# Boundary layer theory.

```Schlichting, H. (1979). Boundary Layer Theory.
Part A. Fundamental laws of motion for a viscous fluid
1. Outline of fluid motion with friction. Real and perfect fluids. Viscosity. Compressibility.
The Hagen-Poiseuille equations of flow through a pipe. Principle of similarity; the Reynolds
and Mach numbers. Comparison between the theory of perfect fluids and experiment.
2.
Outline of boundary-layer theory. The boundary-layer concept. Separation and vortex
formation. Turbulent flow in a pipe and in a boundary layer.
3.
Derivation of the equations of motion of a compressible viscous fluid (Navier-Stokes
equations). Fundamental equations of motion and continuity applied to fluid flow. The rate
at which a fluid element is strained in flow. Relation between stress and rate of deformation.
Stokes’ hypothesis. Bulk viscosity and thermodynamic pressure. The Navier-Stokes
equations.
4.
General properties of the Navier-Stokes equations. Derivation of Reynolds’ principle of
similarity from the Navier-Stokes equations. Frictionless flow as “solutions” of the NavierStokes equations. The Navier-Stokes equations interpreted as vorticity transport equations.
The limiting case of very large viscosity (very small Reynolds number). The limiting case of
very small viscous forces (very large Reynolds number). Mathematical illustration of the
process of going to the limit R   .
5.
Exact solutions of the Navier-Stokes equations. Parallel flow. Other exact solutions.
6.
Very slow motion. The differential equations for the case of very slow motion. Parallel flow
past a sphere. The hydrodynamic theory of lubrication. The Hele-Shaw flow.
Part B. Laminar boundary layers
7. Boundary-layer equation for two-dimensional incompressible flow; boundary layer on
a plate. Derivation of boundary-layer equations for two-dimensional flow. The separation of
a boundary layer. A remark on the integration of the boundary-layer equations. Skin friction.
The boundary layer along a flat plate. Boundary layer of higher order.
8.
General properties of the boundary-layer equations. Dependence of the characteristics of
a boundary layer on the Reynolds number. “Similar” solutions of the boundary-layer
equations. Transformation of the boundary-layer equations into the heat-conduction. The
momentum and energy-integral equations for the boundary layer.
9.
Exact solutions of the steady-state boundary-layer equations in two-dimensional
motion. Flow past a wedge. Flow in a convergent channel. Flow past a cylinder;
symmetrical case (Blasius series). Flow in the wake of flat plate at zero incidence. The twodimensional laminar jet. Parallel streams in laminar flow. Flow in the inlet length of a
straight channel. The method of finite differences. Boundary layer of second order.
10. Approximate methods for the solution of the two-dimensional, steady boundary-layer
equations. Application of the momentum equation to the flow past a flat plate at zero
incidence. The approximate method due to Th. Von Kármán and K. Pohlhausen for twodimensional flows. Comparison between the approximate and exact solutions. Further
11. Axially symmetrical and three-dimensional boundary layers. Exact solutions for axially
symmetrical boundary layers. Approximate solutions for axially symmetric boundary layers.
Relation between axially symmetrical and two-dimensional boundary layers; Mangler’s
transformation. Three-dimensional boundary layers.
12. Thermal boundary layers in laminar flow. Derivation of the energy equation.
Temperature increase through adiabatic compression; stagnation temperature. Theory of
similarity in heat transfer. Exact solutions for the problem of temperature distribution in a
viscous flow. Boundary-layer simplifications. General properties of thermal boundary layers.
Thermal boundary layers in forced flow. Thermal boundary layers in natural flow.
13. Laminar boundary layers in compressible flow. Physical considerations. Relation
between the velocity and the temperature fields. The flat plate at zero incidence boundary
layer with non-zero pressure gradient. Interaction between shock wave and boundary layer.
14. Boundary-layer control in laminar flow. Methods of boundary-layer control. Boundarylayer suction. Injection of a different gas (Binary boundary layers).
15. Non-steady boundary layers. General remarks on the calculation of non-steady boundary
layers. Boundary-layer formation after impulsive start of motion. Boundary-layer formation
in accelerated motion. Experimental investigation of the starting process. Periodic boundarylayer flow. Non-steady, compressible boundary layers.
Part C. Transition
16. Origin of turbulence I. Some experimental results on transition from laminar to turbulent
flow. Principles of the theory of stability of laminar flows. Results of the theory of stability
as they apply to the boundary layer on a flat plate at zero incidence. Comparison of the
theory of stability with experiment. Effect of oscillation free stream on transition.
17. Origin of turbulence II. Effect of pressure gradient on transition in boundary layer along
smooth walls. Determination of the position of the point of instability for prescribed body
shape. Effect of suction on transition in a boundary layer. Effect of body forces on transition.
Effects due to heat transfer and compressibility. Stability of a boundary layer in the presence
of three-dimensional disturbances. The influence of roughness on transition. Axially
symmetrical flows.
Part D. Turbulent boundary layers
18. Fundamentals of turbulent flow. Introductory remarks. Mean motion and fluctuations.
Additional, “apparent” turbulent stresses. Derivation of the stress tensor of apparent
turbulent friction from the Navier-Stokes equations. Some measurements on fluctuating
turbulent velocities. Energy distribution in turbulent streams. Wind-tunnel turbulence.
19. Theoretical assumptions for the calculation of turbulent flows. Fundamental equations.
Prandtl’s mixing-length theory. Further assumptions for the turbulent shearing stress. Von
Kármán’s similarity hypothesis. Universal velocity-distribution law. Further development of
theoretical hypotheses.
20. Turbulent flow through pipes. Experimental results for smooth pipes. Relation between
law of friction and velocity distribution. Universal velocity-distribution laws for very large
Reynolds numbers. Universal resistance law for smooth pipes at very large Reynolds
numbers. Pipes of non-circular cross-section. Rough pipes and equivalent sand roughness.
Other types of roughness. Flow in curved pipes and diffusers. Non-steady flow through a
pipe. Drag reduction by the addition of polymers.
21. Turbulent boundary layers at zero pressure gradient; flat plate; rotating disk;
roughness. The smooth flat plate. The rotating disk. The rough plate. Admissible roughness.
22. The incompressible turbulent boundary layer with pressure gradient. Some
experimental results. The calculation of two-dimensional turbulent boundary layer.
Turbulent boundary layers on aerofoils; maximum lift. Three-dimensional boundary layers.
23. Turbulent boundary layers in compressible flow. General remarks. Relation between
velocity and temperature distribution. Influence of Mach number; laws of friction.
24. Free turbulent flows; jets and wakes. General remarks. Estimation of the increase in width
and of the decrease in velocity. Examples. Diffusion of temperature in free turbulent flow.
25. Determination of profile drag. General remarks. The experimental method due to Betz.
The experimental method due to Jones. Calculation of profile drag. Losses in the flow