1. Introduction
Reacting flow models play an important role in the simulation of many physical and engineering problems, such as the pollutant transport process in water and air, heat conduction process in flowing fluids, chromatography column in reactors [
1], and high-speed eddy current in electromagnetic fields [
2]. A reacting flow model is often composed of a group of convection dominated partial differential equations (PDEs) with non-linear source terms [
2,
3,
4,
5], which usually accompanies autocatalytic reactions. Through mutation, the automatic catalyst is transformed into another form, which can simultaneously conduct autocatalytic reaction and ultimately result in competition between the palingenetic and original automatic catalyst [
6]. By accurately solving these reacting flow models, we can not only analyze the reaction problems in chemistry, physics, electromagnetics and fluids, but can also design proper reaction units and optimize the schemes for process control. Therefore, developing an accurate and efficient numerical method is of great significance to promote the development of these disciplines.
The PDEs in reacting flow models are usually convection dominated, i.e., the transport of reactants is more affected by the flow than the diffusion and reaction processes, and the time scale for convection is smaller than those for diffusion and reaction [
7]. It is well known that convection dominated PDEs present various difficulties to numerical methods, especially when discontinuities or moving steep fronts are presented [
1,
7,
8]. In order to clearly observe the differences among different discretization schemes, Lim et al. [
1] reviewed fourteen competitive spatial discretization methods developed for convection terms and analyzed their accuracy, temporal performance and stability. The discretization methods were classified by fixed stencil (traditional upwind and central schemes), adaptive stencil (ENO (essentially non-oscillatory) schemes) and weighted stencil (weight ENO (WENO) schemes) types. It is found that the ENO and WENO methods are efficient for tracking a shock because they avoid crossing discontinuities in the interpolation procedure with a relatively small number of mesh points, thus they are generally considered to be the most powerful tools for convection dominated problems. Wang and Hutter [
4] compared a series of numerical schemes for one-dimensional convection-diffusion problems and Alhuamaizi [
7] gave a comparative evaluation of different finite difference methods for a convection-dominated reaction problem with three reactants. The evaluated methods include first-order accurate difference schemes like upstream difference (well-known as upwind) and Lax-Friedrichs, second-order methods like central, Lax-Wendroff, Fromm and Beam-Warming, third-order upwind and QUICK, and the high-resolution techniques, including the total variation diminishing (TVD) methods with flux-limiters, flux-corrected transport (FCT), monotone upstream scheme for conservation laws (MUSCL) and WENO. Numerical experiments have well proved that first-order accurate solutions are likely to produce unacceptable numerical diffusion, while higher order numerical methods easily result in unphysical oscillation (Gibbs phenomena) with various levels of accuracy loss [
4,
7]. By increasing the grid number and hence decreasing the grid Peclet number in first-order methods, numerical diffusion can be resolved, but high computational costs will be caused in the process. Wang and Hutter [
4] pointed out that the modified TVD Lax-Friedrichs scheme with the Superbee limiter [
9,
10,
11] (MTVDLF-Superbee) is the most competent method for convection dominated problems with a steep spatial gradient of the variables. Alhuamaizi [
7] concluded that high-resolution techniques such as FCT, MUSCL and WENO schemes and TVD with flux limiters, are all efficient for tracking steep moving fronts and are essential for cases which use small numbers of grid points. In terms of both accuracy and computing time, the Superbee flux limiter is found to be the most appropriate method for simulating the sharp fronts of the reactor model.
All of the above numerical methods are based on the finite difference or finite volume discretization, which belong to Eulerian approaches using grids fixed in space. Grid-based high-resolution approaches can eliminate excessive numerical diffusion without causing unphysical oscillations, but they can hardly deal well with numerical accuracy and computational efficiency at the same time, and hence are not suitable for long-term and large-scale transport processes with multicomponent reactants. In addition, the implementation process is somewhat tedious and involves many intermediate variables. The objective of this paper is to develop a Lagrangian particle algorithm based on a smoothed particle hydrodynamics (SPH) method for convection-reaction equations to enhance both the numerical accuracy and computational efficiency.
Smoothed particle hydrodynamics is a fully Lagrangian, or meshless particle method invented in the late 1970s by Lucy [
12] and Gingold and Monaghan [
13] to solve unbounded astrophysical flow problems. The basic idea of the method is to represent a continuous fluid with a group of interacting particles, each of which carries various physical quantities such as mass, density, position, velocity, concentration, etc. It consists two steps of approximation known as kernel approximation and particle approximation [
14]. As a build-in feature of the SPH method, the adaptive nature makes it very attractive and it can easily handle problems involving large deformations and highly irregular geometries. In principle, as long as the number of particles is sufficient and the particle distribution is not too irregular, the approximations will be accurate and the process can be properly described. On the basic theory of SPH, the readers are referred to the recent reviews [
15,
16,
17,
18,
19].
Regarding to the application of SPH to transport problems, Gadian et al. [
20] used SPH to extend the developments in finite difference and finite-element to the modelling of atmospheric fluid flow. The particle method was used to describe the 2D convection. Zhu and Fox [
21] presented the application of SPH to tracer diffusion in porous media under steady state and transient conditions. Based on the multiphase SPH framework, Adami et al. [
22] proposed a Lagrangian particle method for the simulation of multiphase flows with surfactant. The insoluble surfactant on an arbitrary interface geometry as well as interfacial transport such as adsorption or desorption were simulated. Szewc et al. [
23] proposed a new variant of SPH for the simulation of natural convection phenomena in a non-Boussinesq regime. Orthmann and Kolb [
24] applied SPH to convection-diffusion simulations of incompressible fluids. A temporal blending technique was proposed to reduce the number of particles in the simulation. It greatly reduced the error introduced in the pressure term when changing particle configurations and enabled larger integration time-steps in the transition phase. Using SPH as a discretization tool on uniform Eulerian grids, Danis et al. [
25] investigated transient and laminar natural convection in a square cavity. To obtain a divergence-free velocity field, the incompressibility was strictly enforced by employing a pressure projection method. Although the SPH method has been successfully applied to simulate transport processes, it is mainly used for solving fluid dynamics with a focus on capturing the free-surface and moving interface [
15,
18,
22]. Its application to diffusion is restricted to the transport process in porous media [
21], where the flow velocity is relatively slow.
To show the advantages of the proposed Lagrangian particle algorithm, its numerical results and computational performance are compared with those of four typical Eulerian methods, including the upstream difference scheme (UDS), Lax- Friedrichs scheme (LFS), Lax-Wendroff scheme (LWS) and MTVDLF-Superbee method.
The structure of this paper is organized as follows. The reacting flow model governed by the one-dimensional convection-reaction equations is introduced in
Section 2. Traditional Eulerian methods including UDS, LFS, LWS and MTVDLF-Superbee are presented in
Section 3.
Section 4 describes the Lagrangian particle algorithm (SPH) and its application to the autocatalytic reaction model with multicomponent reactants. Numerical results are presented in
Section 5 with detailed comparison between the proposed Lagrangian particle scheme and the Eulerian methods. Finally, some discussions and concluding remarks are drawn in
Section 6.
3. Traditional Eulerian Methods
Many numerical methods have been applied to the convection-reaction equations. The finite difference and finite element methods are more classical, but the finite volume methods (FVMs) [
26] are more widely used due to their conservative properties. The basic strategy of FVMs is to write the differential equation in conservation form, integrate it over small regions (called “cells” or “finite volumes”) and convert each such integral into an integral over the boundary of the cell by means of the divergence theorem.
Common schemes of FVMs for convection-reaction equations include the first-order accurate methods like the upwind and Lax-Friedrichs (LF) schemes, the second-order accurate methods, such as the second order upwind, Lax-Wendroff, Beam-Warming and Fromm schemes, and the high-order accurate QUICK and the modified TVD Lax-Friedrichs scheme with different slope limiters [
2,
7,
8]. To better compare and analyze the numerical performance of the proposed Lagrangian particle algorithm, four typical Eulerian methods are studied as the comparisons.
By discretizing the spatial and temporal derivatives in Equation (3) using the finite volume method, we obtained
where
are the numerical fluxes on the cell interfaces at
, and
and
are the time and space step size, respectively. Mean value theorem is applied to the reaction term
. Many schemes have been developed and the difference between them just depends on how to define the numerical convective fluxes.
The convective flux of the upwind scheme is given by
with
,
and
. The characteristic velocity
is given by
The convective flux of the Lax-Friedrichs scheme is
The upwind and LFS are first-order accurate in both space and time.
The convective flux of the Lax-Wendroff scheme (LWS) is based on Taylor series expansion of the space derivative term [
5,
27] and can be written as
This scheme has second-order accuracy in both space and time.
In the modified TVD Lax-Friedichs (MTVDLF) scheme, the convective flux is given by
and the modified dissipative limiter [
28] is
where the characteristic speed
is obtained from the Rankine-Hugoniot jump condition
and
and
have been considered to form a linear piecewise reconstruction for each interface. The former is obtained from the left-side cell
and the latter is due to the right-side cell
. The specific expressions are
where the slop limiter is defined as
and
is a function of
, which represents the ratio of the consecutive gradients, i.e.,
There are various choices for the function
in practice [
29]. If
is defined by the upper boundary of the second-order TVD region, the so-called Superbee limiter [
4,
29] is resulted as
4. Lagrangian Particle Algorithm (SPH)
The smoothed particle hydrodynamics (SPH) method [
30,
31] has been well-known in fluid simulation problems since its proposition by Lucy [
12] and Gingold and Monaghan [
13]. It is a fully Lagrangian, meshless method in which an approximation technique with a smoothing kernel function is introduced to estimate the spatial derivatives in the governing equations of fluid dynamics. Different from traditional mesh-based methods, the SPH method uses a set of particles without predefined connectivity. It simulates complex flow-related problems by solving the hydrodynamic equations and tracking the particle trajectories. Each particle has characteristic properties of mass, density, velocity, position and others, depending on the specific problem. In the light of mesh independence among individual particles, the number of particles in the calculation domain can be increased or decreased to satisfy the solution requirements for complex problems.
Regarding to the basic idea, there are two main steps in the SPH method, namely, kernel approximation and particle approximation, which are presented in this section.
4.1. SPH Kernel Approximation
The integral representation for any physical quantity
can be written as
where
is the domain containing
x and
is the Dirac delta function defined as
Replacing
by a kernel function
, then Equation (22) can be converted into the SPH kernel approximation form as
where
represents the value approximation,
is known as the smoothing length and it determines the radius of the compact support of
. The kernel
should satisfy the following requirements:
- (i)
normalization condition: ;
- (ii)
compact support: , if , where is a scale parameter and is called nuclear radius;
- (iii)
nonnegativity: , if ;
- (iv)
symmetry: ;
- (v)
monotone decline: is monotonically decreasing with respect to , which represents decreasing interaction;
- (vi)
Dirac delta function property: .
The step of kernel approximation is equivalent to the smoothing and denoising of the original signal.
In order to obtain the approximation of the derivative of a function, we used
to replace
in Equation (23) and get
where
is the gradient operator. According to the divergence theorem, (24) can be transformed into
Using the compact support property of the kernel, when the support domain of
was located inside the problem domain, we obtained
. Therefore, for a point with kernel support in the problem domain, formulation (25) can be written as
which is the standard expression for the derivative approximation in SPH.
4.2. SPH Particle Approximation
In the particle algorithm, the continuum medium was partitioned into a finite number of
parts that are referred to as particles. Particle approximation is a process of transforming the integral in kernel approximation into a discrete summation form of particles in the support domain. Each particle carries a mass, density, velocity, concentration, and other properties depending on the specific problem. Here using
to represent the volume of particle
, then its mass can be calculated by
where
is the density of particle
.
Applying particle approximation to the integrals in kernel approximations (23) and (26), the SPH approximation of
at
can be written as
and the SPH approximation of the derivatives can be obtained as
With the anti-symmetry property of the kernel gradient, i.e.,
, we can obtain the mostly used derivative approximation in SPH as
or simply
where
,
, which denotes the kernel derivatives taken with respect to the coordinates of particle
and
is the smoothing length of particle
.
4.3. Lagrangian Particle Autocatalytic Reaction Model
With the definition of material derivative, the convection-reaction equation (2) can be written in Lagrangian form as
Applying the SPH approximation to Equation (32), we get
Contrary to Euler approaches that use fixed grids in space, in the Lagrangian particle scheme, the particles move according to the flow velocity
With a time integration algorithm, the semi-discrete formulations of the Lagrangian convection-reaction Equation (34) and the particle motion (35) can be marched until the specified simulation time. For simplicity, the explicit Euler method is applied in this paper, which is one-step and first-order accurate.
Since the location of each particle changes with time, adding particles at the inlet and deleting particles at the outlet is required to mimic the flowing medium. A new particle is added when the most leftmost particle gets across the inlet boundary, and a particle is deleted when it gets across the outlet boundary. It is worth noting that at most one particle needs to be added at one time step for the one-dimensional reaction problem considered herein. This condition is equivalent to the CFL condition in grid based methods and it must be satisfied to ensure the continuity of the flowing medium.
5. Numerical Results
In this paper, the kinetic parameters in the autocatalytic reaction model were taken as
,
and
, and the dimensionless flow velocity is
. These parameters have been well studied by Alhumaizi [
7]. In all the following simulations,
is taken as the time step size. The space step size is
, where
N denotes the number of particles for Lagrangian particle method and grid points for grid-based methods.
The concentration of species B at
simulated by the upwind scheme (UDS) with different grid numbers is shown in
Figure 1. It is seen that with an increased grid number the UDS results converge to a particular solution, which is similar to the conclusion that the limitation of UDS for solving convection-diffusion equation converges to the theoretical solution [
3]. Since there is no exact solution for the autocatalytic reaction model considered herein, the UDS solution with grid number
is taken as the reference.
The grid number
is employed for LFS, LWS, MTVDLF-Superbee and the proposed Lagrangian particle algorithm in the following simulations.
Figure 2 compares the numerical results for the three reactants at
.
It is clear that UDS has good performance for both species B and C (see
Figure 2b,c), but there is an evident spike close to the discontinuity for reactant A (
Figure 2a), which is an unacceptable numerical defect. Therefore, even though the UDS solution can be taken as a reference, it might not be a good baseline for error analysis as conducted by Alhumaizi [
7]. Reducing the grid size can reduce numerical diffusion in UDS, but it would also increase the computational time [
32]. The amplitude of the spike increases with the decreasing grid size too. LFS causes serious numerical dissipation that completely pollutes the whole solution. LWS can capture the shock front but leads to unphysical oscillations behind the shock for all three reactants, due to which the numerical accuracy remarkably degenerated. The high-resolution MTVDLF-Superbee scheme is capable of yielding accurate solutions for
and
, but it also suffers from unphysical oscillations near the discontinuity for species C as shown in
Figure 2c (the solution of
was not given in reference [
7]). The present Lagrangian particle algorithm succeeded in solving the situations with steep concentration profiles without suffering from any numerical diffusion or unphysical oscillations. The results are almost the same as the reference solutions, but without any spike in the solution of species A.
Figure 3 compares the concentration distribution obtained with different methods for the autocatalytic reaction model at
. Numerical dissipation and oscillations appeared in LFS, LWS and MTVDLF-Superbee are clearly shown, whilst the Lagrangian particle algorithm does not suffer from either numerical dissipation or unphysical oscillations.
The concentration profiles obtained with different methods at four time levels are shown in
Figure 4, together with the reference solutions. The results of LFS and LWS are not shown due to unacceptable numerical dissipation and spurious oscillations. It is clear that, in the reference solutions, there are the same evident spikes near the discontinuity for reactant A (see
Figure 4a). Unphysical oscillations exist in the solutions of the MTVDLF-Superbee scheme for species C (see
Figure 4c), and the oscillation amplitude increases with an increasing time. The Lagrangian particle algorithm has somewhat profile shifting characteristics as before, but it is capable of eliminating numerical dissipation and suppressing unphysical oscillations.
The front shifting is mainly due to a lack of particles and can be overcome by simply increasing the particle number. When particle number is increased to
, there is no profile shifting anymore as shown in
Figure 5 for the reactant concentration of species A. A similar performance holds for species B and C.
To further investigate the effects of flow velocity on numerical oscillations appeared in LWS and MTVDLF-Superbee scheme, additional experiments were conducted for cases with
and
. The time step size and the grid number remain unchanged. The simulation times were increased to 10 and 2, respectively, to ensure the same flow distance of the fluid. The results for three reactants at
are shown in
Figure 6 and
Figure 7, respectively. It is clear that LWS still suffers from unphysical oscillations for all three reactants in both cases with
and
. The oscillations in the case with
are relatively larger than those in the case with
. This is because with increasing flow velocity, the convection-reaction process is becoming more convection dominated. The MTVDLF-Superbee scheme still generates oscillations in
(
Figure 6c and
Figure 7c) in all the two cases, but the amplitudes are much smaller. Slight oscillations appear in
(
Figure 7a) and
(
Figure 7b) for the case with
, whilst no visible oscillations appear in the case with
(
Figure 6a,b).
6. Summary, Conclusions and Future Research
The aim of the paper is to develop a numerical method for convection-dominated reacting flow problems without suffering from the numerical dissipation and unphysical oscillations that have existed in traditional grid-based Eulerian methods. Based on the meshless smoothed particle hydrodynamics (SPH) method, a Lagrangian particle scheme has been proposed for solving the autocatalytic reaction model with multicomponent reactants. For a better illustration of the proposed particle scheme, four typical Eulerian methods were also studied, including the first-order accurate UDS and LFS, the second-order accurate LWS and high-resolution MTVDLF with Superbee limiter.
Numerical results indicated that the MTVDLF-Superbee scheme is capable of yielding more accurate solutions than the other three Eulerian methods. This is consistent with Alhumaizi’s conclusion that the MTVDLF-Superbee scheme is the most appropriate method for simulating the autocatalytic reaction model [
7]. However, MTVDLF-Superbee still suffers from unphysical oscillations near the shock region for a certain component reactant. This will greatly reduce the accuracy of the solution. However, the proposed Lagrangian particle algorithm can successfully solve the autocatalytic reaction model without any numerical diffusion and unphysical oscillations. It is therefore considered to be an effective numerical method for completing the convection-reaction models.
As the reaction flow model considered in this work is governed by the convection-reaction equations, there will be no differential operators in the governing equations after being transformed into the Lagrange system. As a result, there is no need to apply the SPH approximation for derivatives in the proposed Lagrangian particle scheme. However, when the diffusion process needs to be included, derivative approximations will be necessary. This is being worked on and will better illustrate the advantages of the proposed Lagrangian particle scheme.