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Physical characteristics and mathematical modelling of the pulsed ultrasonic flowmeter

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Abstract

A pulsed ultrasonic Doppler flowmeter for detailed measurements of velocity profiles in man is described. The device projects a beam of ultrasound in bursts of 0·4 μs duration, at 5 MHz, into the flow; the back-scattered signals are processed to produce a signal corresponding to the mean velocity over a small region of the flowing stream. The size and shape of this ‘sample volume’ determines the flowmeter sensitivity and accuracy. The velocity profile obtained from this instrument can be shown to be a weighted average of the ultrasonic intensity and the flowfield velocity over the sample volume, and is mathematically described by a convolution integral. A method of probing the ultrasonic beam and describing its characteristics in mathematical terms was developed. Using this model with a system whose velocity profile is parabolic, the pulsed ultrasonic Doppler-flowmeter output could be predicted via the convolution integral. Theoretical flowmeter-output curves were generated from the mathematical model by a digital-computer simulation and verified through experimental profiles in steady laminar flow. The modelling technique is sufficiently general for flowmeter output to be predicted for any general flow-velocity profile in steady or pulsetile flow.

Sommaire

L'article décrit un débitmètre Doppler pulsé aux ultra-sons servant pour les mesures détaillées des profils de vitesse dans l'homme. Le dispositif projette dans le débit un faisceau d'ultra-sons par paquets de 0,4 us de durée (à 5 MHz): les signaux rétrodiffusés sont traités pour donner un signal correspondant à la vitesse moyenne sur une petite zone de l'écoulement. Les dimensions et la forme de ce ‘volume échantillon’ déterminent la sensibilité et la précision du débitm⪻tre. On peut démontrer que le profil de vitesse obtenu avec cet appareil est une moyenne pondérée de l'intensité des ultrasons et de la vitesse du champ de débit sur le volume échantillon, et ce profil est décrit mathématiquement comme intégrale de convolution. Une méthode a été mise au point pour analyser le faisceau d'ultra-sons et décrire ses caractéristiques en termes mathématiques. Si ce modèle est employé avec un système dont le profil de vitesse est parabolique, alors on peut prévoir le signal de sortie du débitmètre Doppler pulsé à ultra-sons au moyen de l'intégrale de convolution. Des courbes théoriques pour les signaux de sortie de débitmètre ont été produites à partir du modèle mathématique par simulation sur calculateur numérique, et elles ont été vérifiées au moyen de profils expérimentaux en écoulement laminaire stable. La technique des modèles est suffisamment générale pour permettre de prévoir les signaux de sortie, de débitmètre aveo n'importe quel profil de vitesse de débit général en écoulement stable ou pulsé.

Zusammenfassung

Ein impulsgesteuerter Ultraschall-Doppler-Strömungsmesser für ausführiche Messungen von Geschwindigkeitsprofilen in Menschen wird beschrieben. Das Gerät projiziert, einen Ultraschallstrahl in Impulszügen von 0,4 us Dauer bei 5 MHz in den Strom hinein die rückwarts gestreuten Signale werden au einem Signal verarbeitet, das der mittleren Geschwindigkeit in einem kleinen Bereich des fliessenden Stromes entspricht. Die Grösse und Form dieser ‘volumetrischen Probe’ sind für die Empfindlichkeit und Genauigkeit des Strömungsmessers bestimmend. Das von diesem Instrument gelieferte Geschwindigkeitsprofil kann als gewogenes Mittel der Ultraschallintensität und der Strömungsfeldgeschwindigkeit durch die volumetrische Probe angesehen werden und wird mathematisch als ein Faltungsintegral beschrieben. Ein Verfahren, den Ultraschallstrahl abzutasten und seine Kenndaten in mathematischen Ausdrücken zu beschreiben, wurde entwickelt. Unter Benutzung dieses Modells für ein System, das ein parabolisches Geschwindigkeitsprofil hat, kann das Ausgangssignal des impulsgesteuerten Ultraschall-Doppler-Strömungsmessers mittels des Faltungsintegrals vorausgesagt werden. Theoretische Strömungsmesser-Ausgangskurven wurden aus dem mathematischen Modell durch Simulieren in einem Digitalrechner erzeugt und durch experimentelle Laminarströmungsprofile im gleichmässigen Strom nachgeprüft. Die Modellierungstechnik ist allgemeingültig genug, um Voraussagen der Strömungsmesser-Ausgangsleistung für jegliche allegmeine Strömungsgeschwindigkeitsprofile in gleichmässigen oder pulsierenden Strömen zu ermöglichen.

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Abbreviations

A :

peak sample volume intensity

A max :

peak sample volume intensity at focal point

C :

steady-state intensity ofI″1 andI′1

c :

acoustic velocity

d :

distance from the transducer face to a point in the sound field

d :

an increment of sampled volume

Δf :

Doppler frequency shift,f s −f r

f r :

return signal frequency

f s :

reference signal frequency

G 1,G 2,G 3 :

sample volume geometric constants

γ1 :

sample volume rise constant

γ2 :

sample volume decay constant

I :

intensity

I 1 :

sample volume intensity forx≤x b including noise level and trigger level effects

I′1 :

sample volume intensity forx≤x b including noise level effect

I″1 :

sample volume intensity fort≤t b with-out noise or trigger level effects

I 2 :

sample volume intensity forx≥x b including noise level and trigger, level effects

I′2 :

sample volume intensity forx≥x b including noise level effect

I″2 :

sample volume intensity fort≥t b without noise or trigger level effects

I max :

intensity on the axis of the sample volume

I n :

noise level intensity

I 0 :

sample volume truncation constant

I ratio :

ratio of the integral of intensitities over the region sampled to the integral of the intensities over the entire sample volume

I s :

signal intensity characteristic of transducer

I′ s :

signal intensity across the diameter of a test vessel

I t :

frequency-meter trigger level intensity

K :

sample volume intensity coefficient

K 1,K 2 :

frequency-meter correction factors

K 3 :

frequency-meter correction factor

m 1 :

sample-volume rise constant (m 1=γ1/c)

m 2 :

sample-volumedecayconstant (m 2=γ2/c)

Q :

transducer figure of merit

Q :

volume flowrate

R :

range co-ordinate along sound beam axis

r :

sample volume radial co-ordinate,\(r = \sqrt {y^2 + z^2 } \)

r o :

sample volume radius

r omax :

maximum sample volume radius or sound beam radius

r :

max. sample volume radius at noise floor

STL :

signal-to-trigger level intensity ratio

T s :

range-gate time delay

t b :

electrical driving burst duration

θ:

angle between the sound source velocity vector and the emitted sound

θ t :

angle between the sound scatterer velocity vector and the sound beam

V :

velocity of a sound scatterer

V max :

max. velocity in a parabolic velocity profile

V s :

velocity of the sound source

V w :

weighted average velocity

V w(ideal) :

weighted average velocity not including frequency meter characteristics

X :

vessel co-ordinate across the diameter (X=0 at the near wall)

x :

sample volume length co-ordinate

x b :

equivalent driving burst length (x b −c.t b )

x 0,x 1,x 2 :

sample volume parameter

x 2n :

sample volume length at the noise floor

Y :

vessel co-ordinate

y :

sample volume co-ordinate

Z :

vessel co-ordinate along the length

z :

sample volume co-ordinate

References

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Authors and Affiliations

  1. Center for Bioengineering, University of Washington, Seattle, Washington, USA

    Jens E. Jorgensen, Daniel N. Campau & Donald W. Baker

Authors
  1. Jens E. Jorgensen
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  2. Daniel N. Campau
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  3. Donald W. Baker
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Jorgensen, J.E., Campau, D.N. & Baker, D.W. Physical characteristics and mathematical modelling of the pulsed ultrasonic flowmeter. Med. & biol. Engng. 11, 404–421 (1973). https://doi.org/10.1007/BF02483686

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