Exercise 5

The Navier-Stokes equation for an incompressible fluid,

 

$$\partial_t {\bf u} = - {\bf u}\cdot \nabla {\bf u} - \nabla p + \nu \nabla^2 {\bf u} + {\bf f}$$ (1)

$$\nabla \cdot {\bf u} =0,$$

is considered in a periodic box. The spectral representation of the velocity field is then,

$${\bf u}({\bf x})=\sum_{\bf k} {\bf u}_{\bf k} e^{i {\bf k} \cdot {\bf x}}.$$ (2)

The velocity field is a real field so we have ${\bf u}^*_{\bf k}={\bf u}_{-\bf k}$.

(1) Show that the incompressibility implies ${\bf u}_{\bf k} \cdot {\bf k} =0$ for all k.

(2) Express the advection term in spectral representation.

Using the incompressibility we get a Poisson equation for the pressure field from taking the divergence of (1),

$$-\Delta p = \nabla \cdot ({\bf u} \cdot \nabla){\bf u}.$$ (3)

(3) Use the spectral representation for the advection term to solve this equation for p.

(4) Finally, derive the spectral form of the Navier-Stokes equation.