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Experimental and numerical investigation of the cavitation-induced choked flow in a herschel venturi-tube
Flow Measurement and Instrumentation 54 (2017) 56–67
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Experimental and numerical investigation of the cavitation-induced choked flow in a herschel venturi-tube
MARK
⁎
S. Brinkhorsta, , E. von Lavantea, G. Wendtb a b
Chair of Fluidmechanics, University of Duisburg-Essen, 47057 Duisburg, Germany Head of Department “Liquid Flow”, Physikalisch-Technische Bundesanstalt PTB, 38116 Braunschweig, Germany
A R T I C L E I N F O
A BS T RAC T
Keywords: Cavitation Choked flow Herschel venturi-tube Cavitating nozzle Liquid flow metering
Due to their simple geometry, cavitating Venturi nozzles (CV) are a long time subject of experimental as well as numerical investigations. However, research mostly focused on certain aspects like the comparison of experimental data with numerical cavitation models or the spray development of diesel injection nozzles, but rarely on the choked flow condition itself, especially with regard to liquid flow measurement. If the pressure decreases due to the local acceleration of the flow to the respective vapor pressure, a choked flow condition similar to the well-known critical flow Venturi-nozzles (CFVN) develops. For the purpose of gaining further insight into the choked flow condition with respect to liquid flow measurement, high-speed camera investigations of a transparent Herschel Venturi-tube configuration, also known as classical Venturitube, were performed. Together with pressure and flow rate measurements, they demonstrated the overall stable flow behavior under choked conditions. With additional numerical investigations, phenomena during the onset of the choked condition were clarified. Furthermore, a simple correlation for the calculation of the actual flow rate during the choked condition, including a temperature correction was proposed.
1. Introduction As a practical realization of the so called traceability requires, hydraulically cavitating Herschel Venturi-tubes were investigated for their highly stable flow rates under choked flow conditions. In addition to accuracy, the traceability of a given measuring device is of essential importance, in particular with respect to the general guarantee of uniformity and comparability of measurements. In gas flow measurement, the so-called critical flow Venturi-nozzles (CFVN) have been widely established as an efficient transfer standard during the last two decades. Comparable solutions for liquid flow measurement did not exist to this day. The analogies observed in experiments with Venturinozzles using gas or liquids as working fluids, already observed in 1930 by Ackeret, led the Physikalisch-Technische Bundesanstalt (PTB) to start detailed investigations of cavitating Venturi-nozzle (CV) flows. The intention was to clarify their applicability in liquid flow measurement comparable with the use of critical nozzles for gas metering. Though Ackeret has already mentioned the experimentally observed analogies between CVs and CFVN in 1930, his work was not focused on flow rate investigations [1]. Numachi et al. investigated different Venturi-tubes, nozzles and orifices under cavitating conditions [2–5]. However, the differential-pressure flow rate measuring
⁎
devices were investigated to obviate incorrect measurements due to effects of possible cavitation at certain small pressure ratios. Nevertheless, the phenomenon of an enforced constant mass flow rate through the nozzles, while cavitation in the nozzle throat occurred, was investigated and described. Still, a well-directed exploitation of this effect for flow rate metering applications has not been considered then. Recently, some experimental work has been published focusing on the use of CVs as liquid flow meters, with special emphasis on the choked flow condition. Ghassemi et al. showed the general possibility of the application of CVs as liquid flow meters [6], while Abdulaziz gained further insights into the cavitation process by using an optical flow visualization method [7]. Rudolf et al. characterized experimentally the cavitation regimes of a Herschel Venturi-tube by evaluating the loss coefficient, thereby revealing a unique behavior during the choked condition, as well as during its onset [8]. Ashrafizadeh et al. performed numerical and experimental investigations of different Venturi-tube geometries designed as flow-control devices [9]. Recent experimental investigations by Schmidt using CVs confirmed the high stability of the flow rate during the choked condition [10]. The investigations mentioned so far were supplemented by the already published numerical investigations of CVs by the authors [11,12]. These investigations were mainly focused on geometry influ-
Corresponding author. E-mail address: [email protected] (S. Brinkhorst).
http://dx.doi.org/10.1016/j.flowmeasinst.2016.12.006 Received 13 July 2016; Received in revised form 15 November 2016; Accepted 12 December 2016 Available online 16 December 2016 0955-5986/ © 2016 Elsevier Ltd. All rights reserved.
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to the Sauer cavitation model the terms modeling the rate of evaporation and condensation were:
Table 1 Properties defined for the numerical simulations, with μl and μv representing the liquid and vapor viscosity respectively.
m˙ +, m˙ − = C
property
value
pressure, inlet (P01) vapor pressure αv, inlet T ρl ρv μl μv number density of seeds initial seed radius turbulent intensity, inlet Cprod , Cdest
3.764 bar (total pressure) 2333.21 Pa 0 298 K 998.34 kg/m3 0.0173 kg/m3 1.001477 mPa s 0.009727 mPa s 1e12 /m3 0.001 mm 1% 1
2 pv − p 3 ρl
(2)
With C = Cprod for evaporation and C = Cdest for condensation and the growth/collapse of a bubble being calculated by the simplified Rayleigh-Plesset equation. The vapor pressure pv is the saturation pressure corresponding to the liquid temperature, p denotes the pressure surrounding the bubble, ρl denotes the liquid density and ρm denotes the density of the vapor-liquid mixture, calculated by:
ρm = αv ρv + (1 − αv ) ρl
(3)
Due to the use of the simplified Rayleigh-Plesset equation, viscous and surface tension effects on the rate of change of the bubble radius were neglected. Turbulence was modeled by the realizable k-ϵ two-layer model, including an all-y+ wall treatment [18]. The chosen turbulence model has already been proven to predict satisfying results for cavitating flow [9,19,20]. Nevertheless, it should be mentioned that no preferred turbulence model, as well as cavitation model, exists in literature. This led to a plethora of cavitation-/turbulence-model combinations. Though the choice of either model usually has a large impact on the numerical results, at least globally matching results between each model can be obtained by carefully adjusting the empirical constants [21]. For time and space discretization, second order accuracy was used. All numerical results presented were obtained with full 3D simulations. For a proper time wise resolution a time step of 10−4 s was chosen. The numerical mesh was modeled based on a grid sensitivity analysis presented in [11] and was created using the CCM+ grid generator. With a grid refinement in the cavitating zone the resulting mesh consisted of about 1 million cells. For all simulations the following boundary conditions (b.c.) have been applied: A stagnation pressure inlet b.c. for defining a constant total pressure at the inlet, while a constant static pressure has been defined at the outlet; at the wall the no-slip b.c. was used.
ences on the stability and accuracy of the flow rate, with regard to flow metering applications. They demonstrated the advantages of Herscheltube configurations compared to a typical ISO 9300 CFVN geometry, with respect to the constancy of the flow rate for different pressure ratios within the cavitation induced choked flow condition Table 1. The recently performed experimental and numerical investigations presented in this paper revealed a new phenomenon during the onset of the choked flow condition, namely the sudden expansion of the vapor cloud. Furthermore, due to high-speed camera observations with up to 140.000 fps, reasons for the steadiness and high stability of the vapor cloud and thus the choked flow rate will be presented. For the numerical simulations, the commercial code CD-adapco STAR-CCM+10.04.011 (CCM+) was used, solving the three-dimensional, Reynolds-averaged Navier-Stokes (RANS) equations for turbulent, incompressible, unsteady, cavitating two-phase flow. The experiments were carried out using the experimentation and water meter test rig (EWZP) of the PTB. 2. Numerical cavitation model The commercial CFD program CCM+, used an interface capturing approach via the Volume-of-Fluid-method (VOF-method) to model the cavitating two-phase flow. The cavitation itself was based on the inertia controlled bubble growth theory obtained by Rayleigh [13]. The capability of this solver has already been extensively validated by different authors [14–17]. Due to the VOF-method treating the vapor and liquid phase as a mixture, only one set of the mass and momentum equations is solved, where the physical properties of the mixture or equivalent fluid were functions of the physical properties of its constituent phases and their volume fractions. In addition to the mass and momentum conservation, an equation had to be solved governing the transport of the volume fraction α between the two phases:
→ ∂αv + ∇ ·(αv → u ) = Γv ∂t
ρv ρl 3αv (1 − αv ) ρm Rb
3. Experimental setup For the experimental investigations, the PTB provided their experimentation and water meter test rig (EWZP), with a schematic representation shown in Fig. 1. The volumetric flow rate (Q) was measured by a magnetic flow meter (MID). Furthermore, the differential pressure across the nozzle (ΔP ) as well as the static pressure (P1) and the temperature at the nozzle inlet were measured. The EWZP has an expanded measurement uncertainty of 0.05%. Details of the meters used can be found in Table 2. For the measurements, the stagnation pressure in front of the nozzle could be controlled by the rotational speed of the pump in combination with the valve position. The valve was located about 50 pipe diameter behind the nozzle. For visual inspection of the cavitation, as well as the use of high-speed camera recordings, the investigated Herschel Venturi-tube was manufactured out of acrylic glass. The dimensions of the experimentally and numerically investigated Herschel-tube are presented in Fig. 2. For the visualization of vapor
(1)
The subscript v denotes the vapor phase, while the definition of the volume fraction of vapor is: αv = Vv / V , with Vv being the volume of vapor inside the control volume V. The phase change due to cavitation is governed by the source term Γv on the right side of Eq. (1), whereby Γv is usually split into two terms Γv = m˙ + + m˙ −. m˙ + denotes the rate of evaporation and m˙ − the rate of condensation. In fact, most cavitation models differ only in the formulation of this source term. In CCM+, the common Sauer cavitation model was used for the calculation of the phase change. The Sauer cavitation model is based on the assumptions of having homogeneously distributed seed bubbles with equal radius Rb. These bubbles remain spherical at all times and the number density of seeds n0 together with the initial seed radius Rb,0 was a model parameter to be set. The two phases were treated as incompressible and slip velocity between bubbles and liquid was neglected. Thus, according
Fig. 1. Schematic representation of the EWZP of the PTB.
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Table 2 Specifications of the used meters. Meter
Range
Uncertainty
ADMAG AE Yokogawa AE202MG E+H deltabar S PMD 235 E+H Cerabar PMC71
0.5–17.5 [m3/h] −1000 to 1000 [mbar] 0–40 [bar]
0.5% 0.1% 0.075%
Fig. 3. Measured volumetric flow rate of the MID as a function of the pressure ratio p2 /P01. Fig. 2. Schematic representation of the investigated Herschel Venturi-tube.
cavity patterns, two cameras were used. A Samsung NX300M for longtime exposure and a Photron FASTCAM SA-Z 2100 K for high-speed recordings of up to 140.000 fps. The measurements were started by opening the valve completely and adjusting the rotational speed of the pump. Thus, the measurement started with the highest possible pressure difference (lowest pressure ratio). Leaving the rotational speed of the pump constant, the valve was incrementally closed, with an appropriate time for each pressure ratio to reach static conditions before the measurements were recorded. To exclude a hysteresis effect, one measurement curve was carried out vice versa. 4. Results
Fig. 4. Measured, non-dimensional volumetric flow rate of the MID as a function of the pressure ratio p2 /P01; close-up of choked flow inception.
As a first step, experimental and numerical results regarding the flow rate for different pressure ratios will be presented. These flow rate investigations will be supplemented by investigations of the evolution of the vapor cavity and the loss coefficients. The combined gained knowledge of these investigations is used to explain a certain new phenomena which is observed and described for the first time. Additionally, a new method for the calculation of the choked mass flow rate will be presented with the advantage of a detailed explanation of what the introduced correction factor actually corrects. The second part of the paper deals with the analysis of the performed high speed camera investigations, enhancing and confirming several already published investigations. Furthermore, new explanations are given for the reason of the high steadiness of the choked flow rate, revealing the minor importance of re-entrant jet phenomena for the investigated Herschel-tube.
pressure ratio instead of a further decrease, while the choked condition sets in. The fact that not all curves showed this behavior could be due to a too coarse resolution of the measured pressure ratios for some of the curves. Maybe this jump only occurs at a very specific pressure ratio that was just not measured or, especially taking a look at the curves exhibiting this feature, it is somehow unique to flow rates around 8 m3/ h or rather to the corresponding total pressure P01. The total pressure at the inlet, P01, for the experiments and the numerical case is listed in Table 3. The values always represent the choked condition. The dashed lines mark the inlet total pressure range for which the “jump” was observed. Whereby special emphasis is made for the word observed, as it was already denoted that some measurement curves probably had a too coarse resolution of the measured pressure differences to exhibit the actual “jump”. For the purpose of explaining the sudden jump, as well as the whole transition from cavitation inception to the choked flow condition, Fig. 5
4.1. Flow rate In Fig. 3, the measured volumetric flow rates of the MID were plotted against the pressure ratio p2 / P01. Qmax denotes the maximum choked flow rate, p2 the static pressure behind the Herschel-tube and P01 the total pressure at the inlet of the Herschel-tube. For all measured curves, the choked condition sets in at a pressure ratio p2 / P01 ≈ 0.875 − 0.885, characterized by the flow rate remaining constant even for further decreases of the pressure ratio. For pressure ratios above 0.88 the flow rates tended to zero. In Fig. 4, the volumetric flow rates were presented in a dimensionless form, obtained by dividing each curve by its maximum flow rate. Thus, the curves fell together in the choked region but they still deviated in the non-choked region. For pressure ratios above 0.88 there was a clear tendency of higher flow rates with increasing Qmax. Within the close-up view of Fig. 4, a certain behavior of some curves became visible. The curves Qmax = 7.99, 8 and 8.01 m3/h exhibited a sudden “jump” or discontinuity, meaning an unexpected increase of the
Table 3 Total pressure P01 at the inlet of the Herschel-tube, for the experiments and the simulation for the choked condition.
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Fig. 7 denoted the same pressure ratios as in Figs. 5 and 6. All three cases showed a similar behavior regarding the cavity length as well as the sudden expansion of the cavity. That was especially interesting recalling the fact that no adjustment of the cavitation model constants had taken place. A physical reason for the sudden expansion of the vapor cavity was found by analyzing the numerically predicted, instantaneous pressure fields of the pressure ratios 3 and 4. Pressure contours of the two pressure ratios were opposed in Fig. 8, each on a 2D section going through the center line of the Herschel-tube. The upper half represented the numerically investigated pressure ratio 3 before the sudden expansion and the lower half the pressure ratio 4 just after the sudden expansion. For clarification, ⊕ and ⊖ denoted regions of higher and lower pressure along the wall. A relative unstable, alternating distribution with the lowest pressure inside the vapor cavity, a pressure rise behind it due to the reentrant jet, again a small region of lower pressure at the intersection of the throat and the diffuser and finally the steady pressure rise within the diffuser could be seen before the sudden expansion took place. In contrast, the pressure distribution after the sudden expansion, represented in the lower half of Fig. 8, resembled a simplified, more stable situation. After the sudden expansion, there was only the low pressure region inside the cavity and the pressure rise in the diffuser section. Hence, the sudden expansion was a result of an complex pressure field stabilizing itself. This was also confirmed by taking a look at the pressure distribution along the wall of the Herschel-tube for the two pressure ratios 3 and 4. In Fig. 9, the abscissa denoted the axial position along the center line, with 0 m marking the beginning of the cylindrical throat section and 0.0112 m its end. For pressure ratio 3 ( p2 / P01 = 0.8833), the zigzag distribution within the throat section was obvious, while the distribution after the sudden expansion was constant along the throat wall with a single sharp increase behind the vapor cavity at the beginning of the diffuser section. This physical explanation of the sudden extension of the vapor cavity gives a hint on whether or not this phenomenon is present in all measured curves. As the evolving flow field should always be similar, at least if flashing is neglected, the sudden extension should always take place, at least in this Herschel-tube geometry. Yet, the question if the sudden extension always leads to the “jump” seen in Fig. 4 remains unclear without further experimental investigations. On the other hand, the simplification of the alternating pressure field should lead to a decrease of the pressure loss, thus the lack of this phenomenon in most of the measurement curves could be an experimental issue. For measurement purposes it would be of advantage to be able to calculate the maximum flow rate in the choked condition. As explained by the authors in detail in [22], the constant mass flow rate within the choked condition could be calculated by the introduction of a correction factor Ccav into the continuity equation. Recently, Kim [23] had proposed a similar approach for flashing flows.
Fig. 5. Measured, non-dimensional volumetric flow rate for the measured curve Qmax = 7.99 m3/h and the simulated curve Qmax = 8.40 m3/h .
presented the measured curve Qmax = 7.99 m3/h and a curve with Qmax = 8.40 m3/h , obtained by numerical simulations using CCM+. First of all, the numerically obtained results were in quite good agreement with the experimental values, especially considering the fact that the numerical cavitation model was just based on the CCM+ standard values. This could also be one reason for the slightly larger deviation of the simulated pressure ratio denoted as point 2. One has to keep in mind that at this pressure ratio, as the vapor cavity is still developing within the throat, even small deviations of the numerically predicted cavity length will have a large influence on the flow within the throat and thus the flow rate. In Fig. 5 two significant stages of the cavitation were marked, the cavitation inception as well as the point where the vapor cavity had grown to the end of the cylindrical throat section, starting to extend into the diffuser. The numerically predicted and experimentally measured pressure ratio for the cavitation inception corresponded very well. Interesting was the fact, that for both, simulation and experiment the choked flow condition did not correlate with the cavitation inception but rather with the vapor cavity reaching the end of the cylindrical throat section. This was also illustrated by the pictures shown in Fig. 6, that corresponded to the numbered pressure ratios in Fig. 5. The pictures on the left side were captured by a Samsung NX300M and, due to the round cross section of the Herschel-tube, represent a plan view of the forward 180°-section of the cylindrical liquid-vapor interface. The figures on the right side represented the instantaneous volume fraction of vapor on a 2D plane through the center line with the image section showing only a part of the throat section. The pressure ratios numbered as 1 denoted in Fig. 5 the cavitation inception and were shown in Fig. 6a and b. Fig. 6a marked the visible onset of cavitation, with the first small cavitation bubbles indicated by the arrows. The picture on the left was taken using a shutter speed of 1/180 s. For both, experiment and simulation, the cavitation set in close to the beginning of the cylindrical throat section. Across points 2 and 3 (Fig. 6c–f), the vapor cavity extended steadily to about 1/3 of the throat section, with the pictures on the left side now using a shutter speed of 3 s, thus representing time-averaged vaporcavities. One should draw attention to the fact that point 3 denoted, for both experiments and simulation, the pressure ratio just before the choked flow condition sets in. Equally, the pressure ratio numbered as 4 denoted the first one in the choked condition. According to Fig. 6g and h, between pressure ratio 3 and 4 the vapor cavities had suddenly extended from 1/3 of the throat section until its end and even into the diffuser section. This was further illustrated in Fig. 7, where the maximum length of the vapor cavity Lcav was plotted as a function of the pressure ratio for three different Qmax. In the two experimental cases, Lcav was obtained by analyzing pictures with an exposure time of at least 3 s. For the numerical simulations, the maximum length was obtained by analyzing the vapor volume fraction along the wall. The numbering in
m˙ = ρl Vth,∞ Ath Ccav
(4)
With ρl being the liquid density, Ath the cross section of the throat and Vth,∞ the average, inviscid throat velocity calculated by the Bernoulli eq. with the assumption of the throat pressure being equal to the vapor pressure:
Vth,∞ =
2(P01 − pv ) ρl
(5) ⋆ Ccav
= 0.8693, obtained arbitrarily for the A correction factor of measurement curve Qmax = 8.01 m3/h , in combination with a temperature correlation was able to predict all measured maximum mass flow rates within a 0.35% deviation. ⋆ Ccav = Ccav ·
59
ρl ρl, ref
(6)
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(a) Point 1
(b) Point 1
(c) Point 2
(d) Point 2
(e) Point 3
(f) Point 3
(g) Point 4 Fig. 6. Vapor cavity for the measured curve Qmax = 7.99
(h) Point 4 m3/h
(left) and the simulated curve Qmax = 8.40
The calculation via the inviscid throat velocity, as well as the introduction of an correction factor is of course arbitrary and thus scientifically not very satisfying. But, the following investigation of the measured and simulated loss coefficients provided new and interesting information about what the correction factor actually corrects.
ζ=
m3/h
(right) for the numbered points marked in Fig. 5.
2( p1 − p2 ) ρl Vth2
(7)
The throat velocity Vth was calculated for the choked condition by Eq. (5) and in the non-choked condition obtained by the measured volumetric flow rate. In Fig. 10, the loss coefficient for all measured curves was plotted as a function of the pressure ratio. For pressure ratios smaller 0.89, which means for the choked condition, all curves fell together on one single line. The sudden decrease of the loss coefficient at the beginning of the choked condition is caused by the calculation now being based on the inviscid Bernoulli approach, but only partially as will be shown. The
4.2. Loss coefficient Based on the investigations of Rudolf et al. [8], experimental and numerical results regarding the loss coefficient will be presented. For the loss coefficient, the following definition will be used in accordance with Rudolf et al.: 60
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Fig. 11. Loss coefficient ζ as a function of different definitions of the cavitation number σ for the measurement curve Qmax = 8.01 m3/h and the simulated case. Fig. 7. Maximum length of the vapor cavity Lcav in terms of the throat diameter dth as a function of the pressure ratio for two measured and one simulated curve.
Fig. 12. Loss coefficient ζ as a function of different definitions of the cavitation number σ for the measurement curve Qmax = 8.01 m3/h and the simulated case, additionally including the a velocity correction.
curves again deviated slightly as the flow rates did in Fig. 4. The slight increase of the loss coefficient just before the sudden expansion takes place at pressure ratios of about 0.89, was consistent with the results of Rudolf et al.. The increase of the loss coefficient could be a reason of the complex pressure field shown in Fig. 8, while the drop of the loss coefficient could be due to the simplification of the flow field after the sudden expansion. Instead of the pressure ratio, the loss coefficient can also be plotted against the cavitation number σ. For this purpose, three different definitions have been used, to determine their applicability:
Fig. 8. Pressure contours for the simulated pressure ratios ③ ( p2 /P01 = 0.8833) [upper half] and ④ ( p2 /P01 = 0.8767) [lower half] with Qmax = 8.40 m3/h ; ⊕ and ⊖ denote regions of higher and lower pressure.
σp =
σth1 =
Fig. 9. Pressure along the wall for the simulated pressure ratios ③ ( p2 /P01 = 0.8833) and
p2 − pv p1 − p2
(8)
2( p1 − pv ) ρl Vth2
(9)
④ ( p2 /P01 = 0.8767 ) with Qmax = 8.40 m3/h .
σth2 =
2( p2 − pv ) ρl Vth2
(10)
Where σp was a cavitation number based solely on pressure differences [24] and σth1,2 were either based on the inlet or the outlet pressure as the reference pressure. Vth was calculated in the same way as describes for the loss coefficient. The loss coefficient was plotted against the three different definitions of σ in Fig. 11 for the measurement curve Qmax = 8.01 m3/h . Additionally, the results obtained by the numerical simulations for Qmax = 8.40 m3/h were plotted, denoted by the dashed lines. In the non-choked and non-cavitating case, the agreement between both curves was quite good. Strikingly, even in the numerical case the distinct behavior of a small increase of the loss coefficient shortly before the choked condition sets in, followed by a decrease of the same amplitude was predicted. For the choked condition, a constant vertical shift of about 8.7% between the numerically predicted and experimentally calculated ζ values was visible for the σp curve. As the numerically predicted loss coefficient was always based on the viscous throat velocity, the only difference between the σp curves for the choked condition was the used velocity for the calculation of the loss coefficient. It was either inviscid for the experimentally estimated values or viscous for the numerically
Fig. 10. Loss coefficient ζ as a function of the pressure ratio p2 /P01.
main reason for the decrease of the loss coefficient, while choking sets in, is the simplification of the pressure field already described in Section 4.1. Thus, Fig. 10 is the evidence that all measured curves indeed show the simplification of the pressure field and with that the sudden jump of the vapor cavity. For the non-choked condition, the loss coefficients of the different 61
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Fig. 13. High-Speed Photographs taken by the FASTCAM system at 50.000 fps, covering a time of Tref = 13 ms ; σp = 7.21; Qmax = 9.09 m3/h ; Video link.
numerically predicted throat velocity and the inviscid throat velocity, calculated via Eq. (5). Increasing the throat velocity in the choked condition for the experimental values, calculated via Eq. (5), by 4.26% for both the calculation of ζ and σth2 resulted in a much better agreement between the experimental and numerical curves. This was displayed in Fig. 12, where two additional curves based on the velocity correction were indicated. Of course, one has to remember, that Eq. (5) itself was based on imperfect assumptions such as using the vapor pressure as the respective throat pressure and using the density of the liquid as the respective throat density. Another advantage of the velocity deviation estimation was the possibility to apply it to the correction factor Ccav for the calculation of the choked mass flow rate, Eq. (4). As Ccav corrects the product ρl Ath Vth,∞ by 13.07% and Vth,∞ itself was defective by 4.26% the remaining 8.8% had to account for the correction of ρl Ath . This means, that the influence of choosing the liquid density as the reference density
Fig. 14. Cloud cavitation scheme by Stanley et al. [26]; with A denoting the fixed vapor cavity, B separated and convected bubbles and C the re-entrant jet.
predicted values. Thus, with the 8.7% shift, an estimation of the deviation caused by the usage of the inviscid calculation via Eq. (5) could be realized:
ζexp =
2( p1 − p2 ) 2 ρl Vth ,∞
·1. 087 ≈
Vth2 , sim·1. 087 ≈ Vth2 ,∞ Vth, sim·1. 0426 ≈ Vth,∞
2( p1 − p2 ) 2 ρl Vth , sim
= ζsim
(11)
The above estimation showed a 4.26% deviation between the viscous, 62
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of the sheet cavity, a series of pictures taken by the FASTCAM SA-Z 2100 K system will be shown. The pictures of Fig. 13 were taken at 50.000 fps for Qmax = 9.09 m3/h . For choosing a proper time range to display the global steadiness of the sheet cavity, the experimental results of a convergent-divergent section by Stutz and Reboud [25] were considered. For Qmax = 9.09 m3/h , the resulting inlet velocity Vin was 4.905 m/s and analyzing the pictures resulted in the maximum length of the vapor cavity Lcav being equal to 17.55 mm. According to the measurements of Stutz and Reboud, the ratio of Vin / Lcav should result in a periodic frequency of cloud shedding of about 75 Hz or a period of about 13 ms. In that sense, the following pictures of Fig. 13 covered a period of Tref = 13 ms. For a better visual comparison of the sheet cavity length, a thin dashed line was inserted into all pictures of Fig. 13 at the exact same position. Fig. 13a – d represent the first 0.1 ms of the whole 13 ms period. Regarding the global behavior of the sheet cavity, there have only been minor movements, while at the same time the cavity closure region showed a highly complex three-dimensional behavior. The further development of the sheet cavity during the 13 ms period, displayed in Fig. 13e–i, still showed a very steady global behavior. Due to the larger time increments, there was more movement visible within the cavity closure region, while the overall length was still very constant. This can also be checked in the Video, linked in the description of Fig. 13, which covers an even larger time span.1 As can be seen in Fig. 13, the sheet cavity clearly had two liquidvapor interfaces. Obviously, an inner interface between the liquid core flow and the vapor as well as an outer interface between the vapor cavity and a liquid film that existed between the cavity and the nozzle wall. Otherwise, the light reflections and the clearly visible deformations of that outer interface would not be visible. These results support the recently published cloud cavitation scheme by Stanley et al. [26] which is schematically shown in Fig. 14. According to it, between most of the wall and the cavitation cloud a thin liquid film exists, while the cavity itself is only fixed to the wall at its inception point. Considering the small height of the liquid film, the extremely sharp liquid-vapor interface and the fact that most CFD cavitation studies use Y+ values above 30, it is hardly surprising that this feature hasn't been captured numerically via the common VOF-method so far, at least to the authors knowledge. For the determination of the actual velocity inside the vapor cavity, pictures taken at 120.000 fps were analyzed. As an example, the three pictures in Fig. 15 represent such an analysis. Due to the increase of the fps, the image section decreased, with the left side now barely starting at the inception point and the right side almost covering the closure region. For the estimation of the velocity inside the vapor cloud, clearly identifiable structures were traced. Such structures were denoted in Fig. 15a with ①-④. However, only structure ① actually moved in main flow direction during the presented sequence. Structure ② and ③ represented wave like deformations on the outer interface, while structure ④ seemed to be a void in the bubbly structure at the closure region. This three structures were apparently closer to or on the outer interface (closer to the nozzle wall) while structure ① was passing behind those foreground structures, thus had to be either inside the cavity or on the inner interface. This means there had to be a wide difference in the velocities across the vapor cloud in radial direction, which is consistent with experimental and numerical results of different authors [25,27,28]. On the other hand, no experimental information was existing on the axial velocity distribution of certain structures within the vapor cavity. Fig. 16 represented the final results of this investigation, obtained by the analysis of structures within the cavity. For the analysis, the throat section was divided into three equal parts, as denoted in the
Fig. 15. High-Speed Photographs taken by the FASTCAM system at 120.000 fps; σp = 7.21; Qmax = 9.09 m3/h .
Fig. 16. Analyzed velocities within the vapor cavity based on the throat velocity Vth,∞ for the three different throat sections l /lth ; Vth,∞ = 30 m/s , σp = 7.21, Reth=343126 .
has an higher impact on the accuracy of the results than the difference in inviscid or viscous throat velocity. The inviscid velocity clearly overestimates the actual throat velocity, if it would remain liquid. Due to the cavitation though, the volumetric flow rate within the throat increases, leading to a higher than all-liquid-flow velocity.
4.3. High speed camera investigations 1
For the purpose of showing the high stability and global steadiness 63
https://youtu.be/lvVdTAMbtH4
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Fig. 17. High-Speed Photographs taken by the FASTCAM system at 120.000 fps; σp = 7.21; Qmax = 9.09 m3/h .
vapor cavity was shown. The front of the bubbly cloud was denoted in Fig. 17a with the white line at the closure region. Additionally, a prominent wavy region on the outer liquid-vapor interface in the middle of the throat was encircled. Each picture was separated by a time span of 0.25 ms, so in contrast to the fast downstream movement of the structure in Fig. 15, the now shown upstream movement was much slower. From Fig. 17a to c, the bubbly cloud slowly moved upstream in the direction of the cavitation inception point. During the same time, the wavy interface structure somewhat compressed to one single wave that stood more or less still. From Fig. 17c to d, the bubbly cloud front again moved a considerable way upfront, while the interface wave still barely moved. With increasing time, the bubbly cloud front continued to move in upstream direction but the whole progress slowed down (Fig. 17e- f). The same was valid for the wave structure, it continued to move in upstream direction to about the first quarter of the throat section (Fig. 17f) until it vanished. In Table 4, the evaluated velocities between each picture of Fig. 17 were listed. The highest velocity was estimated to 11.4 m/s, so about 1/ 3 of the throat velocity of 30 m/s. This gave an explanation for the overall steadiness of the sheet cavity. As Keil et al. [29] reported, the transition from a steady sheet cavitation to an unsteady cloud cavitation with a periodic shedding of vapor cavities depends on the ratio of the re-entrant jet velocity Vjet to the sheet growth velocity Vs. The sheet growth velocity was already estimated in Fig. 16 to about 0.7–0.9 times the throat velocity of 30 m/ s, therefore 21–27 m/s. The peak re-entrant jet velocity was estimated
Table 4 The evaluated velocities of the bubbly cloud front of Fig. 17. Figure
Bubble Cloud Front Velocity
17 a → b 17 b → c 17 c → d 17 d → e 17 e → f
6.4 m/s 11.2 m/s 10.4 m/s 6.4 m/s 3.2 m/s
lower right side of Fig. 16. Within each section, the velocities of ten clearly identifiable structures were determined. The average velocity, as well as the highest and lowest velocity indicated by the error bars, were then plotted based on the inviscid throat velocity, calculated via Eq. (5). The determined velocities showed an obvious increase with increasing throat length. The high scattering of the determined velocities in the first section was probably due to the small thickness of the sheet cavity in that region. While the velocities in the middle section were quite similar, the last section again showed a higher scattering, probably due to the strong coupling with the closure region directly following. The results were in agreement with similar investigations presented by Stanley et al. [26], likewise having velocities in the order of the throat velocity. However, Stanley et al. made no distinction between the axial location inside the throat of their tracked velocities, which explains the high scattering of their determined velocities. In Fig. 17, the event of a bubbly cloud moving upstream inside the 64
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Fig. 18. High-Speed Photographs taken by the FASTCAM system at 140.000 fps; σp = 7.21; Qmax = 9.09 m3/h ; Video link.
such as the far wavier interface structure at the beginning of the cavity, denoted in Fig. 18a by ①. Additionally, in the bracketed area of Fig. 18a the vapor cloud will collapse, possibly due to a re-entrant jet. In Fig. 18b, the collapse had slowly set in, with the cavity closure region within the bracket starting to move upstream. Furthermore, a tornadolike vortex structure appeared in the upper closure region (②) and was drawing surrounding bubbles into its core. The vortex structure persists throughout the whole shown time interval (Fig. 18 b → g = 32 ms) until it finally decayed in Fig. 18g. In Fig. 18c, the upstream moving closure region could not be clearly distinguished from the background vapor structure of the rear
in Table 4 to 11.2 m/s. Thus the ratio Vjet / Vs ≈ 0.53 − 0.41 gave the explanation for the steadiness of the sheet cavity for this investigated operating point.
steady sheet regime = transition =
Vjet Vs
Vjet Vs
<1
= 1 ↔ Re=Recrit
unsteady cloud regime =
Vjet Vs
>1
(12)
Fig. 18 represented a compilation of pictures taken at 140.000 fps. The presented compilation featured several interesting phenomena, 65
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Fig. 19. High-Speed Photographs taken by the FASTCAM system at 140.000 fps; σp = 7.21; Qmax = 9.09 m3/h ; Video link.
narrow region and not simultaneously across the whole cavity closure region. This was elucidated by the cavity closure region in the upper and background region still being in place. Due to this, the upstream movement of the re-entrant jet region had to be additionally slowed down by the obviously still downstream flowing neighboring flow. The shown compilation was another explanation for the global steadiness of the vapor cavity. Unsteady re-entrant jet phenomena may occur in the investigated case, but, as the analysis of the video material confirmed, they are neither periodic, nor a global phenomenon but rather a local one with minor influence. Nevertheless, they demonstrate the high three-dimensional, unsteady behavior in the closure region of
180°-section of the cylindrical vapor cavity. Instead, another structure was visible in the encircled region denoted with ③. Throughout the remaining compilation, this structure, although right in front of the reentrant jet region, was not moving. In Fig. 18d, 2/140 ms after Fig. 18c, a sharp interface of the pushed back cavity closure region could be seen, denoted by the white bracket. This sharp interface represented a so-called condensation shock wave [30] that basically stood still for the remaining 10 ms of the shown compilation. However, the most interesting observation of Fig. 18 was, that the whole cavity collapse phenomenon took place in a circumferential very 66
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venturidüse, Forsch. Ing. -Wes. 30 (1964) 86–93. [5] F. Numachi, R. Kobayashi, Verbesserung der venturidüse hinsichtlich der kavitationsbeeinflussung, Forsch. Ing. -Wes. 31 (1964) 60–65. [6] H. Ghassemi, H.F. Fasih, Application of small size cavitating venturi as flow controller and flow meter, Flow Meas. Instrum. (2011) 406–412. [7] A. Abdulaziz, Performance and image analysis of a cavitating process in a small type venturi, Exp. Therm. Fluid Sci. (2014) 40–48. [8] P. Rudolf, M. Hudec, M. Gríger, D. Štefan, Characterization of the cavitating flow in converging-diverging nozzle based on experimental investigations, EPJ Web of Conferences 67, 2014. [9] S. Ashrafizadeh, H. Ghassemi, Experimental and numerical investigation on the performance of small-sized cavitating venturis, Flow Meas. Instrum. 42 (2015) 6–15. [10] A.J. Schmidt, Quantitative measurement and flow visualization of water cavitation in a converging-diverging nozzle, Master thesis, Kansas State University, USA, 2016. [11] S. Brinkhorst, E. von Lavante, G. Wendt, Numerical investigation of cavitating herschel venturi-tubes applied to liquid flow metering, Flow Meas. Instrum. 43 (2015) 23–33. [12] S. Brinkhorst, E. von Lavante, G. Wendt, Numerical investigation of effects of geometry on cavitation in herschel venturi-tubes applied to liquid flow metering, in: International Symp. on Fluid Flow Measurement. [13] L. Rayleigh, On the pressure developed in a liquid during the collapse of a spherical cavity, Philos. Mag. 34 (1917) 94–98. [14] A. Oprea, N. Bulten, Cavitation modelling using RANS approach, WIMRC 3rd Int. Cavitation Forum 2011 (2011). [15] X. Margot, S. Hoyas, A. Gil, S. Patouna, Numerical modelling of cavitation: validation and parametric studies, Eng. Appl. Comput. Fluid Mech. 6 (2012) 15–24. [16] E.M. Bennett, Cavitation cfd using star-ccm+ of an Axial Flow Pump with Comparison to Experimental Data, 2014. [17] G. Vaz, D. Hally, T. Huuva, N. Bulten, P. Muller, P. Becchi, J.L.R. Herrer, S. Whitworth, R. Macé, A. Korsström, Cavitating flow calculations for the e779a propeller in open water and behind conditions: Code comparison and solution validation, in: 4th International Symp. on Marine Propulsors. [18] CD-adapco, User Guide, 10.04 edition, 2015. [19] X. Zhang, J. Zhu, L. Qiu, X. Zhang, Calculation and verification of dynamical cavitation model for quasi-steady cavitating flow, Int. J. Heat Mass Transf. 86 (2015) 294–301. [20] J. Kozák, P. Rudolf, D. Štefan, M. Hudec, M. Gríger, Analysis of pressure pulsations of cavitating flow in converging-diverging nozzle, in: Proceedings of the 6th IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems. [21] M. Morgut, E. Nobile, I. Biluš, Comparison of mass transfer models for the numerical prediction of sheet cavitation around a hydrofoil, Int. J. Multiph. Flow. 37 (2011) 620–626. [22] S. Brinkhorst, E. von Lavante, G. Wendt, Experimental investigation of cavitating herschel venturi-tube configuration, in: Proceedings of the 17th International Flow Measurement Conference. [23] Y.-S. Kim, A proposed correlation for critical flow rate of water flow, Nucl. Eng. Technol. 47 (2015) 135–138. [24] W.H. Nurick, Orifice cavitation and its effect on spray mixing, J. Fluid Eng. 98 (1976) 681–687. [25] B. Stutz, J. Reboud, Experiments on unsteady cavitation, Exp. Fluids (1997) 191–198. [26] C. Stanley, T. Barber, G. Rosengarten, Re-entrant jet mechanism for periodic cavitation shedding in a cylindrical orifice, Int. J. Heat Fluid Flow. 50 (2014) 169–176. [27] E. Goncalvès, Numerical study of unsteady turbulent cavitating flows, Eur. J. Mech. - B/Fluids 30 (2011) 26–40. [28] J. Decaix, E. Goncalvès, Investigation of three-dimensional effects on a cavitating venturi flow, Int. J. Heat Fluid Flow. (2013) 576–595. [29] T. Keil, P.F. Pelz, J. Buttenbender, On the transition from sheet to cloud cavitation, in: Proceedings 8th International Symp. on Cavitation, Singapore. [30] P. Tomov, K. Croci, S. Khelladi, F. Ravelet, A. Danlos, F. Bakir, C. Sarraf, Experimental and numerical investigation of two physical mechanisms influencing the cloud cavitation shedding dynamics, 2016. Working paper or preprint.
the cavity. In Fig. 19, the further development of the above described cloud collapse and condensation shock wave of Fig. 18 was displayed. Starting from Fig. 19a, the encircled, pushed back closure region began to develop to its original location again and extended in normal flow direction. While the condensation shock wave was in Fig. 18g a single sharp interface, positioned in the lower region of the picture, it moved a little bit into the center, implying a radial velocity component. Furthermore, the single sharp interface split into two smaller condensation shock waves, denoted in Fig. 19c by the two white lines. The wave front extended to its original position in Fig. 19g and the whole cloud collapse event was completed. To the authors knowledge, the above described circumferential narrow cavity collapse has never been mentioned in literature. This could be because of two reasons: most experiments are either run on 2dimensional, planar nozzle geometries, or use nozzle geometries with larger diffuser angles which are favorable for flow separation, which in turn amplifies the re-entrant jet strength [12]. 5. Conclusion In the presented work, a combined experimental and numerical approach for the investigation of the choked flow condition in cavitating Herschel-tubes was presented. The combined approach gave further insight into the onset of the choked condition, where a sudden expansion of the vapor cavity was revealed by the experiments and could be explained by the numerical simulations. Due to a complex pressure field, the vapor cavity will extend steadily from the beginning of the throat to about 1/3 of the investigated throat length. At this point, an alternating pressure field has developed insight the nozzle throat that collapses and by doing so enforces a sudden extension of the vapor cavity into the beginning of the diffuser section. Furthermore, explanations for the steadiness of the sheet cavity could be presented by the usage of a high speed camera with up to 140.000 fps. While the high speed investigations revealed the high unsteadiness especially within the cavity closure region, they also revealed the minor effect of certain re-entrant jet occurrences at least for the investigated case. The high speed investigations revealed not a single re-entrant jet phenomena that emerged evenly over the entire circumference. The extremely three-dimensional flow behavior in the cavity closure region probably prohibits this, which in turn leads to local, circumferential narrow re-entrant jet phenomena, with less influence on the whole vapor cavity. References [1] J. Ackeret, Experimentelle und theoretische untersuchungen über hohlraumbildung (kavitation) im wasser, Tech. Mech. und Thermodyn. 1 (1930). [2] F. Numachi, M. Yamabe, R. Oba, Cavitation effects on the discharge coeffcient of the sharp-edged orifice plate, J. Basic Eng. 82 (1960) 1–6. [3] F. Numachi, R. Kobayashi, S. Kamiyama, Effect of cavitation on the accuracy of herschel-type venturi tubes, J. Basic Eng. 84 (1962) 351–360. [4] F. Numachi, R. Kobayashi, Einfluß der kavitation auf die durchfluß zahl der
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Experimental and numerical investigation of the cavitation-induced choked flow in a herschel venturi-tube
MARK
⁎
S. Brinkhorsta, , E. von Lavantea, G. Wendtb a b
Chair of Fluidmechanics, University of Duisburg-Essen, 47057 Duisburg, Germany Head of Department “Liquid Flow”, Physikalisch-Technische Bundesanstalt PTB, 38116 Braunschweig, Germany
A R T I C L E I N F O
A BS T RAC T
Keywords: Cavitation Choked flow Herschel venturi-tube Cavitating nozzle Liquid flow metering
Due to their simple geometry, cavitating Venturi nozzles (CV) are a long time subject of experimental as well as numerical investigations. However, research mostly focused on certain aspects like the comparison of experimental data with numerical cavitation models or the spray development of diesel injection nozzles, but rarely on the choked flow condition itself, especially with regard to liquid flow measurement. If the pressure decreases due to the local acceleration of the flow to the respective vapor pressure, a choked flow condition similar to the well-known critical flow Venturi-nozzles (CFVN) develops. For the purpose of gaining further insight into the choked flow condition with respect to liquid flow measurement, high-speed camera investigations of a transparent Herschel Venturi-tube configuration, also known as classical Venturitube, were performed. Together with pressure and flow rate measurements, they demonstrated the overall stable flow behavior under choked conditions. With additional numerical investigations, phenomena during the onset of the choked condition were clarified. Furthermore, a simple correlation for the calculation of the actual flow rate during the choked condition, including a temperature correction was proposed.
1. Introduction As a practical realization of the so called traceability requires, hydraulically cavitating Herschel Venturi-tubes were investigated for their highly stable flow rates under choked flow conditions. In addition to accuracy, the traceability of a given measuring device is of essential importance, in particular with respect to the general guarantee of uniformity and comparability of measurements. In gas flow measurement, the so-called critical flow Venturi-nozzles (CFVN) have been widely established as an efficient transfer standard during the last two decades. Comparable solutions for liquid flow measurement did not exist to this day. The analogies observed in experiments with Venturinozzles using gas or liquids as working fluids, already observed in 1930 by Ackeret, led the Physikalisch-Technische Bundesanstalt (PTB) to start detailed investigations of cavitating Venturi-nozzle (CV) flows. The intention was to clarify their applicability in liquid flow measurement comparable with the use of critical nozzles for gas metering. Though Ackeret has already mentioned the experimentally observed analogies between CVs and CFVN in 1930, his work was not focused on flow rate investigations [1]. Numachi et al. investigated different Venturi-tubes, nozzles and orifices under cavitating conditions [2–5]. However, the differential-pressure flow rate measuring
⁎
devices were investigated to obviate incorrect measurements due to effects of possible cavitation at certain small pressure ratios. Nevertheless, the phenomenon of an enforced constant mass flow rate through the nozzles, while cavitation in the nozzle throat occurred, was investigated and described. Still, a well-directed exploitation of this effect for flow rate metering applications has not been considered then. Recently, some experimental work has been published focusing on the use of CVs as liquid flow meters, with special emphasis on the choked flow condition. Ghassemi et al. showed the general possibility of the application of CVs as liquid flow meters [6], while Abdulaziz gained further insights into the cavitation process by using an optical flow visualization method [7]. Rudolf et al. characterized experimentally the cavitation regimes of a Herschel Venturi-tube by evaluating the loss coefficient, thereby revealing a unique behavior during the choked condition, as well as during its onset [8]. Ashrafizadeh et al. performed numerical and experimental investigations of different Venturi-tube geometries designed as flow-control devices [9]. Recent experimental investigations by Schmidt using CVs confirmed the high stability of the flow rate during the choked condition [10]. The investigations mentioned so far were supplemented by the already published numerical investigations of CVs by the authors [11,12]. These investigations were mainly focused on geometry influ-
Corresponding author. E-mail address: [email protected] (S. Brinkhorst).
http://dx.doi.org/10.1016/j.flowmeasinst.2016.12.006 Received 13 July 2016; Received in revised form 15 November 2016; Accepted 12 December 2016 Available online 16 December 2016 0955-5986/ © 2016 Elsevier Ltd. All rights reserved.
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to the Sauer cavitation model the terms modeling the rate of evaporation and condensation were:
Table 1 Properties defined for the numerical simulations, with μl and μv representing the liquid and vapor viscosity respectively.
m˙ +, m˙ − = C
property
value
pressure, inlet (P01) vapor pressure αv, inlet T ρl ρv μl μv number density of seeds initial seed radius turbulent intensity, inlet Cprod , Cdest
3.764 bar (total pressure) 2333.21 Pa 0 298 K 998.34 kg/m3 0.0173 kg/m3 1.001477 mPa s 0.009727 mPa s 1e12 /m3 0.001 mm 1% 1
2 pv − p 3 ρl
(2)
With C = Cprod for evaporation and C = Cdest for condensation and the growth/collapse of a bubble being calculated by the simplified Rayleigh-Plesset equation. The vapor pressure pv is the saturation pressure corresponding to the liquid temperature, p denotes the pressure surrounding the bubble, ρl denotes the liquid density and ρm denotes the density of the vapor-liquid mixture, calculated by:
ρm = αv ρv + (1 − αv ) ρl
(3)
Due to the use of the simplified Rayleigh-Plesset equation, viscous and surface tension effects on the rate of change of the bubble radius were neglected. Turbulence was modeled by the realizable k-ϵ two-layer model, including an all-y+ wall treatment [18]. The chosen turbulence model has already been proven to predict satisfying results for cavitating flow [9,19,20]. Nevertheless, it should be mentioned that no preferred turbulence model, as well as cavitation model, exists in literature. This led to a plethora of cavitation-/turbulence-model combinations. Though the choice of either model usually has a large impact on the numerical results, at least globally matching results between each model can be obtained by carefully adjusting the empirical constants [21]. For time and space discretization, second order accuracy was used. All numerical results presented were obtained with full 3D simulations. For a proper time wise resolution a time step of 10−4 s was chosen. The numerical mesh was modeled based on a grid sensitivity analysis presented in [11] and was created using the CCM+ grid generator. With a grid refinement in the cavitating zone the resulting mesh consisted of about 1 million cells. For all simulations the following boundary conditions (b.c.) have been applied: A stagnation pressure inlet b.c. for defining a constant total pressure at the inlet, while a constant static pressure has been defined at the outlet; at the wall the no-slip b.c. was used.
ences on the stability and accuracy of the flow rate, with regard to flow metering applications. They demonstrated the advantages of Herscheltube configurations compared to a typical ISO 9300 CFVN geometry, with respect to the constancy of the flow rate for different pressure ratios within the cavitation induced choked flow condition Table 1. The recently performed experimental and numerical investigations presented in this paper revealed a new phenomenon during the onset of the choked flow condition, namely the sudden expansion of the vapor cloud. Furthermore, due to high-speed camera observations with up to 140.000 fps, reasons for the steadiness and high stability of the vapor cloud and thus the choked flow rate will be presented. For the numerical simulations, the commercial code CD-adapco STAR-CCM+10.04.011 (CCM+) was used, solving the three-dimensional, Reynolds-averaged Navier-Stokes (RANS) equations for turbulent, incompressible, unsteady, cavitating two-phase flow. The experiments were carried out using the experimentation and water meter test rig (EWZP) of the PTB. 2. Numerical cavitation model The commercial CFD program CCM+, used an interface capturing approach via the Volume-of-Fluid-method (VOF-method) to model the cavitating two-phase flow. The cavitation itself was based on the inertia controlled bubble growth theory obtained by Rayleigh [13]. The capability of this solver has already been extensively validated by different authors [14–17]. Due to the VOF-method treating the vapor and liquid phase as a mixture, only one set of the mass and momentum equations is solved, where the physical properties of the mixture or equivalent fluid were functions of the physical properties of its constituent phases and their volume fractions. In addition to the mass and momentum conservation, an equation had to be solved governing the transport of the volume fraction α between the two phases:
→ ∂αv + ∇ ·(αv → u ) = Γv ∂t
ρv ρl 3αv (1 − αv ) ρm Rb
3. Experimental setup For the experimental investigations, the PTB provided their experimentation and water meter test rig (EWZP), with a schematic representation shown in Fig. 1. The volumetric flow rate (Q) was measured by a magnetic flow meter (MID). Furthermore, the differential pressure across the nozzle (ΔP ) as well as the static pressure (P1) and the temperature at the nozzle inlet were measured. The EWZP has an expanded measurement uncertainty of 0.05%. Details of the meters used can be found in Table 2. For the measurements, the stagnation pressure in front of the nozzle could be controlled by the rotational speed of the pump in combination with the valve position. The valve was located about 50 pipe diameter behind the nozzle. For visual inspection of the cavitation, as well as the use of high-speed camera recordings, the investigated Herschel Venturi-tube was manufactured out of acrylic glass. The dimensions of the experimentally and numerically investigated Herschel-tube are presented in Fig. 2. For the visualization of vapor
(1)
The subscript v denotes the vapor phase, while the definition of the volume fraction of vapor is: αv = Vv / V , with Vv being the volume of vapor inside the control volume V. The phase change due to cavitation is governed by the source term Γv on the right side of Eq. (1), whereby Γv is usually split into two terms Γv = m˙ + + m˙ −. m˙ + denotes the rate of evaporation and m˙ − the rate of condensation. In fact, most cavitation models differ only in the formulation of this source term. In CCM+, the common Sauer cavitation model was used for the calculation of the phase change. The Sauer cavitation model is based on the assumptions of having homogeneously distributed seed bubbles with equal radius Rb. These bubbles remain spherical at all times and the number density of seeds n0 together with the initial seed radius Rb,0 was a model parameter to be set. The two phases were treated as incompressible and slip velocity between bubbles and liquid was neglected. Thus, according
Fig. 1. Schematic representation of the EWZP of the PTB.
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Table 2 Specifications of the used meters. Meter
Range
Uncertainty
ADMAG AE Yokogawa AE202MG E+H deltabar S PMD 235 E+H Cerabar PMC71
0.5–17.5 [m3/h] −1000 to 1000 [mbar] 0–40 [bar]
0.5% 0.1% 0.075%
Fig. 3. Measured volumetric flow rate of the MID as a function of the pressure ratio p2 /P01. Fig. 2. Schematic representation of the investigated Herschel Venturi-tube.
cavity patterns, two cameras were used. A Samsung NX300M for longtime exposure and a Photron FASTCAM SA-Z 2100 K for high-speed recordings of up to 140.000 fps. The measurements were started by opening the valve completely and adjusting the rotational speed of the pump. Thus, the measurement started with the highest possible pressure difference (lowest pressure ratio). Leaving the rotational speed of the pump constant, the valve was incrementally closed, with an appropriate time for each pressure ratio to reach static conditions before the measurements were recorded. To exclude a hysteresis effect, one measurement curve was carried out vice versa. 4. Results
Fig. 4. Measured, non-dimensional volumetric flow rate of the MID as a function of the pressure ratio p2 /P01; close-up of choked flow inception.
As a first step, experimental and numerical results regarding the flow rate for different pressure ratios will be presented. These flow rate investigations will be supplemented by investigations of the evolution of the vapor cavity and the loss coefficients. The combined gained knowledge of these investigations is used to explain a certain new phenomena which is observed and described for the first time. Additionally, a new method for the calculation of the choked mass flow rate will be presented with the advantage of a detailed explanation of what the introduced correction factor actually corrects. The second part of the paper deals with the analysis of the performed high speed camera investigations, enhancing and confirming several already published investigations. Furthermore, new explanations are given for the reason of the high steadiness of the choked flow rate, revealing the minor importance of re-entrant jet phenomena for the investigated Herschel-tube.
pressure ratio instead of a further decrease, while the choked condition sets in. The fact that not all curves showed this behavior could be due to a too coarse resolution of the measured pressure ratios for some of the curves. Maybe this jump only occurs at a very specific pressure ratio that was just not measured or, especially taking a look at the curves exhibiting this feature, it is somehow unique to flow rates around 8 m3/ h or rather to the corresponding total pressure P01. The total pressure at the inlet, P01, for the experiments and the numerical case is listed in Table 3. The values always represent the choked condition. The dashed lines mark the inlet total pressure range for which the “jump” was observed. Whereby special emphasis is made for the word observed, as it was already denoted that some measurement curves probably had a too coarse resolution of the measured pressure differences to exhibit the actual “jump”. For the purpose of explaining the sudden jump, as well as the whole transition from cavitation inception to the choked flow condition, Fig. 5
4.1. Flow rate In Fig. 3, the measured volumetric flow rates of the MID were plotted against the pressure ratio p2 / P01. Qmax denotes the maximum choked flow rate, p2 the static pressure behind the Herschel-tube and P01 the total pressure at the inlet of the Herschel-tube. For all measured curves, the choked condition sets in at a pressure ratio p2 / P01 ≈ 0.875 − 0.885, characterized by the flow rate remaining constant even for further decreases of the pressure ratio. For pressure ratios above 0.88 the flow rates tended to zero. In Fig. 4, the volumetric flow rates were presented in a dimensionless form, obtained by dividing each curve by its maximum flow rate. Thus, the curves fell together in the choked region but they still deviated in the non-choked region. For pressure ratios above 0.88 there was a clear tendency of higher flow rates with increasing Qmax. Within the close-up view of Fig. 4, a certain behavior of some curves became visible. The curves Qmax = 7.99, 8 and 8.01 m3/h exhibited a sudden “jump” or discontinuity, meaning an unexpected increase of the
Table 3 Total pressure P01 at the inlet of the Herschel-tube, for the experiments and the simulation for the choked condition.
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Fig. 7 denoted the same pressure ratios as in Figs. 5 and 6. All three cases showed a similar behavior regarding the cavity length as well as the sudden expansion of the cavity. That was especially interesting recalling the fact that no adjustment of the cavitation model constants had taken place. A physical reason for the sudden expansion of the vapor cavity was found by analyzing the numerically predicted, instantaneous pressure fields of the pressure ratios 3 and 4. Pressure contours of the two pressure ratios were opposed in Fig. 8, each on a 2D section going through the center line of the Herschel-tube. The upper half represented the numerically investigated pressure ratio 3 before the sudden expansion and the lower half the pressure ratio 4 just after the sudden expansion. For clarification, ⊕ and ⊖ denoted regions of higher and lower pressure along the wall. A relative unstable, alternating distribution with the lowest pressure inside the vapor cavity, a pressure rise behind it due to the reentrant jet, again a small region of lower pressure at the intersection of the throat and the diffuser and finally the steady pressure rise within the diffuser could be seen before the sudden expansion took place. In contrast, the pressure distribution after the sudden expansion, represented in the lower half of Fig. 8, resembled a simplified, more stable situation. After the sudden expansion, there was only the low pressure region inside the cavity and the pressure rise in the diffuser section. Hence, the sudden expansion was a result of an complex pressure field stabilizing itself. This was also confirmed by taking a look at the pressure distribution along the wall of the Herschel-tube for the two pressure ratios 3 and 4. In Fig. 9, the abscissa denoted the axial position along the center line, with 0 m marking the beginning of the cylindrical throat section and 0.0112 m its end. For pressure ratio 3 ( p2 / P01 = 0.8833), the zigzag distribution within the throat section was obvious, while the distribution after the sudden expansion was constant along the throat wall with a single sharp increase behind the vapor cavity at the beginning of the diffuser section. This physical explanation of the sudden extension of the vapor cavity gives a hint on whether or not this phenomenon is present in all measured curves. As the evolving flow field should always be similar, at least if flashing is neglected, the sudden extension should always take place, at least in this Herschel-tube geometry. Yet, the question if the sudden extension always leads to the “jump” seen in Fig. 4 remains unclear without further experimental investigations. On the other hand, the simplification of the alternating pressure field should lead to a decrease of the pressure loss, thus the lack of this phenomenon in most of the measurement curves could be an experimental issue. For measurement purposes it would be of advantage to be able to calculate the maximum flow rate in the choked condition. As explained by the authors in detail in [22], the constant mass flow rate within the choked condition could be calculated by the introduction of a correction factor Ccav into the continuity equation. Recently, Kim [23] had proposed a similar approach for flashing flows.
Fig. 5. Measured, non-dimensional volumetric flow rate for the measured curve Qmax = 7.99 m3/h and the simulated curve Qmax = 8.40 m3/h .
presented the measured curve Qmax = 7.99 m3/h and a curve with Qmax = 8.40 m3/h , obtained by numerical simulations using CCM+. First of all, the numerically obtained results were in quite good agreement with the experimental values, especially considering the fact that the numerical cavitation model was just based on the CCM+ standard values. This could also be one reason for the slightly larger deviation of the simulated pressure ratio denoted as point 2. One has to keep in mind that at this pressure ratio, as the vapor cavity is still developing within the throat, even small deviations of the numerically predicted cavity length will have a large influence on the flow within the throat and thus the flow rate. In Fig. 5 two significant stages of the cavitation were marked, the cavitation inception as well as the point where the vapor cavity had grown to the end of the cylindrical throat section, starting to extend into the diffuser. The numerically predicted and experimentally measured pressure ratio for the cavitation inception corresponded very well. Interesting was the fact, that for both, simulation and experiment the choked flow condition did not correlate with the cavitation inception but rather with the vapor cavity reaching the end of the cylindrical throat section. This was also illustrated by the pictures shown in Fig. 6, that corresponded to the numbered pressure ratios in Fig. 5. The pictures on the left side were captured by a Samsung NX300M and, due to the round cross section of the Herschel-tube, represent a plan view of the forward 180°-section of the cylindrical liquid-vapor interface. The figures on the right side represented the instantaneous volume fraction of vapor on a 2D plane through the center line with the image section showing only a part of the throat section. The pressure ratios numbered as 1 denoted in Fig. 5 the cavitation inception and were shown in Fig. 6a and b. Fig. 6a marked the visible onset of cavitation, with the first small cavitation bubbles indicated by the arrows. The picture on the left was taken using a shutter speed of 1/180 s. For both, experiment and simulation, the cavitation set in close to the beginning of the cylindrical throat section. Across points 2 and 3 (Fig. 6c–f), the vapor cavity extended steadily to about 1/3 of the throat section, with the pictures on the left side now using a shutter speed of 3 s, thus representing time-averaged vaporcavities. One should draw attention to the fact that point 3 denoted, for both experiments and simulation, the pressure ratio just before the choked flow condition sets in. Equally, the pressure ratio numbered as 4 denoted the first one in the choked condition. According to Fig. 6g and h, between pressure ratio 3 and 4 the vapor cavities had suddenly extended from 1/3 of the throat section until its end and even into the diffuser section. This was further illustrated in Fig. 7, where the maximum length of the vapor cavity Lcav was plotted as a function of the pressure ratio for three different Qmax. In the two experimental cases, Lcav was obtained by analyzing pictures with an exposure time of at least 3 s. For the numerical simulations, the maximum length was obtained by analyzing the vapor volume fraction along the wall. The numbering in
m˙ = ρl Vth,∞ Ath Ccav
(4)
With ρl being the liquid density, Ath the cross section of the throat and Vth,∞ the average, inviscid throat velocity calculated by the Bernoulli eq. with the assumption of the throat pressure being equal to the vapor pressure:
Vth,∞ =
2(P01 − pv ) ρl
(5) ⋆ Ccav
= 0.8693, obtained arbitrarily for the A correction factor of measurement curve Qmax = 8.01 m3/h , in combination with a temperature correlation was able to predict all measured maximum mass flow rates within a 0.35% deviation. ⋆ Ccav = Ccav ·
59
ρl ρl, ref
(6)
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(a) Point 1
(b) Point 1
(c) Point 2
(d) Point 2
(e) Point 3
(f) Point 3
(g) Point 4 Fig. 6. Vapor cavity for the measured curve Qmax = 7.99
(h) Point 4 m3/h
(left) and the simulated curve Qmax = 8.40
The calculation via the inviscid throat velocity, as well as the introduction of an correction factor is of course arbitrary and thus scientifically not very satisfying. But, the following investigation of the measured and simulated loss coefficients provided new and interesting information about what the correction factor actually corrects.
ζ=
m3/h
(right) for the numbered points marked in Fig. 5.
2( p1 − p2 ) ρl Vth2
(7)
The throat velocity Vth was calculated for the choked condition by Eq. (5) and in the non-choked condition obtained by the measured volumetric flow rate. In Fig. 10, the loss coefficient for all measured curves was plotted as a function of the pressure ratio. For pressure ratios smaller 0.89, which means for the choked condition, all curves fell together on one single line. The sudden decrease of the loss coefficient at the beginning of the choked condition is caused by the calculation now being based on the inviscid Bernoulli approach, but only partially as will be shown. The
4.2. Loss coefficient Based on the investigations of Rudolf et al. [8], experimental and numerical results regarding the loss coefficient will be presented. For the loss coefficient, the following definition will be used in accordance with Rudolf et al.: 60
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Fig. 11. Loss coefficient ζ as a function of different definitions of the cavitation number σ for the measurement curve Qmax = 8.01 m3/h and the simulated case. Fig. 7. Maximum length of the vapor cavity Lcav in terms of the throat diameter dth as a function of the pressure ratio for two measured and one simulated curve.
Fig. 12. Loss coefficient ζ as a function of different definitions of the cavitation number σ for the measurement curve Qmax = 8.01 m3/h and the simulated case, additionally including the a velocity correction.
curves again deviated slightly as the flow rates did in Fig. 4. The slight increase of the loss coefficient just before the sudden expansion takes place at pressure ratios of about 0.89, was consistent with the results of Rudolf et al.. The increase of the loss coefficient could be a reason of the complex pressure field shown in Fig. 8, while the drop of the loss coefficient could be due to the simplification of the flow field after the sudden expansion. Instead of the pressure ratio, the loss coefficient can also be plotted against the cavitation number σ. For this purpose, three different definitions have been used, to determine their applicability:
Fig. 8. Pressure contours for the simulated pressure ratios ③ ( p2 /P01 = 0.8833) [upper half] and ④ ( p2 /P01 = 0.8767) [lower half] with Qmax = 8.40 m3/h ; ⊕ and ⊖ denote regions of higher and lower pressure.
σp =
σth1 =
Fig. 9. Pressure along the wall for the simulated pressure ratios ③ ( p2 /P01 = 0.8833) and
p2 − pv p1 − p2
(8)
2( p1 − pv ) ρl Vth2
(9)
④ ( p2 /P01 = 0.8767 ) with Qmax = 8.40 m3/h .
σth2 =
2( p2 − pv ) ρl Vth2
(10)
Where σp was a cavitation number based solely on pressure differences [24] and σth1,2 were either based on the inlet or the outlet pressure as the reference pressure. Vth was calculated in the same way as describes for the loss coefficient. The loss coefficient was plotted against the three different definitions of σ in Fig. 11 for the measurement curve Qmax = 8.01 m3/h . Additionally, the results obtained by the numerical simulations for Qmax = 8.40 m3/h were plotted, denoted by the dashed lines. In the non-choked and non-cavitating case, the agreement between both curves was quite good. Strikingly, even in the numerical case the distinct behavior of a small increase of the loss coefficient shortly before the choked condition sets in, followed by a decrease of the same amplitude was predicted. For the choked condition, a constant vertical shift of about 8.7% between the numerically predicted and experimentally calculated ζ values was visible for the σp curve. As the numerically predicted loss coefficient was always based on the viscous throat velocity, the only difference between the σp curves for the choked condition was the used velocity for the calculation of the loss coefficient. It was either inviscid for the experimentally estimated values or viscous for the numerically
Fig. 10. Loss coefficient ζ as a function of the pressure ratio p2 /P01.
main reason for the decrease of the loss coefficient, while choking sets in, is the simplification of the pressure field already described in Section 4.1. Thus, Fig. 10 is the evidence that all measured curves indeed show the simplification of the pressure field and with that the sudden jump of the vapor cavity. For the non-choked condition, the loss coefficients of the different 61
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Fig. 13. High-Speed Photographs taken by the FASTCAM system at 50.000 fps, covering a time of Tref = 13 ms ; σp = 7.21; Qmax = 9.09 m3/h ; Video link.
numerically predicted throat velocity and the inviscid throat velocity, calculated via Eq. (5). Increasing the throat velocity in the choked condition for the experimental values, calculated via Eq. (5), by 4.26% for both the calculation of ζ and σth2 resulted in a much better agreement between the experimental and numerical curves. This was displayed in Fig. 12, where two additional curves based on the velocity correction were indicated. Of course, one has to remember, that Eq. (5) itself was based on imperfect assumptions such as using the vapor pressure as the respective throat pressure and using the density of the liquid as the respective throat density. Another advantage of the velocity deviation estimation was the possibility to apply it to the correction factor Ccav for the calculation of the choked mass flow rate, Eq. (4). As Ccav corrects the product ρl Ath Vth,∞ by 13.07% and Vth,∞ itself was defective by 4.26% the remaining 8.8% had to account for the correction of ρl Ath . This means, that the influence of choosing the liquid density as the reference density
Fig. 14. Cloud cavitation scheme by Stanley et al. [26]; with A denoting the fixed vapor cavity, B separated and convected bubbles and C the re-entrant jet.
predicted values. Thus, with the 8.7% shift, an estimation of the deviation caused by the usage of the inviscid calculation via Eq. (5) could be realized:
ζexp =
2( p1 − p2 ) 2 ρl Vth ,∞
·1. 087 ≈
Vth2 , sim·1. 087 ≈ Vth2 ,∞ Vth, sim·1. 0426 ≈ Vth,∞
2( p1 − p2 ) 2 ρl Vth , sim
= ζsim
(11)
The above estimation showed a 4.26% deviation between the viscous, 62
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of the sheet cavity, a series of pictures taken by the FASTCAM SA-Z 2100 K system will be shown. The pictures of Fig. 13 were taken at 50.000 fps for Qmax = 9.09 m3/h . For choosing a proper time range to display the global steadiness of the sheet cavity, the experimental results of a convergent-divergent section by Stutz and Reboud [25] were considered. For Qmax = 9.09 m3/h , the resulting inlet velocity Vin was 4.905 m/s and analyzing the pictures resulted in the maximum length of the vapor cavity Lcav being equal to 17.55 mm. According to the measurements of Stutz and Reboud, the ratio of Vin / Lcav should result in a periodic frequency of cloud shedding of about 75 Hz or a period of about 13 ms. In that sense, the following pictures of Fig. 13 covered a period of Tref = 13 ms. For a better visual comparison of the sheet cavity length, a thin dashed line was inserted into all pictures of Fig. 13 at the exact same position. Fig. 13a – d represent the first 0.1 ms of the whole 13 ms period. Regarding the global behavior of the sheet cavity, there have only been minor movements, while at the same time the cavity closure region showed a highly complex three-dimensional behavior. The further development of the sheet cavity during the 13 ms period, displayed in Fig. 13e–i, still showed a very steady global behavior. Due to the larger time increments, there was more movement visible within the cavity closure region, while the overall length was still very constant. This can also be checked in the Video, linked in the description of Fig. 13, which covers an even larger time span.1 As can be seen in Fig. 13, the sheet cavity clearly had two liquidvapor interfaces. Obviously, an inner interface between the liquid core flow and the vapor as well as an outer interface between the vapor cavity and a liquid film that existed between the cavity and the nozzle wall. Otherwise, the light reflections and the clearly visible deformations of that outer interface would not be visible. These results support the recently published cloud cavitation scheme by Stanley et al. [26] which is schematically shown in Fig. 14. According to it, between most of the wall and the cavitation cloud a thin liquid film exists, while the cavity itself is only fixed to the wall at its inception point. Considering the small height of the liquid film, the extremely sharp liquid-vapor interface and the fact that most CFD cavitation studies use Y+ values above 30, it is hardly surprising that this feature hasn't been captured numerically via the common VOF-method so far, at least to the authors knowledge. For the determination of the actual velocity inside the vapor cavity, pictures taken at 120.000 fps were analyzed. As an example, the three pictures in Fig. 15 represent such an analysis. Due to the increase of the fps, the image section decreased, with the left side now barely starting at the inception point and the right side almost covering the closure region. For the estimation of the velocity inside the vapor cloud, clearly identifiable structures were traced. Such structures were denoted in Fig. 15a with ①-④. However, only structure ① actually moved in main flow direction during the presented sequence. Structure ② and ③ represented wave like deformations on the outer interface, while structure ④ seemed to be a void in the bubbly structure at the closure region. This three structures were apparently closer to or on the outer interface (closer to the nozzle wall) while structure ① was passing behind those foreground structures, thus had to be either inside the cavity or on the inner interface. This means there had to be a wide difference in the velocities across the vapor cloud in radial direction, which is consistent with experimental and numerical results of different authors [25,27,28]. On the other hand, no experimental information was existing on the axial velocity distribution of certain structures within the vapor cavity. Fig. 16 represented the final results of this investigation, obtained by the analysis of structures within the cavity. For the analysis, the throat section was divided into three equal parts, as denoted in the
Fig. 15. High-Speed Photographs taken by the FASTCAM system at 120.000 fps; σp = 7.21; Qmax = 9.09 m3/h .
Fig. 16. Analyzed velocities within the vapor cavity based on the throat velocity Vth,∞ for the three different throat sections l /lth ; Vth,∞ = 30 m/s , σp = 7.21, Reth=343126 .
has an higher impact on the accuracy of the results than the difference in inviscid or viscous throat velocity. The inviscid velocity clearly overestimates the actual throat velocity, if it would remain liquid. Due to the cavitation though, the volumetric flow rate within the throat increases, leading to a higher than all-liquid-flow velocity.
4.3. High speed camera investigations 1
For the purpose of showing the high stability and global steadiness 63
https://youtu.be/lvVdTAMbtH4
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Fig. 17. High-Speed Photographs taken by the FASTCAM system at 120.000 fps; σp = 7.21; Qmax = 9.09 m3/h .
vapor cavity was shown. The front of the bubbly cloud was denoted in Fig. 17a with the white line at the closure region. Additionally, a prominent wavy region on the outer liquid-vapor interface in the middle of the throat was encircled. Each picture was separated by a time span of 0.25 ms, so in contrast to the fast downstream movement of the structure in Fig. 15, the now shown upstream movement was much slower. From Fig. 17a to c, the bubbly cloud slowly moved upstream in the direction of the cavitation inception point. During the same time, the wavy interface structure somewhat compressed to one single wave that stood more or less still. From Fig. 17c to d, the bubbly cloud front again moved a considerable way upfront, while the interface wave still barely moved. With increasing time, the bubbly cloud front continued to move in upstream direction but the whole progress slowed down (Fig. 17e- f). The same was valid for the wave structure, it continued to move in upstream direction to about the first quarter of the throat section (Fig. 17f) until it vanished. In Table 4, the evaluated velocities between each picture of Fig. 17 were listed. The highest velocity was estimated to 11.4 m/s, so about 1/ 3 of the throat velocity of 30 m/s. This gave an explanation for the overall steadiness of the sheet cavity. As Keil et al. [29] reported, the transition from a steady sheet cavitation to an unsteady cloud cavitation with a periodic shedding of vapor cavities depends on the ratio of the re-entrant jet velocity Vjet to the sheet growth velocity Vs. The sheet growth velocity was already estimated in Fig. 16 to about 0.7–0.9 times the throat velocity of 30 m/ s, therefore 21–27 m/s. The peak re-entrant jet velocity was estimated
Table 4 The evaluated velocities of the bubbly cloud front of Fig. 17. Figure
Bubble Cloud Front Velocity
17 a → b 17 b → c 17 c → d 17 d → e 17 e → f
6.4 m/s 11.2 m/s 10.4 m/s 6.4 m/s 3.2 m/s
lower right side of Fig. 16. Within each section, the velocities of ten clearly identifiable structures were determined. The average velocity, as well as the highest and lowest velocity indicated by the error bars, were then plotted based on the inviscid throat velocity, calculated via Eq. (5). The determined velocities showed an obvious increase with increasing throat length. The high scattering of the determined velocities in the first section was probably due to the small thickness of the sheet cavity in that region. While the velocities in the middle section were quite similar, the last section again showed a higher scattering, probably due to the strong coupling with the closure region directly following. The results were in agreement with similar investigations presented by Stanley et al. [26], likewise having velocities in the order of the throat velocity. However, Stanley et al. made no distinction between the axial location inside the throat of their tracked velocities, which explains the high scattering of their determined velocities. In Fig. 17, the event of a bubbly cloud moving upstream inside the 64
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Fig. 18. High-Speed Photographs taken by the FASTCAM system at 140.000 fps; σp = 7.21; Qmax = 9.09 m3/h ; Video link.
such as the far wavier interface structure at the beginning of the cavity, denoted in Fig. 18a by ①. Additionally, in the bracketed area of Fig. 18a the vapor cloud will collapse, possibly due to a re-entrant jet. In Fig. 18b, the collapse had slowly set in, with the cavity closure region within the bracket starting to move upstream. Furthermore, a tornadolike vortex structure appeared in the upper closure region (②) and was drawing surrounding bubbles into its core. The vortex structure persists throughout the whole shown time interval (Fig. 18 b → g = 32 ms) until it finally decayed in Fig. 18g. In Fig. 18c, the upstream moving closure region could not be clearly distinguished from the background vapor structure of the rear
in Table 4 to 11.2 m/s. Thus the ratio Vjet / Vs ≈ 0.53 − 0.41 gave the explanation for the steadiness of the sheet cavity for this investigated operating point.
steady sheet regime = transition =
Vjet Vs
Vjet Vs
<1
= 1 ↔ Re=Recrit
unsteady cloud regime =
Vjet Vs
>1
(12)
Fig. 18 represented a compilation of pictures taken at 140.000 fps. The presented compilation featured several interesting phenomena, 65
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Fig. 19. High-Speed Photographs taken by the FASTCAM system at 140.000 fps; σp = 7.21; Qmax = 9.09 m3/h ; Video link.
narrow region and not simultaneously across the whole cavity closure region. This was elucidated by the cavity closure region in the upper and background region still being in place. Due to this, the upstream movement of the re-entrant jet region had to be additionally slowed down by the obviously still downstream flowing neighboring flow. The shown compilation was another explanation for the global steadiness of the vapor cavity. Unsteady re-entrant jet phenomena may occur in the investigated case, but, as the analysis of the video material confirmed, they are neither periodic, nor a global phenomenon but rather a local one with minor influence. Nevertheless, they demonstrate the high three-dimensional, unsteady behavior in the closure region of
180°-section of the cylindrical vapor cavity. Instead, another structure was visible in the encircled region denoted with ③. Throughout the remaining compilation, this structure, although right in front of the reentrant jet region, was not moving. In Fig. 18d, 2/140 ms after Fig. 18c, a sharp interface of the pushed back cavity closure region could be seen, denoted by the white bracket. This sharp interface represented a so-called condensation shock wave [30] that basically stood still for the remaining 10 ms of the shown compilation. However, the most interesting observation of Fig. 18 was, that the whole cavity collapse phenomenon took place in a circumferential very 66
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venturidüse, Forsch. Ing. -Wes. 30 (1964) 86–93. [5] F. Numachi, R. Kobayashi, Verbesserung der venturidüse hinsichtlich der kavitationsbeeinflussung, Forsch. Ing. -Wes. 31 (1964) 60–65. [6] H. Ghassemi, H.F. Fasih, Application of small size cavitating venturi as flow controller and flow meter, Flow Meas. Instrum. (2011) 406–412. [7] A. Abdulaziz, Performance and image analysis of a cavitating process in a small type venturi, Exp. Therm. Fluid Sci. (2014) 40–48. [8] P. Rudolf, M. Hudec, M. Gríger, D. Štefan, Characterization of the cavitating flow in converging-diverging nozzle based on experimental investigations, EPJ Web of Conferences 67, 2014. [9] S. Ashrafizadeh, H. Ghassemi, Experimental and numerical investigation on the performance of small-sized cavitating venturis, Flow Meas. Instrum. 42 (2015) 6–15. [10] A.J. Schmidt, Quantitative measurement and flow visualization of water cavitation in a converging-diverging nozzle, Master thesis, Kansas State University, USA, 2016. [11] S. Brinkhorst, E. von Lavante, G. Wendt, Numerical investigation of cavitating herschel venturi-tubes applied to liquid flow metering, Flow Meas. Instrum. 43 (2015) 23–33. [12] S. Brinkhorst, E. von Lavante, G. Wendt, Numerical investigation of effects of geometry on cavitation in herschel venturi-tubes applied to liquid flow metering, in: International Symp. on Fluid Flow Measurement. [13] L. Rayleigh, On the pressure developed in a liquid during the collapse of a spherical cavity, Philos. Mag. 34 (1917) 94–98. [14] A. Oprea, N. Bulten, Cavitation modelling using RANS approach, WIMRC 3rd Int. Cavitation Forum 2011 (2011). [15] X. Margot, S. Hoyas, A. Gil, S. Patouna, Numerical modelling of cavitation: validation and parametric studies, Eng. Appl. Comput. Fluid Mech. 6 (2012) 15–24. [16] E.M. Bennett, Cavitation cfd using star-ccm+ of an Axial Flow Pump with Comparison to Experimental Data, 2014. [17] G. Vaz, D. Hally, T. Huuva, N. Bulten, P. Muller, P. Becchi, J.L.R. Herrer, S. Whitworth, R. Macé, A. Korsström, Cavitating flow calculations for the e779a propeller in open water and behind conditions: Code comparison and solution validation, in: 4th International Symp. on Marine Propulsors. [18] CD-adapco, User Guide, 10.04 edition, 2015. [19] X. Zhang, J. Zhu, L. Qiu, X. Zhang, Calculation and verification of dynamical cavitation model for quasi-steady cavitating flow, Int. J. Heat Mass Transf. 86 (2015) 294–301. [20] J. Kozák, P. Rudolf, D. Štefan, M. Hudec, M. Gríger, Analysis of pressure pulsations of cavitating flow in converging-diverging nozzle, in: Proceedings of the 6th IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems. [21] M. Morgut, E. Nobile, I. Biluš, Comparison of mass transfer models for the numerical prediction of sheet cavitation around a hydrofoil, Int. J. Multiph. Flow. 37 (2011) 620–626. [22] S. Brinkhorst, E. von Lavante, G. Wendt, Experimental investigation of cavitating herschel venturi-tube configuration, in: Proceedings of the 17th International Flow Measurement Conference. [23] Y.-S. Kim, A proposed correlation for critical flow rate of water flow, Nucl. Eng. Technol. 47 (2015) 135–138. [24] W.H. Nurick, Orifice cavitation and its effect on spray mixing, J. Fluid Eng. 98 (1976) 681–687. [25] B. Stutz, J. Reboud, Experiments on unsteady cavitation, Exp. Fluids (1997) 191–198. [26] C. Stanley, T. Barber, G. Rosengarten, Re-entrant jet mechanism for periodic cavitation shedding in a cylindrical orifice, Int. J. Heat Fluid Flow. 50 (2014) 169–176. [27] E. Goncalvès, Numerical study of unsteady turbulent cavitating flows, Eur. J. Mech. - B/Fluids 30 (2011) 26–40. [28] J. Decaix, E. Goncalvès, Investigation of three-dimensional effects on a cavitating venturi flow, Int. J. Heat Fluid Flow. (2013) 576–595. [29] T. Keil, P.F. Pelz, J. Buttenbender, On the transition from sheet to cloud cavitation, in: Proceedings 8th International Symp. on Cavitation, Singapore. [30] P. Tomov, K. Croci, S. Khelladi, F. Ravelet, A. Danlos, F. Bakir, C. Sarraf, Experimental and numerical investigation of two physical mechanisms influencing the cloud cavitation shedding dynamics, 2016. Working paper or preprint.
the cavity. In Fig. 19, the further development of the above described cloud collapse and condensation shock wave of Fig. 18 was displayed. Starting from Fig. 19a, the encircled, pushed back closure region began to develop to its original location again and extended in normal flow direction. While the condensation shock wave was in Fig. 18g a single sharp interface, positioned in the lower region of the picture, it moved a little bit into the center, implying a radial velocity component. Furthermore, the single sharp interface split into two smaller condensation shock waves, denoted in Fig. 19c by the two white lines. The wave front extended to its original position in Fig. 19g and the whole cloud collapse event was completed. To the authors knowledge, the above described circumferential narrow cavity collapse has never been mentioned in literature. This could be because of two reasons: most experiments are either run on 2dimensional, planar nozzle geometries, or use nozzle geometries with larger diffuser angles which are favorable for flow separation, which in turn amplifies the re-entrant jet strength [12]. 5. Conclusion In the presented work, a combined experimental and numerical approach for the investigation of the choked flow condition in cavitating Herschel-tubes was presented. The combined approach gave further insight into the onset of the choked condition, where a sudden expansion of the vapor cavity was revealed by the experiments and could be explained by the numerical simulations. Due to a complex pressure field, the vapor cavity will extend steadily from the beginning of the throat to about 1/3 of the investigated throat length. At this point, an alternating pressure field has developed insight the nozzle throat that collapses and by doing so enforces a sudden extension of the vapor cavity into the beginning of the diffuser section. Furthermore, explanations for the steadiness of the sheet cavity could be presented by the usage of a high speed camera with up to 140.000 fps. While the high speed investigations revealed the high unsteadiness especially within the cavity closure region, they also revealed the minor effect of certain re-entrant jet occurrences at least for the investigated case. The high speed investigations revealed not a single re-entrant jet phenomena that emerged evenly over the entire circumference. The extremely three-dimensional flow behavior in the cavity closure region probably prohibits this, which in turn leads to local, circumferential narrow re-entrant jet phenomena, with less influence on the whole vapor cavity. References [1] J. Ackeret, Experimentelle und theoretische untersuchungen über hohlraumbildung (kavitation) im wasser, Tech. Mech. und Thermodyn. 1 (1930). [2] F. Numachi, M. Yamabe, R. Oba, Cavitation effects on the discharge coeffcient of the sharp-edged orifice plate, J. Basic Eng. 82 (1960) 1–6. [3] F. Numachi, R. Kobayashi, S. Kamiyama, Effect of cavitation on the accuracy of herschel-type venturi tubes, J. Basic Eng. 84 (1962) 351–360. [4] F. Numachi, R. Kobayashi, Einfluß der kavitation auf die durchfluß zahl der
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