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20.5 Panel Method

Panel methods are a classical approach to solving flow problems that is still very useful today when finding solutions to even very complicated geometries. You represent a body as a series of panels.

A few definitions first: we say a source panel of strength λ \lambda (units m/s) is defined by:

ϕp=12π∫λln⁡(r)ds \phi_p = \frac{1}{2 \pi} \int \lambda \ln (r) \dd s

=12π∫λln⁡((xp−x)2+(yp−y)2)ds = \frac{1}{2 \pi} \int \lambda \ln \left( \sqrt{ (x_p - x)^2 + (y_p - y)^2}\right) \dd s

λ=2uλ \lambda = 2 u_\lambda

Figure 20.2

Approximate a body as a polygon of N sides. Each panel has a uniform source λi \lambda_i . Define angle of flow relative to the panel normal βi \beta_i . To define boundary conditions we say the boundary must be a streamline, which means that the normal velocity at each panel is zero uni=0 u_{n_i} = 0 . So we need to balance the free stream flow and the sources from all other panels.

So, at each panel i i the normal velocity comes from the contribution from the panel, the contribution from all other panels, and this must equal the free flow normal −u∞cos⁡βi - u_{\infty} \cos \beta_i

λi2+∑j≠iλj2π∫j∂∂niln⁡(rij)dsj=−u∞cos⁡βi \frac{\lambda_i}{2} + \sum_{j \neq i} \frac{\lambda_j}{2 \pi} \int _j \pdv{}{n_i} \ln ( r_{i j}) \dd s_j = - u_{\infty} \cos \beta_i

This gives N equations and N unknowns (λi \lambda_i ). We can solve them to compute

uti=λi2tan⁡βi u_{t_{i}} = \frac{\lambda_i}{2} \tan \beta _i

ϕ(x,y)=∑iϕi(x,y)→v=−∇ϕ \phi(x, y) = \sum_i \phi_i (x, y) \rightarrow \vec v = - \grad \phi

This method is applicable for non-lifting bodies. Without circulation (vertices) there can't be any lift. We can modify the model by including a term that corresponds with circulation, we can allow for lift.

Define a distributed vortex panel of strength γ \gamma (units m/s):

γ=lim⁡Δs→01Δs∮v⋅dl=2uγ \gamma = \lim_{\Delta s \rightarrow 0} \frac{1}{\Delta s} \oint \vec v \cdot \dd \vec l = 2 u_{\gamma}

dv=γds2πr \dd v = \frac{\gamma \dd s}{2 \pi r}

ϕp=12π∫γdsθ=12πtan⁡−1(yp−yyp−x)γds \phi _p = \frac{1}{2 \pi} \int \gamma \dd s \theta = \frac{1}{2 \pi} \tan ^{-1} \left( \frac{y_p - y}{y_p - x} \right) \gamma \dd s

Figure 20.3

Apply the boundary condition to the vortex panels un,i=0 u_{n, i} = 0

∑j≠iNγj2π∫∂∂ni[tan⁡−1(yi−yjxi−xj)]dsj=0 \sum_{j \neq i} ^N \frac{\gamma_j}{2 \pi } \int \pdv{}{n_i} \left[ \tan ^{-1} \left( \frac{y_i - y_j}{x_i - x_j} \right) \right] \dd s_j = 0

This gives N unknowns γi \gamma_i and only N-1 equations. The last equation comes from the Kutta condition, which says that the tangential velocity at the trailing edge has to be the same from the top as from the bottom of the airfoil. That is to say, the flow must detach at the trailing edge:

ut1=−utN u_{t_1} = - u_{t_N}

Once we've solved for γi \gamma_i and λi \lambda_i we can compute what the flow velocity is

uti=λi2tan⁡βi+γi2+∑j≠iγj2π∫∂∂ti[tan⁡−1(yi−yjxi−xj)]dsj u_{t_i} = \frac{\lambda_i}{2} \tan \beta_i + \frac{\gamma_i}{2} + \sum_{j \neq i} \frac{\gamma_j}{2 \pi} \int \pdv{}{t_i} \left[ \tan ^{-1} \left( \frac{y_i - y_j}{x_i - x_j} \right) \right] \dd s _j

ϕ(x,y)=∑iϕi(x,y)→v(x,y)=−∇ϕ \phi(x, y) = \sum_i \phi_i (x, y) \rightarrow \vec v(x, y) = - \grad \phi

The coefficient of pressure is

Cp=p−p∞12ρu∞2=1−ut2u∞2 C_p = \frac{p - p_{\infty}}{\frac{1}{2} \rho u_{\infty}^2} = 1 - \frac{u_t ^2}{u_\infty ^2}