Example : Viscous Flow Script Comments

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This example shows the application of FlexPDE to problems in viscous flow.

The Navier-Stokes equation for steady incompressible flow in two cartesian dimensions is

dens*(dt(U) + U*dx(U) + V*dy(U)) = visc*del2(U) - dx(P) + dens*Fx
dens*(dt(V) + U*dx(V) + V*dy(V)) = visc*del2(V) - dy(P) + dens*Fy

together with the continuity equation

div[U,V] = 0,

where U and V are the X- and Y- components of the flow velocity, P is the fluid pressure, dens is the fluid density, visc is the fluid viscosity, and Fx and Fy are the X- and Y- components of the body force.

In order to derive a third equation for the Pressure variable, we differentiate the U-equation with respect to X and the V-equation with respect to Y. Using the continuity equation to eliminate terms, we get

del2(P) = 2*dens*(dx(U)*dy(V) - dy(U)*dx(V))

Although this equation is consistent with the continuity equation, it does not enforce it. However, since div[U,V] = 0, we are free to add it at will to the pressure equation. A negative value of div[U,V] implies the destruction of material, so we need a positive pressure to oppose the flow. This implies a modified pressure equation

del2(P) = 2*dens*(dx(U)*dy(V) - dy(U)*dx(V)) + L*(dx(U)+dy(V))

where L is a "large" number chosen to enforce "sufficient" compliance with the material conservation equation.

Setting U and V equal to zero in the U and V equations to reflect the conditions on a no-slip boundary, we get

dx(P) = visc*del2(U)

dy(P) = visc*del2(V)

These relations can be used to specify the natural boundary condition for the pressure equation. The normal component of the gradient of P is

n<dot>grad(P) = nx*dx(P) + ny*dy(P)

where nx and ny are the direction cosines of the surface normal.

The problem posed here shows flow in a 2D channel blocked by a bar of square cross-section. The viscosity has been chosen to produce a Reynold's number of approximately 20. This seems to be the practical upper limit or Reynolds number for steady-state solutions with FlexPDE.

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.

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