The sensitivity of transonic flow past a Whitcomb airfoil to deflections of an aileron is studied at freestream Mach numbers from 0.81 to 0.86 and vanishing or negative angles of attack. Solutions of the Reynoldsaveraged NavierStokes equations are obtained with a finitevolume solver using the kω SST turbulence model. The numerical study demonstrates the existence of narrow bands of the Mach number and aileron deflection angles that admit abrupt changes of the lift coefficient at small perturbations. In addition, computations reveal freestream conditions in which the lift coefficient is independent of aileron deflections of up to 5 degrees. The anomalous behavior of the lift is explained by interplay of local supersonic regions on the airfoil. Both stationary and impulse changes of the aileron position are considered.
1. Introduction
The correct prediction of the effectiveness of wing control surfaces (ailerons and spoilers) is of major importance in the process of aircraft design. The development of numerical methods enables accurate simulation of transonic flow over control surfaces with fixed deflection angles
[1
,
2]
. A number of studies examined transonic flow over timedependent flaps, as well as aeroelastic behavior of airfoils and wings
[3

5]
. However, the flow sensitivity to small perturbations in various bands of the angle of attack and Mach number has not been subject to detailed analysis.
The upward deployment of an aileron or spoiler flattens the profile in the vicinity of the aileronairfoil juncture or even makes it locally concave. In the 2000s, a number of numerical studies demonstrated a high sensitivity of transonic flow to variations of freestream parameters when the airfoil comprises a flat or nearly flat arc. The sensitivity is caused by the interaction of two supersonic regions that arise and expand on the arc as the freestream Mach number increases. The expansion followed by a coalescence of the supersonic regions crucially changes pressure distributions and aerodynamic loads on the airfoil. This phenomenon was scrutinized for a number of symmetric profiles
[6

7]
, as well as for the asymmetric J78 airfoil whose upper surface is nearly flat in the midchord region
[6
,
8]
. Also the instability of closely spaced supersonic regions was examined for a Whitcomb airfoil with a deflected aileron at the Reynolds number Re=5.6×10
^{6}
[9]
.
In this paper, we study transonic flow past a Whitcomb airfoil with aileron deflections at the vanishing or negative angles of attack, which are typical for a descending flight of civil and transport aircraft, at Re=1.4×10
^{7}
. The emphasis is laid on the flow physics and freestream conditions that admit anomalous behavior of the lift coefficient.
2. Problem formulation
We consider a fully turbulent 2D flow past an airfoil given by the expressions
where
x
and
y
are nondimensional Cartesian coordinates, and
y_{whit}
(
x
) refers to the Whitcomb integral supercritical airfoil
[10]
. The last term in (1b) shifts the rear part of the airfoil upward, simulating an aileron rotation at a small angle θ, as illustrated in
Fig. 1
. The airfoil is placed at the center of a lenstype computational domain, bounded by two circular arcs, Γ
_{1}
:
x
(
y
)= 105−(145
^{2}
− y
^{2}
)
^{1/2}
and Γ
_{2}
:
x
(
y
)= −105+(145
^{2}
− y
^{2}
)
^{1/2}
, −100≤
y
≤ 100. The width and height of the domain are 80 and 200, respectively. We set the length
L
_{chord}
of the airfoil chord to 2.5 m.
On the inflow part Γ
_{1}
of the boundary, we prescribe stationary values of the angle of attack α, freestream Mach number M
_{∞}
˂1, and static temperature
T
_{∞}
=223.15 K. On the outflow boundary Γ
_{2}
, we impose the static pressure
p
_{∞}
=26,434 N/m
^{2}
. The above values of
T
_{∞}
and
p
_{∞}
are respective to the standard atmosphere at a height of 10 km. The noslip condition and vanishing flux of heat are used on the airfoil. The air is assumed to be a perfect gas whose specific heat at constant pressure is 1004.4 J/(kg K) and the ratio of specific heats is 1.4. We adopt the value of 28.96 kg/kmol for the molar mass, and use the Sutherland formula for the molecular dynamic viscosity. Initial data are parameters of the uniform freestream, in which the turbulence level was set to 0.2%.
3. A numerical method
Solutions of the RANS equations were obtained with the ANSYS 13 CFX finitevolume solver based on a highresolution discretization scheme for convective terms
[11]
. We employed an implicit secondorder accurate backward Euler scheme for the timestepping. Computations were performed on hybrid unstructured meshes of about 4×10
^{5}
cells, which were clustered in the boundary layer, in the
Sketch of the Whitcomb airfoil with an aileron deflection at a positive angle θ.
Sketch of the computational mesh in a vicinity of the airfoil.
wake, and in the vicinities of the shock waves. The nondimensional thickness y
^{+}
of the first mesh layer on the airfoil was less than 1 (see
Fig. 2
). We used the standard
k
ω Shear Stress Transport turbulence model, which reasonably predicts aerodynamic flows with boundary layer separations from smooth surfaces
[12]
.
The solver was verified by computation of solutions for a few benchmark problems and comparison with experimental and numerical data available in the literature.
Figure 3
shows good agreement of the lift coefficient
C
_{L}
(α) calculated for a RAE 2822 airfoil at −1≤α, deg≤3 with results obtained in
[13

17]
. Also the solver was used for the simulation of an oscillatory transonic flow past a 18% thick circulararc airfoil at zero angle of attack and Re=1.1×10
^{7}
. The amplitude of lift coefficient oscillations at M
_{∞}
=0.75 was 0.35. This agrees well with the value of 0.37 obtained numerically in
[18]
using the SpalartAllmaras and Baldwin Lomax turbulence models.
Lift coefficient versus the angle of attack for a test case of transonic flow over the RAE 2822 airfoil at M_{∞}=0.73, Re=6.5×10^{6}. The references to numerical studies are accompanied by the information on turbulence models used.
Convergence of the lift coefficient with mesh refinement for airfoil (1) at the aileron deflection angle θ=4 deg, angle of attack α=−0.8 deg, and Reynolds number Re ≈1.4×10^{7}.
4. Results and discussion
In the case of stationary boundary conditions, steadystate solutions were obtained by the global timestepping in 2 to 6 seconds. The solutions yield a flow field, as well as aerodynamic forces on the airfoil. This makes it possible to calculate the lift coefficient
C
_{L}
=2
F
/ρ
_{∞}
U
_{∞}
^{2}
S
), where
F
is the normal force,
U
_{∞}
is the freestream velocity module, and
S
is the wing area in plaform. For the simulation of 2D flow we used 3D meshes with one cell of length
L_{z}
=0.01 m in the
z
direction, so that
S
=
L
_{chord}
×
L_{z}
=0.025 m
^{2}
.
Figure 4
displays plots of the lift coefficient versus m
_{∞}
calculated on three different meshes at the angle of attack α=0.8 deg and the aileron deflection angle θ=4 deg. Evidently, the mesh of
Lift coefficient as a function of the aileron deflection angle θ and Mach number M_{∞} for transonic flow past airfoil (1) at Re ≈1.4×10^{7}: (a) α=0, (b) α= −0.8 deg.
Streamlines in a vicinity of the trailing edge at M_{∞}=0.86, α= −0.8 deg, θ=0.
Evolution of the local supersonic regions in transonic flow past airfoil (1) at the increasing angle θ of the aileron deflection and M_{∞}=0.86, α= −0.8 deg, Re =1.4×10^{7}. Mach number contours: 1 – M=1.0, 2 – M=1.125, 3 – M=0.875.
386,360 cells provides a good accuracy of the solution.
Figure 5
illustrates the obtained dependence of the lift coefficient
C
_{L}
on two parameters, M
_{∞}
and θ, for airfoil (1) at the angles of attack α=0 and α= −0.8 deg. As seen from
Fig. 5
a, if α=0 and M
_{∞}
˂ 0.838, then a deflection of the aileron from zero to three degrees entails a considerable fall of the lift coefficient. This is caused by a rapid shrinking of the supersonic region on the upper surface and expansion of the supersonic region on the lower surface of the airfoil. If α=−0.8 deg, then abrupt changes of
C
_{L}
take place at larger values of M
_{∞}
and 3.5 ˂ θ, deg ˂5, so that the surface
C
_{L}
(M
_{∞}
,θ) comprises a slit at 0.846< M
_{∞}
≤0.86 (see
Fig.5
b)
At the same time
Figs. 5
a and
5
b show that, when M
_{∞}
˃0.855, the lift coefficient is independent of aileron deflections up to 5 and 4 degrees, respectively. This can be explained again by the interplay of local supersonic regions. Indeed,
Figure 6
shows that at θ=0 the airfoil’s trailing edge resides in a stagnation zone above the streamline separated from the lower surface of the airfoil. That is why moderate upward deflections of the aileron do not influence the flow field. With increasing aileron deflection angle from 0 to 4 deg, the supersonic regions slightly expand on both surfaces (see
Fig. 7
), that is why the lift coefficient persists. If the angle θ further increases, then the supersonic region on the upper surface shrinks, while that on the lower surface expands. As a consequence, the static pressure drops on the lower surface, and the lift coefficient drops.
Lift coefficient as a function of time for airfoil (1) at α=0, Re≈1.4×10^{7}, and periodic impulse changes of the deflection angle θ: (a) M_{∞}=0.83, (b) M_{∞}=0.85.
Also we considered a timedependent aileron deflection θ(
t
) that switches between 0 and 3 deg every 0.6 s, with a period of 1.2 s.
Figure 8
a shows a calculated dependence of the lift coefficient on time at M
_{∞}
=0.83 and α=0. An increase of M
_{∞}
to 0.85 results on a crucial reduction of the amplitude of lift coefficient oscillations (see
Fig.8
b) in accordance with
Fig.5
a exhibiting
C
_{L}
versus stationary variations of θ.
5. Conclusions
For the airfoil at hand, there exist adverse freestream conditions that admit abrupt changes of the lift coefficient at small variations of the aileron deflection angle. Conversely, there exist conditions in which a response of the lift coefficient to aileron deflections is anomalously weak, so that the aileron fails to control the lift. Both anomalous phenomena are caused by the interplay of local supersonic regions on the airfoil.
Acknowledgements
This research was performed using computational resources provided by the Computational Center of St. Petersburg State University (http://cc.spbu.ru). The work was supported by the Russian Foundation for Basic Research under grant no. 130800288.
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