1. Critical Flow at Pressurized and Free Surface Conduits - by Jerzy Mroz
Alternative to Moody Chart - Mroz Charts
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Jerzy Hubert Mroz, monograph book published in 2016. Book Description - download
Presented book parts numbering of equations, tables, figures and references, refers to the numbering encountered in the main body book.
The author proposes a redefinition of critical flow conditions that become universal for pressurised flows, free-surface flow conditions in any shape of channel cross-sectional areas. In fact, the Darcy-Weisbach formula gives the head loss over the length in either a circular pipe or open channel in the same way. The crucial obstacle in the use of the Moody diagram is that one needs to know a priori the relative roughness of the channel, so in fact the information on surface structural property should be known in advance. In practical applications this is not possible. The author proposes how to tackle this problem effectively and seriously with diagrams presented in the book. The book is factual, hard-hitting and definitely provides new, never before published material.
Prof. Dr. Hab. Pawel Rowiński, Institute of Geophisics, vice-President of Polish Academy of Sciences PAN
2. Table of Contents
- I. Synopsis I
- I.1 Critical flow formula and modified Froude numberI
- I.2 Friction factorII
- I.3 Modified Froude number as weir structure coefficient of dischargeVII
- I.4 Sharp-crested weir with triangular control sectionVIII
- I.5 Sharp-crested weir with circular control sectionIX
- I.6 Sharp-crested weirs with trapezoidal and rectangular cross-sectionsIX
- I.7 Full width circular weirs and spillwaysX
- I.8 Critical flow in compound channelsX
- 1 Introduction1
- 1.1. Critical flow at pressurized non-parallel sided duct2
- 1.2 Semi-circular cross-section: free surface flow11
- 1.3 Circular cross-section: pressurized flow11
- 2 Critical Flow at Bounded Conduits12
- 3 Circular cross-section conduit pressurized and free-surface critical and normal flow conditions203
- 4 Experimental Verification of Modified Froude Number in Pressurized Uniform Flows26
- 4.1 Experimental verification performed for a pressurized test pipe 26
- 4.2 Experimental verification performed for pressurized test pipe and non-Newtonian fluid, peat bath (Grabno) and water flow.28
- 5 Relationship between Transmitted Flux Power and Modified Froude Number 30
- 6 Free Surface Flow–Modified Froude Number Experimental Verification 32
- 6.1 Sample computations and results.34
- 7 Stable Flow Criterion, Vedernikov Number, and Modified Froude Number at Neutral Stability40
- 8 Experimental Verification of Modified Froude Number as the Coefficient of Discharge for a Weir Structure43
- 8.1 Sharp-crested weir with triangular control section45
- 8.2 Sharp-crested weir with circular control section46
- 8.3 Sharp-crested weirs with trapezoidal and rectangular cross-sections47
- 8.4 Full width circular weirs and spillways48
- 9 Design Equations for Pressurized Pipe52
- 10 Analogy between Compressible and Open-Channel Flows55
- 11 Critical flow in compound channels.57
- 12 Critical Flow Analysis66
- 12.1 Analysis of critical flow parameters in rectangular cross-section channels66
- 12.2 Analysis of critical flow parameters in trapezoidal cross-section channel70
- 12.3 Analysis of critical flow parameters for circular cross-section channels72
- 12.4 Analysis of critical flow parameters for egg-shaped cross-section channels78
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4. Critical flow formula and modified Froude number
The condition h=v2 / g associated with the concept of critical flow was first given by J.B. Belanger in 1828. The condition related to the concept of minimum specific energy and critical flow was later introduced in English by B.A. Bakhmeteff in 1932. To measure the degree of rapidity and tranquility of flow, Bakhmeteff used the nation of kinetic flow given by the factor λ=2v2 / 2gh. Investigating the Venturi flume, F.V.A.E Engel noticed in 1933 that the original formula for the Froude number Fr=v / √gh is not a suitable criterion for the state of flaw. He found that an expression somehow similar to the Froude number, B=v / √2gRh (called the Boussinesq number) provides a suitable criterion. P.K. Kundu and J.M. Cohen demonstrated in 2002 that the flow may be characterized by energy E as well as flux kinetic energy F. More specifically, E denotes the energy (specific energy head, H) per unit horizontal area and F the flux (flux kinetic energy head; HFL) per flow cross-sectional area perpendicular to the direction of propagation. Alternatively, F denotes the flux per unit area and E energy per volume. Bakhmeteff ′s critical flow condition h=v2 / g, given with specific energy head H, which roughly satisfies the rectangular channel flow condition, is not adopted if the flow cross-sectional area is compound, and has no physical meaning if the flow is pressurized. Kay Melvin (2008) states that ′The significance of specific energy beyond its simple definition is not so obvious and many engineers still struggle with it′. The critical flow condition, Rh=v2 / 2g, derived form the flux kinetic energy head, HFL, assumption, holds for pressurized flows and free-surface flows of any channel profile.
The main conslusions of the paper are summarized below. There is great utility in expressing the formulas for critical flow as
Rhc = v2 / 2g (1.18)
A3 / P = Q2 / 2g (1.19)
and the modified Froude number (subscripted by M) as
Fr2M = v2 / 2gRh (1.17)
Fr2M = Q2 ⋅ P / A3 ⋅ 2g (1.19.1)
The Darcy-Weisbach formula, expressed with a modified Froude number, gives the friction factor f in terms of the modified Froude number and the friction (or energy) slope function f(FrM , S). The v / ux(f) relation allows the determination of shear velocity. The modified Froude number with the Colebrook-White formula allows the calculation of the equivalent roughness height e separately for each open-channel and pressurized conduit flow problem. A definite concise set of formulas, applicable to open-channel and pressurized resistance problems, is assembled and displayed in a resistance diagram.
5. Friction factor
The friction factor f was the focus of interest in the Process Report of the Task Force on Friction Factors in Open Channels of the Committee on Hydromechanics of the Hydraulic Division [Friction Factors in Open Channels, J. of Hydr Div, March, 1963] (Ref. 53)
′ The report represents approximately four years of work by The Task Force Committee, It may have been hoped by some, when The Task Force was organized, that a definitive, concise set of formulas applicable the most open-channel resistance problems could be assembled. At least, it was hoped that something similar to the resistance diagrams now used for steady flow in uniform pipes would be made available. It should be stated from the outset that these hopes could not be realized at that time. The principal obstacles were the wide range of surface roughness sizes and types encountered in practical channels (from smooth concrete linings to boulder-strewn canyons). ′
The idea of presenting a friction factor and a resistance diagram for any surface irrespective of size of roughness and type encountered in practical channels, rivers, and pipes pressurized, is outlined below.
The Darcy-Weisbach formula,
Sf = 0.25 ⋅ f ⋅ v2 / 2gRh (3.1)
and modified Froude number
Fr2M = v2 / 2gRh (1.17)
combine to yield a friction slope and friction factor, as shown below:
Sf = 0.25 ⋅ f ⋅ Fr2M (3.4)
which can be rearranged to give.
1 / √f = FrM / 2√Sf , f(FrM , Sf ) (2.6)
For f = 8 ⋅ (ux / v)2, v / ux ( f ) the relation is given as v / ux = √8 / f .
The Colebrook-White formula
1 / √f = -0.86 ⋅ ln [e / 14.84 ⋅ Rh] (2.12)
and Eq. 2.6 yield the modified Froude number,
FrM = -1.72 ⋅ √Sf ⋅ ln [e / 14.84 ⋅ Rh] (2.13)
FrM = v / ux ⋅ √0.5 ⋅ Sf (2.3)
and Eq. 2.13 yields the universal velocity distribution log-law,
v = -2.432 ⋅ ux ⋅ ln [e / 14.84 ⋅ Rh] (2.14)
which may also be given as
v = - 1x ⋅ ux ⋅ ln [e / 14.84 ⋅ Rh] (2.15)
where x = 0.411 is obtained the same as von Kármán constant. Eq. 2.14 then yields the ′equivalent roughness height′,
e = 14.84 ⋅ Rh ⋅ 2.72(-0.411 ⋅ v / ux) (2.17)
The principal difference between the Moody Diagram and the author′s diagram is that, to obtain the friction factor f (Re, e / Rh) with a Moody Diagram, the equivalent roughness height e must be known. According to the Task Force, this is the principal obstacle because of the wide range of unknown roughness size and times encountered in principal channels. The friction factor formula given in Eq. 2.6 as f (FrM, S) allows the construction of the chart, which with the f (v / ux) relation, and Eq. 2.17, allows the calculation of the so-called equivalent rougness height, e. Indeed, the total boundary, as well as the internal and spill resistance drag factor, can be evaluated separately for each open or pressurized conduit flow problem. Examples are given with charts; see Figs. 6.4 and 6.5.
Fr2M = 4Sff ; 1 / √f = FrM2√Sf (2.6)
Note that Eq. 2.6 has a straightforward physical meaning and an analytical expression for f (FrM, Sf) in contrast to the commonly used Colebrook-White semi-empirical expression f (Re, e / Rh), for which various investigators have attempted to obtain an analytical one. Obtained from an empirical fit to pressure-drop data of pipe flows, the Colebrook-White equation
Implicit: 1 / √f = -0.86 ⋅ ln [2.51Re√f + e14.84 ⋅ Rh] (2.7)
is valid for the entire non-laminar range of flow. A difficulty with its use is that it is an implicit expression in its dependence on f and the uncertainty in the equivalent roughness e assumption. The Moody chart is a graphical representation of this equation. Introducing into Eq. 2.7 Reynolds number with velocity given by Eq. 1.17, Reynolds number is obtained as
Re = 5.66 ⋅ FrM⋅R1.5h ⋅ g0.5v (2.8)
and the assumption for friction factor f with Eq. 2.6 gives the explicit in a dependence equation.
Explicit: 1 / √f = -0.86 ⋅ ln [0.222 ⋅ vR1.5h ⋅ S0.5f ⋅ g0.5 + e14.84 ⋅ Rh] (2.9)
6. Alternative to Moody chart Mroz charts - Examples
Re ≈ 8,4 ⋅ 104,e/D ≈ 1,4 ⋅ 10-2 → f ≈ 0,043
Critical flow parameters obtained from the experiment and the Colebrook–White formula for FrM = 1.0 are listed in Table 4.1, row 27.
Given: D = 0.098 m, L = 26.9 m, Rh=D / 4 = 0.0245 m, A = 0.00754 m2
Measured: hf = 0.287 m, Q = 0.0052 m3 /s
Calculated: vc = Q / A = 0.689m/s, Sf = hf / L = 0.0107, Eq. 2.3,
ux = √Rn ⋅ sj ⋅ g = 0.0506 m/s, v/ux = 13,62 → f ≈ 0,043
Verification is performed for a pressurized plastic (Genova) test pipe, diameter D = 0.043 m, Rh = 0.01075 m, and length L = 7.5 m. The peat bath of density p = 1041.0 kg / m3 was at temperature T = 313 K (40 C). The measured mass discharge was M = 0.5914 kg / s (Q = 5.68 ⋅ 10-4 m3 / s) and the velocity v = Q / A = 0.391 m/s. With these values, other parameters become fixed: Eq. 1.17, FrM = 0.85; Eq. 2.6, f = 3.28; Eq. 2.3, ux = 0.25 m/s; v/ux = 1.56; pressure head loss hf = 4.455 m; friction slope Sf = 0.594; t = p ⋅ g ⋅ Sf ⋅ Rh = 65.2 N / m2; equivalent roughness height, Eq. 2.17 e = 0.084 m (Fig. 6.4).
Verification: with the Darcy–Weisbach formula, v = 0.39 m/s, Q = 5.66 ⋅ 10-4 m3 / s. The same test pipe experiment was repeated with water at temperature T = 293 K (20 C) and density p = 998.2 kg/m3. The measured discharge rate was Q = 5.66 ⋅ 10-4 m3/s and the velocity v = 0.391 m/s; values for derived parameters are then: Eq. 1.17, FrM= 0.85; Eq. 2.6, f = 0.031; Eq. 2.3, ux = 0.0236 m/s; v / ux = 16.53; pressure head loss hf = 0.042m; friction slope Sf = 5.6 ⋅ 10-3; t = 0.557 N / m2; equivalent roughness height, Eq. 3.16 e = 1.78 ⋅ 10-4 m.
Verification: with Darcy–Weisbach formula, v = 0,39 m / s; Q = 5.66 ⋅ 10-4 m3/s. Inserting the Reynolds number given by Eq. 2.8 into the Blasius formula, f = 0.3164 / RE0.25, the friction factor is obtained,
f = 0.205 ⋅ v0.25Fr0.5M⋅ R0.375h⋅ g0.125.(4.1)
For Rh and FrMvalues given above, and assuming v = 1.006 ⋅ 10-6m2/s, a value for the friction factor od f = 0.029 is obtained. The Blasius formula for Re = v ⋅ D / v = 1.29 ⋅ 104 gives f = 0.03 . Both results are compatible with friction factor f = 0.031, obtained using Eq. 2.6. For a test pipe with very smooth walls, the equivalent roughness height e = 1.79 ⋅ 10-4 m is indeed nil, in contrast to a non-Newtonian fluid, peat bath, where e = 0.084 m is obtained that is nearly twice the diameter length, D = 0.043 m. This fictitious ‘roughness height’ is produced as a consequence of internal resistance that the flow has to overcome. The pressurized pipes experiments form part of the research at the Institute of Water Supply and Hydraulic Engineering Warsaw University of Technology. (Komarzeniec, Mroz, Tichonczuk 1996; in Polish).
Darcy-Weisbach: v = 2.353 m / s, Q = 52.3 m3 / s
Chezy-Manning: v = 2.363 m / s, Q = 52.5 m3 / s
Note that e ≅ 2.3hm is physically impossible, and confirms that the so-called ‘equivalent roughness height’, e, represents the total flow drag due to boundary, internal, and spill resistance that the flow has to overcome, as has been demonstrated with the peat bath (Grabno) pressurized flow analysis.
A presentation of the Cache Creek flow parameters is given in Figs. 6.4–6.6.
Given: vm = 2.353 ms; Rhm= 1.16m;SEm = 2.95 ⋅ 10-2;
Calculated: FrM = 0.493
For and FrM and SEm → fm = 0.49; for fm → vm / uxm ≅ 4.0 → ux = 0.588 ms;
for vm / uxm = 4.0 and Rhm = 1.16 m → em ≅ 3.25m
If em is assumed to be known (as in the Moody diagram) and Rhm is given:
em → Rhm → vm / uxm → fm
for em ≅ 3.25 m → Rhm ≅ 1.16 m → vm / u xm ≅ 4.0 → fm ≅ 0.49 .
If SEm is also known:
fm → SEm → FrM
fm ≅ 0.49 → SEm ≅ 2.95 ⋅ 10-2 → FrM ≅ 0.49
Hence the average velocity is obtainable from Eq. 1.17:
vm = FrM ⋅ √2gRhm ≅ 2.35 m / s so for vm / uxm ≅ 4.0. → ux ≅ 0.588 m / s
or em → Rhm → vm / uxm< ≅ 4.0 and with ux = √Rhm ⋅ SEm ⋅ g ≅ 0.588 m / s
vm = 4 ⋅ 0.588 ≅ 2.35 m / s.
Cache Creek v / ux ≅ 4.0 → f = 0.49
Table 4.1, row 27 v / ux = 10.85 → f = 0.043
Peat Bath v / ux = 16.53 → f = 3.28
Cache Creek For Rhm → M = f / n2 ≅ 75.0 → n = 0.08
7. Stable flow criterion, Vedernikov number, and modified Froude number at neutral stability
Although a theoretical study of roll waves is outside the scope of this text, the Vedernikov stability criterion with modified Froude number assumption is the focus of the present analysis. Vedernikov (1945) developed a stability criterion for flow in open channels. The Vedernikov number associated with such flows is defined as
Ve = β ⋅ θ ⋅ Fr (7.1)
where β is an exponent of the hydraulic radius Rh in general formula for uniform flow: v = C ⋅ Rβh ⋅ S0.5E. Hence, β = 2.0 for laminar flow, β = 2 / 3 for turbulent flow if the Chezy-Manning equation is used, and β = 0.5 for turbulent flow if the Darcy-Weisbach equation is used. The channel shape factor θ is defined as θ = 1 - Rh ⋅ dP / dA and the Froude number as Fr = v / √g ⋅ h. If Ve < 1 then stable flow prevails; if Ve > 1 unstable flow exists. Natural stability occurs in flow with Ve = 1. Examining the natural stability limit, Chen (1995), Ponce and Porras (1995), and Ponce and Simons (1997), using different approaches, concluded the necessity of redefining the Froude number at the Vernikov criterion. Inserting into Eq. 7.1 the modified Froude number, FrM = v / √2g ⋅ Rh given with hydraulic radius, the sole wetted cross-section shape parameter enables the shape factor θ to be substituted in the Vernikov criterion, Eq. 7.1. Thus, the modified Vernikov number is given as
VeM = β ⋅ FrM (7.2)
For VeM = 1, FrM becomes the modified Froude number at neutral stability
FrMs = 1 / β (7.3)
which for various flows at natural stability takes values:
laminar flowβ = 2.0,FrMs = 0.5,
turbulent flowβ = 2 / 3,FrMs = 1.5,
turbulent flowβ = 0.5FrMs = 2.0
From different approaches, the same, very close results were obtained by Chen (1995) and other early researchers available in Chen (1995)
laminar flowFrs = 0.577,Frs = 0.527,Frs = 0.5
turbulent flowFrs = 1.589,Frs = 1.524,Frs = 1.5
turbulent flowFrs = 2.0
In conslusion, Chen (1995) states that all natural stability Froude number expressions, Frs , are reduced to 2.0. A stable flow analysis (Henderson 1996) suggested that stability can arise in a long channel from the slope and resistance of the channel, and the characteristics of flow. Roll waves are commonly observed to occur on steep slopes, where Frs > 2.0. Many early roll-wave researchers cited by Chen (Cheng-lung 1995) such as Jeffreys (1925), Thomas (1940), Keluegan and Patterson (1940), Dressler (1949, 1952), Dressler and Pohle (1952), Lighthill and Whitham (1955) and Escoffier (1961) already obtained Frs = 2.0. Assuming the above, if the modified Froude number FrM = 2.0 is accepted as the natural stability limit, then the natural stability criterion may be presented with the modified Froude number given by Eqs. 1.17, 2.3, 2.6, 2.10, 2.13 and 2.20.
Recalling Eq. 2.6 Fr2M = 4Sf / f, with the modified Froude number at natural stability, FrMs = 2.0, implies that the flow at natural stability occurs when the Darcy-Weisbach friction factor f is equal to the friction (or energy) slope,
Sf = forSf = 4fF (7.4)
Hence, the flow is stable when Sf < f and unstable when Sf > f.
The above confirms the analysis of Henderson (1996, p. 342) that instability can arise from the slope when the resistance of the channel, and the characteristics of flow. Roll waves occur in wide channels on steep slopes where Fr > 2.0. A detailed analysis of the flow characteristics is obtained for FrM given by eq. 2.10. As flow for FrM > 2.0 is fully rough, Eq. 2.13 then represents the flow characteristics at natural stability with fair agreement. From Eq. 1.17 with FrMs = 2.0 gives the stability criterion,
Rh = v2 / 8g (7.5)
Cited in Chow (1959; p. 581) [H. A. Thomas: The propagation of Waves in steep Prismatic Conduits, Proceedings of Hydraulic Conference, State University of Iowa, Studies in Engineering, Bulletin 20, March. 1940, pp. 214-229] presents an analytical and experimental study of pulsating flow. The results of the study indicate that for pulsating flow to occur in a wide rectangular channel, the channel slope must be more than four times the critical slope Ss = 4Sc , or the velocity must be more than twice the critical velocity, vs = 2vc . An examination of Eq. 2.6 with FrM = 1.0 and FrM = 2.0 gives Ss = 4Sc and Eq. 1.17 with FrM = 1.0 and FrM = 2.0 gives vs = 2vc , the same values as indicated in Thomas′ study.
8. Experimental Verification of the Modified Froude Number as the Coefficient of Discharge for a Weir Structure
The relationship between the modified Froude number and the weir structure overflow discharge is considered. Given the control section [c - s] at the weir crest (Fig. 8.1), the hydraulic radius - discharge [Rh - Q] equation with Eq. 1.17 is introduced as
Q = [FrM = cD] ⋅ A ⋅ √2g ⋅ Rh (8.1)
For Rh = A / P, where A and P are the wetted area and perimeter for weir overflow, the [Rh - Q] equation, Eq. 8.1, yields
Q = [FrM = cd] ⋅ √2g ⋅ A1.5 / P0.5 (8.2)
The above equation has the same form for various throat types and geometries.
The following asumptions are made:
- The upstream boundary of the control section, the flow conditions, and weir geometry were dependent on the value of the water level height h above the crest elevation
- The values for the wetted cross-sectional area A, wetted perimeter P, and hence the hydraulic radius Rh are adequate for all these conditions
- The hydraulic radius Rh is a shape factor of the measuring structure.
- The modified Froude number FrM as coefficient of discharge, i.e., the [FrM = cD] assumption, states that the coefficient of discharge becomes a dimensionless physical value describing flow conditions during wier overflow.
In the literature, a head-discharge [H - Q] equation is applied, assuming the upstream energy head H1 (water level h1) as an independent flow parameter at a control section with fictitous wetted area A1 = A + ΔA. It is common practice to express the [H - Q] euqation with h1 as the height of the water level at an assumed control section A1 = A + ΔA. To compare the [H - Q] equation with the [Rh - Q] equation, the water level h in Eq. 8.1 is replaced by water level h1. Hence, Eq. 8.1 is then expressed as:
Q = cD ⋅ A1 ⋅ √2g ⋅ Rh1 (8.3)
The overflow characteristics of triangular, trapezoidal, rectangular, and circular sharp-crested weirs, as well as cyrindrical weirs and spillways, were investigated in the laboratory. A horizontal channel of rectangular cross-section of width b = 0.249 m was used. With water at normal temperature, the discharge was measured by a 90° V-notch weir. The flow depths h and h1 were measured using point gauges. The purpose of this section is to demonstrate experimentally that the modified critical flow formula Rh = v2 / 2g, Eq. 1.18, is applicable along with Eq. 8.2 as weir structure coefficient of discharge [FrM = cD]. For a more exact verification of assumptions to Eq. 8.2, a continuation of laboratory test is welcome.
For Rh = 0.5 ⋅ h ⋅ sin (α / 2), A = h2 ⋅tg(α / 2), and P = 2h/ cos(α/2), Eqs. 8.2 and 8.3 yield an [ Rh - Q ] equation
Q = [FrM = cd ] ⋅ √g ⋅ sin(α / 2) ⋅ tg(α / 2) ⋅ h2.5 (8.4)
For α = 90°, values of h = h1 and corresponding discharges Q are available in Table 5.3 of Bos (1989; pp. 162, 163). With Eq. 8.4 and Table 5.3, a value cD = 0.529 for the coefficient of discharge obtained from that of the [ H - Q] equation, Eq. 8.4, yields
Q = 1.393 ⋅ h2.51 (8.5)
The Kindsvater and Carter semi-empirical [ H - Q ] euation (Bos 1989),
Q = ce ⋅ 8 / 15 ⋅ √2g ⋅ tg(α / 2) ⋅ h2.51 . (8.6)
for α = 90° and ce = 0.585 gives
Q = 1.382 ⋅ h2.51 . (8.7)
It can be seen that discharge values given by Eqs. 8.5 and 8.7 differ insignificantly ( ). Equation 8.4 with water level h, measured over the weir crest, yields a [Rh - Q ] equation with coefficient of discharge [FrM = cD] = 0.557 and therefore
Q = 1.467 ⋅ h2.5 (8.8)
For A = D0.6 ⋅ h1.4 and Rh = 0.4 ⋅ D0.2 ⋅ h0.8 (Hager 1999), P = 2.5 D0.4 ⋅ h0.6 is obtained. Equation 8.2 yields the [ Rh - Q ] equations
Q = [FrM = cD ] ⋅ √0.8g ⋅ D0.7 ⋅ h1.8] ; (8.9)
for h = h1[ Rh - Q] equation is given as
Q = cD ⋅ √0.8g ⋅ D0.7 ⋅ h1.81] . (8.10)
To obtain coefficient of discharge cD from Eq. 8.10, discharge values are calculated using the [H - Q] equation of Staus and von Sanden (1926), (Bos 1989; pp. 167–169),
Q = ce ⋅ 4 / 15 ⋅ √2g ⋅ D2.5 ⋅ ω = ce ⋅ Φi ⋅ D2.5 . (8.11)
A circular weir of diameter D = 0.2 m was investigated. Values of ω, φi coefficient of discharge ce given as a function of filling ratio h1 / D are available in Tables 5.6 and 5.7, (Bos 1989). Equation 8.10 implies cD = 0.55 for 0.04 m < h1 < 0.9 ⋅ D, and cD = 0.52 for h1 > 0.9 ⋅ D.
From tests done in the laboratory and Eq. 8.9 for water levels h, measured over the crest [ FrM = cD ] = 0.62, the coefficient of discharge is obtained. A comparison of discharges Q is calculated below using Eqs. 8.9–8.11:
Eq. 8.9 with [ FrM = cD ] = 0.62, D = 0.2 m, and h = 0.065 m,
Q = 4.11 ⋅ 10-3 m3 / s, (-0.5 %),
Eq. 8.10 with cD = 0.55, D = 0.2 m, and h1 = 0.07 m, (h1 / D = 0.35),
Q = 4.165 ⋅ 10-3 m3 / s, (+0.8 %),
Eq. 8.11 with (h1 / D = 0.35), φi = 0.3866 (Table 5.6), and ce = 0.597 (Table 5.7)
Q = 4.13 ⋅ 10-3 m3 / s, (+0.0 %).
Laboratory tests were conducted for fully contracted weirs of p1 = 0.2 m and width b = 0.06 m for a trapezoidal shape, and b = 0.12 m for rectangular shape. The side inclinations of the weir crest are suitably m = 0.7 and m = 0.0 . Foor A = (mh + b) ⋅ h and P = 2h (m + 1)0.5 + b, Eq. 8.2 yields [Rh - Q ] equation
TrapezoidalQ = [ FrM = cD ] ⋅ √2g ⋅ [(mh + b)h]1.5[2h(m + 1)0.5 +b]0.5 (8.12)
Values of coefficient of discharges for the trapezoidal weir obtained from laboratory tests gave cD = 0.570 for water level h1, and [ FrM = cD ] = 0.714 for water level h. For a rectangular control section, Eq. 8.12 with sides slope m = 0.0 reduces to from
RectangularQ = [ FrM = cD ] ⋅ √2g ⋅ (b ⋅ h)1.5(2h +b)0.5 (8.13)
The corresponding values for the coefficient of discharges are cD = 0.432 for h1 and [ FrM = cD ] = 0.519 for h.
Hydraulic radius-discharge [ Rh - Q ], Eq. 8.13, and head-discharge, [ H - Q ], Eq. 8.14 (Bos 1989),
Q = cD1 ⋅ b ⋅ √g ⋅ [(2 / 3)] ⋅ h1]1.5 (8.14)
are examined for several configurations of circular weirs and spillways tested by Hager (1985), Chanson and Montes (1998), Fawer (1937), cited in Chanson (1998) and the author’s manuscript (2008; unpublished). See Fig. 8.2. Experimental observations are reported in Table 8.2.
Values for the coefficient of discharges given with Eq. 8.14 and formulas available in Table 8.2, calculated for a circular weir of D1 = 0.0888 m, b = 0.3 m, and water depths h1 = 0.09 m for Q = 20 ⋅ 10-3 m3 / s, to h1 = 0.15 m for Q = 45 ⋅ 10-3 m3 / s (Hager 1985; Appendix II, Table 3), are comparable with values cD = 1.33 ÷ 1.48 shown curve A if Fig. 8.2, Eq. 8.14. Curve A in the graph shows an increase in the coefficient of discharge with increasing discharge.
Experimental observations, curve B of Fig. 8.2, demonstrates explicitly that critical flow [ FrM - cD ] ≅ 1.0, Eq. 8.13, occurs at the control section of the weir crest for Q ≅ 20 ⋅ 10-3 m3 / s, and is constant with increasing discharge to Q = 45 ⋅ 10-3 m3 / s (Table 8.1).
For Q = 0.02 [m3 / s] and D = 0.0888m; h = 0.1595 - 0.0888 = 0.0707 [m] ; b = 0.3 [m]
Hager (1989; for see Fig. 5, Appendix 2, Table 1)
A = b ⋅ h = 0.0212 [m2]; P = 2h + b = 0.4414 [m] ; Rh = 0.04805 [m]
ν = Q / A = 0.9423 [m] ; [FrM = cD ] = ν / √2g ⋅ Rh = 0.97 [–] ; b = 0.3 m .
These results are presented in Table 8.2 and compared with previous studies where a satisfactory agreement is noted. The curve C presented with the [Rh - Q] equation of Eq. 8.13, and plotted as a function of water depth h1, gives coefficient of discharge values of cD ≅ 0.66 for Q ≅ 0.02 m3 / s and h1 ≅ 0.09 m through to cD ≅ 0.8 for Q ≅ 0.045 m3 / s and h1 ≅ 0.15 m
The overflow characteristic of a spillway, Fig. 8.4, ( R ≅ 0.02 m, b = 0.249 m, p1 = 0.252 m, α ≅ 52°50′), was examined for Q = 0.003 m3 / s and corrensponding [ Rh - Q ] equation.
The [ Rh - Q ] equation, Eq. 8.13, for h ≅ 0.026 m, measured at the spillway crest, gives [ FrM = cD ] ≅ 0.71.
The spillway crest overflow is subcritical. For water depth hs ≅ 0.012 m, measured perpendicularly to the spillway slope, and downstream of the spillway crest, the flow conditions give FrM ≅ 2.17, specifically, the flow is supercritical. Assuming FrM = 1.0, Eq. 8.13 yields critical depth hc ≅ 0.02 m, located (by trial) at l ≅ 0.045 m downstream from the spillway crest.
Experimental observations of weir overflows demonstrate that the modified Froude number, FrM = ν / √2g ⋅ Rh, given as coefficient of discharge FrM = cD, yield weir overflow conditions with fair agreement. For more exact verification, a continuation of laboratory tests confirming the hydraulic radius–discharge [ Rh - Q ] equation are welcome.
9. Critical flow in compound channel
An analysis of compound channel flows by application of the modified Froude number FrM is given by Blalock and Sturm in a (1981) paper ′Minimum Specific Energy in Compound Channel′ and its closure (Blalock and Sturm, 1983). The measured data (Fig. 7, p. 709. Table 1; p. 710 of paper and Table 7; p. 486 of the closure) was converted from BG and EE units to SI units Table 11.1 and Fig. 11.1. The purpose of analysis was to show that the modified Froude number FrM correctly identifies the occurence of critical flow points in a compound channel.
A summary of the author′s conceptual procedures is presented as below:
Critical flow formula, Eq. 1.1.19, A3 / P = Q2 / 2g and run x datum (Table 11.2) for Qx = 0.04713 m3 / s, Q2x / 2g = 1.132 ⋅ 10-4 m. The left-hand side of this equation, A3 / P, is solved for y by the trial-and-error prodecure until for yx = 0.167 m, A3 / P = 1.09 ⋅ 10-4 is nearly equal to Q2x / 2g. For Rhx = 0.0382 m and vx = 0.883 m/s, FrMx = v / √2g ⋅ Rhx ≅ 1.02 (Table 11.3) is obtained.
Friction factors f (Mroz - Mr) and fc (Blalock & Sturm - B - S) are obtained with Eq.2.6
1 / √f or fc = FrM or Fc2√S (2.6)
velocity vD - W is given by the Darcy-Weisbach (D - W) formula,
vD - W = √8gf or fc ⋅ √Rh ⋅ S,
from which the discharge QD - W is obtained
QD - W = A ⋅ vf or fc
The critical Froude number for the main channel flow of the experimental section was equal to one at two different depths, thereby indicating that there is more than one critical depth. Because multiple critical depths are possible, it is necessary to determine their values to compute correctly the profile of the surface flow of water. Blalock and Sturm (1981) cited and outlined difficulties associated with several available methods for critical depth computations, presented by Patrice and Grant (1978), the Soil Conservation Service (1976), the US Army Corps of Engineers (1982) and the US Geological Service (1976). Chaundry (2007; pp. 75,76) states, ′Blalock and Sturm defined a Froude number Fc which locates the points of minimum specific enegry although they did not present procedure for determining the critical depths.′ A critical depth analysis is given with modified Froude number FrM and Blalock and Sturm′s Fc numbers for compound flow cross-section (Table 1.2).
The Darcy-Weisbach velocity formula vD - W is given with friction factors Eq. 2.6 f(Mr) and fc(B - S), the Froude number for a compound flow cross-section Fc . In contrast, the modified Froude number FrM gives overbank discharges QcDW (runs 1,4,2,3) that deviate significantly from the measured discharges Q (Table 12.2). Run X with depth yx is obtained under the assumption
Qx = (Q3 + Q10) ⋅ 0.5 = 0.04713[m3 / s] and Sx = (S3 + S10) ⋅ 0.5 = 0.002013.
The critical flow formula, Eq. 1.19, A3 / P = Q2x / 2g gives Q2x / 2g = 1.132 ⋅ 10-4. For Qx = 0.04713 m3 / s, the expression A3 / P = 1.132 ⋅ 10-4 is obtained, whis is then solved for y by a trial-and-error procedure until for yx = 0.167 [m], A3 / P = 1.09 ⋅ 10-4 is nearly equal to Q2x / 2g. The specific energy head, E = y + v2 / 2g = 0.210 [m], is calculated as the mean of all runs. The equation for the flux kinetic energy head HFL for each run is given as the mean of HFl = 5 ⋅ Fr2M ⋅ Rh, Eq. 1.20 and HFl = 5 ⋅ v2 / 2g, Eq. 1.21
Moreover, for each run, depth y, energy head E, and flux kinetic energy head HFL versus modified Froude number FrM are given in Table 11.3. For depth yx, the Froude number of the compound flow cross-section Fc ≅ ∼ 1.08 is obtained from Fig. 11.2. An analysis of the compound channel flow by application of the modified Froude number FrM versus depth y, specific energy head E, and flux kinetic energy head HFL are given in Table 11.8.3 and Figs 11.2, and 11.3b.