{ LOWVISC.PDE }
{ **************************************************************
This example is a modification of the VISCOUS.PDE problem, in which the
viscosity has been lowered to produce a Reynold's number of approximately
40. This seems to be the practical upper limit or Reynolds number for
steady-state solutions of Navier-Stokes equations with FlexPDE.
We have included four elevation plots of X-velocity, at the inlet, channel
center, obstruction center and outlet of the channel. The integrals presented
on these plots show the consistency of mass transport across the channel.
************************************************************** }
title 'Viscous flow in 2D channel, Re > 40'
select errlim = 0.005
variables
u(0.1)
v(0.01)
p(1)
definitions
Lx = 5 Ly = 1.5
Gx = 0 Gy = 0
p0 = 2
speed2 = u^2+v^2
speed = sqrt(speed2)
dens = 1
visc = 0.04
vxx = (p0/(2*visc*(2*Lx)))*(Ly-y)^2 { open-channel x-velocity }
rball=0.25
cut = 0.05 { bevel the corners of the obstruction }
penalty = 100*visc/rball^2
Re = globalmax(speed)*(Ly/2)/visc
initial values
u = 0.5*vxx v = 0 p = p0*x/(2*Lx)
equations
u: visc*div(grad(u)) - dx(p) = dens*(u*dx(u) + v*dy(u))
v: visc*div(grad(v)) - dy(p) = dens*(u*dx(v) + v*dy(v))
p: div(grad(p)) = penalty*(dx(u)+dy(v))
Boundaries
region 1
start(-Lx,0)
load(u) = 0 value(v) = 0 load(p) = 0
line to (Lx/2-rball,0)
value(u)=0 value(v)=0 load(p)= 0
line to (Lx/2-rball,rball) bevel(cut)
to (Lx/2+rball,rball) bevel(cut)
to (Lx/2+rball,0)
load(u) = 0 value(v) = 0 load(p) = 0
line to (Lx,0)
load(u) = 0 value(v) = 0 value(p) = p0
line to (Lx,Ly)
value(u) = 0 value(v) = 0 load(p) = 0
line to (-Lx,Ly)
load(u) = 0 value(v) = 0 value(p) = 0
line to close
monitors
contour(speed)
plots
contour(u) report(Re)
contour(v) report(Re)
contour(speed) painted report(Re)
vector(u,v) as "flow" report(Re)
contour(p) as "Pressure" painted
contour(dx(u)+dy(v)) as "Continuity Error"
elevation(u) from (-Lx,0) to (-Lx,Ly)
elevation(u) from (0,0) to (0,Ly)
elevation(u) from (Lx/2,0) to (Lx/2,Ly)
elevation(u) from (Lx,0) to (Lx,Ly)
end