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Unformatted text preview: Chapter 4 303 Differential Relations for a Fluid Particle Solution: Here we have only the single of California Davis
University ‘onedimensional’ convective acceleration: Department of Mechanical o 2Vo2
du
u
2 x 2V and Aerospace Engineering
x
dt
For L 6 u x Vo 1 L L L 1 L Ans. (a) E du 2(10)2
ft NG 103 Fluid Mechanics
2x
10 ,
1
400(1 4x ), with x in feet
s
dt
6 /12
6 /12
Fall 2011
Professor Kennedy
2 (12 g’s); at x L 0.5 ft, du/dt 1200 ft/s2 (37 g’s). Ans. (b)
400 ft/s
Homework 3 Solutions and Vo At x 0, du/dt P4.3 A twodimensional velocity field is given by
V (x2 – y2 x)i – (2xy y) j in arbitrary units. At (x, y) (1, 2), compute (a) the accelerations ax and ay, (b) the
velocity component in the direction
40 , (c) the direction of maximum velocity, and
(d) the direction of maximum acceleration.
Solution: (a) Do each component of acceleration:
du
u
u
u
v
(x 2 y 2 x)(2x 1) ( 2xy y)( 2y) a x
dt
x
y
dv
v
v
u
v
(x 2 y 2 x)( 2y) ( 2xy y)( 2x 1) a y
dt
x
y
At (x, y) (1, 2), we obtain ax 18i and ay 26j Ans. (a) (b) At (x, y) (1, 2), V –2i – 6j. A unit vector along a 40 line would be n
sin40 j. Then the velocity component along a 40 line is
V40 V n 40 ( 2 i 6 j) (cos 40 i sin 40 j) 5.39 units cos40 i Ans. (b) (c) P4.4 A simple flow model for a twodimensional converging nozzle is the distribution
x
y
_______________________________________________________________________
u
U o (1
)
v
Uo
w0
L
L (a) Sketch a few streamlines in the region 0<x/L<1 and 0<y/L<1, using the method of
Section 1.11. (b) Find expressions for the horizontal and vertical accelerations. Chapter 4 313 Differential Relations for a Fluid Particle ________________________________________________________________________ P4.16 Consider the plane polar coordinate velocity distribution C
K
v
vz
0
r
r
where C and K are constants. (a) Determine if the incompressible equation of continuity
is satisfied. (b) By sketching some velocity vector directions, plot a single streamline for
C = K. What might this flow field simulate?
vr Solution: (a) Evaluate the incompressible continuity equation (4.12b) in polar coordinates:
1
(r v r )
rr 1
r (v ) 1
C
(r )
rr r 1
r ( K
)
r 0 0 0 Ans.(a ) Incompressible continuity is indeed satisfied. (b) For C = K, we can plot a representative
streamline by putting in some velocity vectors and sketching a line parallel to them: 1.2
1.0
0.8
0.6
0.4
0.2
0.0 O  0.2
0.2 314 0.1 Solutions Manual 0.0 0.1 0.2 0.3 0.4 Fluid Mechanics, Fifth Edition The streamlines are logarithmic spirals moving out from the origin. [They have axisymmetry
about O.] This simple distribution is often used to simulate a swirling flow such as a tornado. 4.27
P4.17
An excellent approximation for the twodimensional incompressible
laminar boundary layer on the flat surface in Fig. P4.17 is
u U (2 y 2 y3 y4 3 4 ) for y , where C x1/ 2 , C constant Fig. P4.17 320 Solutions Manual Fluid Mechanics, Fifth Edition Solution: For this (gravityfree) velocity, the momentum equation is
u V
x v Solve for 334 V
y p, or:
2 p
Solutions o (2xy i
Manual o [(2xy)(2yi)
3 ( y 2 )(2xi 2yj)]
p
x 2y ), or:
FluidjMechanics, Fifth Edition
o 2xy 2 p Ans. 4.47 P4.28
Consider the incompressible flow field of Prob. P4.6, with velocity components
y
u = 2y, v = 8x, w = 0. Neglect gravity and assume constant viscosity. (a) Determine whether
Solution: With u 3y and v 2x, we
this flow satisfies the NavierStokes equations. (b) If so, find the pressure distribution p(x, y) if
v/ y 0 0 0, OK.
may check u/ x
tFind the streamlines from u o.
he pressure at the origin is p
/ y 3y
and v – / x 2x. Integrate to find
Solution: In Prob. P4.6 we found the accelerations, so we can proceed to NavierStokes:
51°
32
39°
y x 2 Ans.
x
u
u2
p
p
p
2
v)
gx
u
(u
[ 0 (8 x)(2)]
0 0;
16 x
Fig. P4.47
x
y
x
x
x
Set
0, 1, 2, etc. and plot some
v
v
p
p
2
streamlines at right: flowyaround cornerspof g
v)
v
(u
[(2 )(8) 0]
0 0;
16 y
y
x
y
y
y
halfangles 39 yand 51 . QNoting that
.5
P2 p /( xCy )
4.48
onsider the following twodimensional incompressible flow, which clearly satisfies
0 in both cases, we conclude Yes, satisfies Navier  Stokes. Ans.(a )
continuity: (b) The pressure gradients areUo constant, may easilycintegrate:
u simple, so we v Vo onstant
Find the stream function (r, ) of this flow, that is, using polar coordinates.
p
p
dp
dy , coordinates16 xtream function dy quite easy: y 2 ) const
16 y is
8 ( x2
Solution:dx In cartesian or : p
the s dx
x
y
u
/ y Uo and v – / x Vo or: 2
Uoy – Vox constant
y2 )
Ans.(b)
If p (0, 0) po , then p po 8 ( x
But, in polar coordinates, y rsin and x rcos . Therefore the desired result is This is an exact solution, but it isrnsinBernoulli’s equationstant flow is rotational.
(r, ) Uo ot – Vor cos
c on. The Ans.
P4.29 Consider a the stream function
P4.49 Investigate steady, twodimensional, incompressible flow of a newtonian fluid
with – y2), K constant. u –2xy, v y2 – x2, and w 0. (a) Does this flow satisfy
K(x2 the velocity field Plot the streamlines
cn the full xy mass? (b) Find the stagnation
i onservation of plane, find any pressure field p(x, y) if the pressure at point (x 0, y 0)
ipoints, and ainterpret what the flow could
s equal to p .
represent.
Solution: u y The velocities are given by
2Ky; v x 2Kx This is also stagnation flow, with the streamlines turned 45 from Prob. 4.48. Fig. P4.49 ...
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This note was uploaded on 12/18/2011 for the course ENG 103 taught by Professor Chattot during the Fall '08 term at UC Davis.
 Fall '08
 Chattot
 Aerospace Engineering

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