The venturi that stops listening downstream

Not half a venturi took apart the story that a wing lifts because the air above it is squeezed through a channel. The trouble was never the physics — continuity and Bernoulli are exact — but the channel: over a wing, the upper wall is a streamline whose position is part of the answer. It made one concession. In a real venturi, with brass walls somebody machined, the areas are data, and the same chain of argument runs forward from the drawing to the pressure without circularity. The story is true of the device it was borrowed from.

This essay stays inside that device and follows the argument as far as it goes. It goes further than the textbook version admits, because it contains a limit. Squeeze the flow harder and the throat pressure keeps falling — until it reaches the one pressure a liquid cannot be taken below without turning to vapour. Past that point the venturi stops behaving like a meter. It passes a flow rate set only by what is upstream of it, it ignores what is downstream entirely, and it does so for a reason that turns out to be the liquid counterpart of a nozzle choking at Mach one.

In a venturi with walls, the argument runs forward from data #

The conventions are these. The meter is a 25 mm pipe that narrows over 20 mm to a 5 mm throat — a diameter ratio β of 0.2 — and widens again over 100 mm. It carries water at 20 °C, density 998.2 kg/m³ and vapour pressure 2.34 kPa. All pressures are absolute. Two coefficients stand in for the viscous parts of the flow that one-dimensional theory cannot compute: a discharge coefficient of 0.98 on the throat velocity, and a diffuser that recovers 85 per cent of the pressure drop between pipe and throat and loses the rest.

With those, continuity and Bernoulli give the throat pressure and the downstream pressure directly:

$$p_1 - p_t = \tfrac12\rho\Big(\frac{Q}{C_d A_t}\Big)^2(1 - \beta^4), \qquad p_2 = p_t + \eta\,(p_1 - p_t).$$

This is the textbook meter, used exactly within the hypotheses Bernoulli’s equation needs: steady flow along a streamtube whose walls are given, with the friction confined to the two coefficients. From 5 bar upstream and 4.8 bar downstream it passes 0.315 L/s; at 4.59 bar downstream, 0.451 L/s with the throat down at 2.26 bar. Each step down in the outlet pressure buys more flow, and the throat pressure falls faster than the outlet’s, because the diffuser gives back only part of what the contraction took.

The throat has a floor #

The throat pressure is an absolute pressure, and it cannot fall below the liquid’s vapour pressure. That is the liquid’s version of the floor Nothing sucks set for a gas: a gas cannot be taken below zero, and a liquid cannot be taken below the pressure at which it boils at its own temperature, because at that pressure it stops being all liquid. For water at 20 °C the floor is 2.34 kPa, a fortieth of an atmosphere.

The flow rate at which the throat reaches the floor follows from the same equation with $p_t = p_v$:

$$Q_{\text{choke}} = C_d A_t\sqrt{\frac{2(p_1 - p_v)}{\rho(1 - \beta^4)}}.$$

From 5 bar upstream that is 0.608 L/s, with the water through the 5 mm throat at 31.0 m/s. The downstream pressure at which it happens is $p_v + \eta(p_1 - p_v)$: 4.25 bar.

The approach to that floor is not gradual, and the shape of it matters to anyone sizing a meter. The throat’s pressure drop grows as the square of the flow, so the last part of the flow rate spends most of the throat’s margin. Half the choked flow, 0.304 L/s, leaves the throat at 3.76 bar and needs only 0.19 bar across the whole meter. Seven tenths takes the throat to 2.56 bar; nine tenths, 0.547 L/s, takes it to 0.97 bar — already below atmospheric — with the outlet at 4.40 bar, only 0.15 bar above the knee. A meter sized so that its largest expected flow is nine tenths of its choked value is therefore running with a throat under vacuum, where dissolved air is already coming out of the water, and is one small change in outlet pressure away from a different regime.

Below a critical downstream pressure the flow stops listening #

Lower the downstream pressure below 4.25 bar and the argument that has worked so far has nothing left to act on. More flow would need a lower throat pressure, and the throat cannot go lower.

The flow rate rises as the downstream pressure falls, reaches a knee, and is flat from there to vacuum: 0.470 L/s from 3 bar upstream with the knee at 2.55 bar, 0.608 L/s from 5 bar with the knee at 4.25, 0.720 L/s from 7 bar with the knee at 5.95. Everything to the left of each knee — a downstream pressure of 4 bar, 1 bar, 0.1 bar — delivers the same flow. The extra pressure drop does not vanish. It is spent downstream of the throat, where a cavity of vapour forms and then collapses as the diffuser brings the pressure back up, which in the profile figure appears as the diffuser recovering much less than its design fraction. That collapse is the violent process The bubble that hammers computed for a single bubble, repeated continuously in a diffuser.

The flatness is what makes the device useful. A venturi run beyond its knee holds its flow rate fixed against any disturbance on the outlet side, with no moving parts and no control system, which is why cavitating venturis are used as passive flow limiters in propellant feed lines and test stands where a downstream pressure is expected to wander.

It is worth being exact about what goes wrong when a venturi meter is run there by accident, because the obvious guess is not it. The meter reads the difference between the upstream and throat pressures, and past the knee both the flow and that difference are fixed — so the reading is still, to first order, the right flow. What has changed is authority. A valve downstream of the meter, opened or closed partway, no longer changes the flow at all until the outlet pressure climbs back above the knee, and an operator watching a steady reading while turning a handwheel is watching the symptom. The second thing that changes is the calibration: a vapour cavity occupies part of the throat, so the discharge coefficient the meter was calibrated with no longer applies to the flow it is metering.

Where the extra pressure drop goes #

Past the knee, the whole difference between the upstream and downstream pressures is lost, and it is lost in a few centimetres of diffuser. From 5 bar to 1 bar downstream, 0.608 L/s through a 4 bar drop is 243 W of mechanical power, against the 45 W the same flow loses at the knee itself, where the drop is only 0.75 bar. The extra 198 W is dissipated in the formation and collapse of the cavity. Almost none of it shows as heat in the water — spread through the flow, 4 bar of loss warms water by less than a tenth of a kelvin — but concentrated in the collapse it is the power behind cavitation noise and erosion, and it is why a cavitating limiter is built with a diffuser of hard material and expected to wear.

The knee sits exactly at the diffuser’s recovery #

The three knees in that figure are one knee, and collapsing them shows where it is.

Plot the flow as a fraction of its choked value against $(p_2 - p_v)/(p_1 - p_v)$, and every upstream pressure falls on one curve for each diffuser: flat below a ratio equal to the recovery, falling as $\sqrt{(1 - r)/(1 - \eta)}$ above it. The critical ratio is exactly the recovery — 0.70, 0.85 or 0.90 — because the diffuser is the only thing between a throat at vapour pressure and the outlet. A better diffuser pushes the knee closer to one, so a well-designed cavitating venturi holds its flow with the outlet within about ten per cent of the upstream margin, and a sharp orifice with no recovery at all would choke only with its outlet at the vapour pressure itself.

The comparison with a gas is direct. An isentropic nozzle with no diffuser chokes when the downstream pressure falls to 0.528 of the stagnation pressure, and a sonic venturi with a good diffuser chokes much closer to one, by exactly the same mechanism of recovery. In a gas the throat is held at the sonic condition; in a liquid it is held at the vapour pressure. A choked nozzle with friction shows how far the idealised gas picture moves when the walls are real; the liquid picture here has the same kind of dependence on a diffuser whose loss has been reduced to a single number.

Choked, the flow answers only to the upstream margin #

Past the knee the flow rate depends on the upstream pressure through the margin over vapour pressure, and on nothing else.

It grows as the square root of $p_1 - p_v$: 0.470, 0.608 and 0.720 L/s at 3, 5 and 7 bar, in the ratio the square roots require. The vapour pressure enters as a subtraction, so the liquid’s temperature matters. Water at 80 °C has a vapour pressure of 47.4 kPa and a density of 971.8 kg/m³, and from 5 bar it chokes at 0.588 L/s, 3.3 per cent below cold water. Near atmospheric upstream pressure the difference is much larger: at 1.2 bar the hot water’s margin is 72.6 kPa against the cold water’s 117.7, and it chokes about 20 per cent lower. A cavitating limiter set up with cold water and run with hot delivers less than it was set for, by an amount the square root makes easy to compute and easy to forget.

The same relation sizes a limiter. To hold 0.5 L/s of cold water from 4 bar upstream, the throat has to carry that flow at the velocity a 3.98 bar margin over vapour pressure can produce, 28.2 m/s, with a discharge coefficient of 0.98 — a throat of 4.8 mm, almost independent of the pipe it sits in, since for a small throat the $1 - \beta^4$ factor is within a fraction of a per cent of one. The pipe, the downstream plumbing and whatever wanders in it do not enter.

The same floor limits a pump’s suction for the same reason. The pressure at a pump’s inlet eye falls below the suction pressure by the local velocity head, and when it reaches vapour pressure the pump’s flow stops responding to the discharge side in just the way this throat’s does — which is what the inlet-group curve in the specific speed a pump spends its life at tracks as the operating point moves. A pump’s net positive suction head is this essay’s upstream margin over vapour pressure, written as a length of liquid.

A throat carrying a trace of vapour is supersonic for its own mixture #

Why no signal from downstream reaches the flow above a cavitating throat has an answer more specific than “the throat cannot go lower”, and it is the connection to a choked gas nozzle made literal.

A liquid with a little gas or vapour in it has a sound speed far below either phase’s, because the mixture has the liquid’s inertia and the vapour’s compressibility — the result Slower than either of them computed for air in water, with a minimum of 23.8 m/s. Vapour at 2.34 kPa is far more compressible than air at atmospheric pressure, and the effect is correspondingly stronger. Taking the mixture as frozen — no condensation or evaporation during the passage of a wave — its sound speed is 175 m/s at a vapour fraction of one in ten thousand, 55.8 m/s at one in a thousand, 17.7 m/s at one per cent and a minimum of 3.53 m/s at half and half.

The throat velocity at choking from 5 bar is 31.0 m/s. It exceeds the mixture’s sound speed as soon as the vapour fraction passes 0.33 per cent. A cavitating throat carrying even a trace of vapour is therefore supersonic for the two-phase fluid it carries, and a pressure disturbance from downstream cannot travel upstream through it — exactly the statement that makes a gas nozzle choke at Mach one. The venturi story that began as continuity and Bernoulli for an incompressible liquid ends at a compressible-flow result, reached because the liquid stops being incompressible at the one place the story squeezed it hardest. Allowing the phase change the frozen assumption forbids lowers the mixture’s sound speed further, so the conclusion only strengthens.

The cavitation number at onset belongs to the meter #

The onset of choking can be written as a cavitation number, the downstream margin over vapour pressure divided by the throat’s dynamic pressure, and the result is independent of the pressures and the fluid.

It equals $\eta(1 - \beta^4)/C_d^2$: 0.884 for this meter’s β of 0.2 and recovery of 0.85, 0.728 with a recovery of 0.70 and 0.936 with 0.90. At a diameter ratio of 0.6 the same recoveries give 0.634, 0.770 and 0.816, because a larger throat leaves more of the pipe’s velocity head in the balance. A venturi operated below its number is choked, whatever its size and whatever the upstream pressure — which makes the number the one figure of merit a limiter’s designer needs and a meter’s user should know to stay above.

The closed forms against a marched balance #

The throat pressure was computed a second way, by marching the differential momentum balance $dp/dx = -\rho V\,dV/dx$ through the contraction with the cone’s own slope, and it matches Bernoulli’s closed form to 3.4 × 10⁻¹³ of the upstream pressure. A bisection on that march for the flow rate at which the throat first reaches vapour pressure finds the choked formula’s value to 9.1 × 10⁻¹¹. The flow rate on either side of the knee agrees to 2.9 × 10⁻¹², the pressure profile ends on the downstream pressure to 10⁻¹⁶ in both regimes and touches the vapour pressure only when choked, and the mixture sound speed returns the liquid’s 1482 m/s with no vapour and the vapour’s own with no liquid. The calculation refuses an upstream pressure below vapour pressure, reverse flow, a throat wider than its pipe and a diffuser that recovers everything.

What the one-dimensional meter cannot show #

The cavity itself. Its length, its unsteady shedding, the noise it makes and the erosion its collapse causes are all outside a model that represents the cavity only by the recovery it destroys. Real cavitating venturis show a plateau that is nearly but not exactly flat, because the cavity’s size changes the effective geometry a little.

Dissolved gas. Water holds air, and air comes out of solution at pressures well above the vapour pressure, forming gaseous cavities that can begin to choke a throat before the vapour pressure is reached — the same process that breaks a siphon that has run for a day.

Heat. Evaporating liquid cools the throat and lowers its local vapour pressure, a thermodynamic suppression that is negligible for cold water and large for hot liquids and cryogens, where it moves the knee.

Constant coefficients. The discharge coefficient and the diffuser recovery vary with Reynolds number and with the state of the diffuser’s boundary layer, and a diffuser downstream of a cavity is not the diffuser it was designed as.

Still open: whether a choked throat also blocks a transient #

The plateau is a steady statement: a downstream pressure that has changed and stayed changed does not alter the flow. A pressure wave is a different thing. A valve slammed shut downstream sends a water-hammer surge back up the line, and a column that separates at vapour pressure makes that surge worse, not better. Whether a cavitating throat — supersonic for its own mixture — stops such a wave from reaching the pipe upstream, how much of it is reflected and how much passes as the cavity collapses, is the next calculation: the method of characteristics carried through a throat whose local sound speed drops by two orders of magnitude and recovers again within a hundred millimetres.