2.1 The LEISA aircraft

The LEISA configuration is a generic swept-back transport aircraft configuration that was designed at DLR Braunschweig [Reference Wild13] and has been used as a reference aircraft in various research projects [Reference Pott-Pollenske, Wild and Bertsch15]. Figure 1 shows sketches of the aircraft from three perspectives. Essential data regarding size, range and wing geometry are presented in Tables 1 and 2. With a maximum take-off weight of $85,290\text{kg}$ and a maximum payload of $25,000\text{kg}$ , the LEISA configuration is slightly larger than the Airbus A320-200.

Table 1. Weights and range of the LEISA configuration [Reference Wild13]

Table 2. Geometrical parameters of the LEISA wing [Reference Wild13]

Figure 1. Three-view drawing of the LEISA configuration [Reference Heinze, Haupt and Woidt16].

Figure 2 shows the geometry of the wing, including the relevant control surfaces. The high-lift system originally consists of a leading-edge slat over the whole span with a cut-out at the engine position and a single-slotted flap at the trailing edge from the wing root to the aileron [Reference Wild13]. Here, the slat is replaced by a droop nose (DN) divided into five segments; at the trailing edge of the high-lift flap, four separately controllable plain flaps (TEF) are placed. Combined with the aileron, these movables can be used for load alleviation [Reference Ullah, Lutz, Klug, Radespiel and Wild14]. However, within the scope of this study, only the outermost trailing-edge flap and the aileron are considered. This simplifies the present task to evaluate the potential for load alleviation. Similar concepts performed well as discussed by Disney [Reference Disney8], Ramsey et al. [Reference Ramsey and Lewolt9] and White [Reference White11]. The geometrical parameters of the used control surfaces are presented in Table 3.

2.2 Considered manoeuvres

The selection of the considered manoeuvres is based on the CS-25 [1]. According to these requirements, quasi-steady as well as unsteady manoeuvres need to be taken into account for the certification of an aircraft. In this study, only quasi-steady manoeuvres are discussed since Hodges et al. [Reference Hodges and McKenzie7] and Ramsey et al. [Reference Ramsey and Lewolt9] show that loads resulting from this kind of manoeuvre are critical for most parts of the wing structure.

For symmetrical manoeuvres, the evaluation of certification specifications for the LEISA configuration yields the manoeuvring envelope sketched in Fig. 3. Regions with deployed high-lift system at low speeds are neglected here. The manoeuvring envelope is a function of altitude. $\mathrm{M}_{S1}$ denotes the stall Mach number, the lowest Mach number at which steady flight with a load factor of $n_z=1$ is possible. $\mathrm{M}_{S1}$ increases with increasing altitude because of decreasing stagnation pressure, and so does the manoeuvring Mach number $\mathrm{M}_A$ , the lowest Mach number where the maximum load factor of $n_z=2.5$ can be reached. The cruising Mach number $\mathrm{M}_C$ ( $0.80$ ), maximum operating Mach number $\mathrm{M}_{MO}$ ( $0.85$ ) and dive Mach number $\mathrm{M}_D$ are independent of altitude. Based on this envelope, four pitching manoeuvres at different Mach numbers and altitudes are chosen, as also displayed in Fig. 3. Additional parameters of manoeuvre flight are presented in Table 4.

Table 3. Geometrical parameters of the control surfaces used for load alleviation

Figure 2. Slightly distorted view of the LEISA wing including the control surfaces of the load alleviation system.

Figure 3. Manoeuvring envelope at different altitudes and considered pitching manoeuvres.

Table 4. Overview of considered manoeuvres

Manoeuvre 1 is a nose-up pitching manoeuvre at medium altitude $h=6,000\text{m}$ , maximum load factor $n_z=2.5$ and maximum operating Mach number $\mathrm{M}_{MO}=0.85$ .

Manoeuvre 2 describes a nose-up pitching manoeuvre at typical cruising altitude $h=11,000\text{m}$ , also at $\mathrm{M}_{MO}=0.85$ . The maximum achievable load factor at this altitude is $n_z=1.69$ .

For manoeuvre 3, which is also a nose-up pitching manoeuvre, the manoeuvring Mach number $\mathrm{M}_A$ and maximum load factor $n_z=2.5$ are chosen. Since also a low Mach number should be investigated, a low altitude of $h=500\text{m}$ is chosen here. The manoeuvring Mach number at this altitude was initially estimated as $\mathrm{M}_A=0.50$ , slightly above the true value of $\mathrm{M}_A=0.48$ .

Manoeuvre 4 is a nose-down pitching manoeuvre at cruising Mach number $\mathrm{M}_C=0.80$ . Since the desired load factor $n_z=-1$ cannot be achieved at cruising altitude, a lower altitude of $h=6,000\text{m}$ is chosen.

For all manoeuvres, the centre of gravity is assumed to be at a forward position at $x_{CG}=8.5\% \: MAC$ . In this case, e.g. during a nose-up pitching manoeuvre, the tailplane generates high negative lift and the wing yields high positive lift, resulting in high manoeuvre loads. A critical loading condition should be considered, so the maximum aircraft weight of $MTOW=85,\!290\text{kg}$ is chosen for all four manoeuvres. Technically, the case with maximum payload, maximum fuel inside the fuselage and the inboard wing tanks, but with empty outboard wing tanks would be even more critical in terms of loads. Because the centre of gravity of the outboard wing tanks is located further outboard than the centre of lift distribution, the decreasing effect on wing bending moment resulting from inertia forces on the fuel inside the tanks is stronger than the increasing effect on wing bending moment resulting from the additional lift required. However, as part of the load alleviation concept, it is assumed that those fuelling conditions are avoided, making the case with maximum total weight critical.

Other parameters needed for the simulation of the respective manoeuvres, e.g. control surface deflections, angle-of-attack or pitching rate, are determined on the basis of the methods discussed in the following Section 3.

3.1 Approach

The manoeuvres described in Section 2.2 are calculated on the basis of CFD simulations and handbook formulas. For better efficiency, only the wing and fuselage of the LEISA aircraft are considered in the CFD simulations, whereas the effect of the tailplane is considered via handbook methods. For a pitching manoeuvre, the contribution of the tailplane to the lift and pitching moment has to be known to determine the angle-of-attack at which a certain load factor is achieved. Thus, handbook methods serve the purpose of providing the manoeuvre flight parameters needed for the actual CFD simulations. Furthermore, on the basis of handbook calculations and some CFD simulations serving as nodes, the wing loading within the whole flight envelope is interpolated or extrapolated. This approach has the advantage that it combines high precision for the simulation of specific manoeuvres provided by Navier–Stokes methods and the high efficiency for calculations within the whole flight envelope as an advantage of handbook methods. The method was developed within the scope of this study and, to the authors’ best knowledge, has not been applied anywhere else yet.

3.2 Manoeuvre calculation

The pitching manoeuvre is treated as quasi-steady, meaning unsteady processes during the initiation of the manoeuvre are neglected. The angle-of-attack and pitching rate are therefore considered as constant during the manoeuvre, and all forces and moments acting on the aircraft are balanced. This equilibrium of forces and moments is the basis for the calculation methods presented in the following.

The forces and moments acting on the aircraft are sketched in Fig. 4. Note that forces in circumferential direction of aircraft motion are not further considered. Also the contribution of thrust $F_T$ in the radial direction is neglected. Balancing the forces in the radial direction and the pitching moments around the centre of gravity yields

(1) \begin{eqnarray} \sum F_{r,tot} = &F_C + F_M - ( F_{LW} + F_ {LF} ) - F_{LT} = 0 \ ,\end{eqnarray}

(2) \begin{align} \sum M_{yCG,tot} &= M_{yW} + M_{yF} + M_{yT} \nonumber\\ & \quad - (F_{LW} + F_{LF}) \cdot x_{MAC25W-CG} - F_{LT} \cdot x_{MAC25T-CG} = 0\end{align}

where the moments acting on wing and fuselage are evaluated at 25% of mean aerodynamic chord of the wing and the moment acting on the tailplane is evaluated at 25% of mean aerodynamic chord of the tailplane. Hence, $x_{MAC25-CG}$ is the lever arm of respective forces at the centre of gravity.

Figure 4. Forces and moments acting on the aircraft during the pitching manoeuvre.

Evaluation of the weight $F_M=mg$ and centrifugal force $F_C=mVq$ is trivial. From the definition of the load factor $n_z=\frac{F_{L,tot}}{F_M}=\frac{F_{LW}+F_{LF}+F_{LT}}{F_M}$ , one obtains

(3) \begin{eqnarray} q = (n_z - 1) \frac{g}{V}\end{eqnarray}

for the pitching rate.

A pitching manoeuvre requires a certain angle-of-attack dependent on the desired load factor, Mach number and altitude. The angle-of-attack is calculated iteratively by means of the method illustrated in Fig. 5. In the following, this approach is described in more detail.

Figure 5. Iterative calculation of the pitching manoeuvre based on a combination of handbook and Navier–Stokes methods; blocks and arrows inside the dashed box are part of the actual iteration; other blocks have to be evaluated only once for each manoeuvre.

Firstly, the equilibrium of forces from Equation (1) is written in coefficient form as

(4) \begin{eqnarray} \frac{n_z mg}{q_\infty S_W} - ( C_{LW} + C_{LF} ) - C_{LT} \frac{q_T}{q_\infty} \frac{S_T}{S_W} = 0 \ .\end{eqnarray}

It is assumed that $\frac{q_T}{q_\infty}=0.99$ . This factor takes into account that the tailplane is located in the wake of the wing and is therefore exposed to a lower stagnation pressure. From Equation (4), the iteration rule for the lift coefficient of wing and fuselage is derived as

(5) \begin{eqnarray} (C_{LW} + C_{LF})^{i+1} = (C_{LW} + C_{LF})^i + \beta \cdot \left( \frac{n_z mg}{q_\infty S_W} - (C_{LW} + C_{LF})^i - C_{LT}^i \frac{q_T}{q_\infty} \frac{S_T}{S_W} \right) \ .\end{eqnarray}

This corresponds to the damped Newton’s method with $\beta=0.1$ as damping coefficient.

As the initial condition for the iteration,

(6) \begin{eqnarray} (C_{LW} + C_{LF})^0 = \frac{n_z mg}{q_\infty S_W}\end{eqnarray}

is chosen. To evaluate Equation (5), the lift coefficient of the tailplane $C_{LT}$ first needs to be determined. Considering the equilibrium of moments from Equation (2) written in coefficient form as

(7) \begin{eqnarray} &C_{MyW} + C_{MyF} + C_{MyT} \dfrac{q_T}{q_\infty} \dfrac{S_T}{S_W} \dfrac{MAC_T}{MAC_W} - \nonumber\\[5pt] &(C_{LW} + C_{LF}) \cdot \dfrac{x_{MAC25W-CG}}{MAC_W} - C_{LT} \dfrac{q_T}{q_\infty} \dfrac{S_T}{S_W} \cdot \dfrac{x_{MAC25T-CG}}{MAC_W} = 0,\end{eqnarray}

where the moments acting on the wing and fuselage are normalised using the mean aerodynamic chord of the wing and the moment acting on the tailplane is normalised using the mean aerodynamic chord of the tailplane, the remaining unknowns are $C_{MyW}$ , $C_{MyF}$ , $C_{LT}$ and $C_{MyT}$ . For the present study, the geometrical parameters of the tailplane are provided by Wild [Reference Wild17].

At this point $C_{MyW}$ , $C_{MyF}$ , the respective angle-of-attack and all other forces and moments acting on the wing–fuselage configuration are determined by the use of Navier–Stokes methods. For this purpose, CFD simulations at certain Mach numbers and different angles of attack were carried out as described in Section 3.3. Since CFD results are available for specific Mach numbers only, the effect of compressibility needs to be corrected to obtain also values at other Mach numbers. Effects on the aerodynamics of the fuselage are neglected; only the forces and moments acting on the wing are adjusted. For this, the Prandtl–Glauert transformation as described by Schlichting et al. [Reference Schlichting and Truckenbrodt18] (pp 171–173) is used. Strictly speaking, this transformation applies to two-dimensional flow only, but according to Göthert [Reference Göthert19], it is a good approximation for wings with high aspect ratio, too. After correcting the simulation results at the next highest Mach number and those at the next lowest Mach number, linear interpolation regarding the Mach number of the particular manoeuvre is performed between both cases, as illustrated in Fig. 6. That way, information from simulations at different Mach numbers is considered as well as nonlinear effects of compressibility. If the Mach number lies outside of the range covered by simulations, the Prandtl–Glauert-corrected results from the simulations at the nearest Mach number are applied directly.

Figure 6. Determination of the aerodynamic coefficients as a function of Mach number.

Besides the steady aerodynamics of the wing–fuselage configuration, the effect of pitching rate is examined. Accordingly, the forces and moments from the steady CFD simulations are adjusted on the basis of quasi-steady simulations that are also described in Section 3.3. With

(8) \begin{eqnarray} \vec{C}_{WF} = (C_{LW}, \: C_{DW}, \: C_{QW}, \: C_{MxW}, \: C_{MyW}, \: C_{MzW}, \: C_{LF}, \: C_{DF}, \: C_{QF}, \: C_{MxF}, \: C_{MyF}, \: C_{MzF})^T\end{eqnarray}

as a vector of the force and moment coefficients of wing and fuselage, one obtains

(9) \begin{eqnarray} \vec{C}_{WF} (\alpha) = \vec{C}_{WF,q=0} (\alpha) + \frac{\partial \vec{C}_{WF}}{\partial q} q \ .\end{eqnarray}

$\frac{\partial \vec{C}_{WF}}{\partial q}$ must be determined in preliminary quasi-steady CFD simulations. However, for most parts of this study, the effect of pitching rate on the aerodynamics of wing and fuselage is neglected. In those cases, this step can be omitted, leading to

(10) \begin{eqnarray} \vec{C}_{WF} (\alpha) = \vec{C}_{WF,q=0} (\alpha) \ .\end{eqnarray}

Now that the results from CFD simulations are prepared, the angle-of-attack required for a specific manoeuvre as well as all integral forces and moments acting on the aircraft during this manoeuvre can be linearly interpolated regarding the lift coefficients of wing and fuselage. Thus, $C_{MyW}$ and $C_{MyF}$ are now known and can be inserted into the equilibrium of moments in Equation (7).

To find the relation between the two remaining unknowns $C_{LT}$ and $C_{MyT}$ , the aerodynamics of the tailplane are examined. From formulas stated by Schlichting et al. [Reference Schlichting and Truckenbrodt18] (pp. 70, 405), the slope of the lift curve of the tailplane can be derived as

(11) \begin{eqnarray} \frac{d C_{LT}}{d \alpha_T} = \left( \frac{d C_{LT}}{d \alpha_T} \right)_{\varphi = 0} \text{cos} \varphi_{25T} = \frac{2 \pi \Lambda_T \cdot \text{cos} \varphi_{25T}}{\sqrt{(1-\mathrm{M}^2) \Lambda_T^2 + 4}+2} \ .\end{eqnarray}

For simplification, it is assumed that the tailplane aerofoil is symmetrical so that the lift force acts approximately on its $\frac{c}{4}$ -point. Hence,

(12) \begin{eqnarray} \frac{\partial C_{MyT}}{\partial \alpha_T} = 0\end{eqnarray}

follows for the slope of the moment around the $\frac{c}{4}$ -point and

(13) \begin{eqnarray} C_{My0T, \eta_e = 0} = 0\end{eqnarray}

follows for the zero-lift moment coefficient without elevator deflection $\eta_e$ .

Besides the angle-of-attack, also the deflection of the elevator contributes to the equilibrium of forces and moments. According to Schlichting et al. [Reference Schlichting and Truckenbrodt18] (pp. 440, 446), the respective lift coefficient can be considered by adjusting the angle-of-attack:

(14) \begin{eqnarray} \frac{\partial \alpha_T}{\partial \eta_e} = \left( \frac{\partial \alpha_T}{\partial \eta_e} \right)_{incompressible} = - \frac{2}{\pi} \left( \sqrt{\lambda_e ( 1 - \lambda_e ) } + \text{arcsin} \sqrt{\lambda_e} \right) \ .\end{eqnarray}

The change in zero-lift moment coefficient is calculated, also according to Schlichting et al. [Reference Schlichting and Truckenbrodt18] (p. 446) [Reference Schlichting and Truckenbrodt20] (p. 430), via

(15) \begin{eqnarray} \frac{\partial C_{My0T}}{\partial \eta_e} = \frac{1}{\sqrt{1-\mathrm{M}^2}} \left( \frac{\partial C_{My0T}}{\partial \eta_e} \right)_{incompressible} = \frac{-2 \sqrt{\lambda_e ( 1 - \lambda_e )^3}}{\sqrt{1-\mathrm{M}^2}} \ .\end{eqnarray}

The angle-of-attack at the tailplane results from superposition:

(16) \begin{eqnarray} \alpha_T = \alpha + \epsilon_T + \alpha_w + \alpha_q \ .\end{eqnarray}

Here, $\epsilon_T$ is the angle of incidence of the tailplane. By adding $\alpha_w$ , the downwash induced by the wing is taken into account. It can be calculated via

(17) \begin{eqnarray} \alpha_w = - \frac{C_{LW}}{\pi \Lambda_W} \left[ 2+ \left( \frac{b_W}{4 \cdot x_{MAC25T-MAC25W}} \right)^2 (1-\mathrm{M}^2) \right]\end{eqnarray}

where $x_{MAC25T-MAC25W}$ denotes the distance between the wing and tailplane [Reference Schlichting and Truckenbrodt18] (p 407).

$\alpha_q$ is an additional angle-of-attack that results from the pitching motion of the aircraft. As shown in Fig. 4, the aircraft is suspended at its centre of gravity. Due to the rotation, the tailplane is moved downwards with a velocity of $q \cdot x_{MAC25T-CG}$ . This is equivalent to an additional angle-of-attack

(18) \begin{eqnarray} \alpha_q = \text{arctan} \left( \frac{q \cdot x_{MAC25T-CG}}{V} \right) \ .\end{eqnarray}

Finally, the lift coefficient of the tailplane

(19) \begin{eqnarray} C_{LT} = \left( \alpha_T - \frac{\partial \alpha_T}{\partial \eta_e} \eta_e \right) \frac{d C_{LT}}{d \alpha_T}\end{eqnarray}

and its moment coefficient

(20) \begin{eqnarray} C_{MyT} = C_{My0T, \eta_e = 0} + \frac{\partial C_{My0T}}{\partial \eta_e} \eta_e + \frac{\partial C_{MyT}}{\partial \alpha_T} \left( \alpha_T - \frac{\partial \alpha_T}{\partial \eta_e} \eta_e \right) = \frac{\partial C_{My0T}}{\partial \eta_e} \eta_e\end{eqnarray}

can be evaluated. Equations (19) and (20) are inserted into the equilibrium of moments in Equation (7). Two unknowns remain: the angle of incidence $\epsilon_T$ and the elevator deflection $\eta_e$ .

$\epsilon_T$ is determined before the actual manoeuvre is calculated. For this purpose, the calculation method described in this section is applied to steady flight with $n_z=1$ under the constraint $\eta_e=0$ . Thus, the aircraft is trimmed for level flight under the respective conditions, and the manoeuvre is then initiated by deflecting the elevator.

At this point, the last remaining unknown $\eta_e$ can be determined by solving Equation (7). Thereby, the lift coefficient of the tailplane $C_{LT}$ is known and can be inserted into the iteration rule in Equation (5). The iteration is then repeated until the lift coefficient of the wing–fuselage configuration changes by less than $10^{-5}$ .

Besides the angle-of-attack and the pitching rate that are used for further, more precise calculations based on Navier–Stokes methods, the combination of handbook methods and CFD simulations described in this section also yields the integral force and moment coefficients of the wing. These can be used for estimating the wing-root loading already at this point, allowing for an efficient calculation of manoeuvre loads within the whole flight envelope.

Because the moments in the global coordinate system of the aircraft are not very meaningful regarding design of the wing structure, they are transformed into the coordinate system of the wing as illustrated in Fig. 7. From this sketch, the respective transformation can be derived as

(21) \begin{eqnarray} \begin{pmatrix} C_{MB} \\ C_{MT} \\ C_{MS} \end{pmatrix}_{root} = \begin{pmatrix} \text{cos} \varphi_e\;\;\;\;\; & -\text{sin} \varphi_e\;\;\;\;\; & 0 \\ \text{sin} \varphi_e \text{cos} \nu_e \;\;\;\;\; & \text{cos} \varphi_e \text{cos} \nu_e \;\;\;\;\; & \text{sin} \nu_e \\ -\text{sin} \varphi_e \text{sin} \nu_e \;\;\;\;\; & -\text{cos} \varphi_e \text{sin} \nu_e \;\;\;\;\; & \text{cos} \nu_e \end{pmatrix} \cdot \begin{pmatrix} C_{Mx} \\ C_{My} \\ C_{Mz} \end{pmatrix}_{root} \ .\end{eqnarray}

Here, $\varphi_e$ denotes the wing sweep at the elastic axis and $\nu_e$ is the dihedral at the elastic axis. For simplification, it is assumed that the elastic axis is described by the straight connection line between the point at 45% chord at the wing root and the point at 45% chord at the wing tip.

Figure 7. Wing-root moments in the global coordinate system of the aircraft and in the coordinate system of the wing.

Additional to the aerodynamic forces, engine thrust and inertial forces contribute to the wing loading. Engine thrust is neglected here, and the effect of inertial forces on torsional and swing moment is not considered either. However, inertial forces have a strong load alleviating effect on the wing bending moment. This can be taken into account by evaluating the expression

(22) \begin{eqnarray} M_{B,root} = (M_{B,root})_{aero} - n_z m_W g \cdot \frac{y_{CGW}-\frac{d_F}{2}}{\text{cos} \varphi_e},\end{eqnarray}

where $m_W$ is the weight of the wing including engine and fuel, $y_{CGW}$ is the spanwise position of its centre of gravity and $d_F$ denotes the diameter of the fuselage. For the present study, $m_W$ and $y_{CGW}$ are taken from the structural model that was designed by Heinze et al. [Reference Heinze, Haupt and Woidt16] for an aircraft very similar to the LEISA aircraft.

3.3 CFD simulation

Steady simulation

To determine the aerodynamics of the wing–fuselage configuration for the relevant Mach numbers, steady simulations at different angles of attack are carried out. A total of 20 cases (not including the cases beyond $C_{Lmax}$ that were disregarded) were used to create a database that is accessed during the calculations discussed in the previous Section 3.2. Amongst others, those calculations yield the angle-of-attack required for a specific manoeuvre. Neglecting the effect of pitching motion on wing–fuselage aerodynamics, steady simulations of the manoeuvres described in Section 2.2 can then be performed. Note that these results also include the spanwise distribution of manoeuvre loads.

For the simulations, the DLR TAU-Code is used. The DLR TAU-Code solves the Navier–Stokes equations on hybrid unstructured grids using a finite volume method as described by Schwamborn et al. [Reference Schwamborn, Gerhold and Heinrich21] and Heinrich et al. [Reference Heinrich, Dwight, Widhalm and Raichle22]. Assuming symmetrical flow, three-dimensional RANS simulations of an idealised half model of the rigid LEISA configuration without engines and without empennage are carried out. For turbulence modelling, the SAO version of the one-equation SA model from Spalart and Allmaras is applied. In this implementation, the so-called $f_{t2}$ laminar suppression term is eliminated from the standard model, which is a reasonable approach for fully turbulent flows as stated by Allmaras et al. [Reference Allmaras, Johnson and Spalart23]. For the discretisation of convective fluxes, the second-order central scheme is used, while the chosen dissipation scheme relies on scalar dissipation. The simulation domain including boundary conditions and the respective grid is shown in Fig. 8(a) and (b), while the surface grid on the wing and fuselage is illustrated in (c). It features some refinement in the region of the wing where the shock is expected. Figure 8(d) contains the boundary-layer grid at the wing section $\eta = 0.647$ for the case with load alleviation, thus with the trailing-edge flap deflected by $\xi = -15^\circ$ . The boundary layer grid consists of 50 cells in normal direction; the height of the first cell is on the order of $y^+ \approx 1$ , as exemplarily shown for the simulation of manoeuvre 1 in Fig. 9(a).

Figure 8. Domain and grid used for CFD simulations.

Figure 9. Distribution of $y^+$ on the upper surface of the wing for manoeuvre 1 from simulations on the original grid and on the fine grid.

One simulation usually takes about $2.5\text{h}$ (solver wall-clock time) on 100 CPUs of the TU Braunschweig Phoenix-Cluster, while generally a convergence level of $res_\rho < 5 \cdot 10^{-7}$ is reached for the studied manoeuvres. Only for manoeuvre 1 with active load alleviation ( $res_\rho \approx 4.5 \cdot 10^{-6}$ ) and for manoeuvre 2 ( $res_\rho \approx 4.7 \cdot 10^{-5}$ ) are the residuals higher due to the combination of high Mach number and deflected control surfaces or high angle-of-attack. However, the results presented in Section 4.3 reveal that the loads for manoeuvre 2 are not critical, so the higher residual here is not problematic.

For verification, simulations of manoeuvres 1 and 3 were additionally carried out on a grid that is almost eight times finer than the original grid. For manoeuvre 1, Fig. 9(b) exemplarily illustrates that this is equivalent to $y^+ < 0.25$ at the wing surface. A comparison of some integral quantities from the original grid solution and from the fine grid solution is presented in Table 5. Here, the deviations between both grids for manoeuvre 3 ( $\mathrm{M}=0.50$ ) are small. However, the deviations for manoeuvre 1 ( $\mathrm{M}=0.85$ ) are significantly higher and require some discussion.

Table 5. Deviations of the original grid solution from the fine grid solution for manoeuvres 1 and 3

Figure 10 shows the pressure distributions from simulations on both grids at two spanwise positions. At $\eta=0.9$ , one can see that the shock is slightly blurred on the original grid, which leads to a lower lift coefficient compared with the fine grid. Comparing the two wing sections at $\eta=0.1$ and $\eta=0.9$ reveals that this effect is much stronger at the outboard wing than at the inboard wing. This explains why the wing-root bending moment is lower for the original grid. Further, it results in an increased pitching moment coefficient for the swept-back wing. Since the shock and consequently the local lift loss are located in the rear region of the wing section, also the higher wing-root torsional moment for the original grid can be explained.

Figure 10. Pressure distributions at two spanwise positions for manoeuvre 1 from simulations on the original grid and on the fine grid.

All in all, the deviations are considered to be acceptable here. The goal of this study is not to provide highly accurate numbers that can be used for aircraft design, but rather to present a methodology, allowing for an efficient calculation of flight manoeuvres, and to provide insight into the fundamental physics of manoeuvre loads and load alleviation. And these fundamental flow physics are also captured by the original grid, as shown in Figs 10 and 11. Also, it is important to note that the angle-of-attack was not adjusted for the fine grid simulations. That way, the computational effort is kept at a reasonable level and direct comparability between the CFD simulations on the two grids is achieved. However, if the angle-of-attack were adjusted, this would reduce the differences between the simulations. Here, the angle-of-attack would be reduced for the simulations on the fine grid, as otherwise the lift coefficient and thereby the load factor are too high for the specific manoeuvre.

Figure 11. Pressure distribution and separated regions on the upper surface of the wing for manoeuvre 1 from simulations on the original grid and on the fine grid.

Beyond the issue of grid convergence, also the basic approach of using RANS methods to predict transonic flows near buffet onset needs to be discussed at this point. As no wind-tunnel data for the LEISA aircraft are available, the results from the Sixth AIAA CFD Drag Prediction Workshop summarised by Tinoco et al. [Reference Tinoco, Brodersen, Keye, Laflin, Feltrop, Vassberg, Mani, Rider, Wahls, Morrison, Hue, Gariepy, Roy, Mavriplis and Murayama24] are consulted. For the comparable case of the NASA Common Research model at $\mathrm{M}=0.85$ and $C_L=0.5$ , it is generally confirmed that RANS simulations are capable of capturing the fundamental flow physics observed in wind-tunnel tests, even on coarser grids. Further, the study justifies the neglect of the nacelle and pylon by showing that they have only a small effect on the spanwise distributions of lift and pitching moment.

In general, the computed lift is slightly too high and the computed pitching moment is too negative when compared with wind-tunnel experiments. These deviations could partially be related to errors in the reference wind-tunnel data, but the main reason according to Tinoco et al. [Reference Tinoco, Brodersen, Keye, Laflin, Feltrop, Vassberg, Mani, Rider, Wahls, Morrison, Hue, Gariepy, Roy, Mavriplis and Murayama24] is the systematic prediction of excessive aft loading in the simulations. Especially at the outboard wing, the computed pressure distributions show an excessive loading in the rear wing region that leads to an overestimation of lift in those parts of the wing. Based on the fact that most of the participants of the Sixth AIAA CFD Drag Prediction Workshop predicted excessive aft loading, one must assume that this could also be an issue in the present simulations. Here, this would mean that the computed wing bending moments are too high and the computed wing torsional moments are too low.

Further, Tinoco et al. [Reference Tinoco, Brodersen, Keye, Laflin, Feltrop, Vassberg, Mani, Rider, Wahls, Morrison, Hue, Gariepy, Roy, Mavriplis and Murayama24] describe that the shock positions in the simulations become increasingly uncertain at higher angles of attack. This is probably related to buffet, and it is questionable whether steady RANS methods are able to adequately predict the flow in this regime. During the selection of the manoeuvres considered in this study, care was taken to ensure that no buffet occurs in any of them. However, the determination of buffet onset was also based on steady RANS simulations, so this might be a source of uncertainty. Another assumption that might cause some error is the neglect of static aeroelastic deformations of the wing. This aspect is examined further in Section 4.1.

One of the main issues that Tinoco et al. [Reference Tinoco, Brodersen, Keye, Laflin, Feltrop, Vassberg, Mani, Rider, Wahls, Morrison, Hue, Gariepy, Roy, Mavriplis and Murayama24] identified as impairing the validity of RANS simulations is the prediction of premature separation at the junction between wing and fuselage. In the respective simulations, this led to an early lift break compared with wind-tunnel data. However, this does not seem to be a problem in the present simulations as no separation in this specific region and no lift break at low angles of attack are observed.

Although the present simulations might not be accurate enough to use the results in aircraft design, the expected level of uncertainty is considered to be acceptable for the actual purpose of capturing the general trends and understanding the underlying physics.

Quasi-steady simulation

Besides the steady simulations, quasi-steady simulations are performed to investigate the effect of pitching motion on the aerodynamics of the wing–fuselage configuration during the manoeuvre. To identify the angle-of-attack required for a specific, quasi-steady manoeuvre by use of the calculations methods described in Section 3.2, the dynamic derivatives of the model first need to be determined. For this purpose, simulations with and without pitching rate are carried out. From the results, the derivatives are calculated via

(23) \begin{eqnarray} \frac{\partial \vec{C}_{WF}}{\partial q} = \frac{\Delta \vec{C}_{WF}}{q}\end{eqnarray}

which can be inserted into Equation (9).

The grid itself remains unchanged in the quasi-steady simulation, but in this application the grid is moved to simulate aircraft motion. As already illustrated in Section 3.2, the trajectory of a pitching manoeuvre can be approximated by a circular path. To reproduce this correctly in the simulations, the grid is suspended at the centre of gravity of the aircraft and moved along a circular path as shown in Fig. 12. Then, the rotational speed $\omega$ is equivalent to the pitching rate q. The radius of rotation R is chosen based on the criterion that the inflow velocity resulting from grid rotation equates the desired inflow velocity for the specific manoeuvre.