Control Volume, Fluid Mechanics

In summary: That is, the pressure in the control volume is just the sum of the pressure on the walls of the control volume and the atmospheric pressure.
  • #1
Kqwert
160
3

Homework Statement


Water flows over a dam as depicted below. Across section 1 and 2 we have a hydrostatic pressure distribution. Calculate the force on the dam.
6FYsP3D.png

The Attempt at a Solution


My question is regarding control volume. The solution manual has given us this:

3vSiT3q.png


The two hydrostatic forces are understandable, as well as an unknown force Fk inside the CV. But what about the pressure acting on top and at the bottom of the CV? At the top the pressure must be P0, but what about at the bottom? Shouldn't this be an unknown pressure from the water? And when do we have to consider the weight of the fluid, i.e. mg in our CV analyses?
 

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  • #2
The control volume analysis for this kind of problem typically focuses only on the horizontal direction, and the vertical pressure forces acting on the top and bottom do not affect the horizontal force balance (remember, force is a vector). If you apply the macroscopic momentum balance to the control volume for the vertical direction, then you get the weight of the water.
 
  • #3
Thank you, Chestermiller. Really appreciate your help here!

Would this be the forces in the vertical direction? For the same CV as above (I reckon I might miss a contact force from the dam?)

RyEuW59.png


I also assume these would have to sum to zero as we only have flow in the horizontal direction (using sigmaF = rho*q*v,out - rho*q*v,in = 0)
 

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  • #4
Kqwert said:
Thank you, Chestermiller. Really appreciate your help here!

Would this be the forces in the vertical direction? For the same CV as above (I reckon I might miss a contact force from the dam?)

View attachment 225974

I also assume these would have to sum to zero as we only have flow in the horizontal direction (using sigmaF = rho*q*v,out - rho*q*v,in = 0)
The macroscopic momentum balance in the vertical direction is F= mg, where F is the upward force that the base of the control volume (including the dam) exerts on the fluid within the control volume, and mg is the weight of the fluid in the control volume.
 
  • #5
Chestermiller said:
The macroscopic momentum balance in the vertical direction is F= mg, where F is the upward force that the base of the control volume (including the dam) exerts on the fluid within the control volume, and mg is the weight of the fluid in the control volume.
Thank you! but what about the atmospheric pressure P0? Is it canceled by F?
 
  • #6
Kqwert said:
Thank you! but what about the atmospheric pressure P0? Is it canceled by F?
For incompressible fluids, it is easier(and better) to work with gauge pressures.
 

Related to Control Volume, Fluid Mechanics

1. What is a control volume in fluid mechanics?

A control volume is a specific region in a fluid flow system that is chosen for analysis. It is often represented by a closed surface in which fluid enters and exits, and its properties can be measured and analyzed to determine the behavior of the fluid within the system.

2. How is the control volume used in fluid mechanics?

The control volume is used to analyze the flow of fluids in a system by applying the principles of conservation of mass, energy, and momentum. By observing the inflow and outflow of the fluid through the control volume, the effects of external forces and internal processes can be studied.

3. How does the control volume differ from a control mass?

A control mass is a specific amount of fluid that is chosen for analysis, while a control volume is a specific region in a fluid flow system. In a control mass, the mass of the fluid remains constant, while in a control volume, the mass can change due to inflow and outflow.

4. What are the types of control volumes in fluid mechanics?

The two main types of control volumes in fluid mechanics are fixed control volumes and moving control volumes. A fixed control volume is stationary, while a moving control volume moves with the fluid flow. Other types include open and closed control volumes, depending on the boundaries of the volume.

5. How is the control volume analysis used in real-world applications?

Control volume analysis is used in various real-world applications, such as designing aircraft wings, analyzing the flow of blood in the human body, and studying the flow of water in hydraulic systems. It is also used in industries like aerospace, automotive, and energy to optimize fluid flow and improve efficiency.

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