Dimension Analysis in Fluid mechanics

Dimension Analysis in Fluid mechanics

1. subject : Applied fluid mechanics DIMENSIONAL ANALYSIS BVRIT Prepared by: Ravaliya Nirmal

2. Learning Objectives Introduction to Dimensions & Units Use of Dimensional Analysis Dimensional Homogeneity Methods of Dimensional Analysis Rayleigh’s Method

3. Learning Objectives Buckingham’s Method Model Analysis Similitude Model Laws or Similarity Laws Model and Prototype Relations

4. Many practical real flow problems in fluid mechanics can be solved by using equations and analytical procedures. However, solutions of some real flow problems depend heavily on experimental data. Sometimes, the experimental work in the laboratory is not only time-consuming, but also expensive. So, the main goal is to extract maximum information from fewest experiments. In this regard, dimensional analysis is an important tool that helps in correlating analytical results with experimental data and to predict the prototype behavior from the measurements on the model. Introduction

5. Dimensions and Units In dimensional analysis we are only concerned with the nature of the dimension i.e. its quality not its quantity. Dimensions are properties which can be measured. Ex.: Mass, Length, Time etc., Units are the standard elements we use to quantify these dimensions. Ex.: Kg, Metre, Seconds etc., The following are the Fundamental Dimensions (MLT) Mass Length kg m M L Time s T

6. Secondary or Derived Dimensions Secondary dimensions are those quantities which posses more than one fundamental dimensions. 1. Geometric Area V o l ume m 2 m 3 L 2 L 3 2 . K ine m a tic Velocity Acceleration 3. Dynamic m/s m/s 2 L/T L/T 2 L.T -1 L.T -2 a) Force N ML/T M.L.T -1 b) Density kg/m 3 M/L 3 M.L -3

7. Probl e ms Find Dimensions for the following: Stress / Pressure Work Power Kinetic Energy Dynamic Viscosity Kinematic Viscosity Surface Tension Angular Velocity Mome n tum 10.Torque

8. Use of Dimensional Analysis Conversion from one dimensional unit to another Checking units of equations (Dimensional Homogeneity) Defining dimensionless relationship using Rayleigh’s Method Buckingham’s π-Theorem Model Analysis

9. Dimensional Homogeneity Dimensional Homogeneity means the dimensions in each equation on both sides equal.

10. Rayeligh’s Method To define relationship among variables This method is used for determining the expression for a variable which depends upon maximum three or four variables only.

11. Rayeligh’s Method To define relationship among variables This method is used for determining the expression for a variable which depends upon maximum three or four variables only. Methodology: Let X is a function of X 1 , X 2 , X 3 and mathematically it can be written as X = f(X 1 , X 2 , X 3 ) This can be also written as X = K (X 1 a , X 2 b , X 3 c ) where K is constant and a, b and c are arbitrarily powers The values of a, b and c are obtained by comparing the powers of the fundamental dimension on both sides.

12. Rayeligh’s Method To define relationship among variables This method is used for determining the expression for a variable which depends upon maximum three or four variables only. Methodology: Let X is a function of X 1 , X 2 , X 3 and mathematically it can be written as X = f(X 1 , X 2 , X 3 ) This can be also written as X = K (X 1 a , X 2 b , X 3 c ) where K is constant and a, b and c are arbitrarily powers The values of a, b and c are obtained by comparing the powers of the fundamental dimension on both sides. Problem: Find the expression for Discharge Q in a open channel flow when Q is depends on Area A and Velocity V. Solution: Q = K.A a .V b  1 where K is a Non-dimensional constant Substitute the dimensions on both sides of equation 1 M L 3 T -1 = K. (L 2 ) a .(LT- 1 ) b Equating powers of M, L, T on both sides, Power of T, Power of L, -1 = -b  b=1 3= 2a+b  2a = 2-b = 2-1 = 1 Substituting values of a, b, and c in Equation 1m Q = K. A 1 . V 1 = V.A

13. To define relationship among variables This method is used for determining the expression for a variable which depends upon maximum three or four variables only. Rayeligh’s Method Methodology: Let X is a function of X 1 , X 2 , X 3 and mathematically it can be written as X = f(X 1 , X 2 , X 3 ) This can be also written as X = K (X 1 a , X 2 b , X 3 c ) where K is constant and a, b and c are arbitrarily powers The values of a, b and c are obtained by comparing the powers of the fundamental dimension on both sides.

14. To define relationship among variables This method is used for determining the expression for a variable which depends upon maximum three or four variables only. Rayeligh’s Method Methodology: Let X is a function of X 1 , X 2 , X 3 and mathematically it can be written as X = f(X 1 , X 2 , X 3 ) This can be also written as X = K (X 1 a , X 2 b , X 3 c ) where K is constant and a, b and c are arbitrarily powers The values of a, b and c are obtained by comparing the powers of the fundamental dimension on both sides.

15. This method of analysis is used when number of variables are more. Theorem: If there are n variables in a physical phenomenon and those n variables contain m dimensions, then variables can be arranged into (n-m) dimensionless groups called Φ terms. Explanation: If f (X 1 , X 2 , X 3 , ……… X n ) = 0 and variables can be expressed using m dimensions then f (π 1 , π 2 , π 3 , ……… π n - m ) = 0 where, π 1 , π 2 , π 3 , … are dimensionless groups. Each π term contains (m + 1) variables out of which m are of repeating type and one is of non- repeating type. Each π term being dimensionless, the dimensional homogeneity can be used to get each π term. π denotes a non-dimensional parameter Buckingham’s π -Theorem

16. Selecting Repeating Variables: Avoid taking the quantity required as the repeating variable. Repeating variables put together should not form dimensionless group. No two repeating variables should have same dimensions. Repeating variables can be selected from each of the following properties. Geometric property  Length, height, width, area Flow property  Velocity, Acceleration, Discharge Fluid property  Mass density, Viscosity, Surface tension Buckingham’s π -Theorem

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20. For predicting the performance of the hydraulic structures (such as dams, spillways etc.) or hydraulic machines (such as turbines, pumps etc.) before actually constructing or manufacturing, models of the structures or machines are made and tests are conducted on them to obtain the desired information. Model is a small replica of the actual structure or machine The actual structure or machine is called as Prototype Models can be smaller or larger than the Prototype Model Analysis is actually an experimental method of finding solutions of complex flow problems. Model Analysis

21. Similitude is defined as the similarity between the model and prototype in every aspect, which means that the model and prototype have similar properties. Types of Similarities: Geometric Similarity  Length, Breadth, Depth, Diameter, Area, Volume etc., Kinematic Similarity  Velocity, Acceleration etc., Dynamic Similarity  Time, Discharge, Force, Pressure Intensity, Torque, Power Similitude or Similarities

22. 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 B D r m m m  L L P  B P  D P r 2 L A P A m  r 3 L V P V m  where L r is Scale Ratio

23. 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 P V m  V r a P a m  a r where V r is Velocity Ratio where a r is Acceleration Ratio Also the directions of the velocities in the model and prototype should be same

24. 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 P F m  F r where F r is Force Ratio 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.

25. Types of Forces Acting on Moving Fluid 1. Inertia Force, F i 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

26. Types of Forces Acting on Moving Fluid Inertia Force, F i Viscous Force, F v It is equal to the product of shear stress due to viscosity and surface area of the flow.

27. Types of Forces Acting on Moving Fluid Inertia Force, F i Viscous Force, F v Gravity Force, F g It is equal to the product of mass and acceleration due to gravity of the flowing fluid.

28. Types of Forces Acting on Moving Fluid Inertia Force, F i Viscous Force, F v Gravity Force, F g Pressure Force, F p It is equal to the product of pressure intensity and cross sectional area of flowing fluid

29. Types of Forces Acting on Moving Fluid Inertia Force, F i Viscous Force, F v Gravity Force, F g Pressure Force, F p Surface Tension Force, F s It is equal to the product of surface tension and length of surface of the flowing

30. Types of Forces Acting on Moving Fluid Inertia Force, F i Viscous Force, F v Gravity Force, F g Pressure Force, F p Surface Tension Force, F s Elastic Force, F e It is equal to the product of elastic stress and area of the flowing fluid

31. Dimensionless Numbers V Lg InertiaForce Gravity Force  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, R e = Froude’s number, F e = E u l e r ’ s n u mb e r , E u = e 4 . W e b e r ’ s n u mb e r , W = 5 . M a c h ’ s nu m be r , M = V p /  InertiaForce P r e s s u r e F o rc e  InertiaForce V SurfaceTensionForce  /  L    Viscous Force Inertia Force   VL or  VD Inertia Force  V E l a s t i c F o r c e C

32. 2 2 L V T T L L  ρ 3 V  ρ 2 L V  ρ

33. Model Laws The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. 1. Reynold’s Model Models based on Reynolds’s Number includes: Pipe Flow Resistance experienced by Sub-marines, airplanes, fully immersed bodies

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

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

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

37. Model Laws The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity. Reynold’s Model Froude Model Law Euler Model Law Weber Model Law Mach Model Law Mach Model Law is applied in the following cases: Flow of aero plane and projectile through air at supersonic speed ie., velocity more than velocity of sound Aero dynamic testing, c) Underwater testing of torpedoes, and d) Water-hammer problems

38. 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 m m m V L ρ V L ρ  V L t r r r  Time Scale Ratio  Velocity, V = Length/Time  T = L/V t a r V r r  Acceleration Scale Ratio  Acceleration, a = Velocity/Time  L = V/T     R R e e p m 

39. If the gravity force is predominant, the models are designed for dynamic similarity based on Froude number. Froude Model Law L r T r  S c ale R a t i o f or T i m e  g L m g L P m p V m  V p L r V r  V e l o c i t y S c ale R a t i o      F F e e p m  T r  S c a l e Ra t i o f o r A cce l e r a t i o n  1 r 2. 5 Q  Scale Ratio for Discharge  L r 3 F r  S c a l e R a t i o f o r F o r c e  L r F r  S c a l e R a t i o f o r P re ss u r e I n t e n s i t y  L r 3. 5 P r  S c a l e R a t i o f o r P o w e r  L r

40. Ref e ren c e Chapter 12 A Textbook of Fluid Mechanics and Hydraulic Machines Dr. R. K. Bansal Laxmi Publications