Temporal non-modal instability of isothermal viscous mixing layer.

The spectrum of the stability problem of unbounded flows consist of a continuous part and a finite number of discrete eigenmodes. Only the discrete spectrum can be obtained correctly from a discrete numerical model. Since our analysis is numerical we have to avoid consideration of the continuous spectrum. To do this we restrict our analysis to the perturbations that grow inside the mixing zone and decay far from it. Details >>

Mathematical formulation and numerical procedure

The problem is described by a system of Orr-Sommerfeld (OS) and Squire equations, that are written for cross-flow velocity and vorticity components. The linear stability problem is reduced to the eigenvalue problem in a usual way. Non-modal growth takes place due to the existence of non-orthogonal eigenvectors. Details >>

Extracting discrete spectrum

We need an independent test to ensure that we did not exclude any important eigenmode. For this we perfomed a comparison of the growth function GE(t) with the evolution of kinetic energy norm ||q(t)||E. Details >>

Results:

Condition for no energy growth

The results show that there is a possibility of transient energy growth in linearly stable mixing layer flows. The variational method has been used for calculation of critical energetic Reynolds number. Details >>

Non-uniqueness of optimal initial perturbation

We obtained several optimal vectors, corresponding to three different techniques, which yield the same energy growth at a given time. We used variational technique and non-modal analysis: SVD or Cholesky decomposition to decompose the Gram matrix. Details >>

Non-modal growth in energy and enstrophy norms

Growth functions

The non-modal growth for both kinetic energy and enstrophy norms is studied for various Re,a and b. Details >>

The results show that the largest transient growth takes place at values of the wavenumbers for which flow is stable. It is found that at Re=20 a=0.7 one observes no non-modal growth in energy norm and starting of a growth in the enstrophy norm. This may be important for understanding and control of mixing at low Re numbers.

Optimal vectors

Profiles of optimal initial vectors for both kinds of norms are compared by means of all velocity and vorticity components. Details >>

Development of the 2D optimal vector in time for a=1.5 and for Re=100, Re=1000, Re=10000. Amplitude development: Re=100, Re=1000.

The most profound growth is attained by oblique three dimensional waves that propagate at the angle of 45⁰ with respect to the base flow. Velocity profiles and developmint of this 3D optimal vector in the time for both norms are here.