Flow past a cylinder - Projection Method - Boundary Conditions

Question

I plan to write a code that solves the flow past a cylinder and try to see the Von-Karman vortex street. I will solve the 2-D viscous, incompressible Navier-Stokes using the projection method. The following figure demonstrates a subset of my intended boundary.

I am sort of confused about the decoupling of velocity and pressure for the projection method. Since we decouple pressure and velocity how should, for example, the boundary condition for pressure be at the surface of the cylinder? The problem is that we have a whole new set of boundary conditions to set. How are they supposed to be determined? Would I not have to know the physics of the flow beforehand to prescribe these conditions?

https://i.sstatic.net/QGYdm.jpg

Answer 1

The answer depends on the physical boundary condition for the velocity field. Since you are using the incompressible Navier-Stokes equations, in contrast to the Euler equations, the condition $\mathbf{u} \cdot \mathbf{n} = 0$ at the cylinder's surface underdetermines the problem. In contrast, a typical boundary condition for viscous flow past a cylinder is $\mathbf{u} = 0$ at the boundary; this is the no-slip condition.

Physically, of course, there is no boundary condition on pressure, but the nature of the projection method requires us to furnish a numerical boundary condition nonetheless. Assuming a no-slip cylinder boundary, the appropriate boundary condition on the pressure update $\phi$ is then $$\frac{\partial \phi}{\partial \mathbf{n}} = 0.$$ In other words, the pressure update $\phi$ should have a Neumann boundary condition across the cylinder boundary.

To see why this is the case, consider the following. In the projection method, given a value $\mathbf{u}^n$ for the velocity field at time step $n$, we typically solve the Navier-Stokes equations using a lagged pressure term to obtain $\mathbf{u}^*$. We then step forward by projecting out the resulting incompressible part, $$\mathbf{u}^{n+1} = \mathcal{P}\mathbf{u}^*.$$ Using the Helmholtz decomposition to write $\mathbf{u}^* = \nabla \phi + \mathbf{v}$, where $\mathbf{v}$ is divergence-free, we can rewrite the update as $$\mathbf{u}^{n+1} = \mathbf{u}^* - \Delta t \nabla \phi^{n+1}.$$ Taking the divergence of both sides and appealing to the Helmholtz decomposition again, we obtain $$\Delta t \Delta \phi^{n+1} = \nabla \cdot \mathbf{u}^*,$$ which is the equation we solve to determine the pressure correction.

At a no-slip boundary, we can take the dot product of both sides of the velocity update with the boundary normal vector $\mathbf{n}$ to obtain $$\mathbf{u}^{n+1}\cdot\mathbf{n} = \mathbf{u}^*\cdot\mathbf{n} - \Delta t \mathbf{n}\cdot\nabla \phi.$$ But since $\mathbf{u}^{n+1}$ necessarily obeys the no-slip boundary condition, we have $\mathbf{u}^{n+1} = 0$ at the boundary. Furthermore, since $\mathbf{u}^*$ was obtained by enforcing the no-slip boundary condition, we have that $\mathbf{u}^* = 0$ at the boundary as well. This leaves us with the constraint that, at the boundary, $$\frac{\partial \phi}{\partial \mathbf{n}} = \Delta t \mathbf{n}\cdot\nabla \phi = 0.$$

For more details, I suggest taking a look at [1]. This paper discusses the projection method in a finite-difference formulation, but the pressure update is handled within a finite element framework, and the analysis should apply to your situation.

[1] A. Almgren, J.B. Bell, and W. Szymczak, A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput. 17 (1996), pp. 258-369.

Answer 2

I'm late to this discussion, but in fact there is a well-posed boundary condition for the pressure in the incompressible Navier-Stokes equation, which is (almost) derived from the equations themselves. For fluid motions subject to the no-slip condition $\mathbf{u}=0$ on the boundary and with body force $\mathbf{f}$, the boundary condition for pressure takes the form $$ \frac{\partial p}{\partial \mathbf{n}} = -\nu \mathbf{n}\cdot( \nu \nabla\times\nabla\times \mathbf{u}) +\mathbf{n}\cdot\mathbf{f}. $$ This boundary condition appears in work on spectral element methods in the 1990s, and recently has been shown to be well-posed. What I mean by this is that (i) it necessarily holds for any
regular solution of the incompressible Navier-Stokes equations, and (ii) the Navier-Stokes equations remain well-posed if one replaces the divergence-free condition $\nabla\cdot\mathbf{u}=0$ by the Poisson equation for the pressure with the boundary condition above.

The fact that the boundary condition is necessary follows from the vector identity $$ -\nabla\times\nabla\times \mathbf{u} = \Delta\mathbf{u}-\nabla\nabla\cdot\mathbf{u}. $$ Of course $\nabla\cdot\mathbf{u}=0$ for the exact Navier-Stokes solution, but that does not mean $\nabla\cdot\mathbf{u}$ can be neglected in the well-posedness theory, or in intermediate steps in projection methods.

A deeper reason for the curl-curl term to appear is that its contribution to the pressure gradient arises exactly from the failure of the Laplacian to commute with the Helmholtz decomposition.

See [1] below for a thorough discussion of these points, how to implement the pressure boundary condition in a finite-element discretization, and a derivation of 3rd-order time-discrete projection methods based on the pressure-Poisson equation with this boundary condition.

[1] J.-G. Liu, J. Liu, R.L. Pego, Stable and accurate pressure approximation for unsteady incompressible viscous flow, J. Comp. Phys. 229 (2010) 3428-3453.

Answer 3

The pressure-like variable of the family of projection approaches is actually a mathematical variable for which the relation to physical pressure depends completely on what projection method you choose as well as what time-integration scheme you have picked for the provisional momentum equation. This means that the 'pressure' boundary condition should not be based on physical arguments. It must instead be formulated to be consistent with your numerics, particularly how you treat the velocity boundary conditions on intermediate steps.

A couple of good references:

David Brown , Ricardo Cortez , Michael Minion, Accurate Projection Methods for the Incompressible Navier-Stokes Equations,JCP, 168, 464–499 (2001).

J.-L. GUERMOND, P. MINEV, J. SHEN, An Overview of Projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, 195 (2006) 6011--6045.