Sample Problems > Applications > Fluids > 2d_piston

{ 2D_PISTON_MOVINGMESH.PDE

This problem models the flow of a

perfect gas in a compressor

cylinder. The initial gas pressure

is chosen as 1e-4 Atm, to show

interesting swirling. The

boundaries of the domain are

moved according to the oscillation

of the piston, while the interior

mesh is tensioned within the

moving boundaries. This results

in a mixed Lagrange/Eulerian

model, in that the mesh is moving,

but with different velocity

than the fluid.

}

TITLE "Piston"

COORDINATES

Ycylinder

SELECT

regrid=off { disable the adaptive mesh refinement }

painted { paint all contours }

DEFINITIONS

my_ngrid = 30 { later applied to the NGRID control and smoother }

stroke = 8 { cylinder stroke cm }

rad = 4 { cylinder bore radius cm }

zraise = 1 { raised height of piston center }

rraise = 3*rad/4 { radius or raised piston center }

gap = 2 { piston/head clearance }

gamma = 1.4

rho0 = 0.001

P0 = 100 { initial pressure (dyne/cm2) = 1e-4 Atm }

visc = 0.15 { kinematic viscosity, cm^2/sec }

rpm = 1000 { compressor speed }

period = 60/rpm { seconds }

vpeak = (pi*stroke/period)

{ velocity profile: }

vprofile =vpeak*sin(2*pi*t/period)

{ the piston shape: }

zpiston = if r<rraise then zraise else zraise*(rad-r)/(rad-rraise)

{ the time-dependent piston profile: }

zprofile = zpiston+0.5*stroke*(1-cos(2*pi*t/period))

ztop = stroke+gap+zraise { maximum z postion }

VARIABLES

rho(threshold = rho0/10) { gas density }

u(threshold = vpeak/10) { radial velocity }

v(threshold = vpeak/10) { axial velocity }

P(threshold = P0/10) { pressure }

vm(threshold = vpeak/10) { mesh velocity }

zm=move(z) { mesh position }

DEFINITIONS

{ sound speed }

cspeed = sqrt(gamma*P/rho)

cspeed0 = sqrt(gamma*P0/rho0)

{ a smoothing coefficient: }

smoother = cspeed0*(rad/my_ngrid)

evisc = max(visc,smoother)

SELECT

ngrid= my_ngrid

INITIAL VALUES

rho = rho0

u = 0

v = 0

P = P0

EULERIAN EQUATIONS

{ Eulerian gas equations: (FlexPDE will add motion terms) }

rho: dt(rho) + dr(rho*u*r)/r + dz(rho*v) = smoother*div(grad(rho))

u: dt(u) + u*dr(u)+v*dz(u) + dr(P)/rho = evisc*div(grad(u))-evisc*u/r^2

v: dt(v) + u*dr(v)+v*dz(v) + dz(P)/rho = evisc*div(grad(v))

P: dt(P) + u*dr(P)+v*dz(P) + gamma*P*(dr(r*u)/r+dz(v)) = smoother*div(grad(P))

vm: dzz(vm)=0 { balance mesh velocities in z only }

zm: dt(zm)=vm { node positions - move only in z }

BOUNDARIES

{ use a piston and compression chamber with beveled edge, to create a swirl }

REGION 1

START(0,zraise)

value(u)=0 value(v)=vprofile value(vm)=vprofile dt(zm)=vprofile

line to (rraise,zraise) to (rad,0)

value(u)=0 nobc(v) nobc(vm) nobc(zm)

line to (rad,stroke+gap)

value(u)=0 value(v)=0 value(vm)=0 dt(zm)=0

line to (rraise,ztop) to (0,ztop)

value(u)=0 nobc(v) nobc(vm) nobc(zm)

line to close

{ add a diagonal feature to help control cell shapes at upper corner }

FEATURE start(rraise,zraise) line to (rad,stroke+gap)

TIME 0 TO 2*period by 1e-6

PLOTS

for t=0 by period/120 to endtime

{ control the frame size and data scaling to create a useable movie

- the movie can be created by replaying the .PG7 file and selecting

EXPORT MOVIE, or we could add PNG() commands here to create it directly }

grid(r,z) frame(0,0, rad,ztop)

contour(rho) frame(0,0, rad,ztop) fixed range(0,0.01) contours=50 nominmax

contour(u) frame(0,0, rad,ztop) fixed range(-500,500) contours=50

contour(v) frame(0,0, rad,ztop) fixed range(-550,550) contours=50

contour(P) frame(0,0, rad,ztop) fixed range(0,2600) contours=50 nominmax

vector(u,v) frame(0,0, rad,ztop) fixed range(0,550)

contour(cspeed) frame(0,0, rad,ztop) fixed range(0,600)

contour(magnitude(u,v)/cspeed) frame(0,0, rad,ztop) fixed range(0,1.5)

history(vprofile/vpeak,zprofile/stroke) range(-1,1)

report(vpeak) report(stroke)

history(globalmax(P), globalmin(P))

history(integral(P))

history(globalmax(rho), globalmin(rho))

history(integral(rho))

history(deltat)

END

2d_piston_movingmesh

2d_piston_movingmesh