12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

DRAG REDUCTION ON A TRANSONIC WING

Maurizio Quadrio
Dipartimento di Scienze e Tecnologie Aerospaziali
Politecnico di Milano
via La Masa 34, 20156 Milano, Italy
maurizio.quadrio@polimi.it

Alessandro Chiarini
Dipartimento di Scienze e Tecnologie Aerospaziali
Politecnico di Milano
via La Masa 34, 20156 Milano, Italy
alessandro.chiarini@polimi.it

Jacopo Banchetti
Davide Gatti
Dipartimento di Scienze e Tecnologie Aerospaziali
Institute for Fluid Mechanics
Politecnico di Milano
Karlsruhe Institute of Technology
via La Masa 34, 20156 Milano, Italy
Kaiserstr. 10, 76131 Karlsruhe, Germany
jacopo.banchetti@polimi.it
davide.gatti@kit.edu

Antonio Memmolo
High Performance Computing Department
CINECA-Interuniv. Cons., 40033 Bologna, Italy
antonio.memmolo@cineca.it

Sergio Pirozzoli
Dipartimento di Ingegneria Meccanica e Aerospaziale,
Sapienza Univ. Roma, Via Eudossiana, Roma, Italy
sergio.pirozzoli@uniroma1.it

Abstract

additional contributions besides the viscous friction, such aspressure drag, parasitic drag, separation, lift-induced drag andwave drag, and what ultimately matters is reducing the overalldrag and not only the friction contribution. Only over the lastyears some researchers have begun considering how the reduction of turbulent skin-friction affects the other drag contributions. For example Banchetti et al. (2020) applied stramwise
traveling waves of spanwise forcing (Quadrio et al., 2009)
on the incompressible turbulent flow past a bump to investigate the effect of skin-friction reduction on pressure drag.
They found that a distributed reduction of friction influences
the pressure field largely reducing the associated drag contribution, increasing by one half the net energetic benefits.
Later Nguyen et al. (2021) studied the effect of a temporally
spanwise-oscillating pressure gradient in a channel flow withtransverse bars at the wall. They observed that pressure dragis reduced almost as the friction drag, but that in this case thenet energy budget remains slightly negative.

Flow control for reducing skin-friction drag in the turbulent regime is applied to a transonic airfoil to improve its
global aerodynamic performance. The study relies on direct numerical simulations of the compressible turbulent flow
around a supercritical airfoil at Reynolds and Mach numbersof Re∞ = 3 × 105 and M∞ = 0.7. The control is applied ona portion of the airfoil suction side only. However, besides
locally reducing drag, the control modifies the position andthe intensity of the shock wave and significantly improves the
aerodynamic efficiency of the airfoil by increasing the lift andslightly decreasing the total drag. The increase of the aerodynamic efficiency implies that the airfoil can achieve the desired
lift at a lower angle of attack and, therefore, with a much lower
drag. Estimates of the benefits on a full aircraft are provided
and indicate that substantial savings are possible, even whenthe energy cost of an active control is considered. These results
suggest that skin-friction drag reduction should no be considered as a goal only, but also as tool to modify and control the
global aerodynamics of complex flows.

The airplane is one of the applications where flow control mat play a key role. Here the goal is not a decrease of
the drag only, but the enhancement of the lift/drag ratio, i.e.
the aerodynamic efficiency. Recently Atzori et al. (2020) usedhigh-fidelity incompressible Large Eddy Simulations to study
the effect of uniform blowing and suction on a NACA4412 airfoil at a Reynolds number based on free-stream velocity andchord length of Re = 200, 000 and observed an increase of the
aerodynamic efficiency up to 11%. Albers & Schröder (2021)
considered the same airfoil and the same Re, but controlled theflow with spanwise travelling waves of wall-normal deformation. As in their previous work based on a different airfoil,
they found that the control favourably modifies both the friction and pressure drag, yielding a global increase of the aerodynamic performances. Similarly, Kornilov (2021) estimated a
net power saving up to 8% for a NACA0012 airfoil controlled

Introduction
Flow control for reducing turbulent skin-friction drag has
arouse the interest of scholars over the years, because of efficiency and environmental reasons. However, to date only few
strategies have been deployed in applications, owing to an often critical cost/benefit ratio. Most research for skin-friction
drag reduction has taken place in parallel flows, where dragis entirely due to friction. Unfortunately, friction drag reduction of both passive (e.g. riblets) and active techniques (e.g.
spanwise forcing) are proportional to the portion of the surface covered by the drag-reducing device.
In more complex flows the aerodynamic drag contains

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
is longer as it extends for 0.2c ≤ x ≤ 0.78c and the forcing
amplitude is larger, i.e. A = 0.684. Note that in viscous units
for C2 these parameters correspond to A+ ≈ 6.6, ω + ≈ 0.06
and κx+ ≈ 0.013 that is not far from the optimum for the incompressible channel flow at a friction Reynolds number ofReτ = 200.
Four DNS simulations have been carried out for the nocontrol, C1 and C2 cases using the baseline grid (as shownin the following the simulation dealing with C2 has also been
repeated at a different angle of attack). Two further simulations for the no-control and C2 cases have been carried out ona finer grid for validation purpose. The baseline grid consistsin Nx × Ny × Nz = 4096 × 512 × 256 points (≈ 536 millions ofpoints) with an uniform distribution in the spanwise direction
and a hyperbolic distribution in the tangential and wall normal
directions to increase the resolution close to the airfoil and in
the wake. The finer grid is obtained by increasing the number of points of a 1.5 factor in the three directions, yielding
Nx × Ny × Nz = 6144 × 768 × 384, corresponding to approximately 1.8 billions of points. As verified with an a posteriori
check, at the wall the requirements of ∆x+ < 10, ∆y+ ≤ 0.5
and ∆z+ < 5 for a fully resolved DNS (Hosseini et al., 2016)
are satisfied. The time step of the simulations is kept constant, i.e. ∆t = 1.5 × 10−4 and ∆t = 1 × 10−4 for the baseline
and fine grid respectively, and has been chosen to maintainthe maximum Courant-Friedrichs-Lewy number below unity.
After reaching statistical equilibrium the simulations are advanced for further 40c/U∞ to accumulate the flow statistics.
Hereinafter all quantities are made dimensionless with c andwith the free stream values.

with low-amplitude blowing/suction.
All the above mentioned works have considered the flow
in incompressible or subsonic regime. However, following thework by Mele et al. (2016) there are reasons to suspect that afurther advantage of reducing skin friction resides in the ability to interact with the position and the strength of the shock
wave generated over a wing in the transonic regime. Since civil
aircraft cruise and military aircraft maneuver in the transonicregime, this would be of paramount importance in assessing
the effectiveness of flow control for aeronautical applications.
Unfortunately, reliable simulations of the flow over a transonic
airfoil with flow control for skin-friction drag reduction areless widespread. In this work we present the first direct numerical simulation (DNS) of the compressible turbulent flow
over an airfoil in the transonic regime, where flow control forskin-friction reduction is applied. We explore to what extent
a localised control for skin-friction reduction alters the aerodynamic performances of the airfoil. The results are also extrapolated to the entire airplane to provide an estimate of thepotential savings.

Methods
We investigate the transonic flow around the V2C airfoil,
a supercritical airfoil designed by Dassault Aviation in the context of the European research program TFAST. The Reynolds
and Mach numbers of the flow are set to Re∞ = U∞ c/ν∞ =3 × 105 and M∞ = U∞ /a∞ = 0.7; c is the airfoil chord and U∞ ,
ν∞ and a∞ are the free-stream velocity, kinematic viscosity andsound speed. The angle of attack is α = 4 deg, which corresponds to the maximum aerodynamic efficiency of the profileat this Re∞ and M∞ . The DNS code (Memmolo et al., 2018)solves the compressible Navier–Stokes equations for a calorically perfect gas and is based on a second-order finite volume method, which switches to third-order WENO scheme
using a modified Ducros sensor (Ducros et al., 1999). Timeadvancement is carried out with the low-storage, third-order
Runge–Kutta scheme. At the farfield, characteristics-based
non-reflective boundary conditions are used (Poinsot & Lele,
1992), while periodicity is enforced in the spanwise direction.
A C-type mesh with radius of 25c is employed, with the outflow placed at 25c from the trailing edge. The domain extends
for 0.1c in the spanwise z direction which is enough to ensure
decorrelation of all the flow structures (Zhang & Samtaney,
2016; Hosseini et al., 2016). The incoming flow is laminarand, following Schlatter & Örlü (2012), the transition to turbulence is enforced at x = x f = 0.1c via a volume force (tripping)on both sides of the airfoil.
The streamwise-travelling waves of spanwise velocity(Quadrio et al., 2009) are applied on a fraction of the suction side of the airfoil only. The spanwise component of thevelocity at the wall ww enforced by the control is:
ww (x,t) = f (x)A sin(κx x − ωt);Results
Instantaneous and mean fields
Figure 1 characterises the mean and instantaneous fields.
The isosurfaces of the imaginary part λci of the complex conjugate eigenvalues pair of the velocity gradient tensor (ℑ(λci =100) are used to visualise the vortical structures of the uncontrolled case. The background colour map is for the Mach number of the uncontrolled case, while the three lines denote themean sonic line M = 1 for the non-control (red), C1 (blue)
and C2 (green) cases. The flow becomes immediately supersonic at the leading-edge of the airfoil and is laminar up tox = 0.1 where the tripping is applied. The flow remains supersonic up to x ≈ 0.5, where it is abruptly recompressed by
the shock wave. As detailed by the sonic lines, the forcing
moves the shock wave downstream delaying the recompression and enlarging the supersonic region. The streamwise andvertical dimensions of the bubble increase from Dx = 0.475
and Dy = 0.355 for the non-control case to Dx = 0.485 and
Dy = 0.365 for C1 and to Dx = 0.516 and Dy = 0.425 for C2.
The downstream shift of the shock wave is accompanied by anincrease of its intensity. To be quantitative, the pressure jumpacross the shock at y = 0.2 increases from ∆p = 0.1205 forthe non-control case to ∆p = 0.1363 and ∆p = 0.1673 for C1
and C2. Consistently with the enlargement of the supersonicregion the maximum Mach number increases from M = 1.087
for the non-control case to M = 1.093 for C1 and M = 1.116
for C2. Overall, these changes are consistent with a decreased
friction in the actuated region, that leads to an increase of thesupersonic flow speed.
Figure 2 visualises the effect of the control on the boundary layer developing over the suction side of the airfoil. The
instantaneous velocity components is plotted over a curved
surface at the first grid point off the wall, i.e. y ≈ 3 × 10−5 ,(1)A is the maximum forcing amplitude, κx and ω are the spatial and temporal frequencies and f (x) is a smoothing function(Yudhistira & Skote, 2011). In this work, two forcing configurations are considered, hereinafter referred to as C1 andC2, corresponding to different forcing strengths. In both cases
the controlled portion of the suction side starts after the tripping and extends downstream the shock wave. For C1 theactuated region extends for 0.3c ≤ x ≤ 0.78c and A = 0.5,
ω = 11.3 and κx = 161. For C2, instead, the actuated region

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1. Isosurfaces of the swirling strength ℑ(λci ) = 100 in the no-control case, coloured with the kinetic energy k with a white-tored colormap in the range 0 ≤ k ≤ 1. The background colour map is for the mean Mach number in the range. Lines are the sonic line
M = 1 for reference (red), C1 (blue) and C2 (green).

Figure 2. Instantaneous streamwise velocity component at the first grid point off the wall over the suction side, for no-control (top),
C1 (middle) and C2 (bottom). The symmetric blue-to-red clourmap is for −0.1 ≤ u ≤ 0.1; the blue colour is for negative values andthe red colour for positive values. The red lines mark the boundaries of the actuated region.

as a proxy of the wall friction. A pattern of alternating lowand high-speed streaks suddenly appears at x = 0.1 where thetripping is applied. In the non-control case the intensity of
the fluctuations decreases at x ≈ 0.46, due to the interactionof the boundary layer with the shock wave, before increasing again. The control clearly produces spanwise oscillationsin the developing streaks. A decrease of the intensity of thefluctuations indicates that the control strategy is effective in
reducing skin-friction. For C2 the streaks almost disappear atx ≈ 0.3 indicating that the flow almost relaminarise before theshock wave. After the shock wave (x ≈ 0.5) few spots withnegative u < 0 are observed for the uncontrolled case. For thecontrolled cases, instead, the local backflow after the shock ismore intense, in particular for C2, indicating that due to theinteraction with the shock wave the boundary layer separates
and gives origin to a small recirculating region.

where pw (x) is the mean wall pressure and τw = µ t̂t · ∂ u /∂ n|w
is the mean viscous stress at the wall; here t̂t is the tangential
unit vector, µ is the dynamic viscosity and ∂ /∂ n denotes thewall-normal derivative. The lines refer to the simulations carried out with the baseline grid, while the symbols are for the
results computed with the finer grid for the non-control andC2 cases. The results from the two grids are perfectly overlapping for the pressure coefficient. For the friction coefficient,
instead, the results indicate that the baseline grid is slightly
under-resolved at the leading edge. This suggest that the usual
requirements for fully-resolved DNS based on the cell spacing in viscous units (Hosseini et al., 2016) are not adequate.
Note, however, that the changes between the uncontrolled andthe controlled cases obtained with the baseline grid are wellvalidated by the finer grid. In the following the focus is on thesuction side only as the changes on the pressure side, where
there is no actuation, are marginal.

Wall friction and pressure

After the peak at the leading-edge, the friction coefficient
quickly decreases in the laminar boundary layer up to x ≈ 0.1
where the trace of the tripping is visible. Moving downstream,
the friction coefficient slowly increases before decreasing because of the shock-wave/boundary-layer interaction. For the
non-control case the local maximum of c f occurs at x ≈ 0.42,
while the local minimum is observed at x ≈ 0.47. Downstream

The mean friction and pressure coefficients, c f and c p , areshown in figure 3. They are defined asc f (x) =2τw (x)2(pw (x) − p∞ ), cp =
ρ∞U∞2
ρ∞U∞2

(2)

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

·10−2

3
2

1.5
Ref
C1
C2

0.5
−cp

1
cf

0

0

−0.5

−1
−2

Ref
C1
C2

1

−1
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x

−1.5

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x1

Figure 3. Friction coefficient c f (left) and pressure coefficient c p (right). Reference and C2 results obtained on the finer grid areshown with symbols. In the pressure side the green symbols are not visible as they are covered by the red ones.

Aerodynamic forces and extrapolation

the shock wave, after a partial recovery, c f slowly decreases
due to the mild adverse pressure gradient to eventually become
negative just before the trailing edge where the flow separates.
In the non-control case, although small backflow regions weredetected in the instantaneous field in figure 2, the c f remains
positive just after the shock wave, indicating the absence of amean recirculating region. In the controlled cases the spanwise forcing is effective in reducing friction in the actuatedportion on the surface. As expected a short spatial transientis observed where drag reduction develops (Quadrio & Ricco,
2004; Skote, 2012). Consistently with the visualisations in figure 2, for both C1 and C2 the combined action of the control
and of the SBLI lead to negative c f after the shock wave, denoting a mean backflow region that extends for 0.46 ≤ x ≤ 0.49
for C1 and for 0.46 ≤ x ≤ 0.54 for the stronger C2 forcing. A
final remark deals with the position of the local minimum of c f
immediately downstream the shock wave. The position of thelocal c f minimum follows the downstream shift of the shock
wave: it is placed at x = 47, 0.48 and 0.51 for the non-control,
C1 and C2 cases respectively.

The control-induced changes of the pressure and friction
distributions positively affect lift and drag. Table 1 compares
the lift and drag coefficients for the uncontrolled and controlled cases; the ∆1 and ∆2 columns show the control-induced
relative changes for C1 and C2 respectively. The friction andpressure contributions to the total drag are computed separately and reported as Cd, f and Cd,p . For C1 and C2 the control reduces the friction drag by 7.2% and 13.4%, that are verylarge numbers considering that the forcing is applies only ona limited portion of the suction side. The effect of the control on the pressure drag is different in the two cases. Indeed,
Cd,p decreases by a relative 2.4% for C1 and increases by arelative 5.5% for C2. Note, however, that the combination of
the friction and pressure contributions leads to an overall dragreduction for both controlled cases: for C1 Cd decreases bya relative 4.5%, while for C2 by a relative 0.8%. However,an additional favourable effect of the control is the increaseof the lift coefficient. For C1 the increase of Cℓ is marginal(+1.5%), as detailed by the small changes in the pressure distribution shown in figure 3. For C2, instead, the pressure distribution is largely modified and Cℓ significantly increases bya relative 11.3%. Overall, the control leads to an increase theaerodynamic efficiency of the airfoil in both cases, quantified
by 6.8% for C1 and 13.5% for C2.

We now consider the pressure coefficient. After the sharp
expansion at the leading-edge, c p features a flat plateau that extends down to the shock compression. Moving further downstream, the pressure coefficient progressively increases and at
the trailing edge c p = 0. Unlike c f , the control action modifies
the pressure even outside the actuated region. In particular,
two distinct effects are observed to modify the c p distribution,
both leading to an increase of the suction. (i) The compression
associated with the shock wave is delayed. (ii) The expansion
at the leading edge is more intense leading to a plateau withlower c p . The presence of the recirculating region in the controlled cases, indeed, mitigates the adverse pressure gradient in
the area close to the shock (see the milder slope of −c p in correspondence of the pressure recovery before the shock-induced
compression). As a result, the shock wave moves downstream
and enlarges the supersonic bubble. This leads to an increaseof the velocity within the bubble and, therefore, to a more intense expansion in the fore part of the airfoil. Both these effects are more evident in the C2 controlled case, that has been
designed on purpose to produce an evident recirculation after
the shock wave. In a certain way, the effect of the control on c p
can be assimilated to that of a slight increase of the free-stream
Mach number, but on the suction side only.

An increase of the aerodynamics efficiency implies thatthe required lift can be obtained at the cost of a lower drag.
In other words, when comparing the uncontrolled and controlled cases at constant lift coefficient, the above described
control-induced increase of the lift turns into a reduction of
the angle of attack and, therefore, of the total drag. To estimate
the achievable ∆Cd we have computed the Cℓ − α and Cd − αmaps for the uncontrolled case with auxiliary RANS simulations (not shown) carried out with a modified version of thesame code used for the DNS and using the Spalart-Allmaras
turbolence model. In the following only the C2 case is considered for conciseness. By assuming in first approximationthat the control-induced relative changes of the aerodynamic
forces remain constant for small changes of α, we estimatethat for C2 the angle of attack that leads to the reference lift
is α = 3.45 deg. A further DNS simulation has been carriedout for these parameters to evaluate the control performances.
As shown in the last columns of table 1 the resulting lift coefficient is Cℓ = 0.730 (slightly less than expected), but thedrag coefficient decreases to Cd = 0.0210, corresponding to a
relative 15% of drag reduction.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Lift and drag coefficients (Cl , Cd ) of the airfoil, and splitting of drag coefficient into friction and pressure contributions (Cd, f ,Cd,p ), for uncontrolled, C1 and C2 flow cases. ∆ stands for relative change, and the last two columns refer to the C2 case computed foran angle of attack α = 3.45◦ (see text).

Uncontrolled

C1

∆1

C2

∆2

C2 (α = 3.45◦ )

∆2

Cl

0.740

0.751

+1.5%

0.825

+11.3%

0.730

-1.3%

Cd

0.02470.0236-4.5%

0.0245-0.8%

0.0210-15.0%

Cd, f

0.00820.0076-7.3%

0.0071-13.4%

0.0074

-9.7%

Cd,p

0.01650.0161-2.4%

0.0174+5.5%

0.0136-17.6

Cl /Cd

29.7

31.7

+6.8%

33.7

+13.5%

34.8+17.2%

efficiency. In the most effective considered control, the lift increases by a relative 11.3%, while the total drag decreases bya marginal 0.8%, corresponding to an increase of the aerodynamic efficiency of 13.5%. An increase of the aerodynamic
efficiency means that in the controlled case the required lift isachieved at a lower angle of attack and, therefore, with a significant reduction of the total drag. We have measured that thisleads to a total drag reduction of 15%. Scaling these results tothe entire aircraft we have estimated that, in reference flight
conditions, this may lead to a total drag reduction of about
8.5%. The fact that control can be applied locally to achieve alarge, global benefit is of paramount importance for practicalfeasibility and benefit/cost ratio.
We conclude discussing the limitations of the work. One
should be aware of the challenge of finding actuators capable to meet the required specification with an acceptable energy efficiency. However, the general idea we propose is valid
for any skin-friction drag reduction technique, including passive strategies, e.g. riblets. Moreover, the present study is
not indicative of the maximum achievable gain and a detailed
investigation is deserved. Indeed, the available information
for incompressible channel flow and flat boundary layers can
not be scaled up straightforwardly to an airplane, where thegoal is enhancing the aerodynamic efficiency and not reducing only the friction drag. It should be also considered thatwhile the M∞ value here considered is representative of anaircraft the Re∞ is not. Therefore, a serious design attemptshould consider larger Reynolds numbers. Luckily, we knowthat spanwise forcing (Gatti & Quadrio, 2016) and, more in
general, skin-friction drag reduction techniques remain effective at larger Re. Nevertheless, a larger Re∞ would lead to adifferent target flow that certainly requires different details in
the optimisation.

These benefits can be tentatively scaled up to the full aircraft to estimate the potential power savings in aeronautical applications. We consider the wing-body configurration DLR-F6
defined in the second AIAA CFD prediction workshop (Laflin
et al., 2005), with reference flight conditions M∞ = 0.75 andRe∞ = 3 × 106 . The reference lift coefficient is CL = 0.5 thatis obtained at an angle of attack of α = 0.52 deg at the cost of
CD = 0.0295. In the following we try to estimate the achievable drag reduction when the more effective C2 control is applied on the wings. Because of the limited available data,
the following assumptions are made: (i) the wing is responsible for the entire lift of the aircraft and, excluding the liftinduced drag contribution, for 1/3 of the total drag; (ii) thecontrol-induced changes of Cℓ and Cd do not vary along thewing span and do not change with α, M∞ and Re∞ , so thatthe values reported in Table 1 are used. Following the sameline of reasoning as for the airfoil and using the CL − α andCD − α maps available in https://aiaa-dpw.larc.
nasa.gov/Workshop2/DPW_forces_WB_375 we estimate that applying the C2 control on the wings, the reference
CL is obtained at an angle of attack of α = 0.0125 deg at thecost of a total drag of CD = 0.0272, corresponding to a dragreduction of approximately 8.5%. The additional small benefit of the direct skin-frition reduction leads to about 9% of
total drag reduction for the entire aircraft. Also the actuation
power is considered. Under the idealised assumption of actuation with unitary efficiency, it is equal to the power transferredto the viscous fluid by the boundary forcing. Using the DNS
data, for the time-average actuation power we have measured
5.5 × 10−4 ρ∞U∞ . As a result, because of the localised actuation area (it is approximately 1/4 of the wing surface and 1/12of the aircraft surface) the actuation power is estimated to beabout 1% of the overall power expenditure of the aircraft.

Concluding discussion
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