CHAPTER-I

SIMILARITY TECHNIQUES

1.1 History: It is just over 100 years since Helmholtz (1873) introduced the dimensional

analysis approach in which the similarity parameters were developed from the equations

of motion. Over the past century a number of notable models, Reynolds (1885) was not

only the first to demonstrate the importance of the similarity parameter, CρU/μ , later

called the Reynolds number, on the onset of turbulence in pipe flow in 1983, he was also

the first to have developed a model of a river estuary reported in 1987.During this same

nineteenth century period fluid flow models received a great impetus with the

introduction of towing tanks, wind tunnels and water tunnels and the successful models

of boats, turbines, windmills and steam engines. After Fourier, Rayleigh (1892) initiated

the concept of dimensional analysis in the well known pi theorem. This method

dominated engineering studies of fluid flow models until Birkhoff (1950) formalized the

method first suggested by Helmholtz (1873) using the concepts of the mathematical

invariance of the equations of motion under a group transformation. The concept of self-

similar flows and models was introduced by Blasius (1908) by his solution of a boundary

layer on a flat plate, but the method appeared to involve good fortune as well as intuition

in the works of those who found self-similar solutions in boundary layers and

compressible flows until Morgan (1952) , following the concepts of Birkhoff showed

that the boundary layer equations were invariant under certain group transformations that

reduced the number of independent variables by one.

Traditional appears to have the central role in the study of similarity and models.

The practice of naming dimensionless fluid flow parameters after contemporary

researchers has persisted over the years since 1919 when Weber first named the

Reynolds, Froud and Cauchy numbers. Today we have about 300 similarity numbers

identified for the fluid flow systems. A few of these numbers remain anonymous, a few

have more than one name and some names apply to more than one parameter. The list

1

grows each year as people investigate more complex fluid systems, so that it is necessary

to consult a comprehensive list of dimensionless groups before presenting new results.

The century of work on similarity and fluid flow models is so enormous that it is

virtually impossible to review it thoroughly so that there is a real danger that methods and

applications will be rediscovered The depth and breadth of understanding in the area is

reflected in the books by Kline (1965) and Gukhman (1965) where the authors have

presented the concepts and reviewed and literature. Kline in his book makes a

comparative study to show that a “Fractional analysis” of the equations of motion yields

the appropriate parameters directly and rigorously, but it requires slightly more

mathematical sophistication than does “dimensional analysis” using the pi theorem. In

spite of the wide spread understanding of similitude there is some need to state the

method more completely and explicitly for fluid flows and to show how the method may

be used for complex flow systems with a large number of similarity parameters as well as

simple flow systems. In discussing dynamic similarity Lighthill (1950) referred to the five

similarity parameters of compressible flow as an “enormous number” which has proved

to be of only restricted value for high speed flows. On the other hand, complex flow

systems are not uncommon in nature, hence there is a real need to present empirical data

from such systems concisely and in a form that permits quantitative comparisons of

various parameters and hence physical effects. Such a form necessitates the use of

similarity parameters.

1.2 Basic concepts: A model of a system is a replica of the original system. The model is to the

original system as a map is to the countryside. The points on the map are related to the

distance between places in the countryside by a map scale.

The model, which is the basis of all predictions, may be physical, mathematical or

purely intuitive as based on previous analytical and physical experience. The physical

model of the system, called the prototype, has a one to one functional relationship

between it and some of the properties of the prototype. The mathematical model, which

2

may be based on empirical data or which may be deduced from physical principles, has a

similar one to one relationship between the symbols of the model and the properties of

the physical system. The magnitude of the discrepancy between the predictions of the

model and the behavior of the prototype is the absolute error of prediction. Since errors

are associated with all measurements and errors between the model prediction and

prototype behavior can never be zero, however, the discrepancies may be reduced to the

magnitude of uncertainty associated with the measurement. Under these circumstances

the model is considered to be exact.

The formulation of the mathematical problem is always important in the

presentation of the behavioral characteristics of fluid flow systems. The reasons for this

initial step is trivial for analytical solutions, but for physical modeling and for presenting

information on a system behavior this statement requires some justifications. In the

subsequent discussion it will be shown that the mathematical model is the rational basis

of all physical modeling and for the presentation of empirical results of the behavior of

complex systems.

1.3 General Transformation Methods in Flow Theory: 1.3.1 General Concepts:

An appropriate change of variables is probably one of the most useful methods

available for solving the partial differential equations of mathematical physics and

engineering boundary value problems. The most general criterion for a transformation is

simply to change given problem into a simpler problem in some sense or other; that is,

either to a form which will yield to more standard solution techniques, or possible to a

form which has been previously solved in connection with a related or similar problem.

Hodograph transformations and conformal transformations are well-known methods for

transforming given problems into form which yield to classical techniques (such as

separation of variables). Generally, various transformation techniques can be classified

into three groups: transformation only of the dependent variables, transformation only of

the independent variables and the mixed transformations of the both independent and

dependent variables. However, all three groups have a common goal: to find a relation,

3

or more specifically, a basis of comparison, between different physical (or mathematical)

problems. This broad concept of comparison is the definition of the term “generalized

similarity.”

Generalized similarity might be applied in fluid mechanics to attempt to answer

such questions as “Is there any similarity (i.e., basis of comparison) between

compressible and incompressible flow problems, axisymmetric and planar flows, or in

general, any more complicated and a less complicated flow?” by contrast, the usual term

“similarity” as used by Birkhoff (1950), Morgan (1952), Hansen(1964), Abbott and Kline

(1960), and others is defined in terms of independent variables of a problem. Thus it may

ultimately refer to a physical similarity within a given problem, such as similarity of

velocity or temperature profiles.

Much of the previous work on similarity was motivated by the desire to develop

simple methods for reducing the number of independent variables. Out of this previous

work came the realization that each of the proposed methods was based on an assumed

class of transformations and the recognition that more general classes of transformations

might lead to the solution of a wider class of problems.

1.3.2 Typical Transformations Used in Fluid Mechanics:

As a mean of developing some insight into the question of “the most general class

of similarity transformations” and the concept of “generalized similarity,” Table 1.1

compiles of examples of as many different types of known transformations as could be

found to present the field of fluid mechanics.

4

TABLE 1.1 Various Transformations.

Transformations Transformed Independent Variables

Similarity ξ (x) =

Meksyn-Gortler

Von Mises

Crocco

Mangler

Stewartson

orodnitsym

of high speed Flow

(i) Prandtl-Glauett

ξ (x) = x, η (y) = By

nsonic

By com ons in the table, an interesting conclusion can be

made: all of the transformed independent variables are of the general form ξ = ξ (x) and

x, η (x,y) = yγ (x)

∫ 0

x

ξ(x) = u1 (x) dx, η(x,y) = u1 (x) y/√x

ξ (x) = x, η (x,y) = ψ (x,y)

ξ (x) = x, η (x,y) = u (x,y)

ξ (x) = r20 (x) dx, η(x,y) = r0 (x) y ∫

0

x

ξ (x) = a1 (x) p1 (x) dx ∫ 0

x

η (x,y) = a1 (x) p (x,y) dy ∫x

0

ξ (x) = p1 (x) dx, η (x,y) = p (x,y)dy ∫ x x

0

∫ 0

D

Similarity rules

(ii) Gothert

(iii) Von Karman Tra

paring the various transformati

η = η (x, y) (i.e., one of the new variables is a function of only one of the original

variables). This realization raises the question “Is the general transformation ξ (x) and η

5

(x, y) the most general class of transformation that need be considered for physical

problems?” In an attempt to answer this question, it will first prove useful review the

mathematical theory of transformations.

1.3.3 The Primitive Transformation:

The following theorem can be found in the mathematical literature of general

THEO

f a region R in the x, y-plane onto a Region R' in the ξ, η -plane

Diff tion is o

ormations have an important interpretation and application

the presentation of deformation or motions of continuously distributed systems, such

as fluid

ractical” transformations of interest for solving physical

roblems should be one to one, that is, have a unique inverse (except possibly at a finite

= ξx ηy – ξy ηx

transformation theory.

REM: An arbitrary one to one continuously differentiable Transformation ξ = ξ

(x, y), η = η (x, y) o

can be resolved in the neighborhood of any point Interior to R into one or more

continuously differentiable “primitive” Transformations, provided that throughout the

region R, the Jacobian

J (ξ, η) = ∂ (ξ, η) ∂ (x, y)

f the form ers from zero. A “primitive” transforma

ξ = ξ (x) , η = η(x,y)

Now one to one transf

in

s. For example, if a fluid is spread out at a given time over a region R and then is

deformed by motion, the motion of the fluid is described by the coordinates in the

physical R-plane. If the fluid motion in R is characterized by the coordinates ξ, η. The

one to one character of the transformation obtained by bringing every point x,y into

correspondence with a single point ξ, η is simply the mathematical expression of the

physically obvious fact that the fluid motion in the physical R-plane must remain

recognizable after transformation to the transformed R'-plane, i.e., that the corresponding

motions remain distinguishable.

Physically, since most “P

p

6

numbe

ompleteness, the following two properties of the primitive

transformations are noted:

ation

ξ = ξ (x) , η = η(x,y)

d its Jacobian

0, y0), then in the neighborhood of P the

transformation has a unique inverse, and this inverse is also a primitive transformation of

the sam

e transformations, the sense of rotation in the x, y-plane preserved or

versed in the ξ, η -plane according as the sign of the Jacobian is positive or negative,

mary, the primitive transformation appears to be the most general class of

ansformations that need be considered, purely from a mathematical viewpoint, as long

as it is

To illustrate the concept of the relationship between the mathematical model and

t to take the case of a simple example such as spring –

mass-

r of singular points), it appears the primitive transformations or the resolutions in

the primitive transformations, should be considered in the search for “the most general

class of transformations.”

For the sake of c

(i) If the primitive transform

is continuously differentiable, an

J (ξ, η) = = ξ

Differs from zero at a point P(x

e type.

(ii) For primitiv

re

respectively.

In sum

tr

required that a unique inverse of a transformation must exist.

1.4 Models and Similarity:

the physical model it is convenien

damper system for which an analytical solution exists and a good understanding of

the motion is known. The differential equation of motion of mass, m, for any initial

disturbance is found from Newton’s second law to be

x ηy – ξy ηx∂ (ξ, η) ∂ (x, y)

7

+ + m d (1.1)

nt distance of the mass, f is the damping constant, k

spring constant and t is time. The most common situation is to consider some known time

const

(1.2)

ntal equation of modeling. When is the motion of a

ilar to the p

everything in the model system similar to the prototype system. The

eneral answer to this question is based on the concepts of mathematical equality and

physica

ndamped natural frequency , x0 is the initial

e special position at the equilibrium rest position of the mass, t1 is

amping factor or damping ratio. Applying the transformation as given by equation (1.3)

to equa

2x f

dx

Where x is the displaceme is the

raints at time, t1;

Now we pose the fundame

model spring-mass-damper system sim rototype? The trivial answer to this

question is to make

g

l analogy. If the terms in the equation of motion have the same relative size with

respect to each other at each instant then the two physical systems may be similar

provided the initial disturbances of the two systems are similar also. It is assumed in these

physical systems for both the model and the prototype. The most convenient method of

comparing the relative size of terms is to put each term of the equation of motion in to a

non-dimensional form without changing the final prediction of the model. This leaves the

physical interpretation of each term unchanged. To do this, the following non-

dimensional terms are introduced here,

X = (x = x1) / (x0 = x1)

T = (t - t1) ωn

2ςωn=f/m (1.3)

Where ωn = (k/m)1/2 is the u

displacement , x1 is th

the instant in the time when the initial conditions are defined, and ς = f /[2 (mk)1/2] is the

d

tions (1.1) and (1.2) gives,

dt2 dt = 0 kx

dx dt x (t = t1) = x0 , (t = t1) = ν0

8

(1.4) + + = 0 d2X

2 ς dX

ilar to equations (1.1) and (1.2), we

say that these equations are invariant under transformations (1.

model and the prototype will be similar when they ave identical equation of motion and

= time scale factor

rm of the equations of motion, (1.1)and

.2).

, x0, x1, t1 and v0. Changing any one of these parameters will change the resulting

otion. Graphically demonstrating the effect of a variation in the magnitude of each one

size of the spring-mass-damper system at all. Equations (1.4) and (1.5) could just likely

(1.5)

Since the solution of equations (1.4) and (1.5) is sim

3). The motion of the

h

initial conditions provided these equations are written in an non-dimensional form.

Under these circumstances the motion of the model xm(t) will differ from the motion of

the prototype yp(t) by a constant scale factor.

xm – x1m = Sd (yp – y1p) ; Sd = (x0 – x1) / (y0 – y1) = displacement scale factor.

tm – t1m = St (tp – t1p); St = ωnp /ωnm

From these simple scale factor equations we can predict the motion of the prototype from

the measurements of the motion of the model.

The non-dimensional form of the equations of motion (1.4) and (1.5), have some

important advantages over the dimensional fo

(1

(i) It should be noted that equations (1.1) and (1.2) have seven dimensional parameters,

m, f, k

m

of these parameters either from the analytical solution or from the experimental

measurements presented a time consuming task. On the other hand, equations (1.4) and

(1.5) have only two dimensionless parameters, ς and V0.

(ii) The dimensionless form of equation (1.4) and (1.5) do not reflect any overall physical

dT2 dT X

dXdT X (T = 0) = 1 , (T = 0) = V

(x0

ν0

0-x ) 1

9

be the model of the voltage or current in an electrical circuit of an inductor, a resistor and

a capacitor or it could be a model of an acoustical inductor, capacitor and damper system.

pothesis:

well set equations

M{gi [mi (xi) ]}, p {hi [pi (yi )]}

hat results from the transformation,

pi – p1i = (p0i – p1i) βi

1i) η j (1.6)

f the governing system of equations are equal provided the system operate in stable

operties and dimensions of

ero depending on the original coordinates and properties selected and αi ,βj , ςj and ηj are

the dimensionless variables.

There are a number of different physical systems whose motion can be modeled by

equations (1.4) and (1.5). This fortune state of affaires means that the engineer can select

the most convenient kind of physical models to study. To facilitate modeling of this kind

very powerful general methods have been developed notably the digital and analog

computers in addition to purely mathematical analysis. For fluid systems no such

universal techniques are available.

On the basis of this sample example the concept of modeling a simple system

with a known, well set, exact mathematical model can now be generalized for any

physical system, in the following hy

Given a physical model, M, with a set of measurable properties {mi} in a

space{x } of a physical prototype P with properties {p } in a space {yi i i} and an exact

mathematical model is given by the system of

For both the model and the prototype, then the physical model is exact when the

dimensionless parameters

G (m0i , x0j) H (p 0i , y0j )

T

m – mi 1i = (m0i – m1i ) αi ;

xi – x1i = (x0i – x1i) ςi ; y – yi 1i = (y0i – y

O

mode, where m0i , x0j , poi and y0j are characteristic constant pr

a system, m , x1i 1j, p1i are biases in the properties and dimensions that may or may not be

z

10

The concept of modeling as expressed in the above hypothesis is dependent on the

invariance of the system of governing equations under a linear group transformation. The

major difficulty with this approach to modeling knows when a mathematical model of a

physical model is exact. This difficulty can not be resolved directly since it would require

the construction of both the physical model and the prototype and the performance of

extensive repeated tests and iterative adjustment to finally approach the condition of an

exact model. Such a procedure would negate the prime purpose of modeling.

Consequently modeling is unlikely to be exact in most practical situations. However, the

attempted formulation of the exact mathematically well-set models of physical systems is

the only general method available to reduce errors between the model and the prototype

to a minimum.

1.5 Fluid Flow Similarity:

The equations of motion at any point ( t , x ) for a simple fluid system such as an

ideal gas may be written in the differential form ;

(1.7a)

(1.7b)

(1.7c)

Where ρ is density, V is velocity, p is pressure, τ is the sheer stress tensor, f is

the body force vector, U is internal energy and

q is the heat flux vector. Introducing

characteristics constants for each of the independent and dependent these equations

be transformed in to the dimensionless form;

can

(1.8a)

= + ρ ∂t ∂ V V ρ V

Δ Δ= τ- p Δ + - ρ f

Vq := τ+= + ρ ∂

∂t U

U ρ V

Δ

-p Δ V

Δ

V -

= 0 ∂p

+ ( ρ V ) Δ

1 1

∂t

∂p1

= 0 (ρ V ) 1 Δ

∂t1 + S

11

(1.8b)

(1.8c)

the Froude number, E is the Eckert number, γ is the ratio of specific heats and Pe is the

Peclet number. For compressible flows of ideal gasses it is usually convenient to replace

e E

And

E = (γ – 1) M2

Where M is the Mach number. For free convection of gases the Froude number is

replace

Eu = σ

ere is the Weber number and

Where the primes denote the dimensionless form of the terms in equations (1.6)

and S is the Stronghal number, Eu is the Euler number, Re is the Reynolds number, Fr is

th uler number and the Eckert number by

d by

Fr = Re2 /Gr

Where, rG is the Grashof number. For the motion of bubbles, drops or thin films of

liquid and for the study of cavitations the Euler number is written

and

Wh We σ is the cavitations number. For each of

less parameters some physical interpretation of the number can be

ted as actual local force ratio or energy ratios. In spite of this the equations

ions (1.8) does make it possible to compare the relative sizes of terms

nal to the local force

ratios and energy ratios in a given flow region. On the basis of this certain terms can be

ρ1f 1

the above dimension

given in terms of ratios of characteristics forces or energy fluxes; however these are not

to be interpre

in the form of equat

in the equations for various parts of the flow regime since the size of each dimensionless

number may be selected so that they are approximately proportio

1 Fr

-=1 τ

Δ1 1 Re = + ρ1

1 ∂t1 ∂ V1

ρ1 V1 V 1Δ 1Δ-E1 u p1+S

-Δ1 V1

=1 τ

γRe E

+ ∂

∂t1U1

ρ1 S ρ1V1 Δ1 U1 = γ E Δ1 p1

V1- + : V1

1 q 1 pe

Eu = γM2

1

Eu = We

1

12

omitted from the equations as being negligible compared to other terms. This is the basis

f approximation theory which permits simplification of general equations to the point

that a t l r e

o

heoretica esult still accurately predicts xperimental results. In order to do this it

may be necessary to introduce slightly modified dimensionless parameters other than that

suggested by equation (1.8). This is the case for flow in a boundary layer, but no

modification is necessary to drop the viscous terms for flow outside a boundary layer.

In spite of what now appears to be a hopeless complex situation a well set

problem for a general fluid flow situation has not been established. If a count of the

number of unknowns in the above equations is made it will be discovered that there are

far more unknowns than equations, hence, several additional equations must be

introduced. Equilibrium thermodynamics permits us to introduce equations of state such

as; p (ρ .T), Δh = Cp ΔT where T is temperature, h is enthalpy and Cp is specific heat.

The Fourier heat conduction law q k T= − ∇ is used in all problems of heat conduction

and convection where k is the thermal conductivity. The constitutive equation relates the

stress and the rate of strain in fluid motion. For an incompressible Newtonian fluid the

sheer stress becomes, ( )2τ μ= ∇ . None of these equations introduced a new non-

dimensional number directly, however, new numbers can be come important with large

property variations. For example viscosity, μ , may vary with temperature to such an

extent that its effect must be included in the model i.e. μ = μ0 (T/T )n , where T0 0 is a

reference temperature. In this case the dimensionless number “n” becomes a new

parameter that should be modeled along with the other non-dimensional numbers and the

momentum and energy equations b upled or interdependent.

Finally we may have a balance between unknowns and equations so that by

introducing the appropriate in and boundary conditions for each variable

introduction of initial and boundary conditions will simply add an equal number of

dimensionless parameters that must be matched between the model and the prototype.

For fluid flows the matching of the boundary conditions can be very difficult. For

ecome co

itial conditions

we could in theory have a well set problem. It was noted previously that, in general, the

13

example microscopic roughness of a flow surface such as the inside of a pipe or the

outside of a wing may have a profound effect on the resultant flow. In the case of the

pipe it may lead to a very large increase the lift on the wing. Even on a free boundary

uch as the inlet flow to a control volume the variations in the heat and mass transfer

from s

from a prototype, and that is, the flows must be kinematical similar or the

reamlines must be geometrically similar between the model and the prototype.

s

urfaces. To account for this microscopic boundary conditions in our non-

dimensional formalism we must introduce new dimensionless variables such as the

relative roughness, є/D, of solid surfaces and the intensity of turbulent fluctuations, I =

[v 2 21 /v ]1/2, for the inlet form, where v0 1 is the average velocity fluctuation about the mean

velocity v0.

Even after we have a model that is similar in geometry to the prototype and we

have matched all the non-dimensional parameters, the initial and boundary conditions,

the flow in the model and in the prototype can still be quite different. The reason for this

apparent contradiction of our mathematical intuition is that fluid flows can have more

than one stable configuration of their streamlines for any given set of boundary

conditions. Practical advantages are taken of this fact in a distable fluidic amplifier

where the jet flow can attach itself to either of two adjacent walls. Consequently we must

impose yet other restrictions on the formalism that must be followed when a flow model

is designed

st

Finally, under certain conditions laminar flows become unstable and a non-

measurable disturbance can grow in size so that the whole flow may become turbulent,

there is an element of uncertainty about any unstable flows. Under these circumstances it

is impossible in principle to develop an exact model of a prototype in spite of all the

precautions that are taken initially. Most flow situations of engineering significance

involve laminar flow or both laminar and turbulent flow. It is these flow problems that

involve the most uncertainty.

14

1.6 Se

ting results. This procedure

hich was initiated by Blasius (1950) for flow over a flat plate has been formalized by

organ(1952) for the general case. This is stated below in the form.

tial differential equations Φδ = 0 (δ = 1,2,………n) in which xi

(i = 1…

the condition that

stem of equation are invariant under 1.10 than for α1 ≠ 0 the independent and dependent

varia

lf Similarity: A particular case of similarity which is of considerable importance in fluid flow

occurs when each part of the flow is similar to any other part. Such a flow, which occurs

in boundary layers and some compressible flows is said to be self-similar or just similar

by most authors. A flow that is self similar can not only be modeled as suggested

previously but the number of independent variables is reduced by one by selecting new

variables so that the further economy is affected in presen

w

M

For a system of par

…….m) are the independent variables and yδ(δ = 1,………n) are the dependent

variables that is invariant under the transformation ,

⎯x1 = aα1 α2 x = a xr r (r = 2…..m) (1.9) And

⎯yδ = aαδ yδ (δ=1..n) (1.10)

Where a ≠ 0 and is real and α α1, 2 ……….. are to be determined by

sy

bles may be taken to be

( )1

2...r

rr

xn r mx

αα

= =

1

( ) ( ) 12 1 1........ .... /m mf n n y x x x

δαα

δ δ=

The application of this transformation can be made clear by a simple example,

Stoke ms proble for the impulsively started infinite plate. The equation of the motion is 2

2t yυ=

∂ ∂

Where u is fluid speed, t is time, y is the distance normal to the plate and υ is the

kinematics viscosity.

u u∂ ∂

15

The bou e ndary conditions ar

u ( t > 0 , y) = U1

u ( t, y → ) = 0

e initial condition is

u

t = am t, ⎯ = an y, ⎯U = as U

n is invariant under this transformation provided

m = 2n

able taken to be

∞

And th

( t = 0 , y ) = 0

The transformation group consists of

⎯ y

The equation of motio

s - m = s - 2n or

The new independent vari

yt

η =

And the dependent variable remains u when s = 0. The new boundary conditions become

U (η = 0) = U1

U (η→ ) = 0

Wher fferential equation

∞

e the equation of motion is now the ordinary di

2

2 2dy dyν1 0d U dUη+ =

self similar or universal velocity profile as a The solution of this equation yields a

function ofη .

A special case of self-similarity occurs when the flow system is subjected to

periodic time forcing functions or boundary conditions. Such a flow will be periodically

xample of such a motion would occur when a spring mass system

d c forcing function for a long period of time.

may be explicitly

stated as follows:

1. The model and the prototype must be geometrically similar.

2. the dimensionless parameters of

self-similar in fully developed flow regions where transient effects in time and space are

negligible. A simple e

is subjecte to a periodi

In summary, the general conditions for exact large scale fluid modeling

16

(a) the equation of motion

(b) the property relations

n of Different Similarity Techniques:

(c) the initial conditions

(d) The boundary conditions (micro and macro) must be equal between

the model and the prototype.

3. The flows must be kinematical similar.

4. The flows should be stable.

1.7 IntroductioGeneral Concepts:

The concept of similarity solutions is very old and dates hack to such well-known

workers as Bu geman (1931). Originally, the term similarity ckingham (1914) and Brid

analysis implied a procedure for finding some information about the solution of

particular phys t of a complete mathematical answer. In the past ical problem usually shor

work of such analysts as Birkhoff (1950) and Sedov (1969), similarity solutions have

incorporated rigo es which result in solutions in the rous mathematical techniqu

enginee articular importance, the more recent ring sense, that is, numerical answers. Of p

work has yielded solutions to nonlinear partial differential equations which have been

intractable to m re standard solution techniques. In fact, exact solutions of the boundary-o

layer equations of fluid mechanics are almost universally based on similarity methods.

The various similarity solution methods can be broadly characterized as

techniques which employ transformations of variables or parameters, or both. Typically

transformations may linearized a problem (for example the Kirchhoff and hodograph

transformations), reduce partial differential equations to ordinary differential equations

(for example the Blasius similarity transformations), transform the system to one already

solved (such as the Mangler transformation), or perform some other reduction of

mathematical complexity. Sedov (1959) formulates a mathematical theory of similarity

on the basis that two phenomena are similar, if the assigned characteristics of the other

by a simple conversion, which is analogous to the transformation from one system of

units of measurement to another. Thus, a certain analogy exists between the theories of

17

dimensional analysis as formulated by Buckingham (1914) and Bridgeman (1931) and

the geometric theory of invariants relative to a transformation of variables, a fundamental

theory in modern mathematics and physics. Some authors have even found it convenient

to define classes or types of similarity. Kline (1965) defines "external similitude" as the

similarity between problems of given class (a transformation in terms of parameters

alone) and "internal similitude" as a similarity characterized by a mathematical

relationship between points inside a single system of a given class (for example, a

transformation which reduces the number of independent variables).

Inherent in any general theory of similarity, however, should be the recognition of

specifying, in some sense or other, the complete physical problem to be analyzed. In

fact, the term "similarity" implies a comparison between two or more complete and

recognizable systems.

Thus, it is of prime importance to formulate a "complete" mathematical

description of the physical problem under consideration before proceeding with a

similarity solution. Gukhman (1965) characterizes a complete problem as one which is

expressed in terms of the governing equations together with a body of information termed

the conditions for uniqueness of solution which may contain not only boundary and

initial conditions in the usual sense, but also possible auxiliary physical considerations

such as conservation requirements. To be sure, it is not always possible to prescribe the

uniqueness conditions a priori for all problems that engineers are called on to consider.

In fact, it appears that much of the work that has been done in the past to develop

similarity analyses has been motivated by an initial statement of completeness, or lack of

it. For example, the early work of Buckingham (1914) and Bridgeman (1931) on

dimensional analysis requires only recognition of the pertinent variables which apply,

without any statement being made concerning the governing equations. Later workers

such as Morgan (1952), Hansen(1964), Krzywoblocki(1963) and Wecker and Hayes

(1960) investigated similarity methods by considering the governing equations first, and

only examining the boundary and initial conditions as a later step, if at all. Another group

of workers developed similarity methods by starting with a complete mathematical

18

formulation and, thus motivated, to examine less complete (and more general) problems;

Coles (1962), Abbott and Kline (1960) and Gukhman (1965).

An examination of these earlier works show that the initial problem statement, as

far as assumed completeness, determined to a large extent the kind of mathematical

approach employed. The more information that was known, the more direct was the

method developed for finding a similarity solution and, at the same time, the less general

were both the methods and the conclusions (as regards "general solutions"). It is not

suggested that this dichotomy is necessarily bad. The more general techniques, such as

group theory methods (contact Transformation and Lie transformation groups), have

produced powerful theorems and yield results with a minimum of mathematical busy-

work. On the other hand, the group theory methods are difficult for the average engineer

to follow because their motivation is mathematical, not physical, and this has inhibited

their wide use. Also, somewhat amazingly, the more powerful mathematical techniques

have been to a degree more restrictive in some of their aspects (such as the "class of

assumed transformations") than the less elegant methods.

1.8 Similarity Methods: Survey of the literature will revolve number of methods for carrying out similarity

analysis of differential equations. All these methods can divide into following two

groups:

I Direct Methods

(1) free parameter

(2) separation of variables

(3) dimensional method (This method is beyond the scope of our research

work)

II. Group-theoretic Methods

1.8.1 Free Parameter Method:

In this method it is assumed that the dependent variables of partial differential

equation can be expressed as product of two functions; one function of all independent

19

variabl

ent variable that has not occurred in the first function has to be taken in this

For ex dependent variable is φ and the independent

variables are x1

φ ( x , x , x . y) = φ( x1, x2, x3,…… xn,y) F(η) (1.11)

here

n,y) (1.12)

Becaus

lim φ ( x1, x2, x3,…xn,y) = φ ( x1, x2, x3,… xn) (1.14)

transformed to boundary conditions on F(η).

lim F(η) = 1

here, η0 = η ( x1, x2, x3,………. xn,y0) (1.15)

e except one, the other function is assumed to depend on single parameter η,

where η is obtained from transformations involving independent variables. The

independ

transformation.

ample, let us assume that the

, x2,………………. xn,y then φ can be expressed as

,…. xn1 2 3

W

η = η( x1, x , x ,…………x2 3

e η is not specified in the beginning it is called a free parameter method.

Choice of φ and η depends on the nature of the problem under consideration. Say, the

boundary conditions on φ in (1.11) which depends on y are given by

φ ( x1, x , x2 3,………… xn,y0) = 0 (1.13)

y y1

Now the Choice of φ and η should be such that the boundary conditions on φ can

be easily

e.g. φ ( x1, x , x ,……… x2 3 n,y0) = 0 F (η ) = 0 0

lim φ ( x1, x , x ,………x2 3 n, y) = φ ( x , x , x1 2 3,…………xn)

y y1

η η1

W

= η ( x , x , xAnd η1 1 2 3,………. xn,y ) (1.16) 1

20

Under such conditions the transformed equation might reduce to ordinary

ifferential equation. It all depends upon how the conditions on φ get transformed on φ

and η.

η corresponding to the boundary

. This is essential for ordinary

analysis to

hich no characteristic length occurs. For example,

for y = h1

= Constant= c2 for y = h2

η

φ η

then it is necessary that

F(η ) = c when η = h f(x)

Sometimes by change of origin, we may achieve the above requirements, say

η' = h' f(x)= 0

Now, η' has turned to a constant value but η'2 may not be constant. It may happen that

y h' = ∞

= 0 and ∞.

d

Sometimes it may happen that the partial differential equation does not get reduce

to ordinary differential equation. Also the value of

conditions should be transferred to the constant values

differential equation in η. This particular requirement limits similarity

problems in w

Let φ (x, y) = Constant= c1

φ (x, y)

We also assume that,

= y f(x)

and (x, y) = F( )

1 1 1 1

F(η ) = c when η2 2 2 = h2f(x)

η and η will turn out to be constant if and only if h , h and f(x) are constant or h , h1 2 1 2 1 2

have very special values.

h h' = 0, h h' . So that, 1 1 2 2

1 1

η' = h' f(x) 2 2

1

2

Then condition on F(η) will be for η

21

Such thing will happen only for problem where there is no characteristics length.

In general we can say that problems with boundary values associated with finite, nonzero

alues (there by indicating a characteristics dimension) will not have similarity solutions

because

ethod is most useful in the field of fluid flow and heat transfer.

r

inning one has to examine the Boundary Conditions and accordingly

e dependent variable. The function should

e such that the examined Boundary Conditions are changed to constant Boundary

Conditi

termined specifically.

ble Technique:

v

of the requirement that boundary conditions specified on F(η) need to be defined

for constant values of η.

Concluding Remarks:

Free parameter m

Dependent variable is the main tool of the procedure of carrying out a free paramete

analysis. In the beg

should try to set transformation function for th

b

ons. The function occurring in free parameter method normally either

exponentials or power functions, otherwise exact similarity solution may not be found.

Also as long as characteristic lengths are manifest in boundary conditions specified at one

or more points in the finite part of the plane, we can expect that a suitable form for the

parameter η cannot be de

1.8.2. Separation Varia

Birkhoff (1960) gas suggested an alternative method for obtaining similarity

solutions. It is named as a ‘Separation of Variable Technique’. This method chiefly

depends upon some unknown transformation function indicated by possible similarity

that will take place in the problem. The transformed partial differential equation may be

solved by using well-known separation of variable technique. While determining the

unknown functions, the care should be taken to see that it also satisfies the boundary

conditions.

Unlike certain degenerate case this method and one parameter group method are

many times equivalent. Also to some extent separation variable method is similar to the

free parameter method.

22

To apply this method, one has to assume the general form of similarity variable to

begin w

re all auxiliary conditions are not satisfied.

an excellent example for this method.

x x (ξ , η)

Ψ by a classical “separation

f variable technique”.

For thi

Ψ = H (ξ) F(η) (1.17)

m to a common constant and the resulting ordinary differential

s will be solved by usual methods. Here, we shall change the original boundary

ndary conditions on H (ξ) and F (η) and their derivatives. Further

nditions must be applied at constant values of ξ and η.

ith and there by make all substitution in the given equation. Before finding a

specific similarity variable the examination of resulting differential equations and

boundary conditions are essential.

This approach has been greatly developed by Abbot and Kline (1960). The

separation of variable method is well suited in case of (i) well-posed problems (ii)

equation whe

The problem of obtaining a solution for viscous boundary layer equations along a

flat plate Ames (1965) is

Now, let us see the method in detail. Say in some problems the independent

variables are x , y and the dependent variable if Ψ. First we select transformations of

independent variables

y y (ξ , η)

Then we shall try to solve the differential equation in

o

s we take,

Substituting this Ψ in the transformed equation, we shall come across the function

of ξ and η. Equate the

equation

conditions to the bou

more the boundary co

23

Conclu

riable method is similar to the free parameter method and to some

uate it to a well-known group theoretic method. In this method, we do

ot make any assumption except the fact that the dependent variable is separable. In this

onstant values of the similarity variable will occur. On the other hand, no steps are taken

to insu

l theories of

ontinuous transformation groups that were introduced and treated extensively by Lie

(1890), Lie and Scheffers (1891) in the later part of the last

century. Subsequently, the books by Cohen

uite extensively the general theories involved in the similarity

olutions of partial differential equations applied to engineering problems. More recently

04, 2010, 2011) have extended these techniques for the

treatme

ding Remarks:

Separation of va

extent one can eq

n

method the form of the similarity variable is first chosen so that no problem regarding the

c

re that the dependent variable has an appropriate form with consistent with the

boundary conditions.

1.8.3. Group Theory Method:

The foundation of group theoretic methods is contained in the genera

c

(1875), Lie and Engel

(1931), Campbell (1963), Eisenhart (1933),

Ovsjannikov (1982), Bluman and Cole (1974), Olver (1986), Bluman and Kumei (1989)

have contributed greatly to the development and clarification of many of Lie’s theories,

particularly, its applications to the invariant solutions of differential equations. In the

literature of engineering and applied sciences, the works of Birkhoff (1950), Morgan

(1952), Hansen (1964), Na, Abbott and Hansen (1967), Na and Hansen (1971), Ames

(1965, 1972) gives q

s

Timol et al (1986, 1988, 1998, 20

nt of various flow problems including MHD and Electro MHD boundary layer

flow of Non-Newtonian fluids.

The concept of mathematical similarity was first introduced by Helmholtz (1873)

through the dimensional analysis approach. Later on importance of similarity parameters

in physical science was demonstrated by Reynolds (1900), Buckingham (1914) and

Blasius (1908). Sedov (1959) developed the concept of similarity in a more meaningful

way till Morgan (1952) following the concept of Birkhoff (1950) showed that the

boundary layer equations were invariant under certain group of transformations that

24

reduce the number of independent variables by one. Hansen (1964) introduced the

concept of similarity through four different types of methods.

The century of work on similarity and fluid flow models are enormous and are

reflected in book by Sedov (1959), Hansen (1964), Kline (1965), Gukhman (1965),

eshadri and Na (1985), Na (1967, 1971), Rogers and Ames (1989), Bluman and Kumei

(1989)

oup of transformations of

rdinary differential equations based on the infinitesimal properties of group. Also it was

shown

S

, Bluman and Cole (1974). Sedov (1959) has constructed the theories of similarity

via dimensional analysis. Hansen (1964) has discussed the different well-known

similarity methods and has given bibliography in the field of fluid mechanics. Boltzmann

(1894) transformed the diffusion equation to a non-linear ordinary differential equation

with the help of transformation, known as Boltzmann Similarity Transformation. Lie

(1881) analyzed the flat plate problem with the help of similarity transformation. He has

tried to construct a general integration theory for ordinary differential equations under

transformations. Lie introduced the study of continuous gr

o

how a first order differential equation with the help of the group leads

immediately to quadrature and for a higher order equation (or system) leads to a

reduction in the order. Thus, for partial differential equations at least a reduction in the

number of independent variables is sought and in favorable cases a reduction to ordinary

differential equations can be seen. Thus, Lie reduces the number of independent variables

by the invariance property of the group. The resulting solutions are connected with the

usual similarity solutions of partial differential equations. If a system of partial

differential equation is invariant under a Lie group of transformations, one can find,

constructively, special solutions, called similarity solutions or invariant solutions, which

are invariant under some subgroup of the full group admitted by the system. This

application of Lie group was discovered by Lie (1881) which was first come to

prominence in the late 1950s through the work of the Soviet group of at Novosibirsk, led

by Ovsjannikov (1962, 1982). The investigations of the similarity analysis of differential

equations by Lie group are an outgrowth of a continuing study of ways in which partial

differential equations associated with problems of physical interest may be simplified

through transformations of independent and dependent variables. In fact, the basic ideas

25

date back to the last century and are found in the work of the mathematician Sophus Lie

(1881). The great advantage of Lie’s method is that they do not depend on the differential

equations being linear. Consequently, Lie’s theory can be considered to provide the only

widely applicable, systematic treatment available for non-linear differential equations.

The theory of continuous groups was first applied to the solution of partial

differential equations by Birkhoff (1950), who used a one-parameter group. Morgan

(1952) proved a theorem, which established the condition under which the number of

variables can be reduced by one. Morgan’s theorem was later extended by Michal (1952)

to similarity transformations, which reduce the number of independent variables by more

than one. One of the difficulties of the approach by Birkhoff (1950) and Morgan (1952) is

that it is based on particular given groups of transformations. A group has to be arbitrarily

assumed and Morgan’s theorem can then be used to establish whether or not the

differential equation transforms conformally under this particular group. If it does, the

similarity solutions will then exist and the similarity variables can then be taken as the

functionally independent invariants of this group.

This method is quite simple and straightforward. A set of invariants of groups is obtained

by sol

tively.

1

ving a set of simultaneous equations. These equations suggest the similarity

variables.

The outline of general theory of this method is given below.

Consider

φj = 0 (j =1,2,…….., n)

is the given set of partial differential equations, in which,

Xi ( i =1,2,…….., m)

Yj ( j = 1,2,……., n)

are the independent and dependent variables respec

Let a group Γ consisting of a set of transformations be defined as

26

(1.18)

i j (j = 1,….,n)

Where the parameter a ≠ 0 is real and αr, γj are to be determined from the

conditions that the system be constant conformally invariant (including absolutely)

Γ1 [i.e. A set of functions φj (xi) is said to be ‘conformally invariant’ under a one

j = 1,2,……n

ly the sa ⎯xi as the φj are of the xi.

and ‘absolutely invaria r all j].

Γ

We suppose x1 is the independent variable to be eliminated. Let here are two

α1≠ 0, the invariants Γ1 are

If α1 = 0 and the sim

Γ

⎯x1 = aα1 x1 , ⎯xr = a αr x (r = 2,…,m) r

Γ1 : γ j y = a y

∑

under

parameter (a) group.

⎯x xi if φ (⎯xi j i) = fj (⎯x , a) φi j (x1); i = 1,2,……m

Where the φ (⎯x ) are exact me functions of thej i

These functions will be said to be ‘constant conformally invariant’ if fj (⎯xi, a) are

independent of the x nt’ if f =1 foi j

This requirement gives rise to a set of simultaneous equations, nontrivial solutions

of which generate similarity variables, which are invariants of 1 .

cases. If

ηr = xr/x1βr , βr= αr / α1 (r = 2,……., m)

γ j/α1/ fj (η , η ,…….., η2 3 m) = [yj ( x , x1 2, ………. xm) ] x1 (1.19)

(j =1,2,….,n)

ultaneous equations have nontrivial solutions, then we may choose a

group 2 consisting of,

27

⎯x1 = x1 + log a, ⎯xr = aαr xr , yj = a γ j yj

and the

ηr = xr /exp (αr x1) (r =2,…………,m)

fj (η2, η3,…….., ηm) = [yj ( x1, x2, ………. xm) ]/exp (γ j x1) (j = 1,2,…..n) (1.20)

nsformation group. If x, y are

dependent variables and if u is the dependent variable we might take the

d find an absolute invariant, which is a function of the independent variables alone. For

η = y xt

yxt = YXt

also fin

ariant,

Xs

F (η) x -s

new dependent variable.

invariants of the group are

We illustrate the above, for one parameter tra

in

transformation.

X = an x m Y = a y p ⎯u = a u

an

example,

For absolute invariant

d the second absolute invariant g which select in such a way that g involves the

dependent variables u. For example,

g = u xs

For absolute inv

u xs = ⎯u

Then set,

g = F(η)

Then u =

and F (η) is the

28

Lastly putting the transformation for u into the given differential equation and

using η the given partial differential equation can easily transformed to an ordinary

tion.

dent or independent variables are contained in the given differential

then a group of independent variables η1, η2,… are found from the original

and are one less in number. Here, ηi are absolute invariants.

nts gi is sought which involves the

ui h (x1,….., xm).

here ui is the dependent variable. Then we equate function gi to a function

ui = Fi (η1, η2, ….., )/ hi (x1,….., xm)

is the d

pply. One

ick a transformation and proceed. The boundary condition, choice for

e of this method is that in

reducing the number of independent variables by one it is possible to obtain a new system

dvantage to employ group theory method is

that boundary conditions are not considered in any way up to the whole analysis is

differential equa

If more depen

equations

independent variables

Also for each dependent variable, absolute invaria

dependent variable. A good choice is gi =

W

Fi (η , η1 2, ….., η ). If gm-1 i = ui h (x1,….., xm) then,

ependent variable transformation. Substituting the various transformations in to

the given system of equations should lead to a new system with the number of

independent variables reduced by one.

Concluding Remarks:

Main advantage of group theory method is that it is rather simple to a

has merely to p

various functions etc. are not required. Another advantag

of partial differential equations. The first disa

complete. Second disadvantage is that there is uncertainty in selecting a proper

transformation group. In short one can achieve the reduction of independent variable

easily.

29

1.9 Ge

s describing the two-dimensional flow of a laminar incompressible boundary

yer:

uation in terms of the stream function;

For convience in the development, it is assumed that all of the variables are non

imensional. The objective is now to consider very general transformation of variables

d see what conditions must be met by this transformation so as not to violate any

nown physical properties of the problem.

Here, we begin by specifying a transformation of variables. For the independent

d form; thus the form is

ecified simply as

neralization of Similarity Transformations: The question of the most general class of transformations will now be examined

from a more physical view point. Some mathematics will be involved, of course, but the

physical problem will be kept near at hand and frequent reference to it will be made

throughout the analysis.

Because in what follows, by definition, involves a particular physical problem, it

will be convenient to introduce such a problem. As an example, consider the following

equation

la

0 +

(1.21)

or an equivalent single eq

(1.22)

d

an

k

variables, no restrictions at present will be placed on the assume

sp

ξ = ξ (x, y) , η = η (x, y) (1.23)

∂u ∂x =

∂v ∂y

2v u + - ∂ u

∂y2∂u ∂x

∂u ∂y

dp dx - =

- ∂ψ 2

∂x ∂x∂y ∂2

- ∂ ψψ ∂y2

∂ψ 3

∂y ∂ ψ∂y3 = dp

dx-

30

The dependent variable ψ (the stream function) is also transformed to a new

function, say Ψ. Previous work on fluid flow transformations has often assumed that the

two stream functions, ψ and Ψ, should be the same at corresponding points and hence

that streamlines is one plane are transformed in to streamlines in the other. However, this

striction is found to be unwarranted in many problems and will be avoided here.

Instead

(ξ ,η) = g (x, y) ψ (x , y) (1.24)

It should be recognized that this assumption is not trivial and other forms could be

conside

onvenience in interpreting the physical terms after transformation, transformed

velocit

g to consider an argument proposed by Coles (1962). Coles’ argument is based

n the a priori requirement that the transformed flow outside of the shear layer (i.e. for

large values of y) should conform both physically and formally to the original flow. Thus,

re

, the relationship between ψ and Ψ will be specified in some special sense by the

form

Ψ

red. Nevertheless, Eq. (1.24) represents a more general case than usually

employed and attention will be restricted to this form here.

The next step is to carry out the transformation of Eqs. (1.23) and (1.24) on the

left-hand side of Eq. (1.22). Application of the transformations (1.23) and (1.24) to the

boundary-layer equation is not new and was considered previously by Coles (1962) and

others.

The results of the transformation of Eq. (1.22) are given in Table 1.2 For

c

ies U, V have been defined by

U ∂Ψ

The transformed side of Eq. (1.22) given in Table 1.2 is essentially un-

manageable in its present form. While it is unclear at the present time what conditions are

either necessary or sufficient to ensure any particular mathematical behaviour, it is

interestin

o

∂η = V ∂Ψ ∂ξ= -

,

31

he argu

rivative terms and the U2 terms become at, most

nctions of ξ for large y, whereas the remaining terms behave either like y or y2. His

onclusion is to require the η and η2 terms to vanish identically, which can be

ccomplished by requiring that g = g(x), ξ = ξ(x), and ηy = γ (x). At the present time,

about

es, since the physical flow is bounded for large y and is, in fact; at most a function

of x, then the transformed side of Eq. (1.22) must behave in a similar fashion. His

argument is then that the substantial de

fu

c

a

all that can be said is that this assumed form is one possible form of a

transformation which will preserve a certain sense of physical correspondence between

the given and transformed flow.

In summary, the postulated general transformation, based on the physical mode, is:

Ψ (ξ ,η) = g (x) Ψ (x , y)

ξ = ξ(x), η = y(γ (x)). (1.25)

32

33

gy

g2ηyy-

ηy

g yy

- gy

g2 y 2 ηyU

TABLE 1.2 Use of similarity Transformations

ransformation: Ψ (ξ ,η) = g (x, y) ψ (x , y); ξ = ξ(x, y), η = η (x, y),

T

U

∂Ψ ∂η = V

∂Ψ ∂ξ= -

,

= - - ∂2ψ

∂y2∂ψ∂x∂x∂y

∂2ψ ∂ψ ∂y ∂3ψ

∂y3 dp dx

= U ∂U∂ξ + V ∂η

∂U J (ξ,η)

g2(ηy - ξy)

-

gy

g2ξyy

ξy

g yy

- gy

g2 y 2 ξyV - +

+ g

J (ηy / g,η) g

U2 J(ξy/ g,η) + J((ηy / g, ξ) UV

J(ξy/ g, ξ) g

V2+ -

- ∂U ∂ξ

ξyηy

g y 2

ξy

g y

ηy gy ξy ηy g22

ηy

g ξy

y

+ - +

+ + ∂V ∂ξ

ξy

g y ξy

gy ξ2y

g2 ξ2

y

g

y

-

+ - + 2 ∂2U

∂ξ2ξ2

yηy

g ξ2

y

g ηy ∂η

ξ3y ∂2V ∂ξ2

+ - ∂2V ∂η2 + 2

ξyη2y

g η2

y

g ξy g

η3y ∂2U

∂η2

+ +

gy

g yy

Ψ -ηy ∂U ∂η

∂V ∂η ξy

J ( g , η ) g3 -ηy

∂U ∂ξ

∂V ∂ξ ξy

J ( g , ξ ) g3+

- ΨU J(ηy, g) + gJ (gy / g, η)

g3

ΨV J(ξy, g) + gJ (ξy / g, ξ)

g3+

+ g2

J(gy / g2,g) Ψ2

+ - - ηy

g y ηy

η2y

g

y

∂U g2

gy 2yη

∂η

34

It is of interest to note that the resulting form of the independent variables is a

rimitive transformation, but it has been shown by considering a particular problem and

lying on physical arguments that it may be possible to restrict the form of η to be linear

nction of y for this particular class of problems (boundary-layer flows). Thus, the two

pproaches, mathematical and physical, have lead to the same conclusion regarding a

ostulated “most general class of transformation.” Of course it is necessary, as previously

ointed out; to emphasize that there may be some cases of practical interest which will lie

utside the realm of the mathematical conclusions presented here. However, it has been

und that all of the cases listed in Table 1.1 are satisfied by the transformation of Eq.

2.5).

It is now worthwhile to return to Table1.1 for a moment. Recall that from

Table1

er this question.

p

re

fu

a

p

p

o

fo

(

.1, it was noted that a wide class of different transformations were all primitive

transformations. Further, note that one of these transformations, the similarity

transformation, has been shown to yield to simple general analysis for its derivation for

particular problems. In fact, there are two types of similarity analysis that are founded

primarily on simple transformation theory: the free parameter method of Hansen (1964)

and the separation of variables method of Abbott and Kline (1960). It is thus of interest to

speculate on the following question: Would it be possible to derive all of these

“generalized similarity” transformations by the same technique that is used to derive the

similarity transformation? The Generalized Similarity Analysis was developed as attempt

to answ

This new method is based on a single assumption concerning the admissible class

of transformations of the independent variables, namely that the assumed transformation

should be one-to-one (that is, have a unique inverse). Here, it is shown that this

requirement would lead to the primitive transformation, and further that a particular

primitive transformation may be obtained from physical arguments for a particular class

of problems, namely ξ = ξ(x), η = yγ (x) for the boundary-layer flows. Of course the

35

generality of the present ideas goes beyond a particular case, such as boundary-layer flow

analysis and should be applicable to a wide class of problems.

1.10 The Generalized Similarity Analysis:

ion

for the Mangler transformation is that the axisymmetric boundary-layer equations are to

here are three distinct steps to the generalized similarity analysis. These steps are:

(a) Th

new

dependent variables (ξ , η) is required to satisfy the state requirement; for example, that

conditions for the given equation are required to be

tisfied when expressed in terms of the transformed variables.

The development of the generalized similarity analysis was motivated as an

extension of the method of finding similarity variables (i.e., the reduction of the number

of independent variables) to the problem of finding a transformation of variables which

will convert a given physical problem in to an alternate problem under certain prescribed

conditions. For example, the prescribed condition for a similarity solution is that the

number of independent variables must be reduced. By contrast, the prescribed condit

be transformed into the planar form of the equations, and so forth.

T

e general mathematical theory of transformations states that any continuous one-

to-one transformation can be resolved into one or more primitive transformations of the

form

ξ = ξ(x), η = η (x, y)

Thus, a primitive transformation form is assumed a priori, where it is recognized

that the general analysis has the possibility of being repeated more than once, depending

on the particular problem at hand.

(b) The given equation, transformed under a primitive transformation to the

in

for a similarity analysis, the transformed equation should be a function only one of the

new variables.

(c) Simultaneously the boundary

sa

36

These three steps will be shown to completely and uniquely determined the

explicit form of the new variables ξ (x) and η (x, y) if, in fact, the original problem and

associated boundary conditions do admit a generalized similarity solution.

s an example of the method, the classical Blasius flat plate boundary-layer problem will

ndary-layer equations for a steady, two dimensional laminar flow with a

ero pressure gradient (Blasius flat plate flow) can be written in the following form in

the boundary

statement complete, it is necessary to state the given

irement for the transformation x, y ξ, η:

Here, we require that under the transformation, the given equation and boundary

iable).

The three steps of the generalized similarity analysis are carried out in order as

llows:

ξ = ξ (x) and η = η(x, y).

A

be solved to give a relatively simple motivation of the basic ideas.

Example

The bou

z

terms of the stream function:

(1.26)

(All variables are dimensional, where ν is the kinematic viscosity) with

= ∂x∂

2

conditions

(1.27)

To make the problem

requ

conditions, (1.26) and (1.27), are to reduce to a function of a single independent variable

(i.e. , the similarity var

fo

(a) Assume a primitive transformation

y ∂2

- ∂ ψψ ∂ψ ∂y2y

∂ψ 3

∂ ∂x ν ∂ ψ∂y3

∂ψ∂x = 0 = y= 0: ∂ψ

∂y 0 ,

Constant = uo . y ∞ : ∂ψ ∂y

37

Since a

nd further, for this case of a similarity

olution, ξ can be assumed in the simple form ξ = x without loss of generally because the

final re

Ψ (ξ,η) = g(x) ψ (x, y)

endent variables becomes

(1.28)

here primes are used to indicate total derivatives. Substituting these results in to Eq.

.26) yields:

boundary-layer problem is under consideration, it is sufficient to choose the more

specific transformation for η as η = y γ(x), a

s

sult, by definition, is to be a function of the single variable η(x, y), and not both ξ

and η. For the dependent variable ψ (x, y), the transformed variable Ψ (ξ,η) is assumed

in the form

As discussed in section 1.9. Thus, the assumed transformation for the dependent

and indep

Ψ (ξ,η) = g(x) ψ (x, y)

ξ = x, η = y γ (x)

Performing the transformation (1.28) on the given Eq. (1.26) yields the following results:

= ∂

W

(1

ψ ∂y

∂Ψ ∂η

ηy g

= ∂2ψ ∂2Ψ η2

y ∂y2 ∂η2 g

∂3

= ψ ∂y3

∂3Ψ 3 η y ∂η3 g

+ g' η g2

Ψ =

∂ψ 1 ∂Ψ ∂Ψ ∂x + x

g∂η∂ξ -

g

= ∂2ψ ∂x∂x

∂Ψ ∂η

η- g' y g2

ηxy g ∂ξ∂η

∂2Ψ ηy g+ + ∂η2

∂2 ηΨ yηx g

-

+- ηy

1 ∂3Ψ

∂η3 ηxy η2

y g g g' ∂Ψ

∂x

2

+ - ∂ξ∂Ψ 1

gηy∂η2∂2Ψ

∂η ∂Ψ

∂ξ∂η ∂2Ψ

g' g

Ψ 1 ηy

∂2Ψ ∂η2 = 0

ν

38

with boundary conditions:

b) Imposing the stated condition of similarity by requiring that

Ψ = Ψ (η)

hat is, that the transformation must reduce the number of variables, and noting that ηy =

(x), the transformed equation becomes:

(1.29)

ith boundary conditions

(1.30)

dary conditions and the stated requirement that

q. (1.29) must be function only of η. Examining the last boundary condition (1.29)

yields

η = η (x, 0) = 0: ∂Ψ ∂η ηy (x, 0) = 0; - ηx (x,0) = 0 g'

gΨ + ∂ξ∂Ψ

∂η ∂Ψ

+

(

T

γ

w

(c) so far the functions g(x) and γ (x) are unspecified. However, only particular forms of

these functions will satisfy both the boun

E

The only way that this can be true is if

γ(x) = u0 g(x).

- Ψ' 2 g ' g2- γ'

γ2g g' g2γ + ν Ψ''' Ψ Ψ '' = 0

γ(x) g(x)

Ψ' (∞) constant = u0 .

η = 0 : Ψ' γ = 0, + = 0 - g' gΨ Ψ' γ

∞ : η = η (x, y) y (x, y ) . η ∞ ∂ΨConstant = u0 ∂η g2

u0 . γ

η ∞ : g Ψ'

39

Also, the only way that Eq. (1.29) can be a function of the single variable η is if

the coefficient of Ψ and its derivatives are constant. Hence, letting

0 is an undeterminable constant of integration, and letting the arbitrary constant c1

take the value c1 = -ν/2 yields

g = [ν u0 (x + x0)]-1/2

ent of Ψ' 2 :

h inal form of similarity solution is:

(1.31)

similarity solution of the Blasius problem by carrying out

e three steps of the generalized similarity analysis.

It has been shown that the primitive transformation appears to be the most general

provide a transformation with a unique inverse. This

sult was obtained from the general mathematical theory of transformations, but was

lso supported by a physical argument for such cases as boundary-layer problems which

g'

Then integration this ordinary differential equation yields

g = [-2c1 u0 (x + x0)] -1/2

where x

γ = u0 [ν u0 (x +x0)]-1/2

Evaluating the coeffici

T us this coefficient is identically zero and the f

In summary, it has been shown that a unique choice for the transformation

variables can be found for the

th

Concluding Remarks:

class of transformations necessary to

re

a

showed that the transformation

= u0g3 g' g2

1 γ constant = c1 . =

= γ' γ2g

u0g' u0γg2

' gγ = c and = c1

g2 1

ΨΨ'' 1 2 + Ψ''' = 0

Ψ (0) = Ψ' (0) = 0, Ψ' (∞) 1 .

40

Ψ (ξ, η) = g(x) Ψ (x, y)

ξ = ξ(x), η = yγ (x)

is possibly sufficient to ensure proper behaviour of the equations.

he generalized similarity analysis was then introduced to solve a wide class of

problem

A few comments can be made concerning the role of the function g(x) appearing

f the dependent variable for the boundary-layer equations. In

certain

rther, for the case of comparing compressible and

useful, results: Stewartson chooses g = constant and

ces of g for the difficult

roblem of compressible-incompressible transformations of the boundary-layer equations

for turb

on, or more appropriately,

otivated by a physical description of a problem which is in some sense complete.

Depend

T

s. The technique is illustrated by Blasius similarity problem; however all of the

cases given in Table 1.1 can be derived in similar fashion.

in the transformation o

cases, the value of g(x) will be uniquely determined by the problem. For example,

g(x) will be uniquely determined by the problem. For example, g(x) is proportional to √ x

and √ ξ (x) for the similarity and Meksyn-Gortler transformation, respectively. However,

in some cases, the choice for g is arbitrary; for example, g may take on any value for the

von Mises transformation. Fu

incompressible forms of the boundary-layer equations, different choices for g lead to

different , but nevertheless

Dorodnitsyn chooses g = ηy = γ (x). Coles discusses the significan

p

ulent flow.

In summary, the ideas presented here are based

m

ing on the particular case under consideration, completeness may imply

knowledge of all necessary boundary conditions, or possibly only a statement of a

particular requirement of the transformation. In any case, a fairly specific problem

formulation is implied.

In the next chapter, a different technique will be extended which focuses attention

on a more narrow application; the simple similarity problem (in the sense of reducing the

number of independent variables). This technique, the group theory method, being less

41

encumbered by statements of broad generality, will prove to yield very elegant and

powerful mathematical results for finding similarity solutions for a wide range of

applications.

42