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

DNS STUDY OF TURBULENT FLOW THROUGH A CHANNELROUGHENED BY CIRCULAR-ARC RIBS
Weijian Xiong, Jinglei Xu
Jiangsu Province Key Laboratory of Aerospace Power System
Nanjing University of Aeronautics and Astronautics
Nanjing, 29th Yudao Street, China
xwj@nuaa.edu.cn, xujl@nuaa.edu.cn

S. V. Mahmoodi-Jezeh, Bing-Chen Wang
Department of Mechanical Engineering
University of Manitoba
Winnipeg, MB, R3T 5V6, Canada
mahmoosv@myumanitoba.ca, BingChen.Wang@umanitoba.ca

ABSTRACT

were massively influenced by the roughness in the outer region,
which further led to a reduction in the streamwise characteristic length scales of the flow. For the d-type roughness flow, it
is observed that most of the previous studies appearing in theliterature utilitzed transverse square or triangular-shaped ribs
(Coleman et al. 2007; Leonardi et al. 2007; MacDonald etal. 2018). For the square or triangle ribs, the flow separation is typically triggered by the sharp edges of ribs. This is
in sharp contrast to the circular-arc ribs under current study,
whose geometry is contoured continuously. Hence, the flow
over circular-arc ribs exhibits a more complex separation behavior, which further affects the vortex dynamics and coherent
flow structures of the turbulent boundary layer.
In view of these interesting physical features, we aim
at performing a comprehensive high-fidelity DNS study ofcircular-arc-rib roughened channel flows of different pitch-toheight ratios. The results of the ribbed channel flows are compared against those of a smooth channel flow. Our study focuseson the effects of the pitch-to-height ratio on the position andcharacteristics of flow separation, statistical moments of the
turbulent velocity field, budge balance of Reynolds stresses, aswell as flow anisotropy. Also, we aim to investigate the characteristic scales and organization of coherent structures in theseparation regions in both spatial and temporal spaces.

Turbulent flows through a channel roughened withcircular-arc ribs of different pitch-to-height ratios (P/H = 3.0,
5.0 and 7.5) at a fixed Reynolds number of Reb = 5600 are
studied using direct numerical simulations (DNS). It is interesting to observe that in contrast to the classical channel
flow with square bars, the flow separation point in a channel roughened with circular-arc ribs varies. The influencesof the pitch-to-height ratio on the flow separation, statistical
moments of the velocity field, budget balance of Reynolds
stresses and anisotropy of turbulence stresses are investigated.
The dynamics of coherent structures are studied by examining characteristics of the instantaneous velocity field, swirlingstrength and spatial-temporal two-point auto-correlations. It
is also observed that the streamwise characteristic length of
hairpin structures becomes increasingly shortened as the pitchto-height ratio decreases.

INTRODUCTION
Turbulent boundary-layer flows over walls roughened bytransverse circular-arc ribs have important applications to engineering problems, such as turbine blade cooling, heat exchangers and ship hull (Nagano et al. 2004). In this class offlows, large separation region arises from the continuous arc
surfaces, and the position of the flow separation points is sensitive to the pitch-to-height ratio associated with eddy motionsof different spatial and temporal scales. These features greatly
complicate the flow physics, making turbulence characteristics
considerably different from those of a classical 2-D rough-wallboundary-layer flow over a surface with square bars.
In the current literature, extensive experimental and numerical simulations are dedicated to investigations into thephysics of 2-D rough-wall boundary-layer flows. Krogstad etal. (1992) studied the distributions of longitudinal turbulence
intensity and Reynolds shear stresses over a rough wall in bothinner and outer regions of a boundary layer. They showed that
ejection and sweep events near the rough surface are dominantphysical phenomena in terms of not only their intensity but
also their frequency of occurrence. Krogstad et al. (1993)
also conducted a wind-tunnel experiment to study the effectsof the k-type roughness on the coherent structures of a turbulent boundary layer. They found that turbulence structures

TEST CASE AND NUMERICAL ALGORITHM
Figure 1 shows the schematic diagram of the computational domain and coordinate system. Here, x, y and z denotethe streamwise, wall-normal and spanwise coordinates, and u,v and w represent the corresponding velocity components, respectively. The streamwise length is set to L x = 6δ, whereδ is the half channel height. In spanwise direction, the domain length is set to Lz = 2πδ. The pitch and height of thecircular-arc ribs are P and H, respectively. For all three ribbed
channel flow cases considered, the radius of the arc is kept
constant, i.e. R = 0.2δ. Cases P1, P2 and P3 are comparedhere to investigate the effects of the pitch-to-height ratio (for
P/H = 3.0, 5.0 and 7.5, respectively) with a common blockage
ratio of Br = H/2δ = 0.1. Due to the fixed domain size, cases
P1, P2 and P3 consist of 10, 6 and 4 periods, respectively. The
Reynolds number is fixed at Reb = 2δUb /ν = 5600 to keepa constant mass flow rate, where Ub denotes the bulk mean

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1.
Ly = 2 d
Flow

y

H

x

R

z
x¢

Lz = 2pd

Geometry and flow parameters of the test cases

Test case

Reb

Reτ R

Reτ S

Cp R

Cf R

P1

5600

288

208

0.01430.00539

P2

5600

369

222

0.03360.00181

P3

5600

406

227

0.04090.00177P
Lx = 6 dFigure 1. Schematic diagram of the computational domain
and coordinate system. The origin of the absolute coordiante
system [x, y, z] is located at the inner bottom corner of thewindward face of the first rib, and the origin of the relative
streamwise coordinate x 0 is located at the windward face ofeach rib, which is defined to facilitate the analysis of a repeatedrib period.

(a)streamwise velocity. In order to identify the rib effects, anadditional case of a smooth channel flow that follows Kim etal. (1987) has been simulated for the purpose of comparison.
The flow field is fully developed and periodic boundary conditions are applied to both streamwise and spanwise directions.
A no-slip boundary condition is imposed on all solid surfaces.
The DNS is performed using a spectral-element code socalled “Semtex” contributed by Blackburn & Sherwin (2004).
This code is developed using C++ and FORTRAN programming languages, and parallelized using message passing interface (MPI) libraries. All physical quantities are expanded
into the spectral space using Fourier series in the spanwise (z)direction, and 240 Fourier modes are used in each test casewith a grid resolution of 5.3 ≤ ∆+z ≤ 5.9 (measured in wall
unit based on uτS , the average friction velocity over the top
smooth wall). Quadrilateral spectral-elements are used for discretization in the streamwise and vertical directions. For eachelement, a 4th-order Gauss-Lobatto-Legendre Lagrange polynomial is used. Case P3 has the finest grid resolution with
Nx × Ny × Nz = 1200 × 161 × 240 nodes. The streamwise grid
resolutions varies for 2.6 ≤ ∆+x ≤ 4.4, and the maximum valueof ∆+y of the first grid off the walls is 0.20 in all tested cases.
To satisfy the demanding requirement of DNS on grid resolutions, the ratio of the grid size over Kolmogorov
p scale η islimited to 2.17 ≤ max(∆/η) ≤ 3.43, where ∆ = 3 ∆ x ∆y ∆z . A
second-order time-splitting scheme with three sub-steps is used
for time integration. Overall, the discrete solution to the governing equations based on the Semtex code is highly accurate
and the DNS results are of a spectral accuracy. The computercode has been used by our group for conducting DNS studiesof rotating elliptical pipe flows (Rosas et al. 2021), and squarechannel flows (Fang et al. 2017).
For each simulated ribbed flow case, 600 instantaneous
flow fields over 35 large-eddy turnover times (LETOTs, defined as δ/uτR ) were used for collecting statistics once theflow becomes fully-developed and statistically stationary after
a precursor simulation. Here, uτR denotes the average friction
velocity over the bottom rib-roughened wall. The simulations
were conducted using the WestGrid (Western Canada ResearchGrid) supercomputers, and approximately 4 TB data have been
generated for all three simulated cases. For each test case,approximately 100,000 CPU hours were spent for solving thevelocity field and for collecting the flow statistics.

(b)

(c)

Figure 2. Time-averaged streamlines with contours of nondimensional mean streamwise velocity hu/Ub i for all test cases.
(a), (b) and (c) for cases P1, P2 and P3, respectively.

streamwise wall friction velocities of the roughened bottom
wall and smooth top wall, respectively. The method for
calculating the value of uτR follows the approach of Ismail et al. (2018) for a 2-D ribbed turbulent channel flow,
2 = D + D . Here, D and D are the mean pressure
viz. uτRpνpν
and viscous drags on the rough wall, determined as∫1
Dp =hpidy ,
(1)
ρL x ∂Ω R
and



∫
ν
∂hui ∂hvi
∂hui
Dν =
+
dx − 2dy,(2)
L x ∂Ω R∂y∂x∂x
respectively. In these equations, subscript ‘∂Ω R ’ denotes
the boundary of the rough bottom wall, and h·i denotes averaging over time and over the spanwise direction. In table 1, the pressure and viscous drag coefficients are definedas CpR = D p /(Ub2 /2) and C f R = Dν /(Ub2 /2), respectively, forthe rough wall. From table 1, it is clear that as the pitch-toheight ratio increases from 3.0 to 7.5, both values of ReτR andReτS increase monotonically, indicating an increase of friction
drags on both sides of the channel. As the pitch-to-height ratio
increases, it is interesting to observe that the trend of CpR and
C f R is opposite of each other. Furthermore, the ratio of CpR
to C f R increases monotonically, as a result of augmentation of
the recirculation bubble size behind the ribs.
The mean flow structures of the test cases are shown infigure 2 using the contours of non-dimensional mean streamwise velocity hui/Ub and the time-averaged streamlines. The
significant mean flow vortices are marked with A, B and C,

RESULTS AND DISCUSSIONS
Table 1 summarizes key flow parameters for three ribbed
flow cases. Here, two friction Reynolds numbers (ReτR =δuτR /ν and ReτS = δuτS /ν) are defined based on the mean

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
P1

which represent the large recirculation bubbles behind the ribs
(A), the small secondary vortices located at the corners of thewindward and leeward sides of the ribs (B and C, respectively).
D and R indicate the detachment and reattachment points of themean flow, respectively. Accordingly, θ D and θ R denote theangles corresponding to the positions of the detachment andreattachment points of the mean flow on the circular-arc ribs.
The detachment (D) and reattachment (R) points are defined as
the positions where the local mean wall shear stress vanishes,
i.e. hτw i = 0. From figure 2, it is clear that with an increasing
value of P/H, the reattachment point moves upstream progressively, while the detachment point is stable. As a result, thedistance between points D and R increases monotonically andthe flow accelerates near the rib crest as the pitch-to-height ratio increases. This leads to a larger pressure difference between
the windward and leeward of a rib, which further increases theform drag dramatically. The mean flow pattern of case P3 discussed here is typical of the k-type rough-wall flow. For cases
P1 and P2, the mean flow skims over the ribs, which is typicalof a d-type rough-wall flow.
To assess the effect of the pitch-to-height ratio on theseparation vortices, flow separation indicator introduced by
Simpson (1981) can be studied, which is defined as
γ p = hH(u)i ,
(3)
where
(
1, if u > 0
H(u) =(4)0, otherwise

P2
P3

P/H
p

P/H

(a) profiles of γ p along the rib sur- (b) profiles of γ p within the cavity
facebetween two ribs

Figure 3. Distribution of the mean flow separation indicatorγ p on the bottom wall. ‘’, ‘×’ and ‘4’ in panel (a) denote
the reattachment, detachment point of the mean flow and theincipient detachment point, respectively. The vertical red, blueand black dashed lines in panel (b) demarcate the windward
corner of cases P1, P2 and P3, respectively.

is the unit step function of the instantaneous streamwise velocity u. The value of γ p is calculated as the first grid off therough bottom wall, and the flow features separation if γ p < 0.5.
Figure 3 shows the azimuthal and streamwise distribution ofγ p on the bottom wall. In figure 3(a), the reattachment anddetachment points of the mean flow (R and D in figure 2, respectively) are marked using ‘’ and ‘×’, respectively, and theincipient detachment (ID) points (where γ p = 0.99) are marked
using ‘4’. The displacement thickness of the boundary layer
begins to increase rapidly after the ID point (Simpson 1989).
From figure 3(a), it is obvious that the values of γ p for all cases
are all above 0.5 at the reattachment point, which is in sharp
contrast to that of the classical backward-facing step flow inwhich γ p is less than 0.5 at the reattachment point (Simpson
1989). The value of γ p over 0.5 at the reattachment point
indicates the existence of a large-scale intermittent backward
flow around this position due to the flow impingement on therib. As the value of P/H increases, the value of γ p at thereattachment point deviates from 0.5 gradually, showing theenhanced flow impingement onto the rib surface. With anincreasing value of P/H, the ID point and detachment pointof the mean flow both move upstream on the rib surface, andthe backward flow becomes much weaker associated with theoccurance of the lowest value of γ p and the proportion of rib
surface of γ p < 0.5. Consequently, as the value of P/H increases, the angle between ID point and detachment point of
the mean flow increases monotonically, which is ∆θ = 9.49◦ ,13.47◦ and 15.68◦ for cases P1, P2 and P3, respectively. It is
understood that with a wider rib spacing, the turbulent motions
becomes stronger and the separation events occur less intensively on the rib surface. From figure 3(b), it is apparent that
there are two valleys in the profile of γ p in case P3, corresponding to vortices A and C in figure 2(c). The peak locatedaround x 0 /δ = 1.25 between the two valleys results from a reattachment event. In cases P1 and P2, there is only one valley
between two ribs, which corresponds to the large recirculation bubble. Compared to case P3, the secondary dettachment
vortex C changes to a small reattachment vortex in cases P1

above the rib

S. C.
P1
P2
P3

〈 〉

〈 〉

above the rib(a) profiles of hu i at the leeward (b) profiles of hu i midway beof ribs (x 0 /δ = 0.4)
tween ribs

Figure 4. Vertical profiles of non-dimensionalized mean
streamwise velocity at the leeward of ribs and the midspan
between two adjacent ribs. S. C. denotes the smooth channel
test case.

and P2 (or, as the value of P/H decreases from 7.5 to 5.0 and
3.0). It is obvious that an increase of the pitch-to-height ratio
enhances instantaneous reattachment events between the ribs
remarkably.
Figure 4 shows the profiles of mean streamwise velocity
at the leeward corner of ribs (x 0 /δ = 0.4) and at the midspan
between two adjacent ribs (x 0 /δ = 0.5, 0.7 and 0.95 for cases
P1, P2 and P3, respectively). It is observed that the peak of hui
move towards the top wall of the channel as the value of P/Hincreases, although the density of the roughness elements decreases simultaneously. Figure 5 shows the spatial distribution
and profile of TKE. From figures 5(a) and (b), it is apparent
that TKE is generated intensively in the shear layer and transported downstream. A wake region is observed after the flowin the shear layer impinges onto the downstream rib and rushes
upward to the channel center. Owing to the presence of theshear layer and wake region, the profiles of TKE at the leeward
corner exhibits a dual-peak pattern, as is evident in figure 5(c).
The roughness elements enhance the TKE level dramaticallyin a monotonic manner as the value of P/H increases. Furthermore, the TKE level is significantly lower under the rib heightcompared with the smooth channel scenario.
The study of rib effect on the turbulence energy transfer
can be refined by examining the transport equation of Reynolds
stresses, which is expressed as
Hi j = Pi j + Πi j − εi j + Di j ,(5)where Hi j , Pi j , Πi j , εi j and Di j represent the convection,
production, pressure-strain, viscous dissipation, and diffusion
terms, respectively, defined as3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
H22
P22
22

- 22
D22

(a) TKE contours of case P1(b) TKE contours of case P3

above the rib

above the rib(a) smooth channel(b) case P1(c) case P2(d) case P3

S. C.
P1
P2
P3

P/H

P/H

(c) profiles of TKE at the leeward (d) profiles of TKE at the midspanof ribs
between two adjacent ribs

Figure 5. Contours and profiles of TKE at the leeward ofribs (x 0 /δ = 0.4) and the midspan between two adjacent ribs
(x 0 /δ = 0.5, 0.7 and 0.95 for cases P1, P2 and P3, respectively).
The vertical blue dashed lines demarcate the rib crest. S. C.
denotes the smooth channel test case.

Hi j = huk i
Pi j = −hui0uk0 i∂hui0u 0j i

∂hu 0j i
∂ xk

,

∂ xk
− hu 0j uk0 i

Di j = −∂hui0 i

∂ xk
0 !+

,

Πi j =,∂hui0u 0j uk0 i!
∂hp 0u 0j i

+ν

∂ xk
∂ 2 hui0u 0j i
∂ xk ∂ xk−

0 0
1 ∂hp ui i
+
ρ
∂ xj

figure 6(a), it is evident that the budget balance near the smooth
top wall of the ribbed flow cases is similar to that of the smoothchannel flow. From figures 6(b)-(d), it is obvious that all budget
terms enhance monotonically as the value of P/H increases.
The magnitude of the diffusion term D11 is comparable to thatof ε11 near the rib crest for the d-type roughness flow (or cases
P1 and P2), but is greatly enhanced and becomes the dominant
sink term in case P3 (at the highest P/H value tested).
To facilitate a deep understanding of the rib effects on flow
anisotropy, the anisotropy invariant maps (AIM) introduced by
Lumley & Newman (1977) are studied. AIMs provides aneffective description of the local flow topologies associatedwith the Reynolds stress anisotropy tensor, defined as
hui0u 0j i 1(11)bi j = 0 0 − δi j .
huk uk i 3
The Reynolds stress anisotropy tensor has three invariants as asecond order tensor, one zero (i.e., I = bii = 0) and two nontrial invariants (i.e., II = bi j b ji and III = bi j b jk bki ). There arethree distinct characteristic states of turbulence associated withthe so-called Lumley triangle: the axisymmetric turbulence
state where one of the eigenvalue of the Reynolds stress tensor
is either larger (III > 0) or smaller (III < 0) than the other two
which are precisely equal in magnitude, and the two-component
turbulence state in which one eigenvalue of the Reynolds stresstensor vanishes more rapidly than the other two.
Figure 7 compares the AIMs of the Reynolds stressanisotropy tensor at the leeward corner of the ribs (x 0 /δ = 0.4)
for the test cases. The AIMs in the top half of the channel for
all cases are directly compared in figure 7(a). From figure 7(a),it is evident that the turbulence states of cases P1, P2 and P3 in
the top half of the channel show a similar trend like the smoothchannel flow, in the sense that a two-component turbulence
state approaches the one-component turbulence state (II = 2/3
and III = 2/9) close to the wall, and becomes an axisymmetricturbulence state with a large eigenvalue in the log-law region
owing to the rapid increase of hu 0u 0 i in vertical direction. By
comparing figures 7(b)-(d) with 8(a), it is seen that the shapesof AIMs in the bottom half of the channel are totally different due to the complex flow characteristics in the separation(6)0
1 0 ∂ui ∂u jp+
ρ∂ x j ∂ xi
* 00+∂ui ∂u j,
εi j = 2ν
∂ xk ∂ xk*

Figure 6. Budget balance of hu 0u 0 i at the leeward corner ofribs (x 0 /δ=0.4). All of the terms are non-dimensionalized by
Ub3 /δ. The vertical blue dashed line demarcates the rib crest.

(7)
(8)
(9)

∂ xi

(10)

,

respectively. Figure 6 shows the vertical profiles of the budget
terms of hu 0u 0 i at the leeward corner of ribs (x 0 /δ = 0.4), nondimensionalized by Ub3 /δ. Compared with the smooth channel
flow, the magnitude of all the budget terms is greatly enlarged
on the rough-wall side, as is evident in figures 6(b)-(d). A
distinct difference in the bugdet terms between the smooth androughened channels is that at the leeward corner of ribs, all
the budget terms decay to zero at the wall as a result of the
low turbulent fluctuation level in that region. In the shear layersightly above the rib crest, the production term P11 enhances
dramatically and becomes the dominant source of hu 0u 0 i. To
balance the high level of prodution, the dissipation term ε11 ,
diffusion term D11 and pressure-strain term Π11 reach theirminima as the sink terms in the same region. Furthermore, an
M-shaped pattern of the convection term H11 is observed near
the rib crest, indicating that the turbulence energy generated
by the shear layer near the rib crest is spread both ways in thevertical direction and in the downstream region of the rib. The
convection term H11 reaches its negative minimum value inthe wake region (for 0.3 < y/δ < 0.4) above the rib, where thebudget is balanced by P11 as the source term, and Π11 , and ε11
and D11 as the sink terms. By comparing figures 6(b)-(d) with4

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

1.3

1.5

1.7

0.1

1.9

P1
P2
P3

II

II

1.1

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.00 0.08 0.16 0.24

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.3

0.5

0.7

0.9

(a)

0.00 0.08 0.16 0.24

III

III

(a) AIMs in top half of the channel (b) AIM in bottom half of thefor all cases
channel for case P1
(b)

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.3

0.5

0.7

0.9

0.1

II

II

0.1

0.00 0.08 0.16 0.24

III

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.3

0.5

0.7

0.9

(c)

0.00 0.08 0.16 0.24

III(c) AIM in bottom half of the (d) AIM in bottom half of thechannel for case P2
channel for case P3

Figure 7. Anisotropy invariant map based on II = bi j b ji and
III = bi j b jk bki of the Reynolds stress anisotropy tensor at theleeward corner of the ribs (x 0 /δ=0.4) for the test cases. The
dashed red line represents two-component turbulence state (defined as II = 2/9 + 2III), while the dash-dotted black line indicate axisymmetric turbulence state (defined as II = 6(III/6)2/3 ).

Figure 8. Iso-surface of swirling strength λci , colored with
non-dimensional instantaneous streamwise velocity u/Ub , withbackground contours of instantaneous vertical velocity v/Ub .
(a), (b) and (c) for cases P1, P2 and P3, respectively.

events compose a burst event, which contributes greatly to themagnitude of Reynolds stresses. From figure 8, it is observedthat with an increasing value of P/H, the hairpin packets areless likely to be broken up by the ribs, and more hairpins areconserved in a single packet, which increases the strength of
uniform coherent motions and burst events, and further increases the level of Reynolds stresses.
The spatial-temporal features of coherent motions can bestudied using the streamwise spatial-temporal two-point autocorrelations, defined asregion. At the leeward corner of a rib, the turbulence statenear the ribbed bottom wall (for 0.0 < y/δ < 0.1) resembles a
two-component turbulence state. In consequence of the interaction between secondary vortex B at the leeward corner andthe recirculation bubble A (shown in figure 2), the flow withlow momentum beneath the ribs is strongly mixed with the flow
with higher momentum in the recirculation bubble, causing theflow to move towards an isotropic state near y/δ = 0.15. From
figures 7(b)-(d), the flow evolves rapidly towards the axisymmetric turbulence state with a large eigenvalue in the shear
layer (around y/δ = 0.2) due to the large magnitude of hu 0u 0 i
which is the dominant component of TKE, and then returns
to a quasi isotropic state around y/δ = 0.3 under the influence
from the mixing of the shear layer flow with the bulk flow. It
is understood that the flow mixing can effectively reduce thedegree of flow anisotropy in the separation flow. By comparing figures 7(b)-(d), it is observed that with an increasing valueof P/H, the flow anisotropy within and above the shear layer(above y/δ = 0.2) is weakened, as a result of the enhanced interaction between the flow within the cavities between ribs andthe outer bulk mean flow above them.
Figure 8 shows the iso-surface of the swirling strength λci
colored by the non-dimensionalized instantaneous streamwise
velocity. The presence of hairpin structures are evident. In
the figure, the heads of hairpin vortices with high momentum
are intentionally connected with black dashed lines. Clearly,
the intensity of turbulence motions becomes increasingly enhanced as the pitch-to-height ratio increases. The black dashedlines can be treated as the boundaries of the uniform momentum zones following Adrian et al. (2000). The quasi-periodic
streamwise spacing between these dashed lines shows the sizeof the hairpin packets. The successive ejection and sweep

tsRuu(xr , yr ,ξ,τ)=q

hu 0 (xr , yr ,tr )u 0 (xr + ξ, yr ,tr + τ)i

,

(12)

hu 02 (xr , yr ,tre f )ihu 02 (xr + ξ, yr ,tr + τ)i

where subscript ‘r’ denotes the reference point. Figure 9
ts on the ξ-η phase plane.
demonstrates the contours of Ruu
The reference points correspond to the peaks of TKE at theleeward corner of a rib (i.e., [x 0 /δ, y/δ] = [0.4,0.24], [0.4,0.25] and [0.4, 0.24] for cases P1, P2 and P3 in panels (a),(c) and (e), respectively) and the midpoint between two ribs
(i.e., [x 0 /δ, y/δ] = [0.5,0.24], [0.7, 0.23] and [0.95, 0.20] forcases P1, P2 and P3 in panels (b), (d) and (f), respectively).
From figures 9(a) (c) and (e), it is clear that as the value of
P/H increases, the isopleths become increasingly asymmetric
about the origin as a result of the intensive interaction between
the strong wake in the outer-layer from the previous rib period
and the shear layer induced by the present rib. By comparing
figures 9(a), (c) and (e) with (b), (d) and (f), it is evident that thesize of the isopleths at the rib midspan is smaller than that at theleeward corner, showing a smaller spatial and temporal scales
due to an increased turbulence level. As the value of P/Hincreases, the shape of the correlation structures becomes increasingly shortened and flattened. The streamwise convection

5

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

(b)

(c)

(d)

(e)

(f)

hu 0u 0 i dramatically near the rib-roughened wall. In the vicinityof the rib crest, hu 0u 0 i is dominated by P11 as the source term,
and by D11 , Π11 and ε11 as the sink terms. It is observedthat the diffusion term D11 is greatly enhanced in the k-typeroughness flow (case P3) with an increasing value of P/H. The
AIMs of the Reynolds stress anisotropy tensor reveals that the
mixing of the rib induced vortices and the bulk flow results in aremarkable reduction in the flow anisotropy. Furthermore, theflow over the rib crest tends to become isotropic as the value of
P/H increases.
It is observed that as the pitch-to-height ratio P/H increases, the intensity of the turbulence motions enhances, andaccording to the distribution of the spatial-temporal two-point
ts , the characteristic scale of coherent structures
correlation Ruu
becomes increasingly shortened in the streamwise direction.

REFERENCES
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Figure 9. Contours of temporal and spatial two-point correts at the position corresponding to the peaklation coefficient Ruu
value of TKE at the rib leeward and the rib midspan. (a), (c)and (e): leeward of ribs (x 0 /δ = 0.4) for cases P1, P2 and P3;(b), (d) and (f): midspan between two adjacent ribs (x 0 /δ = 0.5,
0.7 and 0.95) for cases P1, P2 and P3, respectively.
velocity huic of the coherent structures can be calculated by theslope of the isopleths, which is marked using a red solid line infigure 9. It is interesting to see that the value of huic decreases
with an increasing value of P/H at both locations, though the
TKE level is significantly enhanced. From figure 9, it is also
observed that as the value of P/H increases, the characteristc
spatial scale of coherent structures decreases remarkably while
the characteristic time scale remains almost unchanged.

CONCLUSIONSDNS of turbulence flows through a channel roughened
with circular-arc ribs has been performed to investigate thepitch-to-height ratio (P/H) effect on the physics of flow separation, statistics of turbulent velocity field, and the characteristic
scales of coherent structures. With a wider pitch between ribs,
the reattachment point of the mean flow moves upstream progressively, while the detachment point remains stable, leading
to an increase of the form drag on the rough-wall side. The
angle between the incipient detachment point and the detachment point of the mean flow increases monotonically with anincreasing value of P/H, indicating that the separation events
become less intensive on the rib surface with an increased turbulence activities. It is also observed that an increase in thepitch-to-height ratio enhances the instantaneous reattachment
events between two adjacent ribs remarkably.
Owing to the shear layer and wake region triggered by
the ribs, the profiles of TKE at the leeward corner exhibits adual-peak pattern. The presence of the rib elements enhances
the magnitudes of TKE and budget terms of Reynolds stresses
dramatically. The production term P11 increases significantlyin the shear layer near the rib crest, which further augments

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