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Part 3 Acoustic Waves as Manipulative Tools 15 Use of Acoustic Waves for Pulsating Water Jet Generation Josef Foldyna Institute of Geonics of the ASCR, v. v. i., Ostrava Czech Republic 1. Introduction The technology of a high-speed water jet cutting and disintegration of various materials attained considerable growth during the last decades. Continuous high-speed water jets are currently used in many industrial applications such as cutting of various materials, cleaning and removal of surface layers. However, despite the impressive advances made recently in the field of water jetting, substantial attention of number of research teams throughout the world is still paid to the improvement of the performance of the technology, its adaptation to environmental requirements and making it more beneficial from the economic point of view. An obvious method of the water jetting performance improvement is to generate jets at ultra-high pressures. The feasibility of cutting metals with pure water jets at pressures close to 690 MPa was investigated already in early nineties of the last century (Raghavan & Ting, 1991). Such a high pressure, however, induces extreme overtension of high-pressure parts of the cutting system which has adverse effect on their lifetime. An alternate approach, as shown in this chapter, is to eliminate the need for such high pressures by pulsing the jet. It is well known that the collision of a high-velocity liquid mass with a solid generates short high-pressure transients which can cause serious damage to the surface and interior of the target material. The liquid impact on a solid surface consists of two main stages (see Fig. 1). During the first stage, the liquid behaves in a compressible manner generating the so-called ‘‘water-hammer’’ pressures. These high pressures are responsible for most of the damage resulting from liquid impact on the solid surface. The situation shortly after the initial impact of the liquid on the solid surface is illustrated in Fig. 2. After the release of the impact pressure, the second stage of the liquid impact begins. Once incompressible stream line flow is established, the pressure on the central axis falls to the much lower Bernoulli stagnation pressure that lasts for relatively long time. The force distribution on liquid jet impact on the solid surface can be summarized as follows: initially a small central area of the first contact is compressed under a uniform pressure. The magnitude of the impact pressure pi on the central axis is given by = + (1) where v is the impact velocity and ρ1, ρ2 and c1, c2 are the densities and the shock velocities in the liquid and the solid, respectively (de Haller, 1933). 324 Acoustic Waves – From Microdevices to Helioseismology st nd 1 stage 2 stage v v v v c1 c1 Target ρ2 vr vr pi c2 pi c2 ps Fig. 1. Two stages of liquid impact on a solid target Water drop Shock envelope c1 Contact edge β c2 c2 Target Fig. 2. Initial stage of impact between a water drop and a solid target with the contact edge moving faster than the shock velocity in the liquid. The liquid behind the shock envelope is compressed and the target beneath this area subjected to high pressure The magnitude of the impact pressure is independent of the geometry of the drop (Thomas & Brunton, 1970), but the duration of the pressure is affected by the size and shape of the drop. For a sphere or cylinder the corresponding radius or half-width of the contact area R exposed to this pressure is given by = (2) where r is the radius of curvature of the drop or cylinder (liquid mass) in the region of contact (Bowden & Field, 1964). The initial area of contact grows as the impact continues; there is very little reduction in pressure on the surface until appreciable outward flow begins. The outward flow of the liquid becomes possible when the limit of the compressible deformation of the liquid is exceeded. The limit is given by = where β is the liquid/solid interface angle – see Fig. 2 (Hancox & Brunton, 1966). (3) Use of Acoustic Waves for Pulsating Water Jet Generation 325 At this stage there is a rapid fall in pressure along the periphery of contact. As the outward flow continues, the water-hammer compression at the centre of impact is relieved until the maximum pressure acting on the surface is the central stagnation pressure for the incompressible flow. The stagnation pressure is given by = 1 2 (4) When the liquid begins to flow away from the point of impact, there is evidence that the velocity of this tangential flow may be as much as five times the impact velocity (Thomas & Brunton, 1970). The velocity increase is thought to be connected with the shape of the head of the jet. It has been observed that an increase in velocity along the surface occurs only in cases where the jet head is inclined at an angle to the surface. Since spherical drops (and/or spherical heads of a train of pulses of pulsating jet) always provide a sloping interface to a plane solid surface it might be expected that high radial velocities will occur on impact. Therefore, there are additional shear forces associated with the high speed flow across the surface acting on the surface in addition to the normal forces. The shear forces acting on a roughened surface are large enough to cause local shear fractures, even in high strength materials (Hancox & Brunton, 1966). Exploitation of above described effects associated with water droplet impingement on solids in a high-speed water jet cutting technology should lead to considerable improvement of its performance, better adaptation to more and more demanding environmental requirements, and consequently to more beneficial use of the technology also from the economical point of view. Generating sufficiently high pressure pulsations in pressure water upstream the nozzle exit enables to create a pulsating water jet that emerges from the nozzle as a continuous jet and it forms into a train of pulses at certain standoff distance from the nozzle exit. Such a pulsating jet produces all of the above mentioned effects associated with water droplet impingement on solids. In addition, the action of pulsating jet induces also fatigue stress in the target material due to the cyclic loading of the target surface. This further improves the efficiency of the pulsating liquid jet in comparison with the continuous one. Thus, destructive effects of the continuous high-speed water jet can be enhanced by the introduction of high-frequency pulsations in the jet, i.e. by generation of pulsating water jets. Recently, a special method of the generation of the high-speed pulsating water jet was developed and tested extensively under laboratory conditions. The method is based on the generation of acoustic waves by the action of the acoustic transducer on the pressure liquid and their transmission via pressure system to the nozzle. The high-pressure system with integrated acoustic generator of pressure pulsations consists of cylindrical acoustic chamber connected to the liquid waveguide. The liquid waveguide is fitted with pressure liquid supply and equipped with the nozzle at the end. The acoustic actuator consisting of piezoelectric transducer and cylindrical waveguide is placed in the acoustic chamber (see Fig. 3). Pressure pulsations generated by acoustic actuator in acoustic chamber filled with pressure liquid are amplified by mechanical amplifier of pulsations and transferred by liquid waveguide to the nozzle. Liquid compressibility and tuning of the acoustic system are utilized for effective transfer of pulsating energy from the generator to the nozzle and/or nozzle system where pressure pulsations transform into velocity pulsations. The acoustic generator can be used for generation of both single and multiple pulsating water jets (e.g. 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 conditions. Acoustic chamber Acoustic actuator Pressure liquid Pulsating supply jet Liquid waveguide Nozzle Fig. 3. Schematic drawing of the high-pressure system with integrated acoustic generator of pressure pulsations 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 pulsating liquid jet. 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, 1995). 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 areas so far are summarized in following sections. 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 327 Use of Acoustic Waves for Pulsating Water Jet Generation 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 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: =2 ( − ) + ( ) (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: =2 (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: = where (7) 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: −2 (∆ ) − Θ( − ) (∆ ) − ∆ =0 (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: − 2 + ( )Δ − Δ =0 (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: 328 Acoustic Waves – From Microdevices to Helioseismology ( ) = ( ) (10) If following expression is denoted κ: =− + (2 + ) (11) then it can be written +Δ =0 (12) In the frequency domain it is valid that ( ) = / . The solution of (12) can be performed by the implementation of spherical coordinate system (r, φ, υ), see Fig. 4. Now, the wave equation can be written in the following form: Fig. 4. Implementation of spherical coordinate system 329 Use of Acoustic Waves for Pulsating Water Jet Generation + 2 + 1 + 1 cotg + 1 1 sin + =0 (13) Let’s assume the solution of (13) as a product of functions: = ( ) (cos )Φ( ) (14) Then, individual particular integrals can be expressed as follows: Φ = cos (cos ) = + sin (15) (cosυ) + (cosυ) (16) where P, Q are special Legendre polynomials: Z = 1 ( √ = )+ 1+4 ( 2 ( ) + 1) (17) (18) 2.1.2 Transfer matrix The objective is to determine transfer matrix P that can be used in solving pressure and flow pulsations in hydraulic systems in conffuser-type tubes. For this purpose, it is convenient to introduce the mean velocity of the liquid cr in a direction of r using following formula: = 1 2 ( , , ) (19) The solution can be simplified by the assumption that the flow is rotary symmetrical. It can be derived under the above mentioned assumption that functions , will be streamlined to the following form: , . Further, considering that the pressure function p varies only a little with respect to the angle υ, the following relation for the mean velocity cr can be written based on Navier-Stokes equations: =− 2 − 2 (20) The continuity equation and component cr in Navier-Stokes equations expressed in the spherical coordinate system were used in the above mentioned derivation. Withal, effects of dynamic viscosity were neglected. If we will keep considering zero initial conditions, it can be written after the Laplace transformation (20): = ; ( ) = ( ) (21) 330 Acoustic Waves – From Microdevices to Helioseismology =− 2 1+ (22) If all assumptions of the solution are considered, following can be written for Laplace images of both the pressure function and the velocity and with respect to (21): = 1 ( . √ )+ = 0; . = = If we introduce for . = ( ) √ ( ) (23) 1 2 (24) . + (25) the state vector = ( , ), ( , ) (26) = ( , ), ( , ) (27) and for r the state vector the dependence in locations r and r0 can be expressed by means of the transfer matrix: ( , )= ( , ) (28) Then, the matrix P will be derived from (26) and (27) by the elimination of integration constants F, G. If we designate: . = ( ) ( . )− ( ) ( ) . ( (29) ) following relation can be written for matrix P: = . 1 . ( ( . ) ) . ( ( ) ) − . ( . − ) . ( ) (30) ( ) In the frequency domain, = is substituted. Both pressure and flow pulsations of hydraulic systems with conffuser-shaped tubes can be solved on the basis of the transfer matrix (30). Individual elements of the transfer matrix are dependent on values of the speed of sound and the second viscosity. Values of both these quantities depend on the static pressure and the value of second viscosity depends also on the frequency. The values can be determined experimentally using the transfer matrix. 2.1.3 Application of the transfer matrix The transfer matrix derived in the previous section can be used in solving transmission of pressure and flow pulsations in complex hydraulic systems. Such a system can consist of cylindrical and confusser-shaped sections; the system can also be bifurcated.