ME 7310
Project 4 Vorticity-Stream
F\rnction Method
Due: Mon., Dec. 6, 2010 at 6:00pm Considerthe incompressiblelaminar flow in the plane channelshown below. The dimensionsand the boundary conditions (non-dimensionalized)are as shown on the figure. The flow is governedby vorticity €(r,y,t) and streamfunction {t(r,y,l) transport equations(non-dimensionalized) :
0{ ,0u { , 0 r € _ 1 ( a ' € , a ' { \ -
at- a" - au R"\a"r- W)
A21b
A21b '
0 r2 '
aur -
------:-r
--F
s
where u: # and u : -#. The Reynoldsnumber Re: I0, basedon the channelheight h. Use an explicit FTCS schemeto obtain the steady-sfotesolution of theseequations. Take l/, : 50 and Na :20 grid points along z and g, respectively.Take the initial condition similar to the inlet boundary condition. The time step Al satisfiesthe stability criteria: N (# + ua- , / -< ln" and tr*ae4 ( 1. For your report: \ Ar " ^=) 1. Show the steady-stateu-velocity profile along the channel. 2. Compare the computed steady-statefully-developedu-velocity profile with the analytical solution given by:, u :6(A - A'). 3. Visually inspect the evolution of velocity profile from question 1. Is the entrance length predicted by CFD, consistentwith the classicalformula: L.f hx 0.06Re?
4';,N, : I - 8dt*,rto-r* S;,N*-r T?hi,Nv . : (i'Nv
no-sripwall oudet
u.l,j :1
ur,i : o
€r,y: o zfh,i : Ui
L=5
--+
y *l,f
:0
t _7t b it -8 t h ; , 2 * { s i, s {l'l:W
:
lrjgr*l,j
AN",j
:
?,lryr"-l,J
{rv,,,j :6N,-r,j $rv.,j : d,rr.-t,j
I
*
lllV",j
no-slip',nll
