2.1. Reference wall-resolved RANS simulations (Reference wrRANS)

Wall-resolved simulations of the various considered wings are performed using the finite-volume solver from the Siemens STAR-CCM+ software (Siemens 2019). The steady-state RANS equations for incompressible flow and with the Spalart–Allmaras turbulence closure model are solved in fully turbulent mode. The diffusive and convective fluxes, as well as the temporal integration, are discretized using the second-order schemes. A segregated approach with the SIMPLE algorithm is used for the pressure–velocity coupling.

The mesh around the considered wings comprises two regions: a background mesh and an overset mesh (respectively illustrated in black and blue in figure 1). The background mesh has an outer dimension of $20b \times 8b \times 8b$, where $b$ is the wing span. It comprises orthogonal cubic cells having an isotropic dimension of $0.03c$ in the vicinity of the blade. The overset mesh region comprises a 2-D mesh of an NACA-0015 profile that is extruded along the wing span. Here, 450 points are used to discretize the profile and a prism layer is used to ensure a $y^+$ value of 1 on the wing surface. The extruded cells in the middle part of the wing have a spanwise dimension of $0.03c$ (similar to the resolution of the background mesh). Globally, the mesh comprises 45 million cells (20 million cells in the background mesh and 25 million cells in the overset mesh). At the inlet of the domain, a uniform velocity is imposed along with a turbulent viscosity ratio $\mu _t/\mu =0.2$. A uniform static pressure is imposed at the outlet and the lateral boundaries of the domain are symmetry planes.

Figure 1. Mesh used for the wall-resolved RANS simulation: overset mesh (blue) and background mesh (black).

This approach was validated against the experimental data provided by Chow (Reference Chow1994) and Chow et al. (Reference Chow, Zilliac and Bradshaw1997) for the case of a rectangular wing of aspect ratio $AR=1.5$ with NACA-0012 profile at Reynolds number $Re_c = U c / \nu = 4.6\times 10^6$ (Villeneuve et al. Reference Villeneuve, Boudreau and Dumas2019; Villeneuve, Winckelmans & Dumas Reference Villeneuve, Winckelmans and Dumas2021). The results of this validation case are reproduced in figure 2. The agreement between the simulation and the experimental results at $y/b=0.667$ and $y/b=0.833$ is excellent. At the spanwise location $y/b=0.889$, slight deviations are observed near the trailing edge of the extrados. These are due to the formation of the tip vortex on the wing upper surface (Churchfield & Blaisdell Reference Churchfield and Blaisdell2013). Nonetheless, these small deviations have little impact on the section lift and drag coefficients. Additionally, our study considers much larger aspect ratios (from $AR=7.5$ to $AR=15$), which are easier to simulate. Finally, the region of the wing where the loads are evaluated in this study corresponds to a distance to the tip normalized by the chord $d_{tip}/c>0.1$, which is where the numerical results perfectly fit those of the experiment. Therefore, the results of the wall-resolved simulations of the various cases are used here as valid references.

Figure 2. Validation of the wall-resolved RANS methodology: the surface pressure coefficient distributions reported by Chow, Zilliac & Bradshaw (Reference Chow, Zilliac and Bradshaw1997) (black markers) are compared with those of the simulation (red curves) at three spanwise locations (reproduced from Villeneuve, Boudreau & Dumas Reference Villeneuve, Boudreau and Dumas2019). (a) $y/b = 0.667$, (b) $y/b = 0.833$ and (c) $y/b = 0.889$.

The software is also used to find the lift and drag coefficients of the NACA-0015 profile at $Re_c=6\times 10^6$, as a function of the angle of attack; as those polar data are required for the ALM and the lifting line methods. For this study, the airfoil is centred in a rectangular domain of size $100c \times 100c$. The mesh comprises 135 000 cells. The lift and drag coefficients obtained using this methodology are provided in Appendix A, together with a fit valid for moderate angles of attack (up to 10$^\circ$).

2.2. Actuator line method (ALM)

The actuator line method (ALM) is used in a flow solver that performs LES. This is achieved by solving the incompressible Navier–Stokes equations supplemented by a subgrid scale (SGS) model on a Cartesian staggered grid using an in-house developed fourth-order finite differences code (Duponcheel et al. Reference Duponcheel, Bricteux, Manconi, Winckelmans and Bartosiewicz2014; Moens et al. Reference Moens, Duponcheel, Winckelmans and Chatelain2018). The time-stepping is performed using a second-order Adams–Bashforth scheme. The SGS model consists of the regularized variational multiscale (RVM) model (Jeanmart & Winckelmans Reference Jeanmart and Winckelmans2007). For the wing cases, however, the wake is steady and the SGS model does not affect the simulation results.

The influence of the forces exerted by the wing on the flow is represented using the ALM. The aerodynamic forces acting on the wing are computed at the control points of the actuator line using the local flow velocity, sampled from the flow solver, and the lift and drag coefficient of the airfoil. The control points are evenly spaced along the aerodynamic centreline of the wing ($=$ line connecting the aerodynamic centres of the airfoil profiles used to define the wing), by a distance $h_c$ taken equal to the flow solver grid size $h$, or close to it. The effective velocity sampling is performed in two steps. First, the velocity is linearly interpolated from the flow solver grid to a 2-D template plane that is perpendicular to the actuator line and centred on the control point, as depicted in figure 3. Then, the effective velocity at the control point $\boldsymbol {v}_c$ is taken as the weighted average of the values obtained on the template, using weights obtained using a 2-D Gaussian kernel; and thus not as a direct interpolation evaluated at the control point. This is referred to as ‘integral sampling’ (Churchfield et al. Reference Churchfield, Schreck, Martinez, Meneveau and Spalart2017).

Figure 3. Schematic of the 2-D template plane with Gaussian weights ($\sigma /h=2$) used by the ALM for the velocity sampling and the force distribution.

The effective velocity at the control point is then used to compute the aerodynamic forces per unit span acting on the airfoil section, as depicted in figure 4. The wing is here moving horizontally, in the $-\hat {\boldsymbol {e}}_x$ direction, and at velocity $U_\infty$. The angle of attack, $\alpha$, is also assumed moderate so that we remain in the linear regime for the lift coefficient.

Figure 4. Velocities and forces on the ALM control point. Note that the angle of attack is exaggerated on purpose, for clarity of the figure.

First, the relative velocity $v_{rel}$ and the downwash angle $\varepsilon$ in the airfoil plane are obtained at the control point. Defining $v_x = \boldsymbol {v}_c \boldsymbol {\cdot } \hat {\boldsymbol {e}}_x$ and $v_z = \boldsymbol {v}_c \boldsymbol {\cdot } \hat {\boldsymbol {e}}_z$, we obtain

(2.1a,b)\begin{equation} v_{rel} = \sqrt{v_x^2 + v_z^2} \quad \text{and} \quad \varepsilon = \arctan\left(-\frac{v_z}{v_x}\right) \simeq{-}\frac{v_z}{v_x}. \end{equation}

The lift $l_p$ and drag $d_p$ per unit span of the airfoil profile are then found using the polar data obtained from wall-resolved 2-D simulations described in § 2.1 and in Appendix A as

(2.2a,b)\begin{equation} l_p = \tfrac{1}{2} \rho v_{rel}^2 c C_l^p(\alpha_e) \quad \text{and} \quad d_p = \tfrac{1}{2} \rho v_{rel}^2 c C_d^p(\alpha_e), \end{equation}

with $\rho$ the fluid density, $c$ the chord of the profile, and $\alpha _e$ the effective angle of attack, defined as the difference between the geometric angle of attack $\alpha$ and the downwash, $\alpha _e=\alpha -\varepsilon$. The assumption of the linear regime is that $\alpha _e$ remains less than 10$^\circ$. The lift and drag per unit span of the airfoil section, expressed in the vertical and horizontal frame of the wing, then read

(2.3)$$\begin{gather} l = l_p \cos(\varepsilon) - d_p \sin(\varepsilon) \simeq l_p - d_p \varepsilon \simeq l_p, \end{gather}$$

(2.4)$$\begin{gather}d = d_p \cos(\varepsilon) + l_p \sin(\varepsilon) \simeq d_p + l_p \varepsilon , \end{gather}$$

where the approximations are obtained using the facts that the downwash angle is small and that the drag is small compared with the lift. The term $l_p \varepsilon$ correspond to the so-called ‘induced drag’.

The distribution of the lift and drag coefficients along the wing span are then defined based on the mean chord $\bar {c}=S/b$ (with $S$ the wing surface and $b$ the wing span) and as a function of the spanwise location $y$ as

(2.5a,b)\begin{equation} C_l(y) = \frac{l(y)}{\dfrac{1}{2}\rho U_\infty^2 \bar{c}} \quad \text{and} \quad C_d(y) = \frac{{d}(y)}{\dfrac{1}{2}\rho U_\infty^2 \bar{c}} . \end{equation}

The obtained aerodynamic force on each profile ($=$ force per unit span $\times h_c$) must then be distributed on the flow solver grid to act as a source term on the flow equations. This is achieved in two steps. The force is first distributed on the template plane using the same Gaussian weights as those used for the evaluation of the effective velocity. Then, each fraction of that force on the template is distributed on the closest flow grid points using a linear distribution kernel. More information concerning the weights of the template with 2-D Gaussian kernel and its usage are provided in Appendix B.

To ensure a smooth distribution of the forces along the wing when it is not aligned with the mesh, the number of points for the force distribution is taken larger than the number of control points. The wing is divided into smaller segments of length $h_p < h_c$. A new point for the distribution is placed at the centre of each of these new segments. The force per unit span at each of these new points is then interpolated from that at the control points. Finally, the associated force at each added point ($=$ force per unit span $\times h_p$) is distributed over its local 2-D template; and from then onto the nearby grid points.

2.3. Prandtl lifting line (PLL)

The Prandtl lifting line theory allows for modelling the aerodynamics of lifting bodies with large aspect ratio at small angles of attack. In this model, the wing is represented by a singular vortex line whose circulation $\varGamma (y)$ varies along the span, as depicted in figure 5. Due to the variation of the circulation, a planar vortex sheet is shed in the wake with an intensity $\gamma (y)=-({{\rm d}\varGamma }/{{\rm d} y})(y)$. At each spanwise location, it is assumed that the flow is two-dimensional, so that the lift of the section is given by the Kutta–Joukowski theorem,

(2.6)\begin{equation} l_p(y) = \rho U_\infty \varGamma(y), \end{equation}

and also by the profile lift coefficient,

(2.7)\begin{equation} l_p(y) = \tfrac{1}{2} \rho U_\infty^2 c(y) C_l^p(\alpha_e(y)). \end{equation}

The downwash $\varepsilon (y)$, necessary for the computation of the effective angle of attack $\alpha _e(y)$, is obtained as the Biot–Savart velocity induced by the semi-infinite singular vortex sheet,

(2.8)\begin{equation} \varepsilon(y) = \frac{1}{U_\infty}\frac{1}{4 {\rm \pi}}\int_{{-}b/2}^{b/2} \frac{1}{(y-y')} \frac{{\rm d}\varGamma}{{{\rm d} y}'}(y')\, {{\rm d} y}'. \end{equation}

The combination of (2.6)–(2.8) leads to the compatibility equation with the circulation $\varGamma (y)$ as the only unknown:

(2.9)\begin{equation} \varGamma(y) = \frac{1}{2} U_\infty\, c(y)\, C_l^p\left(\alpha - \frac{1}{U_\infty}\frac{1}{4 {\rm \pi}}\int_{{-}b/2}^{b/2} \frac{1}{(y-y')} \frac{{\rm d}\varGamma}{{{\rm d} y}'}(y') {{\rm d} y}'\right). \end{equation}

The solution is found using a Fourier series for $\varGamma (\theta )$, where $\cos(\theta) = y/(b/2)$. The profile drag is then obtained as

(2.10)\begin{equation} d_p(y) = \tfrac{1}{2} \rho U_{\infty}^2 c(y) C_d^p(\alpha_e(y)). \end{equation}

The profile lift and drag are then expressed in the wing frame using (2.3) and (2.4), and the lift and drag coefficients distributions are found using (2.5a,b). In this form, they can be compared with the results obtained using the ALM and/or wall-resolved simulations. The classic PLL method is thus used here. It should be noted that a numerical implementation of the lifting line method also exists (Katz & Plotkin Reference Katz and Plotkin2001), which can be further improved to provide a force distribution closer to that obtained using the vortex lattice method (Li et al. Reference Li, Gaunaa, Pirrung, Meyer Forsting and Horcas2022).

Figure 5. Prandtl lifting line (PLL).

2.4. Mollified lifting line (MLL)

To quantify the effect of the mollification on the lift and drag distributions, a mollified lifting line (MLL) theory can be used (Caprace et al. Reference Caprace, Chatelain and Winckelmans2019). The MLL is similar to the PLL, but the computation of the downwash is modified to account for the regularization of the vorticity field by a Gaussian kernel, as depicted in figure 6. The theory was developed for 3-D, 2-D and one-dimensional (1-D) mollifications, yet only the 2-D mollification is considered here. This way, the MLL is in accordance with the ALM that uses a 2-D Gaussian kernel for the force distribution. In this case, the downwash on the centreline of the mollified lifting line $\varepsilon _{\sigma,0}$ is obtained at each spanwise location by (Caprace et al. Reference Caprace, Chatelain and Winckelmans2019)

(2.11)\begin{equation} \varepsilon_{\sigma,0}(y) =\frac{1}{U_{\infty}} \frac{1}{4 {\rm \pi}} \int_{{-}b / 2}^{b / 2}\left(\int_{-\infty}^{\infty} \frac{1}{\sqrt{\rm \pi}} \frac{{\rm e}^{{-}z^{\prime} / \sigma^2}}{((y-y^{\prime})^2+z^{\prime 2})} \frac{{\rm d} z^{\prime}}{\sigma}\right)(y-y^{\prime}) \frac{\mathrm{d} \varGamma}{\mathrm{d} y^{\prime}}(y^{\prime}) \,\mathrm{d} y^{\prime}. \end{equation}

The ALM used in this paper evaluates the effective velocity as a weighted average, using the weights of the distribution kernel, and not directly as that on the centreline. This approach, also known as integral sampling (Churchfield et al. Reference Churchfield, Schreck, Martinez, Meneveau and Spalart2017), is also included in the present MLL by adding a convolution of the induced velocity with the regularization kernel (Caprace et al. Reference Caprace, Chatelain and Winckelmans2019). The resulting mean downwash $\bar {\varepsilon }_\sigma (y)$ is then given by

(2.12)\begin{align} \bar{\varepsilon}_\sigma(y) = \frac{1}{U_\infty} \frac{1}{4 {\rm \pi}} \int_{{-}b / 2}^{b / 2}\left[\iint_{-\infty}^{\infty} \frac{1}{\rm \pi} \frac{{\rm e}^{-{(z^{\prime 2}+z^{2})}/{\sigma^{2}}}}{((y-y^{\prime})^{2}+ (z-z^{\prime})^{2})} \frac{{\rm d} z^{\prime}}{\sigma} \frac{{\rm d} z}{\sigma}\right](y-y^{\prime}) \frac{{\rm d} \varGamma}{{{\rm d} y}^{\prime}}(y^{\prime})\, {{\rm d} y}^{\prime} . \end{align}

The lift distribution on the MLL is obtained similarly as for the PLL using (2.12) evaluated numerically. The downwash obtained using the mollified vortex sheet is, in general, smaller than that obtained using the singular vortex sheet of the PLL. This results in an overestimation of the lift force distribution on the wing, due to a higher effective angle of attack.

Figure 6. Mollified lifting line (MLL) with 2-D isotropic mollification (Caprace et al. Reference Caprace, Chatelain and Winckelmans2019).

2.5. Vortex lattice method (VLM)

The vortex lattice method (also called lifting surface method) models the potential flow past a wing without thickness by dividing its surface into quadrilateral vortex rings, each consisting of four vortex segments in a closed-loop arrangement, as depicted in figure 7. The wake is represented by horizontal horseshoe vortices that extend to infinity. Each vortex ring has a circulation $\varGamma _{i,j}$, which is obtained by computing the downwash induced by all the elements and imposing no-through flow at the colocation point (located at the centre of each element). A detailed explanation is found from Katz & Plotkin (Reference Katz and Plotkin2001). The force $\boldsymbol {F}_{i,j}$ acting on each vortex segment is then obtained using the circulation $\varGamma _{i,j}$ of the segment, the velocity $\boldsymbol {U}_{i,j}$ measured at the centre of the segment and the segment vector $\boldsymbol {s}_{i,j}$ of norm equal to the segment length,

(2.13)\begin{equation} \boldsymbol{F}_{i,j} = \rho \boldsymbol{U}_{i,j} \times (\varGamma_{i,j} \boldsymbol{s}_{i,j}) . \end{equation}

For each wing section $j$, the lift per unit span is obtained as the sum of the vertical forces along the chord, divided by the length of each contributing segment,

(2.14)\begin{equation} l_j = \sum_{i=0}^{i=n_x} (\boldsymbol{F}_{i,j} \boldsymbol{\cdot} \hat{\boldsymbol{e}}_z)/\lVert\boldsymbol{s}_{i,j}\rVert . \end{equation}

The induced drag per unit span is obtained similarly from the force in the streamwise direction,

(2.15)\begin{equation} d_j = \sum_{i=0}^{i=n_x} (\boldsymbol{F}_{i,j} \boldsymbol{\cdot} \hat{\boldsymbol{e}}_x)/\lVert\boldsymbol{s}_{i,j}\rVert. \end{equation}

The total drag per unit span is then found by adding the profile drag obtained using the polar data and an estimated effective angle of attack. The latter is taken as the geometrical angle of attack minus the downwash angle induced by the trailing vortices, as measured at the wing quarter-chord line. The VLM is here used with $n_x=32$ in the chordwise direction and $n_y=128$ in the spanwise direction.

Figure 7. Vortex lattice method (VLM), here illustrated for a rectangular wing with $AR=7.5$ and $4 \times 30$ elements. The number of elements used in simulations is higher than depicted (32 in the chordwise direction, 128 in the spanwise direction).

4.1. Correction function for the lift coefficient

The previous section has shown that the ALM overpredicts the lift and drag distributions compared with the reference results, due to the mollification and the virtual camber effects. In this section, a correction function $F_l$ for the lift is obtained to account for these effects. Since this function aims at correcting the near-tip behaviour, it is expressed as a function of the distance to the tip $d_{tip}(y)$, normalized by the chord $c$. The distance to the tip is defined as (see figure 8)

(4.1)\begin{equation} d_{tip}(y) = \frac{b}{2} - |y|,\end{equation}

which is valid for both sides of the wing. The correction function for the lift is defined as the difference between the non-corrected lift coefficient distribution predicted by the ALM, denoted $C_l^{ALM,nc}(y)$, and the lift coefficient distribution of the reference, denoted $C_l^{ref}(y)$. For the sake of conciseness, the dependency to $y$ is not written explicitly in the equations that follow. The function $F_l$ reads

(4.2)\begin{equation} F_l\left(\frac{d_{tip}}{c}\right) = \frac{(C_l^{ALM,nc}-C_l^{ref})}{C_l^{ALM,nc}}.\end{equation}

It can then be applied to obtain the corrected lift coefficient $C_{l}^{corr}(y)$ as

(4.3)\begin{equation} C_{l}^{corr} = \left( 1-F_l\left(\frac{d_{tip}}{c} \right) \right) C_{l}. \end{equation}

The application of the correction is here illustrated for the rectangular wing with ${AR=15}$. It is first necessary to converge the correction function $F_l$ using multiple iterations, as the modification of the lift coefficient also modifies the wake and hence the induced velocity. The obtained correction function is depicted in figure 11, and its numerical values are also provided in Appendix C. In this case, three iterations were sufficient to reach convergence. The resulting correction function is maximal near the wing tip, and it decreases with the distance to the tip. The value of the correction is below 1 % for $d_{tip}/c \ge 3.5$. Hence, a correction must be applied on a significant region in the outer part of the wing.

Figure 11. Correction function $F_l$ for the lift coefficient. The final function (in dark blue) is obtained in three iterations.

To verify that the current analysis is valid for any small angle of attack, for which the profile lift coefficient increases linearly, the correction function for the lift is obtained for various small angles. The result of this investigation is shown in figure 12. It is shown that the correction function is essentially the same regardless of the angle of attack. This results from the fact that the effect of the mollification and the virtual camber effect both increase linearly with the angle of attack.

Figure 12. Correction functions $F_l$ obtained for various angles of attack.

The results obtained with the ALM and the MLL after the application of the correction function are displayed in figure 13, and compared with the reference results. As expected, the corrected lift is now virtually identical to that of the reference. However, the distribution of the drag changed importantly, and it now decreases near the tip. This is due to the reduction of the downwash caused by the lift correction and that reduces the induced drag component. It is therefore necessary to also apply a correction function for the drag distribution.

Figure 13. Lift and drag distributions before and after application of the correction function $F_l$ for the lift, for the case of a rectangular wing with $AR=15$ at $\alpha =5^\circ$.

4.2. Correction function for the angle of attack

To correct the drag distribution, it is tempting to define a similar function $F_d$ that would act on the drag coefficient,

(4.4)\begin{equation} F_d\left(\frac{d_{tip}}{c}\right) = \frac{(C_d^{ALM,nc}-C_d^{ref})}{C_d^{ALM,nc}}. \end{equation}

However, the previous analysis (reported in figure 10) has shown that the error on the drag coefficient is mostly related to the induced drag, whose value, given by (2.4), depends on the downwash angle. In fact, provided that the profile drag remains essentially constant in the considered region of the wing, and that the lift coefficient is corrected to match the reference results, then the error on the drag distribution necessarily originates from a misprediction of the downwash. As also shown by Pirrung et al. (Reference Pirrung, Van der Laan, Ramos-García and Meyer Forsting2020), it is therefore preferable to correct the effective angle of attack to obtain the correct drag, while also maintaining a correct value for the lift. This is possible due to the small value of the downwash angle. Following (2.3), the lift of the wing is very close to the lift of the profile $C_l \simeq C_l^p$. Moreover, for small angles of attack, the profile lift coefficient is a linear function: $C_l^p(\alpha ) \simeq ({{\rm d} C_l^p}/{{\rm d} \alpha }) \alpha$. The corrected lift coefficient, given by (4.3), can thus be rewritten as

(4.5)\begin{equation} C_l^{corr} \simeq \left( 1-F_l\left(\frac{d_{tip}}{c} \right) \right) \frac{{\rm d} C_l^p}{{\rm d} \alpha} \alpha_e. \end{equation}

Then the correction function $(1-F_l)$ can then be split into two parts: one that applies to the derivative of the profile lift coefficient, denoted $(1-F_{C_l})$, and the other that applies to the effective angle of attack, denoted $(1-F_{\alpha _e})$. This leads to the following expression:

(4.6)\begin{equation} C_l^{corr} = \left( 1-F_{C_l}\left(\frac{d_{tip}}{c} \right)\right) \frac{{\rm d} C_l^p}{{\rm d} \alpha} \left( 1-F_{\alpha_e}\left(\frac{d_{tip}}{c} \right) \right) \alpha_e . \end{equation}

To ensure that the correction of the lift remains unchanged, the condition that $(1-F_l) = (1-F_{C_l})(1-F_{\alpha _e})$ must be enforced. To complete the correction, it is then only necessary to find the function $F_{\alpha _e}$ that modifies the effective angle of attack.

The value of the downwash is not directly measured from the wall-resolved simulations, but can be obtained using the lift and drag distribution and the profile drag coefficient. In fact, (2.4) allows for expressing the reference downwash $\varepsilon ^{ref}(y)$ as

(4.7)\begin{equation} \varepsilon^{ref} = \frac{(C_d^{ref} - C_d^{p}(\alpha-\varepsilon^{ref}))}{ C_l^{ref} }. \end{equation}

This equation can be solved iteratively. Then, the correction function for the effective angle of attack is defined in a similar manner as for the correction function for the lift (4.2) by

(4.8)\begin{equation} F_{\alpha_e}\left(\frac{d_{tip}}{c}\right) = \frac{((\alpha-\varepsilon^{ALM,nc})-(\alpha- \varepsilon^{ref}))}{(\alpha -\varepsilon^{ALM,nc})} = \frac{(\alpha_e^{ALM,nc}-\alpha_e^{ref})}{\alpha_e^{ALM,nc}}. \end{equation}

Note that for an airfoil without camber, the profile drag coefficient is well approximated by a quadratic function $C_d^p(\alpha ) \simeq C_{d,0}^p + \beta \alpha ^2$ (see Appendix A). In this case, (4.7) can also be solved analytically.

The corrected value of the effective angle of attack $\alpha _e^{corr}(y)$, used to evaluate the lift and drag coefficients, is therefore given by

(4.9)\begin{equation} \alpha_e^{corr} = \left( 1-F_{\alpha_e}\left(\frac{d_{tip}}{c} \right) \right)\alpha_e. \end{equation}

The corrected value of the downwash angle $\varepsilon ^{corr}(y)$, used for the evaluation of the induced drag, is found as

(4.10)\begin{equation} \varepsilon^{corr} = \alpha - \alpha_e^{corr}. \end{equation}

Finally, the corrected value of the drag coefficient $C_d^{corr}$ is obtained from (2.4) as the sum of the corrected profile drag coefficient and of the corrected induced drag,

(4.11)\begin{equation} C_d^{corr} = C_d^{p}(\alpha_e^{corr}) + C_l^{corr} \varepsilon^{corr}. \end{equation}

The decomposition of $F_l$ into $F_{C_l}$ and $F_{\alpha _e}$ for the rectangular wing with $AR=15$ is depicted in figure 14. The numerical values of these functions are also provided in Appendix C. It can be seen that the correction of the angle of attack is mostly necessary close to the tip, until a distance $d_{tip}/c \simeq 1$. This indeed corresponds to the region where the drag of the ALM corrected for the lift coefficient differs from the reference results. Further away from the tip, the correction mostly acts on the profile lift coefficient. The value of that correction only drops below 1 % for $d_{tip}/c \ge 3.5$.

Figure 14. Decomposition of $F_l$ into $F_{\alpha _e}$ and $F_{C_l}$ such that $(1-F_l)=(1-F_{C_l})(1-F_{\alpha _e})$.

The results of the simulations performed by applying the correction functions is displayed in figure 15. Clearly, the correction significantly improves the accuracy of the drag distribution, which is now consistent with the reference results. The lift distribution also remains in good agreement with the reference. We also note that solely correcting the ALM for the mollification effect to reproduce the PLL results would not be sufficient in the present case.

Figure 15. Lift and drag coefficient distributions before and after application of the correction functions for the lift coefficient $F_{C_l}$ and for the effective angle of attack $F_{\alpha _e}$ on the rectangular wing with $AR=15$ at $\alpha =5^\circ$.

Note that applying the correction causes an increase of the lift computed on the last control point of the ALM. This originates from the small spanwise mollification of the ALM caused by the distribution of the forces from the template plane to the closest flow grid points. This slightly reduces the induced velocity near the tip. The MLL is purely based on a mathematical formula without any spanwise mollification, and therefore it does not predict an increase of the lift at the tip. When the correction is applied to the ALM, the effect of the small spanwise mollification becomes more significant, leading to a difference with the MLL at the last control point. However, the MLL and the ALM show a very good agreement at the other control points.

The correction functions were here obtained for a rectangular wing with a high aspect ratio ($AR=15$). However, their application to rectangular wings with lower aspect ratios must also be considered. In the next section, it is verified that the obtained correction functions remain valid for wings down to $AR=7.5$.

7.1. Effect of the mollification size

The mollification size, parametrized by the ratio $\sigma /c$, affects the accuracy of the loads predicted by the ALM importantly. In the previous sections, the size of the mollification was fixed to $\sigma /c=1/4$, as suggested by Martínez-Tossas et al. (Reference Martínez-Tossas, Churchfield and Meneveau2017). However, the mollification must sometimes be enlarged for practical applications. In this section, the effect of a change of the mollification size over the correction functions is considered. Simulations of the rectangular wing with $AR=15$ are performed using various values of $\sigma /c$. The results of these simulations are shown in figure 23. The MLL still provides excellent predictions for the ALM results with the larger mollifications. The lift coefficient further differs from the wall-resolved simulation and PLL results as the size of $\sigma /c$ increases, and the effect of the mollification becomes more significant than that of the virtual camber. The drag coefficient is also affected by the value of $\sigma /c$. This illustrates that the effect of the mollification impacts both the lift and the drag by changing the induced velocity on the wing.

Figure 23. Comparison of the lift and drag coefficients distributions along the half-span of the rectangular wing with $AR=15$ at $\alpha =5^\circ$, computed using various methods. The black markers are the results of the wall-resolved simulations.

The evolution of the integrated $C_L$ and $C_D$ with the mollification size is shown in figure 24. These are obtained using the MLL with various mollifications, up to $\sigma /c=2$. The integrated lift coefficient substantially increases with the mollification size: it is 2 % higher than the reference without mollification (PLL), whereas it reaches 8 % for the largest $\sigma /c$. The integrated drag coefficient decreases with the mollification size, starting from an overestimation of 6.3 % (PLL) to an underestimation of $-5\,\%$.

Figure 24. Evolution of the $C_L$ and $C_D$ with the mollification size for a rectangular wing with $AR=15$: effect of the 2-D mollification (black dots) and of the 3-D mollification (blue squares).

The correction functions obtained by comparing the MLL with the various mollification sizes to the reference results are provided in figure 25, and tables with the numerical values are provided in Appendix C. The functions $F_{C_l}$ for the lift coefficient are very similar for the various mollification sizes. This indicates that this function mostly corrects the effect of the virtual camber. The virtual camber indeed mostly depends on the lift variation along the wing span and of the chord, which remains essentially constant regardless of the mollification size.

Figure 25. Correction functions for (a) the lift coefficient $F_{C_l}$ and for (b) the angle of attack $F_{\alpha _e}$ for various mollification sizes in the case of a rectangular wing with $AR=15$.

The function $F_{\alpha _e}$ for the effective angle of attack evolves importantly with the increase of the mollification size, which indicates that the effect of the mollification can be corrected for both the lift and the drag by adapting the induced angle of attack. This is consistent with the fact that the mollification of the shed vortex sheet alters the value of the induced velocity on the wing, and hence mostly affects the effective angle of attack and not the lift coefficient. The function also extends to a larger distance to the tip $(d_{tip}/c)$ when the mollification size increases. However, in all cases, the product of the correction functions remains lower than 1 % for $(d_{tip}/c) \gtrsim 3.5$.

7.2. Effect of the mollification type

Whereas the 2-D mollification has been shown to lead to a better evaluation of the loads compared with a 3-D mollification by Caprace et al. (Reference Caprace, Chatelain and Winckelmans2019), the 3-D mollification is still used by many authors (Martínez-Tossas & Meneveau Reference Martínez-Tossas and Meneveau2019; Meyer Forsting et al. Reference Meyer Forsting, Pirrung and Ramos-García2019; Dag & Sørensen Reference Dag and Sørensen2020; Kleine et al. Reference Kleine, Hanifi and Henningson2023a). One justification for the use of a 3-D mollification is that the spanwise mollification decreases the force gradient at the tip and produces a smoother tip vortex. In some numerical methods, sharp gradients could potentially be problematic for numerical stability. Therefore, the case of a 3-D mollification is here also evaluated and corrected on the rectangular wing with $AR=15$. The variation of the loads near the tip and the correction functions are compared with those obtained using a 2-D mollification.

For the ALM, the use of a 3-D mollification consists in distributing the forces of each control point using a 3-D Gaussian kernel. For consistency, integral sampling is also used: the velocity at each control point is obtained as a weighted averaged using the same 3-D kernel. The MLL with 3-D mollification is obtained by replacing the expression of the downwash in (2.12) by (see Caprace et al. Reference Caprace, Chatelain and Winckelmans2019)

(7.1)\begin{equation} \bar{\varepsilon}_\sigma(y)=\frac{1}{U_{\infty}} \frac{1}{4 {\rm \pi}} \int_{{-}b / 2}^{b / 2} \frac{(1-{\rm e}^{-((y-y^{\prime})^2 / 2 \sigma^2)})}{(y-y^{\prime})} \frac{\mathrm{d} \varGamma}{\mathrm{d} y^{\prime}}(y^{\prime}) \,\mathrm{d} y^{\prime}. \end{equation}

The integral is also evaluated numerically.

Figure 26 shows the lift and drag distributions computed using the ALM and the MLL with 3-D mollification. The agreement between the MLL and the ALM is again very good. The comparison between the 2-D and 3-D mollifications indicates a clear deviation in lift and drag near the tip. The spanwise mollification reduces the induced velocity at the tip, and the downwash measured on the last control points of the ALM is therefore smaller. As a result, the lift is higher due to the larger effective angle of attack, whereas the drag decreases due to the effect of the downwash on the induced drag. The region of the wing affected by the 3-D mollification grows with the size of the mollification parameter $\sigma /c$. This region is of size $\simeq 2 \sigma$, and the number of control points affected by the spanwise mollification near the tip is thus related to the ratio $\sigma /h$. Interestingly, for the case $\sigma =c/4$, some deviations between the 2-D and 3-D mollifications are also visible in the range $0.425< y/b<0.475$. These arise from the fact that this region presents a larger gradient of the forces along the span, whose value is affected by the spanwise mollification. Figure 24 also presents the integrated $C_L$ and $C_D$ coefficients depending on the 3-D mollification size. They differ more from the reference than those obtained using the 2-D mollification, and this difference further increases with the mollification size.

Figure 26. Lift and drag coefficients on the rectangular wing with $AR=15$ at $\alpha =5^\circ$, obtained using ALM/MLL with 2-D or 3-D mollification, compared with the PLL and the reference wrRANS.

The correction functions are also obtained for the case of 3-D mollification, and are depicted in figure 27. Although the lift and drag distributions are not monotonic near the tip, the correction functions have an aspect similar to those obtained for the 2-D mollification. Specifically, the functions for the lift coefficient $F_{C_l}$ are very similar, regardless of the mollification type and size. Only the last two control points present a small deviation. In contrast, the functions for the effective angle of attack $F_{\alpha _e}$ present significantly higher values at a small normalized distance to the tip. Again, this indicates that the effect of the mollification can be mitigated by changing only the effective angle of attack and the downwash angle.

Figure 27. Correction functions obtained for the rectangular wing with 3-D mollification of different sizes and also compared with those obtained with 2-D mollification.