Open Channel Flow: Non-Uniform Flow

Whenever depth changes along a channel, the local balance among gravity, pressure, inertia, and boundary resistance changes. The correct analysis depends on how fast depth changes and whether discharge remains constant. Energy methods are especially useful for controls and gradual profiles; momentum methods are especially useful across short, highly dissipative transitions such as hydraulic jumps.

• RVF: hydraulic jumps, flow under gates, abrupt transitions, spillway toes, and short control regions.
• GVF: backwater and drawdown curves extending over many flow depths.
• SVF: side weirs, gutters, irrigation laterals, and channels receiving distributed runoff.
• Steady and uniform remain separate descriptors: a time-invariant backwater curve is steady but non-uniform.

At fixed discharge and geometry, the specific-energy curve has a minimum at critical flow. For an energy level above the minimum, two positive depths may exist: a shallow, high-velocity supercritical depth and a deeper, low-velocity subcritical depth. These alternate depths have equal discharge and specific energy; they do not generally have equal momentum function.

• Subcritical, $Fr<1$: $V<c$, so small gravity disturbances can propagate both upstream and downstream. Downstream controls can influence upstream depth.
• Critical, $Fr=1$: $V=c$ and specific energy is minimum for the specified discharge under the stated assumptions.
• Supercritical, $Fr>1$: $V>c$, so small disturbances cannot travel upstream against the flow. Upstream controls govern the downstream supercritical profile.

For fixed $Q$ in a prismatic section, differentiate $E(y)$ with respect to $y$. Because $dA/dy=T$, the condition $dE/dy=0$ reduces to the general critical relation. The result is geometric as well as dynamic: a nonrectangular channel reaches critical flow at the depth where its changing area and top width satisfy the same minimum-energy condition.

Broad-crested weirs, flumes, free overfalls, and suitably shaped contractions may create critical control. If a bed rise or contraction requires a minimum specific energy greater than the energy available to the approaching state, the original state cannot pass unchanged. Upstream depth then adjusts, or discharge changes, until the control can be satisfied. This condition is called choking.

A hydraulic jump converts organized mechanical energy into turbulence, heat, air entrainment, surface waves, and sound, so energy is not conserved between the two uniform sections. Over the short jump reach, however, momentum can be balanced accurately when pressure is hydrostatic at the end sections and bed friction and other external streamwise forces are small relative to momentum and pressure forces.

The boundaries are empirical rather than exact and can shift with geometry, approach turbulence, tailwater, and basin configuration.

The classical GVF equation assumes steady, one-dimensional, slowly varying flow in a prismatic channel; approximately hydrostatic pressure; known roughness; small bed slope; and no significant lateral inflow or outflow. Abrupt controls, strong curvature, hydraulic jumps, and rapidly changing geometry require other formulations.

Write total head as channel elevation plus specific energy. Along the flow direction, energy decreases at the friction slope, while bed elevation changes at the bed slope. For a prismatic section with $\alpha\approx1$, differentiating specific energy with depth gives $dE/dy=1-Fr^2$. Combining these relations yields the GVF equation and shows why the denominator becomes singular at critical flow.

The numerator $S_0-S_f$ indicates whether gravity supplies more or less slope than local resistance demands. The denominator changes sign between subcritical and supercritical flow. Evaluating both signs predicts whether depth rises or falls downstream before any numerical integration is attempted.

• Mild, M: $y_n>y_c$.
• Steep, S: $y_n<y_c$.
• Critical, C: $y_n=y_c$.
• Horizontal, H: $S_0=0$; ordinary positive-slope normal depth does not exist.
• Adverse, A: the bed rises in the flow direction; ordinary positive-slope normal depth does not exist.

• Zone 1: actual depth lies above both $y_n$ and $y_c$.
• Zone 2: actual depth lies between $y_n$ and $y_c$.
• Zone 3: actual depth lies below both $y_n$ and $y_c$.

Typical examples are M1 backwater upstream of a downstream control, M2 drawdown toward a critical control, S2 supercritical flow downstream of an upstream critical control, and S3 flow rising toward steep-slope normal depth. Horizontal and adverse slopes do not have finite normal depth, so only the physically meaningful profile labels are used.

Subcritical GVF is normally computed upstream from a known downstream control because gravity-wave information can propagate upstream. Supercritical GVF is normally computed downstream from an upstream control because information cannot travel upstream against the flow. A hydraulic jump may connect the two regimes, but the GVF equation cannot be stepped through the jump itself.

Standard-step analysis is suited to surveyed natural channels, nonprismatic reaches, changing roughness, bridges, culverts, and structures because the station spacing is known and the unknown depth is solved iteratively. Modern one-dimensional water-surface-profile software extends this same energy-balance framework with contraction, expansion, ineffective-flow, and structure-loss models.