Numerical methods for direct simulation

DNS of reacting flows differs in many ways from DNS of non-reacting situations where the fundamental limitation is determined by the turbulence Reynolds number. The maximum Reynolds number which can be reached without combustion is determined by the number of grid points as described below. The simulation of reacting flows requires a set of balance equations and the selection of chemical kinetics and species transport properties. The initial

flow configuration has to be set-up with great care. The Reynolds number upper limit is also controlled by the number of grid points, but another condition is imposed by the proper resolution of the flame inner structure. The main issues and trade-offs encountered in the field of DNS are listed below. The following aspects are successively considered: formulation, description of chemistry, initial and boundary conditions, dimensions, description of heat losses, algorithm selection.

Formulation: incompressible, low Mach number and compressible

Since multiple formulations may be derived for reacting flows equations (Chapt. 1), the first issue is to choose a form for the conservation equations. Most DNS of non-reacting flows have been performed for incompressible (constant-density) flows. For reacting flows, many options may be retained depending on the acceptable simplifications for a certain application. DNS of reacting flows may rely on the constant density (“thermodiffusive”) approximation, low-Mach number formulations (variable density formulations where acoustic waves are filtered out) or fully compressible flow descriptions (variable density, variable pressure).

While constant density formulations yield interesting information on some mechanisms governing turbulent flames, they are limited by the absence of flame-induced flow modifications due to heat release. In thermodiffusive formulations, temperature variations can be taken into account in the evaluation of reaction rates (see Clavin¹⁰²for applications of this approximation in laminar flame theory and Rutland and Ferziger⁴⁶⁰ for a typical direct simulation) if a modified state equation decouples the temperature from density and pressure field. Under this approximation, the essential stiffness of Arrhenius kinetics is retained but the conversion of chemical species and the corresponding heat release do not influence the density.

A less restrictive formulation relies on the low-Mach number approximation where density may change due to temperature variations but remains independent of pressure. In this formulation, acoustic waves are eliminated so that the numerical time step is no longer limited by the classical Courant-Friedrichs-Lewy (CFL) condition.^ix This formulation is well-suited for low-speed flows, typically encountered in subsonic combustion. Nevertheless, it cannot be retained in studying combustion instabilities where flow field, acoustic field and chemical reaction interact (Chapt. 8). It remains also questionable when pressure gradients are involved in scalar turbulent transport (terms (VI) and (VII) in Eq. (5.116)).

Despite the higher cost, fully compressible formulations have some advantages which ex-plain why they have been used in many DNS of reacting flows: flame/acoustics interaction effects are captured, combustion in high-speed flows may be studied, and an easier treatment of boundary conditions is offered (Chapt. 9). With an explicit time advancement scheme the CFL criterion limits the time step and the computational cost becomes important, especially in low-speed flows. Nevertheless, in practical simulations, this time step is often limited by the

ixThe Courant-Friedrichs-Lewy condition prescribes that an acoustic wave cannot move over more than one computational grid size ∆x during one computational time step ∆t. Accordingly, the possible time step is limited as ∆t ≤ ∆x/c where c is the sound speed. The CFL number is defined as CF L = c∆t/∆x and should remain lower than unity in explicit compressible codes. Implicit compressible codes are generally able to work with higher CF L numbers but require the resolution of an implicit equation for pressure.

chemical reaction rate, and not by acoustic wave propagation, so that the fully compressible formulation does not induce additional cost.

Chemical system selection

In addition to the form retained for the flow field equations (incompressible, low Mach Number or compressible flows), a DNS of reacting flows requires a description of chemistry. Specialists in chemistry argue that a realistic computation of hydrocarbon/air flames requires, at least, twenty chemical species while simulation experts point out that numerical constraints of un-steady computations in three dimensions make such a computation practically impossible. The choice of the chemical scheme essentially depends on the question being investigated. Studies of the folding of premixed flames in low-intensity turbulence may be carried out with the sim-plest chemistry or even with no kinetics at all by simply tracking the flame as a propagating front. On the other hand, investigation of pollutant formation in a turbulent flame requires a reasonably complete chemical scheme. The latter computation is much more realistic (and expensive) but the former may also be quite informative. Valuable results on the development of turbulent flames and their modeling have been deduced from simulations based on front tracking (for example Yeung et al.⁵⁶²) showing that physical intuition is often more important than the largest possible computer resources.

Initial and boundary conditions

The choice of initial configuration and boundary conditions introduces additional difficulties.

Most cold flow simulations are carried out in periodic domains. Reacting flows preclude using periodic conditions at least in one direction (a planar premixed flame separates fresh and burnt gases and a diffusion flame take place between fuel and oxidizer streams) and more refined boundary conditions have to be specified, describing for example inlet and outlet flows (see Chapt. 9). The precision of these conditions should match that of the numerical solver.

Parasitic phenomena may couple inlet and outlet boundaries when the numerical scheme has a very low numerical viscosity. Under these circumstances, numerical waves, traveling on the grid and propagating from outlet to inlet, have been observed (Buell and Huerre⁷², Poinsot and Lele⁴¹⁹ or Baum²⁷). For incompressible or low Mach number formulations (Orlanski³⁸⁴, Rutland and Ferziger⁴⁶⁰ or Zhang and Rutland⁵⁶⁶), the specification of boundary conditions preventing such phenomena requires a fine tuning with the pressure solver used inside the domain. Another solution is to sacrifice a certain part of the computational domain near the outlet, used as a buffer zone where perturbations are damped by a high artificial viscosity before reaching the outlet. For compressible flows, Poinsot and Lele⁴¹⁹ present a method based on characteristic analysis (Thompson⁵⁰⁸) for direct numerical simulation of compressible reacting flows that has been validated in many different situations (see Chapt. 9). This method was later extended to multi-species flows with complex chemistry (Baum et al.³⁰). Higher-order formulations of the same techniques for non-reacting flows may also be found in Giles¹⁸³ or Colonius et al.¹¹¹.

Initial conditions may also be difficult to set and limit the possible configurations to be studied. For example, DNS of turbulence/combustion interactions require the initial specifi-cation of a given turbulent flow field, compatible with the mass and momentum conservation equations and having a prescribed spectrum. At the initial time, this turbulent field is su-perimposed to a laminar flame, generally planar. This situation is clearly non physical and an adaptation time is required. To be relevant, the initial flame has to be issued from a pre-vious one-dimensional laminar DNS. In the absence of forcing, turbulence decays with time whereas the flame front is wrinkled by flow motions. Then, no steady state situation may be reached limiting the possible conclusions of the simulations, generally analyzed after several eddy turnover times. Injection of a well-defined turbulent flow field in the numerical domain is possible but difficult and expensive: see Van Kalmthout and Veynante⁵²⁰ for supersonic incoming flows, and, for subsonic incoming flows Rutland and Cant⁴⁵⁸ (low-Mach number formulation), Guichard¹⁹⁷, Vervisch et al.⁵²⁶(fully compressible formulation).

Two-dimensional versus three-dimensional DNS

Because computer resources are limited, it is often necessary to perform reacting flow simula-tions in two dimensions, thereby missing all three-dimensional effects of real turbulence. This approximation is generally not suitable for non-reacting flows because turbulent fluctuations are intrinsically three-dimensional. For premixed combustion, however, direct simulations (Cant et al.⁸⁶) show that the probability of finding locally cylindrical (2D) flame sheets is higher than the probability of finding 3D spheroidal flame surfaces (§ 5.5.5). Two-dimensional flames appear more probable even though the flow field ahead of these flames is fully three-dimensional. Considering the prohibitive cost of three-dimensional reacting flow computa-tions, two-dimensional simulations remain quite valuable. This is especially true when dealing with “flame/vortex” interactions where the dynamics of a flame front interacting with iso-lated vortex structures is examined (§ 5.2.3). Flame/vortex studies yield useful information on turbulent combustion (see Laverdant and Candel²⁹¹, Poinsot et al.⁴²⁴ or Ashurst¹³) This simple problem is also well-suited for comparisons with experimental investigation (Jarosinski et al.²³⁹, Roberts and Driscoll⁴⁴⁹, Roberts et al.⁴⁵¹, Lee et al.²⁹⁸, Roberts et al.⁴⁵⁰, Driscoll et al.¹⁴⁸, Mantel et al.³³⁴ or Thevenin et al.⁵⁰⁶). To study scalar turbulence transport in premixed flames, Veynante et al.⁵³⁶ have compared two and three-dimensional simulations.

In terms of turbulent transport, no clear difference has been evidenced and the criterion pro-posed to delineate between gradient and counter-gradient regimes is in very good agreement with both DNS. But, on the other hand, for the numerical cost of a single three-dimensional DNS, thirty two-dimensional DNS can be performed allowing the exploration of a large range of physical parameters.

Description of heat losses

Direct numerical simulations of reacting flows should include a description of heat transfer by radiation and convection. Heat losses from the reactive region govern flame quenching processes (Patnaik and Kailasanath³⁹⁰) and thus determine the combustion regime (Poinsot

et al.⁴²²) or the flame evolution near a wall. Detailed models of radiation require complicated descriptions of the spectral characteristics of the main chemical species and the solution of the radiative transfer equations. Simplified assumptions (Williams⁵⁵⁴) have been used up to now to avoid the added complexity of the general formulation (see however Soufiani and Djavdan⁴⁸⁹, Daguse et al.¹²⁴, Wu et al.⁵⁵⁶for realistic radiative models coupled to combustion codes).

Numerical schemes

Multiple numerical algorithms have been used for DNS of reacting flows. To simulate constant density flames, classical incompressible DNS codes developed for cold flows and generally based on spectral methods may be used. These schemes are accurate but limited to periodic boundary conditions. For this reason, recent simulations have relied on alternative higher-order finite difference schemes (Lele²⁹⁹) or mixed schemes which use spectral methods in two directions and a finite difference method along the non-periodic direction (Nomura and Elgobashi³⁷⁸, Nomura³⁷⁷). Spatial derivatives in the non-periodic direction are estimated by higher-order upwind schemes or by compact Pade approximates (Lele²⁹⁹).

Fig. 4.19 summarizes the wide range of formulations used for DNS of turbulent flames (dif-fusion or premixed). For example, three-dimensional constant density codes applied to DNS of cold flows may be used without modification to study the wrinkling of material surfaces by turbulence (Yeung et al.⁵⁶²). The propagation of premixed flames with a specified flame speed in a turbulent flowfield may be studied introducing a transport equation for a field variable G. While the computed flow differs from real flames (no chemistry and in most cases no heat release effects), this formulation provides valuable information on premixed flames for a reasonable computational cost. In the middle of Fig. 4.19, three-dimensional computations performed for premixed flames with simple chemistry (Arrhenius law, variable density, vari-able viscosity) or for diffusion flames with infinitely fast chemistry require about 100 times more CPU hours for each run. At the top of the diagram (Fig. 4.19), complex chemistry computations in 2D or 3D require the largest computer resources: calculations are limited to small box sizes and cannot be repeated systematically. A few direct numerical simulations of H2-O2or CH4-air flames in two or three dimensions with complex chemistry, variable density, and multispecies transport models have been reported in recent years (Patnaik et al.³⁹¹, Baum et al.²⁸, Echekki and Chen¹⁵⁵, Tanahashi et al.⁵⁰⁰, Mizobuchi et al.³⁵⁸, Mizobuchi et al.³⁵⁹, Thevenin⁵⁰⁵).

Most direct simulations of turbulent combustion have been based on regular non-adaptive meshes because turbulent flames move rapidly in the whole domain, exhibit considerable wrinkling and cannot be easily treated with self-adaptive gridding. While adaptive meshes work well in simple geometry (Darabiha et al.¹²⁷, Giovangigli and Smooke¹⁸⁶, Dervieux et al.¹³⁶), there are few examples of DNS applications. Lagrangian methods and especially random-vortex methods (RVM), which are grid free, are not emphasized here because our attention is focused on techniques which solve the Navier-Stokes equations.

Complex chemistry

One-step chemistry

No chemistry

2D 3D

Variable densityConstant density

2D 3D

Variable densityConstant density

2D 3D

Variable densityConstant density

Patnaik and Kailasanath 1988 Katta and Roquemore 1998 Baum et al 1994 Haworth et al 2000 Chen et al 1998 Tanahashi et al 1999 Hilbert and Thévenin 2002 Jiménez et al 2002

Tanahashi et al 1999 (TSFP) Mizobuchi et al (2002, 2004) Thevenin (2004)

Rutland and Ferziger 1991 Poinsot, Veynante, Candel 1991

Trouvé and Poinsot 1994 Rutland and Cant 1994 Zhang and Rutland 1995 Swaminathan and Bilger 1999 Montgomery, Kosaly, Riley 1997

Leonard and Hill 1988 Rutland and Trouvé 1990 Cant, Rutland and Trouvé 1990 ElTahry, Rutland, Ferziger 1991 Rutland and Trouve 1993 Ashurst, Peters, Smooke 1987

Laverdant and Candel 1989

Barr 1990

Katta and Roquemore 1993 McMurtry and Givi 1989

Ashurst and Barr 1983 Osher and Sethian 1987 Ashurst 1987

Ashurst, Kerstein, Kerr, Gibson 1987 Yeung, Pope 1990

Cattolica, Barr, Mansour 1989

NB: non exhaustive list

Figure 4.19: Examples of direct numerical simulations (DNS) of turbulent flames.