SIMILITUDE AND DIMENSION ANALYSIS
DEFINITION AND USES OF SIMILITUDE • Similitude means similarity • it impossible to determine all the essential facts for a given fluid flow by pure theory alone • we must often depend on experimental investigations. • we can greatly reduce the number of tests needed by systematically using dimensional analysis and the laws of similitude or similarity. • For these enable us to apply test data to other cases than those observed. • we can obtain valuable results at a minimum cost from tests made with small-scale models of the full-size apparatus. The laws of similitude enable us to predict the performance of the prototype, which means the full-size device, ...view middle of the document...
Fine powder, because of cohesive forces between the particles, does not simulate the behavior of sand. • Again, in the case of a river the horizontal scale is usually limited by the available floor space, and this same scale used for the vertical dimensions may produce a stream so shallow that capillarity has an appreciable effect and also the bed slope may be so small that the flow is laminar. In such cases we need to use a distorted model, which means that the vertical scale is larger than the horizontal scale. Then, if the horizontal scale ratio is denoted by Lr and the vertical scale ratio by Lr’, the cross section area ratio is LrLr’
• Kinematic similarity implies that, in addition to geometric similarity, the ratio of the velocities at all corresponding points in the flows are the same. The velocity scale ratio is = • and this is a constant for kinematic similarity. Its value in terms of L, is determined by dynamic considerations, as explained in the following section. As time T is dimensionally L/V, the time scale ratio is = • and in a similar manner the acceleration scale 2 ratio is
DYNAMIC SIMILARITY • Two systems have dynamic similarity if, in addition to kinematic similarity, corresponding forces are in the same ratio in both. The force scale ratio is
• which must be constant for dynamic similarity. • Forces that may act on a fluid element include those due to gravity (FG), pressure (FP), viscosity (Fv), and elasticity (FE). Also, if the element of fluid is at a liquid-gas interface, there are forces due to surface tension (FT). If the sum of forces on a fluid element does not add up to zero, the element will accelerate in accordance with Newton's law. We can transform such an unbalanced force system into a balanced system by adding an inertia force F, that is equal and opposite to the resultant of the acting forces. Thus, generally, ƩF = FG + FP + FV + FE + FT = Resultant and FI = - Resultant Thus FG +FP+FV+FE+FT+FI = 0
• These forces can be expressed in the simplest terms as: • Gravity: FG = mg = pL3g • Pressure: FP = (Δp)A = (Δp)L2 • Viscosity: FV = μdu/dyA = µ(V/L)L2 = µVL • Elasticity: FE = EvA = EvL2 • Surface tension: FT = σL • Inertia: FI = ma = ρL3 L/T2 = ρL4T-2 = ρV2L2
• In many flow problems some of these forces are either absent or insignificant. In...