Pharos University ME 259 Fluid Mechanics Lecture # 9

1. Pharos UniversityME 259 Fluid Mechanics Lecture # 9 Dimensional Analysis and Similitude

2. Main Topics • Nature of Dimensional Analysis • Buckingham Pi Theorem • Significant Dimensionless Groups in Fluid Mechanics • Flow Similarity and Model Studies

3. Objectives • Understand dimensions, units, and dimensional homogeneity • Understand benefits of dimensional analysis • Know how to use the method of repeating variables • Understand the concept of similarity and how to apply it to experimental modeling

4. Dimensions and Units • Review • Dimension: Measure of a physical quantity, e.g., length, time, mass • Units: Assignment of a number to a dimension, e.g., (m), (sec), (kg) • 7 Primary Dimensions: • Mass m (kg) • Length L (m) • Time t (sec) • Temperature T (K) • Current I (A) • Amount of Light C (cd) • Amount of matter N (mol)

5. Dimensions and Units • All non-primary dimensions can be formed by a combination of the 7 primary dimensions • Examples • {Velocity} m/sec = {Length/Time} = {L/t} • {Force} N = {Mass Length/Time} = {mL/t2}

6. Dimensional Homogeneity • Every additive term in an equation must have the same dimensions • Example: Bernoulli equation • {p} = {force/area}={mass x length/time x 1/length2} = {m/(t2L)} • {1/2V2} = {mass/length3 x (length/time)2} = {m/(t2L)} • {gz} = {mass/length3 x length/time2 x length} ={m/(t2L)}

7. Nondimensionalization of Equations • To nondimensionalize, for example, the Bernoulli equation, the first step is to list primary dimensions of all dimensional variables and constants {p} = {m/(t2L)} {} = {m/L3} {V} = {L/t} {g} = {L/t2} {z} = {L} • Next, we need to select Scaling Parameters. For this example, select L, U0, 0

8. Nature of Dimensional Analysis Example: Drag on a Sphere • Drag depends on FOUR parameters:sphere size (D); speed (V); fluid density (r); fluid viscosity (m) • Difficult to know how to set up experiments to determine dependencies • Difficult to know how to present results (four graphs?)

9. Nature of Dimensional Analysis Example: Drag on a Sphere • Only one dependent and one independent variable • Easy to set up experiments to determine dependency • Easy to present results (one graph)

10. Nature of Dimensional Analysis

11. Buckingham Pi Theorem • Step 1: List all the parameters involved Let n be the number of parameters Example: For drag on a sphere, F, V, D, r, m, & n = 5 • Step 2: Select a set of primary dimensions For example M (kg), L (m), t (sec). Example: For drag on a sphere choose MLt

12. Buckingham Pi Theorem • Step 3 List the dimensions of all parameters Let r be the number of primary dimensions Example: For drag on a sphere r = 3

13. Buckingham Pi Theorem • Step 4 Select a set of r dimensional parameters that includes all the primary dimensions Example: For drag on a sphere (m = r = 3) select ϱ, V, D

14. Buckingham Pi Theorem • Step 5 Set up dimensionless groups πs There will be n – m equations Example: For drag on a sphere

15. Buckingham Pi Theorem • Step 6 Check to see that each group obtained is dimensionless Example: For drag on a sphere Π2 = Re = ϱVD / μ Π2

16. Significant Dimensionless Groups in Fluid Mechanics • Reynolds Number • Mach Number

17. Significant Dimensionless Groups in Fluid Mechanics • Froude Number • Weber Number

18. Significant Dimensionless Groups in Fluid Mechanics • Euler Number • Cavitation Number

19. Drag = f(V, L, r, m, c, t, e, T, etc.) From dimensional analysis, Vortex shedding behind cylinder Dimensional analysis • Definition : Dimensional analysis is a process of formulating fluid mechanics problems in • in terms of non-dimensional variables and parameters. • Why is it used : • Reduction in variables ( If F(A1, A2, … , An) = 0, then f(P1, P2, … Pr < n) = 0, • where, F = functional form, Ai = dimensional variables, Pj = non-dimensional • parameters, m = number of important dimensions, n = number of dimensional variables, r • = n – m ). Thereby the number of experiments required to determine f vs. F is reduced. • Helps in understanding physics • Useful in data analysis and modeling • Enables scaling of different physical dimensions and fluid properties Example Examples of dimensionless quantities : Reynolds number, Froude Number, Strouhal number, Euler number, etc.

20. Similarity and model testing • Definition : Flow conditions for a model test are completely similar if all relevant dimensionless parameters have the same corresponding values for model and prototype. • Pi model = Pi prototype i = 1 • Enables extrapolation from model to full scale • However, complete similarity usually not possible. Therefore, often it is necessary to • use Re, or Fr, or Ma scaling, i.e., select most important P and accommodate others • as best possible. • Types of similarity: • Geometric Similarity : all body dimensions in all three coordinates have the same • linear-scale ratios. • Kinematic Similarity : homologous (same relative position) particles lie at homologous • points at homologous times. • Dynamic Similarity : in addition to the requirements for kinematic similarity the model • and prototype forces must be in a constant ratio.

21. Dimensional Analysis and Similarity • Geometric Similarity - the model must be the same shape as the prototype. Each dimension must be scaled by the same factor. • Kinematic Similarity - velocity as any point in the model must be proportional • Dynamic Similarity - all forces in the model flow scale by a constant factor to corresponding forces in the prototype flow. • Complete Similarity is achieved only if all 3 conditions are met.

22. Dimensional Analysis and Similarity • Complete similarity is ensured if all independent  groups are the same between model and prototype. • What is ? • We let uppercase Greek letter  denote a nondimensional parameter, e.g.,Reynolds number Re, Froude number Fr, Drag coefficient, CD, etc. • Consider automobile experiment • Drag force is F = f(V, , L) • Through dimensional analysis, we can reduce the problem to

23. Flow Similarity and Model Studies • Example: Drag on a Sphere

24. Flow Similarity and Model Studies • Example: Drag on a Sphere For dynamic similarity … … then …

25. Flow Similarity and Model Studies • Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Pump Head Pump Power

26. Similitude-Type of Similarities • Geometric Similarity: is the similarity of shape. • Where: Lp, Bp and Dp are Length, Breadth, and diameter of prototype and Lm, Bm, Dm are Length, Breadth, and diameter of model. • Lr= Scale ratio

27. Similitude-Type of Similarities • Kinematic Similarity: is the similarity of motion. • Where: Vp1& Vp2 and ap1 & ap2 are velocity and accelerations at point 1 & 2 in prototype and Vm1& Vm2 and am1 & am2 are velocity and accelerations at point 1 & 2 in model. • Vr and ar are the velocity ratio and acceleration ratio

28. Similitude-Type of Similarities • Dynamic Similarity: is the similarity of forces. • Where: (Fi)p, (Fv)p and (Fg)p are inertia, viscous and gravitational forces in prototype and (Fi)m, (Fv)m and (Fg)m are inertia, viscous and gravitational forces in model. • Fr is the Force ratio

29. Flow Similarity and Model Studies • Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Head Coefficient Power Coefficient

30. Flow Similarity and Model Studies • Scaling with Multiple Dependent Parameters Example: Centrifugal Pump(Negligible Viscous Effects) If … … then …

31. Flow Similarity and Model Studies • Scaling with Multiple Dependent Parameters Example: Centrifugal Pump Specific Speed

32. Types of forces encountered in fluid Phenomenon • Inertia Force, Fi: = mass X acceleration in the flowing fluid. • Viscous Force, Fv: = shear stress due to viscosity X surface area of flow. • Gravity Force, Fg: = mass X acceleration due to gravity. • Pressure Force, Fp: = pressure intensity X C.S. area of flowing fluid.

33. Dimensionless Numbers • These are numbers which are obtained by dividing the inertia force by viscous force or gravity force or pressure force or surface tension force or elastic force. • As this is ratio of once force to other, it will be a dimensionless number. These are also called non-dimensional parameters. • The following are most important dimensionless numbers. • Reynold’s Number • Froude’s Number • Euler’s Number • Mach’s Number

34. Dimensionless Numbers • Reynold’s Number, Re: It is the ratio of inertia force to the viscous force of flowing fluid. • Froude’s Number, Fe: It is the ratio of inertia force to the gravity force of flowing fluid.

35. Dimensionless Numbers • Eulers’s Number, Re: It is the ratio of inertia force to the pressure force of flowing fluid. • Mach’s Number, Re: It is the ratio of inertia force to the elastic force of flowing fluid.