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2D V
V
≠ ( ) t Steady
(5) V
V
= ( ) x 1D^ V
V
≠ ( ) t Steady
(6) V
V
= ( x y, ,z) 3D^ V
V
= ( ) t Unsteady
(7) V
V
= ( x y, ,z) 3D^ V
V
≠ ( ) t Steady
(8) V
V
= ( x y, ) 2D^ V
V
= ( ) t Unsteady
Problem 2.
For the velocity fields given below, determine:
(a) whether the flow field is one-, two-, or three-dimensional, and why.
(b) whether the flow is steady or unsteady, and why.
(The quantities a and b are constants.)
Solution
(1) V
V
= ( ) x 1D^ V
V
= ( ) t Unsteady
(2) V
V
= ( x y, ) 2D^ V
V
≠ ( ) t Steady
(3) V
V
= ( ) x 1D^ V
V
≠ ( ) t Steady
(4) V
V
= ( x z, )
See the plots in the corresponding Excel workbook
y c x
For t = 20 s = ⋅
y
c
x
For t = 1 s =
For t = 0 s y =c
y c x
−b
a
⋅t
The solution is = ⋅
ln y( )
−b ⋅t
a
Integrating = ⋅ln x( )
dy
y
− b⋅t
a
dx
x
So, separating variables = ⋅
v
u
dy
dx
−b ⋅ t⋅y
a x⋅
For streamlines =
Solution
V = axi ˆ^ − bty ˆ j
r
A velocity field is given by
where a = 1 s-1^ and b = 1 s-2^. Find the equation of the streamlines at any time t. Plot several
streamlines in the first quadrant at t = 0 s, t = 1 s, and t = 20 s.
Problem 2.
Problem 2.4 (In Excel)
A velocity field is given by
where a = 1 s -1^ and b = 1 s -2^. Find the equation of the streamlines at any time t. Plot several streamlines in the first quadrant at t = 0 s, t =1 s, and t =20 s.
Solution
t = 0 t =1 s t = 20 s (### means too large to view) c = 1 c = 2 c = 3 c = 1 c = 2 c = 3 c = 1 c = 2 c = 3 x y y y x y y y x y y y 0.05 1.00 2.00 3.00 0.05 20.00 40.00 60.00 0.05 ##### ##### ##### 0.10 1.00 2.00 3.00 0.10 10.00 20.00 30.00 0.10 ##### ##### ##### 0.20 1.00 2.00 3.00 0.20 5.00 10.00 15.00 0.20 ##### ##### ##### 0.30 1.00 2.00 3.00 0.30 3.33 6.67 10.00 0.30 ##### ##### ##### 0.40 1.00 2.00 3.00 0.40 2.50 5.00 7.50 0.40 ##### ##### ##### 0.50 1.00 2.00 3.00 0.50 2.00 4.00 6.00 0.50 ##### ##### ##### 0.60 1.00 2.00 3.00 0.60 1.67 3.33 5.00 0.60 ##### ##### ##### 0.70 1.00 2.00 3.00 0.70 1.43 2.86 4.29 0.70 ##### ##### ##### 0.80 1.00 2.00 3.00 0.80 1.25 2.50 3.75 0.80 86.74 ##### ##### 0.90 1.00 2.00 3.00 0.90 1.11 2.22 3.33 0.90 8.23 16.45 24. 1.00 1.00 2.00 3.00 1.00 1.00 2.00 3.00 1.00 1.00 2.00 3. 1.10 1.00 2.00 3.00 1.10 0.91 1.82 2.73 1.10 0.15 0.30 0. 1.20 1.00 2.00 3.00 1.20 0.83 1.67 2.50 1.20 0.03 0.05 0. 1.30 1.00 2.00 3.00 1.30 0.77 1.54 2.31 1.30 0.01 0.01 0. 1.40 1.00 2.00 3.00 1.40 0.71 1.43 2.14 1.40 0.00 0.00 0. 1.50 1.00 2.00 3.00 1.50 0.67 1.33 2.00 1.50 0.00 0.00 0. 1.60 1.00 2.00 3.00 1.60 0.63 1.25 1.88 1.60 0.00 0.00 0. 1.70 1.00 2.00 3.00 1.70 0.59 1.18 1.76 1.70 0.00 0.00 0. 1.80 1.00 2.00 3.00 1.80 0.56 1.11 1.67 1.80 0.00 0.00 0. 1.90 1.00 2.00 3.00 1.90 0.53 1.05 1.58 1.90 0.00 0.00 0. 2.00 1.00 2.00 3.00 2.00 0.50 1.00 1.50 2.00 0.00 0.00 0.
V = axi ˆ^ − bty j ˆ
r
The solution is (^) y c x
−b a
⋅t = ⋅
For t = 0 s (^) y =c
For t = 1 s (^) y c x
For t = 20 s (^) y =c x⋅− 20
See the plot in the corresponding Excel workbook
y
c
x
The solution is =
y c x
b
a
= ⋅ c x
ln y( ) = ⋅
b
a
Integrating = ⋅ln x( )
dy
y
b
a
dx
x
So, separating variables = ⋅
v
u
dy
dx
b x⋅ ⋅y
a x
b y⋅
a x⋅
For streamlines =
v − 6
m
s
v = b x⋅ ⋅y − 6 = ⋅
m s⋅
⋅ × 2 ⋅ m
= × ⋅m
u 8
m
s
u a x = ⋅
m s⋅
⋅ ( 2 m⋅ )
= ×
At point (2,1/2), the velocity components are
The velocity field is a function of x and y. It is therefore 2D
Solution
V = ax^2 i ˆ+ bxy j ˆ
r
A velocity field is specified as
where a = 2 m-1^ s-1^ and b = - 6 m -1^ s-1^ , and the coordinates are measured in meters. Is the
flow field one-, two-, or three-dimensional? Why? Calculate the velocity components at the
point (2, 1/2). Develop an equation for the streamline passing through this point. Plot several
streamlines in the first quadrant including the one that passes through the point (2, 1/2).
Problem 2.
See the plot in the corresponding Excel workbook
y
x + 2
y
x
C y x
B
A
+
+
For the streamline that passes through point ( x , y ) = (1,2)
y
C
x
B
A
The solution is
A
− ln y( )
A
ln x
B
A
+
Integrating = ⋅
dy
−A ⋅y
dx
A x⋅ +B
So, separating variables =
v
u
dy
dx
−A ⋅y
A x⋅ +B
Streamlines are given by =
Solution
Problem 2.
Problem 2.7 (In Excel)
Solution
A = 10
B = 20
C =
x y y y y
Streamline Plot
0.0 0.5 1.0 1.5 2. x
y
c = 1 c = 2 c = 4 c = 6 ((x,y) = (1.2)
The solution is
y
C
x
B
A
Problem 2.8 (In Excel)
Solution
a = 1
b = 1
C =
x y y y y
Streamline Plot
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2. x
y
c = 0 c = 2 c = 4 c = 6
The solution is y
b
a x⋅
+C
See the plots in the corresponding Excel workbook
y C
x
For t = 20 s = −
y C 4 x
For t = 1 s = − ⋅
For t = 0 s x =c
y C
b x
a t⋅
The solution is = −
⋅a ⋅t y
− ⋅ bx
Integrating = ⋅ +C
So, separating variables a t⋅ ⋅y ⋅ dy=− b⋅x ⋅dx
v
u
dy
dx
−b ⋅x
a y⋅ ⋅t
Streamlines are given by =
Solution
Problem 2.
Problem 2.11 (In Excel)
Solution
t = 0 t =1 s t = 20 s C = 1 C = 2 C = 3 C = 1 C = 2 C = 3 C = 1 C = 2 C = 3 x y y y x y y y x y y y 0.00 1.00 2.00 3.00 0.000 1.00 1.41 1.73 0.00 1.00 1.41 1. 0.10 1.00 2.00 3.00 0.025 1.00 1.41 1.73 0.10 1.00 1.41 1. 0.20 1.00 2.00 3.00 0.050 0.99 1.41 1.73 0.20 1.00 1.41 1. 0.30 1.00 2.00 3.00 0.075 0.99 1.41 1.73 0.30 0.99 1.41 1. 0.40 1.00 2.00 3.00 0.100 0.98 1.40 1.72 0.40 0.98 1.40 1. 0.50 1.00 2.00 3.00 0.125 0.97 1.39 1.71 0.50 0.97 1.40 1. 0.60 1.00 2.00 3.00 0.150 0.95 1.38 1.71 0.60 0.96 1.39 1. 0.70 1.00 2.00 3.00 0.175 0.94 1.37 1.70 0.70 0.95 1.38 1. 0.80 1.00 2.00 3.00 0.200 0.92 1.36 1.69 0.80 0.93 1.37 1. 0.90 1.00 2.00 3.00 0.225 0.89 1.34 1.67 0.90 0.92 1.36 1. 1.00 1.00 2.00 3.00 0.250 0.87 1.32 1.66 1.00 0.89 1.34 1. 1.10 1.00 2.00 3.00 0.275 0.84 1.30 1.64 1.10 0.87 1.33 1. 1.20 1.00 2.00 3.00 0.300 0.80 1.28 1.62 1.20 0.84 1.31 1. 1.30 1.00 2.00 3.00 0.325 0.76 1.26 1.61 1.30 0.81 1.29 1. 1.40 1.00 2.00 3.00 0.350 0.71 1.23 1.58 1.40 0.78 1.27 1. 1.50 1.00 2.00 3.00 0.375 0.66 1.20 1.56 1.50 0.74 1.24 1. 1.60 1.00 2.00 3.00 0.400 0.60 1.17 1.54 1.60 0.70 1.22 1. 1.70 1.00 2.00 3.00 0.425 0.53 1.13 1.51 1.70 0.65 1.19 1. 1.80 1.00 2.00 3.00 0.450 0.44 1.09 1.48 1.80 0.59 1.16 1. 1.90 1.00 2.00 3.00 0.475 0.31 1.05 1.45 1.90 0.53 1.13 1. 2.00 1.00 2.00 3.00 0.500 0.00 1.00 1.41 2.00 0.45 1.10 1.
The solution is (^) y C b x
⋅^2
a t⋅
For t = 0 s (^) x =c
For t = 1 s (^) y = C −4 x ⋅^2
For t = 20 s (^) y C x
2
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