Hydraulic control

A hydraulic control is a critical section where a shallow water flow passes smoothly between subcritical flow and supercritical flow. One long-wave characteristic speed relative to the bed vanishes there. Regularity supplies an extra relation between discharge, depth and channel geometry, allowing the control to set the discharge or upstream state. A critical depth alone is not sufficient for a regular transcritical solution: the channel forcing must also be compatible there.

A weir is an obstruction across a channel that can regulate volume flux by creating a hydraulic control. In a shallow water model a sufficiently smooth crest has a critical section, with upstream flow subcritical and downstream flow supercritical under free discharge. Losses, submergence and geometry determine whether this ideal control applies.

For reservoir head $\delta$ above a broad crest, negligible approach velocity, constant reduced gravity, and no head loss, hydraulic control gives $\delta =3h_c/2$ and the displayed discharge. Reflecting the vertical coordinate gives the same result for a light layer escaping beneath a lowered roof. An orifice discharge coefficient is a separate closure and should not be silently inserted into this inviscid weir law.

A light shallow layer flows below a roof of height $H(x)$ and width $W(x)$ through a deeper approximately motionless ambient. Hydrostatic equilibrium and inviscid steady Bernoulli equation give the displayed specific energy for constant volume flux $Q$ and reduced gravity $g^{\prime }$. At fixed position, $E_h=g^{\prime }(1-{\mathrm{Fr}}^2)$ and $h_c=[Q^2/(g^{\prime }{W}^2){]}^{1/3}$. A roof depression raises the minimum admissible energy, just as a bed elevation does for a heavy lower layer.

For two same-height throats carrying the same light-layer volume flux and reduced gravity, the narrower throat has the larger critical minimum energy. A narrower downstream throat can back up and submerge an upstream throat. If the first throat is narrower, it can control discharge and feed a supercritical current through the second; a hydraulic jump followed by a second control is also possible when the available energy drop and intermediate geometry permit it. Equal throat widths are a marginal lossless case. Relative width alone does not fix jump position or mixing in the intervening reservoir.

For loss of fluid carrying its local axial velocity, the inviscid Bernoulli equation head stays constant while discharge decreases. On a flat bed a regular control requires a positive local maximum of $q=Q/b$. In the parabolic-width channel $b=b_0(1+x^2/L^2)$, this maximum occurs at $x_c=-S{L}^2/(Q_0+\sqrt{Q_0^2+S^2{L}^2})$, with $q_c=(Q_0+\sqrt{Q_0^2+S^2{L}^2})/(2b_0)$. Its critical depth is $(q_c^2/g{)}^{1/3}$. A control exists only when the prescribed total head equals the maximum critical-head envelope and the reach includes that point while retaining positive discharge.

For a prescribed positive head-loss rate $\gamma (x)$, the steady energy balance is $(E+H{)}_x=-\gamma$. The accumulated loss is added to the bed elevation when applying hydraulic control compatibility: $H_x+\gamma =h_c{b}_x/b$. A local loss law alone does not determine the accumulated loss between a remote inlet and the control; that integral must also be specified.

For a smooth rectangular channel, conservation of discharge and Bernoulli equation head gives $(1-F^2)h_x=F^{2}h{b}_x/b-H_x$. A regular hydraulic control has $F=1$ and the displayed geometric compatibility condition. At fixed discharge the critical-head envelope is $T(x)=H(x)+3h_c(x)/2$, with $h_c=(Q^2/(g{b}^2){)}^{1/3}$. A simple smooth control requires a local maximum of $T$: $T_x=0$ and $T_{xx}<0$. A local minimum cannot be crossed at the critical head because the real depth branches disappear nearby.