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Dimensional Analysis. A tool to help one to get maximum information from a minimum number of experiments facilitates a correlation of data with minimum number of plots. Can establish the scaling laws between models and prototype in testing. Parameter Dimension s.
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Dimensional Analysis • A tool to help one to get maximum information from • a minimum number of experiments • facilitates a correlation of data with minimum number of plots. • Can establish the scaling laws between models and prototype in testing.
Parameter Dimensions Consider experimental studies of drag on a cylinder Drag (F)depends upon Flow Speed V, diameter d, viscosity m, density of fluid r Just imagine how many experiments are needed to study thisphenomenon completely, It may run into hundreds A dimensional analysis indicates thatCdandReynolds number, Reor theMach numberMcan determine theCdbehaviour thus making itnecessary to perform only a limited number of experiments.
Buckingham Pi Theorem Consider a phenomenon described by an equation like g = g(q1, q2, q3, ………..,qn) where q1, q2, q3, ………..,qn are the independent variables. Ifmis the number of independent dimensions required to specify the dimensions of all q1, q2, q3, ………..,qn then one can come up with a relation like, G(P1, P2, P3, Pn-m) = 0 where P1, P2, P3, Pn-m are non-dimensional parameters. In other words the phenomenon can be described by n-m number of non-dimensional parameters.
Important non-dimensional numbers in Fluid Dynamics • Reynolds Number Re • Euler Number • or Pressure CoefficientCp • Froude Number Fr • Mach NumberM • Cavitation NumberCa • Weber Number We • Knudsen Number Kn
Reynolds Number,Re Ratio ofInertial forcestoViscous forces. Flow at low Reynolds numbers are laminar Flows at large Reynolds numbers are usually turbulent At low Reynolds numbers viscous effects are important in a large region around a body. At higher Reynolds numbers viscous effects are confined to a thin region around the body.
Euler Number or Pressure Coefficient, Cp Ratio ofPressure forcestoInertial Force An important parameter in Aerodynamics
Cavitation Number In cavitation studies, Dp(see formula for Cp) is taken as p - pv where p is the liquid pressure and pv is the liquid vapour pressure, The Cavitation number is given by
Froude Number Square of Froude Number related to the ratio of InertialtoGravity forces. Important when free surfaces effects are significant Fr < 1Subcritical Flow Fr > 1 Supercritical Flow
Weber Number Ratio ofInertiatoSurface Tensionforces. Where sis surface tension
Mach Number Could be interpreted as the ratio of Inertial toCompressibilityforces Wherecis the local sonic speed, Evis the Bulk Modulus of Elasticity. A significant parameter in Aerodynamics. NOTE: For incompressible Flows,c = andM = 0
Similitude and Model Studies For a study on a model to relate to that on a prototype it is required that there be Geometrical Similarity Kinematic Similarity Dynamic Similarity
Lm Geometrical Similarity Physical dimensions of model and prototype be similar Hp Hm Lp
um up vm vp Kinematic Similarity Velocity vectors at corresponding locations on the model and prototype are similar
Ftp Fnp Dynamic Similarity Forces at corresponding locations on model and prototype are similar Ftm Fnm
Problem in Wind Tunnel testing While testing models in wind tunnels it is required that following non-dimensional parameters be preserved. Reynolds Number Mach Number Cd = f (Re, M) But the available wind tunnels do not permit both these numbers to be preserved.
Solution for Wind Tunnel testing Remedy is offered by nature itself At low speeds viscous effects are more important than the compressibility effects. So onlyReynolds numberbe preserved. Cd = f (Re) At higher speeds compressibility effects are dominating. So only Mach number need be preserved. Cd = f (M)
