1. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Perturbation: Background n Algebraic n Differential Equations

2. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Perturbation n Original Equation n Perturbed equation

3. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Perturbation n Change in result (absolute values) vs Change in equation n Perturbed equation n Simple (Regular) Perturbation n Answer can be in the form

4. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Perturbation n Original Equation n Perturbed equation n Two roots instead of one n Roots are not close to the original root

5. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Perturbation n Other root varies from the original root dramatically, as epsilon approaches zero! n Change in result (absolute values) vs Change in equation n Singular perturbation n Answer may NOT be in the form

6. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n Solution n Perturbation-1 n Solution n Regular Perturbation

7. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n Solution n Perturbation-1 n Solution n Regular Perturbation

8. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n Another Regular Perturbation n Perturbation-1a n Solution n Perturbation-1b n Solution

9. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n Perturbation-2 n Exact Solution u (eg using Integrating factors method) n Singular Perturbation

10. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n Singular Perturbation n Can the solution be of the following form? (to satisfy the extra boundary condition?) n No! u Based on the perturbed equation

11. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n At the limit n Method to find solution Transform variables (x,y,  ) n Called “Stretching Transformation” n Zooms in the ‘rapidly varying domain’ n Obtain “inner solution”

12. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n Method to find solution n Inner solution:  Let  =0 and simplify eqn u 2nd order equation, satisfying only one boundary condition (x=0) u one constant remains arbitrary u Valid only near x=0 Outer Soln Inner solutions n Obtain outer solution, for first order equation, satisfying one Boundary Condition (x=1) u Valid everywhere, except near x=0

13. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential Equations n Method to find solution n Match the two solutions in the segment in between, by choosing the remaining constant u Match the value and the slopes Outer Soln Inner solution n Close to the exact solution

14. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Numerical Solution (to BL) n Grid generation n Structured grid vs unstructured grid n Uniform vs non-uniform grid “Real” solution n What about placing more grid everywhere? n More grid points near surface n Similar to “stretching transformation” Approx solution “Real” solution Approx solution

15. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Boundary Layer theory n Situations we have seen so far n Laminar flow in cylinder n Fully developed (entrance effects are negligible) n Steady State n Unsteady State n Again, entrance effects are negligible n Movement of infinite plate, in a semi-infinite medium V0V0

16. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Boundary Layer theory n Flow over cylinder (inviscid) n Flow over sphere (2D, viscous flow) (tutorial problem) n Flow over any other shape (while accounting for no-slip condition and not assuming fully developed flow) is treated with “boundary layer theory” n Inviscid flow (irrotational) n Will NOT satisfy ‘no slip’ condition at the plate Fluid Velocity V 0 n Semi-infinite plate

17. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Boundary Layer theory n Away from plate, inviscid solution is valid (and will satisfy the boundary condition). This is “outer solution” n Near the plate, different solution (including viscosity) will be found using ‘stretching transformation’. [Inner Solution] n Inner solution will satisfy the boundary condition (no slip) n Match both solutions to find the other constant Fluid Velocity V 0 n Semi-infinite plate

18. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Boundary Layer theory n Solid Boundary No Slip Velocity 0 Velocity V 0 0 INF 0 INVISCID FLOW ASSUMPTION OK HERE FRICTION CANNOT BE NEGLECTED HERE

19. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Boundary Layer theory n Solid Boundary 0 INF 0 x   B L thickness 99% Free Stream Velocity B L thickness increases with x What happens to  when you move in x? Momentum Transfer

20. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Boundary Layer theory Draw  vs x x   B L thickness increases with x n Analytical Expression, for velocity vs (x,y), below BL: n Continuity n Navier Stokes Equation y

21. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Boundary Layer theory n Steady,incompressible, two dimensional (semi infinite plate) n Hence 1

22. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n Consider only X and Y equations (2D assumption) n Steady flow, gravity can be incorporated in Pressure term (or assume gravity is in Z direction, for example) n V z =0, V x and V y are not functions of z

23. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n Obtain “order of magnitude” idea n Can be used to ignore small terms (simplify eqn by removing ‘regular’ perturbations) n Can be used to non-dimensionalize equations n example: n Steady State

24. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n Write the NS-eqn in “usual” form, for steady state

25. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n What are the relevant scales for the lengths (eg what are L 1, L 2 in this particular case? x y L 

26. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n What are the relevant scales for the velocity? x y n V x varies from 0 to V o (or we can call it V INF ) n ~ means “Order of ” n Note: Some books show it as n Similarly

27. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n What are the relevant scales for the derivatives? n Note: The sign is not important here x y n Continuity 1

28. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n What are the relevant scales for the derivatives? x y L  Thin Boundary Layer assumption

29. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n Can we approximate pressure drop? n Assume that pressure drop is similar to inviscid flow x y L  From Bernoulli’s eqn Claim: as  -->0,

30. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 N-S Eqn n For the Y component of N-S equation n Each term is small compared to the equivalent in X-eqn n ==>

31. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn (steady state) Unsteady State

32. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn : flow over Flat plate  No pressure Drop  Steady State  2D-flow (Stream Function) x y L   Stretching Transformation (near the boundary)  Non-dimensionalize y

33. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn : flow over Flat plate  Another perspective for the choice of  x y L  Boundary Conditions  If we write the BL eqn in stream function

34. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn : flow over Flat plate x y L 

35. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn : flow over Flat plate x y L  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. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn : flow over Flat plate n Boundary Conditions: n No solution in ‘usual’ form Blasius Solution: Series solution, valid for small  Plot of V x /V INF vs  Note: definition of  may be slightly different in various books (usually by a factor of 2) 1 0 30 For large , asymptotic series that matches with the boundary condition n Numerical values tabulated ( f,f’,f’’... )

37. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn : flow over Flat plate n Blasius Solution n Valid for high Reynolds Number n Re  Local:(  V  /  )  More useful (convenient): (X V  ) (sometimes, this is referred to as “local” Reynolds number) u 10 5 or more n Not valid very near x= 0 (at the point x=0,y=0) n Another way to express boundary layer thickness n Reynolds number high ==> Boundary layer is thin

38. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Prandtl BL eqn : flow over Flat plate n Boundary layer thickness n Drag estimate n Other definitions (for thickness) n Similarity n Effect of pressure variation (Loss of similarity and separation) n Thermal vs momentum Boundary Layer n von Karman method

39. IIT-Madras, Momentum Transfer: July 2005-Dec 2005 References: n Introduction to Mathematical Fluid Dynamics by Richard E Meyer n Perturbation methods in fluid dynamics, by Van Dyke n BSL n 3W&R n Fluid flow analysis by Sharpe n Introduction to Fluid Mechanics by Fox & McDonald