I believe that the question cannot be answered without considering the numerical method that you are using. The exact solution to your flow problem will (of course) exactly satisfy the equations and the boundary conditions, but depending on your numerical method this may or may not be true for your numerical solution. If you are using a pressure-correction projection scheme, then your corrected (divergence-free) solution will not satisfy the boundary conditions, see the discussion on page 6016 in the paper by Guermond. Also, if you are using a method where the boundary conditions are only weakly imposed (for example a FEM formulation), then there will probably be a non-zero residual at the boundaries.

Article An overview of projection methods for incompressible flows

Dear Aditya Ghantasala 

Generally, the residual should be calculated in the domain, not at the boundaries, especially for the pre-defined boundary value problems like Dirichlet (the first type) boundary conditions. It holds because the governing equations are valid throughout the domain, not at the boundaries.

Cheers

Thank you for your comment Maziar Dehghan.

Yes for convergence checking during the simulation, I only calculate and check for residual inside the domain (not on the Domain).

And I did not understand "It holds because the governing equations are valid throughout the domain, not at the boundaries." 

1. What holds ?

2. So you mean to say that governing equations are valid throughout the domain except on the boundaries. This means as I mentioned, that the pressure calculated and velocity specified on a velocity Dirichlet boundary will not satisfy the Navier-Stokes equations that is both momentum and continuity? I am trying to understand the physical implication of such a conclusion.

Yes. Your statement is right.

May I ask something Maziar?

Boundary conditions for any ODE/PDE is part of the solution according to Hamilton theorem(if I'm not making any mistake). So it means that governing equation is valid on the boundary. Am I Right?

Although as Maziar said, you should calculate the residual on the domain only. Otherwise it is wired!!!. Because at every iteration we specify the values on the boundary (despite of boundary condition type) and these values are not changing.

I believe that the question cannot be answered without considering the numerical method that you are using. The exact solution to your flow problem will (of course) exactly satisfy the equations and the boundary conditions, but depending on your numerical method this may or may not be true for your numerical solution. If you are using a pressure-correction projection scheme, then your corrected (divergence-free) solution will not satisfy the boundary conditions, see the discussion on page 6016 in the paper by Guermond. Also, if you are using a method where the boundary conditions are only weakly imposed (for example a FEM formulation), then there will probably be a non-zero residual at the boundaries.

Article An overview of projection methods for incompressible flows

Dear Navid

You're absolutely right for an exact solution (despite the name of the theory), but a numerical solution should be treated with more considerations as Mr. Ludvig af Klinteberg said. depending on the boundary condition type and the solution procedure, the residual may be computed or neglected at the boundaries. For example, for the second-type of boundary conditions (the Neumann BC) equating the gradient at the boundary equal to zero or for the third type (the Robin BC), the residual at the boundaries should be calculated, since the exact value is not known priorly.

Generally, I prefer to neglect the residual at the boundaries, to avoid any further complications, since it depends on the boundary condition and the solution procedure, and almost doesn't play a key-role.

Ghorbanat

The boundary condition for Navier-Stokes equarions is zero velocity at the contact with solid surface at rest. 

Navier-Stokes equations are valid everywhere, in the domain and also on the surfaces and boundaries, the residuals however could be zero or non-zero (at the boundary faces) depending on the type of the boundary conditions you specify, for example for a subsonic inlet, you can specify up to 4 parameters  (e.g. total pressure, temperatures and flow directions, but static pressure is computed from the inlet domain and iterated, during the iterations, the residuals would be non-zero until the flux balances are reached everywhere (which is the function of the losses in the system, e.g., the shape of  a body or a nozzle determine this). Similar situation on the solid surfaces although for no slip, velocities are zero, but you may want to specify the temperature (in heat conduction cases), hence the Dirichlet boundary condition or you may choose to compute zero heat flux, i.e. adiabatic wall boundary condition, Neumann boundary condition. Still, residual will not be zero on the wall because static pressure will be iterated on and depends on the flow domain (adjacent to the surface).  Also for a supersonic inlet, you should specify ALL your primitive boundary conditions, hence the residual for those faces of the inlet boundary condition would be zero and the values won’t change during the CFD iterations.

The residual is calculated where the equations are discretized. Although you can discretize the boundaries of your problem, it is not usual. So, the residual at the boundaries is not calculated and only the residual at the field is importante to evaluate the convergence of the solver.