ch02 - Problem 2.1 For the velocity fields given below,...

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2D V V t () Steady (5) V V x = 1D V V t Steady (6) V V xy , z , = 3D V V t = Unsteady (7) V V , z , = 3D V V t Steady (8) V V , = 2D V V t = Unsteady Problem 2.1 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 , = 2D V V t Steady (3) V V x = 1D V V t Steady (4) V V xz , =
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See the plots in the corresponding Excel workbook yc x 20 = For t = 20 s y c x = For t = 1 s = For t = 0 s x b a t = The solution is ln y () b t a ln x = Integrating dy y b t a dx x = So, separating variables v u dy dx = b t y ax = For streamlines Solution j bty i ax V ˆ ˆ = 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 2.4
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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 xyyy 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 24.68 1.00 1.00 2.00 3.00 1.00 1.00 2.00 3.00 1.00 1.00 2.00 3.00 1.10 1.00 2.00 3.00 1.10 0.91 1.82 2.73 1.10 0.15 0.30 0.45 1.20 1.00 2.00 3.00 1.20 0.83 1.67 2.50 1.20 0.03 0.05 0.08 1.30 1.00 2.00 3.00 1.30 0.77 1.54 2.31 1.30 0.01 0.01 0.02 1.40 1.00 2.00 3.00 1.40 0.71 1.43 2.14 1.40 0.00 0.00 0.00 1.50 1.00 2.00 3.00 1.50 0.67 1.33 2.00 1.50 0.00 0.00 0.00 1.60 1.00 2.00 3.00 1.60 0.63 1.25 1.88 1.60 0.00 0.00 0.00 1.70 1.00 2.00 3.00 1.70 0.59 1.18 1.76 1.70 0.00 0.00 0.00 1.80 1.00 2.00 3.00 1.80 0.56 1.11 1.67 1.80 0.00 0.00 0.00 1.90 1.00 2.00 3.00 1.90 0.53 1.05 1.58 1.90 0.00 0.00 0.00 2.00 1.00 2.00 3.00 2.00 0.50 1.00 1.50 2.00 0.00 0.00 0.00 j bty i ax V ˆ ˆ = r The solution is yc x b a t = For t = 0 s = For t = 1 s y c x = For t = 20 s x 20 =
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Streamline Plot (t = 0) 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 0.00 0.50 1.00 1.50 2.00 x y c = 1 c = 2 c = 3 Streamline Plot (t = 1 s) 0 10 20 30 40 50 60 70 0.00 0.50 1.00 1.50 2.00 x c = 1 c = 2 c = 3 Streamline Plot (t = 20 s) 0 2 4 6 8 10 12 14 16 18 20 -0.15 0.05 0.25 0.45 0.65 0.85 1.05 1.25 x c = 1 c = 2 c = 3
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See the plot in the corresponding Excel workbook y c x 3 = The solution is yc x b a = cx 3 = ln y () b a ln x = Integrating dy y b a dx x = So, separating variables v u dy dx = bx y ax 2 = by = For streamlines v6 m s = vb x y = 6 1 ms 2 × m 1 2 × m = u8 m s = ua x 2 = 2 1 2m 2 × = At point (2,1/2), the velocity components are 2D The velocity field is a function of x and y . It is therefore Solution j bxy i ax V ˆ ˆ 2 + = 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).
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ch02 - Problem 2.1 For the velocity fields given below,...

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