Learning Objectives
1. Introduction to Dimensions & Units 2. Use of Dimensional Analysis 3. Dimensional Homogeneity 4. Methods of Dimensional Analysis 5. Rayleigh’s Method Learning Objectives 6. Buckingham’s Method 7. Model Analysis 8. Similitude 9. Model Laws or Similarity Laws 10. Model and Prototype Relations 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 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 kg M Length m L Time s T Secondary or Derived Dimensions Secondary dimensions are those quantities which posses more than one fundamental dimensions. 1. Geometric a) Area m2 L 2 b) Volume m3 L 3
2. Kinematic a) Velocity m/s L/T L.T-1 b) Acceleration m/s2 L/T2 L.T-2 3. Dynamic a) Force N ML/T M.L.T-1 b) Density kg/m3 M/L3 M.L-3 Problems Find Dimensions for the following: 1. Stress / Pressure 2. Work 3. Power 4. Kinetic Energy 5. Dynamic Viscosity 6. Kinematic Viscosity 7. Surface Tension 8. Angular Velocity 9. Momentum 10.Torque Use of Dimensional Analysis 1. Conversion from one dimensional unit to another 2. Checking units of equations (Dimensional Homogeneity)
3. Defining dimensionless relationship using a) Rayleigh’s Method b) Buckingham’s Ï€-Theorem 4. Model Analysis Dimensional Homogeneity Dimensional Homogeneity means the dimensions in each equation on both sides equal. Problems Check Dimensional Homogeneity of the following: 1. Q = AV 2. EK = v2 /2g 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. 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 X1 ,X2 , X3 and mathematically it can be written as X = f(X1 , X2 , X3 ) This can be also written as X = K (X1 a , X2 b , X3 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. 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 X1 ,X2 , X3 and mathematically it can be written as X = f(X1 , X2 , X3 ) This can be also written as X = K (X1 a , X2 b , X3 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.Aa .Vb 1
where K is a Non-dimensional constant Substitute the dimensions on both sides of equation 1 M0 L 3 T-1 = K. (L2 ) a .(LT1 ) b Equating powers of M, L, T on both sides, Power of T, -1 = -b b=1 Power of L, 3= 2a+b 2a = 2-b = 2-1 = 1 Substituting values of a, b, and c in Equation 1m Q = K. A1 . V1 = V.A 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 X1 ,X2 , X3 and mathematically it can be written as X = f(X1 , X2 , X3 ) This can be also written as X = K (X1 a , X2 b , X3 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.
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 X1 ,X2 , X3 and mathematically it can be written as X = f(X1 , X2 , X3 ) This can be also written as X = K (X1 a , X2 b , X3 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. 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 (X1 , X2 , X3 , ……… Xn ) = 0 and variables can be expressed using m dimensions then f (Ï€1 , Ï€2 , Ï€3 , ……… Ï€n - m) = 0 where, Ï€1 , Ï€2 , Ï€3 , … are dimensionless groups. Each Ï€ term contains (m + 1) variables out of which m are of repeating type and one is of nonrepeating type. Each Ï€ term being dimensionless, the dimensional homogeneity can be used to get each Ï€ term. Ï€ denotes a non-dimensional parameter Buckingham’s Ï€-Theorem Selecting Repeating Variables: 1. Avoid taking the quantity required as the repeating variable.
2. Repeating variables put together should not form dimensionless group. 3. No two repeating variables should have same dimensions. 4. 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 Ï€-Theorem Example Example Example 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.
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