Stability of a stratified viscous shear flow in a tilted tube

Abstract

The present investigation is concerned with the effects of viscosity on the stability of a bounded stratified shear flow with Prandtl number Pr≫1. Theoretical results obtained from the solution of the Orr–Sommerfeld equation extended to stratified fluids are compared with experiments performed in a tilting tube filled with water and brine. Theoretical analysis shows that a complete stabilization of the flow field with respect to infinitesimal disturbances is attained, irrespective of the Richardson number J, as the Reynolds number Re decreases below 75. This damping action of viscosity is shown to appreciably reduce the critical Richardson number Jc with respect to the inviscid limit Jc=0.25, even at moderately high Re. On the other hand, the destabilizing action enhanced by viscosity through the diffusion of momentum leads to a viscous mode of instability that may develop if J decreases below a threshold value. An extensive series of experiments has been carried out in a long tilting tube in order to verify theoretical results. The agreement between observations and theory is quite satisfactory. Kelvin–Helmholtz waves grow whenever theoretical unstable conditions are attained. The values of measured wavelengths well correspond to maximum growth rate wave numbers. The comparison between theoretical and experimental results also shows that acceleration plays a stabilizing action.