Zero-k mode in Pseudo-spectral solution of Stokes Flow

Question

I'm trying to solve a Stokes flow problem with a pseudo-spectral method in periodic boundary conditions.

The equations of interest are

$-\nabla^2 \bf{v} + \nabla p = \bf{f} \\ \nabla \cdot \bf{v} = 0$

where $\bf{f}$ is a force vector coming from the gradient of a stress tensor, $\bf{f} = \nabla \cdot \sigma$.

To do so, I use a transverse projection operator (i.e. the Green's function/Oseen tensor) in Fourier space as follows

$\bf{\tilde{v}} = \bf{\tilde{T}} \cdot \bf{\tilde{f}}$

where $\bf{\tilde{T}}$ is given by

$\bf{\tilde{T}} = \frac{1}{k^2} \left [ \bf{I} - \frac{\bf{k}\bf{k}}{k^2} \right ]$

My question is: What is the appropriate way to treat the zero k-mode? In the statement above the solution is undefined for $k = 0$. I think I should be able to obtain the volume averaged velocity from this mode, and I can't think off-hand why it shouldn't be well-defined. (Note this problem is similar to this related question).

Answer 1

You are correct that the $k=0$ mode corresponds to the mean (volume averaged) velocity in the domain. Because the equations for Stokes flow are Galilean invariant they are undetermined up to a constant (the mean flow). Usually this ambiguity is resolved by the boundary conditions, but purely periodic boundary conditions offer no such resolution. Instead, you must prescribe the desired mean velocity. Most often this is zero. In fact, all choices of mean velocity are equivalent since they are related by a simple change from one inertial reference frame to another.