Analytic Investigation of the Increase in Kinetic Energy at the Shear-Free Constant-Pressure Boundary of a Fluctuating Flow

Abstract

Physical and numerical experiments and computations reported in the literature conclude that the kinetic energy of free-survace bounded turbulent flows appears to increase as the free surface is approached from below. There is in the previous literature no satisfactory explanation of the mechanism underlying this feature. In order to provide insight revealing the mechanism, an analytic investigation of an oscillating laminar flow bounded on the lower side by a horizontally moving no-slip wall and on the upper side by a shear-free constant-pressure boundary is presented herein. Although the motion is periodic, rather than turbulent, essential features including momentum diffusion are present and hence the flow is a valid candidate for indicating the mechanism. A mathematical solution for the velocity and the kinetic energy is analyzed to determine the physical mechanism leading to the increase in kinetic energy. The finding is that the relaxation of the shear stress at the free surface produces enhanced viscous acceleration of the fluid in a thin layer adjacent to the free surface. The enhanced acceleration produces a nearly constant kinetic energy in the layer. It is concluded that the effect of the shear-free constant-pressure boundary of a fluctuating flow is to produce a nearly uniform flow in a thin layer adjacent to the surface, and that the effects of the free surface penetrate far into the bulk fluid with exponentially decreasing significance.