The Navier-Stokes Equations

The physics of turbulent flows have been discussed by presenting experimental observations and comparing it to laminar flows. In this chapter, the focus will shift to the governing equations of these flow fields.

The Navier-Stokes (NS) equations were named after Claude-Louis Navier and George Gabriel Stokes. These equations govern the flow motion of a fluid and usually consist of the mass conservation equation and the momentum conservation equation. Sometimes, the momentum conservation equations alone represent the Navier-Stokes equations. 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. In the case of incompressible Newtonian fluid the instantaneous continuity equation is the mass conservation equation, which reads in vector notation as: (1)

$\nabla \cdot \vec{u} = 0$

where $\vec{u}$ is the velocity vector.

In tensor notation, the continuity equation is expressed as $\frac{\partial u_{i}}{\partial x_{i}} = 0$ .

The instantaneous momentum equation using vector notation is written as (2)

$\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u} = \frac{- \nabla p}{ρ} + \frac{\nabla \cdot τ}{ρ}$

where

$ρ$ is the fluid density,

$p$ is the pressure and

$τ$ is the viscous stress tensor. In tensor notation, the incompressible momentum equation is given by (3)

$\frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} = \frac{- 1}{ρ} \frac{\partial p}{\partial x_{i}} + \frac{1}{ρ} \frac{\partial τ_{i j}}{\partial x_{j}}$

The left-hand side of the equation describes acceleration, which includes the unsteady 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 over position. This term can be disregarded in one-dimensional flow and Stokes flow (or creeping flow). For the Newtonian fluid, the viscous shear stress is assumed to be proportional to the shear strain rate. Thus, viscous stresses can be obtained by

In vector notation, $τ = μ (\nabla \vec{u} + {(\nabla \vec{u})}^{T}) = μ \nabla^{2} \vec{u}$

In tensor notation, $τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})$