1. Louisiana Tech University Ruston, LA 71272 Boundary Layer Theory Steven A. Jones BIEN 501 Friday, April 11 2008

2. Louisiana Tech University Ruston, LA 71272 Momentum Balance Learning Objectives: 1.State the motivation for curvilinear coordinates. 2.State the meanings of terms in the Transport Theorem 3.Differentiate between momentum as a property to be transported and velocity as the transporting agent. 4.Show the relationship between the total time derivative in the Transport Theorem and Newton’s second law. 5.Apply the Transport Theorem to a simple case (Poiseuille flow). 6.Identify the types of forces in fluid mechanics. 7.Explain the need for a shear stress model in fluid mechanics. The Stress Tensor. Appendix A.5 Show components of the stress tensor in Cartesian and cylindrical coordinates. Vectors and GeometryVectors and Geometry

3. Louisiana Tech University Ruston, LA 71272 Flow Over a Flat Plate What can we say about this flow? x y  – Boundary layer thickness

4. Louisiana Tech University Ruston, LA 71272 Flow Over a Body x  y UU U0U0

5. Louisiana Tech University Ruston, LA 71272 Continuity & Momentum Use 2-dimensional equations. In contrast to previous derivations, we do not say that terms are zero. We say that they are “small.”

6. Louisiana Tech University Ruston, LA 71272 Continuity Non-dimensionalize so that velocities and lengths are of order 1.

7. Louisiana Tech University Ruston, LA 71272 Exercise If: How do you non-dimensionalize:

8. Louisiana Tech University Ruston, LA 71272 x -Momentum Non-dimensionalize small If inertial terms are important with respect to viscous terms.

9. Louisiana Tech University Ruston, LA 71272 Characteristic Pressure Non-dimensionalize

10. Louisiana Tech University Ruston, LA 71272 y -momentum From y -momentum Small in comparison to

11. Louisiana Tech University Ruston, LA 71272 Characteristic Pressure From y -momentum P is not a function of y, so the pressure in the boundary layer is the pressure in the free stream, which can be determined from a potential flow solution. Take the derivative with respect to x to get something to plug into the x-momentum equation.

12. Louisiana Tech University Ruston, LA 71272 Final x -momentum, B.C.s

13. Louisiana Tech University Ruston, LA 71272 Flow Over a Flat Plate x x Velocity at large y does not depend on x.

14. Louisiana Tech University Ruston, LA 71272 Special Case: Flat Plate Use the stream function:

15. Louisiana Tech University Ruston, LA 71272 Special Case: Flat Plate Assume a similarity solution:

16. Louisiana Tech University Ruston, LA 71272 Special Case: Flat Plate

17. Louisiana Tech University Ruston, LA 71272 Special Case: Flat Plate

18. Louisiana Tech University Ruston, LA 71272 Final Differential Equation Or, more simply This equation is 3 rd order and nonlinear. There is no closed form solution, but it can be solved numerically.

19. Louisiana Tech University Ruston, LA 71272 Final Differential Equation Remember that f is the stream function, so you must take derivatives to get velocity after you solve for f. x -velocity for flow over a flat plate.

20. Louisiana Tech University Ruston, LA 71272 Integral Momentum Boundary Instead of seeking an exact solution to the differential equation, we can integrate the equations for continuity and x -momentum.

21. Louisiana Tech University Ruston, LA 71272 Integral Momentum Boundary The terms in these equations have specific meanings. Becomes zero because velocity becomes constant far from the boundary. Wall shear stress

22. Louisiana Tech University Ruston, LA 71272 Integral Momentum Boundary The terms in these equations have specific meanings. Integrate by parts with

23. Louisiana Tech University Ruston, LA 71272 Integral Momentum Boundary v x becomes U  far from the boundary, and v y (  ) is obtained from the integrated continuity equation. Both velocities are zero at the wall.

24. Louisiana Tech University Ruston, LA 71272 Integral Momentum Boundary Because by Continuity:

25. Louisiana Tech University Ruston, LA 71272 Integral Momentum Boundary In summary: So:

26. Louisiana Tech University Ruston, LA 71272 Integral Momentum Boundary With: If we knew v x, we would be able to integrate for the wall shear stress. It turns out that the exact form of v x is not as important as one might think, and good results can be obtained with a form that looks reasonable and satisfies the boundary conditions.