The critical regime plays a primordial role in the study of gradually varying flows by classifying flow regimes and slopes. Through this work, a new approach is proposed to analyze critical flow regime in an egg-shaped channel. Based on both the definition of Froude number and Achour and Bedjaoui general discharge relationship, a relation between critical and normal depths is derived and then graphically represented for the particular case of a smooth channel characterized by a generating diameter equal to 1 m. The results show the influence of the slope on the frequency of occurrence of the critical regime. At the same time and independently of the flow rate, a very advantageous approach for the calculation of the Froude number has been proposed. The study shows that there are six zones to differentiate the various flow states, namely: on the one hand for steep slopes two subcritical zones interspersed by a supercritical zone and on the other hand for mild slopes a zone corresponding to uniform flow, an area where the flow is probably gradually varied and finally an area where the flow is abruptly varied. Based on the specific energy equation, a validation process concluded that the proposed relationships were reliable.

  • Determination of the frequency of occurrence of the critical regime.

  • The nature of the flow regime can be deduced from the value of the slope.

  • New theoretical approach for the study of the critical regime in an egg-shaped conduit considered as a form with several advantages.

  • The Froude number can be determined independently of the flow rate. Establishment of an abacus to define the flow domain including non-uniform flows.

In nature, water flows in a natural slope whose relief is irregular, creating different types of flows. These flows are classified into two main categories: non-permanent flows whose characteristics vary over time, and permanent flows (French 1985). According to their developments, permanent flows take various aspects: varied or uniform. In practice, uniform flow, with constant geometric and hydraulic characteristics over time and space (Chow 1959; French 1985) is the most studied type. In fact, the study of uniform flow is fundamental in the calculation of various flows (French 1985; Vatankhah 2015). This type of flow is also encountered in pressurized closed channels and at full section. It can also be in subcritical or supercritical regime. Therefore, the critical regime is a particular case of uniform flow where the inertia and gravity forces are in equilibrium. The critical flow plays a significant role in the analysis, design, operation and maintenance of channels (Shang et al. 2019). Critical depth computation in open channel is very important to preserve it from erosion caused by unstable critical flow or high kinetic energy (Petikas et al. 2020). Several authors have attempted to express the relationship to calculate the critical depth for a given shape of closed or open channel. Based on the expression of Froude number, some authors have attempted to propose a practical method for determining critical depth. Chow (1959) presented a methodology for calculating the critical depth by algebraic method for the rectangular shape and by successive approximations or by graphical method for the more complex forms. Similarly, Henderson (1966) established a graph for critical depth evaluation in a circular pipe and in a trapezoidal channel for different side slopes. French (1985) reproduced the 1959 Chow diagram and presented the semi-empirical formulas developed by Straub (1982) to calculate critical depth in open channels in rectangular, trapezoidal, triangular, parabolic, circular, elliptical and exponential shape. Heggen (1991) introduced a correction exponent for velocity distribution in the development of the critical depth relationship in rectangular, parabolic and triangular sections. These relationships have been associated with a diagram. However, for the case of trapezoidal channel, the proposed relationship still remains implicit. Swamee (1993) developed explicit critical depth equations for some shapes of irrigation canals. Wang (1998) used an iterative theory to produce an explicit equation for critical depth in a trapezoidal channel. In addition, Swamee & Rathie (2005) proposed exact analytical equations for critical depth calculation in a trapezoidal channel. However, the developed relationships have the disadvantage of being presented in the form of an infinite series. A particularly interesting study was also proposed by Swamee & Swamee (2008) for the explicit calculation of hydraulic parameters for non-circular sewerage pipes including critical depth. Using curve fitting method, Vatankhah & Easa (2011) have proposed explicit solutions for critical and normal depths in trapezoidal, circular, and horseshoe shaped channels. Vatankhah (2015) obtained an explicit solution for computation of critical depth in semi elliptical channels with an excellent approximation of the incomplete wetted perimeter integral. Shang et al. (2019) present three explicit solutions for critical depth in closed conduits for circular, arched, and egg-shaped sections. To optimize the model parameters, revised particle swarm optimization (PSO) algorithms have been implemented in MATLAB. Using an adaptive cubic polynomial algorithm (ACPA), Petikas et al. (2020) presents a new method for the computation of multiple critical depths in compound and natural channels. Regarding studies discussed previously, only the flow and the geometric parameters are taken into account. However, many other parameters, such as the slope of the channel or conduit, the absolute roughness, and the kinematic viscosity of the flowing liquid can significantly influence the nature of the flow regime. In this regard, these parameters were introduced in the study performed by Nebbar & Achour (2018) concerning the sizing of the rectangular channel under a critical flow regime. Moreover, Achour & Amara (2020) provided new considerations on the critical flow regime in free-surface circular pipe. Indeed, depending on the slope of the pipe, the critical flow regime is not always present in such pipes, as well as for a given diameter and slope, two critical states of the flow can occur for two different flow rates. They also show that there is a slope that generates a single critical state of the flow.

In this context, the present study aims to propose a new approach in the analysis of critical flow and thus to examine the different possibilities of the critical flow state in an egg-shaped conduit. It has been shown in the past that this shape of conduit is suitable for combined sewer sewage networks and present best resistance against traffic loads than usual circular pipes (Regueiro-Picallo et al. 2016). In addition, this type of pipe has also improved hydraulic performance in dry weather conditions of normal operation of combined sewer systems, where a high percentage of the discharge time of the flow is carried by the lower part of the section (Butler & Davies 2010). Under these conditions, egg-shaped pipes have higher flow rates due to their smallest wetted perimeter, reducing particle sedimentation and the operational costs of cleaning sewers (Butler & Davies 2010). The resuspension of sewer sediments during wet weather flows is an important source of the pollution of combined sewer overflows (Suarez & Puertas 2005), and their control is one of the main objectives of the integrated urban water management in urban systems (Regueiro-Picallo et al. 2016).

The egg shape is complex consisting of three different geometric parts. The flow area, the wetted perimeter and the hydraulic radius are expressed according to the geometrical locus occupied by the flow (Table 1). The flow level may then be in one of the following intervals: zone I: , zone II: and zone III: . Where η = y/D represents the filling rate of the conduit, y the depth of the flow in the conduit and D is the generating diameter. Table 1 represents the different expressions of the geometric characteristics of the conduit, where A is the wetted area, P is the wetted perimeter, T is the top width, and Rh is the hydraulic radius. In addition, the functions φ(η), σ(η), ρ(η), φ(η), τ(η), λ(η), , ζ(η), δ(η) and ξ(η) are expressed exclusively as a function of the filling rate η.

Table 1

Some common characteristics of the flow in an egg-shaped conduit (Lakehal & Achour 2014). (Note that to distinguish the normal flow from the critical one, the indices ‘n’ and ‘c’ will be assigned to the relationship)

  
 
 
 
 
 
 
 
  
;  
 
 
 
 
  
;  
 
;  
 
  
 
 
 
 
 
 
 
  
;  
 
 
 
 
  
;  
 
;  
 

η = y/D, conduit filling rate; y, flow depth; D, conduit generating diameter; A, wetted area; P, wetted perimeter; T, top width; Rh, hydraulic radius.

Basic equations

The critical regime corresponds to minimum specific energy which is characterized by a Froude number equal to unity. The Froude number is expressed physically as the ratio of the inertial to gravity forces. At critical flow conditions, Froude number is equal to unity and is written very often in the following form (Shang et al. 2019):
formula
(1)
where α represents the velocity distribution coefficient (generally = 1), Q is the flow rate, g is the acceleration due to gravity, Ac is the critical wetted area and Tc the critical top width.
Furthermore, in a uniform flow occurring in any channel profile, the flow rate can be computed using Achour and Bedjaoui general relationship (2006). This relationship gives the discharge Q as a function of the longitudinal slope S0, the absolute roughness ε, the wetted area A, the hydraulic radius Rh, and the Reynolds number R* corresponding to the shear Reynolds number. The relationship is as follows:
formula
(2)
With:
formula
(3)

The subscript ‘n’ denotes normal characteristics of the uniform flow.

Theoretical development

Normal depth/critical depth relationship

Uniform flow can be found in a subcritical, supercritical or critical regime depending on the importance of its inertia. This importance is reflected either by the value of the Froude number or by opposition of both normal and critical depths. To highlight the type of flow, the calculation of normal and critical depths must be done for the same flow rate.

Eliminating the discharge Q between Equations (1) and (2) and considering that Rh,n = An/Pn, it results:
formula
(4)
where the hydraulic radius Rh,n is expressed as a function of the generator diameter D and the filling rate ηn (Table 1) and the Reynolds number R* in Equation (4) is given by the relation (3). Taking into account hydraulic radius relations (Table 1), Reynolds number Equation (3) takes the following form:
formula
(5)
where Fn,2 functions are regrouped in Table 2.
Table 2

Expression of the Fc, Fn,1 and Fn,2 functions for each zone

ZoneFunction
FcFn,1Fn,2
    
    
    
ZoneFunction
FcFn,1Fn,2
    
    
    
For a full section of the egg-shaped conduit, the flow is governed by the functions of the third zone: and . The Reynolds number Rf* characterizing the full flow section (Achour 2014) is written as:
formula
(6)
Taking into account Equations (5) and (6), Reynolds R* number can be related to Reynolds number Rf* at full section as:
formula
(7)
For the egg-shaped section, the expressions of the wetted perimeter, the wetted area and the hydraulic radius are written according to the filling zones indicated in Table 1. Therefore, as a result the relation (4) becomes:
formula
(8)

The functions Fc, Fn,1 and Fn,2 are expressed exclusively according to the filling rate of the pipe and are grouped in Table 2.

In the results and discussion section, the in-depth study of Equation (8) will allow describing the nature of the flow and some interesting conclusions will be drawn.

Froude number relationship

In any study that attempts to characterize the nature of the flow regime, the Froude number value is calculated as a function of the flow rate and geometric characteristics of the given section. In order to make the calculation of the Froude number easier, a highly interesting approach will be developed and used as an alternative technique. By using the general form of Froude number relationship:
formula
(9)
By combining the Equations (2) and (9), one may obtain:
formula
(10)
From Equation (7) and Table 1, Equation (10) becomes:
formula
(11)
where the functions of Fn,2 are given by Table 2, while Fn,3 functions are expressed as:
formula

In Equation (11), the Froude number is expressed as a function of the slope S0, the generating diameter D of the conduit, the functions Fn,2 and Fn,3 containing the filling rate η, the absolute roughness ε and the Reynolds number Rf* at the full state of the conduit which takes into account the effect of the kinematic viscosity ν. It is worth noting that Equation (11) takes into account all the flow parameters that allow a better representation of the natural phenomenon. It is necessary also to specify that Equation (11) allows the explicit calculation of the Froude number independently of the volume flow rate. Moreover, in the absence of the flow rate value, Equation (11) provides a basis for determining the nature of the flow regime. However, it is necessary to analyze this relationship graphically in order to examine the flow behavior according to the importance of the slope value. To highlight the developed expressions, graphs will be constructed in a Cartesian coordinate system considering a smooth egg-shaped conduit of generating diameter D equal to unity.

Analysis of the normal depth/critical depth relationship

For a better visibility of Equation (8), it is essential to use a graphic representation expressing the influence of the slope in the production of the different types of flows and consequently the number of critical states that can possibly occur in the conduit. For this purpose, as an example, a smooth egg-shaped conduit (ε → 0) with a diameter D = 1 m is considered. Figure 1 shows the variation of the filling rate ηc at the critical state as a function the relative normal depth ηn for longitudinal slope S0 range usually encountered in the practice of the hydraulic engineer.

Figure 1

Variation of ηc depending on ηn for slopes values S0 according to Equation (8) for smooth conduit.

Figure 1

Variation of ηc depending on ηn for slopes values S0 according to Equation (8) for smooth conduit.

Close modal

In Figure 1, the first bisector represents the critical state of the flow corresponding to ηc = ηn. This limit makes it possible to differentiate two domains of flow regimes, namely: the subcritical and supercritical domains. Moreover, one may note that some curves remain below this line. Therefore, no critical condition can be observed in this area and the predominant regime is subcritical.

This can be explained by the fact that for low slopes, the gravity forces prevail over inertia forces regardless the variation in the filling rate. Therefore the developed energy does not allow a transition to the supercritical regime; the limit of this zone is marked by the curve tangent to the first bisector characterized by a slope S0 = 0.00241096 (Figure 2).

Figure 2

Curve of variation of ηc according to ηn tangent to the first bisector. Slope: S0 = 0.00241096, (•) ηn = ηc = 0.199997835 ≈ 0.2, ηc,max ≈ 0.762, ηn(ηc,max) ≈ 0.955.

Figure 2

Curve of variation of ηc according to ηn tangent to the first bisector. Slope: S0 = 0.00241096, (•) ηn = ηc = 0.199997835 ≈ 0.2, ηc,max ≈ 0.762, ηn(ηc,max) ≈ 0.955.

Close modal

By examining the curve ηc = f(ηn) in Figure 2, it is clear that only one critical condition can occur. This state is observed at the point tangent to the first bisector which corresponds to ηc = ηn ≈ 0.2. In addition, it should be noted that the limit curve which gives a single critical state presents a maximum for ηc,max ≈ 0.762 corresponding to ηn(ηc,max) ≈ 0.955.

Beyond this limit curve, the curves intersect the first bisector at two points, thus indicating the existence of two critical states at two different flow rates.

Figure 3 illustrates an example of the two critical states generated for the slope S0 = 0.003.

Figure 3

Variation between ηc and ηn for the slope S0 = 0.003.

Figure 3

Variation between ηc and ηn for the slope S0 = 0.003.

Close modal

Regarding Figure 3, one can observe that the curve ηc = f (ηn) corresponding to the slope S0 = 0.003 intercepts the curve of the critical flow regime (Red line) at two points thus indicating the presence of two critical regimes for this slope.

Figure 3(a) represents the first point of intersection with a low filling rate ηc = ηn = 0.0311162 ≈ 0.031 while the second point of intersection is obtained at a higher filling rate ηc = ηn = 0.618078 ≈ 0.618 as shown in Figure 3(b).

Froude number relation study

In the previous section, a method has been developed from which it is possible to deduce the different states of the flow in the pipe by the combination of both normal and critical flows. At the same time, it is interesting to note that it is easy to characterize flows by an elementary analysis of normal flow exclusively. For this purpose, the variation of the Froude number as a function of the filling rate η for different slopes S0 was represented graphically in Figure 4 according to Equation (11).

Figure 4

Variation curves of Froude number Fr as a function of the filling rate ηn according to Equation (11) for smooth conduit.

Figure 4

Variation curves of Froude number Fr as a function of the filling rate ηn according to Equation (11) for smooth conduit.

Close modal

In Figure 4, the horizontal bold line (in red color) corresponds to Froude number Fr = 1. It should be noted that below this line the flow regime is subcritical, i.e. Fr < 1, while the flow regime is supercritical in the zone above the line, i.e. Fr > 1. For slope values greater than the limit slope, it is noted that, for a given slope and for two different flow rates, the critical regime appears initially for low filling rates then at higher filling rates. On the contrary, for certain values of the slope S0, the curves are entirely in the zone of the subcritical regime thus indicating that no critical regime can occur. In addition, a limit curve characterized by a point of tangency with the horizontal red line indicates the existence of a single critical condition corresponding to the limit slope S0 = 0.00241096 and the filling rate η ≈ 0.2 (Figure 5).

Figure 5

Point of tangency. Slope: S0 = 0.00241096, E/D=0, (•) ηn = ηc = 0.199997835 ≈ 0.2.

Figure 5

Point of tangency. Slope: S0 = 0.00241096, E/D=0, (•) ηn = ηc = 0.199997835 ≈ 0.2.

Close modal
The minimum specific energy leads to a critical regime and consequently it can be used as a method of validating the results obtained in the previous section. The basic equation for specific energy is written:
formula
(12)
In Equation (12), the wetted area A is given by Table 1. By dividing the specific energy of relation (12) by the generating diameter D, one may obtain the dimensionless specific energy: Es* = Es/D. Equation (12) is then reduced to:
formula
(13)
With: ηn = yn/D, and the function Fn,4 is written according to the filling rate as follows:
formula
Considering the filling rate ηn, the relative flow rate Q* is expressed according to relations (2), (7) and Table 1, as:
formula
(14)

Note that the functions Fn,1 and Fn,2 are detailed in Table 2.

The results obtained in the previous section can be validated using relations (13) and (14).

Figure 6 shows the curve form of the relative specific energy ES* variability based on the filling rate ηn. For all the slopes above, the first bisector (Figure 1), or at the right side of Fr = 1 (Figure 4), where the curves have two critical states, the slope S0 = 0.003 was taken as an example of validation process.

Figure 6

Variation curve for relative specific energy based on filling ratio (S0 = 0.003; E/D = 0). (a) ηn = ηc = 0.0311162, ES*min = 0.04169881, Q* = 0.000597196. (b) ηn = ηc = 0.618078, ES*min = 0.846566461, Q* = 0.205214732.

Figure 6

Variation curve for relative specific energy based on filling ratio (S0 = 0.003; E/D = 0). (a) ηn = ηc = 0.0311162, ES*min = 0.04169881, Q* = 0.000597196. (b) ηn = ηc = 0.618078, ES*min = 0.846566461, Q* = 0.205214732.

Close modal

In Figure 6, it appears that the minimum relative specific energy perfectly coincides with the two critical states observed for the slope (S0 = 0.003; E/D = 0) taken as an example.

In order to characterize the boundaries between the different states of uniform flow in an egg-shaped pipe, it would be judicious to plot the curves characterized by the critical states according to the Equation (8) by substituting the « n » index expressing normal flow with that of the critical flow « c ». Therefore, the relation expressing the slope S0 is written:
formula
(15)
Note: functions Fc,2 and Fc,3 correspond in this case to Fn,2 and Fn,3 respectively.

Figure 7 shows the variation of the slope S0 as a function of the critical depth according to relation (15) for a smooth egg-shaped conduit. The dashed line curve expresses the variation of the slope S0 as a function of the maximum Froude number.

Figure 7

Different flow states in smooth egg shaped conduit. (──) The slope changes with the fill rate. (─ ─) Slope variation according to Froude number. (─ · ─) Straight line corresponding to S0 = 0.002411.

Figure 7

Different flow states in smooth egg shaped conduit. (──) The slope changes with the fill rate. (─ ─) Slope variation according to Froude number. (─ · ─) Straight line corresponding to S0 = 0.002411.

Close modal

Through Figure 7 it is possible to deduce:

  • 1.

    Zone I: The depths are weak and the regime is subcritical.

  • 2.

    Zone II: The regime is supercritical and the boundary between zones I and II is marked by passing through a critical regime.

  • 3.

    Zone III: At strong depths, the regime is again subcritical and the transition is made through a second passage by the critical regime.

  • 4.

    Zone IV: Mild slope zone where no critical regime can occur. The flow is uniform and subcritical, bordered by the maximum values of Froude number.

  • 5.

    Zone V: In this area, the uniform flow is improbable, the values of the Froude number are less than unity; the slopes are weak so the flow is, eventually, gradually varied type (backwater).

  • 6.

    Zone VI: In this zone, the uniform flow is also implausible, the values of the Froude number are greater than unity, and the slopes are weak, therefore the flow is suddenly varied (hydraulic jump). The demarcation between zones V and IV denotes the passage by a fictitious critical regime.

It was possible during this work to achieve a functional relationship between critical depth and normal depth for egg-shaped conduit. This relation has been represented graphically for slopes varying between 10−6 and 5.10−2. The analysis of the obtained curves shows the existence of different states of critical flow in the conduit. Indeed, it has been observed that for the limit slope S0 = 0.00241096; E/D = 0 a single critical regime is possible, also for slopes less than this slope, no critical regime can occur and for higher slopes two critical states are possible.

However, a highly interesting relation has been developed in order to calculate the Froude number independently of the flow rate. The graphic representation of this relationship confirmed the observations mentioned above. Still, the curve of variation of the normal depth according to the strong slope on the one hand, and the Froude number according to the mild slopes on the other hand, lead to the establishment of six zones regrouping the different types of flows encountered in the practice. Validation of the obtained results was possible through the analysis of the specific energy for a slope S0 = 0.003.

Finally, it is important to note the need for further study in order to examine the effect of the generating diameter and absolute roughness on critical flow.

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

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