1 Introduction

The flow around wings with moderate to high leading-edge sweep angle and small wing aspect ratio is characterized by leading-edge vortices, which already evolve at moderate angles of attack. In recent decades, mainly slender delta wings with sharp leading edges [2, 8, 20,21,22, 25, 33, 41, 46] or round leading edges with small leading-edge radii [7, 11, 28, 29] have been investigated. The formation of the leading-edge vortices in the context of these configurations has been extensively studied, and the corresponding flow physics are well understood to a large extent and well documented. [13] gives an overview over recent developments in delta wing aerodynamics, which focus more on the unsteady phenomena like shear-layer instabilities, vortex wandering and vortex breakdown. In recent years, the focus has been on configurations with reduced wing sweep and (partially) rounded leading-edge contours. Investigations on such configurations with moderate wing sweep and (partially) rounded leading-edge contours concentrate on the vortex formation and on the stability and control characteristics. Several configurations like the stability and control configuration SACCON [4, 34, 35, 37], the AVT-183 diamond wing configuration [14, 17, 18, 27], the Swing configuration [42], the 1303 configuration [30] or the SAGITTA configuration [16, 19] have been investigated. Considering the SAGITTA diamond-wing configuration, a comprehensive data set with respect to steady aerodynamic characteristics has already been obtained. The unsteady aerodynamic characteristics of such configurations have not been subject to detailed investigations. A flying-wing configuration with a lambda-like wing planform was investigated in more detail, whereas this configuration features considerably complex aerodynamic characteristics [37, 45]. This article focuses on the unsteady aerodynamic characteristics of the SAGITTA diamond-wing configuration. Although the steady aerodynamics of the SAGITTA configuration do not feature significant non-linear characteristics, its unsteady aerodynamics require a detailed investigation. The flow at the SAGITTA diamond wing is dominated by attached flow up to an angle of attack of \(\alpha \approx 8^\circ\). At this point, the flow begins to separate from the round leading edge in the wing tip area. With increasing angle of attack, the flow separation onset moves upstream and the area of separated flow increases and forms an irregular recirculation area with flow reversal. Simultaneously, a pair of partly developed leading-edge vortices evolves in the wing apex region. At high angles of attack, the flow is dominated by flow separation in the outboard wing area and partly developed leading-edge vortices in the wing apex region. A detailed description of the steady aerodynamic characteristics of the SAGITTA configuration is given in [19].

Fig. 1
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Geometric details of the SAGITTA diamond-wing configuration

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Fig. 2
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SAGITTA diamond-wing configuration with geodesic (g) and body-fixed (b) coordinate system

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Fig. 3
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Forced harmonic rigid body motions

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Fig. 4
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Computational grid of the SG-00 configuration

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In the linearized form of the flight mechanics equations of motion, the aerodynamic forces and moments are expressed via coefficients and derivatives [3]. The latter can be distinguished into static, dynamic and control derivatives [32]. The dynamic derivatives are necessary to describe the behavior of the aircraft in consequence of unsteady aircraft motions and unsteady freestream conditions. Knowledge of them is especially important for high angles of attack, where strong non-linear characteristics are expected [31]. They can be obtained from flight testing, wind tunnel testing, CFD computations, or a combined approach that uses data sheets, linear aerodynamic theory and empirical relations [5]. A typical approach to determine the dynamic derivatives uses the response of aerodynamic forces and moments to forced harmonic oscillations [6, 9, 45]. The dynamic derivatives are calculated by processing the resulting temporal aerodynamic force and moment distributions. Different approaches exist based on linear [34] and non-linear assumptions [26]. The oscillations can be performed as a rotary motion about a body-fixed axis, an oscillating motion along a defined axis, or a combination of both, cf. Ref. [9]. The oscillating motions presented in this article are the rotary about a body-fixed axis. The resulting dynamic derivatives can be assigned as damping, cross and cross-coupling derivatives [9]. This article deals with the damping derivatives associated with the pitching, yawing and rolling motion.

2 SAGITTA diamond-wing configuration

The presented investigations are performed on the SAGITTA flying-wing configuration. The SAGITTA configuration is of diamond wing planform type and features a positive leading-edge sweep angle of \(\varphi _\mathrm{le}=55^\circ\) and a negative trailing-edge sweep angle of \(\varphi _\mathrm{te}=-25^\circ\). The wing aspect ratio results in \(\mathrm{AR}=2.01\) and the wing taper ratio in \(\lambda =0.025\). In accordance with former wind tunnel investigations on the SAGITTA diamond-wing configuration at the Chair of Aerodynamics and Fluid Mechanics of the Technical University of Munich (TUM-AER) [19], a root chord of \(c_\mathrm{r}=1\) m corresponding to the root chord of the applied wind tunnel model is chosen. The model exhibits a wing span of \(b=1.029\) m, a tip chord of \(c_\mathrm{t}=0.025\) m, a mean aerodynamic chord of \(l_\mu =0.667\) m, a moment reference point of \(x_\mathrm{mrp}=0.418\cdot c_\mathrm{r}\), and a reference area of \(S_\mathrm{ref}=0.528\,\mathrm{m}^2\). The wing planform properties are additionally summarized in Fig. 1. The SAGITTA configuration features a symmetric NACA 64A0012 airfoil over the whole wing span, whereas the airfoil is slightly modified at the inboard wing section. Within the first twenty percent of the wing half span, the blunt leading edge is replaced by a sharp leading edge. In consequence of the sharp leading edge at the wing inboard section, a geometrically predefined flow separation takes place even at moderate angles of attack. The separated flow in the inboard section rolls up and forms a vortex near the wing surface [21]. To increase the lateral stability of the configuration, a double vertical tail (double V/T) is attached for the first flight campaign of the SAGITTA demonstrator, see Fig. 2. Due to low-observability reasons, the final SAGITTA configuration will not be equipped with vertical tails. Stability and controllability must, therefore, be ensured by means of the equipped control devices. Both the SAGITTA configuration with attached double V/T (SG-VT) and without double V/T (SG-00) are considered in the present numerical investigation.

3 Calculation methodology for the dynamic derivatives

The dynamic derivatives are determined by means of the aerodynamic response to forced harmonic rigid body motions. The considered types of body motion are the rolling, pitching, and yawing motion, see Fig. 3a–c. The rolling motion is a rotation of the geometry about the body-fixed x-axis, the pitching motion is a rotation about the body-fixed y-axis, and the yawing motion is a rotation about the body-fixed z-axis. The center of rotation is the moment reference point \(x_\mathrm{mrp}\). The applied body-fixed coordinate system highlighting the positive rotation directions is illustrated in Fig. 2. The body-fixed coordinate system is designated by the subscript letter b. The forced harmonic oscillations of the rigid body result in a harmonic response of the aerodynamic forces and moments. The harmonic response features the same frequency as the harmonic motion but incorporates a certain phase shift relative to the harmonic motion. This phase shift arises due to the continuous change of the wing position relative to the freestream direction. Since the phase shift is solely a result of the harmonic motion, it can be utilized in order to determine the dynamic characteristics of the aircraft.

Fig. 5
figure 5

\(y^+\)-Distribution of the SG-00 configuration for selected freestream conditions

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Fig. 6
figure 6

Lissajous figures for two signals of same amplitude and frequency and with different phase shifts

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Fig. 7
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Comparison of the steady aerodynamic derivatives of the SG-VT configuration from different data sources for selected freestream conditions

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An aerodynamic derivative can be expressed as a Taylor series expansion including steady and dynamic derivatives and the degree of freedom, cf. [43]:

$$\begin{aligned} C_\nu \left( k\tau \right) =C_{\nu 0}+C_{\nu \zeta }\zeta \left( k\tau \right) +C_{\nu \dot{\zeta }}\dot{\zeta }\left( k\tau \right) +\mathcal {O}\left( \ddot{\zeta }\right) . \end{aligned}$$
(1)

This expression of an aerodynamic derivative serves as basis for the applied calculation methodology. The subscript \(\nu\) denotes an arbitrary aerodynamic force or moment coefficient and \(\zeta\) denotes a degree of freedom, e.g., the angle of attack \(\alpha\). k is the reduced frequency and \(\tau\) the non-dimensional time. They are defined as:

$$\begin{aligned} k=\frac{\omega l_\mu }{U_\infty }, \quad \tau =\frac{t U_\infty }{l_\mu }. \end{aligned}$$
(2)

This approach assumes linear characteristics of the aerodynamic coefficient \(C_\nu\) with respect to the degree of freedom \(\zeta\). The following paragraph describes the relation between the harmonic aerodynamic response and the Taylor series expansion of the aerodynamic coefficient, cf. Ref. [9].

The harmonic motion and its derivative can be expressed by

$$\begin{aligned} \zeta \left( k\tau \right) =\zeta _0+\zeta _\mathrm{max}\sin \left( k\tau \right) , \end{aligned}$$
(3)
$$\begin{aligned} \dot{\zeta }\left( k\tau \right) =\zeta _\mathrm{max}k\cos \left( k\tau \right) . \end{aligned}$$
(4)

Since \(\zeta \left( k\tau \right)\) is a \(2\pi\)-periodic function, the aerodynamic derivatives can be determined by a harmonic analysis. \(C_\nu \left( k\tau \right)\) can then be expressed by a complex amplitude \(\tilde{C}_{\nu j}\) and the complex oscillation \(e^{ik\tau }\):

$$\begin{aligned} C_\nu \left( k\tau \right) =\overline{C}_\nu +\sum\limits _{j=1}^nRe\left[ \tilde{C}_{\nu j}e^{ik\tau }\right] . \end{aligned}$$
(5)

Applying Eulers identity on Eq. 5 and neglecting terms of higher order as well as the imaginary part leads to

$$\begin{aligned} C_\nu \left( k\tau \right) =\overline{C}_\nu +Re\left[ \tilde{C}_{\nu 1}\right] \cos \left( k\tau \right) -Im\left[ \tilde{C}_{\nu 1}\right] \sin \left( k\tau \right) . \end{aligned}$$
(6)

The Taylor series expansion must be rewritten in a next step. This is done by expressing the excitation signal given in Eqs. 3 and 4 as:

$$\begin{aligned} \zeta \left( k\tau \right) =\zeta _0+\zeta _\mathrm{max}e^{ik\tau }, \end{aligned}$$
(7)
$$\begin{aligned} \dot{\zeta }\left( k\tau \right) =ik\zeta _\mathrm{max}e^{ik\tau }. \end{aligned}$$
(8)

Equations 7 and 8 are then substituted in Eq. 1. Considering the real part results in

$$\begin{aligned} C_\nu \left( k\tau \right) = C_{\nu 0}+\left( \zeta _0 +\zeta _\mathrm{max}\cos \left( k\tau \right) \right) C_{\nu \zeta }-\zeta _\mathrm{max}k\sin \left( k\tau \right) C_{\nu \dot{\zeta }}. \end{aligned}$$
(9)

A comparison of the coefficients in Eqs. 6 and 9 enables the expression of the aerodynamic derivatives depending on the harmonics of the aerodynamic response:

$$\begin{aligned} C_{\nu \zeta }=\frac{Re\left[ \tilde{C}_{\nu 1}\right] }{\zeta _\mathrm{max}}, \quad C_{\nu \dot{\zeta }}=\frac{Im\left[ \tilde{C}_{\nu 1}\right] }{k\zeta _\mathrm{max}}. \end{aligned}$$
(10)

\(C_{\nu \zeta }\) represents a steady aerodynamic derivative, e.g., \(C_{my\alpha }\), which can also be obtained from steady investigations at constant freestream conditions. \(C_{\nu \dot{\zeta }}\) represents a dynamic derivative, e.g., \(C_{my\dot{\alpha }}\). The described method can be applied on arbitrary harmonic motions, e.g., rotational, translational, or a combination of both. The exact type of harmonic motion prescribes the resulting (coupled) aerodynamic derivatives. Harmonic rotational motions about a body-fixed axis are required in order to obtain the damping derivatives. The following subsections present the determination of the damping derivatives of the pitching, yawing, and rolling motion. The formulations are also valid for freestream conditions deviating from \(\alpha _0=0^\circ\), \(\beta _0=0^\circ\).

Fig. 8
figure 8

Lissajous figures for the pitching, yawing, and rolling moment coefficient of the SG-VT configuration obtained by the time-accurate distribution of the aerodynamic coefficients and by the first harmonics of the corresponding aerodynamic coefficients for selected freestream conditions

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Fig. 9
figure 9

First harmonics of the pitching moment coefficient as function of the reduced frequency due to a pitching motion \(\left( \Theta _\mathrm{max}=1^\circ \right)\) of the SG-VT configuration for \(\alpha _0=0^\circ\) and \(\beta _0=0^\circ\)

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Fig. 10
figure 10

Pitch-damping derivative of the SG-00 and the SG-VT configuration over the considered angle of attack and sideslip angle regime

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3.1 Pitching motion

The pitching motion about the body-fixed y-axis represents a combination of two degrees of freedom, namely the angle of attack \(\alpha\) and the pitch rate q. The corresponding Taylor series reads

$$\begin{aligned} C_\nu \left( k\tau \right) &= C_{\nu 0}+C_{\nu \alpha }\alpha \left( k\tau \right) +C_{\nu \dot{\alpha }}\dot{\alpha }\left( k\tau \right)\\ &\quad+C_{\nu q}q\left( k\tau \right) +C_{\nu \dot{q}}\dot{q}\left( k\tau \right) . \end{aligned}$$
(11)

The last term on the right side can be omitted, because \(\dot{q}\propto \ddot{\alpha }\). Equation 11 can then be rewritten as:

$$\begin{aligned} C_\nu \left( k\tau \right) =C_{\nu 0}+C_{\nu \alpha }\alpha \left( k\tau \right) +\left( C_{\nu \dot{\alpha }}+C_{\nu q}\right) \dot{\alpha }\left( k\tau \right) . \end{aligned}$$
(12)

This representation of the Taylor series involves the coupled damping derivative related to the pitching motion. For an arbitrary freestream condition considering \(\alpha _0\ne 0^\circ\) and \(\beta _0\ne 0^\circ\), Eq. 12 must be expanded to

$$\begin{aligned} C_\nu \left( k\tau \right) &= C_{\nu 0}+C_{\nu \beta }\beta \left( k\tau \right) +C_{\nu \dot{\beta }}\dot{\beta }\left( k\tau \right) \\&\quad+C_{\nu \alpha }\alpha \left( k\tau \right) +\left( C_{\nu \dot{\alpha }}+C_{\nu q}\right) \dot{\alpha }\left( k\tau \right) . \end{aligned}$$
(13)

In consequence of the pitching motion about an initial angle of attack \(\alpha _0\), the effective angle of attack changes with respect to time. Considering the pitch amplitude \(\Theta \left( k\tau \right)\), the effective angle of attack, and its derivative are

$$\begin{aligned} \alpha \left( k\tau \right) =\alpha _0+\Theta \left( k\tau \right) , \quad \dot{\alpha }\left( k\tau \right) =\dot{\Theta }\left( k\tau \right) . \end{aligned}$$
(14)

The effective sideslip angle \(\beta\) is not affected by the pitching motion and hence reads

$$\begin{aligned} \beta \left( k\tau \right) =\beta _0, \quad \dot{\beta }\left( k\tau \right) =0. \end{aligned}$$
(15)

The harmonic pitching motion is given as \(\Theta \left( k\tau \right) =\Theta _\mathrm{max}e^{ik\tau }\). The corresponding aerodynamic derivatives related to the pitching motion result in

$$\begin{aligned} C_{\nu \alpha }=\frac{Re\left[ \tilde{C}_{\nu 1}\right] }{\Theta _\mathrm{max}}, \quad C_{\nu \dot{\alpha }}+C_{\nu q}=\frac{Im\left[ \tilde{C}_{\nu 1}\right] }{k\Theta _\mathrm{max}}. \end{aligned}$$
(16)

3.2 Yawing motion

The yawing motion is a combination of two degrees of freedom, namely the sideslip angle \(\beta\) and the yaw rate r. The corresponding Taylor series expansion reads

$$\begin{aligned} C_\nu \left( k\tau \right) &=C_{\nu 0}+C_{\nu \beta }\beta \left( k\tau \right) +C_{\nu \dot{\beta }}\dot{\beta }\left( k\tau \right) \\&\quad+C_{\nu r}r\left( k\tau \right) +C_{\nu \dot{r}}\dot{r}\left( k\tau \right) . \end{aligned}$$
(17)

Since the yawing angular velocity \(\dot{\Psi }\) is proportional to the yaw rate r, the Taylor expansion can be rewritten as:

$$\begin{aligned} C_\nu \left( k\tau \right) =C_{\nu 0}+C_{\nu \beta }\beta \left( k\tau \right) +C_{\nu \dot{\beta }}\dot{\beta }\left( k\tau \right) +C_{\nu r}\dot{\Psi }\left( k\tau \right) . \end{aligned}$$
(18)

In order to account for initial freestream conditions deviating from \(\alpha _0=0^\circ\) and \(\beta _0=0^\circ\), Eq. 18 is expanded to

$$\begin{aligned} C_\nu \left( k\tau \right) &= C_{\nu 0}+C_{\nu \alpha }\alpha \left( k\tau \right) +C_{\nu \dot{\alpha }}\dot{\alpha }\left( k\tau \right) \\&\quad+C_{\nu \beta }\beta \left( k\tau \right) +C_{\nu \dot{\beta }}\dot{\beta }\left( k\tau \right) +C_{\nu r}\dot{\Psi }\left( k\tau \right) . \end{aligned}$$
(19)

For an arbitrary initial freestream condition, the effective angle of attack changes with respect to time in consequence of the yawing motion about the body-fixed z-axis. It can be approximated as:

$$\begin{aligned} \sin \alpha \left( k\tau \right)&=\cos \Psi \left( k\tau \right) \sin \alpha _0 +\cos \beta _0\sin \alpha _0\sin \Psi \left( k\tau \right) \nonumber \\&\quad\times(\cos \beta _0\sin \Psi \left( k\tau \right) -\cos \alpha _0\cos \Psi \left( k\tau \right) \sin \beta _0). \end{aligned}$$
(20)

Considering a yawing motion with \(\Psi _\mathrm{max}=1^\circ\) at a freestream condition of \(\alpha _0=16^\circ\) and \(\beta _0=8^\circ\) results in a negligible maximum induced angle of attack of \(\Delta \alpha _\mathrm{max}=0.036^\circ\). The angle of attack is thus written as:

$$\begin{aligned} \alpha \left( k\tau \right) \approx \alpha _0, \quad \dot{\alpha }\left( k\tau \right) \approx 0. \end{aligned}$$
(21)

The effective sideslip angle \(\beta\) is calculated by

$$\begin{aligned} \sin \beta \left( k\tau \right) =\cos \Psi \left( k\tau \right) \sin \beta _0-\cos \alpha _0\cos \beta _0\sin \Psi \left( k\tau \right) . \end{aligned}$$
(22)

Applying the small angle approximation on \(\beta _0\) and \(\Psi\), results in the effective sideslip angle and its derivative given as:

$$\begin{aligned} \beta \left( k\tau \right) =\beta _0-\Psi \left( k\tau \right) \cos \alpha _0, \end{aligned}$$
(23)
$$\begin{aligned} \dot{\beta }\left( k\tau \right) =-\dot{\Psi }\left( k\tau \right) \cos \alpha _0. \end{aligned}$$
(24)

The Taylor expansion finally results in

$$\begin{aligned} C_\nu \left( k\tau \right)&=C_{\nu 0}+C_{\nu \alpha }\alpha _0+C_{\nu \beta }\left( \beta _0-\Psi \left( k\tau \right) \cos \alpha _0\right) \nonumber \\&\quad +\left( C_{\nu r}-C_{\nu \dot{\beta }}\cos \alpha _0\right) \dot{\Psi }\left( k\tau \right) . \end{aligned}$$
(25)

Applying the harmonic oscillation \(\Psi \left( k\tau \right) =\Psi _\mathrm{max}e^{ik\tau }\) results in the following expressions for the aerodynamic derivatives:

$$\begin{aligned} C_{\nu \beta }=-\frac{Re\left[ \tilde{C}_{\nu 1}\right] }{\Psi _\mathrm{max}\cos \alpha _0}, \quad C_{\nu r}-C_{\nu \dot{\beta }}\cos \alpha _0=\frac{Im\left[ \tilde{C}_{\nu 1}\right] }{k\Psi _\mathrm{max}}. \end{aligned}$$
(26)
Fig. 11
figure 11

Imaginary part of the surface pressure coefficient distribution due to harmonic pitching motion \(\left( \Theta _\mathrm{max}=1^\circ \right)\) on the wing upper side of the SG-00 configuration at selected freestream conditions

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Fig. 12
figure 12

Skin-friction lines and surface pressure distribution at \(\beta _0=8^\circ\) [19]

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Fig. 13
figure 13

Skin-friction lines and surface pressure distribution at \(\beta _0=8^\circ\), detail view of the double V/T [19]

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Fig. 14
figure 14

Imaginary part of the surface pressure coefficient distribution due to harmonic pitching motion \(\left( \Theta _\mathrm{max}=1^\circ \right)\) at the double V/T of the SG-VT configuration at selected freestream conditions

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3.3 Rolling motion

The rolling motion features the roll rate p as degree of freedom. The corresponding Taylor series reads

$$\begin{aligned} C_{\nu }\left( k\tau \right) =C_{\nu 0}+C_{\nu p}p\left( k\tau \right) +C_{\nu \dot{p}}\dot{p}\left( k\tau \right) . \end{aligned}$$
(27)

The last term in Eq. 27 can be neglected, since \(p\propto \dot{\Phi }\) with \(\dot{\Phi }\) being the rolling angular velocity. Considering an initial angle of attack and sideslip angle different from zero, the rolling motion results in harmonic variations of the effective angle of attack and sideslip angle. Hence, the Taylor series must be expanded to

$$\begin{aligned} C_{\nu }\left( k\tau \right)&= C_{\nu 0}+C_{\nu p}p\left( k\tau \right) +C_{\nu \alpha }\alpha \left( k\tau \right) +C_{\nu \dot{\alpha }}\dot{\alpha }\left( k\tau \right) \nonumber \\&\quad +C_{\nu \beta }\beta \left( k\tau \right) +C_{\nu \dot{\beta }}\dot{\beta }\left( k\tau \right) . \end{aligned}$$
(28)

The effective angle of attack and sideslip angle due to a body-fixed rolling motion read

$$\begin{aligned} \sin \alpha \left( k\tau \right) =-\cos \alpha _0\sin \beta _0\sin \Phi \left( k\tau \right) +\sin \alpha _0\cos \Phi \left( k\tau \right) , \end{aligned}$$
(29)
$$\begin{aligned} \sin \beta \left( k\tau \right) =\sin \beta _0\cos \Phi \left( k\tau \right) +\sin \alpha _0\cos \beta _0\sin \Phi \left( k\tau \right) . \end{aligned}$$
(30)

Considering small angles, Eqs. 29 and 30 can be rewritten. The effective angle of attack \(\alpha\) and its derivative then result in

$$\begin{aligned} \alpha \left( k\tau \right) =\alpha _0-\sin \beta _0\Phi \left( k\tau \right) , \quad \dot{\alpha }\left( k\tau \right) =-\sin \beta _0\dot{\Phi }\left( k\tau \right) . \end{aligned}$$
(31)

The effective sideslip angle \(\beta\) and its derivative are given as:

$$\begin{aligned} \beta \left( k\tau \right) =\beta _0+\sin \alpha _0\Phi \left( k\tau \right) , \quad \dot{\beta }\left( k\tau \right) =\sin \alpha _0\dot{\Phi }\left( k\tau \right) . \end{aligned}$$
(32)

Substituting these definitions, the Taylor expansion can be rewritten as:

$$\begin{aligned} C_{\nu }\left( k\tau \right)&= C_{\nu 0}+C_{\nu \alpha }\alpha _0+C_{\nu \beta }\beta _0 \nonumber \\&\quad +\left( -C_{\nu \alpha }\sin \beta _0+C_{\nu \beta }\sin \alpha _0\right) \Phi \left( k\tau \right) \nonumber \\&\quad +\left( C_{\nu p}-C_{\nu \dot{\alpha }}\sin \beta _0+C_{\nu \dot{\beta }}\sin \alpha _0\right) \dot{\Phi }\left( k\tau \right) . \end{aligned}$$
(33)

Applying the excitation \(\Phi \left( k\tau \right) =\Phi _\mathrm{max}e^{ik\tau }\), the derivatives result in

$$\begin{aligned}&-C_{\nu \alpha }\sin \beta _0+C_{\nu \beta }\sin \alpha _0=\; \frac{Re\left[ \tilde{C}_{\nu 1}\right] }{\Phi _\mathrm{max}}, \nonumber \\&C_{\nu p}-C_{\nu \dot{\alpha }}\sin \beta _0+C_{\nu \dot{\beta }}\sin \alpha _0=\frac{Im\left[ \tilde{C}_{\nu 1}\right] }{k\Phi _\mathrm{max}}. \end{aligned}$$
(34)
Fig. 15
figure 15

Yaw-damping derivative of the SG-00 and the SG-VT configuration over the considered angle of attack and sideslip angle regime

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Fig. 16
figure 16

Imaginary part of the surface pressure coefficient distribution due to a harmonic yawing motion \(\left( \Psi _\mathrm{max}=1^\circ \right)\) on the wing upper side of the SG-00 configuration at selected freestream conditions

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Fig. 17
figure 17

Imaginary part of the surface pressure coefficient distribution due to a harmonic yawing motion \(\left( \Psi _\mathrm{max}=1^\circ \right)\) at the double V/T of the SG-VT configuration at selected freestream conditions

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4 Numerical approach

4.1 Grid generation

The applied computational grids are of hybrid unstructured type. They are generated with the grid generation software CENTAUR. The software realizes the mesh generation process in three separated steps. In a first step, the surface of the considered geometry is meshed with unstructured surface elements of triangular and/or quadrilateral type. Based on the unstructured surface mesh, a quasi-structured prismatic grid is created by a wall normal extrusion of the surface elements. Finally, the remaining domain is filled with tetrahedral elements. The software allows for local adjustments of several parameters like the grid node density, stretching ratio or first layer thickness.

In line with former numerical investigations of the SAGITTA configuration, the surface mesh is considerably refined towards the wing leading edge [16], see Fig. 4a and detailed view in Fig. 4b. The applied Spalart–Allmaras turbulence model requires a maximum \(y^+\)-level of \(y^+\approx 1\) and a moderate wall normal stretching ratio in the vicinity of the wing surface. This is necessary in order to properly resolve the viscous sublayer of the boundary-layer flow. A first cell height of the prism layer grid of \(h_1=0.004\) mm results in the required maximum \(y^+\)-level. This holds for zero angle of attack and sideslip angle conditions, see Fig. 5a, as well as for the most restrictive case with an angle of attack of \(\alpha _0=16^\circ\) and a sideslip angle of \(\beta _0=8^\circ\), see Fig. 5b. In order to cover the complete boundary-layer flow by the prismatic grid, a number of 38 prism layers is applied. The wall-normal stretching factor is set to \(g=1.235\) for the first 30 prism layers. The last eight prism layers feature a constant cell height. The remaining domain is filled with tetrahedral elements. The tetrahedral elements are considerably refined in the vicinity of the upper wing surface, in order to provide a well resolved grid for the separated vortical flow, see Fig. 4b. The computational domain is restricted by a sphere with a radius of 20 semi wing spans. A grid convergence study performed on the SG-VT configuration and on the SG-00 configuration at steady freestream conditions results in a number of \(14 \times 10^6\) grid nodes for the SG-00 configuration. The SG-VT configuration features \(23\times 10^6\) grid nodes. Complex flow structures at the double V/T require a fine resolution of the computational grid in this area, which entails a significant increase of the number of grid nodes in comparison to the SG-00 configuration.

4.2 Flow solver

The simulations presented in this paper are performed with the TAU-Code, a CFD solver developed by the German Aerospace Center (DLR) Institute of Aerodynamics and Flow Technology [12]. The TAU-Code solves the three-dimensional compressible (unsteady) Reynolds-averaged Navier–Stokes (U)RANS equations. It is optimized for the usage with hybrid unstructured grids and developed with respect to parallel efficiency on high-performance computers. Hence, all simulations have been run in parallel mode at the GCS Supercomputer SuperMUC at the Leibniz Supercomputing Center (LRZ). The CFD code uses a finite-volume scheme and is based on a dual grid approach [39]. The TAU-Code is composed of several modules, which can be executed independently. The most important modules are the preprocessing and solver module. During the preprocessing, the secondary grid required for the dual grid approach is generated. It is computed according to the cell-vertex grid metric from the initial primary grid. Furthermore, the additional coarser grids for the multigrid approach are created during the preprocessing. The solver module solves the (U)RANS equations on the provided dual grid [12].

Fig. 18
figure 18

Roll-damping derivative of the SG-00 and the SG-VT configuration over the considered angle of attack and sideslip angle regime

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Fig. 19
figure 19

Imaginary part of the surface pressure coefficient distribution due to a harmonic rolling motion \(\left( \Phi _\mathrm{max}=1^\circ \right)\) on the wing upper side of the SG-00 configuration at selected freestream conditions

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4.3 Numerical setup and test conditions

The DLR TAU-Code has already been validated for low-aspect-ratio configurations with vortex-dominated flow and (partially) rounded leading edges. This experience [10, 16, 37] is utilized for the numerical setup of the present configuration. For the spatial discretization, a second-order central scheme introduced by Jameson is applied [23]. Using the matrix dissipation scheme, central-difference schemes become closer to upwind biased methods [44]. The temporal discretization is realized by an implicit backwards Euler method with a LUSGS algorithm [24] and a dual time stepping scheme. A three-level 3w multigrid cycle and a point-explicit residual smoother are used in order to accelerate the convergence of the solution. For turbulence modelling of the fully turbulent simulations, the one-equation Spalart–Allmaras model is applied in the SA-neg version, which is able to cope with negative values of the SA viscosity without any negative influence on the numerical solution [1, 40]. In former investigations [15, 16, 19, 36,37,38] for similar configurations the SA-model showed good agreement with experimental results. Especially, the SA prediction of the pitching moment at higher angles of attack was superior to those with two-equation models [37].

The oscillating motions are realized by means of the rigid-body-motion module of the TAU-Code. Periodic motions are described as a combination of polynomial and Fourier series. The mathematical description of a rotational motion reads

$$\begin{aligned} \Xi \left( t\right)&= \sum \limits _{n = 0}^{n = N_{PR}}r_nt^n+c_0+ \sum \limits _{n = 1}^{n = N_{FR}}\nonumber \\&\quad \times \left( c_n\cos \left( n\omega t\right) +d_n\sin \left( n\omega t\right) \right) . \end{aligned}$$
(35)

\(\Xi\) denotes the rotation angle about a body-fixed axis and r, c, d denote the polynomial and Fourier coefficients. \(\omega\) is the angular velocity and defined as shown in Eq. 2. The desired periodic motion can be realized by a corresponding definition of the polynomial and Fourier coefficients. To obtain the harmonics of the aerodynamic forces and moments from the CFD simulations, the time-accurate coefficients of one complete motion period are required. At least two periods of motion are simulated to obtain a converged solution. The first period of motion is needed to settle the convergence. The second period of motion is then evaluated on the convergence and absence of higher harmonics. If the convergence is not sufficient after two periods of motion, two more periods with increased number of inner iterations per physical time step are simulated. The applied standard setup for the rigid body motion defines 1000 inner iterations per physical time step and 100 physical time steps per period. This was not sufficient to obtain a converged solution for certain types of motion at high angles of attack and sideslip angles. The lateral motions (rotation about the x- and z-axis) of the SG-00 configuration at \(\alpha _0=16^\circ\) and \(\beta _0=8^\circ\) required two more periods with 1500 inner iterations. Considering the SG-VT configuration, all motions at \(\alpha _0=16^\circ\) and \(\beta _0=8^\circ\) required two more periods with 1500 inner iterations. The yawing motion at this freestream condition additionally takes a fifth period with 3000 inner iterations. All simulations are performed at low Mach number conditions, which are comparable to the wind tunnel conditions of former steady experimental investigations. This enables a comparison of the steady numerical results with the steady experimental data. The analyses are performed for a Mach number of \(Ma_\infty =0.12\) and a Reynolds number of \(Re_\infty = 1.7\times 10^6\), based on the mean aerodynamic chord of \(l_\mu =0.667\) m. With a reference temperature of \(T_\infty =288.15\) K, this results in a freestream velocity of \(U_\infty =40.83\) m/s. The oscillatory mean angles of attack are \(\alpha _0=\left\{ 0^\circ , 4^\circ , 8^\circ , 12^\circ , 16^\circ \right\}\) and the mean sideslip angles are \(\beta _0=0^\circ\), \(\beta _0=4^\circ\), and \(\beta _0=8^\circ\). The harmonic oscillations are performed for a reduced frequency of \(k=0.1\) (\(f=0.974\) Hz) and with maximum excitation amplitudes of \(\Phi _\mathrm{max}=\Psi _\mathrm{max}=\Theta _\mathrm{max}=1^\circ\). The rotations were performed around the moment reference point \(x_\mathrm{mrp}\), see Fig. 1.

5 Results

5.1 Evaluation of the approach

The described approach towards determining the dynamic derivatives entails several restrictions (e.g., linearity with respect to the reduced frequency and degree of freedom, absence of higher harmonics). Preliminary analyses are performed in order to ensure that these requirements are met. The evaluation comprises the analysis of steady aerodynamic derivatives, Lissajous figures, and the variation of the reduced frequency. Lissajous figures, which are also called Bowditch curves, are shapes created by plotting two harmonic signals over each other. The x-coordinate of the plot is defined by one harmonic signal and the y-coordinate is defined by the other harmonic signal. The signals can be written as \(x=A\sin (at+\Delta \varphi )\) and \(y=B\sin (bt)\), with A and B defining the amplitude of the signal, a and b defining the frequency and \(\Delta \varphi\) defining the phase shift between both signals. Depending on these parameters of both curves, a certain pattern arises. Figure 6 shows the Lissajous figures for two harmonic signals of same amplitude and frequency at different phase shifts. The patterns are of circular, elliptic or linear shape. Considering the CFD simulations, a deviation of the obtained Lissajous curves from these geometrical patterns indicates the presence of higher harmonics. The evaluation of the described requirements is done by means of the SG-VT configuration.

The steady aerodynamic derivatives obtained from the dynamic simulations are compared to the steady aerodynamic derivatives obtained from steady-state numerical simulations and steady-state wind tunnel tests. The experimental data are available from former wind tunnel tests performed at TUM-AER. Selected results of the wind tunnel tests and detailed information on the wind tunnel model are given in Ref. [19]. Figure 7 illustrates some steady aerodynamic derivatives obtained from the dynamic numerical simulations (Dynamic CFD), from the steady numerical simulations (Steady CFD) and from the steady wind tunnel tests (Steady W/T).

Figure 7a shows the pitching moment derivative versus the angle of attack at \(\beta _0=0^\circ\) for the three different data sources. Considering the dynamic motion, the pitching moment derivative is obtained by Eq. 16. The pitching moment derivatives resulting from the steady-state motions are obtained by the difference quotient around the discrete data points. A small deviation between the numerical and the experimental results is visible. The reason for this deviation is discussed in Ref. [19]. The numerical results, however, show good agreement. Only small deviations are observed for angles of attack of \(\alpha _0 \ge 8^\circ\). The rolling moment derivative is presented as function of the angle of attack at \(\beta _0=8^\circ\) in Fig. 7b. The rolling moment derivative is determined from the yawing motion via Eq. 26. All three data sources show good agreement up to an angle of attack of \(\alpha _0=12^\circ\). Slight deviations are visible at \(\alpha _0=16^\circ\). The yawing moment derivative \(C_{mz\beta }\) is shown for a sideslip angle of \(\beta _0=0^\circ\) and \(\beta _0=8^\circ\) in Fig. 7c, d, respectively. Considering the symmetric freestream condition, both the steady and dynamic CFD results show good agreement up to \(\alpha _0=12^\circ\). The experimental results show slight deviations to both numerical results for \(\alpha _0 \ge 4^\circ\). The deviations between the yawing moment derivatives calculated from the different data sources increase for \(\beta _0=8^\circ\). The deviations between the “Steady CFD” and the “Steady W/T” results can be motivated by the explanation given in Ref. [19]. The experimental investigations have been performed without a flow tripping mechanism at the W/T model which would force a laminar-turbulent transition of the boundary-layer flow. Following, it was supposed that areas of laminar flow are also present at the wing. In contrast to that the CFD simulations are run fully turbulent. The flow separation at the round leading edge depends on the boundary-layer flow condition. A turbulent boundary layer flow is less threatened by flow separation than a laminar boundary layer. This would consequently entail certain deviations between both data sources. The discrepancy between the CFD results, however, can be explained by the varying calculation method for the derivative. In case of the “Steady CFD” data, the derivative is determined by the difference quotient between the discrete data points. Due to considered sideslip angles of \(\beta _0=\left[ 0^\circ ,\, 4^\circ ,\, 8^\circ \right]\), the derivative is calculated with \(\Delta \beta _0=4^\circ\). In contrast, the steady derivatives determined by means of the “Dynamic CFD” data are calculated with \(\Delta \beta _0=1^\circ\), since the yawing motion is performed with an yaw angle amplitude of \(\Delta \Psi =1^\circ\). In summary, the presented results evaluate the applicability of the approach by means of steady aerodynamic coefficients.

In addition, the absence of higher harmonics and non-linearities need to be checked. Figure 8 shows the Lissajous figures for the aerodynamic coefficients of selected freestream conditions. The ones marked as “Temporal” are obtained from the time-accurate aerodynamic coefficients. The time-accurate coefficients are plotted versus the excitation (roll, pitch or yaw angle), which results in the corresponding Lissajous figure. The ones marked as “1. Harmonic” are obtained from the first harmonics of the aerodynamic coefficients. The harmonics can be recalculated in a time-accurate signal via Eq. 6,

$$\begin{aligned} C_\nu \left( k\tau \right) =\overline{C}_\nu +Re\left[ \tilde{C}_{\nu 1}\right] \cos \left( k\tau \right) -Im\left[ \tilde{C}_{\nu 1}\right] \sin \left( k\tau \right) . \end{aligned}$$

The obtained time-accurate values are then plotted versus the excitation, which results in the corresponding Lissajous figure. If the solution of the CFD simulation is converged and no higher harmonics are present, both Lissajous curves must fit together and form a circular, linear or elliptical shape. Figure 8a presents the Lissajous figure of the pitching moment coefficient for \(\alpha _0=0^\circ\) and \(\beta _0=0^\circ\). For this flight condition, the assumed absence of higher harmonics and linear characteristics is clearly confirmed. Figure 8b illustrates the Lissajous figure of the pitching moment coefficient for \(\alpha _0=16^\circ\) and \(\beta _0=8^\circ\). Although non-linear characteristics could be expected for such freestream conditions, the applied reduced frequency and excitation angle result in almost linear characteristics without higher harmonics in the aerodynamic response. Slight deviations between the “Temporal” and the “1. Harmonic” curve indicate small non-linearities. Those non-linear effects are small and can thus be neglected. However, if angles of attack of \(\alpha _0>16^\circ\) would be considered, the aerodynamic response would also need to be checked carefully on non-linear effects and higher harmonics. The flow around the diamond wing becomes more dominated by separated flow with increasing angle of attack, which might induce non-linear effects. Considering the yawing motion, the Lissajous figure of the yawing moment coefficient is depicted, see Fig. 8c. It represents a freestream condition of \(\alpha _0=16^\circ\) and \(\beta _0=8^\circ\). The linearity and the absence of higher harmonics can be observed. In the vicinity of the maximum positive excitation, small non-linear effects can be observed, which are negligible. Figure 8d shows the Lissajou figure of the rolling moment coefficient for the rolling motion at \(\alpha _0=16^\circ\) and \(\beta _0=8^\circ\). Both curves show good agreement, which indicates the absence of higher harmonics.

The applied approach furthermore requires linear characteristics with respect to the reduced frequency. This is assumed to be met for \(k<0.2\). The linear characteristics can be checked via the imaginary part of the first harmonic of the pitching moment coefficient, since the imaginary part represents the dynamic part of the harmonic response. The requirements with respect to the reduced frequency are met, if the imaginary part of the coefficient shows linear characteristics and approaches zero for a reduced frequency of \(k\rightarrow 0\). Figure 9 illustrates the first harmonics of the pitching moment coefficient as a function of k, of the SG-VT configuration for a freestream condition of \(\alpha _0=0^\circ\) and \(\beta _0=0^\circ\). Three different reduced frequencies of \(k=0.05\), \(k=0.1\) and \(k=0.2\) are considered. The imaginary part representing the dynamic part of the aerodynamic response shows linear characteristics with respect to the reduced frequency and shows a trend approaching a value of zero for a zero reduced frequency. The real part, which is used to determine the steady derivative, is constant with respect to k. Consequently, the dynamic motion has no influence on the quasi-steady part of the solution. The first harmonics of the pitching moment coefficient demonstrate that the requirements associated with linear characteristics with respect to the reduced frequency are met.

5.2 Damping derivatives

The damping derivatives of the pitching and yawing motion are coupled derivatives, since the corresponding motions are a combination of several degrees of freedom. The roll-damping derivative exhibits additional contributions for freestream conditions of \(\alpha _0\ne 0^\circ\) and \(\beta _0\ne 0^\circ\). The damping derivatives read \(C_{my\dot{\alpha }}+C_{myq}\), \(C_{mzr}-C_{mz\dot{\beta }}\cos \alpha _0\) and \(C_{\nu p}-C_{\nu \dot{\alpha }}\sin \beta _0+C_{\nu \dot{\beta }}\sin \alpha _0\) and are associated with the pitching, yawing and rolling motion, respectively. Considering an increasing angle of attack and/or sideslip angle, the cross and cross-coupling derivatives become important as well, which are, however, not subject of this paper. The damping derivatives are determined for the SG-00 and SG-VT configuration. An evaluation regarding the dynamic stability of the SAGITTA configuration can be made by means of the sign of the corresponding damping derivative. Subsequently, the damping derivatives are presented and observed effects are discussed by means of surface pressure distributions induced by the harmonic motions.

5.2.1 Pitch-damping derivative

Figure 10 shows the pitch-damping derivative of the SG-VT and the SG-00 configuration for the considered angle of attack and sideslip angle range as well as a comparison of both configurations at selected freestream conditions. The pitch-damping derivative and harmonic surface pressure coefficient are presented with respect to a nose-up pitching motion about the body-fixed y-axis. Consequently, a negative pitch-damping derivative represents dynamically stable characteristics in pitch. Considering a dynamically increasing angle of attack, the dynamic motion induces a negative pitching moment, which indicates a stabilizing nose-down motion.

The SG-00 configuration shows dynamically stable characteristics for all considered freestream conditions, see Fig. 10a. This is indicated by consistently negative pitch-damping derivatives. A constant pitch-damping derivative is observed for \(\alpha _0<8^\circ\). Non-linear characteristics with respect to the angle of attack are observed for \(\alpha _0\ge 8^\circ\). Of special interest is the freestream condition \(\alpha _0=8^\circ\), \(\beta _0=0^\circ\), since the pitch-damping derivative exhibits a local maximum at this point. The increasing value at this particular freestream condition indicates an effect slightly destabilizing the configuration. For \(\alpha _0>8^\circ\), the pitch-damping derivative significantly decreases to values even lower than for angles of attack of \(\alpha _0<8^\circ\).

The effects resulting in this characteristic of the pitch-damping derivative at \(\beta _0=0^\circ\) can be explained by means of the imaginary part of the harmonic surface pressure distribution. Similar to the harmonics of the aerodynamic force and moment coefficients, the harmonic surface pressure consists of the real and the imaginary part. It is obtained from the time-accurate surface pressure distribution of one motion period. It can be expressed similar to the formulation given in Eq. 6, whereas the aerodynamic coefficient is replaced by the surface pressure coefficient. Similar to the harmonics of the aerodynamic force and moment coefficients, the DLR TAU-Code provides the harmonic surface pressure with the solution of the rigid body motion simulation. Considering the pitching motion, the imaginary part represents the induced pressure in consequence of the dynamic motion around the pitching axis, see Eq. 16. Figure  11a–c illustrate the imaginary part of the harmonic surface pressure distribution, the skin-friction lines and the body-fixed coordinate system located in the center of rotation of the pitching motion at selected freestream conditions. Attached flow dominates the complete wing surface for \(\alpha _0<8^\circ\). The magnitude of the induced pressure is small but acts on a large area of the wing upper and lower side and results in a nose down pitching moment. For fully attached flow, varying freestream conditions do not affect the pitch-damping derivative. However, the flow starts to separate at the wing tips at \(\alpha _0=8^\circ\). The flow separation is associated with the short wing chord in the wing-tip area, which results in a high aerodynamic load and a significant adverse pressure gradient. With increasing angle of attack the flow separation onset moves upstream and the separated flow forms an irregular recirculation area with flow reversal. The location of the flow separation onset and the exact formation of the flow separation are sensitive to the local incident flow, which changes in consequence of the harmonic pitching motion. The induced surface pressure is thus of higher magnitude in these areas and, consequently, has a significant influence on the derivative. At \(\alpha _0=8^\circ\), the induced pressure in the area of separated flow is dominated by positive values. The location of the induced pressure at the wing tip is behind the c.o.r., which results in a nose-up pitching moment and thus destabilizes the configuration. However, negative pressure levels are dominant in the same area for \(\alpha _0\ge 12^\circ\), which stabilizes this configuration, see Fig. 11b, c.

Taking asymmetric freestream conditions into account requires a short survey on the evolving flow structures at the diamond wing. A detailed discussion of the subsequently regiven description of the flow phenomena is available in Ref. [19]. Considering a sideslip angle of \(\beta _0=8^\circ\) and an angle of attack of \(\alpha _0=8^\circ\) reveals minor changes due to the sideslip angle. The inboard leading-edge vortices are slightly asymmetric, exhibiting a stronger vortex at the leeward side and a weaker vortex at the windward side, cf. Fig. 12a. The most considerable influence of the sideslip angle is observed at the double V/T. A flow separation is present at the suction side of the windward fin, see Fig. 13a. The recirculation area at the trailing edge is clearly noted. There is also an influence on the flow at the wing surface near the wing-fin junction. The flow at the double V/T induces a reduced pressure at the wing surface and the flow separation at the fin results in a small recirculation area at the wing surface. The described effects are hardly seen at the leeward fin, because the incipient flow is more aligned in chordwise direction. With increasing angle of attack, the flow characteristics at the configuration considerably change. Large-scale asymmetric flow phenomena on the wing become obvious, cf. Fig. 12b. The inboard leading-edge vortex at the windward side further weakens in intensity, whereas the one on the leeward side becomes stronger. In the outboard wing section, the separation region becomes larger on the windward side. In contrast to that, the area of irregular flow reversal vanishes on the leeward wing side. Instead of the irregular flow reversal, an additional leading-edge vortex emerging from the round leading-edge contour is observed. This vortex does not exhibit flow reversal along the vortex core axis but is of retarded type. The flow at the double V/T is more aligned in chordwise direction because of the mentioned flow phenomena at \(\alpha _0=16^\circ\). This results in a mitigation of the effects at the double V/T previously described for \(\alpha _0=8^\circ\), see Fig. 13b.

The effect of the asymmetric freestream conditions on the pitch-damping derivatives is analyzed in the following. The observed characteristics are discussed by means of the described flow phenomena at the wing due to asymmetric freestream condition. The effect of \(\beta _0>0^\circ\) is small for angles of attack up to \(\alpha _0=12^\circ\), see Fig. 10a. The small dependency on the sideslip angle can be explained by means of the harmonic surface pressure coefficients. The changes in the flow field due to \(\beta _0\ne 0^\circ\) are small and primarily influence the areas of separated flow for \(\alpha _0\ge 8^\circ\), cf. Fig. 11d, e. The previously described effects stabilizing or destabilizing the configuration are mainly present at the windward wing side. However, the induced pressures are of higher magnitude in comparison to the case with the symmetric freestream condition. The overall effect of the sideslip angle on the pitch damping is thus mitigated. Strong dependency on the sideslip angle, however, is observed at \(\alpha _0=16^\circ\). This can be associated with the described change in the flow field due to the sideslip angle. The formation and thus intensity of this vortex are sensitive to the local incident flow. In consequence, the induced surface pressure due to the rigid body motion is of large magnitude, cf. Fig. 11c, f. The induced positive surface pressure results in a nose-up pitching moment, which decreases the dynamic stability.

The pitch-damping derivative of the SG-VT configuration with attached double V/T is illustrated in Fig. 10b. The SG-VT configuration is dynamically stable for all considered freestream conditions. It features similar pitch-damping characteristics as the SG-00 configuration at \(\alpha _0\le 4^\circ\). Furthermore, no significant deviations between the pitch-damping derivative of both configurations are observed for all considered angles of attack at \(\beta _0=0^\circ\). The influence of the sideslip angle, however, is more significant for this configuration in comparison to the SG-00 configuration. In addition to the effects previously described on the SG-00 configuration, the attached double V/T affects the pitch-damping characteristic. Most significant changes in comparison to the SG-00 configuration are observed at \(\beta _0=8^\circ\) for \(\alpha _0=8^\circ\) and \(\alpha _0=12^\circ\). The double V/T has a significantly stabilizing effect for \(\alpha _0=8^\circ\) and a slightly destabilizing effect for \(\alpha _0=12^\circ\). The flow past the double V/T needs to be considered in more detail. For these cases, a leading-edge vortex and a trailing-edge separation forms at the suction side of the windward vertical tail due to the sideslip angle, cf. [19] and Fig. 14. Both effects are sensitive to the varying incident flow due to the pitching motion. In consequence of that, not only significant pressure changes occur at the surface of the vertical tails itself, but as well on the wing upper side in the vicinity of the wing/vertical-tail junction. The induced pressure due to this effect exhibits negative levels for the angle of attack of \(\alpha _0=8^\circ\) (stabilizing), cf. Fig. 14a, and positive values for \(\alpha _0=12^\circ\) (destabilizing), cf. Fig. 14b.

In Fig. 10c, d, the pitch-damping derivative \(C_{my\dot{\alpha }}+C_{myq}\) and its decoupled parts (\(C_{my\dot{\alpha }}\), \(C_{myq}\)) are shown for the SG-00 and SG-VT configuration for \(\beta _0=0^\circ\) and \(\beta _0=8^\circ\), respectively. \(C_{my\dot{\alpha }}\) is obtained by the harmonic analysis of a heave motion and \(C_{myq}\) is obtained by quasi-steady rotational motions about the pitch axis. The sum of both parts results in the pitch-damping derivative obtained by the pitching motion. For zero sideslip angle, both configurations feature the same pitch-damping derivatives over the angle of attack range, see Fig. 10c. The pitch-damping derivative of both configurations is dominated by \(C_{myq}\). The non-linear characteristics observed in the pitch-damping derivative, however, are mainly associated with the dynamic derivative \(C_{my\dot{\alpha }}\), whereas \(C_{myq}\) features almost linear characteristics. For \(\beta _0=8^\circ\), the pitch-damping derivative is dominated by \(C_{myq}\) as well, which is similar for both configurations, see Fig. 10d. Consequently, the deviations observed between both configurations with respect to the sideslip angle are primarily associated with \(C_{my\dot{\alpha }}\), which is also responsible for the significant non-linear characteristic of \(C_{my\dot{\alpha }}+C_{myq}\) with respect to the angle of attack.

5.2.2 Yaw-damping derivative

The yaw-damping derivative for both considered SAGITTA configurations is presented in Fig. 15. The yaw-damping derivative and corresponding harmonic surface pressure coefficient distributions are presented with respect to a nose-right yawing motion about the body-fixed z-axis. Consequently, dynamic stability in yaw is indicated by negative yaw-damping derivatives. The SG-00 configuration shows constant values at all sideslip angles for \(\alpha _0\le 8^\circ\), see Fig. 15a. For this flight range, the values are approximately zero and thus no damping in yaw is present. A non-linear increase in the yaw-damping derivative is observed for \(\alpha _0>8^\circ\) at all considered sideslip angles, indicating unstable dynamic characteristics in yaw. Highest and thus most unstable yaw-damping derivatives are present at maximum angle of attack and \(\beta _0\le 4^\circ\). The yawing motion features most significant influence on the imaginary part of the harmonic surface pressure in the area of separated flow in the outboard wing region, see Fig. 16. This effect, however, influences the derivative only at \(\alpha _0\ge 12^\circ\), cf. Fig. 16b. At \(\alpha _0=8^\circ\), the area of separated flow as well as the effect of the yawing motion on it is too weak for an impact on the yaw-damping derivative. The yawing moment is primarily influenced by the part of the surface pressure acting normal to the body-fixed xz- and yz-plane. Thus, the induced pressure at the wing leading edge is of interest. At the windward side, the motion induces positive pressure and at the leeward side it induces negative pressure at the wing leading edge. Considering the c.o.r., a nose-right yawing moment is induced, which exaggerates the yawing motion and leads to positive yaw-damping derivatives. The flow separation onset moves upstream with increasing angle of attack, which strengthens this effect, see Fig. 16b, c. The leading-edge vortices at the wing apex, in contrast, are hardly affected by the yawing motion. The induced pressure levels are small and thus their contribution to the damping derivative is minor. Figure 15a indicates negligible influence with respect to the sideslip angle for angles of attack up to \(\alpha _0=12^\circ\). With increasing angle of attack, the instability considerably increases for all considered sideslip angles. At \(\alpha _0=16^\circ\) and \(\beta _0=8^\circ\), however, the growth of instability is much less in comparison to smaller sideslip angles. This effect is associated with the transformation of the flow field at the leeward wing side, cf. Fig. 16f. Significant negative values are induced at the vortex axis in the outboard region of the leeward wing side. However, the affected area of negative induced pressure in the vicinity of the leading edge is considerably reduced in comparison to the case for \(\beta _0=0^\circ\). Furthermore, the magnitude of positive induced pressure levels in the vicinity of the leading edge at the windward wing side is slightly reduced. Both effects lead to the reduction of the induced positive yawing moment.

A remarkable influence of the double V/T on the yaw damping is indicated by the yaw-damping derivatives of the SG-VT configuration, see Fig. 15b. It exhibits dynamic stability in yaw for \(\alpha _0\le 8^\circ\) with exception of \(\alpha _0=8^\circ\), \(\beta _0=8^\circ\). The double V/T stabilizes the configuration for most considered freestream conditions due to the positive induced pressure at the leeward side and the negative induced pressure at the windward side of both vertical tails, which results in a nose-left yawing moment, see, e.g., Fig. 17a, e, f. This is to be expected, because the dynamic yawing motion locally induces a positive velocity in y-direction. This decreases the effective sideslip angle at the vertical tail, which results in the observed changes in the surface pressure distribution. The resulting positive force in y-direction creates a yawing moment which counteracts the yawing motion and thus has a stabilizing effect on the configuration. At certain freestream conditions (\(\alpha _0=8^\circ\), \(\beta _0=8^\circ\); \(\alpha _0=12^\circ\), \(\beta _0=4^\circ\); \(\alpha _0=16^\circ\), \(\beta _0=4^\circ\)), however, the stabilizing effect of the double V/T is not observed. The induced pressure levels at the leeward side of the windward vertical tail are reversed for those cases, see Fig. 17b–d. In consequence, the windward vertical tail has a destabilizing effect and the leeward vertical tail has a stabilizing effect. In combination, the effects cancel each other out and no effect of the double V/T is observed in the yaw-damping derivative. As already described, the asymmetric freestream conditions have a significant influence on the flow around the double V/T. Vortex flow as well as flow separation with flow reversal are present for numerous flight conditions. Similar to the observations at the wing, these flow structures are sensitive with respect to the local incident flow. Especially, for higher angles of attack, the local incident flow at the double V/T is not only changed due to the variation in the freestream direction but also due to the changing flow structures evolving at the upwind wing section like the inboard leading-edge vortices. This results in a complex situation at the double V/T in consequence of the harmonic motion. Hence, at certain flight conditions the expected stabilizing effect on the yaw damping is not observed.

In Fig. 15c, d, the yaw-damping derivative and its separated parts \(( C_{mz\dot{\beta }},\,C_{mzr})\) are shown with respect to the angle of attack for the SG-00 and the SG-VT configuration. \(C_{mzr}\) is obtained by quasi-steady rotational motions about the yawing axis and \(C_{mz\dot{\beta }}\) is obtained by harmonic lateral oscillations. Note that due to the definition of \(\beta\), \(C_{mz\dot{\beta }}\) indicates stable behavior for positive values. The yaw-damping characteristics of both configurations are similar. However, an almost constant offset of \(\Delta (C_{mzr}-C_{mz\dot{\beta }}\cos \alpha _0)\approx 0.15\) is observed between the values of both configurations. The attached double V/T significantly increases the stability of the configuration without introducing additional non-linearities. In case of the SG-00 configuration, both parts feature values of approximately zero up to \(\alpha _0=4^\circ\). With increasing angle of attack, \(C_{mz\dot{\beta }}\) considerably increases, whereas the quasi-steady \(C_{mzr}\) derivative is almost constant with respect to the angle of attack. Considering the SG-VT configuration, \(C_{mzr}\) shows negative values, which are slightly increasing with respect to \(\alpha _0\). The non-linear effects are mainly introduced by \(C_{mz\dot{\beta }}\). It exhibits positive values for angles of attack up to \(\alpha _0=8^\circ\). The derivative decreases non-linearly for a further increase of the angle of attack. The sideslip angle of \(\beta _0=8^\circ\) is considered in Fig. 15d. The yaw-damping derivatives of both configurations exhibit smaller changes with respect to the angle of attack. The quasi-steady derivative \(C_{mzr}\) shows similar values and characteristics for both presented sideslip angles. Hence, the variations of the yaw-damping derivative with respect to the sideslip angle are associated with \(C_{mz\dot{\beta }}\). It shows values of \(|C_{mz\dot{\beta }}|<0.1\), which considerably change with respect to the angle of attack. The \(C_{mz\dot{\beta }}\) characteristics of the SG-00 configuration exhibit the effect of the evolving outboard leading-edge vortex for high angles of attack. The corresponding characteristics of the SG-VT configuration additionally represent the effect of the double V/T, which is especially obvious at \(\alpha _0\ge 8^\circ\).

5.2.3 Roll-damping derivative

The rolling motion is a motion of one degree of freedom, namely the roll rate p. The effective angle of attack and sideslip angle change in consequence of the rolling motion, if freestream conditions deviating from zero angle of attack and zero sideslip angle are considered. This results in additional parts in the roll-damping derivative associated with the angle of attack and sideslip angle. The roll-damping derivative for arbitrary freestream conditions reads

$$\begin{aligned} C_{\nu p}-C_{\nu \dot{\alpha }}\sin \beta _0+C_{\nu \dot{\beta }}\sin \alpha _0. \end{aligned}$$
(36)

The roll-damping derivative and harmonic surface pressure coefficients are presented with respect to a rolling motion about the body-fixed x-axis with right wing tip down and left wing tip up. Negative roll-damping derivatives hence indicate dynamic stability in roll. Figure 18 illustrates the roll-damping derivative of the SG-00 and SG-VT configuration for the considered freestream conditions. The SG-00 configuration features dynamic stability in roll at all considered freestream conditions. The roll-damping derivative is on a constant level of \(C_{\nu p}-C_{\nu \dot{\alpha }}\sin \beta _0+C_{\nu \dot{\beta }}\sin \alpha _0\approx -0.25\) for \(\alpha _0\le 4^\circ\). Increasing the angle of attack up to \(\alpha _0=8^\circ\) results in a slight reduction of the roll-damping derivative. Up to this angle of attack, no influence of the sideslip angle is observed. Figure 19 depicts the imaginary part of the harmonic surface pressure coefficient on the wing upper side of the SG-00 configuration at selected freestream conditions. The downward motion of the right wing tip increases the effective angle of attack at the right wing side and thus induces a negative surface pressure at the wing upper side, see Fig. 19a. On the left wing side, the upward motion induces reversed pressure levels. The induced pressure on the left and right wing side results in a rolling moment moving the right wing tip up. Consequently, the rolling motion is damped by the induced dynamic effects. The effect of the rolling motion on the flow separation at the wing tips at \(\alpha _0=8^\circ\) is responsible for the slight decrease of the roll-damping derivative. Significant dependency on the sideslip angle can be observed for further increased angles of attack (\(\alpha _0\ge 12^\circ\)). The dynamic stability in roll is considerably reduced for \(\beta _0=8^\circ\). This characteristic is associated with the increasing area of separated flow at the windward wing side, see Fig. 19e. Positive pressure is induced in the area of separated flow at the windward wing side, which results in a destabilization in roll. The affected area is small, but the resulting force acts with a large lever arm with respect to the rotation axis. This results in a significant reduction of the magnitude of the roll-damping derivative. Although these flow effects on the windward wing side increase with increasing angle of attack, the roll-damping improves for \(\alpha _0=16^\circ\), \(\beta _0=8^\circ\), see Figs. 18a and 19f. This effect is associated with the evolving leading-edge vortex at the outboard region on the leeward wing side, which counteracts the destabilizing effect of the flow separation on the windward side. The induced positive pressure on the surface in the area of the vortex core is of high magnitude and features a high lever arm. In summary, both effects almost cancel each other out. The SG-VT configuration shows almost the same roll-damping derivatives as the SG-00 configuration, see Fig. 18b. The double V/T exhibits minor influence, since the induced pressure levels at the double V/T are of small magnitude and the lever arm to the rotation axis is small. Consequently, the previously described effects dominate the SG-VT configuration as well.

6 Conclusion and outlook

This article has presented results of numerical investigations of unsteady aerodynamic characteristics of the SAGITTA diamond-wing configuration at low-speed wind-tunnel conditions. The investigations have been conducted with attached double vertical tail (V/T) and without double V/T at several angles of attack and sideslip angles. The pitch-damping, yaw-damping, and roll-damping derivatives have been determined by means of the aerodynamic response to forced harmonic motions. The pitch-damping derivative resulting from the pitching motion shows dynamically stable characteristics in pitch at all considered freestream conditions and for both considered SAGITTA configurations. The attached double V/T influences the pitch-damping derivative for a sideslip angle of \(\beta _0\ne 0^\circ\). Depending on the freestream condition, the double V/T increases or decreases the dynamic stability in comparison with the clean configuration. Non-linear characteristics of the pitch-damping derivative are observed with respect to the angle of attack and sideslip angle. The yaw-damping characteristics are more critical than the pitch-damping characteristics. Without double V/T, the yaw-damping derivative indicates considerable unstable yaw-damping behavior for increasing angle of attack and sideslip angle. The attached double V/T entails a significant improvement of the yaw damping behavior for the majority of the considered freestream conditions. The yaw-damping characteristics with respect to the angle of attack and sideslip angle feature significant non-linearities with increasing angles of attack and sideslip angles. For \(\beta _0>0^\circ\), the quasi-steady \(C_{mzr}\) derivative mainly ensures the lateral stability of the SG-VT configuration. The roll-damping derivative exhibits dynamic stability in roll for both configurations at all considered freestream conditions. Non-linear characteristics are observed with increasing angle of attack and sideslip angle. Both configurations show almost identical roll-damping derivatives, indicating minor influence of the double V/T on the roll-damping characteristics.

The considered damping derivatives show in general a constant value at angles of attack less than \(8^\circ\). Increasing the angle of attack and sideslip angle result in significant non-linear characteristics of the damping derivatives. The non-linearities are associated with occurring flow separations and vortex formation at the blunt leading edge. They exhibit distinct sensitivity with respect to the body motions. The results indicate that approaches using linear aerodynamic theory are not an appropriate tool to determine the dynamic derivatives for such configurations at higher angles of attack and sideslip angles. The effect of the dynamic motions on the flow past the wing requires further investigation. Especially, the influence on the vortex evolving at the rounded leading edge and on the flow past the double V/T is complex and needs to be considered in detail. Furthermore, the harmonic surface pressure distributions indicate the presence of cross and cross-coupling derivatives at higher angles of attack and sideslip angles, which need to be evaluated.