Lecture 24 Notes - EGN 3353C Fluid Mechanics Lecture 24 When

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EGN 3353C Fluid Mechanics Lou Cattafesta MAE Dept. University of Florida Lecture 24 When nondimensionalizing an equation, nondimensional parameters often appear. Example Consider an object falling due to gravity in a vacuum 2 2 dz g dt = − . We will solve for the elevation z two ways: (1) the conventional dimensional approach, and (2) dimensionless approach. Integrating twice and applying the initial elevation 0 z and velocity 0 w Æ 2 00 1 2 zz w t g t =+ Æ dimensional answer If we want to determine ( ) zt , we need to specify 0 z , 0 w , and g (3 parameters). If one of these changes, we have to repeat the calculation! Instead we can nondimensionalize the above equation by 0 z to give 2 0 0 1 1 2 wt zg t z Now define some new dimensionless variables : ** 0 , w == The last term becomes ( ) 0 2 * *2 0 2 0 0 2 1 1 2 1 22 gtz w g z z t gz t w = = . The boxed term is a dimensionless parameter. It is related to the well-known Froude number 0 0 Fr w gz = in Fluid Dynamics.
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EGN 3353C Fluid Mechanics Lou Cattafesta MAE Dept. University of Florida So the solution can be written in dimensional or nondimensional form as follows: 2 00 1 2 zz w t g t =+ or *2 ** 2 1 1 2Fr t zt =+ − Ignoring time, 2 nd equation is only a function of 1 parameter 0 0 Fr w gz = , while the first is a function of 3!
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Lecture 24 Notes - EGN 3353C Fluid Mechanics Lecture 24 When

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