Problem formulation

Governing equations: linearized non-dimensional momentum, continuity equations, and z-component of vorticity :

      Here u, v, w are the complex amplitudes of the disturbance velocity in the streamwise (x), spanwise (y) and vertical (z) direction, respectively.

Scales:  length ~ δ_v, velocity ~ U_max, time ~ , pressure ~ 

Reynolds number:  

Perturbation type:

Solving equations: Orr-Sommerfeld and Squire equations for and  :

where   ,  .

The problem is considered for t > 0.

Domain boundaries:    .

Velocity components:

Vorticity components:

Numerical technique

Discretization: 2^th order central finite difference method. The spectrum and the eigenvectors were computed using the QR algorithm.

Computational domain: 2L, where L=20

Grid: N=1100 nodes; grid strongly stretched near the centerline within the interval  and the uniform grid out of this interval. The stretching function used is  , (s=3).

Growth function computation: Euclidean norm of the vector k(t) defined in the eigenvector decomposition:

     where is the Gram matrix and  S=F^HF – not unique decomposition. We calculated it using either Cholesky decomposition or SVD.

     The growth function is defined by the first singular value σ₁ of the matrix :

Optimal initial vector:

where V(1,·) - first right singular vector.

Validation:  the same value of growth function was achived for each of

- SVD+Cholesky

- SVD+SVD

- variational techniques

- ODE solution of initial value linear problem.