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Perturbation: Background

Perturbation: Background. Algebraic Differential Equations. Perturbed equation. Original Equation. Perturbation. Perturbed equation. Answer can be in the form. Perturbation. Change in result (absolute values) vs Change in equation. Simple (Regular) Perturbation. Perturbed equation.

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Perturbation: Background

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  1. Perturbation: Background • Algebraic • Differential Equations

  2. Perturbed equation • Original Equation Perturbation

  3. Perturbed equation • Answer can be in the form Perturbation • Change in result (absolute values) vs Change in equation • Simple (Regular) Perturbation

  4. Perturbed equation • Original Equation Perturbation • Two roots instead of one • Roots are not close to the original root

  5. Answer may NOT be in the form Perturbation • Change in result (absolute values) vs Change in equation • Other root varies from the original root dramatically, as epsilon approaches zero! • Singular perturbation

  6. Solution Differential Equations SKIP • Perturbation-1 • Solution • Regular Perturbation

  7. Solution • Perturbation-1 • Solution Differential Equations • Regular Perturbation

  8. Differential Equations • Another Regular Perturbation • Perturbation-1a SKIP • Solution • Perturbation-1b • Solution

  9. Differential Equations • Perturbation-2 • Exact Solution • (eg using Integrating factors method) • Singular Perturbation

  10. Singular Perturbation Differential Equations • Can the solution be of the following form? (to satisfy the extra boundary condition?) • No! • Based on the perturbed equation

  11. Differential Equations • At the limit • Method to find solution • Transform variables (x,y,e) • Called “Stretching Transformation” • Zooms in the ‘rapidly varying domain’ • Obtain “inner solution”

  12. Outer Soln Inner solutions • Method to find solution Differential Equations • Inner solution: • Let e =0 and simplify eqn • 2nd order equation, satisfying only one boundary condition (x=0) • one constant remains arbitrary • Valid only near x=0 • Obtain outer solution, for first order equation, satisfying one Boundary Condition (x=1) • Valid everywhere, except near x=0

  13. Outer Soln Inner solution Differential Equations • Method to find solution • Match the two solutions in the segment in between, by choosing the remaining constant • Match the value and the slopes • Close to the exact solution

  14. “Real” solution “Real” solution Approx solution Approx solution Numerical Solution (to BL) • Grid generation • Structured grid vs unstructured grid • Uniform vs non-uniform grid • What about placing more grid everywhere? • More grid points near surface • Similar to “stretching transformation”

  15. Boundary Layer theory • Situations we have seen so far • Laminar flow in cylinder • Fully developed (entrance effects are negligible) • Steady State • Unsteady State • Again, entrance effects are negligible • Movement of infinite plate, in a semi-infinite medium V0

  16. Boundary Layer theory • Semi-infinite plate • Inviscid flow (irrotational) • Will NOT satisfy ‘no slip’ condition at the plate Fluid Velocity V0 • Flow over cylinder (inviscid) • Flow over sphere (2D, viscous flow) (tutorial problem) • Flow over any other shape (while accounting for no-slip condition and not assuming fully developed flow) is treated with “boundary layer theory”

  17. Boundary Layer theory • Semi-infinite plate • Away from plate, inviscid solution is valid (and will satisfy the boundary condition). This is “outer solution” • Near the plate, different solution (including viscosity) will be found using ‘stretching transformation’. [Inner Solution] • Inner solution will satisfy the boundary condition (no slip) • Match both solutions to find the other constant Fluid Velocity V0

  18. Velocity V0 INF INF INVISCID FLOWASSUMPTION OK HERE Velocity V0 No Slip FRICTION CANNOT BENEGLECTED HERE Velocity 0 0 0 Boundary Layer theory • Solid Boundary

  19. INF INF d d 0 0 Boundary Layer theory B L thickness 99% Free Stream Velocity • Solid Boundary What happens to d when you move in x? x B L thickness increases with x Momentum Transfer

  20. d d Boundary Layer theory • Draw d vs x • Analytical Expression, for velocity vs (x,y), below BL: • Continuity • Navier Stokes Equation y x B L thickness increases with x

  21. Hence 1 Boundary Layer theory • Steady,incompressible, two dimensional (semi infinite plate)

  22. N-S Eqn • Consider only X and Y equations (2D assumption) • Steady flow, gravity can be incorporated in Pressure term (or assume gravity is in Z direction, for example) • Vz=0, Vx and Vyare not functions of z

  23. N-S Eqn • Obtain “order of magnitude” idea • Can be used to ignore small terms (simplify eqn by removing ‘regular’ perturbations) • Can be used to non-dimensionalize equations • example: • Steady State

  24. N-S Eqn • Write the NS-eqn in “usual” form, for steady state

  25. d y x L N-S Eqn • What are the relevant scales for the lengths (eg what are L1, L2 in this particular case?

  26. ~ means “Order of ” • Note: Some books show it as y x • Similarly N-S Eqn • What are the relevant scales for the velocity? • Vx varies from 0 to Vo (or we can call it VINF)

  27. Note: The sign is not important here • Continuity 1 y x N-S Eqn • What are the relevant scales for the derivatives?

  28. Thin Boundary Layer assumption d y x L N-S Eqn • What are the relevant scales for the derivatives?

  29. From Bernoulli’s eqn d y Claim:as m -->0, x L N-S Eqn • Can we approximate pressure drop? • Assume that pressure drop is similar to inviscid flow

  30. Each term is small compared to the equivalent in X-eqn • ==> N-S Eqn • For the Y component of N-S equation

  31. Prandtl BL eqn (steady state) Unsteady State

  32. d y x L Prandtl BL eqn : flow over Flat plate • No pressure Drop • Steady State • 2D-flow (Stream Function) • Stretching Transformation (near the boundary) • Non-dimensionalize y

  33. d y x L Prandtl BL eqn : flow over Flat plate • Another perspective for the choice of d • If we write the BL eqn in stream function • Boundary Conditions

  34. d y x L Prandtl BL eqn : flow over Flat plate

  35. d y x L Prandtl BL eqn : flow over Flat plate Some books may have -ve sign, or a factor of 2, in the equation, depending on the definition of Stream function and transformations used

  36. 1 0 0 3 Prandtl BL eqn : flow over Flat plate • Boundary Conditions: • No solution in ‘usual’ form • Blasius Solution: Series solution, valid for small h • For large h, asymptotic series that matches with the boundary condition • Numerical values tabulated (f,f’,f’’...) • Plot of Vx/VINF vs h • Note: definition of h may be slightly different in various books (usually by a factor of 2)

  37. Prandtl BL eqn : flow over Flat plate • Blasius Solution • Valid for high Reynolds Number • Re • Local:( dVr/m) • More useful (convenient): (X V r/m ) (sometimes, this is referred to as “local” Reynolds number) • 105 or more • Not valid very near x= 0 (at the point x=0,y=0) • Another way to express boundary layer thickness • Reynolds number high ==> Boundary layer is thin

  38. Prandtl BL eqn : flow over Flat plate • Boundary layer thickness • Drag estimate • Other definitions (for thickness) • Similarity • Effect of pressure variation (Loss of similarity and separation) • Thermal vs momentum Boundary Layer • von Karman method

  39. References: • Introduction to Mathematical Fluid Dynamics by Richard E Meyer • Perturbation methods in fluid dynamics, by Van Dyke • BSL • 3W&R • Fluid flow analysis by Sharpe • Introduction to Fluid Mechanics by Fox & McDonald

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