River Flow 2020 – Uijttewaal et al (eds)© 2020 Taylor & Francis Group, London, ISBN 978-0-367-62773-7

Three-dimensional flow structures of straight rough-bed channels
with different aspect ratios
Y. Takakuwa
Graduate School of Science and Engineering, Chuo University, Tokyo, Japan

S. Fukuoka
Research and Development Initiative, Chuo University, Tokyo, Japan

ABSTRACT: It is significant in hydraulic engineering to clarify the relationship between flow
structures, channel width and bed roughness. In this paper, three-dimensional numerical experiments were conducted about turbulent flows over rectangular straight rough-bed channels withdifferent aspect ratios (0.36 ~ 1.38) and relative depths on the roughness height (3.63 ~ 12.00).
Characteristics of secondary flows in rough-bed channel and mutual relations of the outer secondary flow, the inner secondary flow and the bottom secondary flow were investigated indetail. The velocity-dip phenomenon occurred when the aspect ratio was larger than 0.84 andthe relative depth was smaller than 4.18, and did not occur when the aspect ratio was smaller
than 0.39 and the relative depth was larger than 4.88.

1 INTRODUCTION
In the field of hydraulic engineering, it is important to clarify the relationship between flow
structures, channel aspect ratio (b/h, b = channel depth; h = channel depth), and relative
depth (h/r, r = roughness height) in order to understand flow resistance and three-dimensional
flow structures.
Hayashi et al. (2006) conducted the direct numerical simulation in smooth-bed channel
with aspect ratio 2.0. They found three kinds of secondary flows in the smooth-bed channel.
Figure 1 shows an outline drawing of the secondary flows by reference to the results ofHayashi et al. (2006). Secondary flows toward a corner at the channel bottom are divided
into the bottom secondary flow along the channel bottom and the outer secondary flow
rising along the sidewall. The outer secondary flow transports slow water from the bottom
corner to the channel center of the free surface. Therefore, the free surface velocity in thechannel center becomes small and the maximum streamwise velocity appears below the freesurface (velocity-dip phenomenon). Simultaneously, the inner secondary flow which rotatesin the opposite direction to the outer secondary flow generates at the corner surrounded by
the free surface and the sidewall.
Nezu & Nakagawa (1993) clarified the generation of the outer secondary flow and bottom
secondary flow irrespective of the aspect ratio by the experiments of smooth channel with the
aspect ratio 2.0 or more, and also clarified the maximum secondary flow velocity was 1.5 ~
3.0% of the maximum streamwise velocity. Greg et al. (1995) showed generation of threekinds of secondary flows in experimental and numerical study using the rectangular smoothchannel where the aspect ratio is less than 1.0. Joung & Choi (2010) found by the direct simulation in the smooth-bed channel with the aspect ratio 1.0 that directions of inner secondary
flow and outer secondary flow were closely related to instantaneous vertical velocity andcoherent structures of the ejection and sweep along the sidewall.
There are few researches concerning effects of the aspect ratio and roughness on secondary
flows in rough-bed channels. Tominaga & Nezu (1986) experimentally showed that the1

Figure 1. Schematic drawing showing bottom, outer and inner secondary ﬂows and the velocity-dip
phenomenon in the smooth-bed channel by reference to the results of Hayashi et al. (2006).

maximum secondary flow velocity became large (roughly 3 to 4% of the maximum streamwise
velocity) in rough-bed channels with different aspect ratios (b/h = 2.00 ~ 8.00) and relative
depths (h/r = 6.67~ 26.3), and that secondary flows were observed even at a transverse position more than 2.5 times channel depth away from the sidewall. Little consideration, however, has been given to effects of aspect ratio and roughness on flow structures overrectangular straight rough-bed channels.
In this study, we performed three-dimensional numerical experiments on steady flows inrectangular straight rough-bed channels with extremely narrow widths (b/h = 0.36 ~ 1.38) andlarge bottom roughness elements (h/r = 3.63 ~ 12.00). First, we investigated effects of the
aspect ratio and the relative depth on scales of secondary flows, and discussed where the maximum streamwise flow velocity occurred. Second, we examined relationships between secondary flows and flows rising or sinking around roughness elements, and clarified threedimensional flow structures near the channel bottom in extremely narrow rough-bed
channels.

2 NUMERICAL METHOD
2.1 Calculation of fluid motion
Fluid motions are calculated using the APM (Fukuoka et al. 2014) in the Cartesian coordinate
system. Basic equations of fluid motions are described as follows. SGS stresses are evaluatedby the Smagorinsky model.
∂ui
¼ 0
∂xi

Dui
1 ∂P
∂ 
¼ gi þ
2 ð þ t ÞSij
Dt
ρ ∂xi
∂xj


∂uj
1 ∂ui
þ
Sij ¼
∂xi
2 ∂xj
p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
t ¼ ðCS ΔÞ2 2Sij Sij

ð1Þ
ð2Þ
ð3Þ
ð4Þ

where xi = Cartesian coordinate; ui = ith component of mass averaged velocity; P = sum of
the pressure and isotropic component of SGS stress; gi = ith component of gravitational2

acceleration; ρ = volume averaged density; ν = kinematic viscosity; νt = SGS turbulent viscosity; Sij = strain rate tensor; CS = Smagorisky constant (=0.173); Δ = grid size.
Advection terms in the Equation 2 are solved by upwind differences of the third orderaccuracy, time advancement is achieved by the SMAC method, and pressure convergence iscalculated by the SOR method. The free-surface variation is analyzed using the VOF method(Hirt & Nichols 1981).
2.2 Methodology of fluid calculation for grids containing the channel surface
Figure 2a shows the methodology of fluid calculation for grids containing channel surfaces,
and these grids are dealt as solid-liquid mixed fluid (Ushijima et al. 2008). The mass andmomentum of solid-liquid mixed fluid are evaluated using the volume fraction of the solid
phase α as shown in Equations 5 and 6.
ρ ¼ αρs þ ð1  αÞ ρfð5Þρui ¼ αρs us;i þ ð1  αÞ ρf uf ;ið6Þwhere suffix s and f denote the solid and the liquid phase, respectively.
The volume fraction of the solid phase influences evaluation of the velocity of mixed fluid
directly, and its precision has large influences on the velocity distribution in the vicinity ofboundaries. The volume fraction of the solid phase is evaluated using the subcell method(see Figure 2b).

3 NUMERICAL EXPERIMENTS AND CALCULATION CONDITIONS
3.1 Concepts of numerical experiments
Three-dimensional numerical experiments were conducted about turbulent flows which havesame discharge per unit width in rectangular straight rough-bed channels with different
aspect ratios (b/h = 0.36 ~ 1.38) and relative depths on the roughness height (h/r = 3.63 ~12.00). Figure 3 and Table 1 show channel schemes and hydraulic conditions, respectively.
In the case 1 (b/h = 0.36), a steady flow (Q = 0.06 l/s) was supplied in the channel with
2 m long, 7 mm width and 1/20 bed slope, following to the experiment by Ancey et al.
(2002). Bottom roughness of the channel was given by cylinders of uniform size. In thepaper of Ancey et al. (2002), the size of cylinder was not clearly described, so that the

Figure 2a. Methodology of ﬂuid calculation for grids containing the channel surface using the one-ﬂuid
model of solid-liquid multiphase. Figure 2b. Concept of the subcell method to evaluate the volume fraction of the solid phase in a computational grid.

3

Figure 3. Channel schemes of case 1 and 3. The origin of the coordinate was set r/3 lower than the topof cylinders.

diameter of cylinders was set to 2r = 8 mm to satisfy the experimental channel depth h =20 m. The origin of the coordinate was set r/3 lower than the top of cylinders.
Case 2 and 3 were analyzed under the similar conditions to case 1. Discharges per unit
width were set equal to that of case 1. Case 4 with small roughness (2r = 3 mm) at the bed andthe same channel width as case 1 was analyzed in order to estimate the effect of roughness
sizes on flow structures.
The velocity distribution at x = 0.20 m was given to the upstream boundary condition in
every step to make the boundary layer develop in the short distance. At the downstream end,
channel depth was controlled by giving the channel depth at the upstream point by one step.
3.2 Calculation conditions and measurements of the turbulent flow
Our previous research (Takakuwa & Fukuoka 2019) showed that flow structures over the permeable fixed rough-bed could be analyzed with sufficient accuracy by the APM under following
conditions. First, the computational grid size Δ must be less than 1/10 of the roughness diameter. Second, the subcell size Δsubcell should be less than 1/6 of the grid size Δ to accurately evaluate the volume fraction of the solid phase including in a computational grid (see Figure 2b).
In this paper, according to these conditions, grid sizes and subcell sizes were set to Δ = Δx = Δy =Δz = 0.3 mm and Δsubcell = 0.05 mm, respectively. Time step for each case was set to 2.0×10-5 s.
Table 1. Channel and hydraulic conditions, where Q = discharge; R = hydraulic radius;h = channel depth; b = channel width; b/h = aspect ratio; r = height of roughness element;
h/r = relative depth on roughness height; ūmax = maximum value of the mean streamwise
velocity; Umean = bulk velocity; U*(R) = friction velocity; Re = Reynolds number on thehydraulic radius and the bulk velocity; Fr = Froude number on the hydraulic radius and thebulk velocity; |svmax|= maximum size of the mean secondary flow velocity.

umax . Umean[m/s] [m/s]U  ðRÞ
[m/s]

Re [-]

Fr
[-]

4.88

0.68

0.44

0.038

1.3E+03

2.58 0.04

0.06

4.18

0.78

0.47

0.049

2.3E+03

2.14 0.07

0.09

1.38 4.0

3.63

0.88

0.59

0.054

3.5E+032.43 0.07

0.08

0.39 1.5

12.00 0.68

0.48

0.038

1.4E+032.81 0.03

0.04

Q
[1/S]

R
h
b
b/h
[mm] [mm] [mm] [-]rh/r[mm] [-]

Case 1

0.06

2.97

19.5

7

0.36 4.0

Case 2

0.11

4.93

16.7

14

0.84 4.0

Case 3

0.17

5.92

14.5

20

Case 4

0.06

2.93

18.0

7

4

SVmax jSVmax j=umax[m/s] [-]

Measurements of turbulent flows in the APM were performed according to Nezu (1977).
The sampling time step was set to 4.0×10-4 s, and the total number of sampling for each point
was 2500 (1 second measurement). It was confirmed that structures of time-averaged flows
were almost the same as those measured for 2 seconds (case 3).

4 RESULTS AND DISCUSSIONS
4.1 Time-averaged flow structures in a cross-section
Figures 4ⅰ and 4ⅱ show cross-sectional distributions of the time-averaged streamwise velocity
and the time-averaged secondary flow velocity at a top section of roughness element of case 1~ case 4. Three kinds of secondary flows appear in case 1, 2, and 4 of rough-bed channels aswell as the secondary flows in a smooth-bed channel (Figure 4ⅱ of case Hayashi et al. 2006).
On the other hand, in case 3 where the relative depth h/r is the smallest, the outer and bottom
secondary flows occur, but no inner secondary flows are formed at corners near the free surface. Focusing on the streamwise velocity, velocity-dip phenomena occur in case 2 and 3, but
not in case 1 and 4 where aspect ratios are quite small. The reasons are described below.
In cases 2 and 3 where aspect ratios are not small compared to case 1 and 4, outer secondary
flows proceed to the channel center after reaching the free surface where the slow water massfrom the bottom corner to the channel center of the free surface is transported as shown in
Figure 1. The free surface flow velocity in the channel center becomes small, and the velocitydip phenomenon occurs.
Figure 4ⅲ shows an outline drawing of the secondary flows in the extremely narrow channel. The outer secondary flow is formed at considerably lower position. Although the weak
upward flow of the outer secondary flow rises along the sidewall, but do not reach the freesurface.
Therefore, in cases 1 and 4, the maximum streamwise velocity appears at the channel centeron the free surface, and velocity-dip phenomena do not occur.
As mentioned above, the scale of the outer secondary flow and the position where the maximum streamwise velocity appears are closely related. As seen in respective cases 1 and 4 of
Figure 4, outer secondary flows expand from the bottom z = 0 to z = 8 and 7 mm. In case 1
where the relative depth (h/r = 4.88) is smaller than that of case 4 (h/r = 12.00), it is probablethat the upward flow along the sidewall is strong and the outer secondary flow spreads torather high position. These outer secondary flows eventually develop to higher positions (case1: 82% of the channel depth, case 4: 67%). In case 3 where the relative depth h/r is 3.63 andthe aspect ratio b/h is 1.38, the outer secondary flow rises along the side wall to the free surface, and the inner secondary flow could not occur.
4.2 Longitudinal variation of secondary flows within a cross-section
Figure 5 shows the schematic drawing of three-dimensional structures near the channelbottom in a rough-bed channel. As shown in solid arrows near the cylinder roughness element,
the flow over bottom roughness element collides with the adjoining roughness element and isdivided into the downward flow sinking into the valley between roughness elements (sinking
flow, position c in Figure 5) and the upward flow rising to a higher position (rising flow, position d in Figure 5). Outer and bottom secondary flows merged with these vertical flows, andsecondary flow structures change longitudinally.
At the position c, the upward component of the outer secondary flow along the sidewall is
weakened by the sinking flow, and the upward flow velocity becomes small as shown by thedashed line in Figure 5. The downward component of the bottom secondary flow near thebottom corner is strengthened by the sinking flow, and the downward flow velocity becomeslarge. In addition, the upward component of the bottom secondary flow near the channel
center is weakened by the sinking flow.

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Figure 4. Mean streamwise velocities and mean secondary ﬂow velocities within cross-sections of top ofroughness element. ⅰ and ⅱ of all cases show distributions of streamwise velocity and secondary ﬂow velocity, respectively. In ⅰ of case 2, broken lines indicate the detail distribution of streamwise velocity. ⅱ ofcase Hayashi et al. (2006) shows the distribution of secondary ﬂows in a smooth-bed channel, cited from
Left half of the Figure 7 of Hayashi et al. (2006) and partially modiﬁed. ⅲ shows a schematic drawing of
secondary ﬂows in the extremely narrow channel.

6Figure 5. Schematic drawing showing three-dimensional structures of time-averaged ﬂows near the channel bottom in a rough-bed channel. Blue solid arrows indicate the ﬂow sinking into the valley between
roughness elements. Red solid arrows indicate the ﬂow rising after hit with a cylinder roughness element.
Dashed arrows indicate outer and bottom secondary ﬂows affected by the sinking ﬂow and the rising ﬂow.

At the position d, the upward components of outer and bottom secondary flows arestrengthened by the rising flow.
 Þ of cases 2 and 4. Contours
Figure 6 shows longitudinal variationsof secondary flows ð
v; w

0:5
 2 . Figure number a ~ d in Figure 6 correspond toindicate the secondary flow velocity v2 ; w
the longitudinal positions on the roughness element of Figure 5. In case 2 with the relative depthof h/r = 4.18, the maximum size jsvmax j of the secondary flow velocity near the roughness elementmax . In case 4 where the relative depth h/r isbecomes 9% of the maximum streamwise velocity u
12.00, the ratio jSVmax j=
umax is 0.04 and is similar to values (jSVmax j=
umax = 0.03 ~ 0.04) of roughbed channels with relative depths (h/r = 6.67 ~ 26.3) conducted by Tominaga & Nezu (1986).
When the relative depth further increases, the ratio jSVmax j=
umax is considered to approach thevalue of the smooth-bed channel (jSVmax j=
umax = 0.015 ~ 0.03, Nezu & Nakagawa 1993).
Vertical ranges of secondary flows are limited to z = 1.1 r and 0.75 r (h/r = 4.18 and 12.00,
respectively) from the top of the roughness element as shown in solid lines.

5 CONCLUSIONS
Following conclusions were obtained.
1. The ﬂow sinking into the valley between roughness elements and the ﬂow rising the roughness
element occur near the bottom in a rough-bed channels. Outer and bottom secondary ﬂows
merged with these vertical ﬂows, and secondary ﬂow structures near the channel bed change longitudinally. The ratio jSVmax j=
umax of the rough-bed channel with the small relative depth becomeslarger than that of the smooth-bed channel. When the relative depth increases and the effect ofroughness elements decrease, the value jSVmax j=
umax approaches that of the smooth-bed channels.
2. Scales of outer secondary ﬂows and bottom secondary ﬂows are spread in the range fromthe top of the roughness element to the height of a radius of the bottom cylinder, and secondary ﬂow structures above that height are almost longitudinally uniform.
3. The velocity-dip phenomenon occurred when the aspect ratio was larger than 0.84 and therelative depth was smaller than 4.18. When the aspect ratio was smaller than 0.39 and therelative depth was larger than 4.88, outer secondary ﬂows did not reach the free surface
and the velocity-dip phenomenon did not occur.
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Figure 6. Longitudinal variation of secondary ﬂows of cases 2 and 4. Streamwise positions ofcross-section views a, b, c, and d correspond to longitudinal positions of Figure 5. Solid lines, brokenlines and dotted lines of cases 2 and 4 show heights that secondary ﬂows change in streamwise direction,heights of roughness top, and heights of the valley between cylinders, respectively.

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