How to turn inviscid data to viscous?

Question

I have used Panel method to calculate Cp vs x/c graph and CL. But as panel method assumes attached flow it does not give an flow separation and thus in CL vs alpha graph I could not find any stall. My question is, I have all the inviscid datas, how do I turn into viscous flow data? I know that I have to involve reynold's number, skin friction coefficient and all to make it stall in CL vs alpha curve. How to I do that? What formulas, equations or method I should use?

Answer 1

You can still use the panel method and implement along with it an integral method for the boundary layer (like it is done in the xfoil software). The lift will be modified with the theory of displacement while for the drag the viscous drag will be computed directly by considering the friction coefficient found thanks to the integral method. One of the most common integral method (and one of the most used at my knowledge) is the Thwaites's method. You can find a reference here and an awesome review about the different methods here. I describe here one of the possible procedure:

• You define the stagnation point on your profile, starting from the pressure coefficient distribution obtained from your panel method
• Strating from the stagnation point the BL that is developing is laminar, hence to compute the friction coefficient (and the displacement boundary layer) you will use initially the Thwaites method. Here I will limit in describing this method. If you want to know more about transition you can couple with Michel criteria and you will find all the information in the hereby reference.

The Thwaites method is an empirical method based on:

The Von Karman integral equation for the BL: $$ \frac{d \theta}{dx}+\frac{\theta}{U_e}\left(2+H\right) \frac{d U_e}{dx}=\frac{1}{2}c_f $$

The shape factor (we define as $\delta^*$ the displacement BL thickness) $$ H=\frac{\delta^*}{\theta} $$

A local Reynolds number $$ Re_{\theta}=\frac{U_e\theta}{\nu} $$

The Thwaites parameter $$ l=\frac{1}{2}Re_{\theta}c_f $$

The Polhausen parameter, that gives an index of the local pressure: $$ \lambda=\frac{\theta^2}{\nu}\frac{d U_e}{dx} $$

The Von Karman equation has three unknowns $($\theta$, $H$, $c_f$ )$, and hence we will need 2 closure equations. The first one is an empirical relation: $$ \frac{U_e}{\nu}\frac{d\left(\theta^2\right)}{dx}=2\left[l-\left(2+H\right)\lambda\right]\approx 0.45-6\lambda $$ By substitution we arrive at the following ODE: $$ \frac{d}{dx}\left(U_e^6\theta^2\right)=0.45\nu U_e^5 $$ It is here that you will use your velocity computed with your panel method. in fact $U_e$ is the sum of the asymptotic velocity and of the perturbation velocity computed with your non viscous method. In this way you will obtain $\theta$ and it will be possible for you to compute the necessary quantity for each point of the discretization. Starting from the stagnation point you will need a first $\theta$. This is approximated throught the following empirical relation: $$ \theta=\sqrt{\frac{0.075 \nu}{U_e/R_0}} $$ For the points that will come after this, after having computed $\theta$ you can obtain $\lambda$ using a second empirical relation and finally the friction coefficient $c_f$: \begin{equation} \lambda=\frac{\theta^2}{\nu}\frac{dU_e}{dx} \end{equation}

\begin{equation} l(\lambda) = \left\{ \begin{array}{rcl} 0.22+1.5\lambda-1.8\lambda^2 & ,& \lambda>0\\ 0.22+1.402\lambda+\frac{0.018\lambda}{0.107+\lambda} & , & \lambda<0 \end{array}\right. \end{equation}

\begin{equation} c_f=\frac{l(\lambda)}{1/2 Re_{\theta}} \end{equation}

once obtained all the local $c_f$ it is possible to obtain the total $c_{d,f}$ by the integration of the local contribution over the profile. Through the computation of the displacement thickness you can start to correct your lift coefficient with the viscous data.

To predict the stall the queation is a bit more complicated. I suggest you to have a look at the paper I proposed you for the transition and to this paper. Even if it is not of the best quality is quite explicative.