Two-dimensional channel flow

Geometry of a two-dimensional channel flow.

The Orr-Sommerfeld equation governs the dynamics of two-dimensional velocity fluctuations around the laminar channel flow,

$begin{array}{rcl} Delta psi_{t} (k_x, y, t) & !! = !! & left( {displaystyle frac{1}{Re}} , Delta^{2} , - , mathrm{i} , k_x , U(y) , Delta , + , mathrm{i} , k_x , U''(y) right) psi(k_x, y, t) [0.5cm] 0 & !! = !! & psi(k_x, pm 1, t) ; = ; psi' (k_x, pm 1, t) end{array}$

where

	—	streamfunction
	—	Reynolds number
	—	streamwise wavenumber
	—	wall-normal coordinate
	—	base flow
$Delta = D^{(2)} - k_x^2$	—	Laplacian
$Delta^2 = D^{(4)} - 2 , k_x^2 , D^{(2)} + k_x^4$	—	‘‘square’’ of the Laplacian

For Re = 10000 (based on the centerline velocity and the channel half-width), the discretized version of the Orr-Sommerfeld equation is obtained using the pseudo-spectral scheme with M = 150 colocation points in the wall-normal direction. Flow fluctuations with k_x = 1 are then advanced in time using the matrix exponential with time step Delta t = 1 and randomly generated initial profile (that satisfies both homogeneous Dirichlet and Neumann boundary conditions). After a transient period of ten time-steps, N = 100 snapshots are taken to form the snapshot matrices and apply the standard DMD algorithm along with its sparsity-promoting variant.

(a) Spectrum of the Orr-Sommerfeld operator (circles) along with the eigenvalues resulting from the standard DMD algorithm (crosses) for the 2D channel flow with Re = 10000 and k_x = 1 . Dependence of the absolute value of the DMD amplitudes on (b) the frequency and (c) the real part of the corresponding DMD eigenvalues.

Performance loss, $% , Pi_{mathrm{loss}} ; mathrel{mathop:}= ; 100 , sqrt{J (alpha) / J (0)}$ , of the optimal vector of amplitudes alpha resulting from the sparsity-promoting DMD algorithm with N_z DMD modes.

Eigenvalues of $F_{mathrm{dmd}}$ (first row) and the absolute values of the DMD amplitudes alpha_i (second row). The results are obtained using the standard DMD algorithm (circles) and the sparsity-promoting DMD algorithm (crosses) with N_z DMD modes.