Moving media - anisotropic speed of sound implementation

[Ambr0K]

Hi everyone.

We're considering an experiment of measuring the temperature (and hopefully the velocity) of water right after it passes through the core of a research nuclear reactor by using a circular array of transducers at the reactor tank edges.

We're considering using k-wave, however we'd really be interested in the doppler effect due to the moving fluid. In the simulations, we're currently changing the medium parameters (speed_of_sound, alpha, density, etc) to change the temperature. However, there appears to be no option, to asses the doppler effect in a moving fluid. Could this be done by introducing the speed of sound as a vector or a tensor?

At a first glance, it appears there is an option to introduce perturbations to "c_ref" in their respective directions in "kspaceFirstOrder.py", but this would probably use this new value of speed of sound in both "x+" and "x-" direction.

We'd appreciate any suggestions on how to tackle this problem.

Best!

Klemen

[waltsims]

Hi Klemen,

This sounds like an exciting problem. Let me see if I understand you set up correctly. Are you trying to estimate the flow velocity with a perpendicular tomographic through transmission measurement of the outlet or with pulse-echo imaging like in medical color Doppler? To the best of my knowledge particles in the water would be the only way to measure the "Doppler effect" (which in ultrasound imaging is essentially the optical flow between a series of measurements of diffuse scattering) since the water itself is anechoic. It is possible to simulate heterogenous velocity distributions and the velocity variation due to temperature in water is well described by this function between 0 and 95 degrees. I may be missing something and would be happy to help if I am.

Happy to discuss more if I can be of more help.

Best, Walter

[Ambr0K]

Dear Walter.

Thank you for the insight. We're still open to methods. Essentially, we have a vertical tank with a 2 m in diameter, and the water flow goes through the core with rough diameter of 80 cm due to natural convection i.e. fission. We're thinking on putting the transducers at the tank's edge. The arrangement of transducers is still open and is subject to simulation results .

Regarding the temperature: thank you for the insight.

Regarding the velocity measurement, I suppose the tomographic-trough transmission measurements would be the way to go. We do have some particles in water (there are some extremophile bacteria growing on the tank), which we're also considering in using with PIV and other techniques (we're testing the Background Oriented Schlieren prototype in a couple of weeks).

Best!

Klemen

[waltsims]

Hi Klemen,

Given the bacteria in the tank, a vertical array would allow you to measure the pulse-echo color Doppler signal similar to PIV with lower resolution.

Measuring the through transmission phase shift due to temperature changes at the outlet might be possible with a ring array.

I have never heard of the setup of measuring the velocity through a ring array though. Is that what you were considering?

You can set time varying anisotropic particle velocities in the medium with the source term in k-wave in matlab. I believe this should be able to simulate convection. I have not yet tested this in python but would be happy to support you if something breaks.

[Ambr0K]

Hi Walter.

The preferred way of measuring the velocity field with an ultrasound array would indeed be with a ring array (or multiple arrays at different elevations) , since counting on bacteria to flow within the pool is not a given (hence the PIV setup is also on the back burner in favour of Schlieren), since we clean the tank on regular basis to reduce the burden on resin ionic-exchangers.

In a multiple-level ring grid, wouldn't there be a delay towards the lower-level ring transducers and an increase intensity in the upper ring, if the flow was upwards? On the other hand, the water is essentially circulating within the tank, so I'm not sure this approach is feasible at all.

[waltsims]

Hi Klemen,

Now I understand. You're discussing a concept similar to an ultrasonic flow sensor. Given what I have seen, staggered arrays might be able to pick up on the delay in the compressional wave as you describe. It looks like this topic was previously discussed here. I am personally not an expert on this topic.

Here is a snippet from Brad Treeby's response:

This is an interesting problem. There are a few different approaches you could take. First, you could extend the code to take into account the non-zero background particle velocity. The equations you'd need to solve are outlined by Coulouvrat in this paper. Note the additional terms in the mass and momentum conservation equations compared to those solved in k-Wave.

Second, you could solve an anisotropic wave equation such as the one described here. If you're not comfortable with coding this up, one option would be to find a general anisotropic elastic wave solver, and use it to just model the compressional wave.

Finally, as Andrew mentioned, in some situations, you may also be able to model an equivalent situation by moving the source rather than the medium as described in The Doppler Effect Example.

You might get the your desired effect with an anisotropic additive velocity source which you interrogate with transmissions from a ring array, but I haven't explored this space before, and therefore cannot speak to the accuracy of doing so given the additional terms the Brad mentions. I hope this helps!

Please let me know what you come up with.

Best, Walter

[Ambr0K]

Dear Walter.

Thank you for the help.

While the easiest way appears to be moving the staggered ring transducer array, I assume this would only work if the fluid was moving one way, which is not the case with us, where we have a sort of circular flow.

We'll be looking towards the implementation with additional terms you suggest. I'm sure we're going to need some help with it.