Question

A NACA 0009 airfoil (data shown in appendix A) is a thin, symmetric airfoil. Use thin airfoil theory to determine (a) the lift slope (b) the coefficient of lift for an angle of attack of 6 degree and (c) the zero-lift angle of attack. Compare each of these predicted values to values read from the NACA plot

Answer 1

LETS FIRST DESCRIBE THE MEANING AND CONCEPT OF NACA 009 AIRFOIL WITH SOME FIGURES AND THEORY

mean (camber) line Ус m(x) chord line 4J mean line model, y(s) vorticity distribution

A simple solution for general two-dimensional aerofoil sections can be obtained by neglecting thickness effects and using a mean-line only section model. For incompressible, inviscid flow, an aerofoil section can be modelled by a distribution of vortices along the mean line. This is a standard potential flow modelling technique which will give quick and reasonable estimates of lift coefficient and moment coefficient. However, as it models inviscid flow, there will be no estimate of drag coefficient.

LIFT SCOPE:

The vortex distribution along the mean line forms a continuous vorticity sheet. So rather than considering the strength of point vortices, we consider the strength of the distribution per unit length, γ(s). The distribution function is assumed to take the following form.

This function is Glauert's approximation and is based on Joukowski transformation results (A₀ term) which mainly covers the effect of angle of attack, plus a Fourier series variation (A₁ terms) to account for camber. It automatically obeys the Kutta condition with zero vorticity at the trailing edge. It is based on a mapped angular position (θ) rather than an exact surface location (s) to allow for ease of integration. The mapping between s and θ is shown below,

ds c/2

The vorticity distribution is thus given as a function of the angular variable (θ) which is related to chordwise position (x) as follows,

where (c) is the chord length. Note that chord-wise position (x) is used instead of distance along the mean line (s) for simplicity and is valid in cases where the camber height is not too large. For typical aerofoils with small camber, the difference between these two distances is negligible. With no camber this model becomes equivalent to the Joukowski mapping of a cylinder to a flat plate aerofoil.

The magnitude of the vortex sheet strength must be calculated to complete the mathematical model.

For thin cambered plate models, a boundary condition of zero flow normal to the surface is applied in order to create an equation that can be solved for the required coefficients (A₀, A₁, A₂,.....) to determine the necessary strength of the vortex sheet.

Given an aerofoil geometry, freestream velocity and angle of incidence, the magnitude of the coefficients (A₀, A₁, A₂,....) is to be found by solving the boundary condition equation, along the surface. In this case the condition of flow velocity normal to the surface can be more easily formulated in terms of horizontal and vertical velocity components.

The ratio of vertical (v) to horizontal (u) velocity at the surface (mean line) must equal the surface gradient (dy_c/dx).

The flow horizontal and vertical velocities are made up of freestream and vortex induced components.

where u_i and v_i are the horizontal and vertical velocities induced by the vortex sheet. Both of these components will be much less than the freestream velocity. Also due to the “flatness” of the section v_i will be much larger than u_i, so for small angles of incidence, the horizontal vortex induced component can be neglected. If these small angle assumptions are made for the incidence, the boundary condition equation becomes ,

The velocity induced vertically (v_i) at any point on the mean line can be found by summing up the effects of small individual segments (ds) of the vorticity distribution.

where (x) is the location at which the induced velocity is being calculated and (s) is the chord-wise location of the vortex element. On integration after substitution for s,x and γ this gives,

Substituting this result for induced velocity into the boundary condition equation gives ,

COEFFICIENT OF LIFT:

The solution for coefficients (A₀, A₁, A₂,...) can now be obtained from this equation. The solution is based on Glauert's integral method. The equation is summed (integrated) along the chord line to find initially coefficient A₀. It is then scaled by cosine multiples and again summed along the chord. Each scaled integration will yield one higher order coefficient. ∫ (boundary condition equation).dθ along chord line produces,

The reversed integral limits result from the inverted nature of the relationship between x and θ due to the mapping. Inverting the limits gives,

And similarly integrating the boundary condition multiplied by cos(nθ) along the chord length gives,

or reversing the limits,

Once the vorticity coefficients are found, the lift of a small element of the vortex line can be predicted from the Kutta-Joukowski law. The complete lift is then found by summing all elements of lift from leading to trailing edge.

Because of the symmetry of the vorticity function only the first two coefficients (A₀, A₁) will have a contribution to this integration. Thus lift coefficient per unit span (S= c x 1) can be found as follows ,

thus

By summing elements of vortex lift which are multiplied by a moment arm based on their distance from the ¼ chord point, the pitching moment coefficient about this point can be found.

Note: Many publications use an alternate mapping of x/c = (1- ½cos(θ)) so that θ=0 at the leading edge. This will reverse the sign of coefficient A₂. This alternate mapping in some cases makes it easier to perform the dy_c/dx integration.

THANK YOU DEAR.

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