EGM 6812, F11, University of Florida, M. Sheplak1/16 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. ExampleInvestigate drag on a sphere, with roughness (i.e. golf ball, high speed) , , ,,,DDFFDuc If we investigate 10 iterations of each parameter, we would have to conduct 610experiments. Instead perform a non-dimensionalization. 6.1 Buckingham -theorem Answer- 22RoughnessRe,,12DDMaCFuDufcDu DStep 1.List all variables , , ,,,DDFFDuc 7nStep 2.Select the primary dimensions M,L,t 3rStep 3.List dimensions of each parameter Parameter F D c u Dimension ML t2 L L M L3 M Lt L t L t Step 4.Select “r” repeating parameters (engineering judgment call!) Don’t choose:, ,,DFc non-repeaters
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EGM 6812, F11, University of Florida, M. Sheplak2/16 Section 6, Dimensional Analysis “parameters that are physically interesting”Do choose , ,u D} make sure that they cover mass, length, and time Step 5.Set up (n-r) groups 1abcDFu D nondimensional choose a,b,c such that the s are nondimensional 24...defu DWhen you do this you end up with drag force12222areadynamicpressuredrag coefficient DDDFFCu DuD 2ratio of roughness to diameterD 31ReuD 41Macu6.2 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: 22Re,,DDFCfMav DDFor Dynamic similarity