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16. Dimensional Analysis

16. Dimensional Analysis. CH EN 374: Fluid Mechanics. Dimensional Analysis. Given a set of n dimensionless groups ( s) describing a system or process: Define one as the dependent How can we use this?. From Last Class.

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16. Dimensional Analysis

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  1. 16. Dimensional Analysis CH EN 374: Fluid Mechanics

  2. Dimensional Analysis • Given a set of n dimensionless groups (s) describing a system or process: • Define one as the dependent • How can we use this?

  3. From Last Class • Used the Pi Method to find two dimensionless groups for flow around a car or other solid object:

  4. Dimensional Analysis • Let’s say we want to find the drag force on a prototype sports car • It would be easier to measure the drag on a smaller model and then scale it up

  5. Geometric Similarity • The full-size prototype and the scale model are geometrically similar • They are the same shape but scaled by some factor • Does that necessarily mean that the ratio of important forces is the same for both?

  6. Dynamic Similarity • In dynamic similarity all of the forces in the smaller flow scale by a constant factor to forces in the larger flow • For the model and prototype to be dynamically similar, all groups must be the same

  7. Dynamic Similarity • If () is the same for both cars, what does that mean about ()? • So if is the same for both, they are dynamically similar.

  8. Problem • We want to predict the aerodynamic drag on a sports car at a speed of 50.0 mph. We build a one-fifth scale model which we place in a wind tunnel. What velocity should we run the air in the tunnel at to make the model similar to a full-size prototype?

  9. Problem • The drag force on the model car in the last problem is measured as the force required to hold the model in place in the wind tunnel and is found to be . Find the drag force on the full-sized sports car at 50 mph. • Reminder:

  10. Non-Dimensionalization • We’ve seen that dimensionless ratios can tell us a lot about a system, process, or problem • We can also non-dimensionalize equations • Can be useful for: • Simplifying equations to make them easier to solve • Comparing different situations

  11. Imagine a ball thrown straight up… • Let’s solve this ODE the “normal” way

  12. But there is another approach! Parameters: • Variables: • Dimensional constants: Primary dimensions of parameters: • Select scaling parameters: • Choose from dimensional constants • We need at least one for each primary dimension. Nondimensionalize variables:

  13. Sidenote: This is the definition your book uses, is common. • We turned several parameters ( into one ()

  14. Problem • The gravitational constant at the surface of the moon is only about one-sixth of that on earth. An astronaut on the moon throws a baseball at an initial speed of 21.0 m/s at a 5 angle from an initial height of 2m. Use this chart to find the time it takes for the ball to hit the ground.

  15. Non-Dimensionalized NS

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