Fluid Mechanics and Hydraulics

Critical depth and specific energy

Specific energy E is the energy head of the flow measured with respect to the channel bed, E = y + \frac{\alpha V^2}{2g}. For a rectangular channel with discharge per unit width q = Q/b and \alpha = 1, it becomes E = y + \frac{q^2}{2gy^2}. For a given discharge, E has a single minimum, and the depth at which this minimum occurs is the critical depth y_c. Setting dE/dy = 0 gives q^2/g = y_c^3, so y_c = (q^2/g)^{1/3} and E_{min} = \frac{3}{2}y_c.

The general criterion, with a kinetic energy correction factor \alpha, is that the Froude number equals one, or equivalently \alpha Q^2 T/(gA^3) = 1. For a general section this is written \alpha Q^2/g = A_c^3/T_c. A dimensional check confirms the form, since Q^2/g and A^3/T both have units of \mathrm{m}^5. For a rectangular section with a correction factor, the condition becomes \alpha q^2/(g y_c^3) = 1.

Depths greater than y_c give subcritical flow (Fr < 1), and depths less than y_c give supercritical flow (Fr > 1). For a specific energy above E_{min}, two alternate depths exist. A raised bed (a step) can force critical flow without changing the upstream depth. If the step is placed so that no head is lost, its height must equal \Delta z = E_1 - E_c, the difference between the upstream and critical specific energies. For a fixed discharge and a required critical depth, the width follows from b = Q/\sqrt{g\,y_c^3}.

A compound section (main channel plus flood plain) has a top width that changes abruptly at the junction. Its E(y) curve can therefore have several stationary points, so one discharge may produce more than one critical depth. For a symmetrical compound section, the maximum number of critical depths is three.

Example: a rectangular channel with q = 2 m^2/s has y_c = (4/9.81)^{1/3} \approx 0.74 m and E_{min} \approx 1.11 m.

Hydrostatic pressure at depth

The hydrostatic pressure in a static liquid increases linearly with depth, p = \gamma h, where \gamma = \rho g is the unit weight. Gauge pressure is measured relative to atmospheric pressure, while absolute pressure adds the atmospheric term, p_{abs} = p_{atm} + \gamma h. Forgetting the atmospheric term is a common error when an absolute value is asked for at depth.

The unit weight is found from the specific gravity as \gamma = SG \times 9.81 kN/m^3 for water-based liquids, with \gamma_{water} \approx 9.81 kN/m^3.

Pressure is the same at every point at the same level in a continuous body of one fluid. A manometer is solved by starting at a point of known pressure and moving through the fluids: going down adds \gamma h for each column, and going up subtracts it. A mercury column of height h contributes \rho_{Hg} g h, which is about 13.6 times the effect of the same height of water.

For a vertical plane surface, the resultant force is F = \gamma \bar{h} A, where \bar{h} is the depth of the centroid. For a rectangular plate extending from the free surface to depth L, the pressure diagram is triangular and F = \frac{1}{2}\gamma L^2 b, acting at 2L/3 below the free surface. A gate hinged at its base is checked by equating the moment of this force about the hinge with the moment the shaft can carry; this gives the maximum depth of water the gate can hold.

When two immiscible liquids are stacked against a wall, compute the pressure at each interface, then find the force on each layer as the area of its pressure diagram times the width. The total horizontal force is the sum of these layer forces. When a tank is hydraulically connected to an elevated reservoir, the pressure at any point below the free surface of the liquid equals \gamma times the height of the reservoir free surface above that point.

Hydraulic jump specific force

A hydraulic jump is a sudden transition from supercritical to subcritical flow, which dissipates energy through turbulence. The specific force (momentum function) of a section is M = \frac{Q^2}{gA} + A\bar{y}, where \bar{y} is the depth of the centroid of the flow area below the free surface. The specific force has units of m^3 and is the same at the pre-jump and post-jump sections, which is the basis for the momentum balance across a jump.

For a rectangular horizontal channel, the sequent depth ratio is \frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right), where Fr_1 is the Froude number of the supercritical upstream flow. Solving this for a given ratio gives Fr_1^2 = \frac{1}{8}\left[(2y_2/y_1 + 1)^2 - 1\right]. For example, a sequent depth ratio of 2 requires Fr_1 = \sqrt{3}.

The depth downstream of a jump, called the sequent depth, is the tail water depth needed to form the jump at the toe of a spillway or sluice. Using the specific unit discharge q = V_1 y_1 and the upstream depth, the tail water depth follows directly from the sequent depth formula.

The energy dissipated in the jump is \Delta E = \frac{(y_2 - y_1)^3}{4y_1 y_2}, and the power dissipated is P = \rho g Q \Delta E. Both are used in design of stilling basins.

A hydraulic jump forms when the channel slope changes from steep to mild. A steep slope carries supercritical flow, and a mild slope carries subcritical flow, so the transition between them must occur through a jump.

Gradually varied flow profiles

Gradually varied flow (GVF) is steady non-uniform flow in which the depth changes slowly along the channel. Its governing equation is \frac{dy}{dx} = S_0\frac{1 - (y_n/y)^N}{1 - (y_c/y)^M}, where y_n is the normal depth and y_c the critical depth. The profile type is found by comparing the depth y with y_n and y_c.

For mild slopes (y_n > y_c), the M1 profile lies above y_n, the M2 profile lies between y_c and y_n, and the M3 profile lies below y_c. For steep slopes (y_n < y_c), the S1 profile lies above y_c, the S2 profile lies between y_n and y_c, and the S3 profile lies below y_n.

The sign of the water surface slope distinguishes the profiles. For M1 and M3 the depth increases or decreases with positive slope, while M2 has a negative water surface slope because the depth falls toward y_c in the flow direction. M3 never has zero slope.

A supercritical stream entering a mild section has depth below y_c, so it follows the M3 profile, approaching the normal depth from below.

When a mild reach is followed by a steep reach, the depth passes through critical depth at the break. The mild upstream reach carries an M2 profile, and the steep downstream reach carries an S2 profile, with no jump at the break because the transition is from subcritical to supercritical flow.

Pipe flow velocity profile

In laminar flow of a Newtonian fluid in a circular pipe of radius R, the velocity profile is parabolic, u = u_{max}\left(1 - \frac{r^2}{R^2}\right), where r is the radial distance from the axis. The mean velocity is half the maximum, V = u_{max}/2, and the discharge is Q = \frac{\pi u_{max} R^2}{2}. This is the Hagen-Poiseuille profile.

The velocity at any radius follows directly from the profile. Using the ratio r/R, the velocity is u = u_{max}\left[1 - (r/R)^2\right], so the velocity at mid-radius is 3/4 of the maximum.

Energy and momentum correction factors account for the non-uniform velocity in the kinetic terms. They are defined as \alpha = \frac{\int u^3 dA}{V^3 A} and \beta = \frac{\int u^2 dA}{V^2 A}. For laminar pipe flow, \beta = 4/3 and \alpha = 2, so the ratio \beta/\alpha = 2/3.

For turbulent pipe flow the profile is flatter than the parabola, and it is commonly approximated by a power law. The correction factors are close to unity in that case.

Dimensionless numbers in fluid flow

A dimensionless number is the ratio of two forces or of two quantities of the same kind, and it allows results from models and prototypes to be compared. The common groups are:

• Reynolds number Re = \rho V D/\mu — the ratio of inertial to viscous forces. It is matched with viscosity effects.
• Froude number Fr = V/\sqrt{gL} — the ratio of inertial to gravity forces. Since V^2/(gl) is dimensionless, it is the square of the Froude number.
• Mach number M = V/c — the ratio of flow speed to the speed of sound. It is the measure of compressibility effects.
• Euler number Eu = p/(\rho V^2) — the ratio of pressure forces to inertial forces.

By the Buckingham \pi theorem, the number of independent dimensionless groups equals the number of variables minus the number of fundamental dimensions. For the drag on a sphere, five variables with three fundamental dimensions give two groups, such as the drag coefficient F_D/(\rho V^2 D^2) and Re.

Pipe head loss and friction factor

The Darcy-Weisbach equation gives the friction head loss in a pipe as h_f = \frac{fLV^2}{2gD}. Substituting V = 4Q/(\pi D^2) gives h_f = \frac{8fLQ^2}{\pi^2 g D^5}. For a fixed flow rate and length, the head loss is proportional to f/D^5, so a pipe with a lower friction factor needs a smaller diameter for the same head loss. When the length, flow rate and head loss are all unchanged, the product f/D^5 must remain constant, so D_2 = D_1(f_2/f_1)^{1/5}.

For pipes in series, the discharge Q is the same in each pipe, and the head loss in each pipe is proportional to fL/D^5. Head losses are therefore compared through this ratio for each pipe.

A pump working against a static head sees a system curve equal to the static head plus the friction head, H_{sys} = H_{static} + kQ^2. The operating point is the intersection of the pump characteristic with this system curve, found by equating the two heads.

Regime channel design (Lacey)

Lacey's regime theory describes an alluvial channel that carries sediment without scouring or silting its bed and banks. The regime relations are:

• Wetted perimeter P = 4.75\sqrt{Q}, with Q in m^3/s. For a regime channel the wetted perimeter equals the width of flow, so the discharge follows from Q = (P/4.75)^2.
• Hydraulic mean depth R = 0.47\left(Q/f^2\right)^{1/3}, where f is the silt factor. For a wide channel the hydraulic mean depth equals the average depth.
• Silt factor f_s = 1.76\sqrt{d}, where d is the mean grain size in millimetres, not centimetres.
• Scour depth D = 0.47\left(Q/f_s\right)^{1/3}, measured below the design (high) flood level, not below the existing riverbed.

The relation P = 4.75\sqrt{Q} gives Lacey's waterway, which is the wetted width used for the opening of a bridge across an alluvial channel.

Parallel pipe flow

Pipes connecting the same two reservoirs in parallel have the same head loss, equal to the difference in water levels. Each pipe therefore satisfies h_f = \frac{8fLQ^2}{\pi^2 g D^5} with the same head, so the discharge in each pipe is proportional to \sqrt{D^5/(fL)}.

When the lengths and friction factors are equal, the discharge scales as Q \propto D^{5/2}. The ratio of discharges is then Q_1/Q_2 = (D_1/D_2)^{5/2}, so a pipe of twice the diameter carries 2^{2.5} \approx 5.66 times as much flow.

In the Darcy-Weisbach form, the head loss is h = fLV^2/(2gD), which gives V \propto \sqrt{D} for equal h, f and L. Multiplying by the area \propto D^2 gives Q \propto D^{5/2} again, so the bigger pipe carries the larger share of the flow.

Channel contraction and critical flow

When a rectangular channel is narrowed without changing the upstream water level, the flow at the contracted section may reach critical depth. At the contraction, the specific energy equals the upstream value, and critical depth at the throat satisfies y_c = \frac{2}{3}E, where E is the upstream specific energy E = y + V^2/(2g).

The maximum discharge per unit width at critical flow is q_c = \sqrt{g\,y_c^3}. The minimum width that just produces critical flow is therefore b = Q/q_c. Narrowing the channel beyond this width (maximum contraction, or choking) does not increase the upstream depth in the ideal case, and the throat flow stays at critical.

Because the contraction width and the discharge per unit width are linked through E, any throat width below the critical value would force upstream conditions to change.

Irrotational and incompressible flow

A two-dimensional velocity field \vec V = u(x,y)\hat i + v(x,y)\hat j is incompressible when its divergence vanishes, \nabla\cdot\vec V = \dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} = 0, which is the continuity condition for constant density. The flow is irrotational when the vorticity vanishes, and for plane flow only the z-component matters: \omega_z = \dfrac{\partial v}{\partial x} - \dfrac{\partial u}{\partial y}. These two properties are independent, so a field can be incompressible and rotational at the same time. For example, u=-y and v=x give zero divergence but \omega_z = 2 \neq 0, which is solid-body rotation in the plane. A field with zero divergence and zero vorticity is both incompressible and irrotational. The curl and divergence are easy to confuse: the z-component of the curl pairs \partial v/\partial x with \partial u/\partial y with a minus sign, while the divergence adds the two derivatives taken in the same direction, \partial u/\partial x and \partial v/\partial y. Interchanging the derivatives gives a wrong expression. For a source, the streamlines are radial lines and the equipotential lines are circles; interchanging these two families converts the source into a free vortex, not a sink.