Helix

{ 

   HELIX.PDE 

   This problem shows the capabilities of FlexPDE in hyperbolic systems.

   We analyze a single turn of a helical tube with a programmed flow velocity.

   A contaminant is introduced into the center of the flow on the input surface.

   Contaminant outflow is determined from the flow equations.

   The contaminant concentration should flow uniformly around the helix.

}

title 'Helical Flow: a hyperbolic system.'

select

ngrid=30    regrid=off { Fixed grid works better in hyperbolic systems }

contourgrid=60       { increase plot grid density to resolve peak }

surfacegrid=60

variables

u

definitions

Rin = 1

Rout = 2

R0 = 1.5

dR = 0.3              { width of the input contaminant profile }

gap = 10              { angular gap between input and output faces }

gapr = gap*pi/180     { gap in radians }

cg = cos(gapr)

sg = sin(gapr)

pin = point(Rin*cg,-Rin*sg)

pout = point(Rout*cg,-Rout*sg)

r = magnitude(x,y)

v = 1

vx = -v*y/r

vy = v*x/r

q = 0         { No Source }

sink = 0              { No Sink }

equations

div(vx*u, vy*u) + sink*u + q = 0

boundaries

region 1

   start (Rout,0)

   value(u) = 0        { We know there should be no contaminant on walls  }

       arc(center=0,0) angle=360-gap   { positive angle on outside }

   nobc(u)        { "No BC" on exit plane allows internal solution to dictate outflow }

       line to pin

   value(u)=0

       arc(center=0,0) angle=gap-360   { negative angle on inside }

   value(u)=exp(-((x-R0)/dR)^4)     { programmed inflow is supergaussian }

       line to (1.2,0) to (1.4,0) to (1.6,0) to (1.8,0) to close  { resolve shape }

monitors

contour(u)

plots

contour(u) painted

surface(u)

elevation(u) from (Rin,0.01) to (Rout,0.01)

elevation(u) from (0,Rin) to (0,Rout)

elevation(u) from (-Rin,0.01) to (-Rout,0.01)

elevation(u) from (0,-Rin) to (0,-Rout)

elevation(u) from pout to pin

end