The Boundary Layer Method In order to calculate the friction drag of an airfoil for a given flow condition (angle of attack, Reynolds number), an analysis of the viscous boundary layer is necessary. From the momentum loss in this small layer on the surface of the airfoil the drag can be derived. As the velocity distribution changes with angle of attack, the drag changes too. Also, the thickness of the boundary layer changes with Reynolds number. The boundary layer module uses the velocity distribution derived by the panel method and performs its calculations based on the formulas presented in [14, 15, 16]. The method is a so called integral boundary layer method, which does not handle laminar separation bubbles or large scale separation (stall). The boundary layer module works best in the Reynolds number regime between 500'000 and 20'000'000. The results of the boundary layer module are also used to correct lift, drag and moment coefficients empirically, if separation occurs. Additionally, a blending to separated, flat plate coefficients is performed for very high angles of attack. The procedure starts at the stagnation point and marches along each surface, integrating simplified boundary layer equations. The integration follows a 2nd order Runge-Kutta scheme with stabilization by automatic step reduction. This can be a bit slow some times, but works more reliable than the simple Newton method used before. During the way towards the trailing edge, the method checks, whether transition from laminar to turbulent or separation occurs. The following empirical transition criteria have been implemented and can be selected by the user: Method Eppler 1[14]

Transition assumed when

Recommendation Re > 1x105 Re > 1x105

Eppler 2 [17]

Michel 1 [35]

Re > 2x106

Re > 2x106

Michel 2 Here, an additional local pressure gradient parameter K is used ("Pohlhausen parameter")

Instability is assumed when K > Kinstability

Re > 5x106

Granville In regions of instability, transition is assumed when K > Ktransition

approximation of n transition can occur when n

Drela e approximation (Xfoil pre 1991)

approximation of n transition can occur when

Drela en approximation (Xfoil post 1991) [36]

Note: Depending on the version of Xfoil, The last constant in the second equation has been changed several times between 0.62 and 0.7.

approximation of n transition can occur when Arnal en approximation by Würz

Note: The coefficients ai and bi can be found in [37] (as well as an interesting discussion of empirical transition models). If laminar separation is detected, the method switches to turbulent flow and continues. When turbulent separation is found, the boundary layer integration is stopped and an empirical drag penalty depending on the length of the separated region is added to the result. Flow State Separation assumed when laminar turbulent The drag is applied by examining the boundary layer parameters at the trailing edge, using the so called Squire-Young formula.

Tabular Output The tables produced on the Boundary-Layer card contain the following columns:

symbol description x/l normalized x-coordinate y/l normalized y-coordinate v/V normalized surface velocity d1 displacement thickness d2 momentum loss thickness d3 energy loss thickness Cf local friction coefficient H12 shape factor d1/d2 H32 shape factor d3/d2 flow state laminar, turbulent, separated the first cell height required for y+=1 (multiplied by 100 resp. in %) y1 This value can be useful for grid generation for Navier-Stokes solvers For abbreviations see the quick reference page.