Fluid Mechanics and Open Channel Flow

Manning's equation for open channel flow

Manning's equation is the standard empirical relation for uniform flow in open channels. In SI units the mean velocity is V = \frac{1}{n}R^{2/3}S^{1/2}, where n is Manning's roughness coefficient, R = A/P is the hydraulic radius (flow area divided by wetted perimeter), and S is the bed slope, which equals the energy slope under uniform flow. From continuity, Q = AV = \frac{1}{n}AR^{2/3}S^{1/2}. The group K = \frac{1}{n}AR^{2/3} is called the conveyance, so under uniform flow Q = K\sqrt{S} and therefore K = Q/\sqrt{S}. The conveyance already contains the roughness and the section geometry, so it can be found directly from the discharge and the bed slope. For a rectangular section of width b and depth y, A = by, P = b + 2y and R = by/(b+2y). When b = 2y, R = y/2. To find the slope, rearrange the equation as S = \left(\dfrac{nV}{R^{2/3}}\right)^2. Slopes asked in per cent must be multiplied by 100. When two channels with the same n, the same bed slope and the same depth are joined, the discharges add, and since Q \propto AR^{2/3}, the sum of AR^{2/3} for the parts equals the value for the combined section.

Dimensionless numbers (Reynolds)

A dimensionless number is a group of quantities whose units cancel, so its value does not depend on the unit system. The Reynolds number, Re = \rho v L/\mu (or vL/\nu), is the ratio of inertial forces to viscous forces, and it governs whether pipe or channel flow is laminar or turbulent. Other common groups are easy to confuse with it. The Froude number compares inertia with gravity, the power number relates power drawn by an agitator to inertial force, and the Biot number compares internal conduction resistance with external convection resistance. Each must be matched to the force ratio it actually represents.

• Puddling index — a ratio that describes the degree of soil puddling, so it has no units.
• Reel index — the ratio of reel peripheral speed to machine forward speed, so it is dimensionless.
• Cone index — penetration resistance expressed as pressure per unit area, so it carries units.
• Performance index — area covered per unit time or per unit energy, so it carries units.

Before assuming that an index is dimensionless, check whether it is a ratio of like quantities.

Gear pump torque and efficiency

A positive displacement pump delivers a fixed volume per revolution, and against a pressure rise \Delta p the ideal (theoretical) torque on the shaft is T_{th} = \dfrac{\Delta p\,V_d}{2\pi}, where V_d is the displacement in m^3 per revolution and \Delta p is in Pa, giving torque in N·m. Friction in a real pump requires more torque, so the actual torque is T_{act} = T_{th}/\eta_t, where \eta_t is the torque efficiency. The pump speed in rpm does not change the torque at a given pressure and displacement. Speed affects the flow rate and the power, which is P = 2\pi N T/60 with N in rpm. When a question gives displacement in cm^3/rev and pressure in MPa, convert both to SI units before computing torque.

Chimney draught pressure difference

Draught is the pressure difference that moves flue gas up a chimney. It arises because the column of hot gas inside the chimney is lighter than the column of outside air of the same height. Draught is often read on a water manometer and quoted as a head of water column, so the pressure difference is \Delta P = \rho_w\,g\,h. For water with \rho_w = 1000 kg/m^3 and g = 9.81 m/s^2, each millimetre of water column corresponds to about 9.81 Pa. Thus a draught of h mm of water gives \Delta P \approx 9.81h Pa. The draught is measured as the difference between the outside air pressure and the gas pressure at the base of the chimney.

Most economical trapezoidal channel section

A channel section is most economical when it carries the maximum discharge for a given area, which is equivalent to giving the minimum wetted perimeter. For a trapezoid this is the best hydraulic section, where the hydraulic radius is R = y/2, half the depth. For side slope z (horizontal run per unit vertical rise), the optimum bottom width is b = 2y\left(\sqrt{1+z^2} - z\right). For a 1:1 side slope, b = 2(\sqrt{2}-1)y \approx 0.828y, and the section is a half-hexagon. The flow area is A = (b + zy)y, the wetted perimeter is P = b + 2y\sqrt{1+z^2}, and the discharge at a given slope and roughness follows from Manning's equation. Always verify that R = y/2 before using the optimum dimensions.

Orifice discharge

A sharp-edged orifice discharging freely into the atmosphere has discharge Q = C_d A\sqrt{2gH}, where A is the orifice area, C_d is the coefficient of discharge (around 0.6 for a sharp-edged orifice), and H is the head measured from the free water surface to the centroid of the orifice. For a rectangular orifice the centroid is at mid-height, so the head is the surface level above the top edge plus half the orifice height. Using the head to the top or bottom edge instead of the centroid is a common error. The coefficient C_d accounts for the contraction of the jet and the losses at the edge, so it is always less than 1 for a real orifice.

Critical depth in open channel flow

Critical flow occurs when the Froude number equals one, F = V/\sqrt{gD} = 1, where D = A/T is the hydraulic depth and T is the top width. For a wide rectangular channel with discharge per unit width q, the critical condition is q^2/(g\,y_c^3) = 1, so the critical depth is y_c = (q^2/g)^{1/3}. At critical depth the specific energy is a minimum, with E_c = 1.5\,y_c for a rectangular section. Critical depth depends only on discharge and geometry, not on the bed slope or the roughness.

Parabolic grassed waterway design

A parabolic waterway section with top width T and depth d has flow area A = \frac{2}{3}Td. When the depth is small compared with the top width, the wetted perimeter is approximately P \approx T + \dfrac{8d^2}{3T}, which gives R = A/P. The design starts from continuity, A = Q/v_{perm}, so that the velocity does not exceed the permissible velocity for the grass lining. Manning's equation with the given roughness and slope is then used to check that the section carries the design discharge. The top width and depth are chosen to satisfy both conditions, and the freeboard is not included in the design depth.