Theoretical investigation of hysteresis loops in free-surface flow under sluice gates for power-law channels

A

the wetted cross-sectional area of the channel (m²)

a

gate opening (m)

a_max

maximum gate opening (m)

a_min

minimum gate close (m)

B

top width of rectangular channel (m)

C

correlation model constant

C_c

contraction coefficient

D

hydraulic diameter (m)

E

specific energy (m)

⁠, ⁠,

Froude number of incident flow, flow immediately upstream of the gate, and flow downstream of the gate, respectively

g

acceleration due to gravity (ms⁻²)

K

correlation model constant

κ

shape factor of the parabolic channel

dimensionless parameter of the parabolic channel

m

shape factor of the triangular channel

dimensionless parameter of the triangular channel

n

Manning's roughness coefficient

P

matching coefficient

Q

flow rate (m³ s⁻¹)

flow per unit of width (m³ s⁻¹ m⁻¹)

q

specific discharge of the rectangular channel (m³ s⁻¹ m⁻¹)

R

correlation coefficient

S

channel bottom slope

T

top width of parabolic and triangular channels (m)

u

flow velocity (m s⁻¹)

incoming flow depth (m)

y₁

flow depth immediately upstream of the gate (m)

y₂

flow depth downstream of the gate (zone of maximum contraction) (m)

α, β

factor shape of a power-law channel

χ

alternate depth ratio

η

aspect ratio

λ

hysteresis loop extent (m)

θ

the angle of inclination of the sidewalls to the horizontal of the triangular channel (°)

ω

conjugate depth ratio

ψ

the filling rate upstream of the gate

In supercritical free-surface flows, the introduction of an obstacle into a channel or the modification of a geometric configuration such as a contraction, bed hump, or sluice gate often induces disturbance in the main flow. It results in a transition from supercritical to subcritical states, a process widely recognized as a hydraulic jump. This phenomenon exhibits a particular behavior; the possibility of two distinct flow configurations arising under identical initial conditions. This non-uniqueness in flow solutions is influenced not only by the geometric characteristics of the obstacle and the incoming flow but also by the dynamic evolution of the flow conditions themselves (Mehrotra 1975; Pratt 1983; Lawrence 1987; Defina & Susin 2003, 2006; Viero & Defina 2017, 2019). This behavior is indicative of the hysteresis effect associated with flows encountering obstacles, known as hydraulic jump hysteresis and the dual curve of the hydraulic system response during a cycle is defined as the hysteresis loop.

The theoretical foundations and experimental validations of hysteresis behavior have been thoroughly documented. Early works on this phenomenon, during the transition from supercritical to subcritical flow, were made by Abecasis & Quintela (1961, 1964) and Quintela & Abecasis (1979), with further empirical and theoretical expansions by Muskatirovic & Batinic (1977), Pratt (1983), Baines (1984), Austria (1987), and Lawrence (1987). These studies collectively affirm that any form of obstruction within a supercritical flow inherently induces hydraulic hysteresis, and straightforward methodologies for its calculation have been proposed.

Using catastrophe theory, Austria (1987) developed a descriptive model to elucidate the hysteresis observed in flows over a sill. The fundamental energy and momentum equations were used together with experimental validations to reinforce the model predictions. Conversely, Lawrence (1987) adopted an experimental approach for classifying flow regimes around a fixed obstruction, building upon Baines’ (1984) earlier work involving a moving barrier. These experimental inquiries have been pivotal in demonstrating the existence of hysteresis loops, where two flow states coexist under identical conditions, thereby revealing the extent of hysteresis.

In a combined theoretical and experimental endeavor, Defina & Susin (2003) examined the flow through a vertical sluice gate within a rectangular channel, and showed that hysteresis loops can occur in both supercritical and subcritical flows, contingent on the Froude number exceeding 0.8. Their subsequent exploration of various obstacles within a rectangular channel further delineated the theoretical boundaries of the hysteresis region, affirming that flow states can be accurately predicted through the Froude number and the geometric attributes of the obstacle, with experimental outcomes aligning with theoretical forecasts, especially when the obstruction does not alter the channel's geometry.

A comprehensive study by Defina & Viero (2010) explored the dynamics of flow through a linear channel contraction, both theoretically and experimentally. Their work introduced a quasi-2D model for predicting shock wave patterns through such contractions. Notably, discrepancies were observed between the theoretical predictions and experimental outcomes, particularly concerning the transition paths between stable flow states. This discrepancy highlighted the existence of two distinct types of hysteresis loops: minor and major, each characterized in detail.

Achour & khattaoui (2013) presented a theoretical investigation of hydraulic jump hysteresis due to the presence of a sill in a rectangular channel. Two configurations of incoming flow were analyzed; one generated by a bottom gate and the other by a steeply sloping spillway. The study aimed to present a theoretical approach for determining the threshold Froude number necessary for the emergence and the disappearance of the hydraulic jump, thereby enhancing the understanding of the hysteresis phenomenon in the design of energy dissipators.

Further investigations carried out by Viero & Defina (2017) on supercritical flow behaviors around obstacles revealed intricate flow patterns, both downstream and upstream, emphasizing the complexity of flow transitions through different obstacle types, such as positive steps and channel contractions.

Subsequent work by the same authors Viero & Defina (2019) focused on the nuances of flow through sluice gates, demonstrating the dual possibility of free and submerged flow states under identical inflow and gate opening conditions. This finding underlined the significance of the hysteresis effect in hydraulic systems, which was further validated by various experimental conditions.

Recent experimental investigations by Dhar et al. (2021) have shed light on the hysteresis effect associated with hydraulic jumps in closed and inclined rectangular conduits. This study demonstrated the effectiveness of applying continuity and momentum equations, along with considerations of turbulent viscosity, to achieve a more nuanced theoretical analysis of the phenomenon. Remarkably, the theoretical predictions were found to align closely with the experimental observations, underscoring the robustness of the analytical framework employed.

Recently, Daneshfaraz et al. (2022) analyzed the hysteresis effects on flows through channels subjected to sudden and gradual contractions. The study focused on the impact of hysteresis on the relative residual energy, flow depth, and longitudinal profile under different varying contraction scenarios. Their findings revealed a pronounced hysteresis behavior, markedly influencing the relative residual energy with an average increase exceeding 50%. Moreover, in scenarios involving a submerged hydraulic jump, the absence of hysteresis was consistently observed across all the experimental setups. A direct correlation was established between the expansion of the hysteresis region and the width of both sudden and gradual contractions, highlighting the relationship between channel geometry and flow dynamics.

The primary objective of this work is to analyze the physical phenomenon of the hysteresis loops and derive explicit equations characterizing this complex flow behavior. Through a methodical parametric study enriched by extensive computational analysis of hysteresis loops, a simplified theoretical model for each channel profile is proposed. These models span a comprehensive range of Froude numbers and channel geometries, facilitating a nuanced understanding of the sensitivity of hydraulic systems to dimensionless parameters. This approach aims to deduce the extent of hysteresis loops for each channel profile, offering valuable insights for practitioners. Specifically, the methodology focuses on determining the maximum gate opening values that ensure the re-establishment of undisturbed flow conditions across the predefined channel geometries, essentially identifying the threshold gate opening required to eliminate the hydraulic jump from the channel. Furthermore, the comparative analysis of hysteresis loops across different channel profiles aims to enhance the understanding of hysteresis behaviors due to changes in flow parameters, thereby contributing to the enhancement of the design process in open channel engineering.

In open channel hydraulics, the concept of alternate depths refers to two specific water depths that a given discharge can achieve under the same specific energy but in different flow regimes. This section presents the calculation of alternate depth ratios within power-law channels.

According to the alternate depths definition, the specific energy is the same, such as ⁠.

For a known downstream Froude number and channel shape (factor β), Equation (7) can be used to calculate the exact alternate depth ratio in a power-law channel. To do so, and for the investigated channel shapes, Equation (7) leads to an algebraic polynomial of nth order, whose solutions can be obtained analytically (up to the fourth order in accordance with the Abel-Ruffini theorem) using classical methods (Pandey et al. 2018) or even numerically. Quadratic, cubic, and quartic equations are obtained for rectangular, parabolic, and triangular-shaped channels, respectively. Since we are dealing with flow depths, the retained values of the polynomial roots should be real, positive, and consistent with the flow regime type.

For a given momentum, the concept of conjugate depths refers to two specific depths at which the same discharge occurs, synonymous with the phenomenon of hydraulic jumps.

Equation (8) is nonlinear (i.e., non-unique in solutions), and its implicit character prevents a direct analytical solution of the conjugate depth. To unify the solutions and produce an explicit equation, Vatankhah (2017) used the asymptotic matching technique, which consists of finding two distinct solutions to the problem, each of which is valid within a certain interval of values of the dependent variable ω. These two solutions are then combined together by a matching coefficient P to form a global solution.

The approximate solution formed by Equations (9) and (10) is used in the present analysis for the determination of the conjugate depths of the hydraulic jump.

Gates are hydraulic structures commonly used for flow regulation, measurement in irrigation projects or on spillway crests, and for controlling upstream water levels. Flow through a vertical sluice gate can be free or submerged, depending on the tailwater depth.

Several analytical and experimental studies have investigated the flow behavior under gates; many of them have found that there are significant deviations between the mathematical models and the experimental measurements (Montes 1997).

One of the most determining factors influencing flow behavior beneath gates is the contraction coefficient C_c (Figure 2(b) and 2(c)). The task of estimating the contraction coefficient has attracted much experimental and theoretical research under free and submerged flow conditions. Despite the comprehensive research dedicated to the contraction coefficient, uncertainties in its calculation persist, attributed to the complex nature of flow behavior under the sluice gate. This complexity is compounded by variables such as turbulence and vortex formation. Among the seminal and most influential contributions, the classical hydrodynamic theories of Cisotti (1908) and Von Mises (1917) stand out. These theories apply potential flow theory through conformal mapping but do not account for gravitational effects (Montes 1997). Following in this theoretical lineage, Pajer (1937) and Marchi (1953) maintained the resolution's philosophical core by employing conformal mapping theory, yet they innovatively incorporated the approximate integration of gravitational effects, specifically the variation in flow velocity along the section downstream from the gate. Furthermore, the application of numerical methods by researchers such as Issacs (1977), Betts (1978), Montes (1997), and Vanden-Broeck (1997) underscores the multidimensional approach to understanding flow contraction. The results of experimental investigations conducted by Benjamin (1956), Rajaratnam & Subramanya (1967), Rajaratnam (1977), Rajaratnam & Humphries (1982), Roth & Hager (1999), Yen et al. (2001), and many others later are to be cited in this regard.

The core idea of the explicit approximation involves the expansion of the implicit Von Mises equation using a Taylor series, leading to a polynomial equation that can be solved analytically. It is evident that with increasing the order of the Taylor series expansion, the deviation between the implicit and explicit solutions diminishes. Nevertheless, the complexity of achieving an analytical solution increases, eventually becoming impracticable. Consequently, to maintain solvability, the Taylor series expansion is truncated to a certain order leading to a cubic equation, thus facilitating the use of trigonometric functions for the solution derivation.

It is important to note that Von Mises' theory was originally formulated for the contraction coefficient for plane problems, making it ideally applicable for wide rectangular channels. Since so far, no other theory exists for the confinement effect (wall effect), the same theory is considered to hold for triangular and parabolic-shaped channels.

Equation (11) reproduces the particular cases for which, when ⁠, the contraction coefficient C_c takes the limit value of ⁠, and when the contraction coefficient takes the upper limit value of 1. However, the present approximate solution (Equation (16)) is valid for values of ψ up to 0.75. Beyond this value, the deviation between the two solutions tends to become significant. This is due to the singularity of the tangent function in the vicinity of ⁠. This singularity introduces inaccuracies in the Taylor series approximation, as it trends toward an infinite vertical asymptote, a behavior unattainable by any finite Taylor series expansion. In the present case and throughout the study, the ratio is always in the range where the contraction coefficient solutions given by the two approaches are very close.

The approach detailed above is applicable for initial flow conditions that are purely supercritical ⁠. In this paper, we have focused on Froude numbers ⁠, to avoid errors in estimating the conjugate depth of the hydraulic jump (y₁), since when the Froude number of the incident flow is less than approximately 2, the hydraulic jump in this case is considered undular, and present secondary waves (Favre waves) and instability of the free-surface flow lead to increased errors as a result. For this reason, the analyses are carried out from a Froude number equal to 2 in the series of hysteresis loop computations so that the formulas obtained in the present study are free from the secondary effects induced by the Froude numbers between 1 and 2.

To study the hysteresis phenomenon within power-law channel flows, intensive numerical computations were conducted with the ultimate goal of developing a simplified approximate model for engineering design purposes. This tool is envisioned to simplify the process of selecting the most suitable channel cross-section for engineering projects, by providing a direct means to evaluate the hysteresis effect without the need for repeated, detailed simulations for every conceivable variation in flow parameters or channel geometries. Such an endeavor traditionally requires significant computational effort and resources.

The methodology followed consists of an exhaustive computational analysis, meticulously carried out across a range of channel profiles and the deduction of approximate mathematical models.

For each channel profile, a detailed computational analysis was performed, starting with the definition of the channel geometry and flow parameters. The analysis began by adjusting the operation range of the sluice gate from an initial opening wider than the water depth of the incident flow ⁠, to a final position where the sluice gate intersects the flow ⁠. The hysteresis curve, which represents the relationship between the water depth upstream of the gate and the gate operational range, was plotted upon completing the gate's closing and opening cycle (Figure 3). The hysteresis curve is obtained when the sluice gate has completed the closing and opening phases.

Next, once the hysteresis curve is obtained, the ratio is calculated and inserted into Equation (16) to calculate the new contraction coefficient. If this last one is equal to the initial C_c used in the calculations, then the obtained curve is the desired one. Otherwise, the initial C_c takes the value of the new C_c calculated by Equation (16), and so on until is obtained. The procedure is repeated until the final hysteresis curve is obtained.

Finally, for each final hysteresis curve obtained, the ratio between the hysteresis loop extent and the discharge per unit of width is taken, and plotted in a new graph as a function of the incident Froude number where ranges from 2 to 12. The procedure is repeated for all channel profiles for a wide range of cross-sectional shape parameters, which are discussed in the following section.

For a rectangular channel, Equation (20) defines the specific discharge ⁠. However, for nonrectangular profiles, it is important to point out that this concept bears no relation to unit flow; it is merely a theoretical concept introduced to facilitate calculations. It can be viewed as a pure normalization technique rather than a physical concept. Table 1 concisely presents the range of dimensionless parameters employed throughout this series of computations. The range of dimensionless parameters, as summarized in Table 1, encompasses the variability in channel shapes and flow conditions, ensuring comprehensive coverage in the present analysis.

Table 1

Range of dimensionless parameters used in the hysteresis loops computation

.	Rectangular channel .	Triangular channel .		Parabolic channel .
Dimensionless parameter
Range

.	Rectangular channel .	Triangular channel .		Parabolic channel .
Dimensionless parameter
Range

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The computation of hysteresis loops in open channels with power-law profiles involves systematic steps summarized as follows:

• 1. Channel shapes (⁠ and ⁠) corresponding to rectangular, parabolic, and triangular channel shapes, respectively, are selected.

• 2. The channel geometry and flow parameters are fixed ⁠. The index 0 refers to the incident flow parameters.

• 3. The maximum opening and closing of the sluice gate and its operating increment are defined.

• 4. Using Equation (17), the aspect ratio η is computed.

• 5. Compute the new flow per unit of width from Equation (20).

• 6. From Equations (18) and (19), the new cross-sectional shape parameters and are computed for triangular and parabolic channel shapes, respectively.

• 7. From Equation (10), the coefficient P is calculated.

• 8. The P value is inserted into Equation (9) to calculate ω.

• 9. Calculate the conjugate water depth upstream of the sluice gate.

• 10. For each gate opening step, calculate by using Equation (4).

• 11. By referring to Equation (16), calculate the contraction coefficient in such a way that one obtains ⁠.

• 12. Following the channel shape, Equation (7) is used to compute the water depth immediately upstream of the sluice gate y₁.

The series of computations from step 10 will be repeated for each sluice gate opening and closing step. Because the approach discussed in this paper requires a large amount of calculation, a computational code written in Matlab was developed to ensure the complete calculation of the aforementioned algorithm for the hysteresis loops.

To obtain a unique general analytical expression for calculating the extent of hysteresis loops for different channel profiles, the curves in Figure 6 must collapse into a single curve to obtain a unique equation that relates the extent of hysteresis loops to the Froude number and the cross-sectional shape parameter for each channel profile. Formally, the self-similarity principle along with the dimensional analysis technique was used (Sedov 1959; Barenblatt 1996) and the procedure has led to the following functional relationships:

• for the rectangular channel,

• for the parabolic channel,

• for the triangular channel.

The first application example is taken from experimental data of Defina & Susin (2003). A flow of with a water depth of 0.65 m is conveyed in a rectangular channel of 0.38 m wide. The maximum gate opening required to restore the initial flow conditions is then to be determined. When the gate is moved down from 1.3 to 0.55 m, then it returns to 1.3 m. Note that the Froude number of the incident flow is ⁠.

It follows that ⁠. From Equation (20), ⁠. Using Equation (10), ⁠, and from Equation (9) one obtains ⁠, which leads to ⁠. Using Equation (4), is calculated for each sluice gate step.

Replacing the value of when in Equation (7) gives the water depth immediately upstream of the gate in order to obtain ⁠. Repeating the process from step 10 for each gate opening one obtains ⁠, which implies ⁠. This value allows the computation of such as ⁠.

Applying the proposed approximate model (see Table 2) ⁠. It results then ⁠. Finally ⁠.

Table 3 summarizes the comparison between the experimental measurements of Defina & Susin (2003) and the proposed theoretical approach. As can be seen, there is a very good agreement between the theoretical results and the experimental observation with a maximum deviation of around 5%.

Let us consider a parabolic-shaped channel described by the function ⁠, with ⁠. The channel slope and the roughness coefficient are and ⁠, carrying a flow of ⁠. The objective is to determine the maximum gate opening value for the hydraulic jump to disappear from the channel when the gate decreases from 2.5 to 1 m and then returns to 2.5 m.

Using Manning's equation, we obtain the normal flow depth ⁠, implying ⁠.

The resulting top width is then ⁠. In the same way as before, the calculation procedure for Application 2 is:

and ⁠. Using Equation (19), ⁠. From Equation (10) ⁠. By substituting the value of P in Equation (9), one obtains which gives ⁠.

When then and according to Equation (7), ⁠. In this case, implying leading to the value of ⁠.

Applying the proposed explicit approximate model (Table 2) ⁠, allowing the determination of the value ⁠. Finally, ⁠, which is the exact value found above.

In this example, a flow is conveyed in a triangular channel with sidewalls making an angle with the horizontal. Furthermore, the channel slope and the roughness coefficient are and ⁠, respectively.

If the sluice gate has been lowered from 2.2 to 0.95 m and then raised again to 2.2 m, then it is required to determine the maximum value of the gate opening necessary to restore the initial flow conditions.

The normal depth is (i.e. ⁠) which results in ⁠.

The sidewall parameter leading to a top width ⁠. From Equation (20) ⁠. Using Equation (18), ⁠. Then, from Equation (10) ⁠. By reporting the value of P in Equation (9), it results in which gives ⁠. Replacing the value of when in Equation (7) gives the water depth immediately upstream of the gate such as ⁠.

When applying the proposed approximate model (Table 2) ⁠. It follows that ⁠, which gives ⁠.

The deviation of the approximate model from the exact value is then

The summary of the results of the computation procedure are listed in Table 4.

Table 4

Summary of results for the Applications 2 and 3

Parameters .	Application 2 (parabolic-shaped channel) .	Application 3 (triangular-shaped channel) .
y0	1.2	1.06
	2.24	2.33
T	2.19	1.778
η	1.2	1.06
	5.02	2.812
	4.8	/
	/	0.472
P	10.48	15.426
ω	0.468	0.526
	2.56	2.01
	5.96	7.95
y1	9.44	10.87
	0.198	0.306
λ	0.994	0.86
amax	2.2	1.92
Solution using proposed approximate models (Table 2)
	0.198	0.31
λ	0.994	0.872
amax	2.2	1.932

Parameters .	Application 2 (parabolic-shaped channel) .	Application 3 (triangular-shaped channel) .
y0	1.2	1.06
	2.24	2.33
T	2.19	1.778
η	1.2	1.06
	5.02	2.812
	4.8	/
	/	0.472
P	10.48	15.426
ω	0.468	0.526
	2.56	2.01
	5.96	7.95
y1	9.44	10.87
	0.198	0.306
λ	0.994	0.86
amax	2.2	1.92
Solution using proposed approximate models (Table 2)
	0.198	0.31
λ	0.994	0.872
amax	2.2	1.932

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This paper presents a theoretical approach for hysteresis loop computation in power-law channels under supercritical flow conditions through a vertical sluice gate. The primary objective was to investigate the hysteretic behavior of the flow considering three channel shapes rectangular, parabolic, and triangular and propose simplified explicit models for hysteresis loop extent computation.

The focus of the study was to understand the hysteretic behavior of flow through a vertical sluice gate, with the specific aim of determining the maximum gate opening required to restore undisturbed flow conditions, thereby eliminating the hydraulic jump from the upstream channel.

By employing energy and momentum equations, expressions for alternate and conjugate depths upstream of the gate were given, leading to second, third, and fourth-degree polynomials for rectangular, parabolic, and triangular channels, respectively.

The extensive computational analysis of hysteresis loops revealed a strong correlation between the maximum gate opening necessary for restoring initial flow conditions and the incident Froude number, as well as the contraction induced by the sluice gate. Additionally, an original approximate solution based on the implicit Von Mises equation for predicting the contraction coefficient was proposed.

Furthermore, the study resulted in the development of explicit simplified models derived from numerical computations of hysteresis loops across a wide range of Froude numbers and dimensionless parameters describing channel geometry. The obtained results show a very satisfactory agreement between the models derived from the suggested approach and the few results found in the literature. Unfortunately, a large gap exists in the specialized literature with very rare experimental results regarding the hysteresis phenomenon in power-law channels for a complete validation.

Overall, this research contributes to the understanding of flow behavior in channels with power-law profiles controlled by a sluice gate. Furthermore, this approach offers a valuable tool for predicting the hysteresis phenomenon in the hydraulic engineering of open channels.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.