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Saikishan Suryanarayanan Engineering Mechanics Unit JNCASR

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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

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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.

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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 [x]. The function transforms as where

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Lie Groups, Lie Series and Condition for Invariance The function is invariant, ie. [x’] = [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.

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Derivation of self – similar solutions (generally obtained by physical insight / trial and error) using Symmetry Group Analysis.

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2D Navier Stokes Equations The Continuity (conservation of mass) and Navier – Stokes Equations (conservation of linear momentum) in two dimensions are

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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 :

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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

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Stream Function Formulation Introducing the stream function 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.

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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 :

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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.

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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

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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

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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/ x / (Re x ) 1/2

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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.

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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)

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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.

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Rotation/ Reflection The Navier stokes equations are invariant under uniform rotation/reflection of the position and velocity vectors where In Cartesian indicial notation, since

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Rotation The invariance is demonstrated as follows :

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Rotation/Reflection In operator form, Rotation about z- axis Rotation about x- axis Rotation about y-axis Applies for compressible flows also.

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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 :

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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 d 2 a/dt 2 = 0

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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

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Dilatation The continuity equation is invariant under the dilatation group transformation as shown below The invariance of the momentum eqn is shown as follows:

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Dilatation In operator form, Aside : Euler Equations (when = 0) admit a two parameter dilatation group.

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Commutator Table X1X2X3X4X5X6X7X8X9 X10C 122 X2000C 166 X6C 177 X7C 188 X82 X1 X2-C 122 X c 292 X2 X3000- X5 X4X6,X7,X8 0 X400X50- X3X6,X7,X8 0 X500-X4X30X6,X7,X8 0 X6-C 166 X60X6,X7,X8 000C 696 X6 X7-C 177 X70X6,X7,X8 000C 797 X7 X8-C 188 X80X6,X7,X8 000C 898 X8 X9-2 X1-c 292 X2000-C 696 X6-C 797 X7-C 898 X80

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Jeffery-Hamel Flow 2D viscous flow through converging/ diverging channel.

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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

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In polar co-ordinates, On substitution into the original PDE, the following ODE is obtained

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Solution The ODE can be solved numerically, to obtain the following self-similar solution.

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Compressible Euler Equations For compressible flow of an inviscid fluid, the conservation equations of mass, momentum and energy are given by

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Time Translation

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Rotations

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Translation

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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, ]

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References Brian J. Cantwell Introduction to Symmetry Analysis. Cambridge University Press.

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