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{  BUOYANT+TIME.PDE

 

 This example is the time-dependent form of the steady-state example

BUOYANT.PDE.

 

 Here we gradually ramp up the heat input to the level given in the

 steady-state problem.

 

 At early times, a single convection cell is established, but at later

 times the bottom of the bowl stagnates and establishes the two-cell

 flow pattern seen in the steady problem.

 

}  

 

TITLE 'Buoyant Flow by Stream Function and Vorticity - no slip'  

 

VARIABLES  

  temp(100)  

  psi(0.001)  

  w(1)  

 

DEFINITIONS  

  Lx = 1   Ly = 0.5  

  Rad = 0.5*(Lx^2+Ly^2)/Ly  

  Gy = 980  

 

  sigma_top = 0.01     { surface heat loss coefficient }  

  sigma_bowl =  1     { bowl heat loss coefficient }  

  k =  0.0004         { thermal conductivity }  

 

  alpha = 0.001       { thermal expansion coefficient }  

  visc = 1  

 

  heatin = min(10,t)  

  t0 = 50  

 

  rho0 = 1  

  rho = rho0*(1 - alpha*temp)  

  cp = 1  

 

  u = dy(psi)  

  v = -dx(psi)  

 

  penalty = 5000  

 

EQUATIONS  

  temp: div(k*grad(temp)) = rho0*cp*(dt(temp) + u*dx(temp) + v*dy(temp))  

  psi:  div(grad(psi)) + w = 0  

  w:    dt(w) + u*dx(w) + v*dy(w) = visc*div(grad(w)) - Gy*dx(rho)  

 

BOUNDARIES  

  region 1  

 

  { on the arc of the bowl, set Psi=0, apply conduction loss to T,

       and apply penalty function to w to enforce no-slip condition. }  

  start(0,0)  

    natural(temp) = -sigma_bowl*temp  

    value(psi) = 0  

    natural(w)=penalty*tangential(u,v)  

    arc (center=0,Rad) to (Lx,Ly)  

 

    { on the top, continue the prior BC for Psi,

       but apply a heat input and loss to T.

       Apply natural=0 BC (no vorticity transport) for w }  

    load(temp) = heatin*exp(-(10*x/Lx)^2) - sigma_top*temp  

    natural(w)=0  

    line to (0,Ly)  

 

    { in the symmetry plane assert w=0, with a reflective BC for T }  

    value(w)=0  

    load(temp) = 0  

    line to close  

 

TIME 0 to 100  

 

MONITORS  

  for cycle=5 { watch what's happening }  

  contour(temp) as "Temperature"  

  contour(psi) as "Stream Function"  

  contour(w)   as "Vorticity"  

  vector(curl(psi)) as "Flow Velocity" norm  

 

PLOTS  

  for t = 1 by 1 to 10 by 10 to endtime  

  grid(x,y)  

  contour(temp) as "Temperature"  painted  

  contour(psi) as "Stream Function"  

  contour(w)   as "Vorticity"  painted  

  vector(curl(psi)) as "Flow Velocity" norm  

  contour(rho) as "Density"  painted  

 

HISTORIES  

history(temp) at (0.1*Lx,Ly) (0.2*Lx,Ly) (0.5*Lx,Ly) (0.8*Lx,Ly)  

        (0.7*Lx,0.5*Ly) (0.04*Lx,0.1*ly) as "Temperature"  

history(u) at (0.1*Lx,Ly) (0.2*Lx,Ly) (0.5*Lx,Ly) (0.8*Lx,Ly)  

        (0.7*Lx,0.5*Ly) (0.04*Lx,0.2*Ly) as "X-velocity"  

history(v) at  (0.04*Lx,0.1*ly) as "Y-velocity"  

 

 

END