Introduction
(This page outlines the overall structure of the course in broad terms.)
Our goal is to calculate the lift, drag and pitching moment of wings in low subsonic flows, which are characterized by freestream velocities much smaller than the speed of sound. This condition is typically met when the Mach number \(M < 0.3\).
Now for any aircraft, there will be three forces and moments about pitch, roll and yaw axes. However, for most of the flight duration of an aircraft, it will be typically be in a cruise mode. The side force will be negligible (This is our first assumption.). Hence, we can concentrate on the forces and moments in the longitudinal plane of an aircraft as a start. Under these assumptions, we are only interested in the lift, drag and the pitching moment. In other words, we will only be concerned with the calculation of the coefficients \(C_L\), \(C_D\) and \(C_M\).
Clearly, we have two approaches to calculate these coefficients:
- Experimental method
- Modelling method (Physical modelling of flow followed by the mathematical solution of the model)
Approach (1) is too expensive, but accurate. We use experiments when we do not know the underlying physics or if the mathematical model is too complex to solve.
Approach (2) is preferred in situations where experiments are prohibitively expensive or difficult to perform. Usually, we use this approach when we have a good understanding of the underlying physics and the mathematical model is simple enough to solve. This is the approach we will take in this course.
Let us assume (for the time being) that we know the governing equations of fluid motion. We will review them later. The governing equations are partial differential equations (PDEs) and are non-linear. Analytical solution of these are only possible for simple geometries. For complex geometries only numerical solution is possible. So we wish to see if they can be simplified under some justifiable assumptions.
I am going to assume some knowledge of airfoil geometry and angle of attack. If you are not familiar with these terms, please see Anderson (2011) section 4.2.
We begin the process of simplification by observing that there is a thin region of flow immediately surrounding the airfoil where most of the viscous effects are concentrated. This region is called the boundary layer. The boundary layer is thin compared to the chord length of the airfoil. So we can assume that the flow outside the boundary layer is inviscid. This is our second assumption.
Secondly, it is known (from experiments) that at high angles of attack the flow separates from the upper surface of the airfoil at some point near the trailing edge. This separation point moves closer to the leading edge with increasing angle of attack. The flow downstream of the separation point is turbulent and viscous effects cannot be ignored in this region. Hence, we will only concentrate on the flows with small to moderate angle of attack such that there is no flow separation. This is our third assumption.
Finally, we will assume that the flow is steady. This is our fourth assumption.
Nonlinear PDEs are typically solved using numerical methods. There are many of them. Please see Anderson (2011) (section 2.17) for a nice introduction.
Using these assumptions and some clever mathematical analysis we will be able to show that the governing equations can be simplified to a set of equations called the Euler equations. These equations are still non-linear and are difficult to solve. However, we can make further simplifications to these equations by assuming that the flow is irrotational. This is our fifth assumption. This assumption is justified by the fact that the flow is incompressible and the Mach number is small. This assumption leads to the Laplace equation which is linear and can be solved analytically. This is the approach we will take in this course.
It turns out that we can use only the kinematics of the flow to calculate what is called circulation around the airfoil. This is a very important concept in aerodynamics. We will see that the circulation is related to the lift generated by the airfoil. This is called the Kutta-Joukowski theorem.
Following this line of reasoning, we will be able to calculate the lift fairly accurately. However, drag over an airfoil is a very complex phenomenon. We will see that the drag is related to the viscous effects in the boundary layer. We will not be able to calculate the drag accurately.
The drag on an airfoil can be divided into two parts: (1) the drag due to the pressure distribution over the airfoil and (2) the drag due to the viscous effects in the boundary layer. The first part is called the pressure drag and the second part is called the friction drag. We will be able to calculate the pressure drag accurately. However, we will not be able to calculate the friction drag accurately. We will see that the friction drag is related to the displacement thickness of the boundary layer. We will also see that the displacement thickness is related to the momentum thickness of the boundary layer. We will be able to calculate the momentum thickness accurately. However, we will not be able to calculate the displacement thickness accurately.
So we will be able to calculate the lift, pitching moment and the pressure drag accurately.
The next part of the course will deal with analysing the boundary layer in order to calculate the drag more accurately. We will see that the boundary layer can be classified into two types: (1) laminar and (2) turbulent. We will see that the laminar boundary layer is unstable and will eventually become turbulent. We will see that the transition from laminar to turbulent boundary layer is a very complex phenomenon. We will see that the turbulent boundary layer is more stable than the laminar boundary layer.
A brief introduction to turbulence and Reynolds averaging of the Navier-Stokes equations will be given.
We will conclude the course with a review of high lift devices and aerodynamics of the delta wing.