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