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

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. Viiflow employs a similar approach compared to XFOIL[1], but is based on a complex variable panel method and 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.
- (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].
- Airfoil cascades can be calculated.

The boundary layer equations derive from the Navier-Stokes Equations by assuming among others that the *change* in 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.