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

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

3. 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

4. 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

5. 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

6. 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)

7. 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

8. Parabolised Stability Equations
Blasius boundary layer

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

10. 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

11. 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

12. Absolute instability of heated jets
Monkewitz & Sohn 1988 C A

13. 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

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

15. Absolute instability of heated jets
Monkewitz et al. JFM 1990

16. 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

17. 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

18. 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

19. 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)

20. 2D example: cylinder wake
Noack & Eckelmann
JFM 1994

21. 3D example: jet in crossflow
Bagheri et al. JFM 2009