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Unformatted text preview: _.___. _ _.—a__. M....—_—__ . _. —__._mu—n—n__——I— 8.64 Consider the polar—coordinate velocity potential a = Brl'zcostllﬂ), where s is a
constant. [3) Determine whether via: 0. If so, [b] ﬁnd the associated stream function
Mr, 49} and {c} plot the full streamline which includes the xaxis (6': Ill} and interpret. Solution: {a} It is laborious, but the velocity.r potential satisfies Laplace’s equation
in polar coordinates: is as is 32¢ .
v2 =—— — —— — at: r :3 '3 1.23 A .
(I) r3r[r3r]+r2 3r[392] l $ r “ﬂ ) as {a} {h} This example is one of the lialnilglr of “corner ﬂow" solutions in Eq. {3.49}. Thus:
yr = armsmaea) Ans. {b} [c] This function represents ﬂow around a 15W“ corner, as shown below. Ans. {c} Flg. PB.E4 3.68 Investigate the complex potential ﬁusction ﬁe} = Umz + m ln[[z +a).'[z — all], where
m and a are constants, and interpret the ﬂow pattern. Flg. Pass Solution: This represents flow past a Rankine oval, with stream Function identical to
ﬂsat given by Eq. {3.29}. ”._.—u  ruu———rq._.._.W—u—u._.— 1 "i." ——wvwW"—ﬁ"" 'rr— — —w§_—Qﬁ£méﬂ arm
__.. _ — “:1 : _.__'I'_TE___‘—.+_.,:I.rr{. . _.. _
_ __ it“ #:3513 .itmg_ ....... _ . 5.4552 {H+1}’+‘*2} f _F ' 1}"'”“2}+{H+1 _
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_ 5.4551 5.455?  PPP 5.4555 5.4555 5.4553 2.? .55 2 2.5 .55 2 2.5 3.111} For full}.r developed laminar incompressible ﬂovtr through a straight noncircular
duct, as in Sec. 6.3, the Navier—Stokes Equation {4.33} reduce to 32a 32a: trip —+— =——= const {U 3y” 822 six where (y, a} is the plane of the duct cross section and x is along the duct axis. Itifu‘avitgpr is
neglected. Using a nonsquane rectangular grid (or, av), develop a ﬁnitecliﬂ’erence model
for ﬂﬂs equation, and indicate hov.r it may.r he applied to solve for ﬂow in a rectangular
duct of side lengths n and it. i+l.j+l i1.§,+1 n+1 an] i+1.ji Flg. PE.11D Solution: An appropriate square grid is shown above. The ﬁnitedifference model is “5+1. j _ 211:4 + “t—t ' “5,141 — 2% + uLj—t 1 I111 2 “+—2w——, or, if nv=nz,
to?) trim) Ii dx
1 (as): up
“HZ; “Ig+1+u,J—1+“mg+ul—r,1— I: E Ans. This is “Poisson’s equation,” it looks like the Laplace model plus the constant “source"
term involving the mesh size my} and the pressure gradient and viscosity. ...
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 Spring '08
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