Navier-Stokes equation

The dynamics of an incompressible Newtonian fluid is determined by the celebrated Navier-Stokes equations (1823), supplemented by the incompressibility condition:

$$\partial_t {\mbox{\boldmath$u$}} + {\mbox{\boldmath$u$}} \cd... ...o}} + \nu \Delta {\mbox{\boldmath$u$}} + {\mbox{\boldmath$f$}}$$ (1.1)

$$\nabla \cdot {\mbox{\boldmath$u$}} = 0$$ (1.2)

where $P$ is the pressure, $\rho$ is the density of the fluid, $\nu = {\mu \over \rho}$ is its kinematic viscosity, and ${\mbox{\boldmath$f$}}$ represents the sum of the external forces per unit mass which sustain the motion.

Let us briefly describe the different terms in Navier-Stokes equation:

• ${\mbox{\boldmath$u$}} \cdot \nabla {\mbox{\boldmath$u$}}$ the inertial, or non-linear term which characterizes Navier Stokes equation, and is responsible for the transfer of kinetic energy in the turbulent cascade.
• $- \nabla P / {\rho}$ the pressure gradients which guarantee the incompressibility of the flow. In absence of external forces they are determined by the Poisson equation
  

  $$\Delta p = - \rho \partial_i \partial_j u_i u_j$$ (1.3)

  
  
  which is obtained taking the divergence of Eq. (1.1).
• $\nu \Delta {\mbox{\boldmath$u$}}$ the dissipative viscous term. It is originated by the Reynolds stresses of the Newtonian fluid, and it is proportional to the viscosity. It is the dominant term in the laminar regime.

The origin of Eqs. (1.1,1.2) is just the conservation of mass and momentum per unit volume:

$${\partial \rho \over \partial t } + \nabla \cdot (\rho {\mbox{\boldmath$u$}}) = 0$$ (1.4)

$$\rho {D u_i \over Dt} = {f_i} + {\partial \mathbb{T}_{ij} \over {\partial x_j}}$$ (1.5)

where $\mathbb{T}$ is the stress tensor of the fluid, which for a Newtonian fluid is linear in the deformation tensor $e_{ij} = 1/2 (\nabla_j u_i + \nabla_i u_j)$, and is given by [4]:

$$\mathbb{T}_{ij}^N = -P\delta_{ij} + \mu \left[ (\nabla_j u_... ...nabla_i u_j) - {2 \over 3} \nabla_k u_k \delta_{ij} \right]\;.$$ (1.6)

The incompressibility assumption is consistent until velocities smaller than speed of sound $c_s$ in the fluid are considered. Since eventual density fluctuations are swept away exactly as sound waves, for small value of the Mach number which measures the ratio between the typical velocities and the speed of sound in the considered fluid, the density can be assumed to be constant in time and space $\rho({\mbox{\boldmath$x$}},t) = \rho$ and the mass conservation (1.4) leads to the divergence-less condition on the velocity field $\nabla \cdot {\mbox{\boldmath$u$}} = 0$. It is common to assume the constant density to be equal to unity, or equivalently to consider dynamical quantities per unit mass of fluid. As an example we will often refer to the square modulus of velocity as kinetic energy.

Because of the presence of a non-linear term in Navier-Stokes equation, the space of its solutions does not have an affine structure, and consequently a generic solution can not be obtained as linear superposition of basic solutions. Moreover, a typical feature of turbulence is the presence of chaos, i.e. the Navier-Stokes equations display a strong sensitivity to initial conditions, which drastically reduces the interest for their exact solutions. For this reason the theory of turbulence has a statistical approach, trying to predict the statistical properties of the flow instead of searching a peculiar analytic solution.