B2.3 Navier-Stokes Equations and Viscous as well as Laminar Flows

The flow of viscous media is described by the Navier-Stokes equations in conjunction with the continuity equation as the equation of momentum conservation:

\frac{\partial (ρ \cdot u_{i})}{\partial t} + \frac{\partial (u_{j} \cdot u_{i})}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} (μ \frac{\partial u_{i}}{\partial x_{j}} + μ \frac{\partial u_{j}}{\partial x_{i}}) - \frac{\partial}{\partial x_{i}} (\frac{2}{3} μ \frac{\partial u_{j}}{\partial x_{j}}) - \frac{\partial p}{\partial x_{i}} + ρ \cdot g_{i}

u_i	Component of the flow velocity
p	Pressure
ρ	Fluid density
μ	Dynamic viscosity
g_i	Component of gravitational acceleration

Navier-Stokes Equations

Only in cases of very low flow velocities can laminar flow be assumed, which can then be solved with minimal computational effort. Direct numerical simulation (DNS) of flows with respect to the Navier-Stokes equations is practically infeasible due to the enormous computational effort, which increases with the cube of the Reynolds number. Therefore, for all practical applications, it is necessary to extend or modify the flow equations. Two fundamental model classes for turbulence modeling have become established, each existing in numerous variants and combinations. The most commonly used turbulence approaches are RANS (Reynolds-Averaged Navier-Stokes) and LES (Large Eddy Simulation).

Parent Chapter

B2. Fluid Mechanical Modeling