Water flows in a channel with uniform curvature RR. The channel has height h=6mm and width w=215mm (normal to the drawing plane) as shown in Fig Q3. The curved part of the channel has a length of l=100mm in the x-direction. You can assume that the channel height is very small compared to the curvature radius. Neglect effects of gravity. Fig Q3: Geometry of curved channel, drawing not to scale. The curvature of the channel is not shown, as too small to be drawn accurately at this scale. Work to 4 significant digits. Enter all values using base units or their combinations, i.e. m, m/s, Pa, N. Do not use multiples as e.g. mm, kPa. You can use values with exponents, such as 0.12e3. a) Assuming that the pressure is uniform on each of the channel walls, determine the pressure difference between the upper and lower wall that is needed if the required net force on the two walls is 2200N pointing downward (in the negative y-direction). Use a sign convention that a positive pressure difference or gradient means the pressure at the higher y-position is higher. b) Your team agree that, from experience, the average velocity of the fluid is v=46ms. Assuming that the velocity is uniform across the channel, and neglecting gravitational effects, determine the curvature R required to generate this force with this flow velocity. Use a sign convention that a positive value indicates that the centre of the radius is above the channel, i.e. the middle of the channel is bent downwards. A negative radius indicates the middle of the channel is bent upwards. c) Your team agree on a radius ofR=100mm for the centreline of the channel. One of your team members points out that velocity is not uniform across the channel. To improve your analysis, assume that the flow in the channel can be split into three streamtubes, of height h/3, each having a constant velocity. The two streamtubes near the walls can be assumed to have average velocity vw=29m/s. For each of the streamtubes, use the streamtube's centreline to determine its radius.
Water flows in a channel with uniform curvature RR. The channel has height h=6mm and width w=215mm (normal to the drawing plane) as shown in Fig Q3. The curved part of the channel has a length of l=100mm in the x-direction. You can assume that the channel height is very small compared to the curvature radius.
Neglect effects of gravity.
Fig Q3: Geometry of curved channel, drawing not to scale. The curvature of the channel is not shown, as too small to be drawn accurately at this scale.
Work to 4 significant digits. Enter all values using base units or their combinations, i.e. m, m/s, Pa, N. Do not use multiples as e.g. mm, kPa.
You can use values with exponents, such as 0.12e3.
a) Assuming that the pressure is uniform on each of the channel walls, determine the pressure difference between the upper and lower wall that is needed if the required net force on the two walls is 2200N pointing downward (in the negative y-direction).
Use a sign convention that a positive pressure difference or gradient means the pressure at the higher y-position is higher.
b)
Your team agree that, from experience, the average velocity of the fluid is v=46ms. Assuming that the velocity is uniform across the channel, and neglecting gravitational effects, determine the curvature R required to generate this force with this flow velocity.
Use a sign convention that a positive value indicates that the centre of the radius is above the channel, i.e. the middle of the channel is bent downwards. A negative radius indicates the middle of the channel is bent upwards.
c)
Your team agree on a radius ofR=100mm for the centreline of the channel. One of your team members points out that velocity is not uniform across the channel. To improve your analysis, assume that the flow in the channel can be split into three streamtubes, of height h/3, each having a constant velocity. The two streamtubes near the walls can be assumed to have average velocity vw=29m/s.
For each of the streamtubes, use the streamtube's centreline to determine its radius.
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