The Navier-Stokes Equations

The Navier-Stokes (NS) equations are the set of equations that govern the motion of a fluid.

These equations consist of the mass conservation equation and the momentum conservation equations for incompressible flows. The NS equations govern the behavior of a viscous fluid by balancing the forces with Newton’s second law and assuming that the stress in the fluid is the sum of a diffusing viscous term and a pressure term. In the case of a Newtonian fluid, the instantaneous continuity equation is defined as (1)

\frac{\partial ρ}{\partial t} + \frac{\partial (ρ u_{j})}{\partial x_{j}} = 0

where $ρ$ is the fluid density and $u_{j}$ is the velocity tensor.

The incompressible momentum equation is given by (2)

\frac{\partial (ρ u_{i})}{\partial t} + \frac{\partial (ρ u_{i} u_{j})}{\partial x_{j}} = - \frac{\partial (p δ_{i j})}{\partial x_{j}} + \frac{\partial τ_{i j}}{\partial x_{j}}

where

ρ

is the pressure,

τ_{i j}

is the viscous stress tensor and

δ_{i j}

is the Kronecker delta. (3)

δ_{i j} = {\begin{matrix} 0 i f i \neq j, \\ 1 i f i = j . \end{matrix}

The left-hand side of the equation describes advection, which includes two terms, the acceleration term and convective term. The right hand side of the equation represents the summation of pressure and the shear-stress divergence terms. The convective term is non linear due to an acceleration associated with the change in velocity as a function of position. This term can be disregarded in one-dimensional flow and Stokes flow, or creeping flow.

For Newtonian fluid the viscous shear stress is assumed to be proportional to the shear strain rate, following Stokes Law. Thus, viscous stresses can be given by (4)

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) - \frac{1}{3} \frac{\partial u_{k}}{\partial x_{k}} δ_{i j}

where $μ$ is the molecular (dynamic) viscosity of the fluid.

Now, you have a full set of the Navier-Stokes equations to govern fluid flows. The next step is how to solve these equations. There are two approaches available: analytical approaches and numerical approaches. Since most flows involve convective accelerations, it is difficult to obtain the analytical solutions of the Navier-Stokes equations due to the non-linearity associated with the convective term and coupled nature of the equations. Although analytical solutions do exist for some simple flow cases with certain approximations, they are mostly limited to laminar flows. These are Couette flow and Hangen-Poiseuille flow. The former is a laminar flow between two parallel plates, one of which is moving relative to the others, and the latter is a laminar flow in a long cylindrical pipe of constant cross section. Therefore, a complete description of a turbulent flow can only be obtained by numerically solving the Navier-Stokes equations (Moin and Mahesh, 1998). These are divided into a numerical approach resolving turbulent flows by Direct Numerical Simulation (DNS) and numerical approaches not resolving turbulence but modeling it.