Saikishan Suryanarayanan
Engineering Mechanics Unit
JNCASR
Outline
 Introduction to Symmetry Analysis
 Lie series, Group operator and infinitesimal invariance condition for
functions.
 Applications to ODEs and PDEs – general remarks
 Boundary Layer
 Using invariance group to identify self-similarity and solve Blasius boundary
layer
 Incompressible Navier Stokes Equation.
 Group of invariance
 Application : Viscous flow in a diverging channel (Jeffery-Hamel
fl0w)
 Compressible Euler Equation
 Group of invariance
Introduction to Symmetry analysis
 Symmetry analysis involves identifying and studying the
group of symmetries leaves a function/ equation invariant.
 Most physical laws have associated symmetry groups, for
eg. the Schrödinger equation is invariant under rotation.
 When a differential equation and Boundary conditions are
invariant under a certain group, one expects the solution
also be invariant under the same symmetry group.
 Identifying the symmetry groups can provide physical
insight and help solve practical problems,
 ODE - Order reduction
 PDE - Reduction in # of dimensions, some cases reduction
to a ODE.
 Better numerical schemes can be developed for solving
ODE/PDE.
Lie Groups, Lie Series and Condition for Invariance
Symmetry groups of differential equations are continuous groups.
Illustration of 1 parameter Lie group –
The transformation x -> x’ is defined as
with
Since the group is continuous wrt. s, the transformation can be
expressed as
Infinitesimal of the group is defined as
Consider a function y [x]. The function transforms as
where
Lie Groups, Lie Series and Condition for Invariance
The function is invariant , ie. y [x’] = y [x] if and only if
Therefore a group can be expressed in form of its infinitesimals.
Another common way of expressing a Lie Group is by
Transformation of the function can be expressed as
The number of operators (Xs) is equal to the number of
parameters of the Lie group.
The condition for invariance of a differential equation is similar to
the condition for invariance of functions, except that it is applied
to the extended tangent space of differential functions.
Derivation of self – similar solutions (generally obtained by
physical insight / trial and error) using Symmetry Group
Analysis.
2D Navier Stokes Equations
 The Continuity (conservation of mass) and Navier –
Stokes Equations (conservation of linear momentum) in
two dimensions are
The Boundary Layer
 In a seminal paper in 1904, Prandtl proposed that the
changes in velocity from the free stream to the velocity
at the surface take place in a thin layer near the wall
termed as the ‘boundary layer’.
.
Figure from : http://www-scf.usc.edu/~tchklovs/Proposal.htm
Boundary Layer Equations
 In the Boundary layer, the changes in the stream-wise
(x) direction are small compared to changes in the wall
normal (y) direction. Consequently (from continuity)
v << u.
 Under this ansatz and under steady state and constant
free stream velocity (zero pressure gradient) the
governing equations are simplified to
Stream Function Formulation
 Introducing the stream function y , defined as
 The Boundary Layer equation and boundary
conditions in terms of stream function are :
 This is a third order PDE with 2 independent variables.
Symmetry group of the governing equation
Consider the dilatation group,
The governing equation transforms as
If a= b – c, the original form is recovered, therefore the
governing eqn is invariant under the two parameter
dilation group :
Symmetry group of the Boundary condition
 The inhomogeneous boundary conditions transform as
 In order to leave the b.c. invariant, there is an additional
constraint on the group parameters, b = 2c
 The equation, the boundary condition are invariant
under the following transformation
 So it is expected that the solution will also be invariant
under this lie group.
Dimensional Analysis and Reduction to ODE
 The stream function can be expressed as a function of the
co-ordinates and problem parameters as :
 A naïve dimensional analysis (using Buckingham Pi
theorem) leads to :
 On the transformation by the symmetry group of the
governing PDE and BC
Dimensional Analysis and Reduction to ODE
 y/x is not invariant to the group transformation,
therefore the solution, should be of the form
 On substitution into the original PDE, it leads to the
following ODE and Boundary conditions
Comments, implications and solution
 Solution is Self- Similar – the velocity profile only gets
stretched from one stream-wise location to another.
 Without using group theory, this problem is solved by
physical insight/ experimental observation of self
similar profiles/ guess work.
 Numerical Solution
to the ODE.
 u/Ue = f [y/d]
d ~ x / (Rex)1/2
Incompressible Navier Stokes Equations
 For incompressible flow, the conservation equations of
mass and momentum are given by
 Note that the above form applies to both two and three
dimensions. Einstein convention is used, dummy
indices are summed over.
 The pressure satisfies a Possion equation obtained by
taking divergence of the momentum equation.
Time Translation
 The Equations are invariant under the following
transformation
 This can be shown as
 The infinitesimal of this group is given by
 Also holds for compressible flows (shown later)
Arbitrary function of time added to pressure
 The Equations are invariant under the following
transformation
 This can be shown as
 In operator form,
 Doesn’t apply for compressible flows.
Rotation/ Reflection
 The Navier stokes equations are invariant under uniform
rotation/reflection of the position and velocity vectors
where
In Cartesian indicial notation,
since
Rotation
 The invariance is demonstrated as follows :
Rotation/Reflection
In operator form,
Rotation about z- axis
Rotation about x- axis
Rotation about y-axis
Applies for compressible flows also.
Non-uniform Translation
 The Incompressible Navier Stokes equations are
invariant not only under constant translation, but also
translations of the form:
 This is demonstrated as follows :
Non – uniform translation
 In operator form, the non-uniform translation about x, y
and z axes are:
 Physically, invariance under non-uniform translation
implies that the incompressible NS equations are valid
even in accelerating frames of reference, provided the
acceleration is irrotational.
 This invariance does not hold for compressible flows,
unless d2a/dt2 = 0
Dilatation
 The full NS equations are invariant under the
following one –parameter dilatation group (note: the
boundary layer equations were invariant under a two
parameter dilatation group)
 The partial derivatives transform as
Dilatation
 The continuity equation is invariant under the dilatation
group transformation as shown below
 The invariance of the momentum eqn is shown as
follows:
Dilatation
 In operator form,
 Aside : Euler Equations (when n = 0) admit a two parameter
dilatation group.
Commutator Table
X1
X2
X3
X4
X5
X6
X7
X8
X9
X1
0
C122 X2
0
0
0
C166 X6
C177 X7
C188 X8
2 X1
X2
-C122 X2
0
0
0
0
0
0
0
c292 X2
X3
0
0
0
- X5
X4
X6,X7,X8
X6,X7,X8 X6,X7,X8
0
X4
0
0
X5
0
- X3
X6,X7,X8
X6,X7,X8 X6,X7,X8
0
X5
0
0
-X4
X3
0
X6,X7,X8
X6,X7,X8 X6,X7,X8
0
X6
-C166 X6
0
X6,X7,X8 X6,X7,X8 X6,X7,X8
0
0
0
C696 X6
X7
-C177 X7
0
X6,X7,X8 X6,X7,X8 X6,X7,X8
0
0
0
C797 X7
X8
-C188 X8
0
X6,X7,X8 X6,X7,X8 X6,X7,X8
0
0
0
C898 X8
X9
-2 X1
-c292 X2
-C696 X6
-C797 X7
-C898 X8
0
0
0
0
Jeffery-Hamel Flow
 2D viscous flow through converging/ diverging
channel.
Jeffery-Hamel Flow
The governing equation for the stream function (2D INS) is
which is invariant under the dilatation group as shown in
the last section
Dimensional analysis of
and choosing
dimensionless parameters that are invariant to the group
transformations leads to
 In polar co-ordinates,
 On substitution into the original PDE, the following
ODE is obtained
Solution
 The ODE can be solved numerically, to obtain the
following self-similar solution.
http://www.cfd-online.com/Wiki/Jeffery-Hamel_flow
Compressible Euler Equations
 For compressible flow of an inviscid fluid, the
conservation equations of mass, momentum and energy
are given by
Time Translation
Rotations
Translation
Dilatation
 Since all terms in all equations have either a single time
derivative or a single space derivative, it can be seen that
they are invariant under the above dilatation group – not
valid if viscosity is present.
 Equations admit additional symmetry groups based on
functional form of F[p,r]
References
 Brian J. Cantwell. 2002. Introduction to Symmetry
Analysis . Cambridge University Press .