Stokes Flow
NekRS provides the option to solve unsteady Stokes flow with constant or variable viscosity. The channel case verifies the Stokes flow solver using the Method of Manufactured Solution (MMS). The problem is solved in a quasi-two-dimensional square domain with edge length 2 and an arbitrary user-specified orientation angle .
Governing equations
The nondimensional unsteady Stokes equations for an incompressible fluid with spatially varying kinematic viscosity are
(1)where is the velocity vector, is the hydrodynamic pressure, is the kinematic viscosity, and is a prescribed forcing function. The nonlinear advection term is omitted from the momentum equation, distinguishing Stokes flow from the full incompressible Navier-Stokes equations.
Manufactured solution
The manufactured steady-state velocity solution is
(2)The manufactured viscosity field is
(3)where is the Reynolds number, is a user-specified velocity scale, is a user-specified viscosity scaling parameter, and and are the velocity components in the and directions, respectively. The spatial variation in viscosity introduces an additional contribution to the viscous operator through . The corresponding manufactured forcing function is
(4)Verification criteria
The simulations are performed using a polynomial order of seven and two geometrical configurations. The first configuration uses the original geometry, while the second rotates the geometry by . Errors are evaluated at ; because the initial condition is the exact solution, this time is short enough for the test to run quickly, but long enough such that any code changes or inconsistencies in the implementation would manifest as errors. The volume-integrated velocity error is defined as
(5)For both geometries, the final-time pass criterion requires the errors in both velocity components to be below .
Figure 1 and Figure 2 present separate -refinement studies for the two configurations. These refinement sweeps are not part of the routine regression tests, which use polynomial order ; the figures showcase spectral convergence of the velocity solution and provide additional evidence of the Stokes flow solver's accuracy.
Figure 1: Spectral convergence under p refinement for the original-geometry channel case; the routine regression test uses polynomial order seven.
Figure 2: Spectral convergence under p refinement for the rotated-geometry channel case; the routine regression test uses polynomial order seven.