Open-Channel Flow: Method

Manning, Critical Flow, Jumps, Profiles, Weirs

Scope

The tool treats steady flow of water in a prismatic open channel: uniform flow by Manning's equation, critical flow and the specific-energy curve, the best hydraulic section, the hydraulic jump, the gradually varied water-surface profile by the direct-step method, and the discharge of sharp-crested, V-notch and broad-crested weirs. Five section shapes are offered: rectangular, trapezoidal, triangular, circular (a pipe or culvert running partly full) and parabolic. Unsteady flow, sediment transport, bends, bridge piers, culvert inlet control, natural channels of irregular section (use the standard-step method, as in HEC-RAS) and flow in a closed conduit under pressure are outside its scope.

Conventions and units

Lengths, discharges and velocities go through the site's unit dictionary (ft, ft³/s, ft/s in USCS; m, m³/s, m/s in SI). The bed slope S₀, Manning's n, the side slope z (horizontal per unit vertical), the energy coefficient α and the weir angle are plain numbers. The bed shear stress is printed as a plain number in lb/ft² or Pa, and the flow areas in ft² or m², because the dictionary carries neither lb/ft² nor a foot-based area display. Depths are measured vertically from the lowest point of the section. In the profile computation x is the distance from the control section, positive downstream, so a profile computed upstream of a dam has negative x. All results are computed in SI and shown in the selected system; the Manning constant is 1 in metre units and 1.486 (= 3.2808^1/3) in foot units, with the same n.

Section geometry

For a trapezoid of bottom width b and side slopes z: A = (b + z y) y, P = b + 2y √(1 + z²), T = b + 2 z y; the rectangle is z = 0 and the triangle is b = 0. For a circle of radius r at depth y, with φ = arccos(1 − y/r) the half central angle, A = r² (φ − sin φ cos φ), P = 2 r φ and T = D sin φ; uniform flow in a pipe is possible only up to y = 0.938 D, where the discharge is greatest (it is 7.6 % more than the full-pipe value because the perimeter grows faster than the area near the crown), and the tool refuses a larger discharge. A parabola is defined by its top width T_p at a reference depth y_p; its width varies as √y, A = ⅔ T y, and the perimeter is the exact arc length (T/2)[√(1 + x²) + ln(x + √(1 + x²))/x] with x = 4y/T. The hydraulic radius is R = A/P, the hydraulic depth D = A/T. The momentum function of the jump needs the first moment of the area about the free surface, A ȳ: b y²/2 + z y³/3 for the trapezoid, (4/15) T y² for the parabola, and for the circular segment r³ [⅔ sin³φ − φ cos φ + sin φ cos²φ], which reduces to 2r³/3 for the half circle.

Uniform flow and Manning's n

In uniform flow the depth is constant and the friction slope equals the bed slope. Manning's formula (1891), V = (1.486/n) R^2/3 S₀^1/2 in foot units or (1/n) R^2/3 S₀^1/2 in metres, is an empirical fit in which the Chézy coefficient grows as the sixth root of the hydraulic radius; it is dimensionally inconsistent, hence the unit-dependent constant. Given the discharge, the normal depth y_n is the root of Q(y) = Q, which the tool finds by bisection because Q(y) increases monotonically with the depth for every open section (up to 0.938 D in a pipe); the passes are tabulated. Given the depth, Q follows directly. The conveyance K = (1.486/n) A R^2/3 gathers the geometry and roughness so that Q = K √S₀; the mean bed shear stress is τ₀ = γ R S₀, the quantity that governs the stability of an erodible lining (Chow Chapter 7). Manning's n is picked from Chow's Table 5-6 (1959), the table every later handbook reproduces: the select writes the table's "normal" value into the field and the minimum and maximum are about 20 % either side (trowel-finished concrete 0.011–0.013–0.015, float finish 0.013–0.015–0.016, unfinished 0.014–0.017–0.020, brick in mortar 0.012–0.015–0.018, cement rubble masonry 0.020–0.025–0.030, riprap 0.020–0.030–0.035, planed wood 0.010–0.012–0.014, new clean earth 0.016–0.018–0.020, weathered earth 0.018–0.022–0.025, clean gravel 0.022–0.025–0.030, earth with short grass 0.022–0.027–0.033, winding earth 0.023–0.025–0.030, dense weeds in a deep channel 0.030–0.035–0.040, smooth rock cut 0.025–0.035–0.040, jagged rock 0.035–0.040–0.050, unmaintained dense weeds 0.050–0.080–0.120, clean straight minor stream 0.025–0.030–0.033, stones and weeds 0.030–0.035–0.040, winding with pools 0.033–0.040–0.045, sluggish weedy reaches 0.050–0.070–0.080, mountain stream with gravel and cobbles 0.030–0.040–0.050, with large boulders 0.040–0.050–0.070, pasture flood plain 0.025–0.030–0.035, scattered brush 0.035–0.050–0.070). White's Table 10.1 gives slightly different averages (asphalt 0.016, brickwork 0.015, clean earth 0.022), which is why his examples are reproduced with his values in the test script rather than with the select. Since Q varies as 1/n, a 20 % uncertainty in n is a 20 % uncertainty in the discharge, far more than any other term; the third digit of a normal depth is for checking, not a claim of accuracy.

Critical flow, Froude number and specific energy

The specific energy E = y + α V²/(2g) (Bakhmeteff 1912) is the head above the channel bottom; α is the Coriolis energy coefficient of the velocity distribution (1.0 for a uniform velocity, 1.05–1.15 in regular channels; Chow's Example 10-1 uses 1.10 and the tool's default follows it, which moves the critical depth of his channel from 2.15 to 2.22 ft). For a fixed Q, E has a minimum at the critical depth, where α Q² T / (g A³) = 1 (Chow Chapter 4; White eq. 10.37); the section factor A³/T rises monotonically with the depth, so y_c is found by bisection, and for a rectangle it is (q²/g)^1/3 with E_min = 1.5 y_c. The Froude number is written Fr = √(α V² T / (g A)) = V / √(g D / α), so that it is unity at the critical depth whatever α; Fr below 1 is subcritical (tranquil) flow, in which a disturbance travels upstream and the control is downstream, Fr above 1 supercritical (rapid) flow, controlled from upstream. Every E above the minimum is carried at two alternate depths, one on each limb of the curve; the tool prints the alternate of the flow depth and draws the curve with the flow depth, its alternate, the critical point and, when a jump is computed, its two depths, whose horizontal separation is the energy the jump destroys. The critical slope S_c is the bed slope at which the normal depth would equal the critical depth, S_c = (n V_c / (1.486 R_c^2/3))²; a slope below it is mild (y_n above y_c), above it steep.

Best hydraulic section

For a given area the discharge is largest when the wetted perimeter is smallest, which for a rectangle means b = 2y (R = y/2), for a trapezoid of fixed side slope z means b = 2y(√(1 + z²) − z), again with R = y/2, and among all trapezoids the half hexagon (z = 1/√3, sides equal to the bottom); the best triangle is the 90° V and the best of all sections is the semicircle (White Section 10.3; Chow Chapter 7). The tool solves the dimensions that carry the entered discharge on the entered slope with the best-section relation imposed, and shows beside them the half hexagon and the semicircle carrying the same discharge, with the saving in area. White's Example 10.4 (5 m³/s, brick, S₀ = 0.001) gives y = 1.27 m and b = 2.53 m, and at the same area the half hexagon would carry 5 % more and the semicircle 8 % more. Lining cost, erosion and freeboard usually decide the final shape; the best section is the reference against which they are weighed.

Hydraulic jump

A supercritical flow forced to become subcritical does so through a jump: a short, violently turbulent rise in depth across which momentum is conserved and energy is lost. With the bed horizontal and friction neglected over the short length, the momentum function M = Q²/(g A) + A ȳ is the same on both sides, where ȳ is the depth of the centroid of the flow area below the surface (Chow Chapter 3); the tool solves M(y₂) = M(y₁) for the conjugate (sequent) depth on the subcritical side of the critical depth, for any of the five sections. For a rectangle this is Bélanger's result y₂/y₁ = ½(√(1 + 8 Fr₁²) − 1) and the energy loss is (y₂ − y₁)³/(4 y₁ y₂); both are printed for comparison. The loss ΔE = E₁ − E₂ and the power γ Q ΔE are what a stilling basin is built to absorb. The form of the jump follows the upstream Froude number, after the Bureau of Reclamation's flume tests (Peterka, Engineering Monograph 25, Figure 9; Chow Chapter 15; White Figure 10.12): undular below 1.7, weak from 1.7 to 2.5, oscillating from 2.5 to 4.5 (a pulsating jet that is hard to contain), steady from 4.5 to 9 (the best stilling-basin range, 45–70 % of the energy dissipated), strong above 9. The length of the jump, from the toe to the point where the surface levels off, comes from the recommended curve of Monograph 25 Figure 7: L_j/y₂ rises from 4.0 at Fr₁ = 1.7 to about 6.1 between Fr₁ = 5 and 10 and falls slowly to 5.45 at 20; the curve was measured on rectangular flumes and is applied to other sections as an estimate. White's Example 10.7 (q = 10 m³/(s·m), y₁ = 1.25 m) gives Fr₁ = 2.285, y₂ = 3.46 m, a loss of 0.625 m (14 % of the upstream energy) and 61.3 kW of power per metre of width.

Gradually varied flow

When the depth changes slowly along the channel the pressure stays hydrostatic and the energy equation gives dy/dx = (S₀ − S_f)/(1 − Fr²), with the friction slope S_f = n² V²/(1.486² R^4/3) taken from Manning's formula as if the flow were locally uniform (Chow Chapter 9; White Section 10.6). The sign of the numerator depends on whether the depth is above or below the normal depth, the sign of the denominator on whether it is above or below the critical depth, which gives the classical classification: the slope letter M (mild, y_n above y_c), S (steep), C (critical, y_n = y_c within 1 %), H (horizontal) or A (adverse), and the zone 1 (above both reference depths), 2 (between them) or 3 (below both). A subcritical profile is controlled from downstream and computed upstream of its control; a supercritical one from upstream and computed downstream. Profiles that approach the normal depth do so asymptotically (M1, M2, S2, S3, C1), so the computation stops 1 % from it; those that run into the critical depth (M3, S1, C3, H3, A3) reach it at a finite distance, where the equation is singular and a jump or a free overfall follows, and the computation stops 1 % short of it; H2 and A2 grow without bound and are followed to the end of the entered reach. The direct-step method (Chow Chapter 10) chooses the depths and solves for the distances: between two depths Δx = ΔE/(S₀ − S̄_f) with S̄_f the mean of the friction slopes at the two ends, exactly the computation of Chow's Table 10-4. The tool spaces the depth steps geometrically toward the limiting depth so that the long, flat tail of the profile is resolved, tabulates every row, interpolates the depth at the end of the reach when the profile is longer than the reach, and plots the surface with the normal and critical depths. Chow's Example 10-1 (the default channel with a dam holding 5 ft, α = 1.10) reaches 3.40 ft, 1 % above the normal depth, 2,375 ft upstream in his table; the exact integration gives 2,385 ft. A coarse depth step misplaces the profile where the depth is near the critical depth: White's Example 10.9 prints x ≈ 230 ft for a rise from 3 to 4 ft on an S3 curve with y_c = 4.27 ft, while the exact integration gives 237 ft; the tool's fine steps reproduce the latter.

Weirs

A weir is a crest over which the flow passes critical, so that the discharge is fixed by the upstream head H alone; it is both a measuring device and a channel control that raises a backwater profile upstream (White's Example 10.13 joins the two: the weir formula gives the depth at the dam and the M1 profile the depth 2 km upstream). For a rectangular sharp-crested weir the ideal relation Q = ⅔ C_d √(2g) L H^3/2 is corrected by experiment. The Kindsvater–Carter method, the one recommended by the Bureau of Reclamation's Water Measurement Manual (eq. 7-1), writes Q = C_e (L + k_b)(H + k_h)^3/2 in foot units with k_h = 0.003 ft, k_b a function of L/B (the crest length over the approach-channel width, from 0.0079 ft at L/B = 0 to −0.003 ft at 1) and C_e = C₁ H/P + C₂ from the manual's Table 7-1 (C₁ = 0.400, C₂ = 3.220 for a suppressed weir, L = B; down to −0.0087 and 3.152 at L/B = 0.2); it covers suppressed and contracted weirs in one expression. Rehbock's formula for a suppressed weir is C_d = 0.611 + 0.075 H/P, i.e. Q = (3.27 + 0.40 H/P) L H^3/2 in foot units as Chow prints it, for H/P up to about 2; White's fit C_d = 0.564 + 0.0846 H/P in Q = C_d L √g H^3/2 is the same relation in another normalisation. Francis's formula Q = 3.33 (L − 0.1 n_c H) H^3/2 (1.84 in metres) accounts for n_c end contractions by shortening the crest 0.1 H each; it is older and less accurate but still the irrigation standard. For a V notch the discharge varies as H^5/2: Kindsvater–Shen, Q = (8/15) C_e √(2g) tan(θ/2) (H + k_h)^5/2, with C_e ≈ 0.58 and k_h ≈ 0.003 ft varying slightly with the angle (the manual's Figures 7-6a and 7-6b, used here through published polynomial fits), covers fully contracted notches of 25 to 100 degrees; the Cone equation Q = 2.49 H^2.48 (ft³/s, ft) is the traditional rating of the fully contracted 90° notch from 0.05 to 4.25 ft³/s; White's eq. 10.60 is the coefficient 0.44 in Q = 0.44 tan(θ/2) √g H^5/2, about 42 % below the ideal 8√2/15 because of the contraction. A broad-crested weir carries critical flow on its crest, so ideally Q = (⅔)^3/2 L √g H^3/2 = 1.705 L H^3/2 in metres or 3.09 L H^3/2 in feet (C_d = 0.544); the boundary layer on the crest reduces the effective head, and White's eq. 10.57 puts C_d = 0.544 (1 − δ*/H)^3/2 with the displacement thickness δ*/L_c = 0.001 + 0.2 √(k_s/L_c) for a round-nosed crest of length L_c and roughness k_s, valid for H/L_c below about 0.7; otherwise enter a coefficient (0.46–0.53 for real crests). Given the discharge, the head is found by bisection. Every weir formula needs a ventilated nappe and a tailwater below the crest; a submerged weir needs a reduction factor the tool does not apply.

Reading the results and their precision

The default case is Chow's trapezoidal channel: 20 ft bottom, 2:1 sides, n = 0.025, S₀ = 0.0016, 400 ft³/s. His Example 6-2 gives the normal depth 3.36 ft and 4.45 ft/s; the critical depth with α = 1.10 is 2.22 ft, the slope is mild, and a dam that holds 5 ft produces the M1 backwater of his Example 10-1, 3.40 ft deep 2,375 ft upstream. The jump from an illustrative 1.0-ft supercritical depth in the same channel (not in Chow) shows the momentum function applied to a trapezoid: the conjugate depth 3.89 ft and the energy lost. The bisection tables show how fast the depths converge and what Q(y) looks like; the specific-energy chart shows at a glance how far the flow is from critical and whether a small change of energy would move the depth a lot (near the nose of the curve) or little. In the profile table the column S₀ − S̄_f shows where the surface is nearly parallel to the bed (small Δx) and where it is far from it. Normal and critical depths are exact for the stated n and geometry; the dominant uncertainty is n itself. The jump length is a chart reading accurate to perhaps 5 %; the weir coefficients are accurate to 1–2 % within their stated ranges and unreliable outside them.

Limits

Prismatic channels of the five shapes only; Manning friction (fully rough turbulent flow, which excludes very smooth linings at low velocity and laminar sheet flow); water as the fluid (the bed shear and the jump power use γ = 62.4 lb/ft³ = 9,807 N/m³); hydrostatic pressure (no curvilinear flow at a sharp break of slope, which is where the profile ends in any case); the jump on a horizontal bed without friction and the jump length from the rectangular-flume curve; the direct step with the mean of the end friction slopes (Chow's form; averaging V and R first, as some texts do, changes a coarse step by 1–2 % and a fine one by nothing); weirs fully ventilated and unsubmerged, with the approach velocity head neglected as the formulas assume (correct the head for it when V₁²/(2g) exceeds a few percent of H); one weir at a time, independent of the channel above it.