V&V » LohCG: Poiseuille flow

Problem setup

The problem domain consists of a rectangular channel with an imposed streamwise pressure gradient. While the flow field is nominally 2D, we compute it in 3D. The length of the channel in the X coordinate direction is 20 units, its height in Y is , and the width in Z is adjusted to contain a single element in the Z direction depending on the mesh resolution.

An example mesh is displayed below. The initial and boundary conditions prescribe a pressure gradient of , thus the analytic solution for the streamwise velocity is

where is the pressure and is the dynamic viscosity which yields the laminar Reynolds number as , defined based on the fluid density and the average streamwise velocity . The flow starts with quiescent initial conditions: . At the Z extremes of the computational domain Dirichlet boundary conditions are set to and no-slip/no-penetration boundary conditions, , are applied at the Y extremes representing the channel walls.

Coarse surface mesh for computing Poiseuille flow, nelem=3K, npoin=1.2K.

Code revision to reproduce

To reproduce the results below, use code revision 291b6f1 and the control file below.

Control file

-- vim: filetype=lua:

print "Poiseuille flow"

-- mesh: poiseuille?tetz.exo

term = 400.0
ttyi = 1000

cfl = 1.0

solver = "lohcg"
flux = "damp4"
stab2 = true
stab2coef = 0.01
soundspeed = 100.0
rk = 4

part = "phg"

mat = { dyn_viscosity = 0.01 }

pressure = {
  iter = 500,
  tol = 1.0e-12,
  --verbose = 1,
  pc = "jacobi",
  bc_dir = { { 1, 2 },
             { 2, 2 } },
  bc_dirval = { { 1, 2.4 },
                { 2, 0.0 } }
}

ic = {
  velocity = { 0.0, 0.0, 0.0 }
}

bc_noslip = {
  sideset = { 3, 4 }
}

bc_dir = {
  { 5, 0, 0, 1, 1 }
}

fieldout = {
  iter = 100000
}

diag = {
  iter = 10,
  format = "scientific",
  precision = 12
}

Run on 32 single CPUs

./charmrun +p32 Main/inciter -i poiseuille3tetz.exo -c poiseuille_lohcg.q -r 10000

Numerical solutions and errors

To estimate the order of convergence, the numerical solution is computed using three different meshes, whose properties of are tabulated below.

Here h is the average edge length.

Plotted below are the computed streamwise velocity profiles sampled in the middle of the domain along a line in the Y direction using three different meshes, obtained after the solution has converged to a stationary state.

The L1 and L2 errors are computed as

where is the number of points sampled across the channel, and are the exact and computed solutions at mesh point , and denotes the volume associated to mesh point .

The tables below display the numerical errors for two different types of damping for the advection term. The order of convergence is estimated using the two finer meshes as

where is a mesh index.

Errors and convergence rates using second-order damping for advection and three-stage Runge-Kutta time stepping.

Errors and convergence rates using fourth-order damping for advection and four-stage Runge-Kutta time stepping.