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Hydrodynamic coefficients of yawed cylinders in open-channel flow
Flow Measurement and Instrumentation 65 (2019) 288–296
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Hydrodynamic coefficients of yawed cylinders in open-channel flow Elisabetta Persi , Gabriella Petaccia, Andrea Fenocchi, Sauro Manenti, Paolo Ghilardi, Stefano Sibilla ⁎
T
Department of Civil Engineering and Architecture, University of Pavia, Pavia 27100, Italy
ARTICLE INFO
ABSTRACT
Keywords: Drag coefficient Side coefficient Hydrodynamic balance Finite Yawed Cylinder Cylinder submergence Large wood
The hydrodynamic response of a cylinder in an open-channel flow is experimentally investigated by considering the effects of horizontal orientation and submergence, the latter ranging from fully submerged to half-submerged cases. Drag and side coefficients are measured with a specific hydrodynamic balance which allows the planar rotation and the variation of submergence of the sample. The instrument design and characteristics are discussed, also estimating the uncertainties of force and hydrodynamic coefficients measurement. The tests are carried out under four submergence ratios (SR, submerged height to water depth), showing that for lower SR the drag and side coefficients increase with respect to tests performed on a fully submerged cylinder, which correspond to the highest SR. The tests performed for a half-submerged cylinder show instead a reduction of the side and drag coefficients, mainly due to the reduced submerged section. The effect of orientation on the two coefficients is different, yet it leads to similar submergence-independent trends. The drag coefficient is maximum for the 90° crossflow configuration, where maximum flow deviation occurs, while the side coefficient is maximum for the 45° orientation, which shows the most asymmetrical flow pattern. Measurements for deep submergence are validated against available literature results, confirming the reliability of the obtained values.
1. Introduction The interaction between cylinders and fluids has been widely investigated in the past with reference to different engineering applications [45], from cables [28,46], to offshore structure design [9,33] or elongated particles in industrial flows [14,44]. The analysis of the flow distribution around cylinders is also performed to study flow interaction with cylindrical bridge piers [1,25] or to understand scour mechanisms around horizontal cylinders [2]. In this paper, the focus is on finite cylinders in open channel flows, floating horizontally on the water surface. The hydrodynamic response of cylinders is especially investigated with reference to the transport of Large Wood (LW, e.g. [4,21]) during floods. Wooden elements are naturally present in riverine environments, positively contributing to the habitat and morphology [16,34]. However, they also represent a source of hydraulic risk, since they can clog bridge openings, hence increasing upstream water level. Standard simulations of flood propagation do not include these effects, even if wood transport could affect the flood-prone areas in various situations (e.g. dam breaks, dam breaches and flash floods, see [22,23]). Some attempts at modelling wood transport in urban areas, where in-line structures are more common, can be found in literature [13,26]. Ruiz-Villanueva et al. [35]
⁎
showed how flooded areas increase when wooden elements are included in hydraulic simulations, highlighting the need to take into account LW transport to provide adequate estimates of the residual risk and of the connected hazard to people and structures. Aiming at including wood transport in the analysis of flood risk, researchers have recently been working on the numerical modelling of cylindrical body transport, disregarding branches and root wads which may be removed by collision, or may not be present due to harvesting or fire [6,7,36]. Numerical simulations have been used to predict the effects of bridge clogging and to design wood trapping structures [12,24], in addition to computing the incipient motion of floating logs [5,8,35] and their trajectories [3,39]. Following a dynamic approach, Petaccia et al. [32] and Persi et al. [29,30] proposed a model in which the motion of floating logs is simulated as the planar displacement of cylinders on the water surface. Log acceleration is computed from the hydrodynamic force exerted on the body by the flow. An adequate estimation of the components of such force, i.e. the drag force in the flow direction and the side force perpendicular to it, can only be obtained if proper drag and side coefficients are assumed. Drag and side coefficients are commonly expressed as functions of the Reynolds number [11], yet several other variables should be
Correspondence to: Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento 38123, Italy. E-mail address: [email protected] (E. Persi).
https://doi.org/10.1016/j.flowmeasinst.2019.01.006 Received 18 May 2018; Received in revised form 14 December 2018; Accepted 6 January 2019 Available online 09 January 2019 0955-5986/ © 2019 Elsevier Ltd. All rights reserved.
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considered, such as the body features, i.e. its slenderness and roughness [19]. Some researchers already studied the drag coefficient of logs, yet they focused on the case of vertical cylinders to resemble planted vegetation [27,37,40]. Past studies on the hydrodynamic coefficients of horizontal cylinders were performed with laboratory [15,41,42] or field experiments on full-sized logs [10,20,38]. Gippel et al. [15] measured the drag coefficient of a finite cylinder as function of cylinder slenderness (in the range 6.6–21), shape and orientation, with Fr = 0.35–0.63 and flow Reynolds numbers Re = 104–105. Shields and Alonso [38] measured the drag of real-sized logs in a natural open channel, using PVC, oak and hackberry samples (L = 1.80 m, D = 0.22 m and slenderness ≈ 8) with Fr = 0.43 and Rep = 2.7 × 105. The variation with orientation, body submergence and blockage ratio was considered in both the papers. Hayashi and Kawamura [17] computed the drag coefficient of a yawed cylinder in a wind tunnel by measuring the pressure distribution on its surface, focusing on the drag coefficient perpendicular to the cylinder axis and showing its independence from the yaw angle. A landmark study for the estimation of the hydrodynamic coefficients for yawed cylinders is the work by Hoang et al. [18], in which the drag, lift and side coefficients for an infinite inclined cylinder (aluminium bar with slenderness = 18–34, variable vertical and horizontal inclination and Re = 103–104) were measured in a wind tunnel. The employed sample was fixed at the channel walls, disregarding tip effect, and was inclined both vertically and horizontally. Although the experimental configuration differs from that of the transport of floating bodies, i.e. finite cylinders floating on the water surface, their results are the only ones available in literature which report a detailed description of orientation effects on the hydrodynamic coefficients. All the above-cited studies refer mainly to fully submerged cylinders, which do not fit to the real conditions of wood transport, in which logs are generally partially submerged, with different degrees of submergence depending on wood density. The main aim of this paper is to increase the knowledge of the forces acting on a floating cylinder, presenting and discussing experimental measurements of the drag and side coefficients of yawed cylinders placed with various submergences. Experiments were performed using a built-on-purpose hydrodynamic balance, whose accuracy and reliability were assessed after a calibration process and a validation against literature data for the submerged cylinder case.
The balance is based on leverages, the fulcrum being a spherical joint. A steel bar (diameter = 0.010 m, length = 0.10 m) is fixed to the fulcrum and is connected to a slenderer aluminium bar (diameter = 0.006 m, length = 0.19 m), to which the sample is screwed. A 3Dprinted block is fixed to the other end of the steel bar and leans against the load cells, which measure the local forces which balance the moment around the fulcrum induced by the hydrodynamic forces acting on the sample. The fulcrum and the load cells are hosted on an aluminium plate, which is connected to two aluminium rails that allow vertical displacement, to change the submergence of the sample. These vertical rails are fixed to a horizontal one, secured to the sides of the flume. In the present experiments, the sample was always kept at the centre of the flume. The length and the diameter of the two metal bars were chosen to minimise sample tilting and bar bending under maximum hydrodynamic force, so that the sample could be always assumed to be horizontal and the bars to be rigid. The aluminium bar is thinner to minimise the effect of its wake on the flow when the cylinder is totally submerged. The 3D-printed block (red block in Fig. 2) acts on four 3D-printed Lshaped indifferent levers (white devices in Fig. 2b), which turn the horizontal forces into vertical ones, as the adopted Phidgets Micro Load Cells (0–780 g) are sensitive only to vertical loads. The load cells are fixed to four independent horizontal bases, to be strained by the Lshaped leverages with a vertical force (Fig. 2b). The drag force is obtained from the force measured by Cell 2, the side force from those measured by Cells 1 and 3, while Cell 0 has mainly a structural role, to avoid any side misplacement of the block for the parallel and perpendicular configurations. The datasheet of the Phidgets Micro Load Cells (0–780 g) is reported in Table 1. 2.1. Test configuration Flow conditions are constant in all experiments (Table 2). The adopted flow depth h = 0.15 m ensures a significant distance of the sample from the channel bottom even for the configuration with the highest submergence. The ratio between the distance from the bottom and the sample diameter varies from 3.7 for the maximum submergence case to 6.3 for the half-submerged configuration. These values ensure a small interaction between bottom shear and the sample in the present experiments. Previous studies investigated ranges out of our interval (7–14 for [42]; 0–3 for [38]; 0–1.96 for [41]). The blockage ratio (i.e. the ratio of the maximum transverse area of the sample to the effective area of the flow) in our flume is ~5%, slightly smaller than the optimal maximum value of 6% proposed by West and Apelt [43]. This further confirms the negligible influence of the channel boundary on our results. Tests are performed under subcritical and fully turbulent vortex V sheet conditions, as per the Froude number (Fr = where V is the
2. Experimental setup Hydrodynamic coefficients (drag, lift or side) are usually measured within wind tunnels [18]. For the case of floating logs, measurements need instead to be carried out in laboratory flumes [15,20], using a hydrodynamic balance. For this purpose, a custom-made hydrodynamic balance was designed, combining reduced costs and adaptability to different configurations. Experiments were performed in a 9.40 m long and 0.49 m wide flume (Fig. 1), with horizontal metallic bottom and Polymethyl methacrylate (PMMA) sides [31]. The discharge is measured upstream through a Thomson weir. To dissipate the energy, turbulence and surface waves of the falling flow, proper obstacles are positioned downstream of the weir (Fig. 1). The water level and hydraulic profile inside the channel are controlled through a sluice gate in the final section. A piezometer measures the local water level at the section where the hydrodynamic balance is installed. The undisturbed flow depth measured prior to plunging the log sample is considered as reference level The adopted hydrodynamic balance (Fig. 2) measures the two horizontal components of the force on a cylindrical sample, which has diameter D = 0.024 m and length L = 0.15 m. These are the drag force parallel to the main flow and the side force perpendicular to it. The lift force cannot be measured by the designed instrument. Omitting this component is justified since the cylinder is negligibly vertically tilted, so that the lift force is significantly lower than the drag and side ones.
gh
undisturbed flow velocity and where g is the gravitational acceleration) VD and the particle Reynolds number (Rep = where D is the cylinder diameter and ν is the water kinematic viscosity). The particle Froude V number has a subcritical value (Frp = gD = 0.83). A vortex shedding
frequency of 3.33 Hz is estimated from the Strouhal number, corresponding to a 0.30 s period. In order to assess the effect of submergence on the drag and side coefficients, four configurations with different submergence ratios (SR) are tested. SR is computed as follows:
SR = 1
htop h
(1)
where htop is the distance between the top of the sample and the flume bottom and h is the undisturbed flow depth. Fig. 3 provides a sketch of the four configurations with the associated submergence ratios. Note that for the half-submerged case the cylinder axis is placed at the 289
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Fig. 1. Sketch of the laboratory flume. The locations of the hydrodynamic balance, piezometers, weir and energy dissipation obstacles (A, metal nets; B, dissipation basins realized with concrete blocks and masonry bricks; C, floating breakwater device) are shown.
undisturbed water level, so that the top of the cylinder is above the water surface (htop > h), resulting in a negative SR. The orientation of the cylinder on the horizontal plane is varied for each configuration within the 0–90° range, i.e. from parallel to the flow to perpendicular, with a 10°-interval. Each test lasts 30 s, 100 times the shedding period. The sampling rate is 0.008 s, resulting in 3750 measures per load cell for each test, from which mean and standard deviation values for practical use are obtained. The tests are repeated 5 times for each configuration. Load cells provide voltage values, to be transformed into the force exerted by the fluid on the cylinder with the obtained calibration curves. For the cases with SR = 0.33 and SR = 0.10, the aluminium bar which holds the sample is partially immersed. To avoid overestimation, the drag force on the immersed bar portion is estimated with experiments without the log sample and is subtracted from the total drag force measured by Cell 2. The forces measured on the immersed part of the bar are 0.024 N and 0.008 N for cases with SR = 0.33 and SR = 0.10, respectively. If the sample is yawed, it tends to displace laterally and rotate the 3D-printed block. Since it is always rotated in the counter-clockwise direction (Fig. 4), the side force is obtained by combining the measurements of Cells 1 and 3. The mean voltages measured by these two cells are first converted to the applied forces according to the calibration laws. The total side force is then obtained as difference between the side force on Cell 1 and that on Cell 3. Finally, the drag and side force coefficients are obtained as:
Table 1 Datasheet of the Phidgets Micro Load Cells (0–780 g) (available on https://www.phidgets.com). Max weight capacity
780 g
Max load Creep Max difference + /- at zero balance Max repeatability error Non-linearity & Hysteresis Temperature effect
936 g 1.6 g/h ± 11.7 g ± 390 mg 390 mg 39 mg/°C
Table 2 Undisturbed experimental flow conditions. Discharge Flow depth Water velocity Blockage ratio Froude number Particle Reynolds number Particle Froude number Strouhal number
CD =
CS =
Q [m3 s-1] h [m] V [m s−1] B [%] Fr [dimensionless] Rep [dimensionless] Frp [dimensionless] St [dimensionless]
0.029 0.15 0.40 4.95 0.33 1E+04 0.83 0.20
FD 1 2 f
FS1 1 2 f
AV 2
(2)
FS3 AV 2
(3) Fig. 2. Hydrodynamic balance: a) downstream view of the balance installed inside the flume; b) detail of the configuration of the load cells (Cell 2 measures the drag force; Cells 0, 1, 3 measure the side force) and of the 3D-printed block; c) scheme of the leverage functioning. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
290
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Fig. 3. Sketches of the tested configurations with corresponding submergence ratio SR. The grey line is the undisturbed water surface h. The position of the cylinder upper profile htop is also shown.
where A = LD is the longitudinal section of the sample, and ρf is the fluid density. Note that the area A = LD is employed as reference area for the computation of the drag and side coefficients for all configurations, regardless of sample submergence and orientation.
such velocity in the standard expression of the hydrodynamic force, the generic hydrodynamic coefficient C is:
C=
2.2. Hydrodynamic balance calibration
8 0.6 tg (18°) 2g h w5/2 = Kh w5/2 15
K2
A W2h2 h w5
(5)
where, beyond known symbols K is the constant obtained from Eq. (4), hw is the water head on the weir measured at the upstream piezometer, F is the generic hydrodynamic force and W is the flume width. The velocity uncertainty is thus derived by the uncertainty in water level measurements both upstream of the weir and inside the flume. The relative uncertainty in the estimation of the drag and side coefficients is then computed according to the standard expression for uncertainty:
The hydrodynamic coefficients are measured under steady conditions. A static calibration approach is hence followed for the hydrodynamic balance. Static forces are applied in axial directions to stress each load cell. The calibration of the instrument is performed by applying known forces in the range F = 0.05–2 N to one end of the sample. Calibration weights are connected to the end of the sample through an inextensible wire and a frictionless pulley, so that the vertical force exerted by the mass pulls the sample in the desired direction (Fig. 5a). The pulley stands alone thanks to a wooden frame and can be placed either parallel or perpendicular to the balance to calibrate all the cells. Due to the positioning of the cells (Figs. 2b and 5b), Cell 2 (measuring drag force) and Cell 1 (measuring side force) are stressed independently, while Cells 0 and 3 (also measuring side force) are stressed simultaneously. The force acting on the sample Fsample is computed as the force measured by the cell Fsample times the ratio between the leverage arms Lc/Ls. Data are acquired through the PhidgetBridge 4-Input USB interface, a 24 bit analog-to-digital converter with 4 channels and 8-ms data rate. The mean and standard deviation values are computed for each test weight. The complete calibration procedure is repeated 5–7 times for each cell to verify measurement repeatability. Linear regression is performed over the aggregate set of points from the calibration runs (e.g. Fig. 6). The calibration equations for the load cells of the hydrodynamic balance are listed along with their determination coefficients R2 in Table 3. The uncertainty in the measurement of the hydrodynamic coefficients depends on the uncertainties in force measurement and in velocity estimation. The discharge is computed through a triangular notchweir equation adapted to the specific weir:
Q=
F 1 2 f
C = C
F C F C F F
2
+
hw C hw C hw h
2
+
h C h C h h
2
(6)
in which the relative uncertainties are to be estimated, while the partial derivatives can be calculated from Eq. (5) as:
C = F
1 1 2 f
C = hw
K2
A W2h2 h w6
5
C =2 1 h 2
(7a)
1 1 2 f
K2
A W2h2 h w4
(7b)
1 K2
5 f A W2h h w
(7c)
The relative uncertainties on water level measurements are computed by dividing the likely direct measurement resolution, δh = 0.001 m, either by the water head on the weir (hw = 0.32 m) or by the flow depth of the experiments (h = 0.15 m), resulting in δhw/hw = 0.3% and δh/h = 0.7% respectively. For the drag coefficient, the relative uncertainty δFd/Fd is derived by dividing the standard deviation obtained in the measurement of the maximum drag force of the whole experimental campaign by the maximum value of the force itself (Fd = 0.34 N). A relative uncertainty δFd/Fd = 7.8% results, the standard deviation being σ(Fd) = 0.027 N. The resulting relative uncertainty in the estimation of the drag coefficient is ∂Cd/Cd = 8.1%. The same procedure is adopted for the side coefficient. A side force relative uncertainty δFs/Fs = 11.3% results
(4)
The mean velocity is then obtained by dividing the discharge by the wetted area of the flume for the undisturbed water level. By substituting
Fig. 4. Free body diagrams on the sample, on the 3D-printed block and on the cells. 291
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Fig. 5. a) Scheme of force application for calibration; b) detail of the balance, with 3D-printed block and positioning of the load cells.
more disturbed by the sample. The alteration becomes more noticeable for the zero-submergence case (Fig. 3c), in which water still flows over the body. The strongest alteration is obtained for the half-submerged cylinder case (Fig. 3d), with surface waves and abrupt changes in the water level. In addition, the flow does not completely overtop the sample. Photographs taken during the experimental campaign are available as Supplementary material and depict the perturbations of the free surface under the tested configurations. The experimental data have been interpolated with sine and cosine functions for the drag and side coefficients, respectively. Since the data are not normalized, nor their variation is totally explained by pure trigonometrical functions, four parameters are included in the equations to modify amplitude and frequency, and to translate the sine and cosine functions both horizontally and vertically. The parameters were determined with the Curve Fitting Toolbox in MATLAB (The MathWorks, Inc.). The effect of submergence on the drag and side coefficients is highlighted by the variation of these parameters, especially between the totally submerged and half-submerged cases. The resulting equations are reported in Table 4, while the interpolation curves are presented in Fig. 9. For the highest submergence SR = 0.33, the drag coefficients are lower than those for SR = 0.10 and SR = 0.0, especially for yaw angles ϑ > 30°. The coefficients for the SR = 0.10 and SR = 0.0 configurations are almost equal, though slightly higher values are obtained for zero submergence for the lowest angles. It is difficult to discern between the two configurations, because data are quite disperse and overlapping (Figs. 7 and 8). In any case, the balance records that the interaction with the water surface affects the drag force, which increases due to the growing disturbance of the free surface, highlighted by higher waves upstream of the sample (see Supplementary material). This effect is reported in literature [38,42] and increases for lower submergence ratios. The force exerted on the sample diminishes significantly for the partially submerged case. This is mainly due to the hydrodynamic force acting on a portion of the volume of the sample, whereas the reference area for the computation of the force coefficients (Eqs. (1) and (2)) is kept constant, equal to LD. The reduction of the drag coefficient is however not proportional to the theoretical halving of the wetted area of the cylinder. Table 5 reports the ratio between the measured drag coefficients for the half-submerged (SR = -0.08) and deeply submerged (SR = 0.33) cases, highlighting an increasing difference between their values for increasing angles. The small difference between the drag coefficients for the four submergence configurations for the lower orientations (0–30°) is not of straightforward interpretation. In those cases, the force measured by the balance is very small, due to the flow-facing area being basically equal to the cylinder base area. It is hence possible that inner friction inside the balance hides the small drag variation with submergence. When the disturbance of the free surface is evident, as in the zero-
Fig. 6. Calibration points for Cell 1 and linear regression line. Table 3 Calibration equations for the load cells of the hydrodynamic balance (y stands for the cell response [mV/V] and x for the applied force [N]) and related determination coefficients. Cell
Equation
0 1 2 3
y y y y
= = = =
0.386x 0.355x 0.322x 0.343x
R2
0.006 0.036 0.053 0.027
0.999 0.999 0.999 0.999
from the maximum measured side force Fs = 0.20 N and the related standard deviation σ(Fs) = 0.023 N. The relative uncertainty in the estimation of the side coefficient is then ∂Cs/Cs = 11.7%. 3. Results and discussion The water surface around the sample changes according to the cylinder orientation and submergence. Alterations of the water surface grow with the yaw angle for any submergence, being maximum for the 90° crossflow configuration. The rising interaction with the water surface is echoed by the values of the drag and side coefficients, shown in Figs. 7 and 8 for the four tested configurations. For all configurations, the drag coefficient increases from the aligned to the crossflow configuration. The side coefficient is maximum for 45° orientation. Its values are almost symmetrically distributed around the maximum and are almost null for 0° and 90° orientations, in which the flow around the cylinder is symmetrical. The residual measured forces for these limit conditions are due to sample vibrations. The effect of submergence is evident from water surface perturbations. A wave just downstream of the bar is observed when the cylinder is deeply submerged (Fig. 3a). A wave upstream of the bar is instead observed for lower submergence (Fig. 3b), in which the water surface is 292
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Fig. 7. Drag coefficient as a function of orientation, measured for a) SR = 0.33, b) SR = 0.10, c) SR = 0.0, d) SR = -0.08. ϑ = 0° stands for aligned flow, ϑ = 90° for crossflow.
cosine functions makes the relation between the variation of the coefficients with orientation and sample area projections explicit. The drag coefficient is proportional to sin ( ) , thus to the transversal projection of the sample, while the side coefficient is proportional to cos ( ) , hence to the longitudinal projection of the sample (refer to Fig. 4 for the reference orientation). Despite that, the sine and cosine functions introduce some inaccuracies, which contrast with the expected physical behaviour, but do not affect the overall comparison of results among tested configurations. They also provide continuous, readily applicable expressions, which can be employed e.g. inside numerical models of LW transport
submergence configuration, a difference can be noticed even at small angles. As regards the variation of the side coefficient with submergence and orientation (Fig. 9b), the interaction with the free surface increases the side force, as for the drag. The small and null submergence cases therefore display higher side coefficients than the deeply submerged case, whereas lower values are obtained for the half-submerged condition. The ratio of the side coefficient measured in the half-submerged case with respect to the measures of the deeply submerged case for the tested orientations is shown in Table 6. The half-submerged side coefficient is 50–70% of that for deep submergence, still not proportional to the halving of the wetted area of the cylinder. The performed interpolation of experimental data with sine and
Fig. 8. Side coefficient as a function of orientation, measured for a) SR = 0.33, b) SR = 0.10, c) SR = 0.0, d) SR = -0.08. ϑ = 0° stands for aligned flow, ϑ = 90° for crossflow. 293
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Table 4 Interpolating functions for the drag and side coefficients for each submergence ratio, along with the relative determination coefficients R2. CD SR = 0.33
( 0.584sin ( 0.546sin ( 0.241sin ( 0.478sin
SR = 0.10 SR = 0.00 SR = −0.08
0.491 0.474
0.457
0.496
) 92.98) + 0.731 94.65) + 0.736 78.16) + 0.385 86.50 + 0.630
R2
CS
0.983
( 0.350cos ( 0.336cos ( 0.149cos (
0.984 0.982
CD ( ) = CD (90) sin3/2
for 50°
)] for 30° 90°
50°
0.305
0.311 0.279
0.968
) 148.38) + 0.309 148.36) + 0.304 166.19) + 0.173 159.32 + 0.276
0.984 0.964 0.954
ϑ [°]
0
10
20
30
40
45
50
60
70
80
90
CD (d ) CD (a )
97%
94%
82%
73%
67%
65%
63%
60%
59%
57%
56%
Table 6 Ratio between the half-submerged SR = -0.08 and the deeply submerged SR = 0.33 side coefficients for the tested orientations. ϑ [°]
10
20
30
40
45
50
60
70
80
CS (d ) CS (a )
67%
61%
59%
58%
57%
56%
55%
54%
58%
= 45°. The differences among the present results and the values from literature may be caused by the different roughness of the sample (PVC in Gippel et al. [15], PVC and wood with bark in Shields and Alonso [38], aluminium in Hoang et al. [18], and smooth wood in the present experiments) and by the different experimental conditions. The increase in the drag coefficient due to the vicinity to the water surface can be analysed by evaluating the wave drag coefficient, Cw, as proposed in Wallerstein et al. [42] and Shields and Alonso [38]. The coefficient is herein computed as the difference between the mean measured drag coefficients for SR = 0.10 and SR = 0.0 and that for deeply submerged cylinder SR = 0.33 in the crossflow case ϑ = 90°. Fig. 11 shows the variation of the wave drag coefficient with respect to the ratio z/D, where z is the distance between the water surface and the cylinder axis. The measurements by Wallerstein et al. [42] are the upper limit for cylinders with slenderness ≈ 7, since the measurement were performed for Frp ≈ 0.5 (PVC logs with D = 0.019 m and L = 0.15 m). The present results and those by Shields and Alonso [38] were both obtained for Frp ≈ 0.8, so that a decrease is expected. This is because wave drag decreases for particle Froude number values approaching unity [42]. However, while in literature wave drag values
(8) (9)
As regards the side coefficients, the comparison with the results of Hoang et al. [18] for an infinite yawed cylinder is given in Fig. 10b for horizontal yaw angles ϑ = 25–155°. To allow comparisons with our results, the function in Hoang et al. [18] for null vertical angle is considered, although this value is out of the tested limits. The adopted function is:
Cs ( ) = 0.575sin (2 )
0.296
Table 5 Ratio between the half-submerged SR = -0.08 and the deeply submerged SR = 0.33 drag coefficients for the tested orientations.
Laboratory or field measurements of the drag and/or side coefficients of a yawed cylinder can be found in literature [15,38,18]. Fig. 10a compares the experimental drag coefficients obtained for the deeply submerged cylinder SR = 0.33 case (slenderness = 6, Fr = 0.33, Re = 104) with those from Gippel et al. [15] and Shields and Alonso [38], corrected to refer to the same LD reference area used for the present measurements. The correction was performed by multiplying the literature drag coefficient by the sine of the angle ϑ, i.e. lumping the effect of projection into the non-dimensional coefficient. The function of the drag coefficient estimated by Hoang et al. [18] for yaw angles ϑ = 30–90° is also plotted. The value of CD(90) is set equal to 0.928, interpolated from the measurements of Hoang et al. [18] at a Reynolds number equal to 104. The considered functions by Hoang et al. [18] are:
CD ( ) = CD (90)[sin3/2 + 0.5sin3/2 (50
0.291cos
0.993
3.1. Comparison with literature results
R2
(10)
The trend of the herein measured drag coefficients follows that of literature experiments, though being slightly higher especially for lower angles. A wide scatter is observed in literature data, especially for the highest yaw angles, and is connected to the different employed samples. The drag coefficients from Hoang et al. [18] cross the lower limit of literature experimental results until ϑ = 50°, and then follow a flattened S-shaped curve. The interpolation of the experimental side coefficients nearly overlaps the literature results. In this case, the side coefficients for ϑ < 30° are plotted as the trend is assumed to be symmetrical around ϑ
Fig. 9. Drag (a) and side (b) coefficients for the four tested configurations; SR is the submergence ratio, as per Fig. 3. 294
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Fig. 10. a) Comparison of the experimentally obtained drag coefficients for the deeply submerged configuration SR = 0.33 with the results of Gippel et al. [15] and Shields and Alonso [38] and the function of Hoang et al. [18]; b) Comparison of the experimentally obtained side coefficients for the deeply submerged SR = 0.33 configuration and the estimate function from Hoang et al. [18].
hydrodynamic coefficients, due to the halving of the wetted area of the cylinder. Such reduction, however, is not proportional to the decrease of the sample wetted area and depends on the sample orientation. Both hydrodynamic coefficients vary with the yaw angle and can be interpolated with sine and cosine functions, which explicate the dependence of the coefficients on the sample orientation. The drag coefficient is minimum when the cylinder is aligned to the flow and reaches a maximum for higher angles. The side coefficient is almost symmetric around the maximum at 45°, being nearly null when the cylinder is aligned or perpendicular to the flow. The experimentally obtained hydrodynamic coefficients for the deeply submerged cylinder case are in good agreement with existing literature studies, showing that the custom-made hydrodynamic balance is a reliable instrument. Thanks to the relative easiness of manufacture, the reduced costs compared to ready-made commercial devices and the good accuracy, the instrument is suitable for experimental measurements related to a variety of fluid-body interaction phenomena. However, when focusing on the effects of submergence and on the wave drag coefficient, it appears that our overall laboratory setting is not accurate enough to detect the small differences due to the surface waves for the two configurations with the lower submergence. These however should not be attributed only to the inaccuracy of the hydrodynamic balance, but also to uncertainties in the sample vertical positioning. The results of this experimental campaign contribute to understanding the hydrodynamic behaviour of finite cylinders in openchannel flows, under different submergence conditions, and could aid numerical modelling of large wood transport. The present results can be applied to full-scale turbulent subcritical flows, assuming the independence of the hydrodynamic coefficients from the particle Reynolds number and the flow Froude number. Additional experiments would be however needed to remove this simplification, especially dealing with the wave drag coefficient, also assessing the effect of the sample slenderness ratio.
Fig. 11. Wave drag coefficients for literature data and for the present experiments as function of the relative distance from the water surface.
decrease with increasing submergence, our measurements show a constant trend, with the value for SR = 0.0 being smaller than expected. A reason for this discrepancy may lay in the accuracy of our laboratory setting, which is not able to distinguish the small differences between the drag coefficients for SR = 0.10 and SR = 0.0, the experimental conditions differing by a vertical displacement of 15.5 mm only. This means that the previously observed differences (Fig. 9) are related both to measurement uncertainties of the balance and to inaccuracies in the sample vertical positioning, so that the results from these two sets of experiments cannot be truly discerned. Some additional insight about the comparison of the different drag coefficients may be obtained estimating the drag force with the specific momentum equation [41]. This method would need measurements of the water levels upstream and downstream of the cylinder, which were not presently performed. 4. Conclusions
Acknowledgments
This paper presents experimental measurements of the drag and side coefficients of a finite cylinder under different submergence and yaw angles. The description and calibration of the custom-made hydrodynamic balance are reported, estimating the measurement uncertainty. Experiments are performed under steady conditions, computing the hydrodynamic coefficients from the measured forces. The results in terms of drag and side coefficients are affected by the sample submergence and show that the interaction with the free surface plays a fundamental role, leading to an increase of the hydrodynamic coefficients from high to zero submergence, as also noticed by existing literature. The half-submerged case presents instead much lower
The authors thank Mr. Pierangelo Bergamaschi, who gave valuable help in the design of the hydrodynamic balance, and Dr. Gianluca Alaimo and Dr. Stefania Marconi from the Proto-Lab of the University of Pavia, who made the 3D-printed components of the instrument. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at doi.org/10.1016/j.flowmeasinst.2019.01.006. 295
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References
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Hydrodynamic coefficients of yawed cylinders in open-channel flow Elisabetta Persi , Gabriella Petaccia, Andrea Fenocchi, Sauro Manenti, Paolo Ghilardi, Stefano Sibilla ⁎
T
Department of Civil Engineering and Architecture, University of Pavia, Pavia 27100, Italy
ARTICLE INFO
ABSTRACT
Keywords: Drag coefficient Side coefficient Hydrodynamic balance Finite Yawed Cylinder Cylinder submergence Large wood
The hydrodynamic response of a cylinder in an open-channel flow is experimentally investigated by considering the effects of horizontal orientation and submergence, the latter ranging from fully submerged to half-submerged cases. Drag and side coefficients are measured with a specific hydrodynamic balance which allows the planar rotation and the variation of submergence of the sample. The instrument design and characteristics are discussed, also estimating the uncertainties of force and hydrodynamic coefficients measurement. The tests are carried out under four submergence ratios (SR, submerged height to water depth), showing that for lower SR the drag and side coefficients increase with respect to tests performed on a fully submerged cylinder, which correspond to the highest SR. The tests performed for a half-submerged cylinder show instead a reduction of the side and drag coefficients, mainly due to the reduced submerged section. The effect of orientation on the two coefficients is different, yet it leads to similar submergence-independent trends. The drag coefficient is maximum for the 90° crossflow configuration, where maximum flow deviation occurs, while the side coefficient is maximum for the 45° orientation, which shows the most asymmetrical flow pattern. Measurements for deep submergence are validated against available literature results, confirming the reliability of the obtained values.
1. Introduction The interaction between cylinders and fluids has been widely investigated in the past with reference to different engineering applications [45], from cables [28,46], to offshore structure design [9,33] or elongated particles in industrial flows [14,44]. The analysis of the flow distribution around cylinders is also performed to study flow interaction with cylindrical bridge piers [1,25] or to understand scour mechanisms around horizontal cylinders [2]. In this paper, the focus is on finite cylinders in open channel flows, floating horizontally on the water surface. The hydrodynamic response of cylinders is especially investigated with reference to the transport of Large Wood (LW, e.g. [4,21]) during floods. Wooden elements are naturally present in riverine environments, positively contributing to the habitat and morphology [16,34]. However, they also represent a source of hydraulic risk, since they can clog bridge openings, hence increasing upstream water level. Standard simulations of flood propagation do not include these effects, even if wood transport could affect the flood-prone areas in various situations (e.g. dam breaks, dam breaches and flash floods, see [22,23]). Some attempts at modelling wood transport in urban areas, where in-line structures are more common, can be found in literature [13,26]. Ruiz-Villanueva et al. [35]
⁎
showed how flooded areas increase when wooden elements are included in hydraulic simulations, highlighting the need to take into account LW transport to provide adequate estimates of the residual risk and of the connected hazard to people and structures. Aiming at including wood transport in the analysis of flood risk, researchers have recently been working on the numerical modelling of cylindrical body transport, disregarding branches and root wads which may be removed by collision, or may not be present due to harvesting or fire [6,7,36]. Numerical simulations have been used to predict the effects of bridge clogging and to design wood trapping structures [12,24], in addition to computing the incipient motion of floating logs [5,8,35] and their trajectories [3,39]. Following a dynamic approach, Petaccia et al. [32] and Persi et al. [29,30] proposed a model in which the motion of floating logs is simulated as the planar displacement of cylinders on the water surface. Log acceleration is computed from the hydrodynamic force exerted on the body by the flow. An adequate estimation of the components of such force, i.e. the drag force in the flow direction and the side force perpendicular to it, can only be obtained if proper drag and side coefficients are assumed. Drag and side coefficients are commonly expressed as functions of the Reynolds number [11], yet several other variables should be
Correspondence to: Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento 38123, Italy. E-mail address: [email protected] (E. Persi).
https://doi.org/10.1016/j.flowmeasinst.2019.01.006 Received 18 May 2018; Received in revised form 14 December 2018; Accepted 6 January 2019 Available online 09 January 2019 0955-5986/ © 2019 Elsevier Ltd. All rights reserved.
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considered, such as the body features, i.e. its slenderness and roughness [19]. Some researchers already studied the drag coefficient of logs, yet they focused on the case of vertical cylinders to resemble planted vegetation [27,37,40]. Past studies on the hydrodynamic coefficients of horizontal cylinders were performed with laboratory [15,41,42] or field experiments on full-sized logs [10,20,38]. Gippel et al. [15] measured the drag coefficient of a finite cylinder as function of cylinder slenderness (in the range 6.6–21), shape and orientation, with Fr = 0.35–0.63 and flow Reynolds numbers Re = 104–105. Shields and Alonso [38] measured the drag of real-sized logs in a natural open channel, using PVC, oak and hackberry samples (L = 1.80 m, D = 0.22 m and slenderness ≈ 8) with Fr = 0.43 and Rep = 2.7 × 105. The variation with orientation, body submergence and blockage ratio was considered in both the papers. Hayashi and Kawamura [17] computed the drag coefficient of a yawed cylinder in a wind tunnel by measuring the pressure distribution on its surface, focusing on the drag coefficient perpendicular to the cylinder axis and showing its independence from the yaw angle. A landmark study for the estimation of the hydrodynamic coefficients for yawed cylinders is the work by Hoang et al. [18], in which the drag, lift and side coefficients for an infinite inclined cylinder (aluminium bar with slenderness = 18–34, variable vertical and horizontal inclination and Re = 103–104) were measured in a wind tunnel. The employed sample was fixed at the channel walls, disregarding tip effect, and was inclined both vertically and horizontally. Although the experimental configuration differs from that of the transport of floating bodies, i.e. finite cylinders floating on the water surface, their results are the only ones available in literature which report a detailed description of orientation effects on the hydrodynamic coefficients. All the above-cited studies refer mainly to fully submerged cylinders, which do not fit to the real conditions of wood transport, in which logs are generally partially submerged, with different degrees of submergence depending on wood density. The main aim of this paper is to increase the knowledge of the forces acting on a floating cylinder, presenting and discussing experimental measurements of the drag and side coefficients of yawed cylinders placed with various submergences. Experiments were performed using a built-on-purpose hydrodynamic balance, whose accuracy and reliability were assessed after a calibration process and a validation against literature data for the submerged cylinder case.
The balance is based on leverages, the fulcrum being a spherical joint. A steel bar (diameter = 0.010 m, length = 0.10 m) is fixed to the fulcrum and is connected to a slenderer aluminium bar (diameter = 0.006 m, length = 0.19 m), to which the sample is screwed. A 3Dprinted block is fixed to the other end of the steel bar and leans against the load cells, which measure the local forces which balance the moment around the fulcrum induced by the hydrodynamic forces acting on the sample. The fulcrum and the load cells are hosted on an aluminium plate, which is connected to two aluminium rails that allow vertical displacement, to change the submergence of the sample. These vertical rails are fixed to a horizontal one, secured to the sides of the flume. In the present experiments, the sample was always kept at the centre of the flume. The length and the diameter of the two metal bars were chosen to minimise sample tilting and bar bending under maximum hydrodynamic force, so that the sample could be always assumed to be horizontal and the bars to be rigid. The aluminium bar is thinner to minimise the effect of its wake on the flow when the cylinder is totally submerged. The 3D-printed block (red block in Fig. 2) acts on four 3D-printed Lshaped indifferent levers (white devices in Fig. 2b), which turn the horizontal forces into vertical ones, as the adopted Phidgets Micro Load Cells (0–780 g) are sensitive only to vertical loads. The load cells are fixed to four independent horizontal bases, to be strained by the Lshaped leverages with a vertical force (Fig. 2b). The drag force is obtained from the force measured by Cell 2, the side force from those measured by Cells 1 and 3, while Cell 0 has mainly a structural role, to avoid any side misplacement of the block for the parallel and perpendicular configurations. The datasheet of the Phidgets Micro Load Cells (0–780 g) is reported in Table 1. 2.1. Test configuration Flow conditions are constant in all experiments (Table 2). The adopted flow depth h = 0.15 m ensures a significant distance of the sample from the channel bottom even for the configuration with the highest submergence. The ratio between the distance from the bottom and the sample diameter varies from 3.7 for the maximum submergence case to 6.3 for the half-submerged configuration. These values ensure a small interaction between bottom shear and the sample in the present experiments. Previous studies investigated ranges out of our interval (7–14 for [42]; 0–3 for [38]; 0–1.96 for [41]). The blockage ratio (i.e. the ratio of the maximum transverse area of the sample to the effective area of the flow) in our flume is ~5%, slightly smaller than the optimal maximum value of 6% proposed by West and Apelt [43]. This further confirms the negligible influence of the channel boundary on our results. Tests are performed under subcritical and fully turbulent vortex V sheet conditions, as per the Froude number (Fr = where V is the
2. Experimental setup Hydrodynamic coefficients (drag, lift or side) are usually measured within wind tunnels [18]. For the case of floating logs, measurements need instead to be carried out in laboratory flumes [15,20], using a hydrodynamic balance. For this purpose, a custom-made hydrodynamic balance was designed, combining reduced costs and adaptability to different configurations. Experiments were performed in a 9.40 m long and 0.49 m wide flume (Fig. 1), with horizontal metallic bottom and Polymethyl methacrylate (PMMA) sides [31]. The discharge is measured upstream through a Thomson weir. To dissipate the energy, turbulence and surface waves of the falling flow, proper obstacles are positioned downstream of the weir (Fig. 1). The water level and hydraulic profile inside the channel are controlled through a sluice gate in the final section. A piezometer measures the local water level at the section where the hydrodynamic balance is installed. The undisturbed flow depth measured prior to plunging the log sample is considered as reference level The adopted hydrodynamic balance (Fig. 2) measures the two horizontal components of the force on a cylindrical sample, which has diameter D = 0.024 m and length L = 0.15 m. These are the drag force parallel to the main flow and the side force perpendicular to it. The lift force cannot be measured by the designed instrument. Omitting this component is justified since the cylinder is negligibly vertically tilted, so that the lift force is significantly lower than the drag and side ones.
gh
undisturbed flow velocity and where g is the gravitational acceleration) VD and the particle Reynolds number (Rep = where D is the cylinder diameter and ν is the water kinematic viscosity). The particle Froude V number has a subcritical value (Frp = gD = 0.83). A vortex shedding
frequency of 3.33 Hz is estimated from the Strouhal number, corresponding to a 0.30 s period. In order to assess the effect of submergence on the drag and side coefficients, four configurations with different submergence ratios (SR) are tested. SR is computed as follows:
SR = 1
htop h
(1)
where htop is the distance between the top of the sample and the flume bottom and h is the undisturbed flow depth. Fig. 3 provides a sketch of the four configurations with the associated submergence ratios. Note that for the half-submerged case the cylinder axis is placed at the 289
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Fig. 1. Sketch of the laboratory flume. The locations of the hydrodynamic balance, piezometers, weir and energy dissipation obstacles (A, metal nets; B, dissipation basins realized with concrete blocks and masonry bricks; C, floating breakwater device) are shown.
undisturbed water level, so that the top of the cylinder is above the water surface (htop > h), resulting in a negative SR. The orientation of the cylinder on the horizontal plane is varied for each configuration within the 0–90° range, i.e. from parallel to the flow to perpendicular, with a 10°-interval. Each test lasts 30 s, 100 times the shedding period. The sampling rate is 0.008 s, resulting in 3750 measures per load cell for each test, from which mean and standard deviation values for practical use are obtained. The tests are repeated 5 times for each configuration. Load cells provide voltage values, to be transformed into the force exerted by the fluid on the cylinder with the obtained calibration curves. For the cases with SR = 0.33 and SR = 0.10, the aluminium bar which holds the sample is partially immersed. To avoid overestimation, the drag force on the immersed bar portion is estimated with experiments without the log sample and is subtracted from the total drag force measured by Cell 2. The forces measured on the immersed part of the bar are 0.024 N and 0.008 N for cases with SR = 0.33 and SR = 0.10, respectively. If the sample is yawed, it tends to displace laterally and rotate the 3D-printed block. Since it is always rotated in the counter-clockwise direction (Fig. 4), the side force is obtained by combining the measurements of Cells 1 and 3. The mean voltages measured by these two cells are first converted to the applied forces according to the calibration laws. The total side force is then obtained as difference between the side force on Cell 1 and that on Cell 3. Finally, the drag and side force coefficients are obtained as:
Table 1 Datasheet of the Phidgets Micro Load Cells (0–780 g) (available on https://www.phidgets.com). Max weight capacity
780 g
Max load Creep Max difference + /- at zero balance Max repeatability error Non-linearity & Hysteresis Temperature effect
936 g 1.6 g/h ± 11.7 g ± 390 mg 390 mg 39 mg/°C
Table 2 Undisturbed experimental flow conditions. Discharge Flow depth Water velocity Blockage ratio Froude number Particle Reynolds number Particle Froude number Strouhal number
CD =
CS =
Q [m3 s-1] h [m] V [m s−1] B [%] Fr [dimensionless] Rep [dimensionless] Frp [dimensionless] St [dimensionless]
0.029 0.15 0.40 4.95 0.33 1E+04 0.83 0.20
FD 1 2 f
FS1 1 2 f
AV 2
(2)
FS3 AV 2
(3) Fig. 2. Hydrodynamic balance: a) downstream view of the balance installed inside the flume; b) detail of the configuration of the load cells (Cell 2 measures the drag force; Cells 0, 1, 3 measure the side force) and of the 3D-printed block; c) scheme of the leverage functioning. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
290
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Fig. 3. Sketches of the tested configurations with corresponding submergence ratio SR. The grey line is the undisturbed water surface h. The position of the cylinder upper profile htop is also shown.
where A = LD is the longitudinal section of the sample, and ρf is the fluid density. Note that the area A = LD is employed as reference area for the computation of the drag and side coefficients for all configurations, regardless of sample submergence and orientation.
such velocity in the standard expression of the hydrodynamic force, the generic hydrodynamic coefficient C is:
C=
2.2. Hydrodynamic balance calibration
8 0.6 tg (18°) 2g h w5/2 = Kh w5/2 15
K2
A W2h2 h w5
(5)
where, beyond known symbols K is the constant obtained from Eq. (4), hw is the water head on the weir measured at the upstream piezometer, F is the generic hydrodynamic force and W is the flume width. The velocity uncertainty is thus derived by the uncertainty in water level measurements both upstream of the weir and inside the flume. The relative uncertainty in the estimation of the drag and side coefficients is then computed according to the standard expression for uncertainty:
The hydrodynamic coefficients are measured under steady conditions. A static calibration approach is hence followed for the hydrodynamic balance. Static forces are applied in axial directions to stress each load cell. The calibration of the instrument is performed by applying known forces in the range F = 0.05–2 N to one end of the sample. Calibration weights are connected to the end of the sample through an inextensible wire and a frictionless pulley, so that the vertical force exerted by the mass pulls the sample in the desired direction (Fig. 5a). The pulley stands alone thanks to a wooden frame and can be placed either parallel or perpendicular to the balance to calibrate all the cells. Due to the positioning of the cells (Figs. 2b and 5b), Cell 2 (measuring drag force) and Cell 1 (measuring side force) are stressed independently, while Cells 0 and 3 (also measuring side force) are stressed simultaneously. The force acting on the sample Fsample is computed as the force measured by the cell Fsample times the ratio between the leverage arms Lc/Ls. Data are acquired through the PhidgetBridge 4-Input USB interface, a 24 bit analog-to-digital converter with 4 channels and 8-ms data rate. The mean and standard deviation values are computed for each test weight. The complete calibration procedure is repeated 5–7 times for each cell to verify measurement repeatability. Linear regression is performed over the aggregate set of points from the calibration runs (e.g. Fig. 6). The calibration equations for the load cells of the hydrodynamic balance are listed along with their determination coefficients R2 in Table 3. The uncertainty in the measurement of the hydrodynamic coefficients depends on the uncertainties in force measurement and in velocity estimation. The discharge is computed through a triangular notchweir equation adapted to the specific weir:
Q=
F 1 2 f
C = C
F C F C F F
2
+
hw C hw C hw h
2
+
h C h C h h
2
(6)
in which the relative uncertainties are to be estimated, while the partial derivatives can be calculated from Eq. (5) as:
C = F
1 1 2 f
C = hw
K2
A W2h2 h w6
5
C =2 1 h 2
(7a)
1 1 2 f
K2
A W2h2 h w4
(7b)
1 K2
5 f A W2h h w
(7c)
The relative uncertainties on water level measurements are computed by dividing the likely direct measurement resolution, δh = 0.001 m, either by the water head on the weir (hw = 0.32 m) or by the flow depth of the experiments (h = 0.15 m), resulting in δhw/hw = 0.3% and δh/h = 0.7% respectively. For the drag coefficient, the relative uncertainty δFd/Fd is derived by dividing the standard deviation obtained in the measurement of the maximum drag force of the whole experimental campaign by the maximum value of the force itself (Fd = 0.34 N). A relative uncertainty δFd/Fd = 7.8% results, the standard deviation being σ(Fd) = 0.027 N. The resulting relative uncertainty in the estimation of the drag coefficient is ∂Cd/Cd = 8.1%. The same procedure is adopted for the side coefficient. A side force relative uncertainty δFs/Fs = 11.3% results
(4)
The mean velocity is then obtained by dividing the discharge by the wetted area of the flume for the undisturbed water level. By substituting
Fig. 4. Free body diagrams on the sample, on the 3D-printed block and on the cells. 291
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Fig. 5. a) Scheme of force application for calibration; b) detail of the balance, with 3D-printed block and positioning of the load cells.
more disturbed by the sample. The alteration becomes more noticeable for the zero-submergence case (Fig. 3c), in which water still flows over the body. The strongest alteration is obtained for the half-submerged cylinder case (Fig. 3d), with surface waves and abrupt changes in the water level. In addition, the flow does not completely overtop the sample. Photographs taken during the experimental campaign are available as Supplementary material and depict the perturbations of the free surface under the tested configurations. The experimental data have been interpolated with sine and cosine functions for the drag and side coefficients, respectively. Since the data are not normalized, nor their variation is totally explained by pure trigonometrical functions, four parameters are included in the equations to modify amplitude and frequency, and to translate the sine and cosine functions both horizontally and vertically. The parameters were determined with the Curve Fitting Toolbox in MATLAB (The MathWorks, Inc.). The effect of submergence on the drag and side coefficients is highlighted by the variation of these parameters, especially between the totally submerged and half-submerged cases. The resulting equations are reported in Table 4, while the interpolation curves are presented in Fig. 9. For the highest submergence SR = 0.33, the drag coefficients are lower than those for SR = 0.10 and SR = 0.0, especially for yaw angles ϑ > 30°. The coefficients for the SR = 0.10 and SR = 0.0 configurations are almost equal, though slightly higher values are obtained for zero submergence for the lowest angles. It is difficult to discern between the two configurations, because data are quite disperse and overlapping (Figs. 7 and 8). In any case, the balance records that the interaction with the water surface affects the drag force, which increases due to the growing disturbance of the free surface, highlighted by higher waves upstream of the sample (see Supplementary material). This effect is reported in literature [38,42] and increases for lower submergence ratios. The force exerted on the sample diminishes significantly for the partially submerged case. This is mainly due to the hydrodynamic force acting on a portion of the volume of the sample, whereas the reference area for the computation of the force coefficients (Eqs. (1) and (2)) is kept constant, equal to LD. The reduction of the drag coefficient is however not proportional to the theoretical halving of the wetted area of the cylinder. Table 5 reports the ratio between the measured drag coefficients for the half-submerged (SR = -0.08) and deeply submerged (SR = 0.33) cases, highlighting an increasing difference between their values for increasing angles. The small difference between the drag coefficients for the four submergence configurations for the lower orientations (0–30°) is not of straightforward interpretation. In those cases, the force measured by the balance is very small, due to the flow-facing area being basically equal to the cylinder base area. It is hence possible that inner friction inside the balance hides the small drag variation with submergence. When the disturbance of the free surface is evident, as in the zero-
Fig. 6. Calibration points for Cell 1 and linear regression line. Table 3 Calibration equations for the load cells of the hydrodynamic balance (y stands for the cell response [mV/V] and x for the applied force [N]) and related determination coefficients. Cell
Equation
0 1 2 3
y y y y
= = = =
0.386x 0.355x 0.322x 0.343x
R2
0.006 0.036 0.053 0.027
0.999 0.999 0.999 0.999
from the maximum measured side force Fs = 0.20 N and the related standard deviation σ(Fs) = 0.023 N. The relative uncertainty in the estimation of the side coefficient is then ∂Cs/Cs = 11.7%. 3. Results and discussion The water surface around the sample changes according to the cylinder orientation and submergence. Alterations of the water surface grow with the yaw angle for any submergence, being maximum for the 90° crossflow configuration. The rising interaction with the water surface is echoed by the values of the drag and side coefficients, shown in Figs. 7 and 8 for the four tested configurations. For all configurations, the drag coefficient increases from the aligned to the crossflow configuration. The side coefficient is maximum for 45° orientation. Its values are almost symmetrically distributed around the maximum and are almost null for 0° and 90° orientations, in which the flow around the cylinder is symmetrical. The residual measured forces for these limit conditions are due to sample vibrations. The effect of submergence is evident from water surface perturbations. A wave just downstream of the bar is observed when the cylinder is deeply submerged (Fig. 3a). A wave upstream of the bar is instead observed for lower submergence (Fig. 3b), in which the water surface is 292
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Fig. 7. Drag coefficient as a function of orientation, measured for a) SR = 0.33, b) SR = 0.10, c) SR = 0.0, d) SR = -0.08. ϑ = 0° stands for aligned flow, ϑ = 90° for crossflow.
cosine functions makes the relation between the variation of the coefficients with orientation and sample area projections explicit. The drag coefficient is proportional to sin ( ) , thus to the transversal projection of the sample, while the side coefficient is proportional to cos ( ) , hence to the longitudinal projection of the sample (refer to Fig. 4 for the reference orientation). Despite that, the sine and cosine functions introduce some inaccuracies, which contrast with the expected physical behaviour, but do not affect the overall comparison of results among tested configurations. They also provide continuous, readily applicable expressions, which can be employed e.g. inside numerical models of LW transport
submergence configuration, a difference can be noticed even at small angles. As regards the variation of the side coefficient with submergence and orientation (Fig. 9b), the interaction with the free surface increases the side force, as for the drag. The small and null submergence cases therefore display higher side coefficients than the deeply submerged case, whereas lower values are obtained for the half-submerged condition. The ratio of the side coefficient measured in the half-submerged case with respect to the measures of the deeply submerged case for the tested orientations is shown in Table 6. The half-submerged side coefficient is 50–70% of that for deep submergence, still not proportional to the halving of the wetted area of the cylinder. The performed interpolation of experimental data with sine and
Fig. 8. Side coefficient as a function of orientation, measured for a) SR = 0.33, b) SR = 0.10, c) SR = 0.0, d) SR = -0.08. ϑ = 0° stands for aligned flow, ϑ = 90° for crossflow. 293
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Table 4 Interpolating functions for the drag and side coefficients for each submergence ratio, along with the relative determination coefficients R2. CD SR = 0.33
( 0.584sin ( 0.546sin ( 0.241sin ( 0.478sin
SR = 0.10 SR = 0.00 SR = −0.08
0.491 0.474
0.457
0.496
) 92.98) + 0.731 94.65) + 0.736 78.16) + 0.385 86.50 + 0.630
R2
CS
0.983
( 0.350cos ( 0.336cos ( 0.149cos (
0.984 0.982
CD ( ) = CD (90) sin3/2
for 50°
)] for 30° 90°
50°
0.305
0.311 0.279
0.968
) 148.38) + 0.309 148.36) + 0.304 166.19) + 0.173 159.32 + 0.276
0.984 0.964 0.954
ϑ [°]
0
10
20
30
40
45
50
60
70
80
90
CD (d ) CD (a )
97%
94%
82%
73%
67%
65%
63%
60%
59%
57%
56%
Table 6 Ratio between the half-submerged SR = -0.08 and the deeply submerged SR = 0.33 side coefficients for the tested orientations. ϑ [°]
10
20
30
40
45
50
60
70
80
CS (d ) CS (a )
67%
61%
59%
58%
57%
56%
55%
54%
58%
= 45°. The differences among the present results and the values from literature may be caused by the different roughness of the sample (PVC in Gippel et al. [15], PVC and wood with bark in Shields and Alonso [38], aluminium in Hoang et al. [18], and smooth wood in the present experiments) and by the different experimental conditions. The increase in the drag coefficient due to the vicinity to the water surface can be analysed by evaluating the wave drag coefficient, Cw, as proposed in Wallerstein et al. [42] and Shields and Alonso [38]. The coefficient is herein computed as the difference between the mean measured drag coefficients for SR = 0.10 and SR = 0.0 and that for deeply submerged cylinder SR = 0.33 in the crossflow case ϑ = 90°. Fig. 11 shows the variation of the wave drag coefficient with respect to the ratio z/D, where z is the distance between the water surface and the cylinder axis. The measurements by Wallerstein et al. [42] are the upper limit for cylinders with slenderness ≈ 7, since the measurement were performed for Frp ≈ 0.5 (PVC logs with D = 0.019 m and L = 0.15 m). The present results and those by Shields and Alonso [38] were both obtained for Frp ≈ 0.8, so that a decrease is expected. This is because wave drag decreases for particle Froude number values approaching unity [42]. However, while in literature wave drag values
(8) (9)
As regards the side coefficients, the comparison with the results of Hoang et al. [18] for an infinite yawed cylinder is given in Fig. 10b for horizontal yaw angles ϑ = 25–155°. To allow comparisons with our results, the function in Hoang et al. [18] for null vertical angle is considered, although this value is out of the tested limits. The adopted function is:
Cs ( ) = 0.575sin (2 )
0.296
Table 5 Ratio between the half-submerged SR = -0.08 and the deeply submerged SR = 0.33 drag coefficients for the tested orientations.
Laboratory or field measurements of the drag and/or side coefficients of a yawed cylinder can be found in literature [15,38,18]. Fig. 10a compares the experimental drag coefficients obtained for the deeply submerged cylinder SR = 0.33 case (slenderness = 6, Fr = 0.33, Re = 104) with those from Gippel et al. [15] and Shields and Alonso [38], corrected to refer to the same LD reference area used for the present measurements. The correction was performed by multiplying the literature drag coefficient by the sine of the angle ϑ, i.e. lumping the effect of projection into the non-dimensional coefficient. The function of the drag coefficient estimated by Hoang et al. [18] for yaw angles ϑ = 30–90° is also plotted. The value of CD(90) is set equal to 0.928, interpolated from the measurements of Hoang et al. [18] at a Reynolds number equal to 104. The considered functions by Hoang et al. [18] are:
CD ( ) = CD (90)[sin3/2 + 0.5sin3/2 (50
0.291cos
0.993
3.1. Comparison with literature results
R2
(10)
The trend of the herein measured drag coefficients follows that of literature experiments, though being slightly higher especially for lower angles. A wide scatter is observed in literature data, especially for the highest yaw angles, and is connected to the different employed samples. The drag coefficients from Hoang et al. [18] cross the lower limit of literature experimental results until ϑ = 50°, and then follow a flattened S-shaped curve. The interpolation of the experimental side coefficients nearly overlaps the literature results. In this case, the side coefficients for ϑ < 30° are plotted as the trend is assumed to be symmetrical around ϑ
Fig. 9. Drag (a) and side (b) coefficients for the four tested configurations; SR is the submergence ratio, as per Fig. 3. 294
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Fig. 10. a) Comparison of the experimentally obtained drag coefficients for the deeply submerged configuration SR = 0.33 with the results of Gippel et al. [15] and Shields and Alonso [38] and the function of Hoang et al. [18]; b) Comparison of the experimentally obtained side coefficients for the deeply submerged SR = 0.33 configuration and the estimate function from Hoang et al. [18].
hydrodynamic coefficients, due to the halving of the wetted area of the cylinder. Such reduction, however, is not proportional to the decrease of the sample wetted area and depends on the sample orientation. Both hydrodynamic coefficients vary with the yaw angle and can be interpolated with sine and cosine functions, which explicate the dependence of the coefficients on the sample orientation. The drag coefficient is minimum when the cylinder is aligned to the flow and reaches a maximum for higher angles. The side coefficient is almost symmetric around the maximum at 45°, being nearly null when the cylinder is aligned or perpendicular to the flow. The experimentally obtained hydrodynamic coefficients for the deeply submerged cylinder case are in good agreement with existing literature studies, showing that the custom-made hydrodynamic balance is a reliable instrument. Thanks to the relative easiness of manufacture, the reduced costs compared to ready-made commercial devices and the good accuracy, the instrument is suitable for experimental measurements related to a variety of fluid-body interaction phenomena. However, when focusing on the effects of submergence and on the wave drag coefficient, it appears that our overall laboratory setting is not accurate enough to detect the small differences due to the surface waves for the two configurations with the lower submergence. These however should not be attributed only to the inaccuracy of the hydrodynamic balance, but also to uncertainties in the sample vertical positioning. The results of this experimental campaign contribute to understanding the hydrodynamic behaviour of finite cylinders in openchannel flows, under different submergence conditions, and could aid numerical modelling of large wood transport. The present results can be applied to full-scale turbulent subcritical flows, assuming the independence of the hydrodynamic coefficients from the particle Reynolds number and the flow Froude number. Additional experiments would be however needed to remove this simplification, especially dealing with the wave drag coefficient, also assessing the effect of the sample slenderness ratio.
Fig. 11. Wave drag coefficients for literature data and for the present experiments as function of the relative distance from the water surface.
decrease with increasing submergence, our measurements show a constant trend, with the value for SR = 0.0 being smaller than expected. A reason for this discrepancy may lay in the accuracy of our laboratory setting, which is not able to distinguish the small differences between the drag coefficients for SR = 0.10 and SR = 0.0, the experimental conditions differing by a vertical displacement of 15.5 mm only. This means that the previously observed differences (Fig. 9) are related both to measurement uncertainties of the balance and to inaccuracies in the sample vertical positioning, so that the results from these two sets of experiments cannot be truly discerned. Some additional insight about the comparison of the different drag coefficients may be obtained estimating the drag force with the specific momentum equation [41]. This method would need measurements of the water levels upstream and downstream of the cylinder, which were not presently performed. 4. Conclusions
Acknowledgments
This paper presents experimental measurements of the drag and side coefficients of a finite cylinder under different submergence and yaw angles. The description and calibration of the custom-made hydrodynamic balance are reported, estimating the measurement uncertainty. Experiments are performed under steady conditions, computing the hydrodynamic coefficients from the measured forces. The results in terms of drag and side coefficients are affected by the sample submergence and show that the interaction with the free surface plays a fundamental role, leading to an increase of the hydrodynamic coefficients from high to zero submergence, as also noticed by existing literature. The half-submerged case presents instead much lower
The authors thank Mr. Pierangelo Bergamaschi, who gave valuable help in the design of the hydrodynamic balance, and Dr. Gianluca Alaimo and Dr. Stefania Marconi from the Proto-Lab of the University of Pavia, who made the 3D-printed components of the instrument. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at doi.org/10.1016/j.flowmeasinst.2019.01.006. 295
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