DNS of Navier–Stokes Equation

Abstract

The governing equations of fluid mechanics are all based on identical fundamental classical principles of dynamics, namely conservation of mass, momentum and energy. Based on these principles, Claude-Louis Navier and George Gabriel Stokes, derived the governing equation for viscous fluids by applying Newton’s second law to fluid motion, along with the assumption that stresses arising in the fluids are due to diffusing viscous effects and the pressure gradient. These governing equations are known as Navier–Stokes Equation (NSE). It is to be noted that there is no fixed version of NSE, an appropriate version and the formulation has to be chosen based on the need and the characteristics of the fluid-dynamical problem to be considered. For example, for low-speed applications without heat-transfer, the appropriate version is the incompressible NSE, without considering the energy equation. For such equations, it can be shown that the kinetic energy of the fluid in a control volume is automatically conserved from the momentum equation. For flows dominated by heat-transfer, an additional energy equation is to considered which essentially determines the temperature at each point. Temperature gradient induces gradient in fluid-density, which in turn affects the flow if gravitational effects are also present. In incompressible NSE, such buoyancy effects due to temperature induced density-gradient are generally modeled via Boussinesq approximation (Sengupta TK, Bhaumik S, Bose R (2013). Phys Fluids 25: 094102, [87]). In contrast, when the flow speed is on the higher side and the compressibility effects are quite dominant, the appropriate governing equations are the compressible verisons of NSE, where all the conservation constraints (mass, momentum and energy) are to be considered.