Estimating whether the flow through a valve or nozzle cavitates

Question

My understanding is that cavitation occurs in the flow of a liquid when the static pressure drops below the vapor pressure, even intermittently. So even if the time-averaged static pressure (what you might measure) is above the vapor pressure, the pressure fluctuations from turbulence or other unsteadiness could be large enough to cause cavitation locally. So comparing the time-averaged static pressure against the vapor pressure isn't enough; you need to add some extra cushion to account for the pressure fluctuations. (This is my interpretation, not having read too deeply into this.)

So, in various books, websites, and journal articles I have seen two different types of dimensionless numbers for estimating whether the flow through a valve or nozzle cavitates. They are generally called the cavitation index or cavitation number. They take one of two forms:

$$\sigma = \frac{p_\text{in} - p_\text{vapor}}{p_\text{in} - p_\text{out}}$$

or

$$\sigma = \frac{p_\text{in} - p_\text{vapor}}{\tfrac{1}{2} \rho V^2}$$

where $p_\text{in}$ is the inlet pressure, $p_\text{out}$ is the outlet pressure, $p_\text{vapor}$ is the vapor pressure, $\rho$ is the liquid density, and $V$ is some characteristic velocity of the flow (say, in the nozzle case, the velocity at the outlet). Some forms of this number are inversions of the numbers above, but these aren't that different.

What is the difference between these parameters? Based on energy conservation you can relate the pressure drop to the flow rate, but typically there is an empirical coefficient added in to account for non-idealities. Is there something else I am missing?

Is one form preferred over the other? Best I can tell whether to use one or the other depends on what sort of data you have (so, for flow over a turbine blade, the velocity form is preferred), but I've seen both even for nozzles.

Where can I get accurate data to predict cavitation based on these numbers? I've tried using some data on atomizer nozzles from various journal articles but generally they use different forms of the cavitation number. Some of the data suggests the flow through the nozzle will cavitate at the pressures I want, but other data for similar nozzles suggests it won't. I'm not sure what the source of the inconsistency is. My understanding could be faulty, the cavitation number model could be too simplistic, the data could be inaccurate, etc.

Answer 1

The difference between the two equations

The cavitation number is the ratio of the static pressure difference to the dynamic pressure difference. So, if you want to use the first equation, you would need to take the pressure using a Pitot tube to measure the total pressure, whereas if you want to use the second equation you will need to measure the freestream velocity, but I would recommend measuring it upstream rather than downstream because of possible effects of acceleration and boundary layer growth. Also, your $V$ should be $V_{in}$ such that it corresponds to the same location where $p_{in}$ is measured, because this equation is derived from Bernoulli's equation which says the energy is conserved along a streamline.

Is one form preferred over the other?

In all my experience working in cavitation research for many years, we have almost always used the latter equation you mentioned (although I have mainly been working in hydrofoils and propulsion systems). The reason is that we could get more accurate non-intrusive velocity measures using Laser Doppler velocimetry (LDV) than by using an intrusive method.

Where can I get accurate data to predict cavitation based on these numbers?

It is difficult to use experimental data to predict the cavitation number because of the differences in things like turbulence intensity and air nuclei content, which are difficult to match in reality with controlled laboratory methods. Traditionally, in my circles, this is done by running some CFD analysis codes on your design. There are two different approaches here: (1) compute the average mean flow using a RANS or LES technique, and (2) using a bubble dynamics code which will model the air nuclei, but requires a flowfield (either from experimental measures or from the from the CFD model). If you use a typical RANS CFD model to compute the flow-field, it should give you the the pressure coefficient which has a very similar definition to the cavitation number:

$$C_P = \frac{P-P_\infty}{\frac{1}{2}V^2}$$

If you are doing some CFD calculation on your nozzle, you should find the location of minimum pressure, and that is the place where cavitation should occur. You can infer the cavitation number from this pressure coefficient as:

$$\sigma = -C_P^{min}$$

where $C_P^{min}$ is the minimum pressure in your nozzle. I explain this in more detail in this paper. However, this will only give you an idea of the time averaged cavitation inception number. Most people don't go to such detail in trying to get such an accurate prediction of cavitation inception, unless it is absolutely critical.

If you want to get a more accurate number, you need to consider that cavitation inception requires three things to happen at the same time: (1) a local area of pressure which is below the vapor pressure of water, (2) an air nuclei which enters into that low pressure region, and (3) the air nuclei must be in the low pressure for a significant enough time that it basically rapidly grows, becomes unstable and hence collapses. The way people have been able to more accurately estimate this is by using a using a Lagrangian method that simulates sending air nuclei through an Eulerian CFD dataset. Some of the real experts in this field are the people at Dynaflow-inc.com. I might suggest taking a look at this paper:

Chahine, G.L. "Nuclei Effects on Cavitation Inception and Noise", 25th Symposium on Naval Hydrodynamics, St. John's, NL, Canada, Aug. 8-13, 2004. PDF here

However, if you don't want to go to all that trouble, I would recommend that you compute an estimate of the pressure fluctuations $p'$ based on the ambient turbulence intensity of your flow, and then subtract this value off of your mean pressure to get a better estimate of cavitation number. You should be able to get this value out of the turbulence model if your using a RANS technique. If you are looking at possible CFD techniques to use, unless you have a lot of money to spend, I might suggest looking into using OpenFOAM.