Aerodynamic moment characteristics of tandem-scheme aircraft

Aerodynamic interference between forward and back wings of tandem-scheme aircraft significantly affects its pitch and roll moments. The interference increases roll stability in a narrow range of sideslip angles; there is a kink on the dependence of roll moment coefficient versus sideslip angle (that is not observed for conventional-scheme aircraft). Directional stability is decreased by a dihedral angle of forward wings and winglets on them but is increased by the same factors for back wings. If back   wings’   bending   is   significant, then aerodynamic interference may affect directional stability as well. The vortex system of tandem-wings at a sideslip angle was modeled incorrectly by the used CFD method (solving RANS), and further research is needed. The analytical and experimental methods show a good agreement concerning moment characteristics.


Introduction
In the beginning of the XXI century aerodynamic scheme "tandem" (with comparable areas of forward wings and back wings) became widespread among UAVs and small piloted aircrafts ( fig. 1) [1][2][3][4].
Its aerodynamics has specifics caused by inevitable wing-wing interference. The tip vortices of the forward wings affect the flow on the back wings, create upwash and downwash ( fig. 2). For a small vertical gap, forward wings also slow down the flow on the back wings and make it turbulent, but this effect might be low for reasonable parameters of configuration. Nevertheless, downwash and upwash always affect total lift coefficient and moment characteristics that concern the stability of aircraft and is of high importance.

Object and methods
For aerodynamic moments determination the following methods were used: 1) The analytical method developed by the author [5,6]; 2) The CFD method (RANS solving with Ansys software) [7]; 3) Experimental research (in certified wind tunnel AT-1 at Antonov Company - fig. 3, 4) was performed for determination of the aerodynamic characteristics of the tandem-scheme UAV model [8].

Pitch static moment
Pitch static moment can be found only with the correct calculation of average downwash angle and therefore back wings` lift coefficient ( fig. 5). The classic analytical method (by Prandtl) considers wingspan ratio, height, and stagger. The proposed method can take into account the wings` dihedral angles and the sideslip angle as well. For simple cases, both methods show the same results. For geometries with dihedral angles, accurate pitch moment prediction by the new method is preferable to achieve minimal balancing losses for maximal lift-drag ratio at cruise angle of attack ( fig. 6).

Roll static moment
The scheme of forces that create a roll moment for tandem aircraft is presented in fig. 7. Roll static moment dependence differs from the one of traditional aircraft. The interference of forward and back wings results in roll static stability increasing for low sideslip angles, but it rapidly decreases after β=5…10° ( fig. 8). This kink on the roll moment coefficient graph was predicted by the proposed analytical method and was observed in wind tunnel test data. Nevertheless, this effect was not detected by the CFD method based on RANS solving.  For an explanation of the effect, we should consider the position of forward wings` tip vortices. Till both forward vortices are inside the span of back wings, the additional rolling stability is provided ( fig. 9a). At some sideslip angle (β=6…8°), the left forward tip vortex intersects the tip section of the back wing ( fig. 9b). At higher sideslip angles, this particular vortex decreases rolling stability, while right forward tip vortex still increases. So after certain sideslip angle (we may call it kink β), the aerodynamic interference has almost no effect on rolling stability, and it can be estimated as for isolated forward and back wings.

Yaw static moment
Yaw static moment depends on wings` dihedral angles stronger than for conventional scheme. As wings are far from the center of gravity so dihedral angles of forward/back wings create strong destabilizing/stabilizing yaw moment ( fig. 12). So winglets on forward wings decrease directional stability as well, but on back wings increase it greatly.

Fig. 12. Yaw moment scheme
For monoplane, the yaw moment is produced mainly by fin and fuselage, so it can be written as where ψ 1 and ψ 2 are forward and back wings` dihedral angles, respectively; , respectively. According to the well-known method for one fin of low aspect ratio [9], this value can be approximated as where we assume that (4) So for the wings with mean chords b 1 and b 2 we can write as for twin-fin tail: Directional stability degrees predicted by the analytical method (from known monoplane characteristics) and in wind tunnel tests in the range of sideslip angles from -15° to +15° are presented in table 1 and fig. 13.  . It was kept into considerations that real dihedral angles of back wings were not 0° and -9°, but +4° and -5° due to bending measured for α=6.6° and β=0°.
Probably method accuracy can be improved if we consider that different loads for left and right back wings result in different dihedral angles, but such measurements were not performed during the wind tunnel tests. The interference of forward and back wings doesn`t affect yaw stability for a perfectly rigid aircraft (or its model). According to fig. 9, forward wings` tip vortices produce roll moment due to different lift on back wings. Indeed it turns to the different induced drags of left and right back wings. But as for conventional scheme aircraft with wings` dihedral angles, this yaw moment is small (not to confuse with wings` swept angles that provide significant directional stability for values of 20 deg and above). Nevertheless, the wingwing interference can change back wings` bending (e.g. under load the average angle is ψ 2 =0°, but ψ 2 right =+3° and ψ 2 left =-3°, and both wings increase yaw stability).

Conclusions
Aerodynamic interference between forward and back wings significantly affects pitch and roll moments of tandem-scheme aircraft.
In a narrow range of sideslip angles, the interference increases roll stability that was predicted by the analytical method and was proved in the wind tunnel test. Outside this range of sideslip angles, the interference has a low effect on roll moment.
Directional stability is decreased by any dihedral angle (positive or negative) of forward wings and winglets on them but is increased by the same factors for back wings. For a perfectly rigid aircraft (or its model), the aerodynamic interference has a low effect on yaw moment. But for flexible back wings, upwash and downwash changing turn to dihedral angles changing that affect directional stability as well as longitudinal.
The vortex system of tandem-scheme aircraft at the sideslip angle is modeled incorrectly by used CFD method (solving RANS). So while analytical and experimental methods show a good agreement in moment characteristics, further research of CFD methods for tandem-wings aerodynamics is needed (e.g. with structured mesh).
Promising future research of tandem-scheme aerodynamics is to determine flight controls efficiency taking into account the wing-wing interference.