Motohiko NOHMI*
Tomoki TSUNEDA*
Wakana TSURU**
Kazuhiko YOKOTA***
*
Technologies, R&D and Intellectual Property Division
**
Saga University
***
Aoyama Gakuin University
CFD analysis was performed on the cavitation surge generated in the contraction channel. The case where pressure waves propagate through the pipeline upstream and downstream of the contraction was compared with those when incompressible assumption was applied. It was found that fluctuations in flow rate and static pressure occurred only downstream of the contraction. In such cases, outlet boundary conditions that can take into account downstream fluid fluctuations are needed.
Keywords: Cavitation Surge, Pressure Wave, Method of Characteristics, Contraction, Orifice
In the previous paper, we introduced Ohashi and Akimoto’s criteria1) as an index to determine whether fluid fluctuation phenomena in a piping system should be treated as waves based on the compressibility of the fluid or as translational motion of a rigid fluid column (water column in the case of water). The validity of these criteria was verified for a simple piping system with oscillation sources and constant static pressure tanks at both ends of the piping.
This paper presents a specific application of these criteria to cavitation surges and a case study of the phenomenon. Cavitation in pumps is not a practical problem because its effect is negligible in normal operation. However, a noise known as a chugging noise and vibrations in the piping system may occur at off-design points2). This phenomenon is known as cavitation surge. This is an important issue to consider, not only for industrial pumps, but also for advanced pumps such as rocket pumps. Many researchers have been working to clarify this in recent years. Pump designers are required to avoid this or to reduce vibration noise as much as possible. A cavitation surge occurring in a piping system is a phenomenon with a frequency lower than the impeller rotation speed of typical turbo type fluid machinery. In addition, since this is an interaction problem with flow, not only in the vicinity of the cavitation point, but also in the surrounding piping system, numerical analysis tends to result in large-scale calculations in terms of both time and space. If the pipes upstream and downstream of the cavitation generation point are long, consideration of pressure waves propagating through the piping system is necessary. Ohashi and Akimoto’s criteria can be used as its threshold1).
L>c/（kf） Must consider compressibility (pressure wave )
L<c/（kf） Possible with incompressible fluid
Here, L is pipe length, c is the sound speed of pressure wave propagating through a piping system, and f is the surge frequency. k is a dimensionless coefficient. 8 to 10 is recommended if the pipe can be considered free-ended or fixed-ended on both sides. If it is free-ended or fixed-ended on one side, halve the above recommended k value. When numerically analyzing waves in liquid, calculations must satisfy the CFL condition using the speed of sound in the liquid. In general, fluid machinery that handles water often has an internal flow velocity of less than 50 m/s. On the other hand, the speed of sound in pure water is about 1500 m/s at normal temperature and pressure. Even taking into account the inclusion of air bubbles and elastic deformation of the pipe, it is usually higher than 1000 m/s. If the conditions of the CFL number 1 are faithfully followed, the time intervals for compressible and incompressible analyses differ by a factor of about 20 to 30 or more (the ratio of the aforementioned speed of sound to the maximum internal flow velocity) for the same computational grid conditions. In other words, taking into consideration wave motion further increases the computational scale of a cavitation surge. Therefore, in CFD of cavitation surges, it would be beneficial if the relevant phenomenon could be studied well in advance and the computational scale could be reduced while ensuring the necessary accuracy. Among many methods, one method has been proposed in which three-dimensional analysis is performed only in the vicinity of the cavitation-generating part, and other pipe system elements are treated as a lumped parameter system or a one-dimensional flow analysis within the pipe. Table 1 shows the classification of calculation methods for cavitation surges, mainly for pump systems3), 4). The numbers in the table are the reference numbers that deal with the various methods. The authors have developed a joint method of 2D/3D cavitation CFD without considering the compressibility of the liquid phase and the analysis of one-dimensional wave motion propagation in a pipe by the method of characteristics. This method has the problem that the acoustic impedance (product of density and speed of sound) is extremely different between the method of characteristics, which takes into account the compressibility of the liquid, and the two-dimensional/three-dimensional analysis domain, which does not consider the compressibility of the liquid phase, and wave reflections occur at the junction of the two domains. However, in the case of a long piping system where cavitation occurs in a relatively small domain, it is considered an efficient calculation method. In this study, as part of the research to evaluate the validity of this method, three types of analysis were conducted for cavitation occurring in the throttle flow path: (1) two-dimensional cavitation analysis only near the throttle, (2) a combination of two-dimensional cavitation analysis near the throttle and method of characteristics analysis of the upstream and downstream pipe sections, and (3) two-dimensional cavitation analysis of the throttle and all upstream and downstream pipe sections. These three analyses were compared.
The analysis target is the two-dimensional throttling flow path shown in Fig. 1(a). This analysis shows that the water flowing in from the inlet is accelerated in the throttle section and generates cavitation, but the cavitation is eliminated before reaching the outlet section due to deceleration in the downstream expansion section and the resulting static pressure recovery. The entire piping system including this throttle is shown in Fig. 1(b). The pipe lengths LU and LD are constant at 1 m each. The two-dimensional analysis domain is divided by a constant two-dimensional grid of Δx = 1 mm and Δy = 0.5 mm. The analysis was performed using the commercial code ANSYS-FLUENT (Ver. 18). The cavitation model is the Schnerr-Sauer model, the turbulence model is the SST k-ω model for steady analysis, and the Reboud corection (order n = 5) is added to this model for unsteady analysis to strengthen the unsteadiness of the cavitation9). When the method of characteristics is used for the pipe section, considering the numerical stability of the solution, the speed of sound is set to 500 m/s, which is about half the normal speed, and the CFL number is set to 0.54).
Fig. 1 Objective contraction with piping system
Boundary conditions and initial conditions:
(1) In the case of two-dimensional cavitation analysis only in the vicinity of the throttle.
The flow velocity at the inlet is constant at 5 m/s and the static pressure at the outlet is constant at 45,000 Pa. The steady-state solution can be found using SST k-ω turbulence model and perform unsteady analysis with Reboud correction.
(2) In the case of a combination of a two-dimensional cavitation analysis and a method of characteristics analysis of the upstream and downstream pipe sections.
In the analysis in (1) above, the initial condition is the result obtained 2 seconds after the start of the unsteady calculation. Using this initial flow field, the upstream and downstream tank static pressures are calculated, taking into account quasi-steady processes and pipe friction. After this, the boundary conditions of the tank section are set to constant static pressure calculated respectively, and the unsteady calculation, applying the method of characteristics is started. For other details of the analysis, please refer to the reference3).
(3) In the case of two-dimensional cavitation analysis over the entire area
The analysis will cover all pipe sections and throttles, but not tank sections. The steady-state solution is obtained using the SST k-ω turbulence model with 1 m/s flow velocity as the boundary condition of the upstream tank section and constant static pressure as the boundary condition of the downstream tank section. The value of the downstream tank section is adjusted accordingly so that the static pressure at the outlet of the throttle in Fig. 1(a) is approximately 45,000 Pa, and this is used as the initial value. In the unsteady analysis, Reboud correction is added to the turbulence model, and three types of analysis are conducted as boundary conditions: (3)-1 Specified inlet flow rate and specified outlet static pressure, (3)-2 Specified inlet total pressure and specified outlet static pressure, (3)-3 Specified inlet total pressure and specified outlet flow rate.
Fig. 2 shows the static pressure variation at the inlet and outlet of the analysis domain, the area averaged flow velocity variation, the cavitation volume in the analysis domain, and the time variation of time derivative of the cavitation volume and the volume flow difference between the inlet and outlet. Since this is a two-dimensional analysis, the units for cavitation volume and volumetric flow rate are m2 and m2/s, respectively. Fig. 2 shows that the upstream velocity and static pressure fluctuate little except for slight spikes, while the downstream velocity fluctuates significantly. The cavitation coefficient σ is 7.7, based on a flow velocity value of 5 m/s and a static pressure value of 99.5 kPa at the inlet of the analysis domain. FFT analysis of the outlet flow velocity waveform at time t = 3 to 6 s shows peaks at 12.7 Hz and 19 Hz. The time derivative of the cavitation volume and the time-varying waveform of the volume flow difference between inlet and outlet are in good agreement, indicating that the law of conservation of mass is well preserved in the analysis. This well preserved conservation of mass is also true for the cases in Sections 3-2 and 3-3 below.
Fig. 2 CFD results analyzing only the contraction
Fig. 3 Instantaneous flow fields at t = 6 s
Fig. 4 shows the static pressure variation at the inlet and outlet near the throttle, the average flow velocity variation, and the cavitation volume variation in the analysis domain. After a transient change of about 0.2 s, the upstream velocity and static pressure fluctuate little, while the downstream velocity and static pressure fluctuate significantly.
After the transient change, the flow velocity is 5 m/s and the cavitation coefficient σ is 7.7 at the throttle inlet. FFT analysis of the outlet flow velocity waveform at time t = 2.5 to 3 s shows peaks at 4 Hz and 234 Hz.
Fig. 4 CFD results analyzing the whole pipeline by using Method of Characteristics
Fig. 5 shows the inlet and outlet static pressure variation, area averaged flow velocity variation, and cavitation volume variation in the analysis domain near the throttle ((a) in Fig. 1) for the specified inlet flow and specified outlet static pressure. After a transient change of about 0.25 s, the upstream velocity and static pressure fluctuate little, while the downstream velocity and static pressure fluctuate significantly. After the transient change, the flow velocity is 5 m/s and the cavitation coefficient σ is 7.2 at the inlet. FFT analysis of the outlet flow velocity waveform at time t = 3 to 6 s shows peaks at 2.3 Hz and 4 Hz.
Fig. 5 CFD results analyzing the whole pipeline: Inlet B.C velocity specified, Outlet B.C. static pressure specified
Fig. 6 shows the instantaneous value of the velocity and void fraction distributions at t = 6 s. Fig. 6 shows that a considerable distance is required for the jet stream generated at the throttle to reattach.
Fig. 6 Instantaneous flow fields at t = 6 s
Fig. 7 shows the inlet and outlet static pressure variation, area averaged flow velocity variation, and cavitation volume variation in the analysis domain near the throttle ((a) in Fig. 1) for the specified inlet total pressure and specified outlet static pressure. After a transient change of about 0.2 s, the upstream velocity and static pressure fluctuate little, while the downstream velocity and static pressure fluctuate significantly. After the transient change, the flow velocity lowers to 4.94 m/s and the cavitation coefficient σ is 7.2 at the inlet. The outlet velocity fluctuates about 3 Hz at time t = 2 to 4 s. Subsequent FFT analysis of the outlet flow velocity waveform at t = 4 to 6 s, shows a peak at 6.5 Hz.
Fig. 7 CFD results analyzing the whole pipeline: Inlet B.C total pressure specified, Outlet B.C. static pressure specified
In the case of the specified inlet total pressure and specified outlet flow, the trend differs significantly from the previous results. It is similar in that the upstream flow velocity and static pressure remain almost unchanged, but the cavitation volume value in the analysis domain monotonically increases, and the cavitation domain extends downstream from the throttle (Figure omitted).
Cavitation surge frequencies for the analysis combining cavitation analysis and the method of characteristics showed variations of 4 Hz and 234 Hz, while frequencies such as 2.3 Hz, 4 Hz, 3 Hz, and 6.5 Hz were calculated for the two-dimensional cavitation analysis for all flow paths. Of these, 234 Hz is almost identical to 250 Hz, the frequency at which a wave of c = 500 m/s reciprocates through a 1 m pipe under the throttle. The difference from 250 Hz may be due to the compliance effect of the cavitation itself. Other frequencies are in the range of 2.3 to 6.5 Hz. Substituting c = 500 m/s, L = 1 m, and k = 8 into Ohashi and Akimoto’s criteria, the threshold frequency is f = 62.5 Hz. This suggests that the compressibility of the liquid phase does not need to be considered in this analysis. The compressibility of the liquid phase around the cavitation is not considered in any of the calculations in this analysis. Future work is needed to confirm whether or not the frequency of cavitation surges changes significantly by taking this into account. The variation in the frequency of the cavitation surge in each analysis may be due to the transient changes after the start of the calculations have not been fully completed, the boundary conditions are different in each case, and there are some differences in the cavitation coefficient σ at the inlet of the throttle. This too needs to be investigated in detail in the future.
In all of the calculations in Figs. 4, 5, and 7, after a short transient change, both the flow velocity and static pressure hardly fluctuate at the upstream of the throttle. The analysis was stable regardless of whether constant flow or constant total pressure was used as further upstream boundary conditions. The occurrence of cavitation surge causes the flow velocity downstream to fluctuate, but cavitation at the throttle keeps the static pressure near the throttle at a constant saturated vapor pressure, which is thought to reduce the transmission of downstream disturbances to upstream. In spite of this situation, if the boundary condition of inlet total pressure and specified outlet flow rate are specified, the actual phenomenon deviates from the calculated one, since no downstream flow velocity fluctuation occurs. We believe that this is the reason why the cavitation volume increased monotonically without any oscillating surge condition when the inlet total pressure minus specified outlet flow was set in this study. The boundary conditions of specified total inlet pressure and outlet flow rate are often used in pump head breakdown analysis because of the convenience of directly specifying NPSHa. However, in the unsteady analysis of pump cavitation surges, it is necessary to confirm the appropriateness of their use by comparing the results with the boundary conditions of specified inlet flow rate and specified outlet static pressure.
Cavitation surge frequency decreases as the pipe length increases3), 11). In this study, as well, the analysis only in the vicinity of the throttle showed a frequency of 12.7 Hz and 19 Hz, which are higher than the other analysis results. Therefore, to calculate the frequency of cavitation surge accurately, it is desirable to match the pipe length to the actual piping system at the actual site as much as possible.
Figs. 3 and 5 show that the flow is not yet uniform in the cross section at the junction position of the two-dimensional analysis and the method of characteristics analysis for the flow downstream of the throttle, and that the junction is appropriate further downstream where the flow becomes uniform. Since the velocity at the boundary used in the method of characteristics is the cross-sectional area averaged velocity from the two-dimensional analysis, the law of conservation of mass is satisfied. However, on the two-dimensional analysis side, a boundary condition of uniform static pressure obtained by the method of characteristics is imposed, which is a source of error. On the other hand, to make the method of connecting incompressible and compressible analysis reasonable, the incompressible domain must be set to an appropriate size. Although a detailed study is a subject for the future, the following equation (1) can be used as one indicator.
Lincomp
≤cΔt
……………（1）
Where Lincomp
is the representative length in the flow direction of the incompressible computational domain, i.e., the direction of pressure wave propagation, and Δt is the time interval. The right-hand side of equation (1) is the length of one division grid of the method of characteristics. Although the speed of sound is infinite in the incompressible analysis domain, if its length is made equivalent to the grid length of the method of characteristics, the calculation errors such as time delay of pressure propagation in the piping system can be minimized.
As described above, although there are many issues to be considered in the future, we have confirmed that the combined analysis of cavitation analysis and the method of characteristics can be performed stably. From the evaluation of Ohashi and Akimoto’s criteria, it is expected that in cases where compressibility of the liquid phase should be considered, the computation load can be greatly reduced by using the method of characteristics and avoiding two- or three-dimensional analysis of the entire domain.
For cavitation surges in a throttle channel, a comparison was made between an analysis considering pressure wave propagation in the piping system and an analysis neglecting the compressibility of the liquid phase. In both analyses, the frequency of the cavitation surge was confirmed to be in the range where the compressibility of the fluid does not need to be considered in Ohashi and Akimoto’s criteria. In this study case, the upstream of the cavitation resulted in little variation in flow rate or static pressure. For such flow fields, it is necessary to use boundary conditions that can consider downstream flow fluctuations.
In this paper, one of the authors (Nohmi) first learned of the evaluation method, which we call Ohashi and Akimoto’s criteria, from reference12). In that reference, a reference by Ohashi in 1970 was cited, and there it was labeled "Ohashi's Criteria". Later, we came across a reference14) by Akimoto in 1972, which contained a similar formula, and decided to use the name Ohashi and Akimoto's criteria in consideration of the achievements of both authors1). Later, we happened to come across a similar classification in a 1982 article by Mitsuno describing the water hammer phenomenon. In that article, the phenomenon is classified in three ways: wave model, vibration model, and rigidity model. In addition, the phenomenon is organized by valve closure time rather than charcteristic frequency, a concept that is in line with the discussions of Ohashi and Akimoto. If four times the valve closure time is taken as one cycle, the range of application of Ohashi's criteria and the rigidity model by Mitsuno coincide perfectly. This suggests that the true originator of this classification and evaluation method may go back even further, but unfortunately, at this time, no details are available. While we recognize that the present is based on the accumulation of ideas and steady efforts of many predecessors, as is true in all academic fields, we will refer to this as Ohashi and Akimoto’s criteria, representing the two authors of the early literature as far as we know.
This paper is a revised version16) of a contribution to the 19th Symposium on Cavitation organized by the Science Council of Japan, held in October 2018.
We would like to express our deepest gratitude to Naoki Kodama for his efforts in implementing complex algorithms and performing numerous analyses in this study.
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