Background of Ultrasonic Doppler Velocimetry

Doppler ultrasonic technique

Doppler ultrasound technique, was originally applied in the medical field and dates back more then 30 years. The use of pulsed emissions has extended this technique to other fields and has open the way to new measuring techniques in fluid dynamics. The term "Doppler ultrasound velocimetry" implies that the velocity is measured by finding the Doppler frequency in the received signal, as it is the case in Laser Doppler velocimetry. In fact, in ultrasonic pulsed Doppler velocimetry, this is never the case. Velocities are derived from shifts in positions between pulses, and the Doppler effect plays a minor role. Unfortunately, many publications, even recent ones, fails to make the distinction, resulting in erroneous system description and fallacious interpretation of the influence from various physical effects.

Functioning principles of pulsed Doppler ultrasound

In pulsed Doppler ultrasound, instead of emitting continuous ultrasonic waves, an emitter sends periodically a short ultrasonic burst and a receiver collects continuously echoes issues from targets that may be present in the path of the ultrasonic beam. By sampling the incoming echoes at the same time relative to the emission of the bursts, the shift of positions of scatters are measured. Let assume a situation, as illustrated in the figure below, where only one particle is present along the ultrasonic beam.

From the knowledge of the time delay Td between an emitted burst and the echo issue from the particle, the depth p of this particle can computed by:

where c is the sound velocity of the ultrasonic wave in the liquid. If the particle is moving at an angle q regarding the axis of the ultrasonic beam, its velocity can be measured by computing the variation of its depth between two emissions separated in time by Tprf:

The time difference (T2-T1) is always very short, most of the time lower than a microsecond. It is advantageous to replace this time measurement by a measurement of the phase shift of the received echo.

where fe is the emitting frequency. With this information the velocity of the target is expressed by:

This last equation gives the same result as the Doppler equation. But one should always be aware that the phenomena involved are not the same. Assume that the particles are randomly distributed inside the ultrasonic beam. The echoes issue from each particle are then combined together in a random fashion, giving a random echo signal. Hopefully, a high degree of correlation exists between different emissions. This high correlation degree is put in advance in all digital processing techniques used in Signal Processing's Ultrasonic Doppler velocimeter to extract information, such as the velocity.

Advantages and limitations

The main advantage of pulsed Doppler ultrasound is its capability to offer spatial information associated to velocity values. Unfortunately, as the information is available only periodically, this technique suffers from the Nyquist theorem. This means that a maximum velocity exists for each pulse repetition frequency (Prf):

In addition to the velocity limitation, there is a limitation in depth. The ultrasonic burst travels in the liquid at a velocity which depends on the physical properties of the liquid. The pulse repetition frequency gives the maximum time allowed to the burst to travel to the particle and back to the transducer. This gives a maximum depth of:

From the above two equations, we can see that increasing the time between pulses (TPRF) will increase the maximum measurable depth, but will also reduce the maximum velocity which can be measured. The maximum velocity and maximum depth are thus related according to the following equation:

Ultrasound scattering

The ultrasonic waves generated by the transducer are more or less confined in a narrow cone. As they travel in this cone they may be reflected or scattered when they touch a particle having a different acoustic impedance. The acoustic impedance is defined by:

where r is the density and c the sound velocity.

If the size of the particle is bigger than the wave length, the ultrasonic waves are reflected and refracted by the particle. In such a case the direction of propagation and the intensity of the ultrasonic waves are affected. But if the size of the particle is much smaller than the wave length an other phenomena appears, which is named scattering. In such a case, a very small amount of the ultrasonic energy is reflected in all direction. The intensity and the direction of propagation of the incoming waves are practically not affected by the scattering phenomena. Ultrasonic Doppler velocimetry needs therefore particles smaller than the wave length.

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