1. Pharos University ME 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 1.Understand dimensions, units, and dimensional homogeneity 2.Understand benefits of dimensional analysis 3.Know how to use the method of repeating variables 4.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: 1.Massm (kg) 2.LengthL (m) 3.Timet (sec) 4.TemperatureT (K) 5.CurrentI (A) 6.Amount of LightC (cd) 7.Amount of matterN (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/t 2 }

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/length 2 } = {m/(t 2 L)} – {1/2  V 2 } = {mass/length 3 x (length/time) 2 } = {m/(t 2 L)} – {  gz} = {mass/length 3 x length/time 2 x length} ={m/(t 2 L)}

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/(t 2 L)}{  } = {m/L 3 }{V} = {L/t} {g} = {L/t 2 }{z} = {L} – Next, we need to select Scaling Parameters. For this example, select L, U 0,  0

8. Nature of Dimensional Analysis Example: Drag on a Sphere Drag depends on FOUR parameters: sphere size (D); speed (V); fluid density (  ); fluid viscosity (  ) 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, , , & 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 π s 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 Π 2 = Re = ϱ VD / μ Π 2

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. 19 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(  1,  2, …  r < n) = 0, where, F = functional form, Ai = dimensional variables,  j = 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 Vortex shedding behind cylinder Drag = f(V, L, r, m, c, t, e, T, etc.) From dimensional analysis, Examples of dimensionless quantities : Reynolds number, Froude Number, Strouhal number, Euler number, etc.

20. 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.  i model =  i 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  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, C D, 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: L p, B p and D p are Length, Breadth, and diameter of prototype and L m, B m, D m are Length, Breadth, and diameter of model. Where: L p, B p and D p are Length, Breadth, and diameter of prototype and L m, B m, D m are Length, Breadth, and diameter of model. Lr= Scale ratio Lr= Scale ratio

27. Similitude-Type of Similarities Kinematic Similarity: is the similarity of motion. Where: V p1 & V p2 and a p1 & a p2 are velocity and accelerations at point 1 & 2 in prototype and V m1 & V m2 and a m1 & a m2 are velocity and accelerations at point 1 & 2 in model. Where: V p1 & V p2 and a p1 & a p2 are velocity and accelerations at point 1 & 2 in prototype and V m1 & V m2 and a m1 & a m2 are velocity and accelerations at point 1 & 2 in model. V r and a r are the velocity ratio and acceleration ratio V r and a r are the velocity ratio and acceleration ratio

28. Similitude-Type of Similarities Dynamic Similarity: is the similarity of forces. Where: (F i ) p, (F v ) p and (F g ) p are inertia, viscous and gravitational forces in prototype and (F i ) m, (F v ) m and (F g ) m are inertia, viscous and gravitational forces in model. Where: (F i ) p, (F v ) p and (F g ) p are inertia, viscous and gravitational forces in prototype and (F i ) m, (F v ) m and (F g ) m are inertia, viscous and gravitational forces in model. F r is the Force ratio F r 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. 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.