Boundary condition for Pressure in Navier-Stokes equation

Question

I am reading the paper, http://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf

I could not really understand the description. Could someone explain a little bit more?

It says, "For the lid driven cavity problem this means that homogeneous Neumann boundary conditions are prescribe everywhere. This implies in particular that the pressure $P$ is only defined up to a constant, which is fine, since only the gradient of $P$ enters the momentum equation."

What does it mean "only defined up to a constant"? By solving the Poisson equation, you obtain the values of $P$ and can calculate its gradient. The constant means this pressure value? And if it's not "only up to constant", what do we get? derivative or integral?

Answer 1

When we say that the pressure is only defined "up to a constant", what we mean is this: If $u,p$ is a solution to the Navier-Stokes equations, then $u,p+c$ for any constant $c$ is also a solution. In other words, the solution is not unique, but the non-uniqueness has a very particular structure: two solutions $u,p$ and $u,p'$ can only differ in a way where $p-p'=const$.

Answer 2

For the lid driven cavity problem, we apply the Dirichlet boundary condition (or, no-slip boundary condition) to the velocity field $\tt U$, that is, $\tt U = U_{\tt Lid}$ on the moving Lid and $\tt U=0$ on the rest of the boundary. Based on this, we should have ${\tt U}^{n+1} = {\tt U}^{n} $ on the boundaries. Hence, if we have $\tt U^{n+1} = U^{n} - \Delta t \nabla P ^{n+1}$ everywhere (as shown in the article) then $$\tt \nabla P ^{n+1} = 0 $$ must be hold on the boundaries (this is so-called the homogeneous Neumann boundary conditions). In addition, if a pressure field $\tt P$ satisfies the momentum equations then $\tt P + const$ also satisfies.