Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 SAE TECHNICAL PAPER SERIES 980782 Modeling of 1-D Unsteady Flows in I.C. Engine Pipe Systems: Numerical Methods and Transport of Chemical Species A. Onorati and G. Ferrari Department of Energetics, Politecnico di Milano Reprinted From: Modeling of SI and Diesel Engines (SP-1330) International Congress and Exposition Detroit, Michigan February 23-26, 1998 400 Commonwealth Drive, Warrendale, PA 15096-0001 U.S.A. Tel: (724) 776-4841 Fax: (724) 776-5760 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 The appearance of this ISSN code at the bottom of this page indicates SAE’s consent that copies of the paper may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay a $7.00 per article copy fee through the Copyright Clearance Center, Inc. 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Printed in USA Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 980782 Modeling of 1-D Unsteady Flows in I.C. Engine Pipe Systems: Numerical Methods and Transport of Chemical Species A. Onorati and G. Ferrari Department of Energetics, Politecnico di Milano Copyright © 1998 Society of Automotive Engineers, Inc. finite-volume shock-capturing schemes, to solve the hyperbolic system of conservation laws. However, the MOC still retains an important role in the modeling of the flow boundary regions, most models using a characteristic-based approach for the quasi-steady boundary conditions [8]. The most common techniques today are certainly the Lax-Wendroff method [4,9] in different forms, with the addition of flux-limiters, and the FRAM algorithm [10]. The improvement of the predicted results due to the substitution of the MOC with the above shockcapturing schemes has been described extensively by various authors [3-5,11,12]. However, the evolution of the models has generally involved not only the numerical techniques adopted, but also the boundary conditions, the pipe-wall heat transfer and friction sub-models, the thermodynamic combustion models, etc. Thus in recent years the prediction of 1-d unsteady flows in engine ducts has become a mature area, even if there are some important, forefront research fields that still need to be deeply explored, to further improve the simulation programs. For example, a careful investigation is necessary for the application of new numerical methods, continuously improved to become less diffusive, more accurate and computationally more efficient. Moreover, the tracking of chemical species along the intake and exhaust pipe systems is a promising research area [3,13,14,15], which has not been fully investigated yet. At present only few numerical models allow for the propagation of chemical species along the engine ducts, whereas this aspect is fundamental for the prediction of engine emissions and catalyst performance. Besides, the modeling of complex engine mufflers, involving perforates and absorptive silencers, can be further improved from an acoustical and fluid dynamic point of view, resorting to acoustically equivalent duct systems [16,17]. Finally, another forefront research field is certainly the coupling of the 1-d simulation of unsteady flows in simple pipes with the multi-d simulation of complex flow regions, typically arising in junctions of ducts, abrupt are changes, catalysts, silencing elements and so on (hybrid models). ABSTRACT The paper describes recent advances in the research work concerning the 1-d fluid dynamic modeling of unsteady flows in i.c. engine pipe systems. A comprehensive simulation model has been developed, which is based on different numerical techniques for the solution of the fundamental conservation equations. Classical (MacCormack method plus TVD algorithm) and innovative (the CE-SE method, the discontinuous Galerkin FEM) shock-capturing schemes have been compared, considering the shock-tube problem and the shock-turbulence interaction problem. Moreover, the tracking of the chemical species along the intake and exhaust duct systems has been investigated, introducing the species continuity equations in the numerical model. The engine test case reported in the paper points out the predicted transport of chemical species in the ducts. In particular, the back-flow of combustion products in the intake pipe and the through-flow of air in the exhaust duct during the valve overlap have been simulated. INTRODUCTION The numerical simulation of i.c. engine wave dynamics in the intake and exhaust pipe systems represents an essential tool to assist the design and development of new engines. The use of fluid dynamic simulation codes throughout the industry has notably increased in the last years, due to the significant advances of numerical models in terms of reliability, accuracy, robustness and flexibility, and to the adoption of user-friendly programs to create input files and post-process the output data. The complete intake and exhaust system simulation is fundamental for a fast optimization of manifolds with fixed and variable geometry, silencer shape and location, valve timing and valve cam profile, level and quality of tailpipe noise. The intense research work of the last decades has resulted in the development of several different 1-d fluid dynamic simulation codes [1-6] (to cite just a few). In most numerical models the traditional mesh-method of characteristics (MOC), which has been the dominant technique up to the mid 1980’s [7], has been replaced by modern upwind and symmetric finite-difference or The above areas are currently being investigated at the Department of Energetics of the Polytechnic University of Milan, and an advanced fluid dynamic model for the sim1 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 servation equations for one-dimensional, unsteady, compressible flows in ducts with variable cross-sectional area. ulation of one-dimensional unsteady flows in engine ducts, named GASDYN, has been developed. This model is based on classical and innovative shock-capturing numerical methods, and allows for the transport of chemical species along the engine ducts. In particular, the paper deals with the application of new numerical techniques to solve the hyperbolic system of conservation equations, such as the Conservation Element - Solution element method (CE-SE) and the discontinuous Galerkin finite element method (FEM), which are compared to classical symmetric shock-capturing schemes, such as the two-step Lax-Wendroff method and the MacCormack predictor-corrector method, with the addition of a TVD (total variation diminishing) algorithm. The advantages of these innovative methods are illustrated, on the basis of numerical results concerning both the shock-tube and shock-turbulence interaction problems and engine test cases. Moreover, the modeling of the advective transport of chemical species along the engine ducts is investigated, discussing the different assumptions which can be made to supersede the classical hypothesis of perfect gas with constant specific heats. The predicted results shown in the paper highlight the propagation of the typical exhaust chemical species (calculated by a thermodynamic combustion model with chemical equilibrium assumption) along the exhaust and intake ducts since the opening of the valves. The occurrence of contact discontinuities, originated by the proximity of two gas regions with different temperature and species concentration, is pointed out in the case of through-flow of cold air in the exhaust system, and of back-flow of hot exhaust gases in the intake system during the valve overlap period. Intake Cyl.1 5 2 6 3 7 4 8 Exhaust C Air filter T Silencer Figure 1. Sketch of a complex engine geometry which can be modeled by the numerical code GASDYN. These form a quasi-linear hyperbolic system of partial differential equations, which can be written as follows [9,13]: SIMULATION MODEL mass: The non-linear fluid dynamic code GASDYN developed by the authors is a comprehensive model for spark ignition and compression ignition multi-cylinder and multi-valve engines, naturally aspirated or turbo-charged. It allows a complete simulation of the wave action from the air inlets to the exhaust outlets, including air filters, junctions of pipes, turbo-chargers and silencers (Figure 1). A user-friendly pre-processor (GASDYNPRE), developed in Visual Basic, allows a simple treatment of input data files and the visualization of the engine outline. ∂ρ ∂ (ρu) ρu dF + + = 0 ∂t ∂x F dx (Eq. 1) momentum: ∂ (ρu) ∂ ( p + ρu2 ) u2 dF + + ρ( + G ) = 0 ∂t ∂x F dx (Eq. 2) energy: ∂ (ρe0 ) ∂ (ρuh0 ) ρuh0 dF + + − ρq = 0 ∂t ∂x F dx The model consists of a set of numerical routines for: (i) the solution of the 1-d conservation laws by means of different numerical techniques; (ii) the modeling of the typical boundary regions, such as valves, open ends, junctions, abrupt area changes, turbines and muffling elements; (iii) the simulation of the combustion process via single zone (Diesel engine) and two-zones (petrol engine) thermodynamic (equilibrium assumption) models; (iv) the evaluation of intake and exhaust radiated noise levels and spectra via a fast Fourier transform approach. (Eq. 3) (p, u, ρ, e0, h0 are pressure, flow velocity, density, stagnation specific internal energy and enthalpy, respectively; F is the cross-sectional area, G=4fwu|u|/2d, d is the duct diameter, fw is the friction factor at the duct-wall, q is the heat transferred per unit mass per unit time). A forth equation must be added to solve the problem, describing the fluid behavior on the basis of different assumptions, which are discussed further on. Moreover, the capability of tracking the chemical species along the pipe system can be achieved by adding N-1 species continuity equations, as shown in [13], being N the number of species advected: GOVERNING EQUATIONS – The fluid dynamic model is based on the numerical solution of the fundamental con2 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 ∂ (ρY j ) ∂t + ∂ (ρuY j ) ρuY j dF = 0, F dx ∂x j = 1, 2 , . . . , N − 1 chemical composition. In this case the gas can be considered a mixture of N ideal gaseous species obeying the state equation: + (Eq. 4) p= which in non-conservative form can be written as: ∂Y ∂t j + u ∂Y j ∂x = 0, j = 1, 2 , . . . , N − 1 where Yj=mj /m is the mass fraction in the control volume for the specie j. These equations are based on the hypothesis of negligible diffusion in the flow, so that the species are simply advected, in the absence of chemical reactions (non-reacting flows). Only N-1 equations of type (4) are needed for N species, since the N-th equation is simply: N −1 j =1 j =1 N cv = ∑ cvj X j ⇒ e( T ) = cv T j =1 ∑ Y j = 1 ⇒ YN = 1 − ∑ Y j . It should be remarked that this model enhancement, enabling the species transport along the pipes, opens the way to further developments of the numerical model, aimed at predicting the engine emissions at the tailpipe outlet. In fact it is possible to extend the simulation to the catalyst from both a fluid dynamic and chemical point of view, allowing for the conversion of the exhaust products; furthermore, the influence of the EGR (exhaust gas recirculation) can be properly accounted for. iii) the third approach [13], the most general, is an extension of the previous one. In fact the dependence of the specific heats on both the gas chemical composition and the gas temperature is taken into account. The molar enthalpy and the internal energy of the j-th specie of the mixture can be expressed by means of the following polynomial relationships [18]: h j (T ) = R ( a1 j T + The species continuity equations (4) can be considered in addition to the conservation laws (1-3) regardless of the hypothesis adopted to describe the fluid properties. Three different approaches may be followed: i) the typical assumption in most numerical models is to consider a perfect gas with constant specific heats, obeying the state equation: ( k = c p / cv = const ), (Eq. 9) In this second case the resulting simulation model is not only able to predict the species transport, but also the influence of the variation of fluid composition along the ducts on the specific heats values cv , cp , on the internal energy e(cv ,T), hence on the propagation of perturbations in the intake and exhaust duct systems. However, the specific heats are considered constant with the gas temperature, whereas their variation in a large range of temperature can be not negligible; (Eq. 6) This approach allows the modeling of reacting flows, by introducing an appropriate source term w j in each equation (4), related to the j-th specie production rate. This is fundamental to take account of the reactions usually occurring in the exhaust gas downstream of the valves, before entering the catalyst. p / ρ = R*T , (Eq. 8) where Xj and Mj are the mole fraction and molar mass of the j-th specie, respectively, and R is the universal gas constant. Each specie constituting the ideal mixture has a constant specific heat cvj (not depending on temperature), so that the Gibbs-Dalton law may be used to determine the global coefficient cv for the mixture of N gaseous components: (Eq. 5) N ρRT ∑ Nj =1 X j M j a2 j 2 T2 + [ a3 j 3 T3 + e j (T ) = h j (T ) − RT = R ( a1 j -1) T + + a4 j 4 T4 + a5 j 5 a4 j a2 j T 5 + a6 j 2 ] 4 T4 + T2 + a3 j 3 a5 j 5 T 5 + a6 j ) T3 + (Eq. 10) in which the coefficients aMj (M=1,2…,6) for each chemical specie have been determined on the basis of the JANAF and NASA data. The numerical procedure implemented in the model can be notably simplified by considering a quadratic relationship for internal energy: (Eq. 7) where R* is the specific gas constant. Generally k and R* assumes two different but constant values in the intake and exhaust systems (i.e. kintake=1.4, kexhaust= 1.3³1.35), due to the different gas temperature and composition. The internal energy is given simply by e=cv⋅T. In this case the resulting model is able to calculate species transport, but the variation in gas composition along the ducts (related to the possible presence of air in the exhaust ducts and of exhaust products in the intake ducts, due to the valve overlap in a four-stroke engine) does not influence the calculation of gas dynamic quantities in pipes; e j (T ) = α1 j T + α 2 j T 2 (Eq. 11) where the coefficients α1 , α2 for the j-th specie can be obtained by matching the fifth order polynomial curve (10) in a prefixed temperature range, for example 300³1500 K, a reasonable range for the gases in the engine pipe systems. The global α1, α2 coefficients for the mixture of N gaseous components are given by: ii) a more general approach is based on the assumption of an ideal gas with specific heats depending on the gas 3 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 N N j =1 j =1 mann solvers [9] are not involved, so that symmetric schemes are computationally more efficient than upwind schemes, generally at the expense of a slightly worse definition of discontinuous regions of the solution. α1 = ∑ α1 j X j , α 2 = ∑ α 2 j X j ⇒ e(T ) = α1 T + α 2T 2 (Eq. 12) In this final case, the resulting simulation model may be used to predict the advective transport of species and the wave motion in the pipes, accounting for the influence of specific heats variations due to the gas composition and temperature. Classical symmetric schemes – The MacCormack and Lax-Wendroff methods implemented in the numerical code have been described in previous papers [5,16], so will not be reported here. It is important to remark that the Davis’ total variation diminishing (TVD) algorithm [21] has been introduced in the model, to render the schemes TVD and thus completely eliminate (unlike the flux-corrected transport (FCT) technique) the numerical overshoots in the proximity of discontinuities, typical of second order schemes. This flux-limiting technique is fundamental in the modeling of chemical species transport, since it avoids the occurrence of numerical instabilities, which cause nonphysical mass fraction values greater than unity or negative, as shown in [13]. The three approaches outlined above have been implemented in the GASDYN simulation code. The complete system of equations (1-4) can be written in conservative form by introducing the vectors W, F, C (the conserved variable, the flux and the source term vectors): ∂ W( x , t ) ∂ F ( W ) + + C( W ) = 0 ∂t ∂x (Eq. 13) Furthermore, in the case of shock-tube calculations, the artificial compression method (AC) [20] has been appended to the TVD algorithm, to ‘compress’ selectively the regions of abrupt gradients in the solution, and improve the definition of shock waves and contact discontinuities. CE-SE method – The Conservation Element-Solution Element (CE-SE) method is a new, nontraditional symmetric technique recently devised (1992) by Chang et al. [22,23] for the solution of the conservation equations with high accuracy and robustness, and has been applied to the modeling of unsteady flows in engine ducts in [24] and [19]. It differs substantially in both concept and methodology from the well-established methods (i.e. finite difference, finite volume, finite element and spectral methods). (Eq. 14) Finally, the global balance of unknowns and equations is: 4+N unknowns, that are ρ, u, p, e + N specie concentrations Yj , and 4+N equations, that are the conservation equations (1), (2), (3), the N-1 specie continuity equations (4), the global species continuity equation (6) and the fluid equation of state (8). NUMERICAL METHODS – Appropriate shock-capturing, conservative methods can be adopted to solve the quasi-linear hyperbolic problem (13). These are able to capture sharp discontinuities in the flow field (like shock waves or contact discontinuities) with second order accuracy or higher. The numerical code GASDYN is based on different shock-capturing numerical techniques, such as the classical symmetric two-step Lax-Wendroff method and the MacCormack predictor-corrector method, with the addition of flux-limiting techniques [5,16], and the new symmetric Conservation Element-Solution Element method (CE-SE) [19]. Furthermore, the discontinuous Galerkin finite element method has been investigated and applied to typical numerical tests [20], to point out the advantages of this new technique. In the near future this method will also be introduced in the fluid dynamic simulation code. j-3/2 j-1 j-1/2 A- j j+1/2 j+1 j+3/2 n+1 A A+ n+1/2 n n-1/2 t n-1 x Figure 2. Subdivision of the x-t plane into CEs (the rectangles) and SEs (the rhombuses) for the application of the CE-SE method. Symmetric shock-capturing schemes represent the best compromise among accuracy, resolution of discontinuities, simplicity and computational time. The same finite difference algorithm is applied in each mesh point, without orienting the scheme on the basis of the information provided by the local fan of characteristic lines. Thus Rie- Moreover, it has been recently extended to 2-d calculations with excellent results [23]. This scheme does not employ characteristic-based algorithms or flux-limiters, thus a simple approach is followed to generate highly accurate oscillation-free solutions, based on Taylor’s 4 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 adopted in the GASDYN model for applications to engine ducts calculations. expansion and weighted averaging. It adopts a subdivision of the space-time plane into rhombic regions, referred to as Solution Elements (SE), in which the solution is approximated, and rectangular regions, called Conservation Elements (CE), in which the conservation laws are satisfied (Figure 2). A detailed description of the method may be found in [22,24,19], whereas an outline is reported in Appendix A1. The CE-SE method has been adopted in the GASDYN code due to its robustness, accuracy and simplicity, which allows a significant reduction of computer run times with engine simulation, with respect to the Lax-Wendroff and MacCormack methods plus the TVD algorithm. Oscillation-free solutions can be achieved simply by assigning a proper value to an adjustable constant c (see appendix A1), which controls the weighted averaging used in the scheme and is not problem dependent. BOUNDARY CONDITIONS – Any of the classical and innovative explicit numerical methods described above can be adopted to evaluate the solution vector Wn+1 at the time t n+1 in the interior points of the ducts. In any case, boundary conditions are needed to evaluate Wn+1 in the pipe ends. The numerical code makes use of typical boundary conditions, based on the classical assumption of quasi-steady flow, resorting to a characteristicbased approach. The mesh-method of characteristics has been applied in the boundary cells to calculate the Riemann variables λin , λout and the entropy level AA at the boundary, by means of compatibility equations [8]. The interface between the shock-capturing schemes and the characteristic variables is administered by an appropriate routine. According to the assumption considered for the fluid behavior, the specific heats of the gas can vary with the chemical composition and the temperature. Hence, in the case of assumptions (ii) and (iii) concerning the fluid properties, the calculation of the Riemann variables at the boundaries is made on the basis of the local values of the ratio of specific heats k=cp /cv , which can be determined at each time step in the boundary nodes by knowing the mole fractions Xj of the species. Discontinuous Galerkin FEM – The discontinuous finite element Galerkin method (discontinuous FEM) is an advanced numerical scheme recently devised [25,26] for the solution of hyperbolic systems of conservation laws. The method has been fully described in [20,25]; due to its complexity, only a brief outline is reported in Appendix A2. The discontinuous FEM is characterized by great robustness, accuracy and good resolution of discontinuities with minimum local artificial viscosity introduced. The high order accuracy (third order or more) is achievable by simply increasing the order of the polynomial functions used to approximate the solution. Besides, the FEM is well suited to model flow regions characterized by strong variation of the cross-sectional area and of friction and heat transfer at duct walls, since this method allows a more flexible local refinement of the mesh, so that adjacent ducts with very different mesh size may be easily treated. Moreover, the FEM is ideally suited to be used on unstructured grids, due to its great geometrical flexibility. This is very important in multidimensional applications, such as with hybrid models coupling 1-d calculations of flows in ducts with 3-d direct modeling of flow boundary regions. On the basis of the steady conservation equations for mass, momentum and energy, a large range of boundaries can be modeled, following well-established procedures developed in the last three decades [8]: valves, open ends, abrupt area changes, junctions of pipes, carburetors, turbines, and so on. In particular, in the GASDYN model the junctions of three or more ducts can be treated: i) by the equal total enthalpy (constant pressure) model recently proposed in [28], which improves Benson's classic constant pressure junction model; ii) by the pressure loss models for T and Y junctions with different geometrical angles and multi-pipe junctions (“collector” or “supplier” type) [29]. Besides, the pressure losses in the bends of the intake and exhaust systems can be accounted for. Dedicated routines determine the friction factor fw of the bent duct from a data-base of measured loss coefficients [30], which depend on the bend geometry and the flow regime. On the other hand, the discontinuous FEM adopts Riemann solvers at element interfaces, hence it has some unattractive characteristics of upwinded schemes. For example, it is computationally more demanding than finite difference schemes: computer run times of FEM and classical finite difference methods have been compared in [20]. Furthermore, the extension to the modeling of chemical species transport (with assumptions (ii) and (iii) previously described) requires appropriate Riemann solvers [27], which differ significantly from the usual solvers devised with the assumption of perfect gas, in terms of complexity and computational efficiency. It has been shown [16,31] that a correct use of the available boundary conditions, together with the proper corrective lengths, allows to model complex silencers resorting to acoustically equivalent duct systems. Typical reactive and dissipative silencing elements, such as expansion chambers with axial side branches, internal orifices and flow reversals, perforated pipes, soundabsorbent linings, have been recently simulated [5,16,32,33]. Up to now, only a preliminary investigation of the FEM has been carried out [20] and the performance of the scheme has been appreciated on the basis of typical numerical tests. In the near future the method will be COMBUSTION MODELS – The simulation of the thermodynamic cycle in the cylinder of a Diesel engine, dur5 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 front in the chamber is followed through the evaluation of the factor flame [35], whereas the effect of chamber shape and spark plug position is taken into account by the approach described in [36]. Cylindrical and pent-roof combustion chambers can be modeled. The concentration of the ten chemical species in the exhaust products, resulting from the calculation with the equilibrium assumption, can be imposed at the valve opening in the pipe ends adjoining the cylinder, to allow for species propagation along the engine ducts due to advective transport. ing the closed valve period, is performed by a single-zone model; briefly, a Wiebe function is adopted for the heat release, whereas the chemical composition is evaluated resorting to a rapid computation technique [34] to solve the equilibrium equations for ten reacting species (H2O, H, H2, CO, CO2, O, O2, OH, N2, NO). The thermodynamic properties of the gas mixture are evaluated via polynomial expressions [18]. For a spark-ignition engine, a two-zone combustion model has been developed. The chemical species considered, as well as the solution technique of the system of equilibrium equations, are the same mentioned above. The propagation of the flame 1.8 400 Exact MCK+TVD 350 300 1.4 Velocity [m/s] Density [kg/m3] 1.6 1.2 1 0.8 Exact MCK+TVD 150 50 (a) 1.8 (b) 400 Exact MCK+TVD+AC 1.6 350 300 1.4 Velocity [m/s] Density [kg/m3 ] 200 100 0.6 1.2 1 0.8 250 200 Exact MCK+TVD+AC 150 100 0.6 50 (c) 1.8 (d) 400 350 Exact CE-SE, c=1.5 1.6 300 1.4 Velocity [m/s] Density [kg/m3] 250 1.2 1 0.8 250 200 Exact CE-SE. c=1.5 150 100 0.6 50 (e) 0.4 0 (f) 0 0.2 0.4 X [m] 0.6 0.8 1 0 0.2 0.4 X [m] 0.6 0.8 1 Figure 3. (a-f) - Shock tube results, case of perfect gas with constant specific heats and constant chemical composition (air). Density and velocity traces have been calculated by the different numerical schemes (MCK+TVD, MCK+TVD+AC, CE-SE with c=1.5). Initial data at t=0 are: ρ1eft=1.2 kg/m3, e1eft=0.8 MJ/kg, u1eft=0; ρright=0.7 kg/m3, eright=0.2 MJ/kg, uright =0; k=1.4. The tube is 1 m long, with 100 meshes; the results refer to the time t=0.5 ms. RESULTS data concerning different engines and applications [5,16,19]. The results shown in this papers are aimed to point out the enhancements of the fluid dynamic simulation model GASDYN related to the innovative numerical methods adopted, and to the tracking of the chemical species along the ducts. The model has previously been validated by comparing predicted results with experimental SHOCK TUBE RESULTS – The shock-tube problem and the shock-turbulence interaction problem [20] are the typical numerical tests which have been adopted to evidence the performance of the numerical techniques described in the previous section. Figures 3(a-f) and 4(a6 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 ing of some adjustable constants [13] which are generally unknown in engine simulations, hence it cannot be used with reliability. The CE-SE method, with two different values of the constant c (c=1.5 and c=2.0, Figures 3(e,f) and 4(a,b) respectively) for the elimination of residual overshoots, certainly provides a better solution (calculations have been carried out with CFL=0.95), which is much less smeared: especially the shock wave is well captured, but also the contact discontinuity and the rarefaction wave are well computed. The finite element method (FEM, Figures 4(c,d,e,f)) gives excellent results especially with the third order accuracy, being both discontinuities captured within few nodes. However, very small (negligible) overshoots are present in the density and velocity traces, since this FEM is not a TVD scheme. 400 1.8 350 1.6 300 1.4 Exact CE-SE, c=2 3 Density [kg/m ] Velocity [m/s] f) illustrate the shock-tube results, showing the calculated density and velocity trace along the tube. In this case the assumption of perfect gas with constant specific heats (not depending on the gas temperature) and constant chemical composition (air, k=1.4) has been considered for the calculations. The tube is 1 m long with 100 cells for the computation; results are plotted at time t=0.5 ms after the break of the diaphragm, located in the middle of the tube (x=0.5 m). The MacCormack scheme with Davis’ TVD (MCK+TVD, Figure 3(a,b)) provides a satisfactory, non-oscillatory solution, compared to the exact one, but the contact discontinuity in the density trace (Figure 3(a)) is rather smeared. A significant improvement is achieved by the use of artificial compression (MCK+TVD+AC, Figure 3(c,d)), even if it requires the tun- 250 Exact CE-SE. c=2 200 150 1.2 1 0.8 100 400 1.8 350 1.6 Density [kg/m 3] 300 Velocity [m/s] (a) 0.6 (b) 50 250 Exact FEM, 2nd order 200 150 Exact FEM, 2nd order 1.4 1.2 1 0.8 100 400 1.8 350 1.6 Density [kg/m 3] Velocity [m/s] 300 250 Exact FEM, 3rd order 200 150 100 Exact FEM, 3rd order 1.4 1.2 1 0.8 (e) 0.6 (f) 50 (c) 0.6 (d) 50 0 0.4 0 0.2 0.4 X [m] 0.6 0.8 0 1 0.2 0.4 X [m] 0.6 0.8 1 Figure 4. (a-f) - Shock tube results, case of perfect gas with constant specific heats and constant chemical composition (air). Density and velocity traces have been calculated by the different numerical schemes (CE-SE with c=2.0, second order FEM, third order FEM ). Initial data at t=0 are: ρ1eft=1.2 kg/m3, e1eft=0.8 MJ/kg, u1eft=0; ρright=0.7 kg/m3, eright=0.2 MJ/ kg, uright =0; k=1.4. The tube is 1 m long, with 100 meshes; the results refer to the time t=0.5 ms. 7 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 Density 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 Exact MCK+TVD (400 cells) (a) (b) Density 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 Exact CE-SE, c=1.5 (200 cells) Exact CE-SE, c=1.5 (400 cells) (d) (c) Density 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 Exact FEM, 2nd order (400 elements) Exact FEM, 2nd order (200 elements) (f) (e) Density 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Exact MCK+TVD (200 cells) Exact FEM, 3rd order (200 elements) Exact FEM, 3rd order (400 elements) (g) 0 (h) 0.2 0.4 X 0.6 0.8 1 0 0.2 0.4 X 0.6 0.8 1 Figure 5. (a-h) - Shock-turbulence interaction results, case of perfect gas with constant specific heats and constant chemical composition (air). Initial data at t=0 in non-dimensional form are:[ρ1eft, (ρu)1eft, (ρe0)1eft]=[3.8571, 10.1419, 39.1668] for x<0.1, [ρright, (ρu)right, (ρe0)right]=[1+0.2sin(50x), 0, 2.5] for xŠ0.1, k=1.4. The results refer to the (non-dimensional) time t=0.18. “exact” solution (solid line in the diagrams) has been calculated by the third-order FEM using 1600 elements, so it can be regarded as the converged solution. The following observations may be carried out: the MCK+TVD scheme excessively smears the solution with 200 and 400 cells (Figures 5(a,b)), introducing a significant diffusion, due to the switching to a first-order scheme near points of extrema even in smooth regions [37]. It is evident that the high frequency oscillations of density are not correctly SHOCK-TURBULENCE INTERACTION – The shock-turbulence interaction problem [20] allows to better appreciate the advantages of the CE-SE method and of the discontinuous FEM. This is a critical test for numerical schemes, since it points out the capability of capturing concurrent high frequency oscillations and shocks. The (nondimensional) density trace in the tube has been calculated resorting to the different numerical techniques, and the results are reported in Figures 5(a-h). The 8 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 previously explained) have a small influence on the speed propagation of the contact discontinuity, whereas they notably influence the internal energy trace along the tube. Results of figure 6(a-c) have been achieved by the MCK+TVD scheme. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Left 5.0 1600 Exhaust gas 0.107 0.135 0.723 0 0.023 0.010 0.002 (a) H 2O O2 N2 (b) CO 2 H2O Mole fractions Xj Time t=1 ms 0 O2 N2 (c) CO 2 H 2O 0.2 0.4 X [m] 0.6 0.8 1 Figure 6. (a-c) -Predicted mole fractions of CO2, H2O, N2, O2 along the shock tube at three times. Initial conditions are reported in Table 1. ENGINE RESULTS – The calculated results concerning a single-cylinder, four-stroke spark-ignition engine (CAGIVA W16) for a motorcycle are reported in this section, to point out the tracking of chemical species by means of the fluid dynamic model. A sketch of the intake and exhaust systems of the engine considered is shown in Figure 7, whereas some engine data are reported in Table 2. Initial conditions and specie mole fractions for the shock tube problem p (bar) T (K) Mole fractions Xj CO2 H2 O N2 O2 CO H2 NO N2 CO 2 Time t=0.5 ms 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 CHEMICAL SPECIES TRANSPORT – The shock tube problem has initially been adopted also to verify the correct prediction of chemical species transport by means of the fluid dynamic model. For example, the initial gas conditions and composition reported in Table 1, with the main species of the exhaust gas and air, can be considered in the shock tube. The model is able to propagate the contact discontinuity related to the difference in composition and gas temperature, as shown in Figure 6(a-c), which reports the calculated mole fractions Xj of CO2, H2O, N2, O2 along the tube, at three different times (mole fractions of CO, H2, NO are not reported in the diagrams as they are too small). O2 Mole fractions Xj A comparison of the computer run times of the different numerical methods has pointed out that classical symmetric schemes (Lax-Wendroff and MacCormack methods plus TVD flux-limiting techniques) are much faster than the FEM [20] (for which the CFL number is 0.2³0.3, to guarantee numerical stability), whereas they are slower than the innovative symmetric CE-SE scheme, since the CFL number is about 1 during the calculation and no TVD algorithm is used to avoid numerical overshoots, enabling about 20% increase in computation speed [19]. Table 1. Time t=0 Mole fractions Xj evaluated, especially with 200 cells. The CE-SE scheme performs much better than MCK+TVD: with 400 cells the predicted density trace is satisfactory, whereas with 200 cells it is not well defined (Figures 5(c,d)). The secondorder FEM with 200 and 400 elements (Figures 5(e,f)) performs only slightly better than the CE-SE method, whereas the third-order FEM (Figures 5(g,h)) definitely provides the best results even with 200 elements only. Certainly the CE-SE method and the discontinuous FEM have shown an high potential with both the numerical tests considered. The good quality of the results achieved can be fundamental in engine calculations, due to the frequent development of shocks, contact discontinuities and high frequency oscillations in the predicted traces of pressure and gas velocity. This aspect may be crucial in tailpipe noise spectrum calculation, for example. Right 1.0 400 Air 0.0 0.0 0.79 0.21 0 0 0 Intake Exhaust Transducer Air filter Silencer 30 Cyl. 200 80 80 510 mm Figure 7. Sketch of the single-cylinder, four-stroke, fourvalves, spark ignition engine (CAGIVA W16) for motorcycle. The calculations carried out have shown that the assumptions made to describe the fluid behavior (perfect gas or mixture of ideal gases, assumptions (i), (ii) and (iii) Pressure transducers have been located along the exhaust system to measure the pressure traces and eval9 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 through-flow of air in the exhaust duct may occur during the valve overlap, as shown in Figure 9(a,b). These phenomena cause the propagation of contact discontinuities in the intake and exhaust ducts, related to the variation in the species concentration and gas temperature. uate pressure spectra by an FFT spectrum analyzer. In particular, a transducer has been located downstream of the cylinder head, in the position indicated in Figure 7; a comparison between the predicted and experimental pressure traces and related spectra is shown in Figure 8(a,b), at an engine speed of 2750 rpm. A complete simulation has been carried out, modeling the silencer with perforated ducts [16,17], the various bends along the ducts, the Y junctions, and so on. Table 2. Air Intake valve single-cylinder, spark-ignition 601 cm3 102 mm x 73.6 mm 132 mm 8.3:1 e.v.o. 92°, e.v.c. 404° a.v.o. 310°, a.v.c. 622° 2750 rpm Engine speed a) 250 1.4 200 (a) 150 100 1.2 50 1 0 a.v.o. (50° bTDC) 0 overlap 90 180 540 720 900 Crank angle [°] 1080 1260 360 450 540 630 720 1440 (b) 1000 160 800 600 140 Predicted, intake valve cross-section Gas temperature [K] 1200 180 Experimental Predicted, CE-SE method 270 Crank angle [°] 0.6 360 e.v.c. (44° aTDC) Exhaust valve -50 180 (a) Intake valve 0.8 0 b) Air + residuals Predicted, intake and exhaust valve cross-sections Gas velocity [m/s] Pressure [bar] Experimental Predicted, CE-SE method Exhaust gas Figure 9. (a,b) - Back-flow of hot exhaust gas (a) and through-flow of air + residual gas (b) during the valve overlap 2 1.6 Exhaust valve Main data of the CAGIVA W16 singlecylinder, four-stroke, four-valve spark-ignition engine. Engine Swept volume Bore x stroke Connecting rod length Compression ratio Valve timing (aTDC) 1.8 Exhaust gas a.v.o. (50° bTDC) (b) a.v.c. (82° aBDC) S.P.L. [dB] 400 120 200 0 100 0 100 200 300 400 500 600 700 800 0 900 1000 Frequency [Hz] 90 180 270 360 450 Crank angle [°] 540 630 720 Figure 10. (a,b) - Predicted gas velocity (a) at the intake and exhaust valve cross-sections. Calculated gas temperature (b) at the intake valve crosssection; the engine speed is 2750 rpm . Figure 8. (a,b) - Comparison between predicted and measured pressure trace (a) and related S.P.L. spectrum (b) in the transducer cross-section (indicated in Figure 7); the engine speed is 2750 rpm. The engine considered has a large valve overlap of 94° (from 50° bTDC to 44° aTDC). At the intake valve opening, the calculated gas velocity trace in the intake valve cross-section at 2750 rpm, shown in Figure 10(a), points out the initial reverse flow, which lasts from 310° to about 360°. Consequently, the predicted gas temperature in the intake valve cross-section (Figure 10(b)) exhibits a strong increase due to the back-flow of hot exhaust gases. The fluid dynamic model allows to simulate the resultant transport of hot exhaust species in the intake duct, as shown in Figure 11(a,c). The initial mole frac- The calculation has been performed by the CE-SE method (mesh length=10 mm, CFL=0.95, constant c=2.0, see appendix A1), achieving a good agreement between the predicted pressure traces and the measured data (Figure 8(a)); consequently, also the predicted sound pressure level (S.P.L.) spectrum in the duct is in good agreement with the measured one. In particular, the simulation code has been used to investigate the species transport during the valve overlap period. In fact, the back-flow of exhaust products in the intake duct, and the 10 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 Mole fraction Xj 0.8 0.7 0.6 0.5 0.4 0.3 0.8 0.7 O2 Intake duct, 350° (valve overlap) N2 0.6 0.5 CO2 (a) H2O 0.4 0.3 0.2 0.2 0.1 0.1 0 0 0 0.04 0.08 0.12 0.16 Mole fraction Xj e.v.c., until a new discharge process occurs. Figure 11(b,d) illustrates the predicted variation of gas composition along the exhaust duct, at 350° and 360° respectively (2750 rpm); only the concentrations of the major species along the exhaust duct are reported. It can be noted that there is a significant mole fraction of O2 in the first tract of the exhaust duct, due to the mass of air escaped from the cylinder, whereas simultaneously CO2 and H2O are nearly absent. At 360°, the calculated contact discontinuity has propagated downstream, as shown in Figure 11(d). The engine test case adopted to validate the model has pointed out a satisfactory tracking of chemical species transport along the ducts. Further enhancements of the simulation code will be carried out in the near future to calculate the transport of species in multi-cylinder engines. The aim is to evaluate the emissions and to extend the 1-d modeling to the catalyst, from a fluid dynamic and chemical point of view. tions of N2 and O2 in the intake duct are equal to 0.8 and 0.2 respectively (air). At 350°, the propagation of exhaust species through the right end of the intake duct becomes evident. In fact the simulation model predicts a contact discontinuity in the species concentration, shown in Figure 11(a,c), which reports four (O2, N2, CO2, H2O) of the total ten species calculated by the thermodynamic combustion model. The O2 concentration approaches zero, whilst the species CO2 and H2O appear in the right part of the duct. However, at 360° (Figure 11(c)) the combustion products in the intake duct have almost disappeared, since they have been aspirated by the cylinder during the subsequent induction process. Similarly, the through-flow of air in the exhaust duct has been predicted by the fluid dynamic model, as illustrated in Figure 11(b,d). Around the end of the valve overlap period, a small part of the air which has entered the cylinder can leave it through the exhaust valve. This small mass of air then remains in the left part of the exhaust duct after the 0 0.2 O2 N2 CO2 0.04 0.4 0.3 0.8 0.7 Intake duct, 360° (valve overlap) 0.6 0.5 CO2 0.4 (c) H2O 0.3 0.2 0.2 0.1 0.1 0 Mole fraction Xj Mole fraction Xj 0.5 N2 0.12 0.16 0.2 Exhaust ducts Cyl. O2 0.08 X [m] Intake ducts 0.6 (b) H2O X [m] 0.8 0.7 Exhaust duct, 350° (valve overlap) O2 Exhaust duct, 360° (valve overlap) N2 CO2 (d) H2O 0 0 0.04 0.08 0.12 0.16 0.2 0 X [m] 0.04 0.08 0.12 0.16 0.2 X [m] Figure 11. (a-d) - Calculated results for the single-cylinder spark-ignition engine (engine speed=2750 rpm). Predicted back-flow of exhaust gas in the intake duct (a,c) during the valve overlap, variation of species mole fractions in the right end of the intake pipe at 350° and 360°, respectively. Through-flow of air in the exhaust duct (b,d) during the valve overlap, variations of species mole fractions in the left end of the exhaust pipe at 350° and 360°, respectively. CONCLUSIONS both classical and innovative numerical methods for the solution of the fundamental conservation equations. Nontraditional shock-capturing schemes, such as the new symmetric CE-SE method and the upwind discontinuous The peculiarity of the 1-d fluid dynamic simulation model GASDYN described in the paper is that it is based on 11 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 Galerkin FEM with third order accuracy have been investigated. Typical numerical tests, such as the shock-tube problem and the shock-turbulence interaction problem, have pointed out the good quality of the results obtained by means of these new techniques. Furthermore, the model allows the tracking the chemical species along the ducts, with the assumption of non-reacting flows and negligible diffusion. The predicted results have well evidenced the variation in species concentration along the ducts of a single-cylinder four-stroke spark-ignition engine, due to the back-flow and through-flow phenomena during the valve overlap, proving the good potential of the numerical code. In the near future it will be further enhanced to predict engine emissions, by modeling the reactions in the exhaust manifold and the action of the catalyst. 12. 13. 14. 15. 16. ACKNOWLEDGMENTS 17. The authors would like to thank G. D’Errico for his help in the calculations concerning the transport of chemical species in the ducts. Thanks are also due to P. Ronca and D. Gini of Lafranconi Silencers (Lecco, Italy) for their indispensable help in the experimental measurements. 18. 19. REFERENCES 20. 1. J. SILVESTRI and T. MOREL, Study of Intake System Wave dynamics and Acoustics by Simulation and Experiment. SAE paper no. 940206, March 1994. 2. T. MOREL, M.F. FLEMMING, and L.A. LAPOINTE, Characterization of Manifold Dynamics in the Chrysler 2.2 S.I. Engine by Measurements and Simulation, SAE paper 900679, 1990. 3. H. ZHANG and S. WIDENER, An Integrated Engine Cycle Simulation Model with Species Tracking in Piping System. SAE paper no. 960077, February 1996. 4. D.E. WINTERBONE, R.J. PEARSON and Y. ZHAO, Numerical Simulation of Intake and Exhaust Flows in a High Speed Multi-Cylinder Petrol Engine Using the LaxWendroff Method. Instn Mech. Engrs Int. Conf. Computers in Engine Technology, Cambridge (U.K.), September 1991. 5. G. FERRARI and A. ONORATI, Determination of silencer performances and radiated noise spectrum by 1-d gas dynamic modelling. XXV FISITA Congress (Beijing, China), paper no. 945135, October 1994. 6. G. P. BLAIR, Design and Simulation of Two-Stroke Engines. SAE, ISBN 1-56091-685-0, 1996. 7. R.J. PEARSON and D.E. WINTERBONE, Calculation of One-Dimensional Unsteady Flow in Internal Combustion Engin e- How Long Should It Take? Instn Mech. Engrs Int. Conf. Computers in Recipr. Engines and Gas Turbines, London (U.K.), January 1996. 8. R.S. BENSON, The Thermodynamics and Gas Dynamics of Internal Combustion Engines, vol. I. Clarendon Press, Oxford, 1982. 9. C. HIRSH, Numerical Computation of Internal and External Flows, vol. I-II. John Wiley & Sons, 1991. 10. M. CHAPMAN, J. M. NOVAK, and R. A. STEIN, Numerical modeling of inlet and exhaust flows in multi-cylinder internal combustion engines, in Flows in Internal Combustion Engines. T. Uzkan editor, ASME WAM, Austin, TX , 1982. 11. A. ONORATI, D. E. WINTERBONE and R. J. PEARSON, A comparison of the Lax-Wendroff technique and the method of characteristics for engine gas dynamic calculations using fast Fourier transform spectral analysis. SAE Int. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 12 Congress & Exp. (Detroit, Michigan), paper n. 930428, March 1993. J. LIU, N. SCHORN, C. SCHERNUS and L. PENG, Comparison Studies on the Method of Characteristics and Finite Difference Methods for One-Dimensional Gas Flow Through IC Engine Manifold. SAE Int. Congress & Exp. (Detroit, Michigan), paper n. 960078, February 1996. R. J. PEARSON and D. E. WINTERBONE, Calculating the effects of variations in composition on wave propagation in gases. Int. J. Mech. Sci. vol. 35, pp. 517-537, 1993. D.E. WINTERBONE and R.J. PEARSON, A Solution of the Wave Equations Using Real Gases. Int. J. Mech. Sci., vol. 34, pp. 917-932, 1992. A. ONORATI, G. FERRARI, An Advanced 1-d Fluid Dynamic Model for the Numerical Simulation of Unsteady Flows in I.C. Engine Duct Systems, ICE97 - International Conference on Internal Combustion Engine, Capri (Naples - Italy), September 1997. A. ONORATI, Nonlinear Fluid Dynamic Modeling of Reactive Silencers Involving Extended Inlet/Outlet and Perforated Ducts. Noise Control Eng. J. vol. 45(1), Jan-Feb 1997. T. MOREL, J. MOREL, and D. BLASER, Fluid dynamic and acoustic modeling of concentric-tube resonators/silencers. SAE paper 910072, 1991. W.C. GARDINER, Combustion Chemistry. Springer-Verlag, New York, USA, 1984. A. ONORATI, Numerical Simulation of Exhaust Flows and Tailpipe Noise of a Small Single Cylinder Diesel Engine. Proc. Small Engine Tech. Conf. SETC ‘95, SAE paper no. 951755, Milwaukee (USA), September 1995. A. ONORATI, M. PEROTTI and S. REBAY, Modelling 1-D Unsteady Flows in Ducts: Symmetric Finite Difference Schemes vs. Galerkin Discontinuous Finite Element Methods. To be published in the Int. J. Mech. Sci, 1997. S. DAVIS, TVD Finite-Difference Schemes and Artificial Viscosity. NASA CR 172373, 1984. S. CHANG and W. TO, A Brief Description of a New Numerical Framework for Solving Conservation Laws-The Method of Space-Time Conservation Element and Solution Element, NASA technical memorandum 105757, 1992. S. CHANG, X. WANG and C. CHOW, The Method of Space-Time Conservation Element and Solution Element Applications to One-Dimensional and Two-Dimensional Time-Marching Flow Problems. NASA technical memorandum 106915, 1995. G. BRIZ and P. GIANNATTASIO, Applicazione dello schema numerico Conservation Element - Solution Element al calcolo del flusso intazionario nei condotti dei motori a c.i.. Proc. 48th ATI National Congress, pp. 233247, Taormina, Italy, September 1993. B. COCKBURN and C. W. SHU, TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework. Math. Comp., vol. 52, pp. 411-435, 1989. F. BASSI and S. REBAY, Discontinuous Finite Element High Order Accurate Numerical Solution of the Compressible Navier-Stokes Equations. 4th Int. Conf. Fluid Dynam., Oxford, in Numerical Methods in Fluid Dynamics. Clarendon Press, Oxford (UK), 1995. R. SAUREL, M. LARINI and J. C. LORAUD, Exact and approximate Riemann solvers for real gases. J. Comp. Phys. vol. 112, 126-137, 1994. J.M. CORBERAN, A New Constant Pressure Model for NBranch Junctions, Proc. Instn Mech. Engrs vol. 206, pp.117-123, 1992. J.F. BINGHAM and G.P. BLAIR, An Improved Branch Pipe Model for Multi-Cylinder Automotive Engine Calculations. Proc. Instn Mech. Engrs vol. 199, pp. 65-77, 1985. D.S. MILLER, Internal Flow Systems. BHRA, Cranfield (UK), 1990. A. ONORATI, Prediction of acoustical performances of Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 muffling pipe systems by the method of characteristics. J. Sound Vib. vol. 171, p. 369, 1994. 32. A. ONORATI, A White Noise Approach for Rapid Gas Dynamic Modelling of I.C. Engine Silencers. 3rd IMechE International Conference "Computers in reciprocating engines and gas turbines" (CET '96), 9-10 January 1996, London. 33. G. FERRARI and A. ONORATI, Modellazione fluidodinamica non-lineare di silenziatori ad elementi perforati e con materiale fonoassorbente. Proc. 51st ATI National Congress, pp. 937-952, Udine, Italy, September 1996. 34. W.D. ERICKSON and R.K. PROBHU, Rapid Computation of Equilibrium Composition: an Application to Hydrocarbon Combustion. A.I.Ch.E. Journal, vol. 32 no. 7, July 1986. 35. M.R. HEIKAL, R.S. BENSON and W.J.D. ANNAND, A Model for Turbulent Burning Speed in Spark Ignition Engines. SAE paper no. 760160, 1976. 36. G.G. LUCAS and M.J.F. BRUNT, The Effect of Combustion Chamber Shape on the Rate of Combustion in a Spark Ignition Engine. SAE paper 820165, 1982. 37. R. J. LEVEQUE, Numerical Methods for Conservation Laws (Birkhauser Verlag, Basel, 1990). APPENDIX A1: THE CONSERVATION ELEMENT - SOLUTION ELEMENT METHOD (CE-SE) The CE-SE method adopts a subdivision of the spacetime plane into rhombic regions, referred to as Solution Elements (SE), in which the solution is approximated, and rectangular regions, called Conservation Elements (CE), in which the conservation laws are satisfied. Each SE is centred at a mesh point (j,n) where j is a whole integer if n is a half integer and viceversa, hence they are staggered over every half time step (Figure A1). j-3/2 j-1 j-1/2 A- (Eq. A4) The Taylor’s expansion of f m has an analogous expression: j j+1/2 j+1 j+3/2 n+1 A ~ n ~ n ∂f ∂f ~ ~ f m ( x , t ; j , n ) = ( f m ) nj + m ( x − x j ) + m ( t − t n ) ∂x j ∂t j A+ (Eq. A5) ~ If the components f m of the flux vector are expressed ~ , then as a function of w , , may m be defined only in terms of . ~ , ~ f m can be introThe numerical approximations w n+1/2 n n-1/2 m t duced in the conservation equations, in which the surface integration form can be converted into a line integration form along the boundary l of the CE(j,n): n-1 x ~ dx − ~ f m dt = ∫w m Figure A1.Subdivision of the x-t plane into CEs (the rectangles) and SEs (the rhombuses). l ( CE ( j , n )) n+1/ 2 The values of (α m ) j , m=1,2,…L are determined in the following way: if A+, A, A- indicate the nodes (xj+1/2 , tn+1/2), (xj , tn+1/2 ) (xj-1/2 , tn+1/2) respectively (Figure A1), then may be calculated by a simple central difference formula: r r ∫ hm ⋅ ds + ∫∫ c m dΩ = 0 Ω (Eq. A2) where l is the boundary of any surface Ω in the x-t plane. Now, for any (x,t) Œ SE(j,n), w m and f m are approxi~ and ~ f m via a first order Taylor’s expansion: mated by w m ~ ( x , t; j, n ) = (σ ) n + (α ) n ( x − x ) + ( β ) n ( t − t n ) w m m j m j j m j (Eq. A6) Proceeding with the line integration, it is possible to /2 , obtain an expression for the values of (σ m ) n+1 j m=1,2,…L. (Eq. A1) where L is the number of elements of the solution vector W (L=N-1+3=N+2, being N-1 the number of species continuity equations included in the equation system (13)). If r the vector hm = ( f m , wm ) is introduced in the x-t space, the integral form of the above equations is: l ~ The treatment of the source terms cm to get the numericm within the surface Ω(CE(j,n)) is cal approximation ~ explained in detail in [24]. The conservation laws (13) may be rewritten as: ∂wm ∂f m + + cm = 0 ∂t ∂x ∫∫ c m dΩ Ω( CE ( j , n )) (Eq. A7) which is valid only if no discontinuity occurs between A+ and A- . In presence of a discontinuity, an alternate formula can be adopted. The numerical analogues of the left and right derivatives in A are: (Eq. A3) where: 13 Downloaded from SAE International by Univ of Nottingham - Kings Meadow Campus, Sunday, August 12, 2018 substituting the expressions of remembering that (Eq. A8) (α m+ ) nj +1/ 2 so that a suitable weighted average of provides a satisfactory estimation of in A: c αm = ( ) n +1/ 2 βm j and if α m+ + α m− ≠ 0, c + α m− α m = 0 if α m+ + α m− = 0 c (Eq. A9) where c is an adjustable constant (0 ≤ c ≤ 3). n+1/ 2 Finally, the evaluation of the ( β m ) j can be performed by means of the conservation equations (eq. A1) which must be satisfied in the SE(j,n+1/2): n +1/ 2 ~ n +1/ 2 ∂~ f ∂w n +1/ 2 m cm ) j + m + (~ =0 ∂t j ∂x j n +1/ 2 ∂~ fm n +1/ 2 = − − (~ cm ) j ∂ x j , (Eq. A11) which allows to determine appropriate expressions for . Once , , are known at the time tn+1/2, the solution is completely determined at this time step, and the procedure can be repeated to march forward in time from tn+1/2 to tn+1 . The method is second order accurate, and explicit in time, therefore the Courant-Fredrichs-Lewy criterion states the time step amplitude: c α m+ α m− + α m− α m+ α m+ and , we have: CFL = (a + u) ∆t ≤ 1 . ∆x (Eq. A12) (Eq. A10) APPENDIX A2: THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD The FEM is based on the weak formulation of the hyperbolic problem (13): b ∫a V ∂W dV b dx + VF ( W ) − ∫ab F ( W )dx = a ∂t dx [ = − ∫ab VC( W )dx ] ∀V k=1 Wh Vh a0φ0 + a1φ 1 k k a0 a1 (Eq. A13) where V(x) is a test function in the interval (a,b). A discrete analogue of equation (A13) is obtained by subdividing the domain into a collection of nonoverlapping elements Ei = { x ∈ℜ : x i ≤ x ≤ x i +1} , i = 1,2,..., N of uniform length h = xi +1 − xi and by considering functions Wh and Vh which are polynomials Pk of degree less or equal to k within each element (Galerkin method). The functions Wh and Vh can be written in general as: (i-1) i l =0 k i j=0 x (i+1) Figure A2.Piecewise linear approximation by discontinuous functions Wh , Vh . The order of accuracy in space is given by O(hk+1); so second-order accuracy is obtained with linear elements, i.e. k=1. With the appropriate substitutions, equation (A13) is equivalent to a system of k+1 vector equations: $ j φ kj ( x ) $ l (t )φ lk ( x ), Vh ( x ) E = ∑ V Wh ( x , t ) E = ∑ W k element (i) (Eq. A14) $ l (t ) and V $ j denote where the expansion coefficients W the degrees of freedom of the numerical solution and of the test function within the element Ei respectively, and k the shape functions φ j are a base for the polynomial k. No global continuity is required for functions P Wh and Vh , which are discontinuous functions across element interfaces (Figure A2). [ ] d x i +1 k k $ dx + φ kj F( Wh ) ∫ φ φ W dt xi j l l + ∫xxi +1 φ kj C( Wh )dx = 0 i x i +1 xi − ∫xii +1 x dφ kj dx 0 ≤ j, l ≤ k F( Wh )dx + (Eq. A15) The scheme is proved to be linearly stable under the condition CFL≤1/(2k+1) [23], where k is the order of the shape functions and k+1 is the order of accuracy of the numerical method. 14