07 Dimensional Analysis

# 07 Dimensional Analysis - 7 Dimensional Analysis and...

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Unformatted text preview: 7 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) ( 29 , , , , , D D F F D u c ε ρ μ = If we investigate 10 iterations of each parameter, we would have to conduct 6 10 experiments. Instead perform a non-dimensionalization. 7.1 Buckingham Π-theorem Answer- { { { 2 2 Roughness Re , , 1 2 D D Ma C F uD u f c D u D ρ ε μ ρ = 142 43 Step 1. List all variables ( 29 , , , , , D D F F D u c ε ρ μ = 7 n = Step 2. Select the primary dimensions M,L,t 3 r = Step 3. List dimensions of each parameter Parameter F D ε ρ μ 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 F c μ ε non-repeaters “parameters that are physically interesting” • Do choose , , u D ρ } make sure that they cover mass, length, and time Step 5. Set up (n-r) Π groups 1 a b c D F u D ρ Π = nondimensional choose a,b,c such that the Π s are nondimensional 2 4 ... d e f u D ερ Π = Π When you do this you end up with } { { drag force 1 2 2 2 2 area dynamic pressure drag coefficient D D D F F C u D u D ρ ρ Π = = 2 ratio of roughness to diameter D ε Π = 3 1 Re uD μ ρ Π = = 4 1 Ma c u Π = = 7.2 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 Π groups Example : 2 2 Re, , D D F C f Ma v D D ε ρ = = For Dynamic similarity model...
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07 Dimensional Analysis - 7 Dimensional Analysis and...

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