# Drag on a Square-Cylinder Array Placed in the Mixing Layer of a Compound Channel

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. State of the Art

_{d}< 685), for different solid-volume fractions, found that the array-averaged drag coefficient increases with solid fraction and is inversely proportional to the cylinder Reynolds number. Earlier, Nepf [16] suggested that for larger Re

_{d}the array-averaged drag coefficient might decrease with the increase of the solid fraction. Ferreira et al. (2009) [17] performed measurements for Re

_{d}larger than 1000 and, by analyzing the terms of the double-averaged momentum conservation equation, showed that it is likely that the average drag coefficient increases with the solid fraction for Re

_{d}> 1000 if the gradient of form-induced stresses also increases.

_{d}and solid fraction have different values of the patch-averaged drag coefficient depending on the upstream areal density of stems and solid fraction. Hence, in infinite arrays, the equivalent array-averaged drag coefficient may be smaller or larger than that of an isolated cylinder, depending on patch uniformity, areal cylinder density and Reynolds number.

_{d}and on the shear parameter K, defined as the shear rate normalized by the cylinder diameter and the flow velocity in the centre-line of the cylinder [30]. The experimental work of [32] has shown that, for Re

_{d}lower than 1600, the drag coefficient C

_{d}decreases with Re

_{d}for non-zero values of the shear parameter. For a given Re

_{d}, C

_{d}was also observed to decrease with the increase of the shear parameter. These results are strictly true for planar and linear shear flows and may not be applicable to cylinders in mixing layers of compound channel flows.

#### 1.3. Objectives

## 2. Theory

#### 2.1. Computing the Bulk Drag Force from an Integral Momentum Balance

_{1}to S

_{6}), and inner, solid, sections (S

_{0}). There are momentum fluxes across the open sections. The forces applied on S

_{0}constitute the reaction of the solid elements on the fluid in the control volume.

**is the ith component of the time-averaged velocity vector, ${{u}^{\prime}}_{i}$ is the ith component of the velocity fluctuations, g**

_{i}**is the ith components of the acceleration due to gravity, P is the time-averaged pressure, ρ is the density of the fluid, T**

_{i}_{ik}is the time-averaged viscous stress tensor and $-\overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{k}}$ is the Reynolds stress tensor. The control volume is represented by V

_{c}while S

_{c}is the total control surface (the union of S

_{1}to S

_{6}and S

_{0}), S

_{0}is the solid part of S

_{c}, that coincides with the surface of the cylinders, and S

_{c}\S

_{0}represents the open control surface (the union of S

_{1}, …, S

_{6}), through which there may be mass and momentum fluxes. The components of the outward pointing normal unit-vectors applied to each boundary are n

**. For the sake of conciseness, Equation (1) is written in Cartesian tensor notation, where the dummy index k is used in the summations inherent to the dot products in the first, fourth, fifth and seventh terms. The last two terms on the right-hand side of Equation (1) represent the force exerted on the flow by the cylinders:**

_{i}**R**is the reciprocal of the bulk drag force applied on the cylinder array (

_{x}**F**), i.e.,

_{D}**R**= −

_{x}**F**and R

_{D}_{x}= |

**R**| = |

_{x}**F**|.

_{D}_{m}is an open control section, a

^{(m)}is a generic time-averaged variable measured in that control section, [ ] is the section-average operator, [a

^{(m)}] is the mean value of a in that section and S

^{(m)}is the area of the control section. Viscous stresses at the open boundaries of the control volume were considered negligible in comparison to Reynolds stresses. Resolving the integral terms, Equation (1) may be written as:

_{x}is estimated after all other terms in Equation (4) are determined experimentally, through acquisition of the three-component instantaneous velocity of the fluid in all open control sections and the free-surface elevation along the outer rim of the control section. If the vertical distribution of pressure is approximately hydrostatic, the calculation of the mean pressure on the surfaces is trivial and the gradients of the free-surface elevation are sufficient for the calculation of the pressure forces. Selecting a very thin prismatic control volume with its main dimension oriented parallel with the channel bottom, placed at an adequate distance above it, so that the effect of the bottom boundary to vertical distribution of velocities is not relevant, allows also for the definition of a single Reynolds number of reference for the assessed drag force [34]. The details of the mathematical derivation of Equation (3) are shown in the Appendix A.

#### 2.2. Drag Coefficient

_{d}, is expressed as:

_{0}is the time-averaged longitudinal velocity that characterizes the undisturbed upstream flow, calculated as a spanwise average within the frontal area of the cylinder.

_{d}is defined as the normalised bulk drag force (R

_{x}) averaged over the total number of cylinders (n

_{c}):

## 3. Experimental Facility, Instrumentation and Experimental Procedure

#### 3.1. Experimental Facility

#### 3.2. Experimental Tests

_{0}between two consecutive cylinders was kept equal to 0.10 m. Spacing and sizing were chosen so as to align with those of the experiments carried out in the context of FlowRes ANR, a project oriented to the investigation of compound-channel flows with transition in bed roughness, described in [39]. The tests were run under uniform-flow conditions in the compound channel. To establish uniform flow in the channel, the total discharge Q was initially distributed in the main channel (Q

_{mc}) and in the floodplain (Q

_{fp}), according to the weighted divided channel method proposed by [40]. The cylinders were emergent under the tested flow conditions.

_{d}= dU

_{0}/v, where $v$ is the kinematic viscosity of the fluid. The Froude number characterizes separately the main channel and the floodplain, and is given by Fr

_{fp}= (Q

_{fp}/(1.40h

_{fp}))/√gh

_{fp}and Fr

_{mc}= {Q

_{mc}/((0.40 + 0.20)(h

_{mc}− 0.10) + 0.05)}/√gh

_{mc}. The relative flow depth h

_{r}is defined as the ratio between the floodplain flow-depth h

_{fp}and the main-channel flow depth h

_{mc}. The normalized velocity difference is λ = (U

_{mc}− U

_{fp})/(U

_{mc}+ U

_{fp}) [27], where U

_{mc}and U

_{fp}are the time-averaged and depth-averaged longitudinal velocities of the flow in the main channel and in the floodplain, respectively. In both flows, λ = 0.35, which indicates that the mixing processes should be dominated by two-dimensional planar interactions, associated to large-scale structures akin to Kelvin-Helmholtz instabilities. However, given that the value of λ is not significantly higher than 0.3, it was expected that three-dimensional effects in the mixing layer would not be negligible.

#### 3.3. Velocity Measurements

_{0}= h

_{fp}/3 from the floodplain bed and at z = h

_{fp}/3 − 0.06h

_{fp}.

_{1}to S

_{4}.

## 4. Results

#### 4.1. Time-Averaged Flow and Main Wake Processes

_{0}−d)/d of the array is equal to 1.22, a value similar to that for which [13] described the relevant wake patterns as the “gap-flow mode”. This wake flow features an alternatively deflecting jet along the streamwise direction between the cylinders and a nested vortex-shedding process within the one formed by the outward edges of two side-by-side cylinders. The streamlines observed in Figure 4 are compatible with this pattern with one notable exception—the formation length of the cylinder adjacent to the interface is smaller than that of the cylinder placed towards the floodplain. This indicates that drag is unevenly distributed in the lateral direction. The observed high velocities of the flow in the near wake and the relative small formation length of all downstream cylinders may indicate limited downstream velocity-deficits and, in turn, decreased bulk drag compared to that on an array with (d

_{0}−d)/d >> 1.5 [13].

#### 4.2. Analysis of Terms in Equation (4)

_{0}, l = 6.25d

_{0}(note that the plan view of the control volume also shown in Figure 4) and h = 0.06h

_{fp}. The momentum fluxes and the Reynolds stresses computed from the velocity measurements at z

_{0}= h

_{fp}/3 (section S

_{5}) and z = h

_{fp}/3−0.06h

_{fp}(section S

_{6}) confirmed that net momentum flux across sections S

_{5}and S

_{6}is negligible and that the vertical variation of the turbulent stresses is small, justifying the application of Equation (4). All values are normalized with powers of the reference velocity U

_{0}estimated for each test. The values of the terms of Equation (4) are shown in Figure 5, Figure 6 and Figure 7.

_{x}U

_{k}at the vertical control sections S

_{1}, S

_{2}, S

_{3}, S

_{4}are presented in Figure 6a,b. The product U

_{x}U

_{k}is also called as mean flux, as it represents the momentum flux of the time-averaged flow per unit mass and area, transported in the streamwise direction. The signs of the terms are chosen so that they represent simple summations in Equation (4).

_{1}and S

_{3}, are observed near the main-channel/floodplain interface, as a consequence of the higher velocities in the main channel. For the chosen locations of S

_{1}and S

_{3}, both tests present rather balanced fluxes and a near zero net momentum flux. This was sought as a compromise, since placing the S

_{1}and S

_{3}very near the cylinder array would increase the uncertainty in all measured variables. This is especially true for the free-surface at S

_{3}, because of the stronger vertical fluctuations generated by the interaction of the cylinder wakes and the lateral fluxes between the floodplain and the main channel.

_{2}and S

_{4}for both tests is shown in Figure 5b. Flow deflection upstream the array (out of the control volume-negative values in Figure 5b), observed in Figure 4, is also discernible here. The maximum values of the convective transport through these sections occurs just upstream the cylinder array. Flow reattachment is observed downstream the array. In both tests it seems that mean momentum transport out of the control volume upstream the array is higher close to the main channel (S

_{4}) than in the inner floodplain (S

_{2}). The different characteristics of the mixing layers of tests SA_03 and SA_04 is visible in the different values of the convective transport downstream the first row of cylinders. The outward convective fluxes upstream the cylinder are not influenced by the nature of the mixing layer. However, beyond the first cylinder row, the convective flux near the interface ${U}_{x}{U}_{y}$, oscillates between positive (inward in this context) and negative (outward) values. This is especially visible in the higher submersion test SA_04. The negative fluxes are much less expressive in the second and third rows, relative to those upstream of the first row. Outward convective fluxes are not seen at boundary ${S}_{2}$, next to the floodplain. This may indicate a perturbation in the array induced by the flow in the main-channel.

_{0}/d = 2.22, compatible with the range 1.5 < d

_{0}/d < 4 proposed by [12] for partially reattaching flow between two consecutive cylinders. Yet, the flow pattern and the convective fluxes at the boundary ${S}_{4}$ seem to indicate that the boundary layer separated from the upstream cylinder does not reattach to the side of the downstream cylinder but is driven inwards in the space between these two consecutive cylinders. It can be hypothesized that the relevant variables are the momentum difference between the main channel and flow in front of the array of cylinders and the space between the cylinders. In that case, a Reynolds number can be formed, ${\mathrm{Re}}_{I}=\frac{\left({U}_{mc}-{U}_{0}\right)\left({d}_{0}-d\right)}{\nu}$. Test SA_03 (lower submergence) is characterized by ${\mathrm{Re}}_{I}=13\times {10}^{4}$ while SA_04 (higher submergence) has ${\mathrm{Re}}_{I}=16\times {10}^{4}$.

_{0}/d < 3 and Re

_{d}~5600–12,800.

_{0}, are shown in Figure 6a,b. The variable $\overline{{{u}^{\prime}}_{x}{{u}^{\prime}}_{k}}$ is also referred as stress, since it expresses the Reynolds stress per unit fluid density. The values of the stresses, both normal and shear, are smaller than those of the convective fluxes. This observation is highlighted by the use of the same scale of the vertical axis in Figure 5a and Figure 6a. The $\overline{{{u}^{\prime}}_{x}{{u}^{\prime}}_{x}}$ stress values in S

_{1}and S

_{3}(Figure 6a) resemble the distribution of the streamwise convective momentum transport (Figure 5a) featuring higher values at the main-channel/floodplain. Stronger stresses near the interface (y/d

_{0}= 0) are typical of compound-channel flows [23], which suggests that these peaks are not due to the presence of the array. The corresponding peak for S

_{3}is larger in both tests, compared to that of S

_{1}. This observation implies that there is an excess in the $\overline{{{u}^{\prime}}_{x}{{u}^{\prime}}_{x}}$ stress, generated by interaction of the wake flow in the near-interface cylinder and the shear flow. This effect seems more pronounced in the case of test SA_03–lower submersion.

_{2}and S

_{4}are seen in Figure 6b. It is noteworthy that the peak values are obtained at the second row of the array, following a turbulence suppression caused by the accelerated flow upstream the first row. It is also relevant to note that in the case of test SA_03 (lower submergence) turbulent transport is much higher on the side of the interface (S

_{4}). In the case SA_04 (higher submergence), turbulent transport is higher in section S

_{4}, relatively to section S

_{2}, just upstream the second row and for $x/{d}_{0}=1$ onwards. Downstream the latter section the characteristics of the compound flow mixing layer become again prevalent, relative to those of the array wake flow. In the space between the first and the second rows, the separated boundary layer from the first cylinder may seem to cross the ${S}_{4}$ section, in the case of test SA_03. However, this is not the case of SA_04, which is line with the hypothesis that, for this test, boundary layer reattachment occurs inside the space between cylinders (see discussion of convective fluxes at ${S}_{4}$). This justifies the stronger difference between shear stresses at ${S}_{2}$ and at ${S}_{4}$, in case of SA_03.

#### 4.3. Total Net Contribution of Terms in Equation (4)

_{1}and S

_{3}is the single major contribution for the drag force. This raises concerns about the accuracy of the estimation of the drag coefficient, given the assumption of the hydrostatic hypothesis.

#### 4.4. Sensitivity Analysis of the Values of the Drag Coefficient Relatively to the Position of Section S_{3}

#### 4.5. Values of the Drag Coefficient and Discussion

_{d}for tests SA_03 and SA_04 are presented in Table 2. The same table summarizes the values of drag coefficients found in other studies, corresponding to different configurations of square cylinders and types of flow. Among these, some employ cylinders in open-channel flows [23], closed water-channel flows [44] and air flows [13]. One case concerns a regular array in open-channel flow [23] and the others regard cylinder pairs arranged in tandem or side-by-side configurations with relative distance between the cylinders d

_{0}/d similar to that of the regular array examined herein. All C

_{d}values in Table 2 are array-average values. As a general trend, the drag coefficient of square cylinders in tandem is smaller than that of isolated cylinders in uniform flow. The drag coefficient of cylinders in a side-by-side configuration may be increased relatively to that of isolated cylinders. The drag coefficients of the tests conducted here (SA_03 and SA_04) are smaller than those of isolated cylinders and also smaller than those of cylinders in tandem for the same cylinder spacing and range of Reynolds number. The larger cylinder spacing reported by [12] features lower values of the drag coefficient. This is attributed to the lower value of the Reynolds number and not discussed in this paper. The discussion below addresses the reason why the drag coefficient of the cylinders in the mixing layer is smaller than that of cylinders in tandem in uniform flow and clarifies the differences between the values obtained for the two tests SA_03 and SA_04.

_{d}, comparatively to those of isolated, tandem or side-by-side cylinders in uniform flows (see Table 2). The array-averaged values of C

_{d}of both SA_03 and SA_04 are closer to those of tandem cylinders than to those of the side-by-side reported by [14] or [13] for the same ranges of the Reynolds number and of the ratio d

_{0}/d. It is thus hypothesized that downstream cylinders, especially those on the inner column or on the side of the floodplain, experience significantly less drag—or even negative drag—than that of the upstream cylinder. In short, it was expected that the array-averaged drag coefficient of the array under investigation would be smaller than that of an isolated cylinder, since each column is likely to behave as a system of cylinders in tandem, even if the side-by-side configuration may attenuate this effect. The additional drag reduction brought about by the mixing layer called for an explanation. It was hypothesized that it should be the effect of convective fluxes that equalize pressures across the second row of cylinders, especially in the column adjacent to the interface, an effect that is stronger in test SA_04.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

α | generic time-averaged variable; |

b | width of the fluid control-volume; |

C | forces-normalization quantity; |

C_{d} | drag coefficient; |

d | width of square cylinder; |

d_{0} | distance between two consecutive cylinders; |

Fr | Froude number; |

F_{D} | time-averaged force exerted by the flow on the cylinders; |

g | gravitational acceleration; |

h | height of the fluid control-volume; |

h_{r} | relative flow depth in a compound channel; |

K | shear parameter; |

l | length of the fluid control-volume; |

n | outward pointing normal unit-vector; |

n_{c} | number of cylinders of an array; |

P | time-averaged pressure of fluid; |

Q | discharge in the channel; |

Re_{d} | Reynolds number based on cylinder’s width; |

${\mathrm{Re}}_{I}$ | Reynolds number based on the velocity difference between the flow in the main- channel and in the array and on the cylinder spacing; |

R | time-averaged force exerted by the cylinders on the flow; |

R_{x} = |F|_{D} | absolute value of the time-averaged drag force on the cylinders; |

S_{c} | total surface of the fluid control-volume; |

S_{m} | open control-section; |

S^{(m)} | area of an open control-section; |

T_{ik} | time-averaged viscous stress tensor; |

t | Time; |

U | time-averaged velocity; |

U_{0} | mean representative velocity of the approaching flow to the cylinder array; |

$-\rho \overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{k}}$ | Reynolds-stress tensor; |

V_{c} | control volume; |

z_{0} | vertical distance of the measuring level from the floodplain bed; |

x, y, z | longitudinal, lateral and vertical directions; |

θ | angle between the channel bottom and the horizontal plane; |

λ | dimensionless shear; |

ν | kinematic viscosity of fluid; and |

ρ | density of fluid |

Subscript or superscript m (m = 1, 2, 3, 4, 5, 6) refers to an open control-section | |

Subscripts mc and fp refer to main channel and floodplain respectively |

## Appendix A

^{(m)}] is the mean value of a in that section. The unit vectors for the arrangement in Figure 1 are n

^{(1)}= (−1, 0, 0), n

^{(2)}= (0, +1, 0), n

^{(3)}= (+1, 0, 0), n

^{(4)}= (0, −1, 0), n

^{(5)}= (0, 0, +1), n

^{(6)}= (0, 0, −1). Applying the directions of the outward unit vectors, the operational equation for R

_{x}is:

**R**, in the x-direction i.e., ${F}_{D}=-{R}_{x}$ and $\left|{F}_{D}\right|=\left|{R}_{x}\right|={R}_{x}$.

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**Figure 1.**General depiction of the control volume in the x-y and y-z planes. Black thick arrow indicates the flow direction.

**Figure 2.**(

**a**) Channel cross-section. Units are expressed in metres; (

**b**) view from downstream to upstream.

**Figure 3.**The grid of the measuring points, applied at two elevations for both (

**a**) SA_03 and (

**b**) SA_04. The flow is from left to right.

**Figure 4.**Distribution of time-averaged streamwise velocities and streamlines at z = z

_{0}for SA_04. Black lines represent the vertical surfaces of the control volume.

**Figure 8.**Net contributions (in percentage) of the terms in the right-hand side of Equation (4) to drag.

**Figure 9.**Net contributions to the drag coefficient for different positioning of the control section S

_{3}.

Test | h_{r} (-) | h_{fp} (m) | Q (ls^{−1}) | Q_{mc} (ls^{−1}) | Q_{fp} (ls^{−1}) | Fr_{mc} (-) | Fr_{fp} (-) | Re_{d} (-) |
---|---|---|---|---|---|---|---|---|

SA_03 | 0.31 | 0.045 | 58.9 | 42.3 | 16.6 | 0.46 | 0.40 | 13,800 |

SA_04 | 0.41 | 0.070 | 95.4 | 63.2 | 32.2 | 0.53 | 0.40 | 18,100 |

Reference | Flow Type | Configuration | Re_{d} | C_{d} | |
---|---|---|---|---|---|

SA_03 | compound-channel flow | 9-cylinder regular array (d_{0}/d = 2.22) | 13,800 | 0.74 | |

SA_04 | compound-channel flow | 9-cylinder regular array (d_{0}/d = 2.22) | 18,100 | 0.58 | |

Robertson (2016) [14] | open-channel flow | infinite regular array (x: d_{0}/d = 5.26, y: d_{0}/d = 2.63) | 9670 | 2.04 | |

Robertson (2016) [14] | open-channel flow | isolated cylinder in uniform flow | 10,000–22,000 | 2.11 | |

Lyn et al. (1995) [44] | closed water-channel flow | isolated cylinder in uniform flow | 21,400 | 2.10 | |

Yen and Liu (2011) [13] | air flow | isolated cylinder in uniform flow | 21,000 | 2.06 | |

Robertson (2016) [14] | open-channel flow | pairs side-by-side (2 < d_{0}/d < 3) | 5600–12,800 | 2.59–3.28 | |

Yen and Liu (2011) [13] | air flow | pairs side-by-side (d_{0}/d = 2.5) | 21,000 | 1.90 | |

Robertson (2016) [14] | open-channel flow | pairs tandem (2 < d_{0}/d <3) | 5600–12,800 | 0.92–1.15 | |

Yen et al. (2008) [12] | air flow | pairs tandem (d_{0}/d = 3) | 900–1200 | 0.50 | |

Yen et al. (2008) [12] | air flow | pairs tandem (d_{0}/d = 1.5) | 900–1200 | 1.15 |

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**MDPI and ACS Style**

Ferreira, R.M.L.; Gymnopoulos, M.; Prinos, P.; Alves, E.; Ricardo, A.M.
Drag on a Square-Cylinder Array Placed in the Mixing Layer of a Compound Channel. *Water* **2021**, *13*, 3225.
https://doi.org/10.3390/w13223225

**AMA Style**

Ferreira RML, Gymnopoulos M, Prinos P, Alves E, Ricardo AM.
Drag on a Square-Cylinder Array Placed in the Mixing Layer of a Compound Channel. *Water*. 2021; 13(22):3225.
https://doi.org/10.3390/w13223225

**Chicago/Turabian Style**

Ferreira, Rui M. L., Miltiadis Gymnopoulos, Panayotis Prinos, Elsa Alves, and Ana M. Ricardo.
2021. "Drag on a Square-Cylinder Array Placed in the Mixing Layer of a Compound Channel" *Water* 13, no. 22: 3225.
https://doi.org/10.3390/w13223225