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Mathematicalbehaviour of PDE’s Arvind Deshpande
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Governing equations of fluid flow(Navier Stokes equations) div ( V ) 0( Mass)t ( u ) p div ( Vu ) div ( gradu) S Mx ( X momentum) t x ( v) p div ( Vv ) div ( gradv) S My (Y momentum) t y ( w) p div ( Vw) div ( gradw) S Mz ( Z momentum) t z ( i ) div ( Vi ) pdivV div (kgradT ) Si ( InternalEn ergy ) tP RT & i CvT ( Equationso fstate )3/7/2012 Arvind Deshpande (VJTI) 2
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Governing equations of fluid flow(Euler equations) div ( V ) 0( Mass)t ( u ) p div ( Vu ) S Mx ( X momentum) t x ( v) p div ( Vv ) S My (Y momentum) t y ( w) p div ( Vw) S Mz ( Z momentum) t z ( i ) div ( Vi ) pdivV Si ( InternalEn ergy ) tP RT & i CvT ( Equationso fstate )3/7/2012 Arvind Deshpande (VJTI) 3
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Comments on governing equations1. They are a coupled system of nonlinear partial differential equations and hence are very difficult to solve analytically.2. For the momentum and energy equations, difference in the conservation form and non conservation form is the just the left hand side.3. Conservation form contains terms on the left side which include the divergence of some quantity.4. In CFD literature, entire block of equations are called Navier-Stoke’s equations.5. All equations for inviscid flow are called Euler’s equations.6. Conservation form of the governing equations provides a numerical and computer programming convenience in that the continuity, momentum and energy equations in conservation form can all be expressed by the same generic equation. This helps to simplify and organize the logic in a given computer program.Governing equations are “bread and butter” of CFD-learn them well.3/7/2012 Arvind Deshpande (VJTI) 4
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General Transport equation inDifferential form ( ) div ( V ) div (grad ) S t Rate of increaseRate of Net rate of flow Rate of increase of φ of φ due toincrease of φ of φ out of the due to diffusion sourcesof fluid fluid elementelement3/7/2012 Arvind Deshpande (VJTI) 5
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General Transport equation inIntegral form dV n.( V )dA n.(grad )dA S dVt CV A A CV Net Rate ofRate of increase Net rate of Net rate of increase creation of φof φ fluid element decrease of φ of φ due to diffusion due to convection across the across the boundaries boundaries3/7/2012 Arvind Deshpande (VJTI) 6
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General Transport equation in Integral form n.( V )dA n.(grad )dA S dV (steady )A A CV t t CVdV dt t n.( V )dAdt t n.(grad )dAdt t CVS dVdt (unstady ) A A 3/7/2012 Arvind Deshpande (VJTI) 7
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Shock capturing In flow fields involving shock waves, there are sharp discontinuous changes in flow field variables, P,ρ, V, T etc. across the shocks. Shock capturing methods are designed to have the shock waves appear naturally within the computational space as a direct result of the general algorithm, without any special treatment to take care of the shocks themselves. Ideal for complex flow problems involving shock waves for which we do not know either the location or the number of shocks. Numerically obtained shock thickness bears no relation whatsoever to the actual physical shock thickness and precise location of the shock is uncertain. Conservation form of the governing equations is suitable for shock capturing3/7/2012 Arvind Deshpande (VJTI) 8
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Shock fitting In shock fitting methods, shocks are explicitly introduced into the flow field solution. Analytical relations are used to relate the flow immediately ahead of and behind the shock and governing equations are used to calculate the remainder of the flow field between the shock and some other boundary such as surface of a aerodynamic body. Shock is always treated as discontinuity and its location is well defined numerically. For a given problem, you have to know in advance approximately where to put the shock waves and how many they are. For complex flows, this can be a distinct disadvantage. For shock-fitting method, satisfactory results are usually obtained for either form of the equations, conservation or non conservation.3/7/2012 Arvind Deshpande (VJTI) 9
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Boundary conditions Real driver for any particular solution Dirichlet boundary condition - Specification of dependent variables along the boundary e.g. For Viscous flow, Wall boundary condition V = Vw at the surface (No slip) For a stationary wall, V = 0 Known wall temperature, T= Tw3/7/2012 Arvind Deshpande (VJTI) 10
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Newmann boundary condition Specification of T q . K derivatives of dependent n n variables along the T q. boundary n n K e.g. 1) if wall temperature is changing due to heat transfer from or to the surface 2) Adiabatic wall T 0 n n3/7/2012 Arvind Deshpande (VJTI) 11
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Robbins ConditionThe Derivative of the dependent variable is given as afunction of the dependent variable on the boundary.3/7/2012 Arvind Deshpande (VJTI) 12
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Inlet and outlet Inlet – Density, velocity and temperature at inlet Outlet – location where flow is approximately unidirectional and where surface stresses take known values. For external flows away from solid objects and for internal flow, at a location where no change in any of the velocity components in direction across the boundary and Fn = -P & Ft = 0 Specified pressure, un 0, T 0 n n3/7/2012 Arvind Deshpande (VJTI) 13
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Other boundary conditions Open boundary u n condition 0 n Symmetry boundary 0 condition n Cyclic boundary 1 2 condition3/7/2012 Arvind Deshpande (VJTI) 14
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Initial conditions Everywhere in the solution region ρ, V and T must be given at time t = 0 The Initial and Boundary conditions must be specified to obtain unique numerical solutions to PDEs Well posed problem3/7/2012 Arvind Deshpande (VJTI) 15
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Partial Differential EquationsClassifications of PDE’s according to order First Order G 0 x y 2 Second Order 0 x 2 y 2 3 2 Third Order 3 x xy x 0 3/7/2012 Arvind Deshpande (VJTI) 16
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Mathematical behavior of PDE’s 2 2 2 a 2 b c 2 d e f g 0 x xy y x y 2 dy dy a b c 0 dx dx If b2-4ac < 0, elliptic equation (Imaginary characteristics) If b2-4ac = 0, parabolic equation (1 real characteristics) If b2-4ac > 0, hyperbolic equation (2 real and distinct characteristics)3/7/2012 Arvind Deshpande (VJTI) 17
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Eigen value method N 2 N Ajk x x H 0j 1 k 1 j kdet [Ajk-λI] = 0 If any eigen value λ = 0, the equation is parabolic. If all eigen value λ ≠ 0 and they are all of the same sign, the equation is elliptic. If all eigen value λ ≠ 0 and all but one are of the same sign the equation is hyperbolic.3/7/2012 Arvind Deshpande (VJTI) 18
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Elliptic PDE Typical Examples are 2 2 2 0 ( Laplace’s Equation – Irrotational flow of x 2 y an incompressible fluid, steady state conductive heat transfer) 2 2and 2 g ( x, y ) ( Poisson’s Equation) x 2 y Note that In both of the eqns, b=0, a=1, c=1 which makes b 2 4ac 4 which is < 0 The solution domain of Elliptic Eqn has closed ended nature 3/7/2012 Arvind Deshpande (VJTI) 19
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Domain of dependence Pictorial Representation of Elliptical Problem3/7/2012 Arvind Deshpande (VJTI) 20
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Elliptic PDE Characteristics are imaginary/complex Information propagates everywhere Equilibrium problems (div grad φ = 0) Boundary value problems Smooth solution Steady state temperature distribution of a insulated solid rod Steady viscous flow Steady, subsonic inviscid flow3/7/2012 Arvind Deshpande (VJTI) 21
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Parabolic PDE A Typical Example is 2 t x 2 ( Heat Conduction or Diffusion Eqn.) divgrad ( ) t Where is positive, real constantIn above eqn. b=0, c=0, a = which makes b 2 4ac 0 The solution advances outward indefinitely from Initial Condition This is also called as marching type problem The solution domain of Parabolic Eqn has open ended nature3/7/2012 Arvind Deshpande (VJTI) 22
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Domain of dependence Pictorial Representation of Parabolic Problem3/7/2012 Arvind Deshpande (VJTI) 23
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Parabolic PDE Information travels in one particular direction (downstream) Time dependent problems which involve significant amount if dissipation Initial-Boundary value problems Smooth solution Unsteady heat conduction Unsteady viscous flows Thin shear layers – Boundary layers, jets, mixing layers, wakes, fully developed duct flows3/7/2012 Arvind Deshpande (VJTI) 24
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Hyperbolic PDE A typical example is 2 2 2 c ( Wave Equation) t2 x 2 2 c 2 divgrad t 2 Where c2 is real constant and always positive In above eqn b = 0, a = c2, c = -1 which makes b2 4ac 4c 2 which is >0 The solution domain of Hyperbolic Eqn has open ended nature Two Initial conditions are required to start the solution ofHyperbolic eqn in contrast with Parabolic eqn where only one Initialconditions is required. 3/7/2012 Arvind Deshpande (VJTI) 25
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Domain of influence Boundary conditions Boundary conditions P(x’,t’) Domain of dependence Initial conditions Pictorial Representation of Hyperbolic Problem3/7/2012 Arvind Deshpande (VJTI) 26
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Hyperbolic PDE Characteristics are real and distinct Information propagates along these characteristics Time dependent problems which involve negligible dissipation Initial-Boundary value problems Solution may be discontinuous Compressible fluid flows at speeds close to or above the speed of sound Steady, inviscid supersonic flow Unsteady inviscid flow3/7/2012 Arvind Deshpande (VJTI) 27
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