System decomposition technique for spray modelling in CFD codes
Introduction
Decomposition of complex systems into simpler subsystems is de facto almost universally used in engineering and physics applications. It allows the numerical simulation to focus on the subsystems, thus avoiding substantial difficulties and instabilities related to numerical simulation of the original systems. Special rules are introduced to incorporate the results of numerical simulation of the subsystems into the general scheme of the simulation of the whole system. To the best of our knowledge, the hierarchy of the decomposition process has been so far fixed for the duration of a process.
As an example of such decomposition we can mention the solutions of ordinary and partial differential equations (ODEs and PDEs) describing spray dynamics in computational fluid dynamics (CFD) codes. Numerical spray modelling is traditionally based on the Lagrangian approach coupled with the Eulerian representation of the gas phase. This permits the decomposition of complicated and highly nonlinear systems of PDEs, describing interactions between computational cells, and the systems of ODEs that govern processes in individual computational cells, including liquid/gas phase exchange and chemical kinetics. The systems of ODEs are usually integrated using much shorter time steps δt (typically 10−6 s) than the global time steps used for calculating the gas phase Δt (typically 10−4 s). Thus the decomposition of ODEs and PDEs is de facto used although its basis has not been rigorously investigated to the best of our knowledge [1], [2].
Further decomposition of the system of ODEs, describing droplets dynamics inside individual computational cells, is widely used. The simplest decomposition of this system of ODEs is based on the sequential solution of individual subsystems comprising this system (split operator approach). In this approach, the solution of each individual subsystem for a given subset of variables is based on the assumption that all the other variables are fixed. The sequence of the solution of individual subsystems is often chosen rather arbitrarily and the results sometimes vary substantially depending on the order in which these subsystems are solved. In the case of a multiscale system, the reliability of this approach becomes questionable altogether, as shown later in Section 4.
We believe that to overcome these problems the multiscale nature of ODEs needs to be investigated before any attempt to solve them is made. This idea could be prompted by the approaches used in [3] for the analysis of the processes in CO2 lasers and the one used in [4] for the analysis of equations describing the autoignition of diesel fuel (the Shell model). Before solving a system of five stiff ODEs describing five temperatures in these lasers, the characteristic time-scales of these equations were analysed [3]. It was shown that two of these equations describe rather rapid relaxation of two temperatures to the third one. This allowed the authors of [3] to replace the solution of five stiff equations by the system of just three non-stiff equations without any significant loss of accuracy. The approach used in [4] was different from the one used in [3], but the ultimate result of reduction of the number of ODEs to be solved, and elimination of the stiffness of the system of equations, remained the same. In mathematical terms in both papers the dimension of the ODE system was reduced. In other words, the system was decomposed into lower dimension subsystems.
A similar system decomposition into lower dimension subsystems have been used in constructing reduced chemical mechanisms based on intrinsic low-dimensional manifolds (ILDM) (e.g. [5], [6], [7]) and computational singular perturbation (CSP) (e.g. [8], [9], [10], [11], [12]). There are many similarities between these methods. They are based on a rigorous scale separation such that ‘fast’ and ‘slow’ subspaces of the chemical source term are defined and mechanisms of much reduced stiffness are constructed. These approaches, however, were developed with a view of application to modelling chemical kinetics. Their generalisation to general CFD codes has not been considered to the best of our knowledge.
A useful analytical tool for the analysis of stiff systems of ODEs, used for modelling of spray heating, evaporation and ignition, could be based on the geometrical asymptotic approach to singularly perturbed systems (integral manifold method) as developed by Gol’dshtein and Sobolev [13], [14] for combustion applications (see also [15]). This approach overcomes some of the earlier mentioned problems and is, essentially, focused on systems of ordinary differential equations of the form:where x and y are n and m-dimensional vector variables, and ε is a small positive parameter (ε ≪ 1). The first subsystem is called a slow subsystem and the second one is called a fast subsystem. In practical implementations of the integral manifold method a number of simplifying assumptions have been made. These include the assumption that the slow variable is constant during the fast processes. This assumption opened the way to analytical study of the processes [16], [17], [18], [19].
These approaches to decomposing systems of ODEs were developed and investigated with a view to application to rather special problems, and were based on a number of assumptions. These include fixing of the decomposition over the whole period of the process, and not allowing its hierarchy to change with time. The underlying philosophy of these approaches, however, seems to be attractive for application to the analysis in a wide range of physical and engineering problems including spray modelling in general computational fluid dynamics (CFD) codes. The development of a rather universal new method of decomposition of the system of ODEs, allowing the change of the nature of decomposition with time (dynamic decomposition), will be the main objective of this paper.
As in the original integral manifold method, our formal approach to decomposition of the system of ODEs is based on the division of system variables into ‘fast’ and ‘slow’. This leads to the division of this system onto ‘fast’ and ‘slow’ subsystems. In contrast to the original version of the integral manifold method, however, linearised variations of slow variables during the time evolution of the fast variables will be taken into account as the first order approximation to the fast manifold. The usefulness of this division depends on whether the ‘fast’ subsystem has lower dimension compared with the ‘slow’ subsystem. The procedure can be iterative and result in a hierarchical division of the original system. For example the ‘slow’ subsystem can, in its turn, be subdivided into ‘slow’ and ‘very slow’ subsystems.
The proposed procedure will be initially focused on the simplest possible subdivision of the original system into two subsystems, and applied to spray combustion modelling. Note that ‘fast’–‘slow’ decomposition in this case can be different for different phase space regions [16], [17] and for different time intervals. Wider range of application of this method is anticipated.
The preliminary description of this method and its applications was given in [20], [21], [22].
The general mathematical description of the method and the assumptions on which it is based are presented in Section 2. The rest of the paper discusses application of this method to a specific problem of modelling of sprays. Basic equations describing these processes are presented in Section 3. In Section 4 the results of applying the new method to numerical modelling of these processes are described and discussed. The main results of the paper are summarised in Section 5.
Section snippets
Decomposition of the system of equations
Let us consider the system, the state of which is characterized by n dimensionless parameters, which are denoted as Zn (n = 1, 2, … , n). The value of each of these parameters for a given place in space depends on time t, i.e. Zn = Zn(t). This dependence can be found from the solution of the system of n equations, which can be presented in a vector form:where
In the general case, a rigorous coupled numerical solution could be found. This may be not practical, when
Basic equations for spray modelling
In this section basic equations used for modelling droplets heating, evaporation and combustion will be summarised. These equations will be presented in the general form, following [23], [24], and in the simplified form (partly following [25]). The latter form of these equations will allow us to perform a direct comparison between the predictions of conventional CFD approach, and the approach described in this paper. A number of processes, including droplet dynamics, break-up and coalescence,
Application
The method described in Section 2 will be applied to simulate polydisperse spray heating, evaporation and ignition. The model on which the analysis is based is chosen to be rather simple, but capable nevertheless of capturing the essential features of the process. We consider three droplets with initial radii 5 μm, 9 μm and 13 μm, respectively. The initial temperatures of all droplets is taken to be equal to 400 K. The gas temperature is taken to be equal to 880 K [1]. The gas volume is chosen such
Conclusions
A new method of numerical solution of multiscale systems of ordinary differential equations (ODEs) is suggested. This is based on a decomposition technique for systems of ordinary differential equations, using the geometrical version of the Integral Manifold Method. The comparative analysis of the values of the right hand sides of these equations, can result in the separation of the equations for ‘fast’ and ‘slow’ variables. The hierarchy of the decomposition is allowed to vary with time.
Acknowledgements
The authors are grateful to the EPSRC (Grant GR/S98368/01), German-Israeli Foundation (Grant G-695-15.10/2001) and the Academic Study Group (UK) for the financial support of this project. This paper was prepared during the visits of VB and VG to the University of Brighton (UK), and the visits of SS and ES to Ben Gurion University of the Negev (Israel). The authors are grateful to both universities for providing hospitality. Our special thanks are to Mrs. Yu. Shramkova for useful discussions of
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