Sample Problems > Applications > Fluids

{ AIRFOIL.PDE

This example considers the laminar flow of an incompressible, inviscid

fluid past an obstruction.

We assume that the flow can be represented by a stream function, PSI,

such that the velocities, U in the x-direction and V in the y-direction,

are given by:

U = -dy(PSI)

V = dx(PSI)

The flow can then be described by the equation

div(grad(PSI)) = 0.

The contours of PSI describe the flow trajectories of the fluid.

The problem presented here describes the flow past an airfoil-like figure.

The left and right boundaries are held at PSI=y, so that U=-1, and V=0.

}

title "Stream Function Flow past an Airfoil"

variables

{ define PSI as the system variable }

psi

definitions

far = 5 { size of solution domain }

psi_far = y { solution at large x,y }

equations { the equation of continuity: }

psi : div(grad(psi)) = 0

boundaries

region 1 { define the domain boundary }

start(-far,-far) { start at the lower left }

{ impose -dy(psi)=U=-1 (outward normal of psi) on the bottom boundary }

natural(psi)= -1

line to (far,-far) { walk the boundary Counter-Clockwise }

natural(psi)=0 { impose dx(psi)=0 on right }

line to (far,far)

natural(psi)=1 { impose dy(psi)=-U=1 on top }

line to (-far,far)

natural(psi)=0 { impose -dx(psi)=0 on left }

line to close { return to close }

start(-0.5,-0.05) { start at lower left corner of airfoil }

value(psi)=0 { specify no flow through the airfoil surface }

arc to (0.0,0.02) to (0.5,0.05) { specify a gentle arc by three points }

arc (center=0.495,0.1) to (0.5,0.15) { a tight arc by two points and center }

arc to (0.075,0.105) to (-0.35,0) { the top arc by three points }

line to close { finally a straight line to close the figure }

monitors{ monitor progress while running }

contour(psi) zoom (-0.6,-0.4,1.4,1.2) as "stream lines"

plots { write hardcopy files at termination }

grid(x,y) zoom (-0.6,-0.4,1.4,1.2)

contour(psi) zoom (-0.6,-0.4,1.4,1.2) as "stream lines" painted

{ show the flow vectors: }

vector(-dy(psi),dx(psi)) zoom (-0.6,-0.4,1.4,1.2)as "flow" norm

surface(psi) zoom (-0.6,-0.4,1.4,1.2) as "stream lines"

end