Numerical methods for compressible multi-component flows

A brief review of both front-tracking and front-capturing methods successfully applied to the compressible multi-component flow problem detailed above is presented below, showing the diversity of approaches available in the literature to tackle this problem.

Front-tracking methods

Front-tracking methods, as already pointed out, avoid pressure oscillations by explicitly tracking discon-tinuities, thus keeping them sharp and removing numerical diffusion. A level set function is usually used to track the position and topology of the front [72,215,223]. However, modifications of the governing equations are often required. Two major types of front tracking methods are,

◦ The Volume of Fluid Model (VOF) was first described by Noh and Woodward [219], formalised by Collela, Glaz and Ferguson (CGF) [67] and is further discussed in [206]. It is essentially equivalent to a multiphase model where the individual species mass and energies are tracked and the momentum is solved for the mixture. The algorithm reconstructs material interfaces within the computational cell using the different components volume fractions, such that the appropriate local thermodynamic properties of the mixture can be recovered. The fluid mixture is thus evolved as a single fluid, and the pressure equilibrium is maintained. However, its implementation is very complex due to the logistics of having to deal with arbitrary shaped interfaces.

◦ The Ghost Fluid Method (GFM) is a level-set based approach which treats interfaces as internally moving boundaries [95,183,222]. It keeps track of two sets of variables, both real and “ghost”.

Real variables correspond to cell variables in the region where they belong, while “ghost” variables correspond to data across interfaces extrapolated from the real ones. The equations are then solved as usual, with the notable exception that ghost variables are now used within the near-interfaces stencils. This effectively transforms the multi-component model into a single fluid type model. Extrapolation in the ghost cells affects conservation, but refinement tests have shown that convergence is obtained [95].

Front-capturing methods

Following on the apparent failure of fully-conservative schemes, numerous non- or quasi-conservative methods have been proposed in the literature for shock-capturing models.

Karni [146] proposed to use a primitive formulation of hyperbolic laws (Euler equations) modified to account for leading order term conservation errors, and further enforcing the convergence on consistent weak solutions. This was further extended by Quirk and Karni [250] by correcting the formulation at shock waves. Jenny et al. [139] altered the energy equation to modify the conservative variables calcu-lation and make it single-fluid like, thus reducing oscilcalcu-lations amplitude. Karni [148] also proposed to solve the Euler equations separately on each side of the interface using a method designed for single-fluid flows, while the interface was being dealt with by a pressure evolution equation derived from the energy conservation.

Abgrall [5] proposed a “quasi-conservative” method, so called due to the extremely small conservation errors generated. The Euler equations augmented by an advection equation are written in a “quasi-conservative” form and solved using a predictor-corrector approach. An advection equation transports a thermodynamic quantity which does not allow pressure oscillations at interfaces, such as 1/γ − 1.

Most of the above presented models were not, or could not be extended to high-resolution state-of-the-art shock-capturing schemes. Shyue [260] used a similar approach as Abgrall [5] albeit transporting species mass fractions instead of 1/γ − 1, and using high-resolution wave propagation methods showing that the pressure equilibrium could be conserved with high-order methods.

Allaire et al. [7] presented a new set of equations re-writing the species mass transport in terms of the species volume fraction, and adding an advection equation for the volume fraction. The system represents an extension of the work of Abgrall [5] and Karni [147] and is closed by computing γ from the species volume fractions.

More recently, building upon Shyue [260] and Abgrall [5] work, Johnsen and Colonius [143] extended the quasi-conservative method to the WENO [184] (Weighted Essentially Non Oscillatory) framework in which the average primitive variables must be reconstructed to prevent spurious oscillations. They also modified the HLLC Riemann solver [278] to treat advection equations. This work is further generalised to all material discontinuities by Johnsen [142] and Johnsen and Ham [144]. However, this methodology has been shown to fail in the case of strong shock waves by Thornber [273].

Another approach has been proposed by Abgrall and Karni [6] for calorically perfect gas, extended to thermally perfect gas by Billet and Abgrall [25], to high-order WENO methods by Houim and Kuo [133]

and to the Discontinuous Galerkin framework by Billet and Ryan [27]. It relies on the computation of two fluxes at interfaces while maintaining a constant specific heat ratio across the stencil. This allows the computation of the Riemann fluxes in a single-fluid like fashion therefore removing pressure oscillations, at the cost of quasi-conserving total energy, as energy fluxes will not be equal any more from the interface left and right sides. This method is usually referred to as the “double-flux” method.

Kawai and Terashima [149] suggested the use of the fully compressible system (Navier-Stokes equa-tions) using compact central differencing schemes and the Localised Artificial Diffusivity (LAD) method [68], thus adding dissipation where needed to remove pressure oscillations. This method has proved its accuracy in shock-bubble type test cases, as well as in Richtmyer-Meshkov instability simulations.

Numerous valuable pieces of work published on this problem are also worth mentioning, such as the work of Wang et al. [289] with the Total Enthalpy Conservation of the mixture (ThCM) model and its extension by Cael et al. [43] and Bates et al. [17] into the Thermodynamically Consistent and Fully Conservative (TCFC) model, Housman et al. [134,135] and his use of the Split Coefficient Matrix method (SCM) of Chakravarthy et al. [53], Marquina and co-authors [81,198], amongst others.

Application to multi-component compressible reacting flows

Simulating multi-component compressible turbulent combustion is a challenging endeavour, and as such, robust and accurate numerical methods are needed. If momentum and energy conservations are im-portant, the most important requirement would be total mass conservation. An even more restrictive condition in the case of reacting flows is the mass conservation of each species, as even trace amounts of radical species (HO2, H2O2, OH, O, H, etc.) do have large effects on the overall combustion process, and need to be accounted for precisely.

Amongst the front-tracking methods presented above, it can be noted that both have already been applied successfully to combustion of premixed slow deflagrations and detonation discontinuities. One could cite the work of Ton et al. [277] for the VOF, and Fedkiw et al. [96] and Desjardins et al. [74] for the GFM. One could also cite Smiljanovski’s work on detonations using level-set methods [262]. How-ever, it can be noted, that although the governing equations are solved in strong conservation form in the GFM method, it is discretely non-conservative due to variables extrapolations across interfaces [6,183].

At the exception of Kawai and Terashima’s method [149] which is fully conservative, all the above front-capturing methods suffer from the conservation of at least one conserved variable. The non-conservative integration scheme of Karni [148], and the quasi-non-conservative method of Abgrall [5] (non-conservative form of the advection equation) suffer from poor mass conservation as shown by Johnsen [141]. Additionally, non-conservative methods do not always predict accurately shock positions, and potentially fail at large Mach numbers, while the quasi-conservative ones revert to a non-conservative formulation in a small number of cells and thus potentially feature similar deficiencies. Hence, it seems complex to apply these methods to viscous reacting flows. To the author knowledge, no such problem has been identified in the literature for the models of Allaire et al. [7] (where each species mass is conserved but each species energy is not necessarily) and Abgrall and Karni [6] (where total energy is not conserved).

If the combustion needs to be solved accurately, the calorically perfect gas EoS might need to be replaced by the more precise thermally perfect gas EoS. Amongst the previously mentioned models, the methods of Shyue [260], Allaire et al. [7], Abgrall and Karni [6] have been demonstrated with mixtures of non-calorically perfect gases, and only the double-flux has been applied to combustion by Billet and co-authors [24–27], Houim and co-authors [133,269] and Lv and Ihme [188–190].

2.4 Conclusion

In this chapter, different fundamental concepts related to premixed laminar flames were introduced (§. 2.1.1). The flame structure was discussed as well as the different characteristics of laminar pre-mixed combustion such as flame speed and flame thickness that are essential parameters in the mod-elling of turbulent combustion. Different properties of turbulence were also discussed (§.2.1.3), and the combustion-turbulence interaction was characterised using the so-called regime diagrams (§.2.1.4).

The three different computational approximation levels of turbulence modelling (RANS, LES and DNS) have been detailed (§.2.2.1), before presenting in some details the different techniques developed to simulate turbulent combustion within the LES framework (§.2.2.3 - 2.2.5). These are the methods relying on the estimation of the flow mixing properties, on the approximation of the flame front as a ge-ometrical surface propagating in a turbulent flow or on the evaluation of flow statistics. Their strengths and weaknesses have been reviewed to highlight the choice made in this work §.7.3.1.

Finally, the modelling of compressible multi-component flows has been reviewed (§.2.3). The failure of the classical numerical methods (Godunov-type methods) derived for compressible flows when applied to multi-component flows and resulting in spurious pressure and velocity oscillations at interfaces between species was explained and demonstrated on a simple test case (§. 2.3.2). The need for moving towards

quasi- or non-conservative methods for the simulations of such flows was also presented. The two families of techniques derived to overcome this particular issue, shock-tracking and shock-capturing methods, have then been presented. Their application to the case of compressible reacting flows is further discussed (§.2.3.3), and the selection of some of the models considered in this work was also highlighted.

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