5.3. Navier-Stokes equation

The Navier-Stokes equation is the same as the Stokes equation in Section 4.4, but with the addition of the inertia term \(\partial_t \vec{u}+\vec{u}\cdot\nabla \vec{u}\). Again, we have to make sure to meet the inf-sup criterion, i.e. we select a Taylor-Hood formulation:

from pyoomph import *
from pyoomph.expressions import *


class NavierStokesEquations(Equations):
     def __init__(self,rho,mu):
             super(NavierStokesEquations, self).__init__()
             self.rho, self.mu= rho,mu # Store viscosity and density

     def define_fields(self):
             self.define_vector_field("velocity","C2") # Taylor-Hood pair
             self.define_scalar_field("pressure","C1")

     def define_residuals(self):
             u,v=var_and_test("velocity") # get the fields and the corresponding test functions
             p,q=var_and_test("pressure")
             stress=-p*identity_matrix()+2*self.mu*sym(grad(u)) # see Stokes equation
             inertia=self.rho*material_derivative(u,u) # lhs of the Navier-Stokes eq. rho*Du/dt
             self.add_residual(weak(inertia,v)+ weak(stress,grad(v)) + weak(div(u),q))

The inertia term is obtained by using the function material_derivative(). Calling this function with any scalar, vectorial or tensorial expressions \(F\) and the velocity \(\mathbf{u}\) will return \(\partial_t F+\operatorname{grad}(F)\cdot\mathbf{u}\).

Warning

The trivial implementation of the Navier-Stokes equation does not allow for high Reynolds numbers. The reason is similar to the problem discussed with the advection-diffusion equation in the previous example. There are improved schemes using SUPG and pressure stabilization to accurately account for the term \(\vec{u}\cdot\nabla\vec{u}\).

We discuss a few examples of the Navier-Stokes equation in the following sections.