空气动力学课件 (11).ppt

CHAPTER 11
PRESSIBLE FLOW OVER AIRFOILS: LINEAR THEORY
Introduction
This chapter mainly deal with the properties of two- dimensional airfoils at Mach number above but below 1, where pressibility must be considered.
Velocity potential equation
Linearized velocity
potential equation
Prandtl-pressibilty correction
pressibilty Correction
Critical Mach number
The area rule for transonic flow
Supercritical airfoils
Figure
Road Map for .
The Velocity Potential Equation
For two-dimensional , steady, irrotational , isentropic flow, a
velocity potential can be defined such that :
The introduction of velocity potential can greatly simplify the governing equations, we can derive the velocity potential equation from continuity, momentum, energy equations:
The continuity equation for steady,two-dimensional flow is :
or
Substituting into it, we get
To eliminate from above equation, we consider the momentum equation :
Since the flow we are considering is isentropic, so
so
We get the velocity potential equation:
In this equation , the speed of sound is also the function of :
()
For subsonic flow, Eq. is an elliptic partial differential
equation. For supersonic flow, is a hyperbolic partial
differential equation. For transonic flow, is mixed type
equation.
Eq. represents bination of continuity, momentum,
energy equations. In principle, it can be solved to obtain
for the flow field around any two-dimensional flow.
The infinite boundary condition is
The wall boundary condition is
Once is known, all the other value flow variables are directly
obtained as follows:
1. Calculate u and v: and
a:
M:
4. Calculate T,p, :