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{  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.

 

 As the input pressure is raised, the disturbance in velocities propagates farther

 down the channel.  The channel must be long enough that the velocities

 have returned to the open-channel values, or the P=0 boundary condition

 at the outlet will be invalid and the solution will not succeed.

 

 The problem computes half of the domain, with a reflective boundary at the bottom.

 

 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.

 

 We have added a variable psi to compute the stream function for plotting stream lines.

 

}  

 

title 'Viscous flow in 2D channel, Re > 40'  

 

  speed2 = u^2+v^2  

  speed = sqrt(speed2)  

  dens = 1  

  visc = 0.04    vxx = -(p0/(2*visc*(2*Lx)))*(Ly^2-y^2) { open-channel x-velocity }  

 

  rball=0.4  

  cut = 0.1   { value for bevel at the corners of the obstruction }  

 

  penalty = 100*visc/rball^2  

  Re = globalmax(speed)*(Ly/2)/(visc/dens)  

 

  w = zcomp(curl(u,v)) ! vorticity is the source for streamline equation

 

initial values  

  u = 0.5*vxx  v = 0  p = p0*(Lx+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))  

then

  psi:  div(grad(psi)) + w = 0 ! solve streamline equation separately from velocities

 

boundaries  

  region 1  

    start(-Lx,0)  

    load(u) = 0   value(v) = 0 load(p) = 0   value(psi) = 0

      line to (Lx/2-rball,0)  

 

    value(u) = 0 value(v) = 0 load(p) = 0  

    mesh_spacing = rball/10 ! dense mesh to resolve obstruction

      line to (Lx/2-rball,rball) bevel(cut)  

            to (Lx/2+rball,rball) bevel(cut)  

            to (Lx/2+rball,0)  

 

    mesh_spacing = 10*rball ! cancel dense mesh requirement

    load(u) = 0 value(v) = 0 load(p) = 0  

      line to (Lx,0)  

 

    load(u) = 0 value(v) = 0 value(p) = p0 natural(psi) = 0

      line to (Lx,Ly)  

 

    value(u) = 0 value(v) = 0 load(p) = 0 natural(psi) = normal(-v,u)

      line to (-Lx,Ly)  

 

    load(u) = 0 value(v) = 0 value(p) = 0 natural(psi) = 0

      line to close  

 

monitors  

  contour(speed) report(Re)  

  contour(psi) as "Streamlines"

  contour(max(psi,-0.003)) zoom(Lx/2-3*rball,0, 3*rball,3*rball) as "Vortex Streamlines"

  vector(u,v) as "flow"   zoom(Lx/2-3*rball,0, 3*rball,3*rball) norm

 

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)  

  contour(psi) as "Streamlines"

  contour(max(psi,-0.003)) zoom(Lx/2-3*rball,0, 3*rball,3*rball) as "Vortex Streamlines"

  vector(u,v) as "flow"   zoom(Lx/2-3*rball,0, 3*rball,3*rball) norm

 

  Transfer(u,v,p)   ! write flow solution as initial values for Coupled_Contaminant.pde

 

end