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 , density of fluid  Just imagine how many experiments are needed to study this phenomenon completely, It may run into hundreds A dimensional analysis indicates that Cd and Reynolds number, Re or the Mach number M can determine the Cd behaviour thus making it necessary to perform only a limited number of experiments.

Buckingham Pi Theorem Consider a phenomenon described by an equation like g = g(q 1, q 2, q 3, ………..,q n ) where q 1, q 2, q 3, ………..,q n are the independent variables. If m is the number of independent dimensions required to specify the dimensions of all q 1, q 2, q 3, ………..,q n then one can come up with a relation like, G(  1,  2,  3,  n-m ) = 0 where  1,  2,  3,  n-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 NumberRe Euler Number or Pressure CoefficientCp Froude NumberFr Mach NumberM Cavitation NumberCa Weber NumberWe Knudsen NumberKn

Reynolds Number, Re Ratio of Inertial forces to Viscous 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 of Pressure forces to Inertial Force An important parameter in Aerodynamics

Cavitation Number In cavitation studies,  p(see formula for Cp) is taken as p - p v where p is the liquid pressure and p v is the liquid vapour pressure, The Cavitation number is given by

Froude Number Square of Froude Number related to the ratio of Inertial to Gravity forces. Important when free surfaces effects are significant Fr < 1 Subcritical Flow Fr > 1Supercritical Flow

Weber Number Ratio of Inertia to Surface Tension forces. Where  is surface tension

Mach Number Could be interpreted as the ratio of Inertial to Compressibility forces Where c is the local sonic speed, Ev is the Bulk Modulus of Elasticity. A significant parameter in Aerodynamics. NOTE: For incompressible Flows, c =  and M = 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

Geometrical Similarity Physical dimensions of model and prototype be similar Lp HpHm Lm

Kinematic Similarity Velocity vectors at corresponding locations on the model and prototype are similar vpvp upup vmvm umum

Dynamic Similarity Forces at corresponding locations on model and prototype are similar Fn p Ft p Fn m Ft m

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 But the available wind tunnels do not permit both these numbers to be preserved. Cd = f (Re, M)

Solution for Wind Tunnel testing Cd = f (M) Remedy is offered by nature itself At low speeds viscous effects are more important than the compressibility effects. So only Reynolds number be preserved. Cd = f (Re) At higher speeds compressibility effects are dominating. So only Mach number need be preserved.