$\newcommand{\dlogdxi}[1]{\frac{\mathrm{d} \log{#1} }{\mathrm{d} \log{\xi}}} \newcommand{\ddxi}[1]{\frac{\mathrm{d}#1}{\mathrm{d}\xi}}$

Inviscid methods approximate the complex flow defined by the Navier-Stokes equations by disregarding viscosity. This leads to the Euler equations. By assuming subsonic flow, the equations can be further simplified to potential flow with compressibility corrections. Viiflows panel method employs the same streamfunction approach that XFOIL[1] uses, with the following differences:

- The source strength is assumed to be constant over each panel.
- The collocation points of the wake are not the panel nodes but the panel-midpoints.
- The Prandtl-Glauert compressibility corrections are used for $c_p, c_L, c_M$.
- (Steady) pitching airfoils can be simulated by changing the boundary conditions at the collocation points from constant free flow to rotating flow. This also necessitates a change in pressure calculation[8].

The boundary layer equations derive from the Navier-Stokes Equations by assuming among others that the *change* in of the flow in one direction is negligible in comparison to the change in flow in the other direction. For example, near the surface of an airfoil the flow accelerates from 0 (at the surface) to its inviscid outer velocity within the boundary layer thickness $\delta$, while the tangential velocity at our current point of interest is basically constant $\pm \delta$ along the surface.

Viiflow employs the same boundary layer equations as in [2], although slightly recast, with modifications defined in [3], [4], [5] and [6]. Specifically, the equations in the variables $\theta,\delta^*, u_e, n, \sqrt{C_\tau}$ over the surface variable $\xi$ read:

$$ \begin{align} % Momentum \dlogdxi{\theta} + \left( H+2+M_e \right) \dlogdxi{u_e} &= \frac{\xi}{\theta} \frac{c_F}{2} & \mathrm{(laminar + turbulent)}\\ % Kinematic Shape Parameter \dlogdxi{H^*} +\left( 2\frac{H^{**}}{H^*} + 1-H\right) \dlogdxi{u_e} &= \frac{\xi}{\theta}\left(2 \frac{c_D}{H^*} - \frac{c_F}{2}\right) & \mathrm{(laminar + turbulent)}\\ % Shear lag \dlogdxi{\sqrt{C_\tau}} + \dlogdxi{u_e} &= K_C\frac{\xi}{2\delta}\left(\sqrt{C_{\tau,EQ}}-\lambda\sqrt{C_\tau}\right) + \xi\left(\frac{1}{U_{e}}\ddxi{U_{e}}\right)_{EQ} & \mathrm{(turbulent)} \\ % Amplification \frac{\mathrm{d} n}{\mathrm{d}\log{\xi}} &= \xi A & \mathrm{(laminar)} \end{align} $$- The closure relations $\delta, H, H^{**}$ are given in [2].
- The wake is handled as in [1], with modifications for blunt trailing edges from [6].
- The closure relations $c_F, c_D, H^*$ are given in [3].
- The factor $K_C$ is a constant value of 5.6 by default, but can be changed to follow RFOIL[5] or the factor of Thomas[9].
- After transition, the initial $C_\tau$ is set to its equilibrium value.
- The equilibrium values with subscript $EQ$, the amplification slope $A$ and the critical momentum thickness Reynolds number ($Re_{\theta_0}$ in [2]) are chosen as defined in [4].
- The parameter $H^*$ is modified to be twice continuously differentiable by smoothing between the different branches.
- The default $G-\beta$ locus constants for the equilibrium values are chosen as in [5].
- The momentum thickness Reynolds number is calculated by assuming a power law for the dependence of viscosity to temperature [7]: $\mu/\mu_{ref}= (T/T_{ref})^\omega$
- The edge Mach $M_e$ and the density change due to edge Mach for the momentum thickness Reynolds number is calculated from isentropic flow relations.
- The parameter $\lambda$ in the lag equation can be set for airfoil and wake differently using the setup.

[1] Drela, Mark. *XFOIL: An analysis and design system for low Reynolds number airfoils.* Low Reynolds number aerodynamics. Springer Berlin Heidelberg, 1989. 1-12.

[2] Drela, Mark, and Michael B. Giles. *Viscous-inviscid analysis of transonic and low Reynolds number airfoils.* AIAA journal 25.10 (1987): 1347-1355.

[3] Nishida, Brian Allen. *Fully simultaneous coupling of the full potential equation and the integral boundary layer equations in three dimensions.* Diss. Massachusetts Institute of Technology, 1996.

[4] Ye, Boyi. *The Modeling of Laminar-to-turbulent Transition for Unsteady Integral Boundary Layer Equations with High-order Discontinuous Galerkin Method.* Thesis TU-Delft (2015).

[5] Van Rooij, R. P. J. O. M. *Modification of the boundary layer calculation in RFOIL for improved airfoil stall prediction.* Report IW-96087R TU-Delft, the Netherlands (1996).

[6] Drela, Mark. *Integral boundary layer formulation for blunt trailing edges.* 7th Applied Aerodynamics Conference. 1989.

[7] Schlichting, Hermann, and Klaus Gersten. *Boundary-layer theory.* Springer, 2016.

[8] van der Horst, Sander, et al. *Flow Curvature Effects for VAWT: a Review of Virtual Airfoil Transformations and Implementation in XFOIL.* 34th Wind Energy Symposium. 2016.

[9] Thomas, J. *Integral boundary-layer models for turbulent separated flows.* 17th Fluid Dynamics, Plasma Dynamics, and Lasers Conference. 1984.