fluid_mechanics

4.1 The Velocity Field 149
E XAMPLE 4.1
Velocity Field Representation
GIVEN A velocity field is given by V 1V 0/2 1 xî yĵ2
where and / are constants.
V 0
FIND At what location in the flow field is the speed equal to
V 0 ? Make a sketch of the velocity field for x 0 by drawing arrows
representing the fluid velocity at representative locations.
SOLUTION
The x, y, and z components of the velocity are given by
u V 0 x/, v V 0 y/, and w 0 so that the fluid speed, V, is
V 1u 2 v 2 w 2 2 1 2 V 0
/ 1x 2 y 2 2 1 2
The speed is V V 0 at any location on the circle of radius / centered
at the origin 31x 2 y 2 2 12 /4 as shown in Fig. E4.1a. (Ans)
The direction of the fluid velocity relative to the x axis is given
in terms of u arctan 1vu2 as shown in Fig. E4.1b. For this flow
tan u v u
V 0 y/
V 0 x/
Thus, along the x axis 1y 02 we see that tan u 0, so that
u 0° or u 180°. Similarly, along the y axis 1x 02 we obtain
tan u q so that u 90° or u 270°. Also, for y 0 we
find V 1V 0 x/2î, while for x 0 we have V 1V 0 y/2ĵ,
y
x
(1)
indicating 1if V 0 7 02 that the flow is directed away from the origin
along the y axis and toward the origin along the x axis as
shown in Fig. E4.1a.
By determining V and u for other locations in the x–y plane, the
velocity field can be sketched as shown in the figure. For example,
on the line y x the velocity is at a 45° angle relative to the x axis
1tan u vu yx 12. At the origin x y 0 so that
V 0. This point is a stagnation point. The farther from the origin
the fluid is, the faster it is flowing 1as seen from Eq. 12. By careful
consideration of the velocity field it is possible to determine considerable
information about the flow.
COMMENT The velocity field given in this example approximates
the flow in the vicinity of the center of the sign shown in
Fig. E4.1c. When wind blows against the sign, some air flows
over the sign, some under it, producing a stagnation point as indicated.
y
2V 0
2
V 0
V 0 /2
V
u
θ
v
(b)
V 2V 0
0
0 x
y
−
V 0
V = 0
x
−2
2V 0
(c)
(a)
F I G U R E E4.1