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Does the speed of the water flowing radially outwards from a water jet impacting a sink depend on how far the tap is opened?

At the moment, I understand that the maximum velocity of the water in the vertical jet is constant regardless of how far the tap is open (as it is accelerated by gravity, and disregarding air-resistance), but the volume flow rate is greater the further the tap is opened.

I'm thinking that a higher volume flow rate results in a greater depth around the bottom of the jet which reduces the effect of friction between the water and the sink as the contact surface of the water has reduced, which would decelerate the water at a lower rate than if it were shallower, but am not sure.

Could someone explain precisely how opening a tap affects the radial speed of the water around the bottom of the jet?

EDIT

I should have clarified that, for my question, the area is variable. What I had in mind was more like a variably sized circular hole beneath a constant-height water tank. I used the word "tap" for simplicity, but I realise now this is a bad substitute, and has caused more confusion than intended.

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  • $\begingroup$ Related: engineering.stackexchange.com/questions/5490/… $\endgroup$ – Chris Mueller Nov 17 '15 at 13:15
  • $\begingroup$ I'm not sure that the velocity of the jet is independent of tap position. Do you have any reference for this assumption? $\endgroup$ – Carlton Nov 18 '15 at 1:07
  • $\begingroup$ @Carlton As far as I know, opening a tap does not change the pressure of the water behind it, just the space through which it can flow - this would increase the flow rate without increasing speed (I am assuming the water does not build up inside the spout (would this make a difference?)). As for the vertical speed after it has left the tap, the water is being accelerated by gravity so they should be the same (ignoring air-resistance). That was my reasoning behind the assumption, but I am not entirely sure. $\endgroup$ – MadCommy Nov 18 '15 at 1:23
  • $\begingroup$ I'm picturing a garden hose in which the stream of water definitely shoots farther (indicating higher velocity) when turned on full vs. half way, etc. Though, off the top of my head I can't come up with a solid explanation as to why. I suspect water building up inside the spout might be the case. $\endgroup$ – Carlton Nov 18 '15 at 1:34
  • $\begingroup$ I think the velocity is indeed greater since the pressure is higher. Bernouli's law. $\endgroup$ – Vince Scalia Nov 18 '15 at 19:12
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The speed at which water leaves the tap is not constant, as it has a finite area, which remains roughly constant, so increased flow does indeed increase the velocity. ($\dot m= \rho\, VA$)

However, we can still ask the question of how the flow profile of water flowing outward from a jet is dependant on flow rate when jet velocity is held constant.

First let's calculate the reynolds number:

$$Re=\frac{\rho V D}{\mu} \approx \frac{1000\frac{kg}{m^3}\,2\frac{m}{s}1\,cm}{900 Pa\,\mu s}\approx 20000$$

So the viscous effects are going to be tiny during the transition, which means we can use bernoulli's equation. The pressure will be at atmospheric, before and after the transition, and the height will be the same, so the velocity will be the same.

In conclusion, the spray will just get thicker, but remain at the same velocity when the flow rate is increased. Of course, far from the jet the viscous effects will begin to matter more as the fan thickness decreases, and the reynolds number drops. Then indeed the thicker flow will remain at high speed for longer.

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    $\begingroup$ I'm not certain that the cross-sectional area of the stream is constant. At low flow rates, the water may not completely fill the spout, instead looking more like open-channel flow. $\endgroup$ – Carlton Nov 19 '15 at 0:53
  • $\begingroup$ Although you (understandably) misinterpreted my question (as corrected in the edit), this answered my question - the spray gets thicker, but does not speed up as a result of a higher flow rate (assuming jet velocity is constant). $\endgroup$ – MadCommy Nov 19 '15 at 1:39
  • $\begingroup$ @Carlton I agree for spouts that don't have an aerator, and even for spouts that do have an aerator, at low flow rates surface tension will dramatically affect the shape and velocity. So the relationship between flow rate, tap area, and velocity a short distance away from the tap becomes more complicated. But as that wasn't the heart of the question, I didn't go into that. $\endgroup$ – Rick Nov 19 '15 at 13:53

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