New Insights into Turbulent and Laminar Flow Relationships Using Darcy–Weisbach and Poiseuille Laws

: Analytical solutions for turbulent and laminar water flow conditions are developed considering the drainage process of a simple tank reservoir, using the Darcy–Weisbach and the Poiseuille laws, respectively. Near the critical value of the Reynolds number, the Darcy–Weisbach and the Poiseuille laws do not match, and there is a gap regarding the possibility of describing the drainage analytically. This gap corresponds to the critical zone of the Moody diagram, where theoretically a constant discharge occurs under a decreasing hydraulic gradient. In the critical zone , this hydraulic behavior of the flow reflects the different energy losses occurring during the drainage, as the laminar flow is a more efficient flow condition than the turbulent one. In natural systems (e.g., springs and karst aquifers), a smooth transition from the turbulent to the laminar water flow occurs due to the heterogeneity in the medium.


Introduction
Laminar and turbulent conditions are the main types of water flow, characterizing both anthropogenic and natural environments or systems (e.g., pipelines, rivers, and groundwater).
When the flow is turbulent and the velocity decreases, a progressive change to laminar flow can occur, and vice versa.This change depends on the hydraulic and geometric characteristics of the liquid and the medium, respectively, and has important consequences on the hydraulic pressure and the flow velocity.
The Darcy-Weisbach equation is the most commonly used hydraulic law to model the water flow under turbulent conditions.Julius Ludwig Weisbach (a German mathematician and engineer) [1], Henry Darcy (a French hydraulic engineer) [2], and others developed this equation for the calculation of hydraulic head losses, h, due to pipe friction [3].In a straight tube with inner diameter d and length L, the flow velocity, v, is related to the pressure lost, ∆P, by the following formula: where f is a dimensionless parameter known as the Darcy friction factor, which depends on the geometric and hydraulic features of the flow, described in the next sections; ρ is the density of the water.Under laminar flow conditions, the Poiseuille law describes the relation between the flow velocity in small conduits and the hydraulic head losses [4].This law comes from the French physiologist-physicist Jean Leonard Marie Poiseuille [4].During the 1880s, Hagen [5] reached similar results to those of Poiseuille on the water flow in cylindrical tubes; for this reason, the Poiseuille law is also known as the Poiseuille-Hagen law.In their review of the Poiseuille law, Sutera and Skalak [5] stated that the results of Hagen were less extensive and less accurate than Poiseuille's.
The general form of the Poiseuille law expresses the velocity of a liquid, v, in a tube with constant diameter, d, and length, L, in relation to the difference in the liquid pressure between the ends of the tube, ∆P: where µ is the dynamic viscosity of the liquid.Equation (2), which was found empirically, can be derived from physically based assumptions [6,7].
The flow type (turbulent or laminar) can be verified by the Reynolds number, R e : where ρ is the density of the fluid.The Reynolds number is more formally defined as the ratio of inertial to viscous stresses.Thus, when the inertial stresses become too large, the viscosity will be unable to suppress eddies, and turbulence develops because the resulting vortical motion is underdamped.It is particularly important to focus on the lower limit of turbulent flow, defined by the lower critical Reynolds number, R e(crit) , when all the turbulences entering the flow from any source will eventually be damped out by viscosity.This lower critical Reynolds number thus sets a limit below which laminar flow will always occur [7].
To investigate the transition between turbulent and laminar flows, a simple tank reservoir with a basal area A 1 (Figure 1) is used as a model to test different hydraulic laws.The reservoir is drained by a hole through a tank tube with a length L and diameter d (with an area A 2 ), allowing the control of the hydraulic conditions and simulation of output velocity. of the Poiseuille law, Sutera and Skalak [5] stated that the results of Hagen were less extensive and less accurate than Poiseuille's.
The general form of the Poiseuille law expresses the velocity of a liquid, v, in a tube with constant diameter, d, and length, L, in relation to the difference in the liquid pressure between the ends of the tube, ΔP: where µ is the dynamic viscosity of the liquid.Equation ( 2), which was found empirically, can be derived from physically based assumptions [6,7].
The flow type (turbulent or laminar) can be verified by the Reynolds number, Re: where ρ is the density of the fluid.The Reynolds number is more formally defined as the ratio of inertial to viscous stresses.Thus, when the inertial stresses become too large, the viscosity will be unable to suppress eddies, and turbulence develops because the resulting vortical motion is underdamped.It is particularly important to focus on the lower limit of turbulent flow, defined by the lower critical Reynolds number, Re(crit), when all the turbulences entering the flow from any source will eventually be damped out by viscosity.This lower critical Reynolds number thus sets a limit below which laminar flow will always occur [7].
To investigate the transition between turbulent and laminar flows, a simple tank reservoir with a basal area A1 (Figure 1) is used as a model to test different hydraulic laws.The reservoir is drained by a hole through a tank tube with a length L and diameter d (with an area A2), allowing the control of the hydraulic conditions and simulation of output velocity.During the drainage, the change from turbulent to laminar flow in the tank tube has to occur in correspondence with a specific water height hcrit, which depends on the geometrical features of the tank tube (L, d, f), and the hydraulic proprieties of the liquid as well.
We will focus on some hydraulic aspects concerning the transition from the turbulent to the laminar flow by employing analytical tools to simulate the drainage of the tank reservoir in Figure 1.
In natural media, such as aquifers, water flow is laminar in minor fractures and the pore matrix.On the other hand, in wider voids such as karst conduits, the water flow can be turbulent.The hydraulic condition of groundwater flow can switch from turbulent to laminar and vice versa due to the function of the flow velocity and the geometry of the During the drainage, the change from turbulent to laminar flow in the tank tube has to occur in correspondence with a specific water height h crit , which depends on the geometrical features of the tank tube (L, d, f ), and the hydraulic proprieties of the liquid as well.
We will focus on some hydraulic aspects concerning the transition from the turbulent to the laminar flow by employing analytical tools to simulate the drainage of the tank reservoir in Figure 1.
In natural media, such as aquifers, water flow is laminar in minor fractures and the pore matrix.On the other hand, in wider voids such as karst conduits, the water flow can be turbulent.The hydraulic condition of groundwater flow can switch from turbulent to laminar and vice versa due to the function of the flow velocity and the Water 2024, 16, 1452 3 of 11 geometry of the medium, which vary in space and time, especially in karst aquifers.For example, the turbulent-to-laminar transition during flow recession is a well-known hydraulic phenomenon observed in karst spring hydrographs [8], but it is not easy to identify in time and space [9].

Tank Reservoir Models
For the tank reservoir shown in Figure 1, the drainage can be described using a specific theoretical hydraulic law, and the discharge-time plots can be found starting from an initial water height in the reservoir, h 0 , at time t 0 = 0, applying the principle of mass and volume conservation [10].The water pressure in the tank tube is directly controlled by the water height when drainage starts (t 0 ), assuming a zero gauge pressure at the end of the tube.

Torricelli Reserovir
The simplest physical conditions occur when the friction forces and viscosity of the flow can be neglected during the drainage; these conditions approximately occur when the tube's diameter is wide, its walls are smooth, and the velocity is limited.Under such simplifications, the velocity of the flow in the tube is expressed in terms of Torricelli efflux velocity: where g is the gravitational acceleration, and h is the height of the water above the tube.In this case, the velocity is independent of the tank-tube length, L.
Considering that the section area of the tank tube is A 2 = (π/4)d 2 , the discharge from the outlet, Q, is as follows: As the discharged water volume must equal the water storage variation in the tank reservoir, the following equation is obtained [10]: Combing Equations ( 5) and ( 6), the general equation describing the drainage at time t is as follows [10]: with constants: For the Torricelli reservoir, the parameter α T has dimensions [L 3 •T -2 ] and expresses the linear decrease in discharge with time; the main consequence of this behavior is the constant slope of the discharge-time plot at any time, which is independent of the initial height h 0 [10].It is proportional to the second power of the hole cross-section A 2 , and it is inversely proportional to the base area, A 1 , of the tank reservoir.
The drainage of the reservoir described by using the Torricelli efflux velocity is a theoretical condition based on the assumption that no energy losses occur during drainage; however, it is a useful reference when comparing different hydraulic laws, which considers the role of friction forces taking place within the liquid (viscosity) or at the flow-tube interface.

Darcy-Weisbach Reserovir
If turbulent flow occurs in the tank tube, Equation ( 1) can be used to model the drainage.If the gauge pressure is zero at the end of the tank tube, the term ∆P can be replaced by ρ•g•h, and the outflow discharge, Q, from the tank tube with section area A 2 is as follows: Equation ( 9) is also known Thrailkill equation [11].
Comparing Equations ( 5) and ( 9), the following condition has to occur: as the Torricelli efflux velocity is the theoretical maximum possible.
The friction factor, f, is a non-constant parameter that depends on the Reynolds number, R e , as shown in the well-known Moody diagram [12] of Figure 2, for different ratios between the average height of the roughness protuberances, ε, and diameter of the tube, d.Furthermore, the Darcy friction factor, f, depends on the flow velocity; this means that the velocity, v, in Equation ( 1) is present also on its right side and causes the well-known complex relationship with the hydraulic head, h.As a consequence of this complication, the discharge-time relationship of turbulent flow can be found by an iterative process [13], and no equation can be found describing the relation between discharge and time.

Darcy-Weisbach Reserovir
If turbulent flow occurs in the tank tube, Equation ( 1) can be used to model the drainage.If the gauge pressure is zero at the end of the tank tube, the term ΔP can be replaced by ρ⋅g⋅h, and the outflow discharge, Q, from the tank tube with section area A2 is as follows: Equation ( 9) is also known Thrailkill equation [11].
Comparing Equations ( 5) and ( 9), the following condition has to occur: as the Torricelli efflux velocity is the theoretical maximum possible.
The friction factor, f, is a non-constant parameter that depends on the Reynolds number, Re, as shown in the well-known Moody diagram [12] of Figure 2, for different ratios between the average height of the roughness protuberances, ε, and diameter of the tube, d.Furthermore, the Darcy friction factor, f, depends on the flow velocity; this means that the velocity, v, in Equation ( 1) is present also on its right side and causes the well-known complex relationship with the hydraulic head, h.As a consequence of this complication, the discharge-time relationship of turbulent flow can be found by an iterative process [13], and no equation can be found describing the relation between discharge and time.The transition zone (Figure 2) defines a region between the completely turbulent (right part) and the laminar flow conditions (left part).This means that a moving fluid under this transient hydraulic stage contains a laminar component; the amount of this component progressively increases toward the left and causes a progressive increase in the friction factor, f, up to the critical zone.For high values of ε/d (and of f ), the width of the transition zone decreases.
Natural tubes in aquifers have much higher friction factors than straight manufactured pipes because the walls are not smooth and the channels are not straight [14].There are very few studies in this field, and most of them refer to the results by Jeannin [15].These studies assume very high values of the friction factor, ranging between 0.1 and 340 for karst conduits [16], which lie outside the Moody diagram.With these high values, the friction factor of natural channels in aquifers could be considered almost independent of the Reynolds number, R e .This characteristic causes an abrupt transition from completely turbulent to laminar flow conditions in natural conduits, almost without the transition zone.
In the interval values f = 0.1 ÷ 340, Equation (10) becomes the following: which is always respected in natural karst conduits, as L >> d.
For the Darcy-Weisbach reservoir, assuming the friction factor f to be independent of R e in natural conduits, Equations ( 6)-( 8) become the following: with constants: Under the assumption of a constant friction factor, the Darcy-Weisbach reservoir also would provide a linear decrease in the discharge with time.
However, unlike the Torricelli law, the drainage of the Darcy-Weisbach reservoir involves an energy loss; the amount of energy lost is inversely proportional to the factor f .
During the drainage, before reaching the laminar flow zone, there is an uncertain zone known as the critical zone where an abrupt decrease in the friction factor value, f, occurs, and no relationship between f and R e exists, approximately in range 2000 < R e < 3000.The lines of the transition zone could be extended towards the left, up to R e = 2000, but these modifications do not solve the disconnection of these lines with the laminar flow zone on the left.The hydraulic behavior of the flow in the critical zone and laminar flow zone will be discussed in the next section.

Poiseuille Reservoir
Under laminar conditions in the tank tube, the flow velocity is described by Equation (2).If the gauge pressure is zero at the end of the tank tube, the term ∆P can be replaced by ρ•g•h, and the discharge, Q, is as follows: Given that the discharged water volume must equal the variation in the tank-reservoir water storage, the following equation is obtained [11]: Combining Equations ( 15) and ( 16), the general formula describing the drainage at time t for the Poiseuille reservoir is as follows [11]: with constants: The parameter α P has the dimension [T −1 ] and expresses the drainage characteristics of the Poiseuille reservoir.Equation ( 16) represents Maillet's formula [17], and the constant α P is known as the recession coefficient, which controls the exponential decay of the discharge with time.
In the Moody diagram, the laminar condition is associated with a single straight line (independently from the different ε/d values), which represents the Darcy friction factor for low values of the Reynolds number.For straight manufactured pipes, laminar conditions occur when R e < 2000 (Figure 2), and the following relationship occurs: Equation ( 19) has been known since the early 1900s but it is unknown who first introduced it [3]; it will be discussed in the next sections.Replacing the friction factor, f, in Equation (1) (Darcy-Weisbach equation) the Poiseuille law (Equation ( 2)) is found.In laminar conditions, the friction factor is physically explained by a combination of other hydraulic factors, and it is inversely proportional to the velocity of the flow (f = 64µ/ρvd).
The drainage process of the tank reservoir (Figure 1) predicted by the Poiseuille law has been compared to that predicted by Darcy's law, filling the tank tube with sand having hydraulic conductivity K [18].

From Turbulent to Laminar Flow
The fixed geometric and hydraulic conditions of the tank reservoir of Figure 1 allow the use of Darcy-Weisbach and Poiseuille laws when gravity drainage occurs.To reduce the complexity of the analytical solutions, a constant friction factor, f, is considered during the drainage, which is an acceptable assumption for the values of ε/d > 0.015.
The small diameters or great lengths of the tank tube favor the conditions imposed by Equation (10); whereas, the wide diameters or short lengths of the tank tube favor the tendency of f ≈ d/L, and turbulent conditions decrease due to a decrease in the friction forces.The latter theoretically became null under Torricelli efflux conditions.However, when the factor d/fL < 1, the drainage is slower than drainage from a Torricelli reservoir, and the initial discharge, Q 0 , is lower as well.
Fixing the Reynolds number at its critical lower value, R e = R e(crit) , the condition for the laminar flow occurs when the flow velocity is v < v crit , with v crit : Combining Equations ( 20) and ( 2), the critical hydraulic gradient can be found as follows: where h P(crit) is the critical height in the reservoir, below which laminar flow occurs following the Poiseuille law under the hydraulic gradient i < i P(crit) in the tube with length L.
If the same value R e(crit) is considered for the turbulent flow (that is, the flow is turbulent for R e > R e(crit) ), combining Equations ( 1) and (20), the critical height above which turbulent flow occurs, h DW(crit) , is obtained as follows: with the critical hydraulic gradient i DW(crit) in the tube with length L.
The ratio between Equations ( 21) and ( 22) provides the following: If h DW(crit) = h P(crit) , Equation ( 23) becomes Equation (19) for R e ≤ R e(crit) , and for R e(crit) = 2000, the friction factor is f = 0.032.This stage corresponds to the extreme lower point of the line describing the laminar flow in the Moody diagram (Figure 2).However, for any curve describing the turbulent flow in the Moody diagram, the friction factor is f > 0.032 for R e(crit) = 2000.Therefore: Equation ( 24) is an obvious energetic condition, as laminar flow is more efficient than turbulent flow.Equation (24) explains the presence of the abrupt decrease in the friction factor, f, in the critical zone, where both models (Darcy-Weisbach and Poiseuille laws) fail to describe the flow velocity.
The effect of this behavior can be seen by simulating the drainage of the tank reservoir in Figure 1, which is described by the discharge and water height-time plots in Figure 3.The initial water height is h 0 = 10 m, the tank-tube diameter d = 2 mm, and friction factor f = 0.05.It has been assumed that the turbulent regime occurs up to R e(crit) = 2000 without the transition zone, which can be neglected for a high value of the friction factor, f, as previously discussed.As L = 1, the x-axis in Figure 3b also provides the values of the hydraulic gradient, i, in the tube.
Initially, the drainage occurs under turbulent flow and provides a linear discharge decrease with time up to a value Q 1 = 0.245 L/min (v = 1.31 m/s), corresponding to a water height h = h DW(crit) = 2.18 m, where the Reynolds number reaches its critical value R e(crit) = 2000.At this stage (t 1 ≈ 14 days), the laminar flow cannot begin, as it would provide a discharge higher than 0.245 L/min (dashed line).Thus, from this time the discharge has to be constant until a water height h = h P(crit) = 1.39 m is reached in the reservoir (from t 1 ≈ 14 to t 2 ≈ 17 days); during this time interval, the flow progressively changes from turbulent to laminar.This reorganization of the flow allows for a reduction in the energy lost during the drainage, as the same discharge (or velocity) in the tube, Q 1 = 0.245 L/min, requires different hydraulic heads to reach laminar condition (h P(crit) = 1.39 m).The flow velocity (v = 1.31 m/s) is constant during the transition flow regime, despite the water height decreases (hydraulic gradient in the tube as well, from i DW(crit) = 2.18 to i P(crit) = 1.39).The relationship between Q and h is shown in Figure 3b; between h DW(crit) =2.18 m and h P(crit) = 1.39, the discharge assumes the same value, and a flow transition occurs.This phase corresponds to the critical zone, and no physical or empirical laws are available to describe this transition.
The critical zone also corresponds to the discontinuity of the friction factor value, f.In the example (Figure 3), f changes from 0.05 to 0.032 in the critical zone, but this discontinuity comes from the need for Equation (1) for R e > R e(crit) and Equation (2) for R e < R e(crit) .However, the use of the friction factor during the laminar flow is superfluous, as it is explained by the combination of other hydraulic factors.
decreases (hydraulic gradient in the tube as well, from iDW(crit) = 2.18 to iP(crit) = 1.39).The relationship between Q and h is shown in Figure 3b; between hDW(crit) =2.18 m and hP(crit) = 1.39, the discharge assumes the same value, and a flow transition occurs.This phase corresponds to the critical zone, and no physical or empirical laws are available to describe this transition.The critical zone also corresponds to the discontinuity of the friction factor value, f.In the example (Figure 3), f changes from 0.05 to 0.032 in the critical zone, but this discontinuity comes from the need for Equation (1) for Re > Re(crit) and Equation (2) for Re < Re(crit).However, the use of the friction factor during the laminar flow is superfluous, as it is explained by the combination of other hydraulic factors.
Figure 4 highlights the main stages discussed in Figure 3, and shows the relationships of the water level with time, h = f(t), and discharge, h = f(Q).In the critical zone, no equation is available to describe the flow.In granular aquifers, the critical Reynolds number is very low and could be also less than 1 [19].Natural channels in bedrock aquifers present an intermediate case between smooth, straight manufactured pipes and the tortuous pathways in granular aquifers, and so they will have intermediate critical Reynolds numbers [14].
Due to the highly variable geometry of the natural channels, surface runoff is associated with a wide range of Re(crit), covering also the left side area of Figure 5.

Conclusions
The draining of a tank reservoir under turbulent and laminar flow conditions has been simulated using the Darcy-Weisbach and Poiseuille laws, respectively.
Simple geometrical and hydraulic features have been assumed for the simulated hydrological system, which have allowed us to investigate the behavior of the flow and, in particular, the shape of the hydrographs obtained in correspondence with the transition from the turbulent to the laminar flow, which occurs during the drainage.
In tortuous and irregular tubes, the friction factor, f, can be considered constant under the entire range of the turbulent conditions of the flow, and the transition zone of the Moody diagram is restricted.Under these conditions, the flow changes from a completely turbulent regime to a laminar flow in a restricted range of Reynolds number values.If a specific value of the Reynolds number is fixed to define the change from the turbulent and

Conclusions
The draining of a tank reservoir under turbulent and laminar flow conditions has been simulated using the Darcy-Weisbach and Poiseuille laws, respectively.
Simple geometrical and hydraulic features have been assumed for the simulated hydrological system, which have allowed us to investigate the behavior of the flow and, in particular, the shape of the hydrographs obtained in correspondence with the transition from the turbulent to the laminar flow, which occurs during the drainage.
In tortuous and irregular tubes, the friction factor, f, can be considered constant under the entire range of the turbulent conditions of the flow, and the transition zone of the Moody diagram is restricted.Under these conditions, the flow changes from a completely turbulent regime to a laminar flow in a restricted range of Reynolds number values.If a specific value of the Reynolds number is fixed to define the change from the turbulent and the laminar flows, R e(crit) , two distinct values of the critical hydraulic head can be found: (i) h DW(crit) for the turbulent flow, and (ii) h P(crit) for the laminar flow, with h DW(crit) > h P(crit) .These behaviors can be expressed also in terms of hydraulic gradients, with i DW(crit) > i P(crit) .
Between these critical values of the hydraulic head, the flow cannot be described either with Darcy-Weisbach or with Poiseuille laws, and this condition corresponds to the critical zone of the Moody diagram.During the drainage, when water height in the reservoir decreases from h DW(crit) to h P(crit) , a progressive change from turbulent to laminar regime has to occur, but the velocity of the flow remains constant; this behavior reflects the different energy losses during drainage, as the laminar flow is more efficient than turbulent flow.As a consequence, in correspondence with the critical value of the Reynolds number, R e(crit) , the Darcy-Weisbach and the Poiseuille laws do not match, and there is a gap regarding the possibility of describing this process analytically.During this stage, theoretically, a constant discharge occurs under a decreasing hydraulic gradient, in contrast with any hydraulic laws.
In natural environments, such as rivers, springs, and groundwater, due to the great heterogeneity of the medium, the behavior of constant velocity (or discharge) in the critical zone should occur asynchronously in time and space, and the change from turbulent to laminar flow can appear smoothed or masked in the relative hydrographs.

Figure 1 .
Figure 1.Cylindrical tank reservoir, with a basal area A1.Water discharge occurs through the tank tube at the bottom, which has a diameter d and length L, starting from an initial water height h0.During the drainage, the flow condition in the tank tube changes from turbulent to laminar in correspondence with the critical water height, hcrit.

Figure 1 .
Figure 1.Cylindrical tank reservoir, with a basal area A 1 .Water discharge occurs through the tank tube at the bottom, which has a diameter d and length L, starting from an initial water height h 0 .During the drainage, the flow condition in the tank tube changes from turbulent to laminar in correspondence with the critical water height, h crit .

Figure 2 .
Figure 2. Friction factor, f, in relation to Reynolds number, Re, for tubes having different roughness characteristics (ε/D) and the main hydraulic flow zones of the Moody diagram (modified from [12]).

Figure 2 .
Figure 2. Friction factor, f, in relation to Reynolds number, R e , for tubes having different roughness characteristics (ε/D) and the main hydraulic flow zones of the Moody diagram (modified from [12]).Based on the flow regime, four zones are identified in the Moody diagram: (i) Completely turbulent zone (ii) Transition zone (iii) Critical zone (iv) Laminar flow zoneThe completely turbulent regime is described on the right side of the Moody diagram, and occurs for a high value of R e (and thus, of the flow velocity), where each straight line refers to a specific value of the ε/d ratio.The complete turbulence rapidly develops for high values of ε/d, whereas it needs very high R e values for smooth conduits.Under the completely turbulent flow regime, the friction factor, f, does not depend on the Reynolds number, R e , nor on the flow velocity.

Figure 3 .
Figure 3. Hydrograph obtained draining the tank reservoir using the Darcy-Weisbach (Re > 2000) and Poiseuille laws (Re < 2000), with the inset showing geometrical and hydraulic features.(a) The turbulent flow (linear decrease) up to t = t1, and laminar flow (exponential decrease) for t > t2.In the time interval t1-t2, discharge (and velocity) is constant, and flow changes progressively from turbulent to laminar (critical zone).The dotted line shows the unrealistic discharge if the Poiseuille law started at t1.(b) The discharge-hydraulic head relationship.

Figure 3 .
Figure 3. Hydrograph obtained draining the tank reservoir using the Darcy-Weisbach (R e > 2000) and Poiseuille laws (R e < 2000), with the inset showing geometrical and hydraulic features.(a) The turbulent flow (linear decrease) up to t = t 1 , and laminar flow (exponential decrease) for t > t 2 .In the time interval t 1 -t 2 , discharge (and velocity) is constant, and flow changes progressively from turbulent to laminar (critical zone).The dotted line shows the unrealistic discharge if the Poiseuille law started at t 1 .(b) The discharge-hydraulic head relationship.

Figure 4
Figure4highlights the main stages discussed in Figure3, and shows the relationships of the water level with time, h = f (t), and discharge, h = f (Q).In the critical zone, no equation is available to describe the flow.

Figure 4 .
Figure 4. Tank reservoir drained by a tube with length, L, and diameter, d (geometrical and hydraulic features as Figure3).Water level controls the flow regimes in the tube, and the graphs show the relationships h = f(t) and h = f(Q); no equation is available in the critical zone, CZ.

Figure 5 .
Figure 5. Relationship between friction factor, f, and critical hydraulic gradient ratio, iDW(crit)/iP(crit) following Equation (23).Straight lines refer to constant critical Reynolds number, Re(crit), with Re(crit = 2000 for straight tubes.Groundwater flow zone is shown by shaded area.

Figure 5 .
Figure 5. Relationship between friction factor, f, and critical hydraulic gradient ratio, i DW(crit) /i P(crit) , following Equation (23).Straight lines refer to constant critical Reynolds number, R e(crit) , with R e(crit) = 2000 for straight tubes.Groundwater flow zone is shown by shaded area.