I Introduction to Viscous Flow

	In AE 3003, you studied incompressible potential flow. You learnt how to analyze inviscid  flow over arbitrary shaped bodies such as airfoils and wings, and compute quantities of interest such as the surface pressure distribution, lift and moment coefficients.

	In AE 3004, the issue of compressibility was introduced. By the time you have completed AE 4001, you will know how to compute the forces and moments over airfoils and wings, in inviscid compressible flow, both for subsonic and supersonic conditions.

Role of Viscous Effects in Lift Loss:

	Although inviscid "potential" flow theories are extremely useful, they do not tell the complete story about the flow over wings, especially at high angles of attack. For example, consider the variation of the lift coefficient Cl vs. the angle of attack, a. 

	At low angles of attack, potential flow theory predicts the lift as a function of a, to a high level of accuracy. As a increases, the theory and measurements begin to deviate more and more, until a situation is reached where lift begins to decrease with any further increase in alpha. The airfoil has stalled.

	This unexpected behavior of the airfoil can not be explained by potential flow theory. In reality, the following events occur, in addition to the potential flow that exists over the airfoil.

	At low angles of attack, a thin viscous region forms over the airfoil, and grows from leading edge to the trailing edge. On the upper surface, where "adverse" pressure gradients exist (dp/dx>0) the boundary layer grows more rapidly. The outer flow sees an equivalent airfoil that has less camber than the original airfoil (due to the disproportionate growth of the boundary layer on the two sides) and that has an open trailing edge. Such an airfoil produces less lift than the original airfoil.

	At even higher angles, he very large "adverse" pressure gradients (dp/dx > 0) that develop on the upper side as the airfoil attempts to generate more lift causes the boundary layer to separate, leading to a major disruption of the flow over the airfoil, and the wing stalls. This explains the loss in lift with increase in angle of attack. Fortunately, in most instances stall occurs gradually, with a slow upward motion of the separation of the boundary layer from the trailing edge to the leading edge. In some instances (for airfoils with small leading edge radius) the stall is rather abrupt. The flow is well behaved at an angle of attack, say 5.5 degrees, but stalls with flow separation at the leading edge at 5.6 degrees because of the high adverse pressure gradients that occur at such sharp leading edges.

	The growth of boundary layer, the phenomenon of stall, and the location of the separation point thus play a critical role. This course attempts to develop theories and methods by which these quantities can be computed, given the pressure distribution over the airfoil.

Roll of Viscous Flow in Drag:

	We next turn our attention to the prediction of 2-D airfoil drag. For closed 2-D geometries, potential flow predicts zero drag at all angles of attack. In reality, the drag coefficient of the airfoil is dramatically affected by viscous effects, as shown below:

	The above figure indicates that all airfoils have drag. This figure states that some shapes have less drag than other. The drag coefficient is seen to first drop with lift, then rise. These phenomena can not be explained on the basis of inviscid potential flow theory.

	Viscous flow theory can explain these phenomena well. At the interface between the solid and fluid, shear stresses develop as the solid attempts to slow down the fluid flowing over it. This shear stress gives rise to a skin friction drag. It is proportional to the product of a quantity called viscosity, and the slope (or gradient) of the velocity profile across the boundary layer.

	If the airfoil is carefully designed, over most of the airfoil the flow is "laminar". Laminar flows have a low velocity gradient at the solid surface. In turbulent flows, there is increased mixing between the parts of the boundary layer. This leads to a large velocity gradient. 

	The key to reducing drag (and drag coefficient) is to keep the flow as laminar as possible over most of the airfoil. This requires keeping the boundary layer as thin as possible, and avoiding adverse pressure gradients as much as possible. Viscous flow theory allows us to compute skin friction coefficient at the solid surface both for laminar and turbulent flows, and allows us to anticipate where transition from laminar flow to turbulent flow will occur.

Roll of Viscous Flow in Moment Coefficient

	Viscosity also affects the pitching moments of airfoils. Consider the pitching moment coefficient of an airfoil about the quarter chord point. Classical potential flow theory states that this coefficient about this point is zero. In reality, a nose down pitching moment develops at high angles of attack, as the lift distribution over the nose part of the airfoil decreases due to viscous effects. At even higher angles of attack, an abrupt increase in the nose down pitching moment occurs, called "moment stall". This is a viscous phenomenon that can not be satisfactorily explained using potential flow theory.

	Viscous effects play a major role in other flows of interest such as flow through compressors, turbines and diffusers and in non-aerospace applications as well. In all these applications, the questions raised by the designer and the engineer are often the same - where will the flow separate? Will the flow be laminar, or turbulent? What is the magnitude of the shear stress at the wall and boundary layer thickness? How fast is heat transferred from the fluid to the solid and vice versa? The answers to these questions require knowledge of viscous flow, the subject matter of this course.
	This course is organized as follows. We first develop the governing equations. Next, we nondimensionalize these equations identifying important nondimensional parameters such as Reynolds number, Mach number, Prandtl number and the ratio of specific heats. Next, some exact solutions to Navier-Stokes equations are given.
	An approximation to the governing equations called the boundary layer approximation, developed by Prandtl is next described. it is shown that this theory may be used (either in its PDE form, or in an approximate integral form) to compute boundary layer characteristics such as boundary layer growth, skin friction and can identify where separation will first occur.
	The course ends with a discussion of transition- when, where and how it occurs. Empirical methods for identifying transition are presented- followed by a brief overview of turbulent flows.