Entrainment due to turbulent shear flow at the interface of a stably stratified fluid

Abstract

An investigation is made of the entrainment due to turbulent shear flow at the interface of a stably stratified fluid between an upper turbulent layer and a lower non-turbulent layer in which the flow is generated by a disc pump in a recirculating channel, first introduced by Odell and Kovasznay. It is found that, based on friction velocity u_* and on mean velocity U, the plots of the entrainment rates as functions of the corresponding Richardson numbers for two-fluid systems and for linearly stratified systems are in agreement and collapse, so that the two systems follow the same entrainment law. The entrainment rates of the present investigation agree with those of Kantha et al., and for the most part with entrainment rates of Kranenburg. The entrainment rates of Moore and Long are also in agreement with those of present study, as are the heat-stratified entrainment rates of Deardorff and Willis.

Using δb for the buoyancy jump and h for mixed-layer depth, limiting values of Ri_* = hδb/u2* ≅ 2000 and Ri_u = hδb/U² ≅ 20 are obtained experimentally as E_* = u_e/u_e and E_u = u_e/U approach zero and turbulent entrainment ceases, in agreement with an argument of Kantha. Also, limiting values of E_* ≃ 0.3 and E_u ≃ 0.03, are found for small values of Ri_*, based on an argument of Tennekes and Lumley. Although no simple power-law dependence of entrainment rates on Richardson number is obtained for the whole range of Ri_*, it was possible to obtain such dependence when the range 15 < Ri_* < 1500 is divided into 3 smaller subranges.