Modeling and Similarity

Example 1: You have been commissioned to conduct a series of experiments to determine the drag on a supersonic aircraft. More particularly, your client would like for you to tell him which system parameters affect drag and whether they increase or decrease the drag. How do we determine whats really important?

Buckingham Pi Theorem

An equation involving k independent variables and r reference dimensions can be reduced to a relationship in k-r independent parameters.

In other words, instead of having to worry about k variables, whats really important in describing the physical phenomena are the k-r parameters.

The k-r parameters are called pi terms.

Buckingham Pi Theorem

The k variables are things like density, pressure, temperature, velocity, viscosity, etc.

The r reference dimensions are either Length L, Mass M, and Time T, or Length L, Force F, and Time T. Pick one system (M, L, T) or (F, L, T) and be consistent.

The number of reference dimensions for a particular problem depends on the dimensions in the relevant variables.

Buckingham Pi Theorem Example: we are interested in the drag over a

submerged sphere. The relevant variables would be density, velocity,

viscosity, diameter, and drag. Therefore, k=5. Reference dimensions

Density = kg/m3 or M/L3. Drag = N or kg(m/s2) or ML/T2. Velocity = m/s or L/T. Viscosity = Ns/m2 or FT/L2 or (ML/T2)(T/L2) or M/LT. Diameter = m or L. Therefore, r=3.

The minimum number of relevant parameters is k-r = 5-3 = 2

Determination of PI terms List all independent variables involved in the

problem Geometry - size, length, etc. Fluid properties density, viscosity, temperature External effects pressure, forces

Express each variable in terms of the basic dimensions (either LMT or LFT) Choose whichever is simplest for the particular

problem.

Determine the number of required PI terms from the Buckingham PI theorem

Determination of PI terms Select a number of repeating variables

The number of repeating variables will be equal to the number of relevant dimensions

The repeating variables should be dimensionally independent i.e., you cant express the dimensions of one repeating variable in

terms of the dimensions of the others. The repeating variables will be used to express the

remaining variables as groups of dimensionless PI terms. Since we usually want to see how one variable is

influenced by another, dont choose either as a repeating variable. (For example, we might wish to see how drag is influenced by velocity.)

For convenience, choose variables having the simplest dimensions.

Determination of PI terms Form PI terms:

where the us are non-repeating variables. Express final form as relationship between PI

terms:

1 2 3

1 2 3

1 1

2 2

.

a b c

a b c

uu

etc

P P PP P P

3 3

1 1 2, ,..., iI3 3 3 3

Example Newtonian fluid flows through a long, smooth-

walled pipe. We are interested in what factors affect the pressure drop per unit length?

1. List variables of interest: pressure drop per unit length, viscosity, density, diameter, fluid velocity.

2. Write variables in terms of their basic dimensions

, , ,lp f D VG P U

Example

2 3

2

3 4

2

1l

F Fp LL LD L

M FTL LFTLLV T

G

U

P

Example

Determine the number of PI terms k=5, r=3 we need 2 PI terms

Select the repeating variables D, V, U are simple, have all relevant dimensions,

are dimensionally independent, and dont use the dependent variable.

Find the required PI terms

Example 2

1 3 2

1 2 11 2

: 0 1: 0 3 4: 0 2

121

cba b c a

l

l l

F L FTp D V L TL LF cL a b cT b c

abc

Dp DV pV

G U

G U GU

3

3

2

2 2 2

1 1 12

: 0 1: 0 2 4: 0 1 2

111

cba b c aFT L FTD V L TL L

F cL a b cT b c

abc

D V VD

P U

PP U U

3

3

2lDp VDV

PG I UU

Notes

Form of the function must be theoretically or empirically determined.

A crude theory may be needed to determine the relevant variables.

Be sure to include all physically important quantities even if they are not variable, e.g., gravity.

Make sure that all variables are independent.

Common Dimensionless Groups

2

inertial forcesRe Reynolds numberviscous forces

inertial forces Weber numbersurface tension

inertial forces Mach numbercompressibility

local inertial force

VD

V LWe

VM cLSt V

UPUV

Z

s Strouhal numberconvective inertial forces

If a problem has only 1 PI term, the PI term should be held constant between the experiment and real situation.

If a problem involves 2 or more PI terms, then one PI term at a time should be varied in the experiment while holding the others constant.

Using this method, the functional dependence between the PI terms can be determined.

Modeling

Similarity and Modeling Model

a lab device used to obtain information about a real/proposed device.

Typically, the model will be reduced scale. How do we design the model/experiment so that the

results will be applicable to the real/proposed device? Similarity.

Kinematic Similarity Model needs to be the same shape, have the same angle of attack, etc. as the actual device.

Dynamic Similarity Flow phenomena (e.g., turbulent/laminar, supersonic/subsonic, etc.) should be the same.

Similarity and Modeling

Validation predictions using model results should be verified, if possible, using the prototype. If the model and the prototype results agree for a few

test cases, the model results can be safely used. This might be done with a variety of models to see if

one model predicts the results for another.

Distorted models sometimes similarity cant be completely achieved. Better than nothing, but Interpret results carefully!

Modeling and Similarity

Assuming that the functional dependence between PI terms is the same for the model and prototype, the following modeling laws apply

1 2 3 1

1 2 3 1

2 2 3 3 1 1

1 1

, ,..., for the prototype

, ,..., for the model

if , , ,

then .

k r

m m m k r m

m m k r k r m

m

II

3 3 3 3

3 3 3 33 3 3 3 3 3

3 3

Example Consider the aerodynamic drag on a thin,

rectangular plate having dimensions w x h. The Buckingham PI theorem yields the

following relationship between PI terms

We would like to design an experiment that uses a model to represent the actual situation. We might want to use water instead of air, or We might wish to use a different size plate.

2 2 ,D w Vw

hw VUI PU

Example Similarity between the model and the

prototype requires that

So, we can choose the height of the model plate, but its width must scale proportionally, and

We can choose the working fluid, but we must increase or decrease the velocity accordingly.

and

and V

m m m m

m m

mm m m

m m

w V ww Vwh h

w ww h Vh w

U UP P

P UP U

Example

Finally, how is the drag measured in the model experiment related to the drag for the prototype? 2 2 2 2

2 22 2

2 2

or

D=

m m

m m

m

m

m

m

mm m m

DDw V w V

DD w Vw V

w V Dw V

U U

UU

UU