Investigation of the Horizontal Motion of Particle-Laden Jets

computation

Article Investigation of the Horizontal Motion of Particle-Laden Jets

Tooran Tavangar 1, Hesam Toﬁghian 2 and Ali Tarokh 3,*

1 Department of Civil and Building Engineering, University of Sherbrooke, Sherbrooke, QC J1K 2R1, Canada; [email protected] 2 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran 1591634311, Iran; h.toﬁ[email protected] 3 Department of Mechanical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada * Correspondence: [email protected]; Tel.: +1-807-343-8048

Received: 15 January 2020; Accepted: 2 April 2020; Published: 8 April 2020

Abstract: Particle-laden jet flows can be observed in many industrial applications. In this investigation, the horizontal motion of particle laden jets is simulated using the Eulerian–Lagrangian framework. The two-way coupling is applied to the model to simulate the interaction between discrete and continuum phase. In order to track the continuum phase, a passive scalar equation is added to the solver. Eddy Life Time (ELT) is employed as a dispersion model. The influences of different non-dimensional parameters, such as Stokes number, Jet Reynolds number and mass loading ratio on the flow characteristics, are studied. The results of the simulations are verified with the available experimental data. It is revealed that more gravitational force is exerted on the jet as a result of the increase in mass loading, which deflects it more. Moreover, with an increase in the Reynolds number, the speed of the jet rises, and consequently, the gravitational force becomes less capable of deviating the jet. In addition, it is observed that by increasing the Stokes number, the particles leave the jet at higher speed, which causes a lower deviation of the jet.

Keywords: particle-laden jet; tracer; Eulerian–Lagrangian; two-way coupling

1. Introduction Particle-laden jet flows have various engineering applications, such as fuel injection in internal combustion engines. In this regard, the jets laden with particles have been investigated by many researchers experimentally and computationally [1–15]. Many approaches have been used to experimentally analyze the jets laden with particles like Phase-Doppler Anemometry (PDA), Laser-Doppler Anemometry (LDA) and high-speed camera image processing. Fundamental experiments on dilute particle jets by the use of LDA can be found in [1–4]. Fan et al. investigated the effect of dispersed phase on the flow characteristics of a coaxial jet with particles laden. They showed when the particle ratio increases, the spreading ratio decreases [2]. Puttinger et al. analyzed the dilute particle-laden flow by using high-speed camera and image processing [5]. Due to the high spatial and temporal resolution of the latter method, it can be employed to study the turbulent flow with a high concentration ratio. This can be used for a wide variety of data and does not depend on the preset parameters. Generally, there are two main methods to simulate particle-laden flow, namely Eulerian–Eulerian and Eulerian–Lagrangian [6]. The former is used when the continuum assumption can be applied for both phases, while the latter is applied when particles constitute dispersed phases and fluid constitutes continuum phases. Both techniques have pros and cons [6]. A major problem of the first method is that some information such as the motion of the individual particles, and the collision between particles are not considered. In addition, applying boundary conditions for the second phase

Computation 2020, 8, 23; doi:10.3390/computation8020023 www.mdpi.com/journal/computation Computation 2020, 8, 23 2 of 13 is challenging. However, the advantage of the Eulerian–Eulerian method is that the same equation and numerical methods are applied to both phases. In the Eulerian–Lagrangian method, more detailed information about the motion of particles and their interaction with the surrounding can be obtained. However, some computational difficulties arise when the number of particles increases or the size of particles decreases. For instance, when the size of particles decreases, the effect of turbulent eddies on particles should be calculated even in smaller sizes; therefore, the cost of the computational method will be increased. Wang et al. [7] simulated the particle-laden jet using the Eulerian–Lagrangian method. In their study, the gas-phase and fluid-phase velocities show good agreement with the experimental results. They investigated the effects of different parameters on the flow behavior, including particle size, Stokes number and mass loading ratio. The Stokes number (Stk = τp,st/τF) is defined as the 2 ratio of the characteristic time of a Lagrangian phase (τp,st = ρp dp/18µ) to a characteristic time of the continuum phase (τF = Djet/Ujet). The Eulerian–Eulerian method was applied to simulate particle-laden jet turbulent flow in Ref [8]. In their research, the major roles of the particle size and loading ratio on the flow characteristic of particle-laden jets are shown. The Eulerian–Lagrangian method was employed by Jebakumar and Abraham [9] in which the changes in centerline velocities and spreading rate were investigated by varying parameters such as fluctuating particle velocity at the jet inlet, gas-phase turbulence intensity. Lue et al. showed the major effect of Stokes numbers on the clustering of particles by means of one-way coupling direct numerical simulation of a turbulent particle-laden jet [10]. The effect of the Stokes number (stk) on the particle motion and turbulent characteristics of particle-laden flow in the channel was investigated by Lee [11]. They found, for Stokes number of 5 that the particles increase the flow turbulence, while increasing the Stokes number of particles leads to decreasing the turbulence. The coupling behavior of solid particles and fluid makes modeling of this process complex. The coupling between solid and fluid phases has a substantial impact on multi-phase flows. If fluid exerts effects on particle motion but no reverse influence exists, the flow is called one-way coupling regime. If both phases have impacts on each other, the flow is called two-way coupling regime. In addition, if the collision between particles becomes important, the flow is called 4-way coupling regime. The Stokes number (Stk) and concentration or loading ratios are important parameters in particle-laden flows to determine the coupling between solid and fluid phases. For more details, the reader is referred to reference [12]. Numerical simulations were carried out by Wang et al. [7] using one-way coupling in order to estimate the evolution of particles in a plane turbulent wall jet. Kazemi et al. [13] performed a CFD simulation with the assumption of two-way coupling between two phases. They applied three different dispersion models. Their results suggested that the particle loading ratio should be considered in choosing the dispersion model. The Eddy Life Time (ELT) model is considered as one of the conventional dispersion models which can be used to investigate particle dispersion resulted from instantaneous turbulent velocity fluctuations [14]. In the present study, two-way coupling simulation of horizontal turbulent jet laden with particles in Eulerian–Lagrangian framework is conducted employing Reynolds averaged Navier–Stokes (RANS) model and ELT model is employed for modeling particle dispersion. The particle trajectory resulting from the simulation is compared with the experimental results reported by Fleckhaus et al. [15]. The heavy jets, caused by the high difference in density between the jet and surrounding fluid, were investigated by some authors to study the mixing characteristics of wastewater discharge [16,17]. However, there are many industrial applications that contain the discharging process of heavy particles and fluid, which form heavy jets. For example, in industrial wastewater and an exhaust gas flow, which in the effect of solid particles on the jet profile of disposal, wastes must be taken into consideration. In this paper, the effects of three non-dimensional parameters are investigated on the characteristics of heavy jets, which result from heavy particles. The influence of Stokes number, mass loading ratio and Reynolds number (Re = UjetDjet/υ), where υ is the Kinematic viscosity of the continuum phase, and mass loading ratio, which is defined mass fraction of the dispersed phase to continuum phase of the jet on the trajectory distribution of jet with particle-laden are investigated. The definition of the Stokes number is based on the Stokes’ relaxation time scale, and this number represents the diameter of the Computation 2020, 8, 23 3 of 13 particles with a density of 2590kg/m3. In addition, turbulence modulation is not considered in this work. It shows that the effect of gravity on the motion of the jet can be seen clearly by the trajectory of particles.

2. Governing Equation In this study, the carrier ﬂow is considered as a continuous phase and particles as dispersed phases, which are simulated using the Eulerian–Lagrangian framework. The two-way coupling is applied to study the interaction between two phases.

2.1. The Eulerian Equations The gas phase is described by the Reynolds averaged Navier–Stokes equations (RANS) for incompressible ﬂow. The continuity and momentum equations can be written as follows [18]:

∂uj = 0 (1) ∂xj

∂ui ∂uiuj 1 ∂P ∂τij 1 + = + + Sui (2) ∂t ∂xj −ρ ∂xi xj ρ

th with ρ being the density of the gas, P the pressure, ui the velocity in the i direction; Sui defines momentum source term because of the interaction between Eulerian and Lagrangian parts; τij denotes the effective stress tensor which can be defined as below: " # ∂ui ∂uj 2 ∂uk τij = (ν + νt) + δij (3) ∂xj ∂xi − 3 ∂xk where ν and νt are kinematic viscosity and turbulent kinematic viscosity, respectively; δ is the Kronecker delta. Here the turbulent viscosity νt is computed using the k-ω SST (Shear Stress Transport) turbulence model [19]. In the present study, a passive scalar equation is added to the solver as a tracer to track the gas phase, to investigate different fluid dynamic behavior between the tracer (particles with zero inertia) and inertial particles, which can be described as follows:

∂c ∂ujc ν ν ∂2c + = + t (4) ∂t ∂xj Sc Sct ∂xj∂xj

In the above equation, Schmidt number (Sc) and turbulent Schmidt number (Sct) are 0.4 and 0.7, respectively.

2.2. The Lagrangian Equations The dispersed phase is composed of parcels. Each parcel is defined by the group of properties including velocity (u), diameter (d), density (ρ), position (x), computational weight (ω). The diameter, density and computational weight of parcels are constant after injection. It is assumed that the drag and gravity forces have a dominant effect compared to other forces. Therefore, other forces are neglected. In the case of the heavy particle, i.e., ρ2 >> ρ1, the volume force applied by surrounding fluid flow can be assumed to be equal to the drag force [20]. The governing equations for the disperse phase can be written as [12,21]:

dxp,i = u (5) dt p,i

dup,i m = F + m g (6) p dt D,i p i Computation 2020, 8, 23 4 of 13

The drag force (FD,i) acts on a particle i which is equal to:

ur,i FD,i = mp (7) τp where relative velocity (ur,i) and particle relaxation time (τp) can be described as:

u = u u (8) r,i s,i − p,i 2 dpρp τp = (9) 18µ fd Relaxation time is applied to show the resistance of a dispersed phase to adjust to continuum phase motion and fd is the correction factor, which is deﬁned as below:

( 0.687 1 + 0.15Rep ; Rep < 1000 fd = (10) 0.44Rep/24; otherwise

Rep is the particle Reynolds number base on the relative velocity (ur) which is deﬁned as:

ρurdp Rep = (11) µ where us,i denotes the instantaneous gas velocity. It is the summation of mean gas flow velocity and fluctuating gas flow velocity. The mean velocity is interpolated at the parcel location from RANS equations; the eddy lifetime (ELT) dispersion model is applied to calculate the fluctuating velocity. In 00 the ELT model, the fluctuating velocity us,i is defined as [13]: r 2k u00 = ξ (12) s,i j 3 where ξj is the normally distributed random number of the standardized Gaussian distribution and k denotes the turbulent kinetic energy. The updated value fluctuating velocity should be calculated after the characteristic lifetime (τe): τe = τL ln(r) (13) − where r is a random number between 0 and 1 and Lagrangian integral time scale (τL) calculated as:

k τ = C (14) L 1 ε where C1 is an empirical factor whose value is adjusted based on the experimental results. In many studies, the value for this parameter is chosen as 0.3, which is also applied in the work of Refs [22,23]. th The momentum transfer term Sui in Equation (2) calculated for l cell in each time step as:

Np h il 1 X S = G x ω F (15) ui [l] l p,i i D,i ∆t i = 1 ∀C [l] where ∆t is the time step, is the volume of the lth cell, G (x) is the Kernel function of the lth cell, ω ∀C l i is the statistic weight of the parcel and FD,i is the drag force applied on the ith parcel. The momentum transfer term is obtained by the summation of product term for all parcels. The Kernel function is considered as a linear weighted interpolation operator in OpenFoam [24]. Computation 2020, 8, 23 5 of 13

3. Analysis and Modeling To solve the governing equations, open-source OpenFOAM package (www.openfoam.org) is employed. In OpenFOAM, the equations of the continuum phase are solved with the discrete cell-centered finite volume method on the discrete computational grid. The boundary condition at the inlet is a fixed velocity profile, zero gradient at the outlet and no-slip conditions are applied for the walls. The pressure is zero gradient for the inlet, and fixed value (0) applied to the outlet. The Concentration field is of a fixed value for the inlet and zero gradients for all others. The turbulent Computation 2020, 8, x FOR PEER REVIEW 5 of 13 intensity is 10% for the inlet, and standard wall function k-ω are used for walls. The parcels are injected atinjected a rate ofat 1,000,000a rate of 1 per,000 second.,000 per Each second. parcel Each will parcel be considered will be considered omitted after omitted reaching after the reaching outlet, andthe inoutlet approaching, and in approaching the wall, they the are wall, considered they are rebounded.considered rebounded. To considerconsider gridgrid independencyindependency for our domain, three didifferentfferent grid sizes (1,100,000,(1,100,000, 700,000,700,000, 400,000)400,000) were compared based on the velocity profileprofile ofof thethe continuumcontinuum phasephase alongalong threethree sectionssections Z/⁄D ==1010,, 20 20 andand 3030. The. The maximum maximum error error was 4% was for 4 the% 700,000-cellfor the 700 grid.,000- Therefore,cell grid. theTherefore, simulations the weresimulation dones using were 700,000done using cells 700 in the,000 domain.cells in the The domain computational. The computational domain and domain its grid and quality its grid are shownquality inare Figure shown1. A in block Figure of the 1. structured A block of grid the is structured applied between grid is the applied jet and cylindricalbetween the enclosure jet and insteadcylindrical of usingenclosure the polar instead grid of inusing order the to polar keep grid the in number order to of keep the grid the number elements of as the low grid as elements possible. Theas low PISO-SIMPLE as possible. (PIMPLE)The PISO-SIMPLE algorithm (PIMPLE) with exterior algorithm loops with is used exterior to solve loops the is pressure-velocityused to solve the couplingpressure-velocity of the Navier–Stokes coupling of the equations Navier–Stokes and increase equations the timeand increase intervals the [25 time]. The intervals time interval [25]. The is determinedtime interval dynamically is determined based dynamically on the Courant based number. on the ThisCourant number number. is kept This less number than 1 for is the kept entire less simulation.than 1 for the Di entireffusion simulation. terms are discretizedDiffusion terms following are discretized the standard following second-order the stan centraldard second differencing-order scheme,central differencing and the convection scheme, terms and arethe discretizedconvection byterms the second-orderare discretized “Gauss by the limited second Linear”-order scheme,“Gauss whichlimited based Linear on” schemeTotal Variation, which Diminishingbased on Total (TVD) Variation method. Diminishing Lagrangian (TVD) equations method. are Lagrangian integrated byequations analytical are methods integrated [26 ]. by Particle analytical tracking methods is performed [26]. Particle using tracking the Face-to-face is performed particle using tracking the procedureFace-to-face approach particle [tracking27]. procedure approach [27].

Figure 1. Computational domain and meshmesh structure.structure.

InIn thethe presentpresent study,study, didifferentfferent casescases areare consideredconsidered (summarized(summarized in Table1 1)) to to investigate investigate thethe eeffectsffects ofof two-waytwo-way couplingcoupling on thethe motionmotion ofof aa particle-ladenparticle-laden jet.jet. Since in all three mass loadings, and three investigated sizes of the particles, the same number of the parcels was injected, didifferenfferentt statistical weights were obtainedobtained forfor thethe particles.particles. The following table presents the statistical weightweight

of the particles. Stokes number () is defined as the characteristic time of Lagrangian (,) to the

characteristic time of the flow (), where each is as the following.

, = (16) = (17) , 18

= (18) The computational weight of parcels in each simulation is calculated as below:

̇ = (19) ̇

Computation 2020, 8, 23 6 of 13

of the particles. Stokes number (Stk) is deﬁned as the characteristic time of Lagrangian (τp,st) to the characteristic time of the ﬂow (τF), where each is as the following.

τp,st Stk = (16) τF

2 ρpdp τp st = (17) , 18µ

Djet τF = (18) Ujet The computational weight of parcels in each simulation is calculated as below:

. Np ω = . (19) Nc . . where Np and Nc are the number of particles and computational parcels respectively that inject per . second during the simulation. Since Nc is one million per second, we have two million parcels after . . two seconds in our simulations. Np is calculated from the injected mass ﬂow rate (minj) as below:

. . minj Np = (20) mp where mp is the particle mass.

Table 1. The parameters of four diﬀerent cases.

Mass Loading Stokes Number (Stk) Reynolds Number (Re) Computational Weight of Parcels 0.53 5 5000 40 1.06 5 5000 80 2.13 5 5000 160 2.13 5 10,000 160 2.13 5 20,000 160 2.13 1 5000 17, 20 2.13 10 5000 55

4. Validation The validation of the simulation results was conducted by comparing the experimental normalized axial velocity observations reported by Fleckhaus et al. [15]. They used glass beads with properties of 3 ρp = 2590 kg/m , dp = 64 µm, Re = 20, 000 and with a mass loading ratio of 0.3. Figure2 presents the predicted velocity of the disperse phase at three locations (Z/D = 10, 20 and 30) along the centerline. The resulting values compare well with the published experimental data in [15]. The results were normalized by the maximum velocity of continuum phases along section Z/D = 10. In Figure2, the results of simulations are compared with the experimental data from [15], which are in a very good agreement. In Figure2, dispersed and continuum phase are indicated using dashed and solid lines, respectively. Two million parcels are injected to reduce the statics errors to an acceptable value to obtain the graphs of the disperse phase for each case. Nevertheless, perturbations can be seen in the diagrams. These high-frequency oscillations are omitted using a high pass filter after applying Fourier transform in the frequency domain. There are different approaches such as filtering, parcel-number-density control algorithms which are used to achieve numerically convergent solutions in Eulerian–Lagrangian Computation 2020, 8, 23 7 of 13

(EL) simulations [21,28,29]. As it is shown in Figure2, the decay rate obtained from the simulation results for the continuum phase is higher than those obtained from the experimental data. It can be seen that at larger distances from the nozzle, the normalized velocity of the dispersed phase falls below that which is presented by the experimental data. In the case of the Lagrangian phase, the results show overestimated value for section Z⁄D = 10. However, the simulation results at Z⁄D = 20, 30 shows lower value comparing experimental result, due to a higher decay rate. The numerical dissipation can be a reasonComputation for 20 the20,high 8, x FOR decay PEER rate REVIEW of velocity proﬁles. 7 of 13

Figure 2. The comparison between normalized axial velocities resulted from simulation and Figure 2. The comparison between normalized axial velocities resulted from simulation and experimental data [15]. Dashed line is the disperse phase, the solid line is continuum phase and experimental data [15]. Dashed line is the disperse phase, the solid line is continuum phase and symbols are experimental data observed by [15]. symbols are experimental data observed by [15]. 5. Results and Discussion 3. Results and Discussion In the present study, to investigate the influence of the presence of particles on the jet motion direction,In the a present passive study, scalar to(c) investigateas a tracer the of concentrationinfluence of the of thepresence continuum of particles phase on is injectedthe jet motion at the nozzledirection (jet), a entrance.passive scalar This scalar() as shows a tracer the of concentration concentration of of a scalarthe continuum tracer which phase comes is injected out from at the jet.nozzle Figure (jet)3 ,entrance. from left This to right, scalar demonstrates shows the concentration the influence of of a mass scalar loading, tracer which Reynolds comes number out from of the the jetjet. and Figure Stokes 3, from number left to of right, the particles demonstrates on the concentrationthe influence of distribution mass loading, of the Reynolds continuous number phase of and the distributionjet and Stokes of number the axial of velocity the particles of the continuumon the concentration phase, respectively. distribution The of jet the velocity continuous profile phase at the and top ofdistribut the jetion is quite of the similar axial velocity to that of of the the spread continuum of the phase, jet tracer. respectively. In comparison, The jet at velocity the bottom profile of the at the jet, wheretop of the particlesjet is quite leave similar this regionto that ofof thethe jet, spread the rate of the of spread jet tracer. of the In jetcomparison, tracer is higher. at the The bottom amplitude of the ofjet, the where velocity the showsparticles a generalleave this form region of the of jet. the However, jet, the rate it cannot of spread present of the the jet distribution tracer is higher. of the flowThe ratheramplitude the extension of the velocity of the shows momentum a general in the form calculation of the jet. domain. However, In this it cannot study, present one aim the is to distributi determineon theof the influence flow rather of the the particles extension on theof the mass momentum motion of in the the continuous calculation phase. domain. The In contour this study, of the one velocity aim is isto showndetermine as line the profiles influence in Figureof the 3particles. Velocity, on unlike the mass concentration, motion of the does continuous not accumulate phase. atThe one contour point. Thisof the is due velocity to the is presence shown of as convection line profiles inherent in Fig inure locations 3. Velocity where, unlike there is concentration flow velocity., Moreover,does not velocityaccumulate does at not one accumulate point. This over is due time to atthe any presence given point of convection in the domain inherent as a in result locations of the where dissipation. there Usingis flow the velocity contour. Moreover, of concentration, velocity does it has not been accumulate shown that over the time released at any flow given is point concentrated in the dom at theain as a result of the dissipation. Using the contour of concentration, it has been shown that the released flow is concentrated at the bottom of the main trajectory line of the jet. This is of critical importance in waste disposition, where it may contain particles that should not be accumulated at one point.

Computation 2020, 8, 23 8 of 13 bottom of the main trajectory line of the jet. This is of critical importance in waste disposition, where it mayComputation contain 2020 particles, 8, x FOR that PEER should REVIEW not be accumulated at one point. 8 of 13

(a) (b) (c)

FigureFigure 3.3. TheThe contours contours of ofjet jettracer tracer scalar: scalar: (a) The (a) left The side left is side for different is for di massfferent loading mass ratio loadings at Re ratios = 5000, at ReStk= =5000, 5. (b) StkThe= middle5. (b) Theone is middle for different one is Reynolds for different number Reynoldss at m_p⁄m_a numbers = 2.13, at m_p Stk⁄ m_a= 5. (c=) The2.13, right Stk =side5. (cis) for The different right side Stokes is for number different at m Stokes p⁄m_a number = 2.13, Re at = m 5000. p⁄m_a The= lines2.13, represent Re = 5000. the The velocity lines contour. represent the velocity contour. This lies in the fact that the downward motion of the particles changes the momentum direction downwardThis lies and in thedeviates fact that from the the downward original path. motion Moreover, of the particles it counteracts changes the the influence momentum of the direction ambient downwardentrainment and toward deviates the fromjet in thethe originallower region path. of Moreover, the jet, which it counteracts increases the the influence rate of spread of the ambientof the jet entrainmenttracer as the towardparticles the leave jet inthe the jet. lower In Fig regionure 4, the of thepath jet, of whichthe motion increases of the the continuum rate of spread phase of is thedrawn jet tracerby connecting as the particles the maximum leave the points jet. In of Figure the trace4, ther. pathFurthermore, of the motion the rate of theof spread continuum of the phase jet tracer is drawn at the bybottom connecting of the thejet increases maximum with points growth of the in tracer. mass Furthermore,loading. Initially the, rate the ofparticles spread and of the jet jet have tracer the at same the bottomvelocity. of However, the jet increases after a while, with growth the particles in mass tend loading. to push Initially, jet in the the down particlesward anddirection jet have as a the result same of velocity.gravitational However, force. after It can a while,be said the that, particles gradually, tend the topush particles jet in have the downwardmore effects direction on the jet as streamline. a result of gravitationalAt higher Stokes force. number, It can be the said accumulation that, gradually, of continuum the particles phase have occurs more eatffects larger on the distance jet streamline. from the Atnozzle. higher This Stokes is due number, to the thepresence accumulation of larger ofparticles continuum, having phase a lower occurs area/volume at larger distance ratio, resul fromting the in nozzle.smaller Thisdrag is forces due toper the volume. presence It can of larger be said particles, that particles having with a lower a larger area diameter/volume ratio,and higher resulting stokes in smallernumber drag will forcesdecay pertheir volume. momentum It can at be larger said thatdistance particless from with the anozzle. larger diameterOn the other and hand, higher particles stokes numberwill exit will the decayjet and their transfer momentum their downward at larger distancesvelocity, as from a result the nozzle. of gravity, On theto the other surrounding hand, particles fluid. willDue exit to this, the jetsmaller and transfer particles their can downward transfer more velocity, downward as a result momentum of gravity, to tothe the fluid. surrounding As a result, fluid. jets Dueladen to wit this,h smaller smaller particles particles will can have transfer a downward more downward trend. Initially momentum (Z/D = to 0) the, the fluid. particles Asa and result, jet have jets ladenthe same with velocity. smaller However, particles will after have a while, a downward the particles trend. tend Initially to push (Z jet/D in= 0),the the down particlesward direction and jet have as a theresult same of velocity.gravitational However, force. after It can a while, be said the that, particles gradually, tend to the push particles jet in thehave downward more effects direction on the as jet a streamline. The decay rate of the axial velocity and the concentration of the continuum phase (tracer) in the direction of the jet (Figure 4) are depicted in Figure 5.

Computation 2020, 8, 23 9 of 13 result of gravitational force. It can be said that, gradually, the particles have more eﬀects on the jet streamline.Computation 20 The20, 8, decay x FOR PEER rate ofREVIEW the axial velocity and the concentration of the continuum phase (tracer)9 of 13 inComputation the direction 2020, 8 of, x FOR the jetPEER (Figure REVIEW4) are depicted in Figure5. 9 of 13 0 0

-2 -2

-4

D -4 / Y D /

Y -6 -6 Stk = 5 -8 Stk = 5 mp/ma=2.13 -8 mmpp//mmaa==21..1036

mmpp//mmaa==10..0563 -10 m /m =0.53 -10 0 5 10 p 1a 5 20 25 30 35 40 0 5 10 15 Z2/0D 25 30 35 40 Z/D Figure 4. The effect of mass loading ratio on the trajectory of the gas-phase jet at Re = 5000. Figure 4.4. The eeffectﬀect of mass loadingloading ratioratio onon thethe trajectorytrajectory ofof thethe gas-phasegas-phase jetjet atat ReRe == 5000.

m /m =0.53 1.2 p a mmp//mma==01..5036 1.2 p a mp/ma=2.13 mp/ma=1.06 1 m /m =2.13 U /U p a 1 gas in Ugas/Uin C 0.8 , n i C 0.8 , U / n i s 0.6 a U g / s

U 0.6 a g

U 0.4 0.4 0.2 C 0.2 C 0 0 5 10 15 20 25 30 35 5 10 15 Z2/0D 25 30 35 Z/D FigureFigure 5.5. TheThe eeffectffect ofof massmass loadingloading ratioratio onon thethe velocityvelocity andand tracertracer concentrationconcentration decaydecay raterate onon thethe trajectoryFiguretrajectory 5. Theofof thethe effect gas-phasegas -ofphase mass jet.jet. loading ratio on the velocity and tracer concentration decay rate on the trajectory of the gas-phase jet. TheThe resultsresults reveal that the impact impact of of mass mass loading loading on on the the gas gas-phase-phase concentration concentration decay decay rate rate is isconsiderably considerablyThe results low. low.reveal However, However, that the the impact decay the decay rateof mass of rate the loading of axial the velocity axialon the velocity gas decreases-phase decreases concentrationwith a rise with in a mass risedecay inloading. rate mass is loading.considerablyThe total The momentum totallow. momentumHowever, of the the of particle thedecay particle ladenrate of laden-jet-jet the becomes axial becomes velocity larger larger, decreases, with with an with anincrease increase a rise inin in massmass mass loading. loading. loading. Meanwhile,TheMeanwhile, total momentum thethe velocityvelocity of ofof the thethe particle gas-phase,gas-phase, laden whichwhich-jet becomes isis deteriorateddeteriorated larger and,and with scattered,scattered, an increase isis compensatedcompensated in mass loading. byby thethe transferMeanwhile,transfer ofof thethe the momentummomentum velocity of fromthefrom gas particlesparticles-phase, totowhich thethe gas.gas. is deteriorated and scattered, is compensated by the transferFiguresFig ureof the 33bb momentumandand6 Fig indicateure 6from indicate that particles the deflectionthat theto the deflection of gas. the jet pathof the is jet diminished path is diminished by increasing by theincreasing Reynolds the number.ReynoldsFigure This number. 3b is and due FigTh toisure an is increase 6 due indicate to inan jetthat increase momentum, the deflection in jet whilemomentum of the the jet gravitational ,path while is diminishedtheforce gravitational is not by strongincreasing force enough is thenot toReynoldsstrong deflect enough thenumber. jet trajectory.to Th deflectis is due Moreover, the to jet an trajectory. increase the increase in Moreover, jet in momentum jet velocity the increase, enhanceswhile inthe jet thegravitational velocity ambient enhances entrainment, force is not the andstrongambient the enough initiated entrainment to radial deflect, and velocity the the jet toward initiated trajectory. the radial jet Moreover, hinders velocity the the toward release increase ofthe the jet in particles hinders jet velocity from the release theenhances jet, which of the the leadsambientparticles to a entrainment reductionfrom the jet, in, the and which spread the leads initiated rate. to However, a reduction radial by velocity decreasingin the toward spread the Reynoldsthe rate. jet However, hinders number, th by thee decreasing release gravitational of thethe forceparticlesReynolds can deviatefrom number, the the jet, the particles which gravitational leadsfrom the to aforce jet reduction and can shift deviate in the the momentum spreadthe particles rate. to the However, from ambient, the by jetgiving decreasing and rise shift to thethe decayReynoldsmomentum rate number,of to the the axial ambient, the velocity. gravitational giving rise forceto the candec ay deviate rate of thethe particlesaxial velocity. from the jet and shift the momentum to the ambient, giving rise to the decay rate of the axial velocity.

ComputationComputation2020 ,208,20 23, 8, x FOR PEER REVIEW 10 of10 13 of 13 Computation 2020, 8, x FOR PEER REVIEW 10 of 13

0 0

-2 -2

-4 -4 D / D / Y Y -6 mp/ma=2.13 -6 mp/ma=2.13 StkS=tk5= 5 -8 Re = 5000 -8 Re = 5000 Re = 10000 Re = 10000 Re = 20000 -10 Re = 20000 -10 0 5 10 15 20 25 30 35 40 0 5 10 15 20 Z/D25 30 35 40 Z/D FigureFigure 6. The 6. The effect effect of Reynolds of Reynolds number number on theon trajectorythe trajectory of the of particle-ladenthe particle-laden jet. jet. Figure 6. The effect of Reynolds number on the trajectory of the particle-laden jet. Figure3c shows that the spread of the jet tracer suddenly increases at the bottom of the jet. FigureFigure 3c shows3c shows that that the thespread spread of theof thejet tracerjet tracer suddenly suddenly increases increases at the at the bottom bottom of theof thejet. jet. Considering Figure7, this increase is due to the particles being released from the jet bottom. These ConsideringConsidering Fig ureFigure 7, this 7, this increase increase is due is due to theto theparticle particles beings being released released from from the thejet bottom.jet bottom. These These particles drag down part of the continuous phase, which increases the depth of the continuous phase particlesparticles drag drag down down part part of the of thecontinuous continuous phase, phase, which which increases increases the thedepth depth of the of thecontinuous continuous phase phase penetration beneath the jet. Moreover, it can be seen that particles move out of the jet faster and penetrationpenetration beneath beneath the thejet. jet.Moreover, Moreover, it can it can be seen be seen that that particles particles move move out out of theof thejet fasterjet faster and and farther away due to the Stokes number rise. This leads to an increase in the depth of the continuous fartherfarther away away due due to the to theStokes Stokes number number rise. rise. This This leads leads to an to increasean increase in the in thedepth depth of the of thecontinuous continuous phase beneath the jet. Nevertheless, since the coupling amongst particles is weaker for higher Stokes phasephase beneath beneath the thejet. jet.Nevertheless, Nevertheless, since since the thecoupling coupling amongst amongst particles particles is weaker is weaker for forhigher higher Stokes Stokes number, they cannot deflect the jet trajectory and gradually depart the jet. In comparison, as Figure8 number,number, they they cannot cannot deflect deflect the thejet trajectoryjet trajectory and and gradually gradually depart depart the thejet. jet.In comparison, In comparison, as Fig as ureFigure 8 8 demonstrates, particles with stronger coupling, due to lower Stokes number, remain in the jet resulting demonstrates,demonstrates, particles particles with with stronger stronger coupling coupling, due, due to lower to lower Stokes Stokes number number, remain, remain in the in thejet jet in jet deflection. resultingresulting in jet in deflectionjet deflection. .

Figure 7. The counter of the continuous phase and position of the particles in Z/D = 20 section with FigureFigure 7.7. TheThe countercounter ofof thethe continuouscontinuous phasephase andand positionposition ofof thethe particlesparticles inin ZZ/D/D == 2020 sectionsection withwith Stokes number 1 and 10 from right to left, respectively. StokesStokes numbernumber 11 andand 1010 fromfrom rightright toto left,left, respectively.respectively.

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-2

-4 D / Y

-6 mp/ma=2.13 Re = 5000 -8 Stk = 1 Stk = 5 Stk = 10 -10 0 5 10 15 20 25 30 35 40 Z/D Figure 8. The effect of Stokes number on the trajectory of particle-laden jet. Figure 8. The effect of Stokes number on the trajectory of particle-laden jet. 6. Conclusions 4. Conclusions The motion of a particle-laden jet was simulated with the particle size of dp = 64 µm and density The motion of3 a particle-laden jet was simulated with the particle size of = 64 and of ρp = 2590 kg/m . The direction of the motion of the jet was traced by adding a tracer to the solver. = 2590 / Thedensity simulation of results were.validated The direction with of available the motion experimental of the jet was data. traced Moreover, by adding the impactsa tracer ofto Stokesthe solver. numbers, The simulation mass loading results ratio were and Reynolds validated number with available were studied. experimental The following data. statementsMoreover, canthe beimpacts inferred of S fromtokes the numbers, results: mass loading ratio and Reynolds number were studied. The following statementsThe total can momentum be inferred of from the particle-ladenthe results: jet becomes larger with an increase in mass loading and the velocityThe total of themomentum gas-phase, of whichthe particle is deteriorated-laden jet and become scattered,s larger is compensatedwith an increase by the in transfermass loading of the momentumand the velocity from of particles the gas- tophase, the gas. which is deteriorated and scattered, is compensated by the transfer of theWith momentum a decrease from in Reynoldsparticles to number, the gas. the gravity force results in deflection of the particles from the jetWith and a shifts decrease the momentumin Reynolds to number, the ambient, the gravity giving force rise toresults the decay in deflection rate of the of the axial particles velocity. from the jetAs and the shifts Stokes the number momentum increases, to the the ambient, particles giving leave rise the jetto fasterthe decay and rate are thereforeof the axial less velocity. capable of bendingAs the the Stokes continuous number phase increases, trajectory the particles of the jet. leave However, the jet faster particles and withare therefore lower Stokes less capable numbers of havebending less the slip continuous velocity than phase the continuoustrajectory of phase the jet. and However, their longer particles presence with in thelower jet resultsStokes innumber greaters gravitationalhave less slip force velocity effects than and the consequently continuous more phase jet deflection.and their longer presence in the jet results in greater gravitational force effects and consequently more jet deflection. Author Contributions: Conceptualization, A.T., and T.T.; methodology, H.T., T.T. and A.T.; software, H.T.; validation,Author Contributions: H.T., and T.T.; Conceptualization formal analysis,, A.T A.T.,., and H.T. T.T and.; methodology, T.T.; writing—original H.T., T.T. and draft A.T. preparation,; software, H.T T.T.;.; writing—review and editing, A.T.; supervision, A.T.; All authors have read and agreed to the published validation, H.T., and T.T.; formal analysis, A.T., H.T. and T.T.; writing—original draft preparation, T.T.; version of the manuscript. writing—review and editing, A.T.; supervision, A.T.; All authors have read and agreed to the published version Funding: This research received no external funding. of the manuscript. Acknowledgments: The authors would like to appreciate Compute Canada for its technical support and to provideFunding: computational This research resources.received no external funding ConflictsAcknowledgments: of Interest: TheThe authors authors declarewould like no conflict to appreciate of interest. Compute Canada for its technical support and to provide computational resources. Nomenclature Conflicts of Interest: The authors declare no conflict of interest. QUANTITY SYMBOL

kg Density ρ m3 Time (s) t Velocity (m/s) u th velocity vector in i direction (m/s) ui Pressure kg/ms2 P

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QUANTITY SYMBOL Gravity m/s2 g

th 2 Gravity in i direction m/s gi Stress tensor kg/ms2 τ

2 Drag Force (kg m/s )FD Momentum Source term kg/m2s2 S Viscosity (kg/ms) µ turbulent viscosity (Kg/ms) µt Mass (kg) m Diameter (m) d Computational weight ω Reynolds number Re Stokes number Stk

( ) 00 ﬂuctuating velocity m/s us,i particle eddy crossing time scale (s) τCross Kinematic Viscosity m2/s υ

Normally distributed random number ξj

Isotropic Lagrangian turbulence time scale (s) τL

Characteristic time of a Lagrangian phase (s) τp,st

Eulerian length scale (m) LE

Characteristic time of a continuum phase (s) τF Schmidt number Sc turbulent Schmidt number Sct

Relative velocity (m/s) ur

Particle relaxation time (s) τp

Instantaneous velocity (m/s) us

Particle Reynolds number Rep indices of x, y, z-direction i

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