Large Eddy Simulation of Particle Wake Effect and RANS Simulation of Turbulence Modulation in Gas-Particle Flows *

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1 Chin. J. Chem. Eng., 15(1) 1 16 (007) Large Eddy Simulation o Particle Wae Eect and RANS Simulation o Turbulence Modulation in Gas-Particle Flows * ZENG Zhuoxiong( 曾卓雄 ) a,b, ZHOU Lixing( 周力行 ) a, ** and QI Haiying( 祁海鹰 ) c a Department o Engineering Mechanics, Tsinghua University, Beijing , China b Department o Mechanical Engineering, Nanchang Institute o Aeronautical Technology, Nanchang , China c Department o Thermal Engineering, Tsinghua University, Beijing , China Abstract The turbulence enhancement by particle wae eect is studied by large eddy simulation (LES) o turbulent gas lows passing a single particle. The predicted time-averaged and root-mean-square luctuation velocities behind the particle are in agreement with the Reynolds-averaged Navier-Stoes modeling results and experimental results. A semi-empirical turbulence enhancement model is proposed by the present authors based on the LES results. This model is incorporated into the second-order moment two-phase turbulence model or simulating vertical gas-particle pipe lows and horizontal gas-particle channel lows. The simulation results show that compared with the model not accounting or the particle wae eect, the present model gives simulation results or the gas turbulence modulation in much better agreement with the experimental results Keywords large eddy simulation, gas-particle low, turbulence modulation 1 INTRODUCTION Turbulence enhancement by the particle wae eect is an important aspect o gas-particle interaction. It is well nown that the wae ormation and shedding o vortices behind particles will induce gas turbulence. Signiicant advances toward the understanding o the mechanism o turbulence enhancement have been achieved. Yuan and Michaelides[1] proposed a semi-empirical mechanistic model, in which the velocity deect in the wae is responsible or the augmentation o gas turbulence and the wor associated with the motion o a particle is responsible or the attenuation o gas turbulence. This model has not been used to predict practical gas-particle lows. Yarin and Hetsroni[] utilized a similar idea, with more detailed description o the wae, but such an analysis has not been incorporated into conventional turbulence models or gas-particle lows. Kenning and Crowe[3] proposed a turbulence modulation model in gas-particle lows based on the wor done by the particle drag and the dissipation based on a length scale corresponding to the inter-particle spacing. Yu and Zhou[4] proposed a turbulence enhancement model taing the particle diameter as the mixing-length, but the predicted results or gas-particle pipe lows gave only qualitative agreement with the experimental results. Fundamental studies on particle-wae induced turbulence are conducted by many investigators. Most numerical predictions o separated lows passing over a sphere are obtained rom Reynolds-averaged Navier-Stoes (RANS) modeling[5]. There is an increasing interest in the use o large eddy simulation (LES) to predict lows past a sphere[6], however, no statistical data o turbulence enhancement are given. In this paper, the gas turbulent lows passing over a single particle is simulated using LES, and the prediction results o turbulence enhancement are compared with the experimental data and RANS modeling results, in order to validate a turbulence enhancement model by the particle wae eect. The proposed turbulence enhancement model is then incorporated into a second-order moment two-phase turbulence model to simulate vertical gas-particle pipe lows and horizontal gas-particle channel lows. The prediction results by the two-phase turbulence models taing and not taing into account the particle wae eect are compared with each other and with the experimental data. LES OF TURBULENT GAS FLOWS PASSING A SINGLE PARTICLE The gas turbulent lows passing over a single particle are simulated using LES. The iltered large-eddy continuity and momentum equations are ρ + ( ρu ) = 0 (1) U i p τ i ( ρ U i ) + ( ρu iu ) = t x x μ x () xi The Smagorinsy-Lilly model[7] is adopted or the sub-grid scale stress τ i, and LES is perormed using a commercial Computational luid dynamics (CFD) sotware Fluent 6.1. For the 3-dimensional computational domain, the Figure 1 The computational domain Received , accepted * Supported by the Major Project o National Natural Science Foundation o China (No ) and the Postdoctoral Science Foundation (No ). ** To whom correspondence should be addressed. zhoulx@tsinghua.edu.cn

2 Large Eddy Simulation o Particle Wae Eect and RANS Simulation o Turbulence Modulation in Gas-Particle Flows 13 size in the streamwise direction is about 30 times o the particle diameter D, and the width in transverse and spanwise directions are about 1 times o the particle diameter. In order to control the mesh distribution, the domain surrounding a particle is divided into our zones that have dierent mesh distribution, so that the grid points are clustered near the surace and in the wae region. A sphere is located at the origin o the three-dimensional Cartesian coordinate system, and the uniorm inlow enters the domain in the direction parallel to the x-axis. Zone Ⅰ is a spherical volume o constant diameter 4D surrounding the sphere, and it is designed to have very ine grids, the size o mesh is set to about D/14, which is the smallest in the entire domain. Zone Ⅱ is a buer region, which is also spherical with the diameter o 9D, the size o mesh in this zone is about D/8. The size o the mesh in Zone Ⅲ and Zone Ⅳ is set to about D/4 throughout the computation. In LES, the time step is taen as 0.001s. For each time step the convergence can be reached ater about 0 iterations. For the numerical procedure, the PISO algorithm[8] is used or p-v corrections, the second order implicit dierence scheme or the time-dependent term, the QUICK (second order) dierence scheme or the convection term and the central dierence scheme or the diusion term are adopted. Running a case in a PC computer with a G EMS memory and double.8g CPU needs about two months to mae the low ield reaching a statistical steady state. 3 LES RESULTS AND DISCUSSION For gas low passing over a particle[9], the inlet air velocity is 1.8m s -1, the turbulence intensity is 0.039, the air inematic viscosity is m s -1, the particle diameter is D= m, the particle Reynolds number is 610. Figs. to 4 show the LES predicted gas time-averaged, root-mean-square (RMS) luctuation velocities and turbulence inetic energy in the region behind the particle on the axis and their comparison with the RANS modeling (using Reynolds stress model) results. For RANS modeling, the air density is 1.g m -1 and the dynamic viscosity is g m -1 s -1. The particle diameter is 1.0mm, the inlet air streamwise velocities is in the range 6 13m s -1, and the particle diameter is in the range o mm. The 3-D computational domain and the setting o mesh are similar to those in LES. It can be seen that the two models give almost the same time-averaged velocity (Fig.). Qualitatively, the predicted gas RMS luctuation velocity and turbulent inetic energy by LES and RANS modeling has similar distribution (Figs.3 and 4 ). Figure 5 gives instantaneous vorticity maps at dierent time instants, showing the asymmetrical vortex structures behind the particle. 4 TURBULENCE ENHANCEMENT MODEL Figures 6 and 7 show the ationship o the turbulence enhancement due to particle wae eect based on RANS modeling results or a single particle. In Fig.6, when the particle size eeps constant, Figure Figure 3 Figure 4 The gas time-averaged streamwise velocity behind the particle on the axis LES; RANS The RMS luctuation velocity behind the particle on the axis Exp.; LES; RANS The turbulence inetic energy behind the particle on the axis LES; RANS such as d p =1.0mm, the magnitude o the turbulence enhancement due to the wae eect increases with the increase o inlet velocity, it is approximately expressed by ΔK U, where Δ K is the dierence between the maximal turbulent inetic energy behind the particle and the inlet turbulent inetic energy. In Fig.7, when the inlet velocity eeps constant, such as U =10 m s -1, the magnitude o turbulence enhancement increases with the increase o particle size roughly in a linear way, ΔK dp. The increase o turbulence intensity by the particle wae eect or dierent inlet velocities and particle sizes can thus be summarized as Chin. J. Ch. E. 15(1) 1 (007)

3 14 Chin. J. Ch. E. (Vol. 15, No.1) Figure 6 (a) t=5.15s (b) t=.85s (c) t=0.58s Figure 5 Vorticity maps Turbulence enhancement or various inlet velocities (d p =1.0mm) calculation; it line we have the turbulence enhancement by a particle as G U d (3) pw p Equation (3) is valid or the eect o a single particle. For practical gas-particle lows with multiple particles, the production term should be directly proportional to number density n p or the particle volume raction: α 3 p = n p π d p /6, so we have α pu pw p p p G n U d The conventional particle source term (production/dissipation term) due to the existence o particles in the gas turbulent inetic energy equation or Reynolds stress equation, in which particles are treated as point sources[10,11], is αρ p p Gp = ( vpivi ) (4) τ rp Perorming dimensional analysis and using the orm given in Eq.(4), we proposed the ollowing semi-empirical turbulence enhancement model by the particle wae eect as p pu Gpw = C ρα (5) τ where the empirical constant C is taen as C=3.0. is the particle axation time: ρpdp τ rp = /3 18μ ( 1 + Rep /6) Re p is the particle Reynolds number: αρ dp U U p Rep =. μ rp d τ rp Figure 7 Turbulence enhancement or various particle sizes (U = 10m s -1 ) calculation; it line ΔK U d Based on the above-obtained results, accounting or the eect o both particle size and inlet velocity, p 5 SIMULATION OF PRACTICAL GAS-PARTICLE FLOWS USING RANS MODELING WITH THE PARTICLE WAKE EFFECT Now, as the second step, the turbulence enhancement model is incorporated into the second-order moment two-phase turbulence model[11,1] or simulating practical gas-particle lows. In the present model, the gas Reynolds stress equation with the particle source terms accounting or the wae eect should be ( αρ uu i j) ( αρ Uuu i j) + = D, ij + P + Π ε + G + G δ (6), ij, ij, ij p, ij pw ij The transport equation o the dissipation rate o gas turbulent inetic energy is ( αρε ) ( αρ Uε ) + = x February, 007

4 Large Eddy Simulation o Particle Wae Eect and RANS Simulation o Turbulence Modulation in Gas-Particle Flows 15 ε Cαρ uul + ε xl 1 ε C ( P + G p ) C αρε + C G τ e where ε1, ε ε3 pw [13] C ε 3 = 1.8, e min ( τ rp, / ε ) τ =. (7) The terms on the right-hand side o Eqs.(6) and (7) are given in Re.[10,11]. The boundary conditions can be ound in Re.[13,14], taing into account the particle collision with rough walls. 6 SIMULATION RESULTS FOR GAS-PARTICLE FLOWS The proposed model is at irst used to simulate vertical gas-particle lows, measured by Tsuji et al.[15]. The pipe inner diameter is D=30.5mm, the mean air velocity is in the range o 8 0m s -1, three inds o particles (0.mm, 0.5mm, 1mm) were used as the particle phase with the particle density 100g m -3, and the mass loading ratio up to 5. The measurements were perormed at a distance o 5.11m rom the entrance. Figs.8 to 10 give the RMS gas luctuation velocities or dierent sizes o particles. It is ound that the results obtained using the model accounting or the particle wae eect are in much better agreement with the experimental results than those obtained using the model not accounting or the particle wae eect in predicting the ollowing phenomena: 0.mm particles attenuate gas turbulence, 0.5mm particles enhance or attenuate gas turbulence at dierent locations, and 1mm particles enhance gas turbulence intensity. Figure 9 Air turbulence intensity o 0.5mm particles experiment m=0, U in =13.4m s -1 ; experiment m=3.4, U in =10.7m s -1 ; m=0, U in =13.4m s -1 ; m=3.4, U in =10.7m s -1 (wae eect); m=3.4, U in =10.7m s -1 (no wae eect) Figure 10 Air turbulence intensity o 1mm particles (U in =13.4m s -1 ) experiment m=0; experiment m=0.6; m=0; m=0.6 (wae eect); m=0.6 (no wae eect) o 5.8m rom the entrance, to ensure almost ully developed low conditions. Figures 11 to 13 show the predicted gas streamwise time-averaged velocity and RMS streamwise and transverse luctuation velocities respectively. The particle wae eect almost has no eect on the gas time-averaged velocity (Fig.11). For the gas RMS luctuation velocities, the present model gives much better simulation results than the model not taing into account the particle wae eect (Figs.1 and 13). Figure 8 Air turbulence intensity o 0.mm particles experiment m=0, U in =13.4m s -1 ; experiment m=0.5, U in =13.1m s -1 ; m=0, U in =13.4m s -1 ; m=0.5, U in =13.1m s -1 (wae eect); m=0.5, U in =13.1m s -1 (no wae eect) The proposed model is then used to simulate horizontal gas-particle channel lows or low roughness, measured by Kussin and Sommereld[16]. The channel is 6m long with height o 35mm, particles have a density o 500g m -3, mean particle diameter o 100μm, mass loading o 0.3, the average inlet velocity is u in =19.7m s - 1. The measurements were perormed near to the end o the channel at a distance Figure 11 Gas streamwise time-averaged velocity experiment; without wae eect; with wae eect Chin. J. Ch. E. 15(1) 1 (007)

5 16 Chin. J. Ch. E. (Vol. 15, No.1) x,y,r coordinates, m α volume raction ε -3 turbulent inetic energy dissipation rate, m s μ molecular viscosity, Pa s ρ density, g m -3 τ e eective character timescale, s τ rp particle axation time, s Superscripts temporal average Subscripts gas phase i,j,,l dimensional index in inlet p particle phase ative Figure 1 Gas streamwise RMS luctuation velocity experiment; without wae eect; with wae eect Figure 13 Gas transverse RMS luctuation velocity experiment; without wae eect; with wae eect 7 CONCLUSIONS (1) The LES o gas turbulent low passing a single particle gives reasonably the wae eect on gas turbulence. () A semi-empirical turbulence enhancement model is proposed (3) For practical gas-particle lows, the model accounting or the particle wae eect can simulate well the gas turbulence modulation, giving much better prediction results than the model not accounting or this eect. NOMENCLATURE C model constant D, d diameter, m G source term in equation H height, m K - turbulent inetic energy, m s m mass loading ratio R radius, m Re Reynolds number t time, s U mean velocity components, m s -1 u, v luctuation velocity components, m s -1 u pressure-strain term REFERENCES 1 Yuan, Z., Michaelides, E.E., Turbulence modulation in particle lows a theoretical approach, Int. J. Multiphase Flow, 18, (199). Yarin, L.P., Hetsroni, G., Turbulence intensity in dilute two-phase lows: the particles-turbulence interaction in dilute two-phase low, Int. J. Multiphase Flow, 0, 7 44(1994). 3 Kenning, V.M., Crowe, C.T., On the eect o particles on carrier phase turbulence in gas-particle lows, Int. J. Multiphase Flow, 3, (1997). 4 Yu, Y., Zhou, L.X., Modeling o turbulence modiication using two-time-scale dissipation models and accounting or the particle wae eect, In: ASME-FED Summer Meeting, Paper FEDSM , Honolulu, Hawaii (003). 5 Johnson, T.A., Patel, V.C., Flow past a sphere up to a Reynolds number o 300, J. Fluid Mech., 378, 19 70(1999). 6 Constantinescu, G.S., Squires, K.D., LES and DES investigations o turbulent low over a sphere at Re=10000, Flow Turb. Combust., 70, 67 98(003). 7 Lilly, D.K., A proposed modiication o the Germano subgrid-scale closure method, Phys. Fluids A, 4, (199). 8 Issa, R.I., Solution o the implicitly discretized luid low equations by operator splitting, J. Comput. Phys., 6, 40 65(1986). 9 Wu, J.S., Faeth, G.M., Sphere waes at moderate Reynolds numbers in a turbulent environment, AIAA J., 3, (1994). 10 Zhou, L.X., Dynamics o Multiphase Turbulent Reacting Fluid Flows, National Deense Industry Press, Beijing (00). (in Chinese) 11 Zhou, L.X., Chen, T., Simulation o swirling gas-particle lows using K-ε-K p and USM two-phase turbulence models, Powder Technol., 114, 1 11(001). 1 Lain, S., Sommereld, M., Turbulence modulation in dispersed two-phase low laden with solids rom a Lagrangian perspective, Int. J. Heat Fluid Flow, 4, (003). 13 Zhang, X., Zhou, L.X., A second-order moment particle-wall collision model accounting or the wall roughness, Powder Technol., 159, (005). 14 Zhou, L.X., Zhang, X., Simulation o sudden-expansion and swirling gas-particle lows using a two-luid particle-wall collision model with consideration o the wall roughness, Acta Mechanica Sinica, 0, (004). 15 Tsuji, Y., Moriawa, Y., Shinomi, H., LDV measurements o air-solid two-phase low in a vertical pipe, J. Fluid Mech., 139, (1984). 16 Kussin, J., Sommereld, M., Experimental studies on particle behavior and turbulence modiication in horizontal channel low with dierent wall roughness, Exp. Fluids, 33, (00). February, 007