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# buoyant+time

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# buoyant+time

{  BUOYANT+TIME.PDE

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

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

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

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)

{ 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 }

natural(w)=0

line to (0,Ly)

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

value(w)=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