Analytical Solution of $k\text{-}ϵ$ Model for Nonuniform Flows

An analytical solution of the standard $k\text{-}ϵ$ model is proposed to study the turbulent characteristics of nonuniform flow. Initially, the most suitable functional forms are assumed for turbulent kinetic energy, $k$, turbulence dissipation rate, $ϵ$, and streamwise velocity, $u$, and the coefficients of the power series of all three variables are determined by solving Reynolds equation and the standard $k\text{-}ϵ$ equation for nonuniform flow. Additionally, to investigate the effect of the damping function on velocity and length scale, an analytical solution is further formulated by incorporating the damping function into the basic equations. The appropriateness of the analytical solution is verified with experimental data for the most accelerated flow and the most decelerated flow. The profile of turbulent energy dissipation increased near the free surface when the damping function was included in the analytical solution. Moreover, depending on the nature of the nonuniform flow, the turbulent energy is dissipated more for accelerated flow than for decelerated flow. The eddy viscosity reduced near the free surface due to the pronounced effect of the damping function. Turbulent mixing is also damped in the accelerated flow and amplified in the decelerated flow. Finally, the effective distributions of $k$, $ϵ$, $u$, and eddy viscosity for nonuniform flow ensured the desirability of the functional form presumed for turbulent kinetic energy, turbulence dissipation rate, and velocity.