# What is appropriate boundary condition for Poisson pressure equation?

I'm doing CFD simulations in unstructured grids. Well, it's a bit different from conventional unstructured grids that are used mainly in FEM or FVM as tetrahedral meshes. Mine is a voxelized mesh of an unstructured surface, which is shown here as an example:

The left image is my unstructured mesh and the right one is the zoom to show the voxels.

As part of a verification for my CFD simulations, which is written based on lattice Boltzmann method (LBM), I want to check the pressure, which I got from ideal gas equation in LBM as:

$$\Delta P = \Delta \rho c_{s}^{2}$$

Where, $$\Delta P$$ is pressure difference ($$\Delta P = P-P_{ref}$$), $$\Delta \rho$$ is density difference ($$\Delta \rho = \rho - \rho_{f}$$), $$\rho_{f}$$ is known fluid density, and finally $$c_{s}$$ is lattice sound wave velocity ($$c_{s} = \hat{c}_{s} \frac{\Delta x}{\Delta t}$$, $$\hat{c}_{s} = \frac{1}{\sqrt{3}}$$, $$\Delta x$$ is mesh size and $$\Delta t$$ is time step).

Conventional way of calculating pressure in FEM or FVM is based on Poisson pressure equation:

$$\nabla^{2} P = - \rho_{f}(\nabla \mathbf{u})^{T} : \nabla \mathbf{u}$$

Where $$\mathbf{u}$$ is velocity of the fluid. I already know the velocity and pressure from LBM, but I want to use just the velocity and put it in Poisson pressure equation to calculate the pressure and compare it to ideal gas equation from LBM. One of the natural Neumann boundary conditions of this Poisson pressure equation is formulated as:

$$\nabla P \cdot \mathbf{n} = \mu \nabla^{2} \mathbf{u} \cdot \mathbf{n}$$

I know this boundary condition is derived from no-slip boundary condition ($$\mathbf{u} = 0$$) at the boundary $$\partial \Omega$$. Also, I know at the outlets, I have the pressures. For example, in the simplest case, I put zero pressure boundary condition at the all outlets by assuming the fact that reference pressure is already considered zero: $$P = 0$$ at $$\partial \Omega_{i}^{outlet}$$, where $$\partial \Omega_{i}^{outlet}$$ is the boundary of $$i$$th outlet. The thing, which I don't know is the pressure boundary condition at the inlet. Initially, in my LBM simulations, I put velocity boundary condition at the inlet and then we could calculate pressure at the inlet plane after simulation is completed. But for extracting pressure from Poisson boundary condition, I don't know what to do at the inlet. I don't want to put calculated pressure from LBM at the inlet cause that somehow will create leakage of pressure information from LBM to this Poisson pressure equation scheme. I can summarize my problem as:

Main equation defined in compuational domain ($$\Omega$$): $$\nabla^{2} P = - \rho_{f}(\nabla \mathbf{u})^{T} : \nabla \mathbf{u}$$

Boundary conditions:

At the wall ($$\partial \Omega$$): $$\nabla P \cdot \mathbf{n} = \mu \nabla^{2} \mathbf{u} \cdot \mathbf{n}$$

At the outlets ($$\partial \Omega_{i}^{outlet}$$): $$P = 0$$

At the inlets ($$\partial \Omega_{i}^{inlet}$$): ?

Update:

I have an idea for finding a pressure boundary condition at the inlet plane. We have Navier-Stokes as:

$$\rho_{f} \frac{\partial \mathbf{u}}{\partial t} + \rho_{f}\mathbf{u} \cdot \nabla \mathbf{u} = -\nabla P + \mu \nabla^{2} \mathbf{u}$$

If we evaluate Navier-Stokes at the inlet boundary and by assuming the fact that at the inlet plane I put Poiseuille flow, we have:

$$\rho_{f} \frac{\partial \mathbf{u}}{\partial t}|_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet} + \rho_{f} \mathbf{u} \cdot \nabla \mathbf{u} |_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet} = -\nabla P |_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet} + \mu \nabla^{2} \mathbf{u} |_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet}$$

Also from Poiseuille flow:

$$\mathbf{u}(\mathbf{r},t)|_{\partial \Omega_{i}^{inlet}} = u_{max}^{i}(t) \Bigg (1 - \frac{|\mathbf{r}-\mathbf{r}_{C}^{i}|^{2}}{R_{i}^{2}} \Bigg ) \mathbf{n}_{i}^{inlet}$$

As a result:

$$\frac{\partial \mathbf{u}}{\partial t}|_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet} = \frac{d u_{max}^{i}(t)}{dt} \Bigg (1 - \frac{|\mathbf{r}-\mathbf{r}_{C}^{i}|^{2}}{R_{i}^{2}} \Bigg )$$

$$\mathbf{u} \cdot \nabla \mathbf{u} |_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet} = 0$$

$$\mu \nabla^{2} \mathbf{u} |_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet} = -\frac{4 \mu u_{max}^{i}(t)}{R_{i}^{2}}$$

Finally:

$$\nabla P |_{\partial \Omega_{i}^{inlet}} \cdot \mathbf{n}_{i}^{inlet} = -\frac{4 \mu u_{max}^{i}(t)}{R_{i}^{2}} - \rho_{f} \frac{d u_{max}^{i}(t)}{d t} \Bigg (1 - \frac{|\mathbf{r}-\mathbf{r}_{C}^{i}|^{2}}{R_{i}^{2}} \Bigg )$$

I appreciate if someone could look at it see if it makes sense or not.