Transient growth in Poiseuille-Rayleigh-Bénard flows of binary fluids with Soret effect

Abstract

The transient growth due to non-normality is investigated for the Poiseuille-Rayleigh-Benard problem of binary fluids with the Soret effect. For negative separation factors such as ψ = −0.1, it is found that a large transient growth can be obtained by the non-normal interaction of the two least-stable-modes, i.e., the upstream and downstream modes, which determine the linear critical boundary curves for small Reynolds numbers. The transient growth is so strong that the optimal energy amplification factor G(t) is up to 102 ~ 103. While for positive separation factors such as ψ = 0.1, the transient growth is weak with the order O(1) of the amplification factor, which can even be computed by the least-stable-mode. However, for both cases, the least-stable-mode can govern the long-term behavior of the amplification factor for large time. The results also show that large Reynolds numbers have stabilization effects for the maximum amplification within moderate wave number regions. Meanwhile, much small negative or large positive separation factors and large Rayleigh numbers can enlarge the maximum transient growth of the pure streamwise disturbance with the wavenumber α = 3.14. Moreover, the initial and evolutionary two-dimensional spatial patterns of the large transient growth for the pure streamwise disturbance are exhibited with a plot of the velocity vector, spanwise vorticity, temperature, and concentration field. The initial three-layer cell vorticity structure is revealed. When the amplification factor reaches the maximum Gmax, it develops into one cell structure with large amplification for the vorticity strength.