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

DIRECT NUMERICAL SIMULATION OF TURBULENT FLOWABOVE RIBLETS UNDER ADVERSE PRESSURE GRADIENT

Satoshi Ikeda
Department of Mechanical Engineering
Osaka University
2-1 Yamadaoka, Suita, Osaka 565-0871, Japan
ikeda@fluid.mech.eng.osaka-u.ac.jp

Kie Okabayashi
Department of Mechanical Engineering
Osaka University
2-1 Yamadaoka, Suita, Osaka 565-0871, Japanokabayashi@mech.eng.osaka-u.ac.jp

Shintaro Takeuchi
Department of Mechanical Engineering
Osaka University
2-1 Yamadaoka, Suita, Osaka 565-0871, Japan
shintaro.takeuchi@mech.eng.osaka-u.ac.jp

ABSTRACT
Spatially-developing boundary layers above smooth andriblet walls are simulated to investigate the drag-reducing
performance of the riblet under an adverse-pressure gradient(APG). Open boundary layer without upper wall is reproduced
using rescaling/recycling method in order to set the blowing
and suction at the upper boundary and simulate APG. All
statistics under the zero pressure gradient (ZPG) are obtained
quantitatively until unphysical discontinuous pressure
fluctuation occurred. The calculated drag reduction by theriblet is in good agreement with the previous experiment.
Under the APG, the separation and reattachment points are
qualitatively reproduced, although the results suggest the lackof resolution in the wall-normal direction.

Drag-reducing performances of the various riblets areintensively investigated by experiments (Bechert et al., 1997).
However, drag-reducing performance of riblets on aircrafts
differs from that measured in an idealized situation inlaboratory. One of the differences is the presence of pressure
gradient. It is reported that the mild adverse pressure gradient(APG) brings favorable effects on performance of riblets(Viswanath, 2002), but the range of values of such APG isremained to be seen. Table 1 lists the review on the effect of
the APG on the riblet performance. In these papers, thepressure gradient is normalized by Clauser pressure gradientparameter
𝑑𝑝 𝛿 ∗
𝛽=
(1)
𝑑𝑥 𝜏𝑤
or
𝑑𝑝 𝜃
𝛽+ =
,
(2)
𝑑𝑥 𝜏𝑤
where 𝛿 ∗ , 𝜃 and 𝜏𝑤 represent displacement thickness,
momentum thickness and wall shear stress, respectively. In
Table 1, we can see that there are large variations among thereports. The minimum 𝛽 which brings worse drag-reducing
performance than zero pressure gradient (ZPG) is reported tobe 0.2, whereas another paper reports that such 𝛽 is between
3 and 3.5. The lower three papers in the table reported that thedeterioration does not occur in the rage of 𝛽 examined. Mostof these reports are experimental studies on the riblet-mounted
airfoil. The concern about such experiments is the difficulty ofadjustment of 𝛽 : the control of 𝑑𝑝/𝑑𝑥 and precise

INTRODUCTION
Reduction of skin friction drag will contribute significantly
to improvement of aerodynamic performance, because about
half of total drag of aircraft at cruising condition is friction
drag. Recently, renewed attention is focused on riblet as arealizable flow control device: a new shape forming methodof riblet on the aircrafts have been developed for practical usethat overcomes long-pending issues (Kurita et al., 2018). As
for introduction of riblet to aircrafts, 2% of total air drag isconfirmed by Airbus flight test (Szodruch, 1991). Effective
spacing of groove of riblets is micron order at cruisingcondition.

Table 1. Review on the drag reduction by riblet under adverse pressure gradient.
 where DR is worse
Authors
 where deterioration starts
max examinedthan ZPG
Sundaram et al. (1996)
1.06
1.5 - 2
2
Subashchander et al. (1996)
1.5
3 - 3.5
3.5
Sundaram et al. (1999)
0.3 (swept wing)
0.4
0.4
Unknown
Squire & Savill (1989)
Unknown
0.05 ()Pulvin & Truong (1990)
0.1
0.2
0.2
Debisschop & Niewstadt (1996)
2.2
Choi (1990)
3.1 ()1

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

𝐿 𝑥 × 𝐿𝑦 × 𝐿 𝑧
𝑁𝑥 × 𝑁𝑦 × 𝑁𝑧
△ 𝑥,△ 𝑦,△ 𝑧

Table2. Domain size, number of grid points, and spatial resolution.
Driver Part
Developing Part
Developing Part(smooth surface)(riblet surface)
125𝜃0 × 100𝜃0 × 38.4𝜃0400𝜃0 × 100𝜃0 × 38.4𝜃0400𝜃0 × 100𝜃0 × 38.4𝜃0200 × 128 × 128
512 × 128 × 128
512 × 128 × 1152
0.625, 0.009~2.86, 0.3
0.78125, 0.002~2.86, 0.3 0.78125, 0.009~2.86, 0.007~0.154

measurement of 𝛿 ∗, 𝜃 and 𝜏𝑤 on the airfoil are difficult. So far,
we cannot grasp even rough estimate of the influence of APG.
In this study, DNS of the flow above the riblet considering
adverse pressure gradient is conducted to investigate theinfluence of APG on the riblet performance. The suction andblowing are imposed at the top of the computational domain
to generate APG. The rescaling/recycling method is used tosimulate the spatially-developing boundary layer. The
advantage of applying blowing and suction at the upperboundary is easiness of adjusting 𝑑𝑝/𝑑𝑥 by controlling thesuction and blowing speed. In addition, 𝛿 ∗, 𝜃 and 𝜏𝑤 can becalculated precisely. In this paper, the results of ZPG and APG
are shown. At the present stage, only the results of the flow
above the smooth surface are shown as APG simulation.

The governing equations are the equation of continuity and
the Navier-Stokes equation of an incompressible flow on a
general curvilinear coordinate system, with the collocatedarrangement of the variables. The velocity-pressure couplingis by the fractional step method. The spatial derivatives are
discretized by central finite differences of second-orderaccuracy. The Adams-Bashforth method of second-orderaccuracy and Crank-Nicolson method are applied for timemarching of the convective and viscous terms, respectively.

OUTLINE OF COMPUTATIONThe numerical simulation of the wall turbulence above thesmooth surface or riblet surface is detailed in the following.
Figure 1 depicts the schematic diagram of the computational
domain. Spatially-developing boundary layer without upper
wall is simulated to set the the blowing and suction at theupper boundary and simulate the adverse pressure gradient.
The domain consists of the driver part (Fig. 1 left) and thedeveloping part (Fig. 1 right). The Reynolds number 𝑅𝑒𝜃
based on the momentum thickness at the inlet of both driverpart and developing part, 𝜃0 , and free-stream velocity at the
inlet, 𝑈∞,0, is set to be 300. The size of the domain, the numberof grid points and the spatial resolution of each part are shownin Table 2. The driver part is set up to provide inflow
conditions for the developing part. Quasi-periodic boundary
conditions based on the rescaling/recycling method by Lundet al. (1998) is applied in the streamwise direction to simulate
an open boundary layer where the fluid is displaced from theupper boundary. The recycle station is located 75 𝜃0
downstream of the inlet. The upper boundary condition ofwall-normal velocity at the driver part and the developing partof ZPG case is based on the streamwise development of thedisplacement thickness (Lund et al., (1998)). The convective
outflow boundary condition is employed at both parts. The
periodic boundary condition is applied in the spanwise
direction, and the no-slip condition is used on the walls. The
velocity profile𝑥𝑐 − 𝑥𝑥𝑐 − 𝑥 2
𝑉𝑡𝑜𝑝 = 𝑉𝑚𝑎𝑥 × √2 × () exp (𝜑 − (
) ),
(3)
𝜎
𝜎
5
9
with 𝑉𝑚𝑎𝑥 = 0.3325 , 𝑥𝑐 = 10 ⁄2 , 𝜎 = 80√2 , 𝜑 = 0.95
(Abe, 2017) is imposed at the upper boundary of thedeveloping part of APG case. Figure 2 shows that the profile
of 𝑉𝑡𝑜𝑝 is blowing and suction velocities. The developing partwith riblet consists of three sections; the smooth surface
section ( 0 ≤ 𝑥 ≤ 9.375𝜃0 ), the transient section ( 9.375𝜃0 ≤𝑥 ≤ 25𝜃0 ) and riblet section ( 25𝜃0 ≤ 𝑥 ≤ 400𝜃0 ). The
triangular riblet with the tip angle of 90 degrees (Fig. 3) is
employed. A total of 36 ribs are arranged in the spanwise
direction. The spacing (pitch) of the riblet groove is set at 17
wall units based on the friction velocity at the inlet, which isreported to be the optimal spacing (Bechert et al., 1997).

Figure 1. Computational domain.

Figure 2. Blowing and suction velocities applied at the upperboundary of the developing part (Abe, 2017).

Figure 3. Computional grid near the riblet.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
RESULTS AND DISCUSSION
Driver Part
Figure 4 represents the time series of the maximum pressurein the computational domain of the driver part. The maximum
pressure increases, and a discontinuous pressure fluctuation
occurs at around 𝑡𝑈∞,0 /𝜃0 = 1500. This may be because thepressure changed to satisfy the continuity equation for theincreased flow rates due to an inappropriate calculation ofmean velocity for rescaling simulation. As a result, at the sametime, the surface-integrated wall shear stress, 𝐷smooth, at thedeveloping part also fluctuates discontinuously (described in
the section of the developing part (ZPG case)). In this paper,
the statistics are sampled at 𝑡𝑈∞,0 /𝜃0 = 400~1500, where
the flow seems to be in quasi-steady state.
Figure 5 represents the turbulence statistics at the inlet ofthe driver part. The profiles are in good agreement with those
calculated using spectral method (Spalart, 1988). Both RMS
and Reynolds stress values are slightly larger than those of thespectral method (Spalart, 1988) at 𝑦 + >50 . In addition, thepeak value of RMS near the wall is different from those ofspectral method (Spalart, 1988). These results may be causedby forming a weighted average of the profiles of the inner andouter layers in the rescaling operation (Lund et al., (1998)).
This could be improved by adjusting the parameters of theweighting function, but we do not pursue further
improvements in this study.
Figure 6 represents the spatial development of momentum
thickness in the streamwise direction. The momentum
thickness of the inlet ( 𝑥/𝜃0 = 0 ) is approximately unity,
which is consistent with the prescribed value. Besides, theslope of 𝜃 in the streamwise direction is about 2.44 × 10−3 inthe present DNS. This value is approximately the same to
2.04 × 10−3 in the literature (Abe, 2017).
Figure 7 shows the pressure coefficient distribution at thewall surface in the streamwise direction. In this paper,pressure coefficient is defined by the following equation:𝑝 − 𝑝0
𝐶𝑝 =
,
(4)
1 2
𝜌𝑈∞,0
2
where 𝑝0 is the time- and spanwise-averaged pressure at 𝑦 = 0
at the inlet of the driver part. The results show that the present
DNS achieves the spatially-developing boundary layer under
the zero pressure gradient.
The results are reasonable until the discontinuous pressure
fluctuation occurs. As mentioned above, the cause of thefluctuations of pressure may be the calculation of the mean
velocity which is used in the rescaling simulation.

(a) Mean streamwise velocity(b) RMS fluctuation(c) Reynolds stress
Figure 5. Turbulence statistics at the inlet of the driver part.

Figure 6. Spatial development of momentum thickness(driver part).

Figure 4. Time series of the maximum pressure (driver part).

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

Figure 7. Pressure coefficient distribution (driver part).

Figure 9. Pressure coefficient distribution.

Developing Part (ZPG case)
As mentioned in the previous section, the pressure
fluctuations occurring in the driver part have a harmful impact
on the physical quantities in the developing part. Therefore, inthis section, as in the driver part, the statistics are sampledfrom 𝑡𝑈∞,0 /𝜃0 = 400 to 1500.
Figure 8 shows the friction coefficient distribution in thestreamwise direction. The empirical model and the previous
DNS results (Spalart, 1988; Abe, 2017) are also plotted forcomparison. Although only in a small range (𝑅𝑒𝜃 =300~550), the friction coefficient distribution of the present
DNS is close to the empirical modeland the results of theprevious studies. Therefore, the time-averaged wall shear
stress is quantitatively reproduced.
Figure 9 represents pressure coefficient distribution on thewall surface. This result shows that the present DNS achieves
the spatially-developing boundary layer under the ZPG evenin the developing part. We can conclude that the simulation in
the developing part under the ZPG is reliable in the renge of𝑡𝑈∞,0 /𝜃0 = 400~1500 . Figure 10 represents the friction
drag, 𝐷smooth and 𝐷rib, calculated by the surface integral of
the wall shear stresses on the smooth and riblet surfaces,
respectively. The friction drag on the riblet surface is always
smaller than that on the flat surface, which indicates theeffectiveness of the riblet. The drag reduction rate calculatedby the time-averaged 𝐷smooth and 𝐷rib equals about 3.1%,
which is almost the same as the experimental value (3.4%) fora triangular straight riblet with a groove spacing of 17 wall
units (Bechert et al., 1997). This implies that the DNS of theflow above the riblet surface under ZPG is also appropriately
conducted.

Figure 10. Comparison of the time series of the friction dragon the smooth and riblet surfaces under ZPG(developing part).

Developing Part (APG case)At the present stage, the results of the flow above thesmooth surface under the APG are presented as a referencecase of the riblet case under the APG. Figure 11 represents thefriction drag, 𝐷smooth,ZPG and 𝐷smooth,APG, calculated by thesurface integral of the wall shear stresses under the ZPG and
APG, respectively. The friction drag under the APG is always
smaller than that under the ZPG, which implies the existenceof separation points. Figure 12 shows the friction coefficient
distribution in the streamwise direction. The separation andreattachment points (i.e. 𝐶𝑓 = 0) are in reasonable agreementwith the previous DNS result (Abe, 2017). However, thevalues in the present DNS are lower than those of the previous
study (Abe, 2017) in other regions. This may be because thedomain length of the 𝑦 direction, 𝐿𝑦 , is 25% larger than thatin the previous DNS: the effect of the blowing and suction at
the upper boundary reaches less on the wall in the present DNS.

Figure 8. Friction coefficient distribution.
Figure 11. Comparison of the time series of the friction on thesmooth surface of ZPG and APG cases(developing part).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
method. The development of the friction coefficient, the dragreducing performance and other statistics under the ZPG were
simulated quantitatively until unphysical discontinuouspressure fluctuation occurred. The distributions of the pressure
coefficient and the friction coefficient under the APG werealso calculated. The separation and reattachment points are
simulated qualitatively, but the results implied the lack of theresolution in the wall-normal direction.
As the next task, the calculation of the mean streamwise
velocity must be revised in order to fix the unphysical pressure
fluctuation in the driver part. In the developing part, gridrefinement in the wall-normal direction is needed. Then, we
will grasp the relationship between the value of Clauserpressure gradient parameter 𝛽 and the strength of the blowing
and suction at the upper boudary. Finally, the drag reduction
effect under the APG will be investigated.

Figure 12. Friction coefficient distribution of APG case(developing part).
Figure 13 represents the pressure coefficient distribution on
the wall. The maximum value of 𝐶𝑝 is smaller than that of theprevious DNS. In addition, the plateau of 𝐶𝑝 (𝑥 =100~200)
observed in the previous DNS does not appear. As mentioned
above, these results are understandable considering that 𝐿𝑦 islarger than that of the previous DNS. However, the fact that
𝐶𝑝 does not recover to zero in the outflow indicates that theresolution is not sufficient for 𝐿𝑦 . Grid convergence of the
distributions of 𝐶𝑓 and 𝐶𝑝 also must be confirmed. Figure 14
shows the 𝛽 distribution up to 𝑥 = 60. The current 𝛽 is toolarge compared to the values in Table 1. This implies that thestrength of the suction in the present study should be
weakened considering an aircraft under cruising condition.

REFERENCES
Abe, H., 2017, "Reynolds-number Dependence of Wallpressure Fluctuations in a Pressure-induced Turbulent
Separation Bubble", Journal of Fluid Mechanics, Vol. 833, pp.
563-598.
Bechert, D. W. et al., 1997, "Experiments on Dragreducing Surfaces and their Optimization with an Adjustable
Geometry", Journal of Fluid Mechanics, Vol. 338, pp. 59-87.
Choi, K. S., 1990, "Effect of Longitudinal Pressure
Gradient on Turbulent Drag Reduction with Riblets", in"Turbulence Control by Passive Means", Coustols, E. ed.,Kluwer Academic, Dordrecht, Netherlands, pp. 109-121.
Debisschop, J. R. and Niewstadt, F. T. M., 1996,"Turbulent Boundary Layer in an Adverse Pressure Gradient:
Effectiveness of Riblets", AIAA Journal, Vol. 34, No. 5, pp.
932-937.
Kurita, M. et al., 2018, "Flight Test of a Paint-riblet for
reducing skin-friction", AIAA Paper, No. 2018-3005.
Lund, T. S. et al., 1998, "Generation of Turbulent Inflow
Data for Spatially-Developing Boundary Layer Simulations",
Journal of Computational Physics, Vol. 140, pp. 233-258.
Pulvin, Ph. and Truong, T. V., 1990, "Riblets in Internal
Flows with Adverse Pressure Gradients", Proceedings of the
IUTAM Symposium on Structure of Turbulence and Drag
Reduction, pp.569-576.
Spalart, P. R., 1988, "Direct Simulation of a Turbulent
Boundary Layer up to 𝑅𝜃 Equals 1410", Journal of Fluid
Mechanics, Vol. 187, pp. 61-98.
Squire, L. C. and Savill, A. M., 1989, "Drag
Measurements on Planar Riblet Surfaces at High Subsonic
Speeds", Applied Science Research, Vol. 46, pp. 229-243.
Subashchander et al., 1996, "Drag Reduction due to
Riblets on a GAW(2) Airfoil", Journal of Aircraft, Vol. 36,
No. 5, pp. 890-892.
Sundaram et al., 1996, "Viscous Drag Reduction Using
Riblets on a NACA 0012 Airfoil to Moderate Incidence",
AIAA Journal, Vol. 34, No. 4, pp. 676-682.
Sundaram et al., 1999, "Viscous Drag Reduction Using
Riblets on a Swept Wing", AIAA Journal, Vol. 37, No. 7, pp.
851-856.
Szodruch, J., 1991, "Viscous Drag Reduction on
Transport Aircraft", AIAA Paper, No. 91-685, pp. 1-8.
Viswanath, P. R., 2002, "Aircraft Viscous Drag Reduction
Using Riblets", Progress in Aerospace Sciences, Vol. 38, pp.
571-600.

Figure 13. Pressure coefficient distribution of APG case(developing part).

Figure 14. Clauser pressure gradient parameter 𝛽 distributionof the region under APG (developing part).

CONCLUSION
Direct numerical simulations (DNS) of spatiallydeveloping wall turbulence above riblet and smooth surfaces
under the zero-pressure gradient (ZPG) and adverse-pressure
gradient (APG) were performed using the rescaling/recycling

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