**2. Acoustic wave propagation in high-pressure system with integrated acoustic generator**

The efficient transfer of the high-frequency pulsation energy in the high-pressure system to longer distances represents one of the basic assumptions for generation of highly effective pulsating water jets with required properties. To achieve that goal, the amplification of pressure pulsations propagating through the high-pressure system is necessary. The amplification can be accomplished by properly shaped liquid waveguide that is used for the pulsations transfer to the nozzle. In addition, maximum effects will be obtained if the entire high-pressure system from the acoustic generator to the nozzle is tuned in the resonance. To be able to study theoretically process of generating and propagation of pressure pulsations in the high-pressure system, both analytical and numerical models of the system with integrated acoustic generator were developed.

### **2.1 Analytical solution**

326 Acoustic Waves – From Microdevices to Helioseismology

rotating) using commercially available cutting heads and jetting tools. Laboratory tests of the device based on the above mentioned method of the pulsating liquid jet generation proved that the performance of pulsating water jets in cutting of various materials is at least two times higher compared to that obtained using continuous ones under the same working

> Acoustic chamber

Fig. 3. Schematic drawing of the high-pressure system with integrated acoustic generator of

waveguide

Pressure liquid

supply Pulsating jet

Nozzle

Acoustic actuator Liquid

However, further improvement of the apparatus for acoustic generation of pulsating liquid jet requires thorough study oriented at determination of fundamentals of the process of excitation and propagation of acoustic waves (and/or high-frequency pressure pulsations) in liquid via high-pressure system and their influence on forming and properties of

Problems related to the generation and propagation of pressure pulsations with frequency in the order of tens of kHz in liquid under pressure of tens of MPa and subsequent discharge of the liquid influenced by the pulsations through the orifice in the air (producing pulsating liquid jet with axial velocity in the order of hundreds meters per second) were not investigated in detail so far. Only partial information on this topic can be found in publications dealing with processes of a fuel injection for combustion in diesel engines (see e.g. Pianthong et al., 2003 or Tsai et al., 1999) and/or underwater acoustics (Wong & Zhu,

Therefore, the research on pulsating water jets was focused recently on the study of fundamentals of the process of excitation and propagation of acoustic waves (highfrequency pressure pulsations) in liquid via high-pressure system and their influence on forming and properties of pulsating liquid jet as well as on the visualization of the pulsating jets and testing of their effects on various materials. Results obtained in above mentioned

The efficient transfer of the high-frequency pulsation energy in the high-pressure system to longer distances represents one of the basic assumptions for generation of highly effective pulsating water jets with required properties. To achieve that goal, the amplification of pressure pulsations propagating through the high-pressure system is necessary. The amplification can be accomplished by properly shaped liquid waveguide that is used for the pulsations transfer to the nozzle. In addition, maximum effects will be obtained if the entire high-pressure system from the acoustic generator to the nozzle is tuned in the resonance. To

**2. Acoustic wave propagation in high-pressure system with integrated** 

conditions.

pressure pulsations

pulsating liquid jet.

**acoustic generator** 

areas so far are summarized in following sections.

1995).

The analytical solution of both pressure and flow oscillation waveforms in the conffusershaped tube with circular cross-section is based on linearized Navier-Stokes equations and wave equation for propagation of pressure wave. The wave equation incorporates both the standard kinematical viscosity and the kinematical second viscosity that is related to the liquid compressibility. Therefore, the irreversible stress tensor *Πij*, on the basis of which the wave equation is derived, can be written as follows:

$$
\Pi\_{lj} = 2\eta c\_{lj} + \delta\_{lj} \int\_0^t \Theta(t-\tau)c\_{kk}(\tau)d\tau \tag{5}
$$

where the function *Θ* (dynamic second viscosity) is related to the voluminous memory, and *cij* represents the tensor of deformation velocity.

In the frequency domain (ω), equation (5) can be written in simplified form verified experimentally:

$$
\Pi\_{lj\omega} = 2\eta c\_{lj\omega} + \delta\_{lj} \frac{k}{\omega} c\_{kk\omega} \tag{6}
$$

whereby *δij* represents Kronecker delta, and *η* dynamic viscosity. It is obvious from (6) that the dynamic second viscosity is frequency dependent. The kinematical second viscosity is then defined using following formula:

$$
\xi = \frac{k}{\rho \omega} \tag{7}
$$

where � represents liquid density.

### **2.1.1 Wave equation**

If one considers linearized Navier-Stokes equations, the wave equation for pressure function can be written using the Laplace operator Δ in the following form:

$$
\frac{
\partial^2 p
}{
\partial t^2
} - 2\gamma \frac{
\partial
}{
\partial t
} \langle \Delta p \rangle - \int\_0^t \rho^{-1} \Theta(t - \tau) \frac{
\partial
}{
\partial \tau
} \langle \Delta p \rangle \, d\tau - \nu^2 \Delta p = 0
\tag{8}
$$

where *γ* is kinematical viscosity, *p* pressure, *t* time and *v* speed of sound in water, respectively.

If Laplace transformation for zero initial conditions is applied in (8), following equation can be obtained:

$$[\mathbf{s}^2 \sigma - \mathbf{s}[2\chi + \xi(\mathbf{s})] \Delta \sigma - \upsilon^2 \Delta \sigma = 0 \tag{9}$$

where *s* represents parameter of the Laplace transformation according to time (*ξ(s)* is the Laplace function of the second kinematical viscosity), and, at the same time, following is valid:

$$L\{p(t)\} = \sigma(\mathbf{s})\tag{10}$$

$$\mathbf{c}^2 = -\mathbf{s}^2[\nu^2 + (2\chi + \xi)]^{-1} \tag{11}$$

$$
\kappa^2 \sigma + \Delta \sigma = 0 \tag{12}
$$

$$\frac{\partial^2 \sigma}{\partial r^2} + \frac{2}{r} \frac{\partial \sigma}{\partial r} + \frac{1}{r^2} \frac{\partial^2 \sigma}{\partial v^2} + \frac{1}{r^2} \text{cotg } \frac{\partial \sigma}{\partial v} + \frac{1}{r^2} \frac{1}{\sin^2 v} \frac{\partial^2 \sigma}{\partial \phi^2} + \kappa^2 \sigma = 0 \tag{13}$$

$$
\sigma = Z(r)W(\cos \upsilon)\Phi(\varphi) \tag{14}
$$

$$
\Phi\_p = A\_n \cos n\varphi + B\_n \sin n\varphi \tag{15}
$$

$$W\_p(\text{cos}\upsilon) = M\_{nm}P\_m^n(\text{cos}\upsilon) + N\_{nm}Q\_m^n(\text{cos}\upsilon) \tag{16}$$

$$\mathbf{Z}\_p = \frac{1}{\sqrt{r}} \left[ F\_m f\_D(\kappa r) + G\_m Y\_D(\kappa r) \right] \tag{17}$$

$$D = \frac{\sqrt{1 + 4m(m+1)}}{2} \tag{18}$$

$$\widetilde{c}\_{r} = \frac{1}{2\upsilon\_{0}} \int\_{0}^{\upsilon\_{0}} c\_{r}(r, \varphi, \upsilon) d\upsilon \tag{19}$$

$$\frac{\partial \widetilde{\mathcal{E}\_r}}{\partial t} = -\frac{2\upsilon\_0}{\rho} \frac{\partial p}{\partial r} - \frac{2\upsilon\_0 \xi}{\rho^2 \nu^2} \frac{\partial^2 p}{\partial r \partial t} \tag{20}$$

$$
\delta s w\_r = \alpha \frac{\partial \sigma}{\partial r}; L\{\tilde{c}\_r(t)\} = w\_r(s) \tag{21}
$$

$$a = -\frac{2v\_0}{\rho} \left( 1 + \frac{s\xi}{\nu^2} \right) \tag{22}$$

$$
\sigma = \frac{1}{\sqrt{r}} [Ff\_{0.5}(\kappa r) + GY\_{0.5}(\kappa r)] \tag{23}
$$

$$m = 0; \; D = \frac{1}{2} \tag{24}$$

$$
\omega\_r = \frac{a}{s\sqrt{r}} \left[ F \frac{\partial J\_{0.5}(\kappa r)}{\partial \chi} + G \frac{\partial Y\_{0.5}}{\partial r} \right] \tag{25}
$$

$$\mathbf{u}^T = [\boldsymbol{w}\_r(\mathbf{r}\_0, \mathbf{s}), \sigma(\mathbf{r}\_0, \mathbf{s})] \tag{26}$$

$$\mathbf{u}^T = [\boldsymbol{w}\_r(\mathbf{r}, \mathbf{s}), \sigma(\mathbf{r}, \mathbf{s})] \tag{27}$$

$$\mathbf{u}(r,\mathbf{s}) = \mathbf{P}\mathbf{u}(r\_0,\mathbf{s})\tag{28}$$

$$\delta = \frac{a}{r} \left[ \frac{\partial f\_{0.5}}{\partial r} \left( \kappa r\_0 \right) Y\_0(\kappa r\_0) - \frac{\partial Y\_{0.5}}{\partial r} (\kappa r\_0) f\_{0.5}(\kappa r\_0) \right] \tag{29}$$

$$\mathbf{P} = \frac{1}{\delta \sqrt{r\_0 r}} \left\| a \frac{\partial f\_{0.5}}{\partial r} \{ \kappa r \} \quad a \frac{\partial Y\_{0.5}}{\partial r} \{ \kappa r \} \right\| \left\| \begin{array}{l} Y\_0 \{ \kappa r\_0 \} & -a \frac{\partial Y\_{0.5} \{ \kappa r\_0 \}}{\partial r} \\\\ -f\_{0.5} \{ \kappa r\_0 \} & a \frac{\partial f\_{0.5}}{\partial r} \{ \kappa r\_0 \} \end{array} \right\| \tag{30}$$

$$\widetilde{\mathfrak{c}}\_{r} = \mathfrak{c}\_{r0} e^{i\omega t}; \mathfrak{w}\_{r0} = \frac{1}{\mathfrak{s} - i\omega} \tag{31}$$

$$p = \lambda c\_r; \ \sigma = \lambda \varkappa\_r \tag{32}$$

Use of Acoustic Waves for Pulsating Water Jet Generation 333

propagation of acoustic waves (pressure pulsations) in high-pressure system and properties of pulsating jet. The high-pressure system consisted of cylindrical acoustic chamber, liquid waveguide provided with high-pressure water supply and the nozzle. To simplify the model, acoustic actuator was substituted by vibrating wall of the acoustic chamber. The fluid flow in the model was solved as 3-D turbulent compressible unsteady flow of water. Water compressibility was taken into account in the numerical model using so called user defined function (UDF). The UDF covers calculations of both water density and speed of

> <sup>=</sup> ���� 1 − �−��� �

(998,2 kg.m-3), *p* and *pop* are real and operating pressures [Pa], *K* represents bulk modulus of water (2.2 . 109 Pa) and *a* is speed of sound in water [m.s-1] determined experimentally. The numerical simulation of a high-pressure system equipped with an acoustic generator was verified by the measurement of pressure pulsations in the high-pressure system upstream to the nozzle exit using dynamic pressure sensors. The pressure waveform in the numerical model was recorded at the same location as the pressure sensor was installed during the laboratory measurement. It was found out that numerical model provides information on the pressure waveform in high-pressure system that is in relatively very good agreement with experimental measurement. Comparison of results of numerical simulation and measurement also proved that the numerical model is able to simulate influence of geometry changes on the amplitude of dynamic pressure accurately and thus also to simulate pressure wave propagation and transmission in the high-pressure system.

Fig. 7. Standing wave amplitudes along longitudinal axis of the high-pressure system

After the verification of plausibility of results of numerical simulation by the laboratory measurement, the model was used in studying of the process of propagation and transmission of acoustic waves in the high-pressure system from the acoustic actuator to the

is water density [kg.m-3], *ρref* is reference water density under normal conditions

� = �� 1���� � 143� (34)

(33)

sound in water related to pressure:

where ρ � =

���� 1 − �� �

Fig. 6. Amplitudes of forced pressure waveforms in the simulated geometry related to the phase angle calculated from the analytical model

### **2.2 Numerical model**

Computational Fluid Dynamics (CFD) models of selected geometrical configurations of the high-pressure system with integrated acoustic generator were created using CFD code ANSYS CFD to simulate numerically the influence of operating and configuration parameters of the acoustic generator and transmitting line on the generation and

Fig. 6. Amplitudes of forced pressure waveforms in the simulated geometry related to the

Computational Fluid Dynamics (CFD) models of selected geometrical configurations of the high-pressure system with integrated acoustic generator were created using CFD code ANSYS CFD to simulate numerically the influence of operating and configuration parameters of the acoustic generator and transmitting line on the generation and

phase angle calculated from the analytical model

**2.2 Numerical model** 

propagation of acoustic waves (pressure pulsations) in high-pressure system and properties of pulsating jet. The high-pressure system consisted of cylindrical acoustic chamber, liquid waveguide provided with high-pressure water supply and the nozzle. To simplify the model, acoustic actuator was substituted by vibrating wall of the acoustic chamber. The fluid flow in the model was solved as 3-D turbulent compressible unsteady flow of water. Water compressibility was taken into account in the numerical model using so called user defined function (UDF). The UDF covers calculations of both water density and speed of sound in water related to pressure:

$$\rho = \frac{\rho\_{ref}}{1 - \frac{\Delta p}{K}} = \frac{\rho\_{ref}}{1 - \frac{p - p\_{op}}{K}} \tag{33}$$

$$a = 2.10^{-6}p + 1432\tag{34}$$

where ρ is water density [kg.m-3], *ρref* is reference water density under normal conditions (998,2 kg.m-3), *p* and *pop* are real and operating pressures [Pa], *K* represents bulk modulus of water (2.2 . 109 Pa) and *a* is speed of sound in water [m.s-1] determined experimentally.

The numerical simulation of a high-pressure system equipped with an acoustic generator was verified by the measurement of pressure pulsations in the high-pressure system upstream to the nozzle exit using dynamic pressure sensors. The pressure waveform in the numerical model was recorded at the same location as the pressure sensor was installed during the laboratory measurement. It was found out that numerical model provides information on the pressure waveform in high-pressure system that is in relatively very good agreement with experimental measurement. Comparison of results of numerical simulation and measurement also proved that the numerical model is able to simulate influence of geometry changes on the amplitude of dynamic pressure accurately and thus also to simulate pressure wave propagation and transmission in the high-pressure system.

Fig. 7. Standing wave amplitudes along longitudinal axis of the high-pressure system

After the verification of plausibility of results of numerical simulation by the laboratory measurement, the model was used in studying of the process of propagation and transmission of acoustic waves in the high-pressure system from the acoustic actuator to the

Use of Acoustic Waves for Pulsating Water Jet Generation 335

Results obtained from the numerical simulation correspond to results obtained from the analytical one even if the numerical model used is not physically accurate in the close vicinity of the nozzle outlet where cavitation occurs. Cavitation model was not implemented in the numerical model with respect to the computational speed. Results of numerical modelling clearly indicate that the geometrical configuration of high-pressure system influences significantly propagation and transmission of pressure pulsations from the acoustic actuator to the nozzle. The amplitude of pressure waves increases towards the nozzle outlet due to the proper shaping of the liquid waveguide – its frustums act as mechanical amplifiers of the acoustic waves. At the same time, the amplitude of pressure pulsations close to the nozzle outlet (where it has crucial influence on the pulsating jet generation) changes significantly with respect to the geometrical configuration of the high-

The use of visualization plays an important role in the study of behaviour of pulsating water jet. It enables not only the examination of characteristics of the jet such as mean velocity and break-up length of the pulsating jet but also to study the morphology and processes of formation of the pulsating jet and development of pulses in the jet. Furthermore, the visualization can be used to validate results obtained from numerical simulation of the

An original method of visualization of pulsating water jets based on the application of stroboscopic effect was elaborated for the above mentioned purposes. The method enables to obtain visual information not only on instantaneous structure of the pulsating jet but also on the mean structure of the jet. In addition, the stroboscopic effect allows observing process of formation of pulsating water jet by the naked eye. Special stroboscope for the pulsating jet visualization was developed where the frequency of stroboscope flashing is controlled by the frequency of pressure pulsations in the high-pressure system measured upstream from the nozzle exit. An example of the mean structure of pulsating jet with pulsating frequency of 20 kHz can be seen in Fig. 9. Exposure time of the photograph was 1/1000 s and the frequency of stroboscope flashing was about 20 kHz, therefore the figure represents

superposition of 20 images of pulsating jet "frozen" by the stroboscope flashing.

Fig. 9. The mean structure of the pulsating water jet generated at 30 MPa (illumination by

An instantaneous structure of the pulsating water jet with the frequency of 20 kHz was studied using the high-speed camera LaVision VC-HighSpeedStar 5 equipped with image amplifier LaVision HighSpeed IRO. The record rate of the high-speed camera was 35 000 frames per second and the gate was set to 1µs. In addition, visualization of instantaneous structure of the pulsating water jet was also performed using Particle Image Velocimetry (PIV) system consisting of LaVision Imager Intense camera and New Wave Research laser,

pressure system.

the stroboscope)

**3. Visualization of pulsating jet** 

process of generation of pulsating jets using CFD methods.

nozzle. An example of the behaviour of amplitudes of standing wave along the longitudinal axis of high-pressure system can be seen in Fig. 7. Figure 8 illustrates forced pressure waveforms in the simulated geometry related to the phase angle.

Fig. 8. Calculated forced pressure waveforms in the simulated geometry related to the phase angle recorded along the longitudinal axis of high-pressure system – from numerical model. (Scale indicates amplitude of dynamic pressure in MPa.)

nozzle. An example of the behaviour of amplitudes of standing wave along the longitudinal axis of high-pressure system can be seen in Fig. 7. Figure 8 illustrates forced pressure

Fig. 8. Calculated forced pressure waveforms in the simulated geometry related to the phase angle recorded along the longitudinal axis of high-pressure system – from numerical model.

(Scale indicates amplitude of dynamic pressure in MPa.)

waveforms in the simulated geometry related to the phase angle.

Results obtained from the numerical simulation correspond to results obtained from the analytical one even if the numerical model used is not physically accurate in the close vicinity of the nozzle outlet where cavitation occurs. Cavitation model was not implemented in the numerical model with respect to the computational speed. Results of numerical modelling clearly indicate that the geometrical configuration of high-pressure system influences significantly propagation and transmission of pressure pulsations from the acoustic actuator to the nozzle. The amplitude of pressure waves increases towards the nozzle outlet due to the proper shaping of the liquid waveguide – its frustums act as mechanical amplifiers of the acoustic waves. At the same time, the amplitude of pressure pulsations close to the nozzle outlet (where it has crucial influence on the pulsating jet generation) changes significantly with respect to the geometrical configuration of the highpressure system.
