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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
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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
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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
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∂ (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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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) ΠSE(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:
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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.
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