What are the complete assumptions and final conclusions from 'Thin Airfoil Theory' in aerodynamics?

Question

I possess expertise in the field of aerospace structures, however I have recently been assigned an 'Aeroelasticity' analysis job at my workplace. In an effort to imbibe the fundamental knowledge which'd lead me to execute my gig in a proper way, I came across the concept of 'Thin Airfoil'.

Although I realized that mutitude of books addresses the mathematical derivation of aerodynamic parameters and coefficients by engaging this thin airfoil concept, but I couldn't still wrap my head around the basic reasons why exactly would we need to introduce this concept and in what way does it emerge favorable for the engineers? What is the complete set of assumptions and the final conclusions which are derived through this theory? Can I reach to similar conclusions and equations if I shy away from employing this concept? Do the real-world airfoils and wings behave exactly in the same fashion as predicted by this theory? What do I gain and where exactly do I compromise in all this?

Answer 1

Thin airfoil theory is based on a laundry list of assumptions -- I may miss some, but here are the big three...

1. Inviscid flow (flow without viscosity / friction)

2. Incompressible flow (fluid medium is incompressible, i.e. liquid, or if a gas, you are near the low Mach number limit where compressibility is not important)

3. Small disturbance, small angles, and the general assumptions that go along with linearizing the governing equations.

Thin airfoil theory was developed during a very active time of aerodynamics (1900-1930 ish). It was the first theory that could do a good job modeling arbitrary airfoils (not just very special cases). Thick airfoil theories would come later -- many of which would only be practical with digital computers.

The main competing theory of the time was 'conformal theory'. I.e. complex analysis (using $\sqrt{-1}$) was used to transform an airfoil into a shape with a known analytical solution (a circle). Different transformations would allow different airfoil-like shapes -- and would thereby provide 'exact' solutions for those airfoils. Unfortunately, you could not transform any shape, only those for which a transformation had been discovered/derived. This would prove to not be very practical for engineering use (though is powerful as a test/check of other methods).

The important thing about conformal theory is that it too relied on 1) and 2) above -- as did the thick airfoil theories that would come soon thereafter.

All of these 2D airfoil theories suffer from d'Alembert's paradox -- theory predicts zero drag on an airfoil, but observation of real airfoils exhibit finite (non-zero) drag. This is due to their ignoring viscoscity.

Boundary layer theory (simple treatment of friction) was developing at the same time, but it was only applied to flat plates at the time -- it would not be applied to airfoils until later and then typically with very cumbersome computations or a modern computer.

Non-incompressible theories (subsonic, supersonic, and eventually transonic) would come along, but would mostly require modern computers.

All that is to say that the unique part of thin airfoil theory was the assumptions from 3), not so much 1) & 2) (as all competing options had the same limitations for some time).

The important outcome of thin airfoil theory is that an airfoil can be decomposed into three flows that can be analyzed separately, but added together (through superposition) to model the complete flow. These three flows are:

A) a thin flat plate at an angle of attack B) a thin curved surface representing the mean camber line C) a symmetrical thickness form

A) tells us that the lift curve slope of an airfoil is $2\pi$ per radian (0.1097 per degree) and the aerodynamic center is at the quarter-chord (c/4).

B) tells us that the camber form determines the zero-lift angle of attack, the design lift coefficient, and the zero lift pitching moment coefficient. This can also be used to calculate the load distribution for a given camber line. Importantly, this step can also be inverted -- starting with a desired load distribution, we can calculate the shape of the mean camber line. This enabled practical airfoil design -- not just analysis.

C) tells us how much the flow accelerates around the airfoil due to the thickness (not just the production of lift). I.e. it will inform the magnitude of the supervelocities around the top and bottom surface, but the resulting pressures cancel out in terms of creating forces. We will be able to design good thickness families (in terms of promoting laminar flow, or delaying compressible drag rise) and then apply those to various camber lines to construct families of designed airfoils.

The approximations introduced by the linearization (3) above) are really quite reasonable in most situations. In practice, thin airfoil theory does a very good job up to pretty thick airfoils.

There are more implications of the use of superposition. We can further use superposition when designing the mean camber line -- two camber lines (say one curved at the front, and one curved aft) can be combined through superposition to achieve a complex camber line with curvature at the front and back.

This principle of aerodynamic superposition is fundamental to aeroelasticity.

First, we will use a similar set of assumptions for the structural model -- we will use linear elasticity theory. This limits us to small disturbances, but it also allows us to use superposition on the structure -- if we calculate the deflected shape of load distribution 1 and 2, we find that the deflected shape of load 1+2 is the sum of the deflected shapes. This idea is central to the development of finite element analysis.

In aeroelasticity, we typically start by finding the natural mode shapes of the structure (from eigen analysis). We then force the aerodynamic tool to model a structure deflecting in that mode shape (first bending, first twisting, second bending, etc). This aerodynamic tool gives us the distribution of pressure increments due to the specified deflection shape.

Just as complex deflections can be formed by superposition of the natural mode shapes, we can calculate the aerodynamic force on a complex deflected shape by superposition of the force due to the natural deflections.

In aeroelasticity, we form a 2nd order ODE -- a spring-mass-damper problem -- in matrix form.

The stiffness matrix is the stiffness matrix from the FEA analysis. The mass matrix is usually the identity matrix (as a result of using the natural modes as our deflections). The damping matrix comes from the aerodynamic tool.

The siffness provides the forces in response to the deflections. The mass provides the forces in response to the accelerations. the damping provides the forces in response to the rate of the deflections.

You then perform linear stability analysis on this ODE to determine the conditions (typically airspeed) at which it goes unstable (flutter boundary).

Obviously there is a lot more to aeroelasticity, but at this point, I think you can see that the linearization and superposition in thin airfoil theory are fundamental building blocks in the foundation of aeroelasticity.

Answer 2

What are the complete assumptions?

Fluid is:

1. irrotational;
2. incompressible and;
3. inviscid.

Under these hypothesis the full Navier-Stokes equations of the fluid dynamics reduce to the simple Laplace's equation:

$\nabla ² \phi=0$.

The Laplace equation is one of the most studied linear partial differential equation since it describes many natural physical phenomena. Knowing the boundary conditions, it can be univocally solved. Being linear, a solution for a complex phenomenon can be obtained summing up several basics solutions.

For an aerodynamic problem the boundary conditions are:

a. velocity is constant at infinity;

b. velocity is zero perpendicularly to the aerodynamic surface (so called non-penetration condition).

The thin airfoil condition introduces two additional simplifications to the problem:

4. the airflow is bidimensional;
5. the airfoil is thin, i.e. the chord is much bigger than the thickness.

Under these additional simplifications both $\phi$ and the boundary condition b. can be expanded as Taylor series and only the first two terms (zeroth-order and first-order) can be kept to do the math and solve the problem.

Due to 1., 2. and 3. Bernoulli's equation applies as well and the aerodynamic characteristics of the airfoil (lift and pitching moment) can be relatively simple solved by hand... and that's why it was introduced in the first place: before any computer was there, the thin airfoil theory provided a method to solve complex aerodynamic problems by hand.

What are the... final conclusions?

Any thin airfoil can be seen as the sum of:

1. a flat plate at a certain angle of attack $\alpha$; this is the zeroth-order solution from the Taylor series;
2. a cambered plate at zero $\alpha$; this is the first-order contribution.

Given the mathematical description of the camber, the aerodynamic lift and pitching moment can be calculated. In particular:

• lift $\rightarrow$ lift due to the zeroth-order (flat plate) is $C_{l_0}=2π\alpha$; lift due to the first-order (camber) depends on the exact equation of the camber but the biggest contribution to the lift comes from the backward part of the airfoil: that's why ailerons, flaps and other moving surfaces are normally located in the last 20% of the chord toward the trailing edge.

• moment $\rightarrow$ if the moment is calculated in respect to the point located at 1/4 of the chord (aerodynamic center) then it doesn't change with $\alpha$ i.e. the contribution due to the zeroth-order is null and only the first-order (camber) contributes to the pitching moment.