9/21/2016

ESO204A, Fluid Mechanics and rate Processes

Dimensional Analysis and Similitude

Simple and powerful qualitative technique applicable to many fields of science and engineering

Chapter 5 of F M White Chapter 7 of Fox McDonald

Dimensional Analysis

If certain physical phenomenon is governed by

where some/all of the fxx, ,.... x  0 12 n variablesx are dimensional

Then the above phenomena can be represented as

 12, ,.... m   0 where all the variables  mn are non-dimensional

The nature of f and ψ may be obtained from experiments

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Dimensional Analysis: Buckingham Pi Theorem

fxx12, ,.... xn  0  12, ,.... m   0 where some/all x where all  are are dimensional non-dimensional

where mnmnk , Minimum number of fundamental dimensions involved: k

Example: fVgh,, 0 n  3 k  2 mnk 1

Pi Theorem: Repeating and non‐repeating variables

x12,xx ,.... n  xr12,xxxxx r ,.... rk ; nr 1 , nr 2 ,.... nrm 

Selection of repeating Construction of Pi‐terms variables:   xxaaa11 x 12 x 13...  x a 1k o They must be 1112nr r r r 3 rk dimensionally

aaa21 22 23 a 2k  2212 xxnr r x r x r 3...  x rk independent ..... o Together, they ..... must include all

aaammm123 a mkfundamental   xx x x...  x mnrmr12 r r 3 rk dimensions

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Experiment shows, for viscous flow fVgh ,,,   0

MLT V 01‐1

g 01‐2n  4 k  2 m  2 h 010We have to select two ν 02‐1(02) repeating variables

Let’s take the repeating variables: gh,

Non‐repeating variables: V ,

LT fVgh ,,,   0 n  4 k  2 m  2 V 1‐1 Repeating variables: gh, g 1‐2 Non‐repeating variables: V , h 10   xxaaa11 x 12 x 13... x a 1k ν 2‐1 1112nr r r r 3  rk

ab a   Vg h 00 1 2 b 112ab a 1 L T LT LT  L  LT V ab 12   1 gh

ab 00 2 1 2a b similarly 2  ghLT LT LT  L  2012ab aab12,  32  2  gh3

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V  fVgh,,,  0 f ,0 1 3 gh gh

V  V  Fr  Fr Froude number   gh gh3 gh Vh Re

We may also write f2 Fr, Fr Re  0 Fr  Fr Re

Frictionless flow: Fr constant

Viscous flow: Fr  Fr Re Experiments are necessary to find the nature of function

Advantages of dimensional analysis

fVgh,,,  0 Fr  Fr Re

V Fr  Fr   gh Re gh3

o Less number of experiments are necessary, as opposed to the dimensional system o Experiments become inexpensive o Data reduction becomes easier, single plot is sufficient to show the results

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Dimensional analysis shows Fr  Fr Re Vfgh  ,,  We would like to do a laboratory experiment now We can vary h and measure V for few 2 cases experiments V Fr  Finally we do a curve‐ Fr Fr   fitting gh Re gh3 *Nature of curve is arbitrary in this case, actual curve may fitted curve look slightly different Fr Re

Fr  Fr Re Conclusions o One Figure is enough, as Vfgh  ,,  opposed to many Figs of dimensional system o Tank size is irrelevant as long as our assumptions 2 experiments aren’t violated V Fr  The plot, developed Fr Fr   from experiments in a gh Re gh3 smaller (model) tank may be used for similar fitted curve bigger (prototype) tank Fr Re

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Similitude: Basic idea behind model testing

 For the present case study Fr   3 gh

Since the relation holds for similar model and prototype tanks

 if  gh33 gh model prototype

then Fr Fr  model  prototype

Model studies (similitude) Certain fluid mechanical phenomenon is governed by

f 12, ,.... n  0 where  i are non-dimensional

When the model is similar to the prototype , in 1,2,... iimodel  prototype

Complete similarity requires

Geometric similarity + Kinematic similarity + Dynamic similarity

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Geometric similarity: length‐scale matching

A model and prototype are geometrically similar if and only if all body dimension in all three coordinates have the same linear scale ratio

All angels, flow direction, orientation with the surroundings must be preserved

Kinematic similarity (velocity‐scale matching) A model and prototype are kinemetically similar if homologous particles lie at homologous points at homologous time

Kinematic similarity requires geometric similarity

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Dynamic similarity (force‐scale matching)

A model and prototype are dynamically similar if ratio of any two forces are same for model and prototype

Dynamic similarity requires geometric, kinematic similarities

Check before model testing Geometric similarity + matching of Pi‐terms

o Geometric similarity depends on proper design, manufacturing, material choice o Proper choice of variables to include necessary fluid dynamical effects o In reality, it is not always possible to attain complete similarity, and forced to work with partial similarity. May happens for more than three dominant forces

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When fluid flows over an object, the object experiences fluid resistance known as ‘drag force’

For incompressible flow with ‘smooth’ objects the drag force is given by

F  fL ,u,,

Conduct dimensional analysis to identify the dimensionless numbers associated with the above phenomenon

A running car experiences fluid resistance known as ‘drag force’ F  fL ,u,,

We are interested to MLT conduct a model study u 01‐1 in a wind tunnel to L 010 know the drag on the prototype ρ 1‐30 11‐2 n  5 k  3 m  2 F μ 1‐1‐1 Repeating: L,u,

a b c ab c 1  FLu  2  Lu 

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