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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, α.
Cl, from AE
Cl from Expt.
Cl
Angle of Attack
At low angles of attack, potential flow theory predicts the lift as a function of α, to a high level of accuracy. As α 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.
Airfoil + Boundary Layer Equivalent Airfoil, that has less camber and an open trailing edge 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:
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.
x
Cp
Cm=0 Theory
Alpha
Cm
Nose-up is +ve
Expt
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.
