Direct Numerical Simulation of Sprays: Turbulent

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reactor are described. Eventually spray ... Such NS resolution is defined as Direct Numerical Simulation. (DNS). DNS solves all .... leads to severe conditions based on the turbulent Reynolds number and the Damkohler num- ber, which is the ...
Direct Numerical Simulation of Sprays: Turbulent Dispersion, Evaporation and Combustion Julien Reveillon CORIA CNRS, University of Rouen, Rouen, France

Abstract Numerical procedures to describe the dispersion, evaporation and combustion of a polydisperse liquid fuel in a turbulent oxidizer are presented. Direct Numerical Simulation (DNS) allows one to describe accurately the evolution of the fully compressible gas-phase coupled with a Lagrangian description in order to describe two-phase flows. Standard coupling is used for the Eulerian/Lagrangian system while some practical issues related to the reactive source terms are addressed by suggesting a fast single-step Arrhenius law allowing one to capture the main fundamental properties of theflamewhatever the local equivalence ratio. Then some basic procedures to describe spray preferential segregation in a turbulent reactor are described. Eventually spray combustion is addressed by first demonstrating the complex interactions caused by the presence of an evaporating liquid phase: definition of various equivalence ratios, apparition offlameinstabilities for a unit Lewis number, etc. Then a history of the development of the existing spray combustion diagrams is presented to display the possibleflamestructures and combustion regimes encountered in spray combustion.

1 Introduction It is generally admitted that the Navier-Stokes (NS) equations offer an accurate description of fluid motion. The basis for these equations is that the fluid under consideration is a continuum. Numerical resolution of the NS equations on a fine computational mesh allows one to capture all the macroscopic structures since all the considered length scales are considerably larger than the molecular length and time scales. Such NS resolution is defined as Direct Numerical Simulation (DNS). DNS solves all the characteristic scales of a turbulentflow^from the Kolmogorov 'dissipative' length scale up to the integral 'energy-containing' length scale. However, if a two-phase flow is considered (gas/liquid for example), the apparition of an interface and a strong variation of density jeopardizes the possibility to achieve a complete DNS of the flow. It is especially true if fundamental physical phenomena, like evaporation or heat transfer, are present at the interface. Then the computational cost of the DNS of the whole flow, including both phases, would sky rocket unless some major assumptions were made. A first possibility is to adopt an interface-tracking approach like the 'volume of fluid' (VOF) method developed by Hirt and Nichols (1981). It is based on the reconstruction of the gas/liquid interface from the time and space evolution of the local volume fraction of liquid. This massconservative procedure is complex and time consuming as far as the interface reconstruction is concerned. Another possibility is to use the level-set procedure of Osher and Sethian (1988),

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Julien Reveillon

which follows the motion of an iso-surface of a specific scalar function that maintains algebraic distances. Even if these simulations are still designed as DNS because no turbulence model is used, from a strict point of view, the results are not an exact resolution of the complete NS equations and some approximations are often necessary. For example, an incompressible formulation is generally used. Then evaporation, heat transfer or even combustion phenomena are difficult to account for. Nevertheless, these methods are very promising as demonstrated very recently by Tanguy and Berlemont (2005) who simulated for the first time the complete atomization of a liquid jet. A second possibility is to give up the idea of a complete DNS of the flow while trying to maintain highly accurate results. This objective seems rather difficult to reach as far as dense flows are concerned. On the other hand, when a dispersed liquid or solid phase is embedded in a gaseous carrier phase, some solutions exist. The principle is to carry out a DNS of the gaseous continuous phase and to model the dispersion of the liquid phase through a Lagrangian or an Eulerian formulation (Gouesbet and Berlemont, 1999). In the framework of DNS, where accurate results are more important than computational cost, Lagrangian modeling of the spray is preferable because every particle (or group of particles) is followed in space and time by the solver, whereas statistical integration of the information is obtained when a Eulerian model is used. It appears however that both procedures are complementary: Lagrangian formulations have to be used for the accurate description the dispersion of few (some millions per processor) particles while, on the other hand, an Eulerian formulation seems to be particularly adapted to complex dispersed or dense flows involving large-scale computations. DNS was first introduced 35 years ago by Orszag and Patterson (1972) and then Rogallo (1981) and Lee et al. (1991) for the simulation of inert gaseous flows. It has since been used in a large range of applications. During the last two decades, DNS of reactive flows has been carried out to study non-premixed, partially premixed and premixed turbulent combustion of purely gaseous fluids (Givi, 1989; Poinsot et al., 1996; Vervisch and Poinsot, 1998; Poinsot and Veynante, 2001; Pantano et al., 2003). DNS has been extended to two-phase flows starting with the pioneering work of Riley and Patterson (1974). Most of the first numerical studies were dedicated to solid particle dispersion (see for instance Samimy and Lele, 1991; Squires and Eaton, 1991; Elgobashi and Truesdell, 1992; Wang and Maxey, 1993, and Ling et al., 1998). More recently, Mashayek et al. (1997), Reveillon et al. (1998) and Miller and Bellan (1999) conducted the first DNS with evaporating droplets in turbulent flows. Since then, DNS of twophase flows have been extended to incorporate two-way coupling effects, multicomponent fuels, etc., and to deal with spray evaporation and combustion phenomena (Mashayek, 1998; Miller and Bellan, 1999,2000; Reveillon and Vervisch, 2005). The objective of this text is to offer a full description of the numerical procedures used to carry out DNS of two-phase dispersed flows. Note that a similar methodology may be used for large-eddy simulations if subgrid turbulence, mixing and dispersion models are added to the general balance equations. Starting from the classical NS equations, specific source terms are added to account for the presence of a dispersed liquid phase whose evolution is described by a Lagrangian solver. The modeling of the liquid phase embedded in a full DNS of the gas phase implies that the droplets are not resolved by the Eulerian solver. They are considered as local point sources of mass, momentum and energy. On the other hand, these source terms are obtained thanks to a fine-scale description of the evolution of droplets that are considered individually by the Lagrangian solver. Another major task concerns the chemical source terms describing

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chemical reactions between the various components. Two major solutions emerge: first, detailed chemistry may be used and leads to an accurate description of the combustion phenomena and the flame structures; however, it increases significantly the computational cost. The other possibility, often adopted in single-phase DNS as well, is to use a one-step Arrhenius law to describe the impact of combustion kinetics from a global point of view. In that case, two major conditions have to be respected: an accurate resolution of the basic features of the flame (propagation velocity, thickness, heat release) with respect to the local properties of the flow (equivalence ratio, stretch). However, basic single-step Arrhenius laws show major drawbacks especially when partially premixed combustion is concerned, which is mainly the case with turbulent two-phase flows. Thus, an adapted single-step kinetics has to be used. In the following, two major parts are developed: first, details concerning the coupling of a complete DNS of a gaseous turbulent flow with a Lagrangian model of the dispersed phase are given. This association is especially useful to study fundamental physical phenomena and to carry out the preliminary development of two-phase flow models. Coupling with the dispersed spray is detailed as well as the specific Arrhenius law allowing one to capture properly the combustion phenomena whatever the local equivalence ratio. Then some applications and analysis procedures are briefly described to demonstrate the ability of DNS to capture every major phenomena present in two-phase flows: dispersion, evaporation and combustion.

2 Direct Numerical Simulation DNS is a powerful way to study turbulent flows. Indeed, the NS equations are resolved with highly accurate and non-dissipative numerical methods that are able to capture all the time and length scales of the flow. No models are necessary to observe the development and the evolution of turbulent structures and all results may be considered to be as close as possible to reality (if the simulations are conducted properly). Occasionally, DNS is even referred to as a 'numerical experiment'. The grid mesh has to be fine enough to capture the smallest scales of the flow. This leads to severe conditions based on the turbulent Reynolds number and the Damkohler number, which is the ratio of the fluid motion time scale to the characteristic reaction time. A large Damkohler number indicates a rapid chemical reaction compared to all other processes. DNS is thus very limited from a technological point of view by the capacity of the current supercomputers. Only configurations with very small dimensions and small turbulent Reynolds numbers may be considered at the present time. Nevertheless, because of the accuracy of the results, it is a useful tool to analyze some specific physical phenomena. One has to be careful as far as the 'DNS' term is concerned. Indeed in many works what is called DNS is a 'direct numerical resolution' of the considered equations without any model but not necessarily with adapted numerical methods and grids. This may lead to severe numerical dissipation and approximations, in which case speaking of DNS is abusive because the results are not a direct outcome of the equations for the physics. Indeed, an implicit numerical filtering exists in between the two. For a complete introduction to DNS of turbulent reactive single-phase flows and the resolved equations, readers may refer to the recent book of Poinsot and Veynante (2005). Many formulations may be derived for the NS equations. The fully compressible equations allow for a complete description of the physics including acoustic phenomena. Apart from having a very small time step mainly based on the sound velocity, the main difficulties lie in a

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good formulation of the initial and boundary conditions that must account for entering and exiting acoustic waves. If acoustic phenomena are negligible, it is possible to select a low Mach number (LMN) formulation with either a constant- or variable-density flow. Then, boundary conditions are straightforward to prescribe. However, an elliptic solver is needed to close the momentum balance equation. For non-reactive flow or by using an implicit scheme for the temperature/species equations, the computational cost may be reduced by a factor of five to ten compared to the corresponding compressible formulation. In the next section the complete compressible equations are given, and then a low Mach number formulation is derived. These sets of equations describe the evolution of the gas phase. They are rapidly introduced because they are widely documented in the CFD literature. Then a derivation of the Lagrangian solver to be resolved simultaneously with the DNS is proposed with a detailed description of the coupling between the carrier phase and the dispersed evaporating droplets. This part ends with a detailed description of a single-step Arrhenius law adapted to the full range of fuel and equivalence ratios. 2.1

Compressible formulation

The carrier phase is a compressible Newtonian fluid following the equation of state for an ideal gas. The instantaneous balance equations describe the evolution of mass p, momentum pU, total (except chemical) energy E^ and species mass fraction. Fp denotes the mass fraction of gaseous fuel resulting from spray evaporation and YQ is the oxidizer mass fraction. The following set of balance equations are solved where the usual notation is adopted: dp ot

dpUk axk

H 7{ = dt dxk dpE, , d(pE, + P)Uk ^ 7 " + ~" 5 "^ ot OXk dpYj^Uk apFp , dpYj^Uk dxk dt

H

:\ dxk

.

,^..

\- T; h vt , dxi dxk d (,^T\ doikUk . ^. Fdxk ~ V^ dxk J~J + ^ + pcoe+e, OXfc V d^k J (J^i 9 / 9FF\ . . OXk V OXk J

(2.2)

-T;

= T;— pD-— + p^o , dxk V ^^k)

,.., (2.3)

(2.5)

with V 9-^7 dxii J) together with the equation of state for ideal gases:

J3

dxk OXk

Jij

^'

,

P=prT. Source terms are present, the coi terms are related to the chemical reaction processes and m, v and e result from a two-way coupling between the carrier phase and the spray. These terms will be discussed in detail later. The sixth-order Fade scheme from Lele (1992) and the Navier-Stokes characteristics boundary conditions (NSCBC) of Poinsot and Lele (1992) or Baum et al. (1994) are usually employed

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to solve the gas-phase transport equations on a regular mesh. The time integration of both the spray and gas-phase equations is done with a third-order explicit Runge-Kutta scheme with a minimal data storage method (Wray, 1990). A third-order interpolation is employed when gasphase properties are needed at the droplet positions. 2.2

Low Mach number approximation

The fully compressible set of equations presented above may be normalized by reference physical quantities. Among them the reference velocity will be defined by wo = M^y/yrTo where M is the Mach number. The normalized compressible NS equations may be written as

1 +1 ^ = 0 , dpUj dt

ot

+

(2.6)

dxi dpUiUj _ _]_dP_ ^ dxj^ = - yM^ ^ ^dxi

ot

axi

dpY^ ^ dpYM dt dXi

y ^

dGij

+ ^dxj, oxi

(2.7) oxf \ oxi J

oxi

d f^dy?\ dxi \ dxi J

(2.9)

The various source terms have been dropped in this intermediary expression and the internal energy {Ei = E^ — U^/2) has been selected to simplify the low Mach number system. A new variable 6 = yM'^ is introduced. If the low Mach number hypothesis is adopted, 6 1). In fact, a constant growth of the velocity may be observed even for equivalence ratios greater than unity, whereas it should reach a maximum value in the vicinity of

, a'

1

Q

~'

o

o

o

; / ° o^^ '^^;

o