In the Navier-Stokes equation, the fundamental physical variable is velocity, which is a function of time and space. Experimentally, most of the conventional velocity measurement techniques such as hot-wire anemometry and LDA are point measurements that obtain a time-varying function at one spatial point. On the other hand, flow visualization techniques give spatial information at one instance of time. It is therefore difficult to obtain quantitative spatio-temporal information about the flow by these means.

The ultrasonic Doppler method (sometimes called the Ultrasonic Doppler Velocity profile method; UVP) enables us, as described below, to obtain a velocity profile along a line instantaneously and subsequently has gathered increasing attention of applied fluid mechanics and fluid engineering investigators. The method has the following advantages:

Since this method can investigate the flow field from a completely different point of view, its development could potentially bring new insights into the physics and engineering of flows.

In this paper, after a brief description of its principle, various details concerning the application of ultrasonic Doppler velocitmetry is presented. At the same time, various examples are introduced for showcasing the capability of this method.

__Ultrasound__ : Sound (ultrasound) is a wave which propagates a medium via mechanical vibration; there are two common modes, being longitudinal and shear waves. The present method uses a longitudinal wave. Naturally, this wave is absorbed and scattered in the medium. There is also reflection and refraction at any interface with a different medium, which changes its propagation mechanism.

The generation of an ultrasonound beam and its characteristics are contained in various textbooks on ultrasound. We only describe matters relevant to this measuring method. Here the sound wave is expressed in terms of pressure field. P_{0 }is the peak pressure, the wave number, *k*, is related to the wavelength *k*=2p /l . l the wave length(l =c/*f*), and *f *a frequency of the wave.

__Reflection and refraction at the interface__ : The interface is defined as the boundary of two media which have different acoustic impedance. The acoustic impedance is a material property and expressed as a product of density and sound speed in the medium (Z=

r c).

As illustrated in Fig.1, the following equation holds for incoming wave P

_{i}, reflected wave P

_{r}, and transmitted wave P

_{t}, when the ultrasonic beam impinges to an interface at an angle of incidence

q .

(1)

A reflection coefficient *R* is defined as the ratio of the pressure of incoming wave to that of the reflected wave

(2)

A transmission coefficient, *T, *is defined as the ratio of the incoming wave to the transmitted wave;

(3)

The refraction of the wave follows Fresnel's law;

(4)

When q _{2}=90 in Eq.(2), total reflection occurs and the angle of the incoming wave in this case is called the critical angle :

q _{1c}=sin-1(c_{1}/c_{2}). (5)

This means that the wave cannot impinge and be transmitted through the interface with any larger angle than this critical angle. When an ultrasound transducer is placed outside of a container wall, inside of which there is some flow, it is necessary to keep the transducer angle smaller than this critical angle. In the case where the ratio of the acoustic impedance of two materials is much larger than 1, the transmission coefficient is extremely low, so that care must be taken for a selection of wall materials.

__Dispersion of the beam :__ An ultrasonic beam has higher directionality at higher basic frequencies, although it has far less coherency compared with a laser beam. Thus beam dispersion is inevitable in practical applications.

The pressure distribution on the beam axis as given by following equation is often used to characterize the ultrasonic beam.

(6)

As illustrated in Fig.2, it oscillates near the transducer to show a number of maxima. It becomes smooth after some distance. The generation of the ultrasonic beam can be understood in terms of a superposition of such waves from a point source on all the position of the transducer's surface. This implies that the beam is not completely formed in this range where there are many strong oscillations. This region is called "near field". Beyond this region, pressure varies smoothly and this is called the "far field".

In the far field, the maximum pressure is always on the beam axis. If we define a divergence angle (divergence angle g _{o}) of the beam as the half width of the pressure distribution (-6dB), g _{o} is expressed using the wave length and the diameter of the transducer by the following equation;

. (7)

__Attenuation__ : A sound wave's energy is transformed during the propagation process (absorption), and it also loses its energy by a scattering off at the surface of impurities in the medium. Both of the above modes of energy loss are classified as attenuation. The following equation expresses the attenuation of the beam along the path of propagation;

p(z) / p_{0} = exp(-a z) (8)

The attenuation coefficient a depends on the material (its crystal structure in solid), basic frequency of ultrasound, temperature, pressure so that attention should be paid in the selection of a medium and materials of flow and structure.

We note that the attenuation increases in proportion to the square of the basic frequency of the ultrasound.

## 2.2 Sound speed and acoustic impedance of various materials

In Table 1, we have given the speed of sound and the acoustic impedance of representative liquids and container materials that are often used in fluid engineering systems and experiments. For the combinations of liquid and structure materials which has a ratio of the acoustic impedance is in the order of 2-3, little problem may arise.

Information theory provides various methods to derive the instantaneous frequency *f*(t ). Here, the one which is generally used for this method is briefly described. In this method, pulse emission and echo reception is repeated. It has the advantage that it does not requires very fast electronics and a well established basis since it has been used since the development of this technique in the medical field.

As illustrated in Fig.5, microparticles suspended in the fluid move while such repetition is made, so that the echo signal at a fixed time after the pulse emission has a slight phase difference. This phase difference is expressed as

f = *f*_{0}T_{d }(11)

The time derivative of this equation is

df /dt = *f*_{0}dT_{d}/dt (12)

Here T_{d} is given using the distance of the moving paricle and the speed of sound

T_{d }= 2z/c (13)

and

dT_{d}/dt = (2/c)dz/dt = 2v/c (14)

Substituting this into Eq.(12),

df /dt = 2v*f*_{0}/c = *f*_{D. }(15)

This is the Doppler equation given in the preceding section.

In the actual system, the pulse repetition frequency (*f*_{prf}) is the most important parameter. This frequency determines the maximum measurable depth.

P_{max }= c/2*f*_{prf }(16)

When one measures large distances, the time until an echo returns to the transducer becomes longer and the pulse repetition frequency has to be smaller.

The maximum measurable velocity is determined from the Nyquist theorem of the repetition frequency

V_{max }= c*f*_{max}/4*f*_{0 }(17)

From the above two equations, the composite measurement limitation by this method is

V_{max} P_{max }< c/8*f*_{0 }(18)

Table 2 lists the typical specification of the system developed. These parameters change with the basic frequency and it is in each case determined corresponding to the configuration and conditions to be measured.

It is not easy to discuss measuring accuracy in general terms, but the following points might be useful for discussion.

__Spatial resolution __: The spatial resolution depends largely on the pulse characteristics of the ultrasound and not on the parameters of electronics. The spatial resolution in the axial direction of the beam is determined by the pulse length. Usually, the ultrasonic pulse takes exactly four cycles of the basic frequency. When the pulse length is larger than this, it should be taken into account; a lower basic frequency results in a larger spatial resolution, The lateral resolution is mainly determined by the beam diameter. When the measuring length is large and the beam dispersion is not negligible, this has to be considered. The measuring volume then has a disk shape with a diameter as mentioned above for a circular beam.

__Time resolution and sampling speed__ : As described below, the velocity profile is obtained by repeating pulse emission and echo reception. Thus, the measuring time is that required for this repetition, namely it depends on the number of repetition. In the system developed so far, the profile is computed immediately after the data acquisition and then on a real time basis so that the sampling time is a sum of times of data acquisition, profile computation and display. The total time changes from system to system, but care should be taken to distinguish two kinds of times; measuring time and sampling time.

__Velocity resolution__ : As given in Eq.(18), this method has a limitation on the maximum detectable velocity. This maximum velocity V_{max} also determines the velocity resolution which is this number divided by the bit length of the digital system used. As a result, one need to lower the maximum velocity or increase the bit length of the system in order to realize a high velocity resolution.

__Accuracy of measured velocity__ : As this method uses frequency information only of the ultrasound, it is seldom that any other frequency change than the Doppler shift is introduced into the echo signal as noise. As long as the frequency characteristics of the electronics is kept precise, the measurement accuracy can be considered to be very high. Typically the UVP monitor displays a fluctuating velocity during measurement. The reason for such fluctuation is considered to be due only to fluctuation of the flow field itself. For instance, when measuring the laminar flow in a circular pipe, one often observes that the instantaneous velocity profile does not show a parabolic distribution. This is mainly because some kind of oscillation remains in the test apparatus and the flow is transient with respect to the time resolution of the measuring system. When one takes the average of the velocity profiles over some time period, one can obtain the expected parabolic distribution.

## 2.6 Notes for practice

Various notes are given here for a practical application of this method to fluid flow measurement.

__Vector components__ : Velocity in a fluid flow is vector. In LDA, the measured velocity component is generally the one normal to the laser beam. In this method, on the other hand, the velocity component along the beam is measured; the projection of the true vector onto the measuring line is detected. (By this reason, the measuring line is inclined to the line normal to the pipe in the example given.)

A similar problem may be faced in optical method such as LDA. One can find good references and practical information from this measurement technique.

__Reflection on the solid wall and free surface__ : By a measurement near the solid wall, acoustic pressure field may be influenced by a reflection of the beam on the wall. As illustrated in Fig.6, the pressure wave by the beam changes its sign by a reflection and it superposes with the incoming wave, resulting in a cancellation of pressure or some other anomaly. As a result, echo is not generated in this region or strongly deformed, which leads to an erroneous results. Moreover, when the wall is not stationary, it induces Doppler shift in the echo, which appears on the profile when the measurement line is aligned such that the reflected beam goes back directly to the transducer. It is therefore important to take sufficient care of the beam alignment when measurement is for the region near the wall. The area or the size where such influence may appear depends on the beam size and pulse width. A good suggestion might be given that several times of the spatial resolution may be a distance to be taken for the safe measurement.

Against the stationary solid wall, it can be well designed such that the reflected wave does not return to the transducer directly by adjusting the aligned angle of the beam to the wall. On the other hand, a free surface changes the inclination angle of the beam non-stationary so that it is necessary to study each profile one by one if it includes any trace of the reflection from the surface.

__Influence of temperature __: When the flowing medium has a temperature distribution inside, there are two problems described below. The first point is that the sound speed depends on temperature. The sound speed is the most important parameter in this method, such that it is used to compute spatial position and velocity. Consequently, spatial distribution of temperature means a considerable influence on those values and their accuracy. The second point is that the ultrasonic beam changes its path (bending) during the propagation due to a continuous changes of acoustic impedance. The reflected echo propagates on the exactly same path and gives velocity profile. The problem however is that one cannot identify the spatial position of the measuring line, which induces a large uncertainty in the accuracy of the spatial position.

A practical problem of the high temperature is a decrease of efficiency. It decreases logarithmically with increasing temperature and it behaves as if the medium has a much stronger attenuation. Furthermore, too high temperature may result in a permanent damage of the transducer itself. It is necessary to take a careful attention about the temperature of the place where the transducer is mounted.

## 3 Applications

Examples of applications are given for each advantages described in the introduction. Since more other examples will be given in this seminar in detail, explanation will be limited to minimum.

## 3.1 Spatio-temporal measurement

Fig.7 shows a measured example of the oscillating pipe flow. The diameter of the pipe is 25mm and a piston is located at one end to generate a monotonic oscillation. Measurement was made with synchronizing its sampling speed to the oscillation frequency and the axial component of the velocity is obtained as a function of time. (Fig.7b) and the phase average was computed over 11 cycles of oscillation. As displayed in Fig.7a, the flow is laminar but shows a parabolic shape only for a very short time period during the oscillation. It is observed that the peripheral region near the wall has the higher velocity than in the central region. Moreover, at one instance (corresponding to the reciprocal position of the piston), the flow direction is opposite for central region and its outer region.

Fig.8 is a result of the measurement of the flow in a cylindrical container simulating the Czokralski crystal growth method. The measured velocity component is the radial component just beneath the seed crystal which is rotating. It shows a typical axisymmetric flow just after the start of rotation of seed crystal. It is observed that the flow becomes oscillating due to a hydrodynamic instability.

By this method, a time resolution cannot be extremely high for the present system, it is not possible to follow a very high speed transient phenomena, but the above example shows that it is quite powerful to the oscillating flow. Generally speaking, as most of the transition due to hydrodynamic instability appear as a change in the spatial distribution but not the temporal distribution, this method is believed to be very useful for investigating flow transition departing from laminar regime. Moreover, if the flow characteristics are spatio-temporal, this method is the best to be used for its investigation.

## 3.2 Flow Mapping

As this method is a line measurement, being different form conventional techniques which are pointwise measurement, flow mapping can be very efficiently performed to study a overall behaviour of the flow field. In two-dimensional case, N beams (measuring lines) can generate N^{2} crossing points, on which points two velocity components can be obtained to form a velocity vector. The N beams can be realized by moving a single transducer or N transducers can be multiplexed electronically.

Fig.9 is an example measure for the coaxial jet (time-averaged). Transducers were aligned with its top surface attached to the free surface with inclination angle of 30°. A characteristic feature of the flow such as a strong flow near the exit nozzle of entrainment is well observed.

Fig.10 is an example in the configuration of two adjacent parallel channels which are connected by a hemicylindrical wall at one end. The measurement was made in the hemicylindrical part. Transducers were set on the outer wall of the cylinder with inclincaiton angle of 15° at various directions. There are 54 measuring lines in total and the number of crossing points in the measured region is 756. Velocity vectors are plotted on these points. Turbulent motion near the separating edge and flow recirculation after the turn is well measured.

## 3.3 Opaque liquids

Applicability of this method to opaque liquids, thanks to a transmission properties of the ultrasound in such liquids, is an extremely important advantages of this method in the field where no flow measurement was possible. These liquids include liquid metals, food staffs, pharmaceutical materials, magnetic fluids, which are not only opaque but also mostly non-Newtonian so that pressure measurement was also not practical for investigating a flow. By the present method, not only velocity distribution but also other physical quantities such as viscosity can be estimated from the measured flow distributions.

Fig.11 is an application to liquid Na in a pipe. Because of the high temperature environment (250°C) and there was not much large freedom in locating the transducer, echo included quite many peaks of reflection from walls and other structures. Consequently one had to carefully treat the dataset, to eliminate noises on one profile by one profile, and the average profile was computed. It shows a good agreement with 1/7 power law in the central region. In the near wall region too, the result agrees quite well with the result of DNS computation.

Fig.12 is an example for the magnetic fluid. A magnetic fluid is a liquid with suspension of micron scale magnetic particles, and it senses the external magnetic field to change its apparent viscosity. In the present example, the liquid is filled in a small square container and shaken on the vibration stand (Sloshing). The transducer is set outside the container wall and the velocity measured is relative to the wall. The distribution naturally changes its shape and the maximum velocity at the measured positions are plotted in Fig.13. When magnetic field is applied horizontally, the flow is suppressed due to an increase of apparent viscosity by increasing the strength of the magnetic field.

As given above, it was not possible to obtain velocity information in these liquids before this method. Therefore, this method is expected to play an important role in this field of investigation.

## 4. Summary

In this presentation, the ultrasonic Doppler method for fluid flow measurement is described. Main emphasis was given to the practical application of the method. This method is a line measurement, being different from any conventional techniques, and has several advantages in scientific and engineering investigations on the fluid flow over any conventional methods. Especially, applicability to opaque liquids and very efficient flow mapping will be important factors in the engineering applications. It has however some limitation such as spatial and time resolution, which are due to physical conditions. In this sense, this method might be complementary to any laser technique. The form of obtained data set is completely different from other techniques. This requires the users to be careful in understanding the results.

## References

J. Krautkrämer & H. Krautkrämer, "Ultrasonic Testing of materials", 4th, Edition. Springer-Verlag, 1990

L. C. Lynnworth, "Ultrasonic measurements for process control", 1989, Academic Press, Inc.

P. Atkinson & J. P. Woodcook, "Doppler ultrasound and its use in clinical measurement", Academic Press, Inc. 1982

Y.Takeda, Velocity profile measurement by ultrasonic Doppler method, Exp. Thermal & Fluid Science, **10 **(1995) 444-453

Y.Takeda, Instantaneous velocity profile measurement by ultrasonic Doppler method, (Invited paper), JSME International Journal, Fluids and Thermal Engineering, Vol.38, No.1, (1995) p8 - 16

Y.Takeda, Instantaneous Velocity Profile Measurement by Ultrasonic Doppler Method, invited paper to the JSME International Journal, **B**, 38 (1995) 8-16

Y.Takeda, Ed., International Symposium of Ultrasonic Doppler method on Fluid mechanics and Fluid Engineering, 9.-11. Sept. 96, PSI-Proceedings, Selected papers to be published in Exp. Fluids.

## Figure captions

- Reflection of ultrasonic beam at the interface.
- Distribution of the acoustic pressure on the beam axis.
- Dispersion of the ultrasonic beam from a disk shaped transducer.
- Working principle (a) arrangement of the flow and ultrasonic beam, (b) received echo, (c) obtained velocity distribution
- Detection of instantaneous frequency
- Reflection of US beam at the interface. The sign of the pressure field is changed on the solid wall resulting in a cancellation of the pressure in the overlapped area.
- Oscillating flow in a pipe. (a) Phase averaged profiles at 90 phases (over 11 cycles, displayed only 45 profiles). (b) Color density plot of the spatio-temporal velocity profile.
- A flow in a Czokralski container. (top) a transient flow after the start of the crystal rotation (bottom) stationary oscillating flow.
- Vector map of coaxial jet.
- Vector map of a return flow in the hemicylindrical region.
- A pipe flow of Na. Re = 70000.
- Sloshing of magnetic liquids. A plot of maximum velocity at each position. A flow is suppressed by the external magnetic field.