SCHOOL OF MECHANICAL ENGINEERING SEMINAR Wednesday, June 6, 2012 at 15:00 Wolfson Building of Mechanical Engineering, Room 206

Ph.D. student of Prof. Alex Gelfgat, School of Mechanical Engineering, Tel-Aviv University

This study is focused on computational modeling of non-modal perturbation growth in both isothermal and stratified viscous mixing layer flows. The current work was initiated and aimed to develop robust computational tools to calculate the initial transient kinetic energy growth, observed in several experiments. The numerical non-modal stability analysis is followed by computational modeling of fully non-linear three-dimensional time-dependent solutions, initial conditions for which are taken as calculated optimal vectors. Thus, the non-linear effect of the optimal excitation is also explored.

The other objective is developing and parallelization of the code with a consequent carrying out of high performance computations. The current study revisits factorization of the incompressible Stokes operator linking pressure and velocity. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down.

After reviewing the mathematical tools and numerical techniques required, we present an analysis of transient growth in a mixing layer model with the tanh base velocity profile. Four independent methods are used to calculate the optimal transient energy growth for specified time horizons and Reynolds numbers from 100 up to at 5000 at different streamwise and spanwise wavenumbers. By comparing results of several mathematical approaches, it is concluded that the non-modal optimal disturbances growth results from the discrete part of the spectrum only. Large energy growth is observed at Reynolds numbers close to the onset of linear instability and the optimal perturbations which lead to this growth are determined. Contrarily to the fastest exponential growth, which is two-dimensional, the maximum growth is attained by oblique three-dimensional waves that propagate at the angle with respect to the base flow. This maximum is found to increase with Re. Finally, full three-dimensional DNS with the optimally perturbed base flow confirms the presence of the structures determined by the transient growth analysis. The time evolution of optimal perturbations is presented and exhibit growth and decay of flow structures that sometimes become similar to those observed at late stages of time evolution of the Kelvin-Helmholtz billows.

In light of previous results, we propose an alternative explanation of the three dimensional largest non-modal growth, which seems to be a common property of plane-parallel shear flows. The three dimensional growth mechanism, is reexamined in terms of the role of vortex stretching and the interplay between the span-wise vorticity and the planar divergent components. For high Reynolds numbers the structure of the optimal perturbations in Couette, Poiseuille, and mixing layer shear profiles is robust and resembles localized plane-waves in regions where the background shear is large. The waves are tilted with the shear when the span-wise vorticity and the planar divergence fields are in (out of) phase when the background shear is positive (negative). A minimal model is derived to explain how this configuration enables simultaneous growth of the two fields, and how this mutual amplification reflects on the optimal energy growth. This perspective provides an understanding of the three dimensional growth solely from the two dimensional dynamics on the shear plane.

Computational modeling of two- and three-dimensional non-modal disturbances growth in homogeneous and stratified viscous mixing layer flows

Helena Vitoshkin