Dimensional and Model Analysis

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Cory Gibbs
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Dimensional and Model Analysis 5.1 Fundamental dimensions 5.2 Rayleigh s and Buckingham s method 5.3 Dimension less numbers and their significance 5.4 Hydraulic similitude 5.5 Type of models 5.

1 Dimensional and Model Analysis 5.1 Fundamental dimensions 5.2 Rayleigh s and Buckingham s method 5.3 Dimension less numbers and their significance 5.4 Hydraulic similitude 5.5 Type of models 5.6 Distorted models 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 Length Time Temperature Current Amount of Light Amount of matter m L t T I C N (kg) (m) (sec) (K) (A) (cd) (mol) 1

Basic for the correctness of any equation.

2 5.1 Fundamental dimensions 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} Principle Of Dimensional Homogeneity Basic for the correctness of any equation. If an equation truly expresses a proper relationship between variables in a physical phenomenon, then each of the additive terms will have the same dimensions or these should be dimensionally homogeneous. 2

{m/(t2l)} { gz} = {mass/length3 x length/time2 x length} ={m/(t2l)} 5.

3 Dimensional Homogeneity 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)} 5.2 Rayleigh s and Buckingham s method Buckingham, E. The principle of similitude. Nature 96, (1915). 3

Example: Drag on a Sphere Drag depends on : sphere size (D); speed (V); fluid density (r); fluid viscosity (m) Buckingham Pi Theorem Step 1: List all the parameters involved Let n be the number of

4 Example: Drag on a Sphere Drag depends on : sphere size (D); speed (V); fluid density (r); fluid viscosity (m) 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 3 Buckingham Pi Theorem List the dimensions of all parameters Let r be the number of primary dimensions Example: For drag on a sphere r = 3 Step 2: Select a set of primary dimensions For example M (kg), L (m), t (sec). Example: For drag on a sphere choose MLt Buckingham Pi Theorem Step 4 Select a set of r dimensional parameters that includes all the primary dimensions Buckingham Pi Theorem Step 5 Set up dimensionless groups πs There will be n m equations Example: For drag on a sphere Example: For drag on a sphere (m = r = 3) select ϱ, V, D 4

Buckingham Pi Theorem Step 6 Check to see that each group obtained is

The pressure drop ΔP per unit length in flow through a smooth circular

D (iii) density of the fluid ρ, and (iv) dynamic viscosity μ.

flow. (b) Using Rayleigh method evaluate the dimensionless parameters.

5 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 Example The pressure drop ΔP per unit length in flow through a smooth circular pipe is found to depend on (i) flow velocity, u (ii) diameter of the pipe, D (iii) density of the fluid ρ, and (iv) dynamic viscosity μ. (a) Using π theorem method, evaluate the dimensionless parameters for the flow. (b) Using Rayleigh method evaluate the dimensionless parameters. Step 1 : tabulate dimensions of the parameters Selecting D, ρ and μ are as repeating variables. 5

Similitude & Dimensional Analysis Modeling and Similitude A

to predict the behavior of the system in some desired respect, e.

drag at high Reynolds number OR Coefficient of drag Example

You can to model drag Pi theorem tells To ensure similarity w/h

6 Similitude & Dimensional Analysis Modeling and Similitude A model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect, e.g. MODEL ANALYSIS Flow around bodies We are often interested in drag at high Reynolds number OR Coefficient of drag Example Consider flow past some plate. You can to model drag Pi theorem tells To ensure similarity w/h and ρvw/µ must be the same in model and prototype. If this is true D/w2ρV2 is the same in model and prototype 6

Types of Similarity Geometric (ratio of length scales the same) Kinematic

situation is: Get all dimensionless variables (Pi groups) the same between model

Photographic enlargement KINEMATIC (ϕm = ϕp) Same flow coefficients Same fluid

7 Types of Similarity Geometric (ratio of length scales the same) Kinematic (velocity structures are the same) Dynamic (ratio forces the same) The best situation is: Get all dimensionless variables (Pi groups) the same between model and prototype. Then all similarities are preserved.. Sometimes hard to achieve it all. Similarity Laws GEOMETRIC Linear dimension ratios are the same everywhere. Photographic enlargement KINEMATIC (ϕm = ϕp) Same flow coefficients Same fluid velocity ratios (triangles) are the same DYNAMIC (ψm = ψp) Same loading coefficient Same force ratios (and force triangles) Energetic (Pm = Pp) Same power coefficient Same energy ratios. 7

8 Forces Acting in a Fluid Flow 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. Eulers s Number, Re: It is the ratio of inertia force to the pressure force of flowing fluid. 8

9 Mach s Number : It is the ratio of inertia force to the elastic force of flowing fluid. 9

10 10

Reduction of Variables Dimensional Analysis

Reduction of Variables Dimensional Analysis 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),