Consider the following steady, incompressible, two-dimensional velocity field: V → = ( u , v ) = ( 4.35 + 0.656 x ) i → + ( − 1.22 − 0.656 y ) j → Generate an analytical expression for the flow streamlines and draw several streamlines in the upper-right quadrant from x = 0 to 5 and y = 0 to y.
Consider the following steady, incompressible, two-dimensional velocity field: V → = ( u , v ) = ( 4.35 + 0.656 x ) i → + ( − 1.22 − 0.656 y ) j → Generate an analytical expression for the flow streamlines and draw several streamlines in the upper-right quadrant from x = 0 to 5 and y = 0 to y.
Solution Summary: The author explains the analytic expression for the flow streamline and draw several streamlines.
Consider the following steady, incompressible, two-dimensional velocity field:
V
→
=
(
u
,
v
)
=
(
4.35
+
0.656
x
)
i
→
+
(
−
1.22
−
0.656
y
)
j
→
Generate an analytical expression for the flow streamlines and draw several streamlines in the upper-right quadrant from x = 0 to 5 and y = 0 to y.
1. Stagnation Points
A steady incompressible three dimensional velocity field is given by:
V = (2 – 3x + x²) î + (y² – 8y + 5)j + (5z² + 20z + 32)k
Where the x-, y- and z- coordinates are in [m] and the magnitude of velocity is in [m/s].
a) Determine coordinates of possible stagnation points in the flow.
b) Specify a region in the velocity flied containing at least one stagnation point.
c) Find the magnitude and direction of the local velocity field at 4- different points that located at equal-
distance from your specified stagnation point.
2. Consider the two-dimensional time-dependent velocity field u(x, t) = (sint, cost, 0),
in the basis of Cartesian coordinates.
a) Determine the streamlines passing through the point x = 0 at the times t =
0, π/2, π and 3π/2.
b) Determine the paths of fluid particles passing through the point x = 0 at the
same times, to = 0, π/2, 7 and 37/2. Hence, describe their motion.
ㅠ
c) Find the streakline produced by tracer particles continuously released at the point
xo = 0 and find its position at t = 0, π/2, π and 37/2. Hence describe its motion.
1. For a velocity field described by V = 2x2i − zyk, is the flow two- or threedimensional? Incompressible? 2. For an Eulerian flow field described by u = 2xyt, v = y3x/3, w = 0, find the slope of the streamline passing through the point [2, 4] at t = 2. 3. Find the angle the streamline makes with the x-axis at the point [-1, 0.5] for the velocity field described by V = −xyi + 2y2j
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