Poiseuille Flow (Traction Boundaries)

This case verifies the traction boundary-condition implementation in NekRS using fully developed Poiseuille flow in a half-channel. The nondimensional channel height is , and periodic boundary conditions are imposed in the streamwise and spanwise directions. A traction boundary condition is imposed at , while a symmetry boundary condition is imposed at .

Governing equation

For steady, fully developed flow, the streamwise momentum equation reduces to

(1)

where is the streamwise velocity, is the Reynolds number, and is the imposed streamwise pressure gradient.

Analytical solution

The analytical velocity profile is

(2)

The corresponding streamwise body force is

(3)

The wall traction is defined using the outward unit normal component at the lower boundary,

(4)

The symmetry condition at is satisfied because the velocity gradient vanishes at the channel centerline,

(5)

Verification criteria

Two simulations are performed; the first uses the channel geometry described above, whereas the second rotates the geometry by . The accuracy of the simulations is assessed by evaluating the volume-integrated error norm of the velocity field with respect to the analytical Poiseuille solution,

(6)