# Summary of Chapter 8

This chapter starts with a discussion of vorticity versus strain rate, the derivation of viscous stresses and of theNavier-Stokes equations for 2-D incompressible flows, fol­lowed by Prandtl boundary layer theory including numerical solution for the bound­ary layer over a flat plate and calculations of boundary layers displacement and momentum thicknesses.

Next, compressible viscous flows are considered and the corresponding Navier – Stokes and boundary layer equations are derived. Determination of drag is also discussed.

Historical and classical works of Blasius solution for a flat plate and Falkner – Skan solution for flow past a wedge, as well as the von Karman Integral Momentum equation (with streamwise pressure gradient) and transformation of boundary layer equations (due to von Mises and also Howarth and Dorodnitcyn) are considered together with comments on flow separation, flow at the trailing edges and three­dimensional boundary layers.

Turbulent boundary layers are not covered in this book.

1.4 Problems

8.5.1

Consider the stream function in the upper half plane

1 3 1 2

ф(л, y) = – y – mxy, y > 0 6 2

where m > 0 is a fixed parameter.

Find the shape and make a sketch of the streamlines of this flow. Some remarkable streamlines correspond to ф = 0. Show that y = 0 could be a solid wall.

Calculate the velocity components (u, v) and show, with arrows, the direction of the flow. Indicate with dots, the location of flow reversal line, u = 0.

Calculate the vorticity w = dv/дx – du/дy and indicate with dotted line the location w = 0.

Calculate the wall shear stress rw(x).

Sketch some velocity profiles u (x, y) atthreefixed x-locations, x = -0.5, x = 0 and x = 0.5.

Why is this not a real viscous flow solution (Hint: look for a pressure field).

References

1. Oswatitsch, K.: Gas Dynamics. Academic Press, New York (1956)

2. Liepmann, H., Roshko, A.: Elements of Gas Dynamics. Wiley, New York (1957)

3. Schlichting, H.: Boundary Layer Theory. McGraw Hill, New York (1979)

4. Howarth, L.: Laminar boundary layers. Fluid Dynamics I. Encyclopedia of Physics, vol. VIII/I. Springer, Berlin (1959)

5. Meier, G. E., et al. (eds.): IUTAM Symposium on One Hundred Years of Boundary Layer Research. Springer, New York (2004)

6. Stewartson, K.: The Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University Press, Oxford (1964)

7. Batchelor, G. K.: An Introduction to Fluid Dynamics. Cambridge University Press, New York (1967)

8. Rosenhead, L. (ed.): Laminar Boundary Layers. Oxford University Press, Oxford (1963)

9. White, F. M.: Viscous Fluid Flow. McGraw Hill, New York (1974)

10. Moran, J.: Theoretical and Computational Aerodynamics. Wiley, New York (1984)