This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nondimensionalize 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 nondimensionalization. 7.1 Buckingham Πtheorem 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 nonrepeaters “parameters that are physically interesting” • Do choose , , u D ρ } make sure that they cover mass, length, and time Step 5. Set up (nr) 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 windtunnel experiment to fullscale? 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...
View
Full
Document
 Fall '09
 RENWEIMEI
 Fluid Dynamics, Thermodynamics, Fundamental physics concepts

Click to edit the document details