Open and traction boundary conditions for the incompressible Navier–Stokes equations

Viscous incompressible flow simulation using penalty finite element method

“…The accuracy of the solution, however, involves specification of the proper boundary conditions [6][7][8]. Using penalty method neither the pressure nor its normal gradient on the boundaries are known until the velocity field is determined.…”

An unconditionally stable rotational velocity-correction scheme for incompressible flows

“…On the other hand, ample evidences have shown that several types of schemes can work properly with approximation spaces that do not satisfy the usual inf-sup condition (such as with equal-order approximation for the velocity and pressure); see e.g. [16,13,17,12,23,22] for the velocity-correction scheme, consistent splitting scheme, and the scheme with explicit treatment of pressure. Our numerical experiments show that the splitting scheme represented by equations (12) and (13) with spectral element discretizations can work properly with equal-order approximations for the velocity and the pressure.…”

“…Aπ a cos ax cos πy sin bt v = A sin ax sin πy sin bt p = A sin ax sin πy cos bt (22) where A, a and b are prescribed constants. The fluid density is assumed to be a unit value, ρ = 1.…”

“…Dirichlet boundary conditions based on the solution (22) are applied on the boundaries of Ω. Initial condition for the velocity field is given by setting t = 0 to the analytic solution.…”

“…With each ∆t value the flow field is simulated in time from t = 0 to t f , and then at t = t f we compute the L ∞ and L 2 errors of the velocity and the pressure on Ω between the numerical solution and the exact solution given by equation (22). In Figure 1 we plot the errors of the x-velocity component and the pressure as a function of ∆t, computed using the new splitting scheme with J = 2 (see equation 3a).…”