06%20Dimensional%20Analysis%20EGM%206812%20F11

06%20Dimensional%20Analysis%20EGM%206812%20F11 - EGM 6812,...

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EGM 6812, F11, University of Florida, M. Sheplak 1/16 Section 6, Dimensional Analysis 6 Dimensional Analysis and Similarity Why? 1. Reduced number of experiments 2. Compact data representation 3. Solve problem fewer times 4. Lab prototype to full scale 5. If we non-dimensionalize the governing equations, we can “simplify” the equations by neglecting certain terms. Example Investigate drag on a sphere, with roughness (i.e. golf ball, high speed)   , , , , , DD F F D u c   If we investigate 10 iterations of each parameter, we would have to conduct 6 10 experiments. Instead perform a non-dimensionalization. 6.1 Buckingham -theorem Answer- 22 Roughness Re ,, 1 2 D D Ma C F uD u f cD uD     Step 1. List all variables   , , , , , F F D u c 7 n Step 2. Select the primary dimensions M,L,t 3 r Step 3. List dimensions of each parameter Parameter F D c u Dimension ML t 2 L L M L 3 M Lt L t L t Step 4. Select “r” repeating parameters (engineering judgment call!) Don’t choose: , , , D Fc  non-repeaters
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EGM 6812, F11, University of Florida, M. Sheplak 2/16 Section 6, Dimensional Analysis “parameters that are physically interesting” Do choose ,, uD } make sure that they cover mass, length, and time Step 5. Set up (n-r) groups 1 a b c D F u D  nondimensional choose a,b,c such that the s are nondimensional 2 4 ... d e f  When you do this you end up with drag force 1 2 2 2 2 area dynamic pressure drag coefficient DD D FF C u D u D    2 ratio of roughness to diameter D   3 1 Re uD   4 1 Ma c u   6.2 Flow Similarity How do we scale data from a wind-tunnel experiment to full-scale? Similarity!!! Geometric Similarity : Same shape between model and full scale Kinematic Similarity : velocity fields about a model and prototype must vary by no more than a constant. Kinematic similarity implies geometric similarity Dynamic Similarity : all forces on a model prototype must vary by no more than a constant scale factor. Dynamic similarity implies kinematic similarity and geometric similarity. To achieve dynamic similarity, we must match all the groups Example : 22 Re, , D D F C f Ma v D D     For Dynamic similarity
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EGM 6812, F11, University of Florida, M. Sheplak 3/16 Section 6, Dimensional Analysis model full scale Re Re vD  model fs Ma Ma v c model full scale DD  6.3 Dimensionless Parameters in Fluids 6.3.1.1 Typical forces in Fluid Dynamics Consider a typical velocity scale “ U ”, length “ L Steady Inertia 3 2 2 2 : SI L F ma L U L t      Viscous Forces ~ F A UL  Pressure 2 ~ p F pA pL   Gravity : 3 ~ g F mg g L Surface Tension ~ st FL Compressibility 2 2 2 ~ c F A a A a L where 2 a  is the bulk modulus of the fluid.
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This note was uploaded on 01/17/2012 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.

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06%20Dimensional%20Analysis%20EGM%206812%20F11 - EGM 6812,...

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