Twodimensional channel flow

Geometry of a twodimensional channel flow.

The OrrSommerfeld equation governs the dynamics of twodimensional velocity
fluctuations around the laminar channel flow,
where
 —  streamfunction 
 —  Reynolds number 
 —  streamwise wavenumber 
 —  wallnormal coordinate 
 —  base flow 
 —  Laplacian 
 —  ‘‘square’’ of the Laplacian 
  

For (based on the centerline velocity and
the channel halfwidth), the discretized version of the OrrSommerfeld
equation is obtained using the pseudospectral scheme with
colocation points in the wallnormal direction. Flow fluctuations with
are then advanced in time using the matrix exponential with
time step and randomly generated initial profile (that
satisfies both homogeneous Dirichlet and Neumann boundary conditions).
After a transient period of ten timesteps, snapshots are
taken to form the snapshot matrices and apply the standard DMD algorithm
along with its sparsitypromoting variant.
