Lattice gas automaton modelling of a vortex flow meter: Strouhal-Reynolds number dependence ∗ Vaidas Jukneviˇcius and Jogundas Armaitis Institute of Theoretical Physics and Astronomy, Vilnius University, Saule˙tekio al. 3, LT-10222 Vilnius, Lithuania Motivated by recent experimental and computational results concerning the three-dimensional structureofvorticesbehindavortexsheddingflowmeter[M.Reiketal., Forsch. Ingenieurwes. 74, 77 (2010)], we study the Strouhal-Reynoldsnumberdependence in the vortex street in two dimen- sionsbehindatrapezoid-shapedobjectbyemployingtwotypesofFrisch-Hasslacher-Pomeau(FHP) 7 models. Ourgeometry is intendedto reproduce the operation of the vortex sheddingflow meter in 1 a two-dimensional setting, thuspreventing the formation of three-dimensional vortex structure. In 0 particular, we checkif theanomalous Reynolds-Strouhalnumberdependencereported for threedi- 2 mensionscanalsobefoundinourtwo-dimensionalsimulation. AswefindthattheStrouhalnumber n is nearly independent of the Reynolds number in this particular setup, our results provide support a for the hypothesis that three-dimensional flow structures are responsible for that dependence,thus J hinting at the importance of the pipe diameter to the accurate operation of industrial vortex flow meters. 0 3 I. INTRODUCTION olates the main operating assumption of the flow me- ] n ter, therefore leading to inaccurate flow measurements y Hydrodynamic theories have been studied for a long in certain regimes. In this paper, we provide an addi- d time, but they still providenew insights in variousprob- tional check if the spatial structure is indeed to blame - u lemsofphysicsandengineering,fromnondissipativecur- for the anomalous relation by numerically studying the l rents in ultracold atomic vapour [1] to stability of tall same problem in a two-dimensional geometry, where the f . buildings [2], in addition to posing some extremely chal- horseshoe vortices cannot form. s c lenging questions along the way [3]. Several branches of The traditional approach to hydrodynamics, namely si the field remain particularly vigorous, including studies solving the Navier-Stokes equations, is considered to be y offlowinstabilities[4]. Besidesbeingoffundamentalim- both analytically andcomputationally complicated. An- h portance, flow instabilities in general, and the renowned alyticalsolutionsofhydrodynamicproblems arepossible p K´arm´an vortex street [5, 6] in particular can be read- onlyinlimitednumberofcasesofflowsatsmallReynolds [ ily observed in everyday life [7], and also has important numbers and in relatively simple geometries. Some hy- 1 practical uses. drodynamic instabilities can be investigatedanalytically v The vortex shedding flow meter stands out as a direct and numerically using dynamical systems approaches. 4 industrialapplicationofthe phenomenonofthe K´arm´an For example, expansion around instabilities may result 5 vortexstreet. Thisdeviceconsistsofabluntobjectposi- in equations that are simpler than Navier-Stokes equa- 7 tionedinsideapipe,andadetectorofvortices. Asliquid tions, but which are nevertheless able to reproduce the 8 0 (or gas) flows through the pipe sufficiently fast, a vortex formation of hydrodynamic patterns [10, 11]. Also, in . street forms downstream from the blunt object. Since some cases,weakly ormoderately turbulentflows canbe 1 thevortexsheddingfrequencyisdependentonthehydro- describedusing the so-calledshell models that consistin 0 dynamic properties of the flow, the signal of the vortex replacing the partial differential equation with a system 7 1 detector can be converted to the velocity measurement of coupled ordinary differential equations via discretiza- : of the flow. This type of a device is uniquely suitable for tion and truncation in Fourier space [10]. However, for v operationin anindustrialsetting, as it is fully contained extremely high Reynolds numbers, because of the large i X inside the pipe, has no moving parts, and is both robust number of relevant degrees of freedom due to the wide r and reliable [8]. rangeofscales,thetraditionaldescriptionsoffullydevel- a It turns out that even in this seemingly mundane set- oped turbulence employ statistical methods [12, 13] and ting of a tested industrial application, novel physics can phenomenological models [14, 15]. be uncovered. In particular, it has been recently sug- An altogether different approach to hydrodynamic gested that the vortex pattern that forms downstream problems are the so-called lattice gas models [16–18]. fromthevortexflowmeterhasathree-dimensionalstruc- These models belong to a wide class of discrete systems ture [9], in the shape of so-called horseshoe vortices. In known as cellular automata. They have a direct physi- turn, this spatial structure alters the flow in the pipe, cal interpretation. Namely, point particles occupy nodes introducingananomalousrelationbetweentheReynolds of a lattice with the possibility to jump from one node and the Strouhal numbers. This anomalous relation vi- to one of its neighbouring nodes in a single time step. In most of these models the particles move with a single speed in one of several directions. Collisions of the par- ticles occur at the nodes, and are executed according to ∗ [email protected] some simple logical rules. Remarkably, if the lattice has 2 propersymmetriesandthecollisionrulessatisfyrelevant evant conservation laws and certain symmetries remain conservation laws (e.g., momentum and energy conser- in tact. vation), then the global behaviour of the system in a The rigid obstacles and impermeable boundaries are coarse-grainedpicture will closely resemble the flow of a introducedbysettingupspecialcollisionrulesdescribing fluid [18, 19]. particle reflection for the nodes at the boundaries. Also Inthiswork,twodifferentrulesetsoftheseven-particle the sources and sinks may be added by special rules of Frisch-Hasslacher-Pomeau (FHP, [19]) model have been particle creation/destructionat some nodes. used in order to simulate the vortex flow meter [9]. These FHP models have been successfully employed to attack diverse problems, including nucleation in super- saturated liquids [20], sand dune growth [21], as well as A. The FHP models flows on curved surfaces with dynamical geometry [22], in addition to shedding insight on various aspects of hy- The FHP models [19] belong to a class of two- drodynamics [18]. Hence, even though more elaborate dimensional lattice gas models based on the two- methods to address fluid dynamics are present (for ex- dimensionaltriangularlattice. Thereareseveralversions ample, the lattice Boltzmann equation [23]), in our case of the FHP models that maintain the same lattice struc- we have employed the FHP lattice gas model in order ture, but differ in collision rules. to minimize the computation effort, while still obtaining The simplest version is the so-called FHP-I model reliable results. where each node has 6 channels corresponding to the 6 The structure of the paper is as follows. Section II directions on the triangular lattice. For our simulations introduces the FHP lattice gas models used for the sim- we have used the FHP-II and FHP-III versions where 7 ulations, together with the definitions of Reynolds and particles at each node may exist – six moving and one Strouhal numbers in this framework. Then, in Section additionalparticleatrest(havingzerovelocity). Besides III, the main results are presented and discussed. Fi- havingahighernumberofpossibleeffectivecollisions,the nally,SectionIVsummarizesthe resultsanddrawssome main feature of the FHP-III model compared to FHP-I conclusions, in addition to discussing several promising and FHP-II models is the property of self-duality. This directions for future investigations. meansthatthedynamicsofparticles(occupiedchannels) is equivalentto the dynamics ofholes (unoccupied chan- nels) and the dynamics, i.e., the collision rules for the II. METHODS OF SIMULATION dual states (with occupied and unoccupied channels ex- changed) are the same as for the original states. In this section we briefly describe the lattice gas au- tomata in general, and the FHP models in particular. Eventhoughseveralexcellentresourcesonthesesubjects are available (see, e.g., Refs. [18] and [17] as well as the B. Averaging and macroscopic observables references within them), we summarize the most impor- tant aspects of the methods employed in order to make The discrete dynamics of the states of nodes on the our discussion self-contained. Bravais lattice described above constitutes the micro- The lattice gas automata consist of discrete nodes ar- scopic dynamics of the model with microscopic quanti- ranged geometrically in a Bravais lattice [17, 18, 26]. ties, such as local density (number of particles at the Since the number of nodes is finite, suitable boundary node), velocities of particles (or local velocity field taken conditions (most commonly, periodic) must be imple- as an averagevelocity of all particles at the given node). mented. Each node has a fixed number of channels that Thesequantitieshavelittletodowithactualmicroscopic canbeeitheremptyoroccupiedbyasingleparticle. The dynamics that takes place in real physical fluids. How- channels point to the nearestneighbouring nodes, there- ever, under appropriate circumstances, the macroscopic fore a particle in each channel is considered to possess a properties of the lattice gas can reproduce the macro- single speed in the directionof the neighbouringnode to scopic properties of real fluids. which that channel is pointing. The time evolution proceeds in discrete steps where The macroscopic observables from the lattice gas sim- eachsinglestepconsistsoftwophases–propagation and ulations areobtainedby spatialandtemporalaveraging. collision. During the propagation phase, the particles Spatial averaging consists in averaging the microscopic move, i.e. the occupied state of a channel in each node states over blocks of nodes. Temporal averaging means is transferredto the channel of the same direction in the thatthevalueofthestateistakenaveragedovermultiple correspondingneighbouringnode. Inthe collisionphase, time steps. the states of each node change locally, according to a Herewehaveusedspatialaveragingover16×16blocks set of fixed rules. In order to reproduce the macroscopic of nodes and for the velocity field. Also, temporal av- propertiesofa physicalfluids, the collisionrules andlat- eraging over 10 time steps has been used unless noted tice geometryareconstructedinsuchawaythatthe rel- otherwise. 3 C. Reynolds and Strouhal numbers on, Fourier analysis and sine function fitting is used to determine the lowfrequencymode fromthe noisysignal, Reynoldsnumberisadimensionlessnumberthatchar- since a direct application of the FFT is often not a good acterizes the flow by showing the relative importance of option, as it requires many steps to obtain reasonably inertialandviscousforces[24]. Itiswidelyusedtoquan- smallerrorsduetofluctuationsofthevelocityfield,which titatively describe different regimes of the flow. in turn come about due to complex flow patterns. The For the FHP-III model, the Reynolds number Re is low frequency mode in the case of K´arm´an vortex street calculated in the following way [18]. First, the density ρ corresponds to the vortex shedding frequency. A piece of the particles on the lattice is measured. Since it is a of raw data and the corresponding sine function fit are number from0 to 7 foreachnode, itis convenientto use provided in Fig. 4. the reduced density d=ρ/7. Because of the self-duality of the FHP-III model, if d > 0.5, then the dynamics of holesinsteadofparticlesisbeingobserved,therefored≡ III. SIMULATION OF A VORTEX FLOW 1−d in that case. Because of this, certain macroscopic METER observables differ from theoretical ones by the density dependent non-Galilean factor [18]: Even though there is a considerable body of knowl- edge concerning vortex formation in flows behind cylin- ders [25], and other highly symmetric objects in 7 1−2d g(d)= . (1) translationally-invariant geometries, the particular case 12 1−d of a prism in a confined setting has not been studied yet up to the best of our knowledge. Since the main goal Another important quantity required in order to cal- of this paper is to simulate the vortex flow meter that culate Re is the kinematic viscosity: is usually placed in a pipe, as in [9], periodic boundary conditions have been used in the x direction being main 1 1 direction of the flow (from the left to to the right in the η = − (2) 28dd¯(1− 8dd¯) 8 figures), and the containment of the flow by the pipe 7 walls has been implemented as impermeable boundaries where d¯=1−d. The Reynolds number is from the top and the bottom (i.e., in y direction). This section presents results from a series of simula- guL tions in several different geometries. First, the velocity Re= . (3) η profile of the steady flow without an obstacle has been obtained in order to test the velocity profile (Fig. 1). where u is the average velocity magnitude and L is the Then, the vortex shedding from a triangle has been im- typical dimension of the obstacle. One readily notices, plemented and visualized in both FHP-II and FHP-III that in order to increase Re, one has to choose a wide models (Fig. 2). Finally, the two-dimensional model of channel, produce high velocity of the flow and optimize a vortex flow meter has been simulated by placing the ∼ d. In the present case Re is maximized at d=0.305. bluntprism-shapedvortexsheddingdevice(Fig.3)inthe The Strouhal number St is another dimensionless flow with two different ratios of obstacle size to channel quantity, characterizing the flow. The function St(Re) width in order to measure the dependence of the vortex providesimportantinformationaboutwhatishappening shedding frequency on the flow velocity and determine at the wake [18]. It is defined by the following equation: the Strouhal-Reynolds number dependence (Fig. 5 and Fig. 6). f ·L St= , (4) u·g A. Velocity profile of the laminar flow where L is again the typical size of the obstacle (in this case, the length of the base of the triangle, see Fig. 3). Before starting the simulation of unsteady flow of the f is the frequency of the wake oscillation produced by vortexstreetbehindanobstacle,thevelocityprofileofan vortex shedding. All the quantities are in naturallattice unobstructed flow has been investigated using the FHP- units (i.e. number of lattice sites and time steps). II model. On a lattice of 120×48 cells (each cell, as The industrial vortex flow meters function under the mentionedbefore,beingablockof16×16nodes)theve- assumption of constant St. If this were the case, the locity component v along the generalflow direction has x frequencyf ofthevortexsheddingwoulddependlinearly been measured. A steady-state velocity profile has been on the flow velocity u. However this turns out not to be determinedforeveryhorizontalblockofcells(coordinate true at least in some regimes of the flow [9]. y rangingfrom1to48)byspatialaveragingoftheveloc- In our case f is determined by the lattice gas hot- ity over cells 40 to 100 in the x direction and temporal wireanemometry[18]–averagingoverablockwherethe averaging over 100 time steps. Two mechanisms of flow local velocity magnitude is recorded at each step. Later induction have been considered. 4 city1.1 20 elo1.0 40 e of v00..89 60 ul0.7 d 80 o0.6 m d 0.5 100 e0.4 z ali0.3 120 m0.2 Nor0.1 140 0.0 0 10 20 30 40 50 160 Coordinate on Y axis 180 200 y -50 0 50 100 150 cit1.1 o vel1.0 DFiatta 20 of 0.9 e 0.8 40 dul0.7 o 60 m0.6 d 0.5 80 e aliz0.4 100 m0.3 Nor0.2 120 0.1 0 10 20 30 40 50 140 Coordinate on Y axis 160 FIG. 1. Left panel: rectangular velocity profile resulting 180 from the fan approach. Right panel: velocity profile in the 200 source/sink case. Circles represent the measured data, and -50 0 50 100 150 thesolid red line is thePoiseuille profile fit. FIG. 2. Wake of a triangular obstacle at source-sink ratio 0.6/0.1 for two versions of FHP. The velocity magnitude is represented by different shades of gray. White bitmap image First, the so-called fan approach [17] has been imple- of the triangular object has been placed by hand on top of mented. Thisapproachconsistsofaverticalzoneof1×48 thecalculatedvelocityfield. Leftpanel: FHP-II.Rightpanel: cells where each particle moving to the left (in the op- FHP-III. posite direction to x) is being reversed with probability 0.001. Using this approach, however, an almost rect- angular velocity profile has been observed (left panel of steps, the expected Poiseuille velocity profile has been Fig. 1), instead of the expected Poiseuille profile [12]. observed (see right panel of Fig. 1). Therefore, the source/sink induction of the flow has been used for fur- Next, we have used the source/sink flow induction ther simulations. mechanism. A source or a sink is a node where each arriving particle is absorbed (destroyed) and new parti- cles moving in all the available directions (6 in the FHP case) are introduced each with some probability [17]. If, B. Vortex shedding in FHP-II and FHP-III for example, this probability is 0.2, then 6×0.2 = 1.2 versions particles at the source/sinknode are createdonaverage. If,onaverage,therearemoreparticlesproducedthande- Thenumericalschemehasbeentestedfurtherbycom- stroyed, then such a node acts as a source, and, if there paring the vortex shedding in the FHP-II and FHP-III aremoreparticlesdestroyedthancreated,anodeactsas models. For this, we have introduced a solid obstacle a sink. shaped as an equilateral triangle in the flow (Fig. 2). We have implemented the source/sink flow induction The creation probabilities of 0.6 and 0.1 of the byintroducingtwoverticalzonesof1×48ofsource/sink source/sink zones have been used, and a simulation of cells at the opposite sides of our system with different 30000 steps has been carried out on a lattice of 120x200 particle creation probabilities (0.5 and 0.4 in this case). blocks. Theabsolutemagnitudeofthevelocitydisplayed After a longer equilibration period of about 20000 time in 100shades of gray(white being highestmagnitude) is 5 FIG. 3. Geometry of thevortex flow meter simulation. shown. NotethattheK´arm´anvortexstreetisclearlyvis- ible in both FHP-II and FHP-III models (left and right panels of Fig. 2, respectively). However, one can also FIG. 4. Dynamics (in time steps) of the velocity perpendic- notice that FHP-III produces more pronounced vortices ular to the channel direction used for determination of the than FHP-II, owing to the lower viscosity and therefore wake oscillation frequency. Here the case of the small vortex a higher Reynolds number [26]. The lower viscosity of shedding device at small flow velocity u = 0.267 (Re= 156) the FHP-III model stems from its expandedset of possi- is shown. The black line represents the output and the red line represents thesine fit for thefrequency 0.0001. ble collisions. In the latter, 76 configurationsparticipate in collisions, as opposed to merely 22 such active con- figurations of the FHP-II model. We refer the reader to case where the boundary effects should play an impor- Refs. [26] and [17], where these collisions are listed ex- tant role (L/D = 0.27∼ 1). Moreover, the number 0.27 plicitly. Forthisreason,weconsidertheFHP-IIIruleset has been read off the geometry of the industrial vortex to be more suitable for the measurementof the Strouhal flow meter investigated in Ref. [9], in order to make a number,asitproducesclearervorticeswithnoadditional contact with the results presented there. In both cases computationaleffort. Thus,theFHP-III modelhasbeen the y component of the velocity v (velocity in the di- used for the simulations of the vortex flow meter. y rection perpendicular to the channel flow direction) has been measured5 cells downstreamfrom the shorterbase ofthe trapezoidthatconstitutes ourvortexsheddingde- C. Vortex flow meter vice. This correspondsto the lattice-gasimplementation of hot-wire anemometry. Vortex shedding has produced The main part of our investigation consists of mea- sine-shapedvariationinv . Anexamplepieceofrawdata y surements of the vortex shedding frequency dependence that has been measured in the simulation is depicted in on the flow rate in the two-dimensional simulation of a Fig. 4 together with the sine function fit from which the vortex flow meter [9] using the FHP-III rule set. The vortex shedding frequency is determined. results have then been used to determine the Reynolds- TheReynoldsnumberReandtheStrouhalnumberSt Strouhal number dependence. have been calculated from the measured flow velocity u Thetwo-dimensionalmodelofavortexflowmetercon- and the vortex shedding frequency f using (3) and (4), sists of a flow in a channel with impermeable walls and respectively. The velocity u has been tuned by changing a trapezoid-shaped obstacle that constitutes the vortex thesource/sinkratiooftheparticle-absorbing/producing shedding device. We have considered two cases differ- zones described in Subsection IIIA. It has been mea- ing in the obstacle to channel size ratio, i.e., the ratio suredbyaveragingacrossthechannelupstreamfromthe between the length L of the longer base of the vortex obstacle. sheddingdeviceandthewidthDofthechannel. Allsim- ulationsusedageometricallysimilarvortexsheddingde- vicewiththe lengthoftheshorterbaseandthe heightof 1. Small vortex shedding device thetrapezoidproportionaltoLandequalto0.225Land 1.1L, respectively. The geometry is depicted in Fig. 3. Here, the general flow direction is indicated by the gray Forthesimulationwithasmallvortexsheddingdevice arrow. where L/D = 0.055, a system of 120×200 cells (each We have investigated a relatively small vortex shed- cell, as before, being a 16× 16 block of lattice nodes) ding device with obstacle to channel ratio L/D = 0.055 hasbeen used. The measuredvortexshedding frequency and a large vortex shedding device with L/D = 0.27. f dependence on the flow velocity u and the computed This particular choice of the two ratios has been made St(Re) dependence are shown in Fig. 5. for two reasons. First, these choices address the two Anapproximatelineardependenceoff onthevelocity opposite physical limits: (i) the transparent situation u has been observed: where the vortices are shed far from the walls of the pipe (L/D = 0.055 ≪ 1), and also (ii) the less intuitive f =−(1±0.1)·10−4+(7.8±0.3)·10−4·u (5) 6 1.0x10(cid:13)-4(cid:13) -4 4.0x10 Data(cid:13) Data 8.0x10(cid:13)-5(cid:13) Linear Fit(cid:13) -4 Linear Fit 3.0x10 6.0x10(cid:13)-5(cid:13) -4 f2.0x10 f(cid:13)4.0x10(cid:13)-5(cid:13) -4 1.0x10 2.0x10(cid:13)-5(cid:13) 0.0 0.0(cid:13) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.25(cid:13) 0.30(cid:13) 0.35(cid:13) 0.40(cid:13) u u(cid:13) 0.35 0.35 0.30 0.25 0.30 St0.20 St0.25 0.15 0.20 0.10 0.05 0.15 0.00 50 100 150 200 250 300 350 0.10 350 400 450 500 550 600 Re Re FIG. 5. Measurements for the small vortex shedding device FIG. 6. Measurements for the large vortex shedding device (L/D = 0.055, see Fig. 3). Left panel: vortex shedding fre- (L/D = 0.27, see Fig. 3). Left panel: vortex shedding fre- quency dependence on flow velocity with linear fit (solid red quency dependence on flow velocity with linear fit (solid red line). Right panel: Strouhal-Reynoldsnumberdependence. line). Right panel: Strouhal-Reynoldsnumberdependence. The uncertainties here arethe errorsinthe least-squares dimensionality of the system. linear fit of the data. Results for the simulation of a large vortex shedding However,onenoticesthatthe lineardependenceisnot device with L/D = 0.27 are shown in Fig. 6. Here, ideal. First of all, it would give a non-zero frequency f a system of 240 × 100 cells has been used. The ratio for u=0. Secondly, one can see a nonlinear trend in the L/D=0.27 has been chosen to be the same as has been data(seeleftpanelofFig.5)whichsuggeststhatStisnot used in Ref. [9] thus allowing a direct comparison of our constant. The latter fact is clearly visible when looking results with the ones presented in Ref. [9] for the three- at the St(Re) dependence computed from the data (see dimensional case. Due to the interaction between the right panel of Fig. 5). vortex street and the channel walls, a very noisy signal has been obtained. The dependence of the frequency f on u might still be considered as slightly increasing (left 2. Large vortex shedding device panel of Fig. 6): The authors of Ref. [9] have observed the decrease f =−(0.1±0.1)·10−4+(1.2±0.5)·10−4·u , (6) in Strouhal number St with increasing Reynolds num- ber Re for Re < 4800 and suggested that the reason but no clear trend in the dependence of St on Re is ap- for this dependence might be related to the formationof parent (right panel of Fig. 6). horseshoevorticesalongthechannelwallsandthethree- dimensional turbulent flow. Bothoftheseeffectsarespecificforathree-dimensional IV. SUMMARY AND CONCLUSIONS geometry, and therefore do not exist in our flat model. Itis thus usefulto study ifthe previouslyreportedtrend In summary, this paper presents three main results in the St(Re) dependence survives given the decreased from a series of two-dimensional hydrodynamic simula- 7 tions using the FHP lattice gas models. tex street is obstructed by the channel walls, only a First, the Poiseuille profile for the laminar flow con- weakdependenceofthevortexsheddingfrequencyonthe fined in a channel has been demonstrated using the flow velocity can be observed (see Eq. (6), Fig. 6) and source/sink method. It has also been shown that the no significant Strouhal-Reynolds dependence has been so-calledfanapproachforinductionoftheflowresultsin found in contrast to the experimental data and hydro- a different, namely, rectangular,velocity profile (Fig. 1). dynamic simulations given in Ref. [9]. Therefore, our Moreover,thevortexsheddingfromatriangularobject two-dimensionalresultssupportthehypothesispresented intheflowhasbeendemonstratedinFHP-IIandFHP-III in Ref. [9], namely, that flow structures particular to the models, exhibiting higher viscosity in the FHP-II model three-dimensionalgeometryareresponsibleforthestrong (Fig.2). Therefore,theFHP-IIImodelhasbeenusedfor St(Re) dependence. further simulations. In future work, it would be interesting to study the The main part of this paper has been the simulation transition from two dimensions to three dimensions, as ofthevortexsheddingfromablunttrapezoid-shapedob- the onset of the strong St(Re) dependence is expected stacle (Fig. 3) in a confined flow. This configurationis a to occur when the extent of the smallest dimension of modelforthevortexflowmeterdescribedinRef.[9]. The the system surpasses the length scale characteristic to Strouhal-Reynolds number dependence was investigated the flow. Therefore, in pipes smaller than the size of in two different obstacle-channel size ratios. the horseshoe vortex (given a certain flow velocity), vor- As already noticed in classical works (see, tex flow meters operate in the accurate linear regime, e.g., Ref. [27]), statistical fluctuations play a prominent whereas when the diameter of the pipe is sufficiently role in lattice gas automaton simulations in general, large, the accuracy of the said flow meters should de- and in turbulence-related problems in particular. Hav- crease. These investigations might lead to a better un- ing this limitation of our method in mind, we have derstanding of the reliable-operation bounds of the in- performed the simulation multiple times in order to dustrial vortex flow meters. investigate run-to-run noise. We have discovered that the differences between runs are appreciable only for the large vortex shedding device case. Hence, we only show the results of different runs for that case (see Fig. 6). V. ACKNOWLEDGEMENTS However, in order to fully ascertain that the results are not dependent on the statistical fluctuations, one should turn to more sophisticatedmethods (see Refs. [28, 29]). It is our pleasure to thank G. T. Barkema for in- Lineardependence(seeEq.(5))ofthevortexshedding troducing us to the FHP models. We also thank frequency on the flow velocity and increasing Strouhal J. Buˇcinskas for his spirited encouragement to publish number with increasing Reynolds number has been our results. J. A. was supported by European Union’s demonstrated (Fig. 5) for the small vortex shedding de- Horizon 2020 researchand innovation programme under vice. the Marie Skl odowska-Curie grantagreementNo 706839 For the large vortex shedding device, where the vor- (SPINSOCS). [1] L.P.PitaevskiiandS.Stringari, Bose-Einstein Conden- wes. 74, 77-86 (2010) sation and Superfluidity (Oxford Univ.Press., 2016) [10] T. Bohr, M. H. Jensen, G. Paladin and A. 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