I would like to solve the Navier-Stokes equations for the unsteady problem of the flow around a circular cylinder. I would like to understand how to write mathematically the boundary condition for the pressure on the surface of the cylinder (essentially the circle). The domain of the problem is shown in the image below with the cylinder having a diameter D and centered in the origin.
I usually solve this problem using the CFD solver OpenFOAM, therefore, I will describe the boundary conditions for this problem which are common among OpenFOAM users. At the inlet the velocity $u$ is $(1,0)$ and zero gradient pressure is imposed. On the sides the so-called slip condition is imposed for velocity and zero gradient is used for pressure, while on the outlet we have zero gradient for velocity and zero pressure. In OpenFOAM, zero gradient pressure is often used for the obstacles in the domain such as the cylinder in this case and a zero velocity is assumed. Whenever a zero gradient condition is mentioned, what is meant is zero gradient in the direction perpendicular to the flow (for example on the outlet only the $x$-derivative is equal to zero for the velocity components $U$ and $V$ that is $U_x = V_x = 0$).
For the case of the cylinder boundary, is it correct to assume that the scalar product between the pressure gradient $\nabla p = (p_x,p_y)$ and the normal vector at the boundary $\boldsymbol{n} = \frac{(x,y)}{0.5D}$ has to be zero, in other words:
$$ \nabla p \cdot \boldsymbol{n} = x p_x + y p_y = 0 $$
Is that the correct formulation of the pressure boundary condition ? and is there another physical condition for the pressure at the cylinder boundary ?
