Hydraulic Behavior Assessment of Type A and Type B Piano Key Weirs from Experimental and Numerical Results ^†

Abstract

Since their development, as a result of an improvement of labyrinth weirs, Piano Key Weirs (PKWs) have been implemented as (a) a flood safety structure for gravity dams, allowing to also increase their storage volume, and (b) in river systems to increase the water level for hydropower or navigation purposes. The rectangular folded crest, consistent with apexes inclined by turns in the upstream and in the downstream direction, turns the PKW a device with a high discharge capacity, especially useful during wet extreme events. Nevertheless, several modifications have been implemented in the PKW geometry, capable to improve and, in some cases, worsen their efficiency. Herein, an experimental and numerical assessment, using the ANSYS Fluent Computational Fluid Dynamics (CFD) software, of the discharge coefficient is presented for two PKW configurations, evaluating the specific discharge over the upstream, downstream, and lateral crests, the velocity in the inlet and outlet keys, and the water surface profile, as well. The investigated configurations are a symmetric type A, designed following the recommended optimal values, and a type B model, with the same geometric features as type A. Results showed that for the specific geometries, the type B is more efficient for lower head; however, once the filling of the outlet key occurs, the type B efficiency is reduced, leading to type A becoming more efficient.

1. Introduction

Labyrinth weirs were developed with the purpose of increasing the crest print length for a given space, which increases the discharge capacity. Afterwards, the Piano Key Weir (PKW) was proposed by Lemperiere and Ouamane [1], aimed at limiting the base of a labyrinth weir. The PKW planform is defined by a rectangular shape, but their apexes are inclined alternately in both upstream and downstream directions. The reduced footprint makes the PKW more suitable to be installed on the top of a dam, working as the spillway structure and increasing the discharge capacity and storage volume.

Due to the complex geometry of the PKW, Pralong et al. [2] defined a specific nomenclature, reported in

Table 1. Fundamental parameters of an entire PKW.

PKW Parameter	Symbol	PKW Parameter	Symbol
Inlet Height	Pi	Inlet Width	Wi
Outlet Height	Po	Outlet Width	Wo
Height of the base	Pb	Wall thickness	Ts
Crest height	P	Inlet Slope	Si
Total Lateral Length	B	Outlet Slope	So
Outlet Length	Bo	Unit Width	Wu = Wi + Wo + 2·Ts
Inlet Length	Bi	Unit Length	Lu = 2·Bh + Wu
Length of the base	Bb	Total developed length	L = Nu·Lu
Number of units	Nu	Total width of the PKW	W = Nu·Wu

.

Likewise, to improve the knowledge of the discharge capacity of a PKW, plenty of authors [3,4,5,6,7,8,9,10,11] have carried out experimental and numerical studies to assess which parameters mainly affect the discharge efficiency of this device. The outcomes of these studies showed that, aiming to maximize the discharge capacity, the following dimensionless ratios should be considered: P/W_u = 1.33, W_i/W_o = 1.25 ÷ 1.5, B_o/B_i ≥ 1. Nonetheless, from an economic viewpoint, an optimal value for the P/Wu ratio can be selected equal to 0.83.

In [1,2,3,4,5,6,7,8,9,10,11], four types of PKW were defined according to their overhangs: type A having symmetric overhangs, type B with a single upstream overhang, type C with a single downstream overhang, and type D which does not present overhangs [12]. After several PKW types, Noui and Ouamane [3] and Cicero and Delisle [13] observed that type B was more efficient than type A [13], whereas type C resulted to be less efficient than the type A. Machiels et al. [14] tested several Bo/Bi ratios and found out that for low heads, the type B configuration (B_o/B_i = ∞) is the most efficient. Nevertheless, this efficiency decreases for a higher head, rendering the type A more efficient. These results contradicted previous studies, proving the necessity of further comparison between the discharge efficiency of type A and type B in different ranges of dimensionless ratios.

Therefore, the purpose of this paper is to assess the hydraulic efficiency of both configurations, PKW_A and PKW_B, depending on the upstream head, aiming at clarifying previous results [3,13,14]. The assessment will evaluate the discharge efficiency of each specific crest, namely, upstream, downstream, and lateral crests by investigating experimentally and numerically two types of PKW. The former, a symmetric type A, PKW_A, designed following the dimensionless ratios presented while considering an intermediate value for P/W_u, and the latter, a type B model, PKW_B, with the same height and crest trace but with a B_o/B_i ratio equal to ∞ (

Table 2. Geometry parameters for PKW_A and PKW_B.

P	0.522 m	Wi	0.255 m
B	1.254 m	Wo	0.170 m
Bo	0.628 m	Wu	0.455 m
Bi	0 m	Lu	2.963 m
Bb	0.626 m	Ts	0.15 m
PKWA	Bo = 0.314 m; Bi = 0.314 m
PKWB	Bo = 0.628 m; Bi = 0 m

).

Specifically, the procedure consisted of testing experimentally and numerically the PKW_A, whereas only numerically investigating the PKW_B type. Coupling the numerical and the experimental approaches allowed collecting information for a wider range of H_up/P.

2. Materials and Methods

2.1. Experimental Setup

The experiments were carried out at the Hydraulic Laboratory of the Department of Civil, Architectural and Environmental Engineering, University of Naples Federico II. The experimental setup test range was fixed to limit scale effects by setting a minimum upstream head H_up > 3 cm [15]. A Perspex test channel 3.6 m long, 0.455 m wide, and 1 m high was built up to carry out the experimental research. The upstream side of the channel was located inside an open tank 4 m long, 1.92 m wide, and a maximum water level of about 0.59 m, with two 1 m long walls of the same width as the channel. The intake is a tank 1 m long and 4 m wide. The pump installed can provide a maximum flow of 80 L/s, establishing an upper limit for the discharge of the experimental tests. The connection between the two tanks was a grid that, in addition to the convergent walls, ensured uniform flow conditions. The longitudinal slope of the channel was equal to zero.

Upstream flow water depth data were taken with a gauge of a ±1 mm reading accuracy while the tank inflow was measured with a diaphragm flow meter with an average value of accuracy of ±1.79%. The accuracy of measurements was estimated using the Error Propagation Method [16], considering the individual variable uncertainty of each measurement method applied in the laboratory. The upstream mean velocity was calculated as:

$$v=\frac{Q}{W·h}$$

(1)

where Q is the inflow (m³/s), W is the width of the channel which corresponds to the PKW width (m), and h is the water level upstream measured with the gauge (m). To calculate the discharge capacity, the discharge coefficient, C_PKW, has been computed using the Poleni discharge equation:

$$C_{PKW}=\frac{Q_{PKW}}{W\sqrt{2g{H}_{up}^3}}$$

(2)

where C_PKW is the discharge coefficient for a PKW (-), Q_PKW is the discharge of a PKW (m³/s) and H_up the head upstream the weir (m), calculated as:

$$H_{up}=h-P+v^2/2g$$

(3)

where P is the PKW height (m) and $v^2/2g$ is the kinetic term (m). A total of 21 flowrates were tested, setting the smallest flowrate of the range to obtain a minimum head above 3 cm. The range of H_up/P experimentally tested was 0.059 ≤ H_up/P ≤ 0.131 for PKW_A.

2.2. Numerical Model

The selected geometries were modelled using Ansys Fluent Software [17]. The multiphase Volume Of Fluid (VOF) was selected to model the interaction between air and water phases. To model the turbulence, the Reynolds-Averaged Navier–Stokes (RANS) equations were selected in addition to the Renormalized Group (RNG) k-ε turbulence model, consistent with Pralong et al. [4] and Crookston et al. [18] studies. A surface tension interaction between the primary phase (water) and the secondary phase (air) with a specific value of 0.072 N/m was accounted for.

The pressure-based solver was selected to solve the simulations, with the implicit formulation. Likewise, First-Order Implicit Transient Formulation was set and the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm was used to model the Pressure–Velocity coupling, established by the RANS equations. As recommended for VOF computations, the Pressure Interpolation Scheme was selected. Second-Order Upwind discretization was implemented to solve Momentum, Turbulent Kinetic Energy, and Dissipation Rate. The Spatial Discretization was based on the Least Square Cell-Based gradient and the Compressive method was selected to solve the Volume Fraction equation. Lastly, the Time Step Size was calculated for each simulation according to the mesh size and the maximum expected velocity that ensured a Courant Number not greater than 1.

In terms of boundary conditions, a uniform velocity distribution at inlet and a static pressure at outlet were set (Figure 1), respectively. For each simulation, the velocity magnitude was set at inlet surface and a relative static pressure equal to 101,325 Pa was set at the outlet surface. The upper surfaces were set in a symmetrical fashion while lateral and bottom surfaces were considered as rigid walls. The numerical domain was discretized into six blocks; therefore, ten matching interfaces were added to solve the non-conformal mesh.

A sensitivity analysis was performed based on the calculation of the Grid Convergence Index (GCI), following the Celik et al. [19] methodology. Four mesh resolutions were considered, by varying the mesh applied in the surrounding area of the PKW (volumes 3, 4, and 5) while maintaining the mesh size constant for volumes 1, 2, and 6. Volume 1 was set with a mesh size equal to 2.5 cm (∼4.8% P) and volume 2 with 5 cm (∼9.6% P). Volume 6 was set with a mesh equal to 5 cm (∼9.6% P) and a refinement corresponding to the bottom of the channel with a variable mesh size resolution. In greater detail, the specific mesh size for volumes 3, 4, and 5, as well as the inflation factor for volume 6 are summarized in

Table 3. Mesh resolutions tested for the Mesh Convergence Method.

Mesh	Volume				Number of Cells	Computational Time
	3	4	5	6
h1	2.0 cm	3.0 cm	2.0 cm	inflation of 2.0 cm	696,799	∼4 days
h2	1.5 cm	2.0 cm	1.5 cm	inflation of 1.5 cm	1,521,601	∼10 days
h3	1.15 cm	1.3 cm	1.15 cm	inflation of 1.15 cm	3,569,820	∼25 days
h4	0.8 cm	1.0 cm	0.8 cm	inflation of 0.8 cm	9,503,531	∼40 days

, with the total number of cells for each mesh and the computational time required to achieve the time convergence. A value of GCI_CPKW²¹ = 0.53% was achieved between mesh h₁ and h₂ in terms of discharge coefficient, thus accounting for mesh h₁ for the simulations.

3. Results

The non-dimensional rating curves C_PKW (H_up/P) of the PKW_A from the experimental and numerical tests are given in Figure 2, with error bars showing the measurement uncertainties. Results showed a good agreement between both set of results, with a mean absolute error, MAE, equal to 3.81%.

Aiming to assess the difference in discharge efficiency between PKW_A and PKW_B, several flowrates were computed, resulting in the rating curve C_PKW (H_up/P) presented in Figure 3. Results show that PKW_B is more efficient than PKW_A for a lower head. Conversely, PKW_A turned to be more efficient at increasing the head. This change in efficiency occurred at H_up/P ≅ 0.3, when the PKW_B discharge coefficient decreases more rapidly than the corresponding discharge coefficient from PKW_A. These results agree with the observations by Machiels et al. [9], which showed that for a higher head, type B started to be less efficient than type A.

4. Discussion

Discharge and Head Distribution over the Upstream, Downstream, and Lateral Crests

For evaluating the variation in efficiency, the specific discharge, q, and head, H, over each single crest was computed for both PKW_A and PKW_B. Results presented in Figure 4a show that for H_up/P ≲ 0.3, the specific discharge over the upstream, downstream, and lateral crests of both PKW_A and PKW_B is comparable. For H_up/P ≳ 0.3, the lateral and downstream crests of PKW_A are characterized by a higher specific discharge, while the upstream crest specific discharge of PKW_B is higher. For all H_up/P values, the specific discharge of the downstream crest is higher, followed by the upstream and the lateral crests’ specific discharges.

Furthermore, the comparison of the total head between the upstream, downstream, and lateral crests of both configurations is shown in Figure 4b. Results show that the head at the downstream crest from the PKW_A is greater, whereas the head on lateral crests is higher for PKW_B. The head at the upstream crest is comparable between both geometries.

Therefore, the upstream crest of PKW_A is more efficient because it discharges more water for a certain head. Regarding the downstream crest, the lower head and the higher discharge of PKW_B result in a higher discharge coefficient. In addition, the lateral crest of PKW_A is more efficient due to the higher discharge and lower head values. Finally, the discharge coefficients of each crest are presented in Figure 5, where it is worth noting that the upstream and downstream crests of both configurations vary narrowly whereas the lateral crest discharge coefficient decreases remarkably, and even more rapidly on the PKW_B.

5. Conclusions

An experimental and numerical investigation was performed to assess the discharge efficiency of two PKW geometries, by studying the discharge capacity and behavior of each individual crest. The geometries tested were a symmetric type A, PKW_A, and a type B model, PKW_B, with the same crest print and height as PKW_A (L, P, W_i, W_o, B_b, B_h). The PKW_A was experimentally and numerically tested while the PKW_B was only tested numerically. The comparison between numerical and experimental results showed the effectiveness of the numerical model. Furthermore, a mesh size of ∼9.6% P can numerically predict the discharge coefficient with reasonable accuracy, obtaining a GCI of 0.53%.

In terms of discharge capacity, the tested PKW_B resulted to be more efficient for lower head (H_up/P ≲ 0.35) than PKW_A; however, when increasing the upstream head, the PKW_A model proved to be more efficient. This change in efficiency can be explained because of the crests’ behavior of both configurations. The downstream crest of PKW_B is more efficient due to the smaller velocities in the inlet section, meaning more favorable inflow conditions, which explains why the discharge coefficient of the tested PKW_B was higher for H_up/P ≲ 0.35. Nevertheless, the discharge efficiency of the lateral crests of the PKW_B is remarkably reduced in comparison with the PKW_A, which results in a change in efficiency at H_up/P = 0.35. For H_up/P > 0.35, the PKW_A is more efficient because the PKW_B is not able to compensate the higher upstream and lateral discharge efficiency of the PKW_A, although the downstream crest is more efficient for all the H_up/P-tested values.