The Momentum Of Tidal Streams

Packo's prediction for the slack water times at Port Phillip Heads.

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The Momentum Of Tidal Streams

Postby packo » Thu, 27 May 2021 11:59 pm

***** Water Momentum in Port Phillip Bay's Tidal Streams *****

Like everything else with mass, moving water has momentum. In Port Phillip Bay, circumstances have conspired to make the effect of the momentum of the tidal streams quite significant. Unfortunately the changes of tidal stream behaviour due to water momentum and inertia effects seem to have eluded a large number of government and non-government players who give some incorrect advice about how slack water occurs at Port Phillip Heads. (This area is generally known to Victorians as simply "The Heads" or "The Rip".)

The discussion below was snipped off the end of an obscure and long post elsewhere in this forum. It is now given its own more prominent place due to its importance in better understanding how tide stream reversal at the Heads really occurs. The discussion is based around the same physics phenomenon, but in a domestic setting that might be more familiar to most readers.


***** Water Hammer *****

In the early parts of my tide work I struggled to comprehend how such slowly moving water near the end of a tidal stream could possibly perform such feats as flowing TOWARDS areas that were up to 40cm or so above its own level. To help others in this regard I will discuss the "water hammer effect" as perhaps the closest everyday example of the relevant physics in action.

Hopefully the modern day prevalence of plastic domestic plumbing, which is slightly elastic, has not diminished the experience of "water hammer" too much. In older houses with rigid metal plumbing, "water hammer" was noticeable where there was a long run of water pipe with a tap at the end. With the tap wide open the water rushes through the pipe. There might be say 1 kg of water in the pipe moving at several knots. This has a certain amount of momentum (mass x velocity).

If the tap is closed quickly the water mass tends to keep flowing, but now into the "dead end" of the closed pipe. A large pressure spike then develops, sounding rather like someone has struck the pipe with a hammer. It is this spike in water pressure, acting back along the pipe, that actually provides the force needed to decelerate the moving water to a stop. The quicker the tap closes the higher the pressure spike will be but its duration will be shorter. A very slow closing of the tap will produce a small pressure spike, but of a sustained duration.

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It is also possible to have "negative water hammer" if the long run of piping occurs downstream of the tap. In that case a negative (low) pressure spike occurs just downstream of the closing tap. In severe cases the negative pressure spike (ie. a partial vacuum) might be intense enough so some water turns into a vapour bubble. This can cause noise and cavitation when it collapses shortly after.

While both forms of water hammer are just a nuisance in the domestic setting, in very large piping systems such as hydroelectric power stations, these effects must be taken very seriously. The opening and closing of the control valves ahead of the turbines in these systems is done with much care to limit the magnitude of the pressure excursions.

Failure to do this can (and has) burst sections of the high pressure piping leading down from the storage pond to the turbine room, or caused vacuum crushing of pipework on the downstream side between the turbine room and the river discharge point. In either case the damage caused is extensive, very expensive to repair, and often life threatening.


***** The Hydraulic Ram Pump *****

My own introduction to the impressive power of controlled "water hammer" was as a young kid playing in the deep fern gully behind the village of Sassafras in the Dandenong ranges east of Melbourne. We are talking way back in the late 1950s when many rural towns didn't have a mains water supply.

Householders lucky enough to sit alongside the gully had this small creek with a reliable water flow, but the bottom of the gully was perhaps 40m below their homes. Farmers along the creek used electric pumps to extract their larger water demands. This wasn't an option for an ordinary home owner because of the expense of getting power down into the gully. There was also the safety factor given that much of it was public land on which naughty little boys like me might be playing.

Enter the "Hydraulic Ram Pump", a genius invention dating from way back in the late 1700s. It was robust, had only 2 moving parts, was self powering, flood proof, and mostly "curious little boy proof".

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A small weir is built across the creek and from it water is guided through a downward sloping metal pipe of about 3m in length and into the pump. One of its moving parts was a spring loaded valve at the end of the supply pipe. That valve opened inwards into the pipe but could be kept open by the force of the spring. Water would thus exit past the valve creating a sort of mini fountain. It then drained back into the downhill section of the creek.

By careful manual adjustment of the initial spring force, you could make it so that when the "water fountain" flow grew close to maximum strength, the drag of water escaping past the valve would overpower the spring and snap the valve closed with a loud "clack". This sound was due to the large water hammer pressure spike that developed as the momentum of the few kilograms of water in the supply pipe suddenly crashes into the end of the now closed off pipe.

The second moving part was a one-way valve that allows some of the high pressure water to escape into a large chamber called "the accumulator". This was partly filled with compressed air. The chamber had an exit in its base consisting of a metal delivery pipe that leads up the side of the gully to the home owner's holding tank some 40m above. As the pressure surge in the supply pipe dies away because the water slows down, the one-way valve closes and so prevents the high pressure water in the accumulator from leaking back into the supply pipe.

As the pressure surge declines further, the "fountain spring" reopens its valve and allows water in the supply pipe to again start accelerating down the pipe. The whole cycle would then begin again. The "clack-clack" pumping cycle would endlessly repeat itself about 20-30 times a minute, each time delivering a small amount of water all the way up to the householder's water tank far above.

So by using a mass of slowly falling water at a pressure of no more than 1 psi, the pump uses deliberate water hammer to amplify the pressure of a small part of that water to over 60psi to boost it a long way uphill. Perhaps only 1-2% of the water passing through the machine is pumped up to that high level with the rest returning to the creek to be used by others.

The "power source" was free and these pumps ran for years without attention. They also demonstrated to me the capabilities of slowly moving water masses to do amazing things by carefully harvesting the momentum in slowly flowing water.


***** "Open Channel" Water Hammer *****

Both the positive and negative pressure forms of water hammer can also occur in open channels where instead of having a closed pipe, we have a "free surface" always open to atmospheric pressure. In this case if a valve closes along the channel the pressure spikes are very much smaller because the water surface is free to rise upwards or fall downwards in response to water pressure variations. In other words the positive and negative pressure surges show themselves as small level rises and level falls on opposite sides of the closing valve.

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The physics of "Open Channel Water Hammer" states that if you close a valve situated mid-way along a uniform channel of flowing water, then the level rise upstream of the valve and the level fall downstream of the valve depend only on the water's initial momentum and the rate at which the valve is closed. The momentum term is the mass of water in the channel times its velocity.

The key point here is the mass term for a uniform channel rises in direct proportion to its length. Therefore even if the velocity in the channel is very low, if you have a long enough channel, then momentum values can be quite high and give a significant level rise and level fall on either side of the closing valve. In the Bay the equivalent channel length is over 15km.


***** Water Momentum In The "Choke Zone" *****

In the case of Port Phillip Bay the ocean is connected to the very extensive "main body" basin via a roughly triangular shaped region extending north east from the entrance for about 15 km and ending just beyond "The Great Sands" region. This area is termed the "choke zone" because it is responsible for controlling water flow into and out of the large main body area of the Bay.

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At the ocean end of this connecting channel, the choke effect is due to it's narrow width, whereas at the northern end the choke effect is because of large areas of shallow depth. In the middle region there is a bit of both. This "water channel" expands from 3km wide at the Heads to around 30km wide at the main body. In cross-sectional area it expands about 4-5 times over its roughly 15km length.

For most of the duration of a tidal stream the amount of water crossing the northern end of the choke zone is 90% or more of the amount entering or leaving through the Heads. Therefore to a reasonable approximation the average velocity across any cross-section of the choke zone is inversely proportional to the area of that cross-section.

This also means that in each kilometre length of the "channel", the water momentum contained in that kilometre is close to that in any other kilometre length. In turn this means that from a momentum perspective the choke zone is roughly equivalent to a uniform channel 3km wide and say 15km long, with a water speed equal to that between the Heads.

The same idea can be extended even further up the Bay because it is not until well over half-way up the Bay that the tidal flux decreases significantly. Our imaginary equivalent uniform channel would then be 3km wide and might extend up to perhaps 30km in length. That length is getting quite significant in that at all points in our imaginary "equivalent momentum channel" the water speed will be about the same as that through the Heads.

Now roughly 2 hours after high or low water at the Heads the "driving force" drops to zero because the ocean and mid-bay levels are briefly equal at that time. (The tide heights in the two regions are changing in opposite ways then, due to the three hour tide delay across the choke zone.) At the equal levels time, the current speed through the heads is known to be in the range from 1 knot to 2 knots for weak and strong tides respectively.

So at this equal levels point, our 3km wide by 30km long "equivalent momentum channel" has several billion tonnes of water just coasting along it at between 1 or 2 knots. The task of bringing this massive stretch of flowing water to a halt to give slack water is immense, calling for large deceleration forces and significant "water hammer slopes" in its free water surface.


***** Port Phillip Heads as a "Control Valve" *****

Whilst there is no physical barrier that comes down to close the opening at the Heads, that region can still be considered as a kind of bi-directional "valve" or "tap". Its "open-ness" or "closed-ness" for flow in a particular direction is controlled by the ocean to mid-bay level difference.

When there is a large forward level drop, the "Heads tap" is fully open for high flow in the forward direction. As the level difference decreases the flow rate decreases and the "tap" is slowly being closed. Note however that even when the level difference falls to zero, the "tap" isn't fully closed because flow is still possible, provided there is something other than gravity to drive it. This is where the water momentum remaining at the equal levels time comes into play. It keeps the flow running in the same direction as before for some further period of time.

Near the end of an outgoing (ebbing) tidal flow and when the ocean tide level is rising rapidly, the Heads acts as a closing "downstream tap" at the southwest end of the choke zone channel. The water hammer effect is then positive and the level of the decelerating water across the choke zone rises higher above the main body level the closer to the Heads we are.

Near the end of an incoming (flooding) tidal flow and when the ocean tide is falling rapidly, the Heads acts as a closing "upstream tap" at the start of the "choke zone channel". The water hammer effect in this case is negative, and the water levels just inside the Heads drop well below those at the northeast end of the choke zone.

In effect the momentum of the northward moving water inside the Bay is "sucking" water northwards and away from the Heads area. This causes the level just inside the Heads to fall. This allows the level here to "follow the ocean level down" despite the fact that the flow is still inwards, and that the main body of the Bay is at a higher and still increasing level. This can continue only for a certain amount of time before the outward force produced by the increasing reverse level difference across the coke zone depletes all the forward water momentum so halting the inflow and producing a brief slack water period.

The reverse ocean - Bay level differences attained by the time the water is halted may reach up to 45cm for tide cycles that feature a large tidal range. The reverse height difference at slack water for the weakest tidal streams with a small tidal range is generally in the 10cm to 15cm range. Even the higher "slack water surface slopes" aren't visible to the naked eye because the height difference is spread over roughly 15km.

At the Heads themselves the water surface slope will be somewhat higher than the average, and may reach a little over 5cm of height per kilometre of distance before a strong stream is halted. At such slack water times the reverse level difference between Rip Bank and Queenscliff is roughly 20cm, with the remaining 25cm being distributed across the remaining 10km of the choke zone.


***** How Long Does The Uphill Flow At The Heads Last? *****

In the domestic plumbing situation the water hammer effects last only a fraction of a second. The pressure surges are very high and the water comes to a halt very quickly. At Port Phillip Heads the "surge" occurs over a much longer time scale. The last of a strong flood stream will run up-slope until the reverse level difference is sufficient to halt the flow. Typically that difference might reach 40cm by the time slack water occurs after a strong tidal flow. How long does that take?

Approaching slack water near the end of a strong flood tide, the fall rate of the ocean level might be typically 55cm/hr. In the main body of the Bay the level is still slowly rising (due to weak inflow) at say 8cm/hr. Combined, this means the reverse level difference is growing at around 63cm/hr, or a little over 1cm per minute. So to build up a 40cm level difference to stop the inflow, around 40 minutes is required. During this time the flow continues inwards despite the fact the "inside" level is growing well above the "outside" level.

The uphill flow phase at the end of an outgoing ebb tide lasts somewhat longer at around 60 minutes. This is because the Bay exhibits a significant asymmetry in that it drains more slowly than it fills. There seems to be two factors that produce this. Firstly the Bass Strait tides just outside the Heads also have an asymmetry in that on average they rise slightly more slowly than they fall. The slower ocean rise means the reverse level difference necessary to stop an outgoing stream grows more slowly with time and therefore takes longer to reach a sufficient value.

A second factor in making the Bay drain more slowly than it fills is connected to the triangular shape of the choke zone. On an outgoing tide this "focussing shape" speeds the water up on its way south. This extracts a little more effort from the driving force than for an incoming tide where the water slows down on its way across the choke zone.

The filling/draining asymmetry shows up in some of the tidal statistics below which were extracted from all the 2020 tide cycle predictions.

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* average FLOOD stream duration (using Packo slacks):-         6hrs 04mins
* average EBB stream duration (using Packo slacks):-           6hrs 21mins

* average delay of Williamstown HIGH tide after Pt Lonsdale:-  2hrs 58mins
* average delay of Williamstown LOW tide after Pt Lonsdale:-   3hrs 38mins

* average delay of "Packo slack" after Pt Lonsdale high tide:- 2hrs 44mins
* average delay of "Cardno/BoM slack" after Pt Lonsdale high:- 2hrs 48mins

* average delay of "Packo slack" after Pt Lonsdale low tide:-  3hrs 22mins
* average delay of "Cardno/BoM slack" after Pt Lonsdale low:-  3hrs 33mins

* average Packo Flood slack reversal rate (slack after flood): -0.41 knots per 10 mins
* average Packo Ebb slack reversal rate (slack after ebb):     +0.32 knots per 10 mins

* average "Packo Flood Slack" time before Williamstown hi tide:-    13mins
* average "Cardno/BoM Flood Slack" time before Williamstown high:-  09mins

* average "Packo Ebb Slack" time before Williamstown low tide:-     16mins
* average "Cardno/BoM Ebb Slack" time before Williamstown low:-     04mins

* Number of Packo Flood Slacks at/after Williamstown high tide:-     nil
* Number of Cardno/BoM Flood Slacks at/after Williamstown high:-     63   

* Number of Packo Ebb Slacks at/after Williamstown low tide:-        nil
* Number of Cardno/BoM Ebb Slacks at/after Williamstown low:-        126   

cheers,
packo



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