# Manufactured solution for pressure based 3d incompressible Navier-Stokes solver with wall boundaries

I already successfully verified my solver (SIMPLE-type FVM-method) with the following manufactured solution (3d Taylor-Green vortex) on the solution domain $[-1,1]^3$ with Dirichlet boundary conditions:

\begin{align*} u &= \frac{1}{2}\sin\left(\pi x\right)\cos\left(\pi y\right)\cos\left(\pi z\right) \\[0.5em] v &= \frac{1}{2}\cos\left(\pi x\right)\sin\left(\pi y\right)\cos\left(\pi z\right) \\[0.5em] w &= -1\cos\left(\pi x\right)\cos\left(\pi y\right)\sin\left(\pi z\right). \end{align*}

I want to use this manufactured solution also for wall boundary conditions on all boundaries. I think this should be possible because wall boundary conditions are essentially Dirichlet boundary conditions with a vanishing normal velocity $\vec{u}\cdot\vec{n}_w$ and as you will note, my manufactured solution exhibits this behavior.

I would expect the same order for the decrease of the discretization error, but slightly different absolute values since my code doesn't discretize Dirichlet and wall boundary conditions in the same way. This is due to the fact, that on wall boundaries the transfer of momentum only is owed to shear stress. This implies that the wall normal stress vanishes. That additional condition is not automatically fulfilled if one would only apply a Dirichlet boundary condition.

However, I am not able to verify my application code using this manufactured solution with wall boundary conditions. Is this because, despite fulfilling the Dirichlet part (vanishing normal velocity), the additional requirement demanding a vanishing normal stress is not met?

Can you point me to a working manufactured solution for wall boundary conditions that I could use in my pressure based 3d incompressible Navier-Stokes solver (SIMPLE with FVM)?