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

ULTRA-FINE ROUGHNESS EFFECT ON TRANSITION DELAY USING
DIRECT NUMERICAL SIMULATION

Shingo Hamada
Department of Aerospace Engineering,Tohoku University
6-6 Aramaki Aza Aoba, Aoba-ku,
Sendai, Miyagi 980-8579, Japan
shingo.hamada.s5@dc.thoku.ac.jp

Aiko Yakeno
Institutute of Fluid Science,
Tohoku University
2-1-1 Katahira, Aoba-ku,
Sendai, Miyagi 980-8577, Japanaiko.yakeno.b4@dc.tohoku.ac.jp

Shigeru Obayashi
Institutute of Fluid Science,
Tohoku University
2-1-1 Katahira, Aoba-ku,
Sendai, Miyagi 980-8577, Japans.obayashi@tohoku.ac.jp

ABSTRACT

tributed roughness, which is under 9[µm], that was below theviscous sublayer in a turbulent state. A reduction in frictionaldrag and turbulent fluctuation intensity of up to 1.9[%] wereachieved. The detailed mechanism of the drag reduction has
not been clarified. We believe that is based on a different
mechanism from that of Riblets, which is based on the suppression of spanwise velocity fluctuations and letting streamwise vortices away from a wall surface. In addition, the useof a micron-sized fiber surface has been also observed to beeffective for the transition delay (Kikuchi et al. (2004)). Here,
we call them the Distributed Micro Roughness (DMR) for a
drag reduction.

Direct numerical simulations (DNSs) were conducted toanalyze the effect of a fine sand-grind rough surface on alaminar-turbulent transition. While wall roughness has often been discussed to increase drag in a turbulent flow state,
this study focuses on its drag reduction effect. We applied
the Volume Penalization Method (VPM) and zonal method toresolve the ultra-fine sand-grind rough surface on a flatplate,in the three-dimensional DNSs. We confirmed the TollmienSchlichting (T-S) wave transition phenomena has been modified on the sand-grind rough surface. T-S wave on the roughsurface broke down to the three-dimensional flow structure thefastest among tested cases, but the turbulent kinetic energy was
kept small the most.

Tameike et al. investigated the effect of a small wavyroughness on the natural transition using two-dimensional direct numerical simulations (DNSs) Tameike et al. (2021).
It was confirmed that the small wavy roughness kept the
Tollmien-Schlichting (T-S) vortices small downstream delaying a transition at a specific roughness wavelength. Recently,
the phenomenon has been confirmed by the three-dimensional
simulations by using an artificial disturbance to cause a T-S
wave upstream Hamada et al. (2020). In this study, as the second step, we analyze the flow of the more practical and complicated roughness, such as a sand-grind roughness as tested inexperiments.

INTRODUCTION
The demand for aircraft transportation is increasing, andthe demand for CO2 emission reduction is becoming more andmore stringent. There is a need to improve fuel efficiency inways such as drag reduction. The most significant aerodynamic drag acting on an aircraft is a frictional drag. To reduce
this, laminar flow control is used to maintain the laminar flow
state longer, and turbulence control is used to reduce frictionaldrag in a turbulent flow state. Riblets, which are micro-grooves
that imitate shark skin, have been studied for many years andhave been shown to reduce friction drag by approximately 8%(Walsh et al. (1989); Walsh (1992)). However, there are someproblems in practical application, such as the maintenance costfor contamination and the fact that the riblets lose their effects
if they are not parallel to the streamwise direction.
The possibility of reducing friction drag on rough surfaces with the finer roughness has been referred to by Tani
(Tani (1988)), which height is below the viscous sublayer(Rek < 6). Such small roughness has been considered negligible from a flow dynamical aspect. Oguri & Kohama (1996)
conducted wind tunnel experiments with the micron-sized dis-Though effects of sand-grind roughness in a channel havebeen investigated experimentally and numerically (Nugroho
et al. (2021); Jelly & Busse (2019); Kuwata & Nagura (2020)),
most of the studies are concerned with the effects in a fullydeveloped turbulent flow and the increase of friction drag.
Their roughness heights are normally higher than the viscous
sublayer, and there are few studies on its use for transition delay and turbulent frictional drag reduction.
Therefore, the purpose of this study is to clarify the effectof such ultra-fine distributed roughness, which effect has beenreferred to and confirmed in experiments to delay the transition and reduce turbulent drag (Tani (1988); Oguri & Kohama(1996); Kikuchi et al. (2004)). In this study, we analyze the ef-

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
fect of sand-grind rough surfaces on a laminar-turbulent transition via the T-S wave on a flatplate.
Fluid

METHODS
Computational Setup

0

In this study, we demonstrate the T-S wave transition
on a flatplate numerically, by using the in-house LU-ADI
three-dimensional compressible Navier-Stokes solver (originally Fujii & Obayashi (1987)). A sixth-order accurate compact scheme is used for the spatial discretization (Lele (1992)),
and the three-step TVD Runge-Kutta method is used for time
integration (Chi-Wang & Osher (1988)). The simulation
schemes are the same as Tameike et al. used in their study(Tameike et al. (2021)).

Roughness height

Solid WallZ
X

Figure 2.
wall.

The Reynolds number based on the boundary layer thickness δs at the inlet boundary is set to 3535. The Mach number
is set to 0.2 to obtain sufficiently large time increments within
which compressibility effects can be neglected.
The boundary conditions are shown in Figure 1. The
time-mean velocity distributions at the inlet boundary are
given by the Blausis solution. At the inlet boundary, an artificial disturbance is added in the wall-normal velocity to induce
a T-S wave. The periodic condition is applied at the spanwise
boundaries. The velocity and pressure are set as free at theoutlet boundary. At the far boundary in the wall-normal direction, the pressure and the streamwise velocity are fixed, andthe wall-normal and spanwise velocities are set as free.

Mask function shape and Grid resolution near the∂ (ρu j )1
∂ ρ ∂ (ρu j )
=−
,
+
−1 χ
∂t
∂xj
φ
∂xj
∂ (ρu j ) ∂ (ρui u j )∂ p ∂ τi j χ+=−+− (ui −U0,i ),∂t∂xj∂ xi ∂ x j
η

∂e
∂ 
+
(e + p)u j∂t ∂ x j∂ (ui τi j )
∂
∂T
χ
κ
−
=
+
(T − T0 ),
∂xj
∂xj
∂xj
ηT

(1)
(2)
(3)

where ρ is the density of the fluid, ui is the flow velocity
U0,i is the solid wall velocity and defined as 0. φ is porosity and is set to 1. κ is the thermal conductivity. τi j is theviscous stress tensor. ν is the permeability and is taken to be
sufficiently small. The νT is the thermal permeability and isassumed to be the same as ν. e is the total energy, p is thepressure, T is the temperature, and T0 is the solid wall temperature. Since temperature change is negligible in this study, thesolid wall temperature is assumed to be the same as the mainstream. The difference in the temperature boundary condition
is not expected to be apparent for low Mach number flows, asin this study. Since χ is a mask function and the VP method
does not require the generation of a body-fitted grid, an arbitrary shape of a solid wall can be reproduced by defining amask function.
Figure 1.

Schematic of the geometry.
χ(x) =

1 (Rigidwall)
0 (Fluid)(4)

As shown in Figure 2, near the surface roughness, theroughness is resolved with a sufficient number of grid points inboth streamwise direction and wall-normal direction. The resolution is approximately 20 points or more per wavelength ofroughness. The VPM is offset from the wall-normal origin andthe edges of the computational grid to allow for the number ofgrid points inside the wall. The VPM tends to accelerate thetransition slightly, possibly because the surface profile is simulated by an orthogonal mesh, which is more liable to generatetiny disturbances from the curved surface.

Volume Penalization Method
In this study, the Volume Penalization Method (VPM) is
used to simulate a three-dimensional rough surface (Liu &Vasilyev (2007), Komatsu et al. (2016)). The VP method forcompressible flow was proposed by Liu and Vasilyev, and theno-slip condition can be given by adding a penalty term to theNavier-Stokes equations. The VP method can be used to analyze walls of arbitrary geometry without the need to generatea grid fitted to a solid wall.

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1.

Zonal Method
To reproduce the shape of a rough surface by an orthogonal mesh, sufficient mesh resolution is required. However, afine mesh width over the entire computational domain would
require a huge computational cost. Therefore, in this study,
a nested grid was used to capture turbulence with high accuracy while minimizing the computational cost. Two zonal grid
distributions are shown in Figure 3.

parameters of rough surfaces.

Sk

ES

hm

hrms

0.395

0.125

0.092

0.028

RESULTS and DISCUSSION
Turbulent kinetic energy
We show a visualization of the iso-surfaces of the second
invariant of the velocity gradient tensor in the instantaneous
field of the three-dimensional simulation with a sand-grind
surface in Figure 5. The flow structure around each roughness
seems to be sufficiently resolved by the present simulation. We
compare the difference in turbulent kinetic energy (TKE) in the
turbulent transition among the two-dimensional wavy surface,
the sand-grind rough surface, and a smooth surface. Figure 6
shows the difference in the spanwise and time-averaged turbulent kinetic energy (TKE, u′i u′i ) in x-z plane, (a) sand-grindroughness, (b) wavy roughness (VP method), (c) wavy roughness (Body Fitted grid) and (d) flat plate. And, figure 7 shows
the maximum TKE in each wall-normal directions. It was
found that the sand-grind rough surface suppressed TKE themost.

In the small zone (red), the grid resolution in the streamwise direction is doubled; the small zone has a viscosity
scale of Lz+ ≈ 50 in the height direction, which captures
the viscous sublayer. Global zone (black) has (Nx , Ny , Nz )= (1454, 151, 284) and the computational domain lengths are(Lx, Ly, Lz) = (56.6, 11.3, 28.3) for global zone.
The small zone has (Nx , Ny , Nz ) = (2843, 201, 137),
and the computational domain lengths are (Lx , Ly , Lz ) =(50.9, 11.3, 0.59). However, Ny = 201 and Ly = 14.1 in the
spanwise direction were used for the smooth surface and thetwo-dimensional wavy rough surface as a comparison. For
the calculation of sand-grind rough surfaces, Ly = 11.3 wasused to keep the computational domain in the span direction
as small as possible, since the higher resolution is needed toreproduce the fine irregularities of the rough surfaces.

Instantaneous flow
The instantaneous snapshots of the three flow fields for
one period of the T-S wave are shown in Figure 8. This represents how the T-S roll vortex is induced at the inlet boundary and propagates in the streamwise direction, and eventually
breaks to become the three-dimensional structure. Comparing the location of the roll vortex breaks down, the sand-grindrough surface is the fastest among the three cases.

Sand-grind roughness shape
The effective slope (ES) and skewness coefficient (Sk) of
the present sand-grind roughness are calculated based on thefollowing equations (Eq. (5) and Eq. (6)). They are summarised in Table 1.

1∂ h(x, z)dxdz.
Lx Lz x z∂xZ Z1(h(x, z) − hm )3 dxdz.
Sk = 3hrms Lx Lz x z

Phase-averaged decomposition

Z Z

ES =

(5)

We apply the phase-averaged decomposition to clarify adifference in the transition process, based on the T-S wave period. The turbulent kinetic energy k′ is divided into the T-S
wave component k̃ and the three-dimensional component k′ ,as shown in the following equation.

(6)

As mentioned above, the effect of roughness in developed turbulent flows can be clarified to some extent by the skewness
and effective slope. The skewness Sk is the bias between thepeaks and valleys of the roughness surface. The effective slope
ES is the ratio of the roughness height to the roughness interval. In this study, skewness Sk = 0.395 and effective slope
ES = 0.125. This parameter is included in the range of theanalysis by Kuwata & Nagura (2020) et al. This value of
Sk = 0.5 is close to that of ES = 0.1, and a downward shift(∆U + = 3 ∼ 4) in the mean velocity distribution is confirmed.
Furthermore, when ES is relatively small, the difference in∆U + by the value of Sk is reported to be small, suggestingthat the difference of Sk in this study is not so important.

k = k̄ + k′ ,(7)′′

k = k̄ + k̃ + k .

(8)Comparing the turbulent kinetic energy k on the threerough surfaces, that on a sand-grind surface has kept small
downstream the most (Figure 9, 10). From the decomposition
statistics, the sand-grind roughness decreases the T-S wave
kinetic energy more than the other two surfaces, although it
increases the three-dimensional kinetic energy the most upstream but downstream.
Intheinstantaneous
visualization,
threedimensionalization occurs fastest on the sand-grind surface.
However, that break seems to make the T-S motion weaken,
which results in the total TKE being small downstream.
The T-S kinetic energy was also suppressed on the wavyrough surface, however, the TKE was larger than that of thesand-grind rough surface. That is because of eddies generated
at the crests of the wavefronts.

The mean height of the sand-grind roughness, hm , is0.092 to reproduce the sand-grind on a scale completely corresponding to that of our wind tunnel experiment Suzuki (2021).
Based on the roughness height, a single sand-grind roughness
data set is not sufficient to cover the full computational domain, so a repeating pattern was used as shown in Figure 4.
The repeated boundaries of the rough surfaces were connectedsmoothly. Although the viscosity unit cannot be strictly determined in the transition process, the viscosity scale is calculated
from the flow downstream, which is converted to the viscosity
scale to give an average height of h+m = 15.5.

CONCLUSION
In this study, the effect of a sand-grind rough surface
on the laminar-turbulent transition was investigated using the3

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

Figure 3.

Figure 4.

Zonal grid distribution for wall turbulence simulation.

Geometry of the rough surface. The dashed line indicates a single unit tile.

Figure 5. Instantaneous iso-surfaces of the second invariant of velocity gradient tensor; Q = 0.001 colored by the streamwise velocity
u in the x-y plane.

Figure 6. The comparison of turbulent kinetic energy. (a) sand-grind roughness, (b) wavy roughness (VP method), (c) wavy roughness(Body Fitted grid) and (d) flat plate.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
dimensional transition by direct numerical simulation. ,vol. 65. APS.
Jelly, T. & Busse, A. 2019 Reynolds number dependence of
reynolds and dispersive stresses in turbulent channel flow
past irregular near-gaussian roughness. International Journal of Heat and Fluid Flow 80, 108485.
Kikuchi, S., Shimoji, M., Watanabe, H. & Kohama, Y. 2004
Control of bypass transition for textile surface. JSME International Journal Series B Fluids and Thermal Engineering
47 (4), 777–785.
Komatsu, R., Iwakami, W. & Hattori, Y. 2016 Direct numericalsimulation of aeroacoustic sound by volume penalization
method. Computers & Fluids 130, 24–36.
Kuwata, Y. & Nagura, R. 2020 Direct numerical simulationon the effects of surface slope and skewness on rough-wall
turbulence. Physics of Fluids 32 (10), 105113.
Lele, S.K. 1992 Compact finite difference schemes with
spectral-like resolution. Journal of Computational Physics
103 (1), 16–42.
Liu, Q. & Vasilyev, O.V. 2007 A brinkman penalization
method for compressible flows in complex geometries.
Journal of Computational Physics 227 (2), 946–966.
Nugroho, B., Monty, JP, Utama, IKAP, Ganapathisubramani,
Bharathram & Hutchins, Nicholas 2021 Non–type behaviour of roughness when in-plane wavelength approaches
the boundary layer thickness. Journal of Fluid Mechanics
911.
Oguri, E. & Kohama, Y. 1996 Drag reduction by micro-sized
distributed surface geometry on a flat plate. Transactions of
the JSME (in Japanese), 62 (597), 1754–1761.
Suzuki, S. 2021 Effect of wall roughness on wall turbulenceby using wind tunnel experiments (in japanese).
Tameike, H., Yakeno, A. & Shigeru, O. 2021 Influence ofsmall wavy roughness on flatplate boundary layer naturaltransition. Journal of Fluid Science and Technology 16 (1),
JFST0008–JFST0008.
Tani, I. 1988 Drag reduction by riblet viewed as roughness
problem. Proceedings of the Japan Academy, Series B
64 (2), 21–24.
Walsh, M. J. 1992 Drag characteristics of v-groove and transverse curvature riblets. AIAA Journal 30 (4), 1119–1122.
Walsh, M. J., Sellers, W. L. & McGinley, C. B. 1989 Riblet
drag at flight conditions. Journal of Aircraft 26 (6), 570–
575.

0.001
sandpaper roughness
wavy roughness (VP)
wavy roughness (BF)
smooth(krms)max

0.00080.00060.00040.0002

0

0

10

20

30

40

50

x

Figure 7. The maximum value of the turbulent kinetic energy
k along the x directon

VPM and zonal method in the direct numreical simulations.
The two key conclusions were obtained: first, the T-S wave
was weakened on two roughness surfaces, not only the wavyrough surface, but also the more complicated random sandgrind rough surface. Second, the breaking up of the T-S wave
was promoted on the sand-grind rough surface, but the total
turbulent kinetic energy was suppressed the most among tested
cases.

ACKNOWLEDGMENTS
Numerical simulations were performed on the ”AFINITY” supercomputer system at the Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University. This research is partially based on a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.

REFERENCES
Chi-Wang, S. & Osher, S. 1988 Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journalof Computational Physics 77 (2), 439–471.
Fujii, K. & Obayashi, S. 1987 Navier-stokes simulations oftransonic flows over a practical wing configuration. AIAA
Journal 25 (3), 369–370.
Hamada, S., Yakeno, A., Obayashi, S. & Nugroho, B.
2020 Small wavy roughness effect on ts wave and three-512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 8. Instantaneous flow fileds: (a)flat plate (b)sand-grind roughness (c)wavy roughness. The isosurface is the second invariantof the velocity gradient tensor Q = 0.001 with countour color u from 0.0 to 0.2.

Figure 10. The contour maps of phase-, tiem-andspanwiseaveraged turbulent kinetic energy. The three-dimensional
component. (a) smooth surface, (b) sand-grind roughness, (c)wavy roughness.

Figure 9. The contour maps of phase-, tiem-andspanwiseaveraged turbulent kinetic energy. The T-S wave component.
(a) smooth surface, (b) sand-grind roughness, (c) wavy roughness.

6