Dimensional Analysis Notes - 2

 Similitude or Similarities The geometric similarity is said to be exist between the model and prototype if the ratio of all corresponding linear dimensions in the model and prototype are equal. 

Geometric Similarity L D D B B L L r m P m P m P    L A A r 2 m P  L V V r 3 m P  is Scale Ratio r where L 

The kinematic similarity is said exist between model and prototype if the ratios of velocity and acceleration at corresponding points in the model and at the corresponding points in the prototype are the same. 

Kinematic Similarity V V V r m P  a a a r m P 

 is Velocity Ratio r whereV is Acceleration Ratio r where a Also the directions of the velocities in the model and prototype should be same The dynamic similarity is said exist between model and prototype if the ratios of corresponding forces acting at the corresponding points are equal Dynamic Similarity F F F r m P 

 is Force Ratio r where F Also the directions of the velocities in the model and prototype should be same It means for dynamic similarity between the model and prototype, the dimensionless numbers should be same for model and prototype. Types of Forces Acting on Moving Fluid 1. Inertia Force, Fi  It is the product of mass and acceleration of the flowing fluid and acts in the direction opposite to the direction of acceleration. 

 It always exists in the fluid flow problems Types of Forces Acting on Moving Fluid 

1. Inertia Force, Fi 2. Viscous Force, Fv  It is equal to the product of shear stress due to viscosity and surface area of the flow.  It is important in fluid flow problems where viscosity is having an important role to play Types of Forces Acting on Moving Fluid 

1. Inertia Force, Fi 2. Viscous Force, Fv 3. Gravity Force, Fg  It is equal to the product of mass and acceleration due to gravity of the flowing fluid.  It is present in case of open surface flow Types of Forces Acting on Moving Fluid 1. Inertia Force, Fi 2. Viscous Force, Fv 3. Gravity Force, Fg 4. Pressure Force, Fp  It is equal to the product of pressure intensity and cross sectional area of flowing fluid  It is present in case of pipe-flow Types of Forces Acting on Moving Fluid 1. Inertia Force, Fi

 2. Viscous Force, Fv 3. Gravity Force, Fg 4. Pressure Force, Fp 5. Surface Tension Force, Fs  It is equal to the product of surface tension and length of surface of the flowing fluid Types of Forces Acting on Moving Fluid 1. Inertia Force, Fi 2. Viscous Force, Fv 

3. Gravity Force, Fg 4. Pressure Force, Fp 5. Surface Tension Force, Fs 6. Elastic Force, Fe  It is equal to the product of elastic stress and area of the flowing fluid Dimensionless Numbers Lg V Gravity Force InertiaForce  Dimensionless numbers are obtained by dividing the inertia force by viscous force or gravity force or pressure force or surface tension force or elastic force. 1. Reynold’s number, Re = 2. Froude’s number, Fe = 3. Euler’s number, Eu = 4. Weber’s number, We = 5. Mach’s number, M = p /  V PressureForce InertiaForce   / L V SurfaceTensionForce InertiaForce     VL VD Viscous Force Inertia Force  or C V Elastic Force Inertia Force  V 2 L 2 V ρ T L L 2 ρ T V L 3  ρ   The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. Model Laws 

1. Reynold’s Model Models based on Reynolds’s Number includes: 

a) Pipe Flow b) Resistance experienced by Sub-marines, airplanes, fully immersed bodies etc. The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. Model Laws 

1. Reynold’s Model 2. Froude Model Law Froude Model Law is applied in the following fluid flow problems: a) Free Surface Flows such as Flow over spillways, Weirs, Sluices, Channels etc., b) Flow of jet from an orifice or nozzle c) Where waves are likely to formed on surface d) Where fluids of different densities flow over one another The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. Model Laws 

1. Reynold’s Model 2. Froude Model Law 3. Euler Model Law Euler Model Law is applied in the following cases: a) Closed pipe in which case turbulence is fully developed so that viscous forces are negligible and gravity force and surface tension is absent b) Where phenomenon of cavitations takes place The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. Model Laws 1. Reynold’s Model 

2. Froude Model Law 3. Euler Model Law 4. Weber Model Law Weber Model Law is applied in the following cases: a) Capillary rise in narrow passages b) Capillary movement of water in soil c) Capillary waves in channels d) Flow over weirs for small heads The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. Model Laws 1. Reynold’s Model 2. Froude Model Law 

3. Euler Model Law 4. Weber Model Law 5. Mach Model Law Mach Model Law is applied in the following cases: a) Flow of aero plane and projectile through air at supersonic speed ie., velocity more than velocity of sound b) Aero dynamic testing, c) Underwater testing of torpedoes, and d) Water-hammer problems If the viscous forces are predominant, the models are designed for dynamic similarity based on Reynold’s number. Reynold’s Model Law   p P P P m ρm Vm Lm ρ V L  V L t r r r  TimeScaleRatio  Velocity, V = Length/Time  T = L/V t V a r r r  Acceleration ScaleRatio  Acceleration, a = Velocity/Time  L = V/T R  R  e m e p  Problems 1. Water flowing through a pipe of diameter 30 cm at a velocity of 4 m/s. Find the velocity of oil flowing in another pipe of diameter 10cm, if the conditions of dynamic similarity is satisfied between two pipes. The viscosity of water and oil is given as 0.01 poise and 0.025 poise. 

The specific gravity of oil is 0.8. If the gravity force is predominant, the models are designed for dynamic similarity based on Froude number. Froude Model Law Lr T  ScaleRatiofor Time  r g L V g L V p P p m m m  Vr Lr F  F   VelocityScaleRatio e m e p  T  Scale Ratio for Accele ration 1 r Lr 2.5 Q  Scale Ratio for Discharge  r Lr 3 F  Scale Ratio for Force  r Lr F  Scale Ratio for Pressure Intensity  r Lr 3.5 P  Scale Ratio for Power  r Problems 1. In 1 in 40 model of a spillway, the velocity and discharge are 2 m/s and 2.5 m3 /s. Find corresponding velocity and discharge in the prototype 2. In a 1 in 20 model of stilling basin, the height of the jump in the model is observed to be 0.20m. 

What is height of hydraulic jump in the prototype? If energy dissipated in the model is 0.1kW, what is the corresponding value in prototype? 3. A 7.2 m height and 15 m long spillway discharges 94 m3 /s discharge under a head of 2m. 

If a 1:9 scale model of this spillway is to be constructed, determine the model dimensions, head over spillway model and the model discharge. If model is experiences a force of 7500 N, determine force on the prototype. 

Problems 4. A Dam of 15 m long is to discharge water at the rate of 120 cumecs under a head of 3 m. Design a model, if supply available in the laboratory is 50 lps 5. A 1:50 spillway model has a discharge of 1.5 cumecs. What is the corresponding discharge in prototype?. If a flood phenomenon takes 6 hour to occur in the prototype, how long it should take in the mode