigenvalue problem (e.g. “eig(L,F)”) usually avoided – Arnoldi method (iterative, focus on limited number of relevant modes)

2D example: cylinder wake Noack & Eckelmann JFM 1994

3D example: jet in crossflow Bagheri et al. JFM 2009