Navier–Stokes and Euler

Run the momentum lab to see how pressure, viscosity and wall motion shape a velocity profile.

Governing equations

Ansys CFX formulates the unsteady Navier–Stokes equations as conservation laws. To see the transport terms clearly, start with a constant-density Newtonian fluid with constant dynamic viscosity. Mass conservation gives $\nabla\cdot\mathbf u=0$ . The momentum balance for velocity component $u_k$ is

\frac{\partial(\rho u_k)}{\partial t} +\frac{\partial}{\partial x_i} \left(\rho u_i u_k-\mu\frac{\partial u_k}{\partial x_i}\right) =-\frac{\partial p}{\partial x_k}+F_k.

$u_k$ is the velocity component whose momentum is being transported. Write one equation for each direction $k$ .
$u_i$ is the velocity component carrying momentum in direction $i$ . The repeated index $i$ is summed over the three spatial directions.
$\rho$ is density, $\mu$ is dynamic viscosity, and $p$ is pressure. The pressure gradient $-\partial p/\partial x_k$ gives the pressure force per unit volume in direction $k$ .
$F_k$ is a body force per unit volume, such as $\rho g_k$ for gravity.

The terms on the left describe local accumulation and net flux. Inside the flux, $\rho u_i u_k$ is advection and $-\mu\,\partial u_k/\partial x_i$ is viscous diffusion for these assumptions.

General transport equation

The same accumulation–flux–source structure can describe transport of a scalar $\phi$ , for example a species mass fraction when the diffusive flux is modeled with a scalar coefficient.

\frac{\partial(\rho\phi)}{\partial t} +\nabla\cdot\left(\rho\mathbf u\phi-\Gamma_\phi\nabla\phi\right) =S_\phi.

Here $\mathbf u$ is the velocity vector, $\Gamma_\phi$ is the diffusion coefficient in this flux form, and $S_\phi$ is the net source per unit volume. The divergence acts on the full vector flux. In index form the flux in direction $i$ is $\rho u_i\phi-\Gamma_\phi\,\partial\phi/\partial x_i$ .

For a steady solution the local accumulation term is zero.

\frac{\partial(\rho\phi)}{\partial t}=0.

The flux and source terms can still vary with position. A steady RANS calculation applies this steady assumption to the averaged equations. The underlying turbulent flow can still fluctuate.

Allow density and temperature to change

For compressible flow, retain the full viscous stress tensor in the momentum equation. Without body forces or external heating, the conservation equations in vector form are

\begin{aligned} \partial_t\rho+\nabla\cdot(\rho\mathbf u)&=0,\\ \partial_t(\rho\mathbf u)+\nabla\cdot(\rho\mathbf u\otimes\mathbf u+p\mathbf I)&=\nabla\cdot\boldsymbol\sigma_v,\\ \partial_t(\rho E)+\nabla\cdot[(\rho E+p)\mathbf u]&=\nabla\cdot(\boldsymbol\sigma_v\cdot\mathbf u-\mathbf q). \end{aligned}

$E=e+|\mathbf u|^2/2$ is total energy per unit mass, $\boldsymbol\sigma_v$ is viscous stress and $\mathbf q$ is heat flux. Close the thermodynamics with an equation of state, for example $p=(\gamma-1)\rho e$ for a calorically perfect gas, and specify material and transport laws. Compressible equations are particularly important when shocks or significant density changes enter the design question.

The inviscid limit

The compressible Euler equations follow by dropping viscous stress and heat conduction. They can represent shocks and inviscid pressure loading. Predictions that depend on skin friction, transition or boundary-layer separation require additional physics. Potential flow goes further by assuming irrotational flow and is useful for some preliminary loading estimates.

How the rest of the library connects

RANS averages the equations, LES filters them, and DNS resolves their turbulent scales directly. Finite volume, finite element and related schemes are ways to discretize equations. A CFD package can offer several combinations of physics and numerics.

Governing equations

General transport equation

Allow density and temperature to change

The inviscid limit

How the rest of the library connects

Sources