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Science in China Series E: Technological Sciences© 2008 SCIENCE IN CHINA PRESSReceived December 25, 2007; accepted February 3, 2008doi: 10.1007/s11431-008-0063-3†Corresponding author (email: chenys@)Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | Aerodynamic simulation of high-speed trains based on the Lattice Boltzmann Method (LBM)WANG YiWei, WANG Yang , AN YiRan & CHEN YaoSong†Department of Mechanics and Engineering, Peking University, Beijing 100871, ChinaAerodynamic simulation of high-speed trains has been carried out by using Lattice Boltzmann Method (LBM). Non-simplified train model was used and the number ofspace grids reached tens of millions. All results under different working conditions reflected the actual situation.Lattice Boltzmann Method (LBM), aerodynamic simulation, engineering applicationBoltzmann equation describes the kinetics of gas based on theory of molecular motion. But be-cause of the complexity of the equation, it is almost incapable of dealing with practical gas ki-netic except deducing a few thermodynamic relations. However, due to the emergence of giant computer, a method of practical value―the Lattice-Boltzmann Method (LBM)[1—3]―has been developed with the help of earlier lattice algorithm since the end of 1980s, which made fluid me-chanics from continuum level to molecular level and has become an advanced method of compu-tational fluid dynamics. After decades’ effort, excellent computational software has been given birth from theoretical LBM, which is now widely applied to the research and development of new products in some main pillar industries[4]. For instance, this computation software of fluid me-chanics is adopted by almost all the automobile manufacturing companies to optimize the aero-dynamic performance of their new models. Nowadays, China’s development of high-speed trains has just started, and the aerodynamic influence plays an important role in this field considering the high speed of trains. In this paper, LBM is used to study the aerodynamic performance of trains running in open air, in tunnel and in strong crosswind, respectively, which provides a tech-nological basis for the design of high-speed trains.Recently, with great development of the high-speed trains, it becomes quite urgent to obtain their aerodynamic performance. For high-speed trains at speed of over 200 km/h, their air drag reaches more than 75% of the total drag[5], which produces a direct impact on the power equip-ment of engines and the collocation of power grid. And what is more, high-speed trains running774 WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783under strong crosswind are very likely to be turned over. As a result, it is both necessary and im-portant to study the aerodynamic performance and evaluate the running safety of high-speed trains.As a key method to study aerodynamic problems, Computational Fluid Dynamics (CFD) has been paid more and more attention to due to its advantage of saving time and money compared to experiment. Nowadays, the most common approach of CFD is solving Navier-Stocks equation based on continuum theory by discretizing it. However, due to the limited capabilities of modern computers, a great number of physical phenomena of fluids, such as turbulence, cannot be real-ized using the direct numerical simulation (DNS) approach. Various physical approximations have to be added. But these physical approximative models are not applicable to practical engi-neering problems concerning flow around cars, trains or other bluff body due to low accuracy of results (such as RANS model [6]) or tremendous quantity of solution process (such as LES/DES models [6]). However, in some area, LBM has an advantage over the computation methods based on Navier-Stocks equations.1 Governing equationsBoltzmann equation is more basic than N-S equation to reflect fluid mechanism [7, 8], from which the LBM method is developed with the help of the progress in computing technologies and the success in lattice algorithms. A standard Boltzmann equation can be described mathematically as follows:(,)(,)(),i i i i f t t t f t C f +∆+∆=+x c x (1) where i f represents the particle distribution function, i C represents the collision term, and i c represents the particle velocity along the direction i [9]. The most common form of the collision term is the so called BGK [10, 11] model: eq(),i f f C f τ−=− (2)where eq f represents equilibrium distribution function, and τ is the relaxation time for the particle distribution to converge to the equilibrium state by collision. It can be proved that the system described by eq. (1) basically satisfies the N-S equation at the macroscopic level under condition that the discrete velocity set satisfies certain symmetry requirements.And the hydrodynamic variables can be obtained by simple summations given below:(,)(,),b i i t f t ρ=∑x x (3)(,)(,),b i i i t f t ρ=∑u x c x (4) 2(,)()(,),b i i i e t f t ρ=−∑x c u x (5) where ρ is fluid density, (,)t u x is fluid velocity, and (,)e t x is energy density.Turbulence is simulated based on the principle of performing very large eddy simulations (VLES)[12] that directly simulate resolvable flow scales while model unresolved scales by using an RNG form of k-epsilon equations with proprietary extensions to achieve VLES time accurate physics.WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783 7752 Numerical model2.1 Vehicle geometryA three-department train model (including the head vehicle, tail vehicle and an intermediate ve-hicle) is considered, which comes from the CAD (computer aided design) data of a real high-speed train (including the bogie and non-simplified doors and windows). The surface gridsgenerated are shown in Figures 1 and 2.Figure 1 Surface grids of the head train.Figure 2 Surface grids of the bogie. 2.2 Computing gridThe Cartesian grid form is adopted to generate the space grid, which comprises nearly 40 million grids when the crosswind is not considered and over 57 million grids when the crosswind is taken into account. Grids near the train, the bogie and the pantograph have been refined, and the mini-mum scale of the grid is 1 cm (see Figure 3).2.3 Computing conditions2.3.1 Working conditionsRunning in open air at speeds of 200, 250 and 300 km/h, respectively. Running in tunnel at speeds of 200, 250 and 300 km/h, respectively.776 WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783Running at a speed of 200 km/h under a strong crosswind of 364 m/s.Figure 3 Sectional plane of space grids.2.3.2 Boundary Conditions [13]. As shown in Figure 4, among the 5 faces of the most outside cube, the left face represents velocity inlet from which the air blows in at the speed of trains when running in the open air, and a modification of piston wind is considered when the trains running in tunnel. The right face represents pressure outlet with the pressure of 101325 Pa, and the other 3 are referred to as the walls of wind tunnel. The gray area represents the floor, at the top of which and under the train, there is a thin slice that has been set as a conveyor which moves at the same speed as the running train. Above the floor, there are two rail tracks which also move at the speed of running trains.Figure 4 Schematic diagram of the model.The wheels are set as rotating walls, which rotate around their corresponding center with their outside edge line speed equal to the speed of running trains.The surface of the train, the pantograph and the bogie part excluding the wheels are set as sta-tionary walls.2.3.3 Modification of piston wind. While the train is running in the tunnel, its piston effect will produce piston wind that moves in the same direction as the train. And the inlet velocity should be set as the speed of the train deducting that of the piston wind. According to the wave propaga-tion theory, the equation that the one-dimensional piston wind must obey can be deduced as 1,2cu p ρ=∆ (6) where ρ is air density, c is the local Mach number, u is the speed of piston wind and p ∆WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783 777is pressure difference of wave source located near the surface of the train. And u can be ob-tained by repeatedly calculating flow field starting with an assumed u and revising it using eq.(6). But in practice, considering the enormous amount of computation, u is attained by using the Dynamic Mesh Method of software FLUENT.3 Computation results3.1 Aerodynamic resultThe drag, lift and lateral force arise from the resultant forces of the corresponding directions. For convenience of analyzing, a dimensionless force coefficient is defined as 2,2F C V S ρ= (7) where F is the resultant force of a direction, ρ is density, V is the speed of the train and S is the reference area. The largest vertical projection, horizontal projection and lateral projection areas are referred to as reference areas for drag, lift and lateral force, respectively.Force conditions of trains under different working conditions are addressed as in Figure 5.Figure 5 The force coefficients and the distribution of force.The study of flow around a Smooth cylinder with high Reynolds number indicates that when the Reynolds number is larger than 5×105, it can be considered that the flow has evolved into a state of self similarity, thus the flow characteristics will become irrelevant to Reynolds number. Flow around a train is much more complicated than around a cylinder, and its critical Reynolds number is smaller as well. The Reynolds number of flow around a train at the speed of 200 km/h has an order of 107, which means the flow has already entered its self similarity state. As a result, Cp, the surface dimensionless pressure coefficient of the train, will not change with the increase in speed, which makes the dimensionless force coefficients of trains at different speeds very close to each other.3.2 Aerodynamic analysis of trains running in the open airThe coefficient of air drag which consists of pressure drag and friction drag is one of the most important characters to measure the aerodynamic performance of trains [14]. The pressure drag arises from pressure difference between the front and back surfaces of every component of the train, as shown in Figure 6. When the surface flow of the train encounters a protruding, the veloc-778 WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783ity of the flow will decrease, which results in the increase of surface pressure. For example, when the flow hits the nose of the train, the highest pressure is produced due to stagnation which makes the velocity of flow nearly zero, and when the flow encounters pantograph and bogie, the pres-sure will also rise under the influence of stagnation. At the contact points of the wheels and tracks, the rotation of wheels gives rise to high pressure in front of these points and low pressure behind them. Near the doors and windows, flow separation and reattachment are noticeable as well, which lead to apparent pressure gradient in these areas, as shown in Figure 7. At the tail of the train, there occurs obvious boundary layer separation as shown in Figure 7 that produces reflux area to form a low pressure that causes pressure difference.Figure 6 Surface pressure distribution curve of train axis.Figure 7 Friction curves and separation points at the tail of the train.When the viscous air is moving along the train, a boundary layer is produced near train surface within which the velocity of flow will increase rapidly from zero on the surface to almost the same as inlet flow at the top of the layer. The thickness of boundary lay is zero at the stationary point and will grow thicker gradually from head to downstream, as shown in Figure 8. Viscous shear stress on train surface is produced due to different velocity layers in boundary layer, the integral force of which in the direction opposite to running train forms the total friction drag of the train.WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783 779Figure 8 Surface boundary layer distribution of the train.Overall, facing the wind directly, the head vehicle is the main drag resource of the train which accounts for half of its total drag. And the drag of tail vehicle is the second important part. Being designed very scientifically, the train in our study has a weak separation at its tail, as shown in Figure 9, while running in open air, because the separation point is postponed to a very later posi-tion as shown in Figure 7, which lessens the drag of tail vehicle to the same level as intermediate vehicle and pantograph. Also facing the wind, the strip-structure pantograph impedes the flow seriously, which makes considerable contribution to total drag and leaves much possibility to im-prove. The direct component of drag from wheels and bogie is relatively small, but their exis-tence can impact the aerodynamic force of body and lead to an indirect influence on the total drag,too.Figure 9 Space streamlines at the tail of the train.Aerodynamic lift is another important aspect to measure the aerodynamic performance of trains. Considering the wheel-rail train system, a positive lift will decrease the friction between wheels and tracks, which can make them “float” in air; the train will even be derailed if the lift is too big. By contrast, a negative lift will increase dynamic axle load of trains, intensify shocks between the train and tracks and fasten the wear of wheels and tracks. According to these, a zero lift of the train is desired. And the lift in this case performs quite well, for all the lift coefficients780 WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783under different conditions have an order of 10―3 or lower (see Figure 5).3.3 Aerodynamic analysis of trains running in tunnelWhile running in tunnel, the train is just like a piston moving in a cylinder, generating a strong piston wind. Restricted by the tunnel, the air in front of train head is compressed intensely to produce a great positive pressure gradient (shown in Figure 10), while the air behind the train rearexpands to produce a great negative pressure gradient (see Figure 10).Figure 10 Pressure contours on the surface of the train and tunnel.The interaction between tunnel and train makes the whole field much more complicated com-pared to the situation when the train runs in open air: Not only the friction lines go upward, but they also curl in large area; the separation phenomena reflected by iso-surface of total pressure coefficient (see Figure 11) is more notable as well; and the wake also performs apparently un-steady and asymmetric, accompanied by the break and shedding of vortexes (see Figure 12).Figure 11 Contrast of zero valued total pressure coefficient iso-surface between running in open air (upper) and in tunnel (lower).It is conspicuous that the drag coefficient will fluctuate with time due to the great chaos of flow field, and the average value of drag coefficient is 1.5 times as much as that in the situation when the train runs in open air (as shown in Figure 5, and the pantograph is not considered in the condition of trains running in tunnel).3.4 Aerodynamic analysis of trains running under strong crosswindThe aerodynamic performance of the train deteriorates greatly while it runs under strong cross-wind [15]. The aerodynamic drag, lift and lateral force increases promptly, with a high rolling mo-ment which brings serious impact on the stability of the train. When the train is running in strongWANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783 781crosswind, the coupled wind leads to a quite different result compared to the condition of running in open air. The air flows around the train to its side rear and produces huge vortexes. In the area near rear and the area between two departments, vortex shedding occurs continuously to result inthe similar cyclical change of some variables (see Figures 13(a) and (d).Figure 12 Velocity-time diagram of wake on a horizontal cross section.Figure 13 (a) Space streamlines; (b) friction lines and velocity distribution of head vehicle; (c) pressure curves on symmetric face and horizontal cross section; (d) velocity-time diagram on the horizontal cross section.The stagnation points which are usually at the nose and at the top of the plate move to the up-wind side. On top surface of the train, the air velocity is the fastest in the fuscous area shown in Figure 13(b), where the surface pressure is quite small and notable separation area can be ob-782 WANG YiWei et al. Sci China Ser E-Tech Sci | Jun. 2008 | vol. 51 | no. 6 | 773-783 served. When the air is passing through the bogie, due to its complicated structure, the flow field behind it exhibits very unstable, and the iso-surface of vorticity seems quite chaos, too. Overall, the pressure on the bottom is much higher than that on the top, which gives birth to a large lift on the train (see Figure 13(c)).Most of the friction lines flow around the train from above, while a small number of them go through from below. In the wake zone near the leeward side of the train, a great deal of separation and reattachment occur and even second or higher order vortices appear (see Figure 14) which make the pressure on the leeward side very low. While on the windward side, the pressure is high because of the stagnation of air. Both of them are the cause of the big lateral force and rollingmoment (see Figure 13(c)).Figure 14 Streamlines and vortexes on vertical cross section of head and rear. 4 ConclusionCompared with traditional CFD method [16, 17] which deals with N-S equation, LBM has a serial of advantages such as saving memory, high efficiency of parallel computation and convenience of using complex geometry, all necessary for dealing with computation of high-speed trains. Pow-erFLOW, the only commercial software that is based on LBM, is adopted in this paper, which, limited to the low-speed aerodynamic area (Mach number <0.5)[18] at present, will soon be appli-cable to more fields such as high-speed, multiphase flow, etc. Not only should the LBM be adopted widely, but it is also necessary to develop new LBM algorithm that can be applied to new fields.We would like to thank EXA Corporation for freely providing the software used in this paper. 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