Application of the two-parametric $k-\omega$ turbulence model for studying the thermal bar phenomenon

In this paper, the phenomenon of the thermal bar in Kamloops Lake (Canada) is studied with a nonhydrostatic mathematical model. A thermal bar is a narrow zone in a lake in temperate latitudes where maximum-density waters sink from the surface to the bottom. Two different turbulence models are compared: the algebraic model of Holland P. R. et al. [1] and the two-equation $k-\omega$ model of Wilcox D. C. [2]. The two-parameter model of turbulence developed by D. C. Wilcox consists of equations for turbulence kinetic energy ($k$) and specific dissipation rate ($\omega$). The mathematical model which includes the Coriolis force due to the Earth's rotation, is written in the Boussinesq approximation with the continuity, momentum, energy, and salinity equations. The Chen–Millero equation [8], adopted by UNESCO, was taken as the equation of state. The formulated problem is solved by the finite volume method. The numerical algorithm for finding the flow and temperature fields is based on the Crank–Nicholson difference scheme. The convective terms in the equations are approximated by a second-order upstream QUICK scheme [10]. To calculate the velocity and pressure fields, the SIMPLED procedure for buoyant flows [11], which is a modification of the well-known Patankar's SIMPLE method [9], has been developed. The systems of grid equations at each time step are solved by the under-relaxation method or N. I. Buleev's explicit method [12]. The turbulence models were applied to predict the evolution of the spring thermal bar in Kamloops Lake. The numerical experiments have shown that the application of the $k-\omega$ turbulence model leads to new effects in the thermal bar evolution.