Mixed-layer deepening in stratified fluid has been studied experimentally in mean-shear-free turbulence generated by an oscillating grid. Conditions were varied over a wide range and both two-layered and constant-gradient fluid systems were considered. It is shown that the mixed-layer deepening rate is represented well by power laws, and when local scaling is used all the data can be collapsed on an entrainment relation E = K Ri−n with n = 1.50±0.05 when Ri [gsim ] 7. This power law suggests that the turbulent kinetic energy is made available for mixing on a buoyancy timescale characteristic of eddy recoil or internal-wave breaking rather than a turbulent-eddy overturning timescale. In the constant-gradient situation internal waves are generated which radiate energy away from the interface. An evaluation of the radiated energy indicates, however, that generally energy radiation does not affect the entrainment rate. The coefficient K therefore has the same value (K ≈ 3.8) in linearly stratified fluid as in the two-layer situation. The interface thickness is found to be a function of stability, but reaches an asymptotic value of h/D = 0.055 when Ri is very large. There is some indication that the interface thickness is also a weak function of Reynolds number.

In the cylindrical coordinate system, a singular perturbation theory of multiplescale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rigid circular cylinder, which is subject to a vertical oscillation. It is assumed that the fluid in the circular cylindrical vessel is inviscid , incompressible and the motion is irrotational, a nonlinear amplitude equation with cubic and vertically excited terms of the vessel was derived by expansion of two-time scales withoutconsidering the effect of surface tension. It is shown by numerical computation that different free surface standing wave patterns will be formed in different excited frequencies and amplitudes. The contours of free surface waves are agreed well with the experimental results which were carried out several years ago.

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