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{ INACTIVE_VARIABLES.PDE

 

   This example demonstrates the use of variables absent in selected regions.

 

   The problem is a modification of LOWVISC.PDE, in which the bottom half of the channel

   has been filled with a solid.

 

   The fluid equations are declared INACTIVE in the solid region, but a temperature

   equation has been added that is active everywhere.

 

   The bottom of the solid is held at temperature = 0, while the fluid has an incoming

   temperature of 1.

 

   We solve the equations in sequence: first the fluid equations, then the temperature.

 

}  

 

title 'Variables inactive in regions'  

 

variables  

   u(0.1)  

   v(0.01)  

   p(0.1)  

   temp(0.1)  

 

definitions  

  Lx = 5       Ly = 1.5  

  Gx = 0       Gy = 0  

  u0 = 0                       { default initial u-velocity }  

  p0 = 0                       { default initial pressure }  

  pin=2                         { inlet pressure }  

  speed2 = u^2+v^2  

  speed = sqrt(speed2)  

  dens = 1  

  visc = 0.04  

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

  k = 0.1                         { default thermal conductivity }  

 

  rball=0.5  

  cut = 0.1       { bevel the corners of the obstruction }  

 

  penalty = 100*visc/rball^2  

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

 

initial values  

  u = u0   v = 0  p = p0  

 

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  

  temp:  div(k*grad(temp)) - u*dx(temp) - v*dy(temp) = 0  

 

Boundaries  

  { bound the entire region, placing temperature boundary conditions }  

  region 1  

    INACTIVE (u,v,p)       { Inactivate the fluid in this region }  

    start(-Lx,-Ly)  

      value(temp)=0       line to (Lx,-Ly)  

      natural(temp)=0     line to (Lx,0)  

      value(temp)=1       line to (Lx,Ly)   { inlet fluid temp = 1 }  

      natural(temp)=0     line to (-Lx,Ly)  

      natural(temp)=-k*dx(temp)   line to close { outlet diffusive temperature flux }  

 

  { overlay the fluid region onto the total domain, including obstruction,  

       and place fluid boundary conditions }  

  region 2  

     u0 = 0.5*vxx  P0=pin*x/(2*Lx)     { initial values in fluid region }  

     K = 0.01                         { fluid thermal conductivity }  

    start(-Lx,0)  

      value(u)=0 value(v) = 0  

    line to (Lx/2-rball,0)  

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

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

          to (Lx/2+rball,0)  

          to (Lx,0)  

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

    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)  

  contour(u) report(Re)  

  contour(v) report(Re)  

  contour(p) as "Pressure" painted  

  contour(temp)  

 

plots  

  contour(u) report(Re)  

  contour(v) report(Re)  

  contour(p) as "Pressure" painted  

  contour(temp)  

  contour(speed) painted report(Re)  

  vector(u,v) as "flow"   report(Re)  

  contour(dx(u)+dy(v)) as "Continuity Error"  

 

end