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Large Eddy Simulation of Premixed Turbulent Combustion Using E Flame Surface Wrinkling Model

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Flow, Turbulence and Combustion 72: 1–28, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 1 Large Eddy Simulation of Premixed Turbulent Combustion Using Flame Surface Wrinkling Model G. TABOR 1 and H.G. WELLER 2 1 School of Engineering, Computer Science and Mathematics, Harrison Building, University of Exeter, North Park Road, Exeter EX4 4QF, U.K.; E-mail: g.r.tabor@ex.ac.uk 2 Nabla Ltd., The Mews, Picketts Lodge, Picketts Lane, Salfords, Surrey RH1 5RG, U.K. Receiv
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  Flow, Turbulence and Combustion 72: 1–28, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands. 1 Large Eddy Simulation of Premixed TurbulentCombustion Using  Flame Surface WrinklingModel G. TABOR 1 and H.G. WELLER 21 School of Engineering, Computer Science and Mathematics, Harrison Building, University of  Exeter, North Park Road, Exeter EX4 4QF, U.K.; E-mail: g.r.tabor@ex.ac.uk  2  Nabla Ltd., The Mews, Picketts Lodge, Picketts Lane, Salfords, Surrey RH1 5RG, U.K. Received 24 April 2002; accepted in revised form 5 May 2003 Abstract. One commonly-used method for deriving the RANS equations for multicomponent flowis thetechnique of conditional averaging. Inthis paper the concept is extended toLES,by introducingtheoperations of conditional filteringandsurface filtering.Propertiesof thefilteredindicator function b are investigated mathematically and computationally. These techniques are then used to deriveconditionally filtered versions of the Navier–Stokes equations which are appropriate for simulatingmulticomponent flow in LES. Transport equations for the favre-averaged indicator function  b and theunresolved interface properties (the wrinkling and the surface area per unit volume) are also derived.Since the paper is directed towards modelling premixed combustion in the flamelet regime, closureof the equations is achieved by introducing physical models based on the picture of the flame as awrinkled surface separating burnt and unburnt components of the fluid. This leads to a set of modelsfor premixed turbulent combustion of varying complexity. The results of applying one of this set of models to propagation of a spherical flame in isotropic homogeneous turbulence are analysed. JEL Codes: D24, L60, 047. Key words: Large Eddy Simulation, premixed turbulent combustion. 1. Introduction PremixedTurbulent Combustion isahighly complex process, but onewhich greatlyaffects everyday life. The quest to understand the physical processes better is con-tinual, and one aspect of it is the search for computational models to describethe processes involved. Such models must of necessity be less detailed than thephysical processes occurring in the system, but should aim to capture the essenceof these processes. In turn, the models can provide a greater understanding of theprocesses involved, and provide us with the ability to predict the behaviour of specific combustion systems. Thus they are of great importance in the design of combustion devices such as Internal Combustion (IC) engines and gas turbines.A working model of turbulent combustion must provide adequate treatments forthe turbulence, the chemical reactions of the combustion (and consequential heat  2 G. TABOR AND H.G. WELLER release), as wellas the mutual interaction ofthese areas, since the combustion altersthe physical properties of the fluid and drives the flow, whilst the flow moves reac-tants and products around and thus influences the combustion. About the simplestpossible model combines a Reynolds Averaged Navier–Stokes (RANS)descriptionof the turbulence with a simplistic model of the combustion which provides amodel for the heat release as a straightforward function of the reactant speciesconcentration (for example, the Eddy Breakup model of Spalding [31]). Numerousimprovements on these simple models have been investigated over the years, inparticular concentrating on improved methods for characterising the species con-centration at apoint (and thus the prediction ofthe heat release) by PDFtechniques,or improved flame modelling.Large Eddy Simulation (LES) of premixed turbulent combustion is an activearea of research. It offers the possibility of significant improvements over RANS,interms ofaccuracy ofthe solution, the ability tohandle counter-gradient diffusion,and the provision of greater information about the turbulent flow field which ren-ders irrelevant some of the modelling assumptions necessary in RANS combustionmodels. The approaches used in LESare based on various ways of computationallyfollowing the flame front. In premixed combustion the flow consists of regions of unburnt reactants and regions of combusted products. The extent of combustionof the gas can be described in various ways in terms of a progress variable takingvalues between 0 and 1, with the extreme values indicating the presence of unburntor fully burnt phases, and the transition between these values marking the flamefront. This can be linked directly to physical properties of the gas, for instance byutilising normalised temperature ( T  ) or product mass fraction ( Y  ): c = T  − T  u T  c − T  u or c = Y  P  Y  P,b . (1)The exact linkage is not important however, and the progress variable can be con-sidered simply as indexing the ammount of combustion, however defined. In thispaper we will use a progress (technically a regress) variable b = 1 − c with b ∈ [ 0 , 1 ] , where 0 represents fully burnt gas and 1 unburnt gas.There is a problem here though. In the flame-sheet regime of high Damköhlerand Reynolds numbers the reaction zone is very thin indeed and the transition inthe progress variable is too sharp to be explicitly resolved on the LES mesh. Oneapproach sometimes employed is the Thickened Flame (TF) approach [7, 10]. Inthe TF method the flame front is artificially thickened by multiplying the thermaland molecular diffusivities by a factor F  and reducing the reaction rate by the samefactor. The result is a thickened flame front with the same laminar flame speed S  l , which can be resolved on the LES computational mesh, and thus its motioncan be calculated without additional SGS modelling. This has several advantages,simplifying the chemical reaction modelling and eliminating the need for ad hocsubmodels for ignition and flame-wall interactions. However it does involve alter-ing the physics of the flame front in a substantial manner. In particular the response  LES OF PREMIXED TURBULENT COMBUSTION 3of the flame to unsteady phenomena and to strain induced by the velocity field ismodified by the thickening procedure [1, 10].The mostcommon approach for LES,known as the G -equation method, is basedon a level-set approach [3, 16, 19, 21, 22]. Here a function G is constructed to havethe property that the zero value isosurface represents the combustion interface. G is not related to the progress variable, so other values of  G have no physical sig-nificance and are merely chosen for computational convenience. A straightforwardtransport equation is then solved for G : ∂ G ∂t  + U . ∇ G = S  T  |∇ G | , (2)where U is the fluid velocity and S  T  the turbulent flame speed, i.e. the rate of propagation of the flame front due to combustion. The challenge in this approachcomes from developing adequate modelling for the turbulent flame speed which isa well-defined quantity that depends on local flow conditions [24]. There are alsonumerical problems with the accurate propagation of  G .The other option for simulation of combustion in this regime is to link theprogression of the flame front to additional physical properties, e.g. geometricproperties of the surface. In the G -equation, surface stretch and curvature effectsare treated by consideration of higher moments of  G . An alternative class of mod-els can be constructed based on solving for variables describing these geometricalparameters [33]. In RANS,the basic ‘laminar flamelet’ models have been extended[8, 9, 20, 27]: the flame front propagates locally as a laminar flame but at thesame time is being wrinkled due to interactions with the turbulence. The flamepropagation speed can be modelled in terms of the laminar flame speed (a knownquantity) and the degree of wrinkling of the flame at the point, given by the flamearea per unit volume  . The system as a whole is described in terms of transportequations for the filtered progress variable and for  . This approach has also beeninvestigated for LES [2, 15]. An alternative RANS model proposed by Weller[34, 35], represents the geometric properties of the flame front in terms of thedensity of wrinkling  , which is the flame area per unit area resolved in the meandirection of propagation. This choice of variable makes the modelling somewhateasier compared with the equivalent equation for  , for instance by separating outa term representing flame annihilation by cusp formation. It also provides for aspectral analysis of the flame-turbulence interaction [36]. This RANS model wasformulated using the technique of Conditional Averaging [11]. The aim of ourcurrent work is to formulate an LES version of this model. In order to do so wemust introduce an analogous techique, that of Conditional Filtering, to derive thetransport equations for a multicomponent system. This technique is the subjectof this paper. Section 2 introduces the concept of Conditional Filtering in LES,and discusses the regularity of the flame surface in relation to the surface filter-ing process introduced as part of the analysis. The effect of filtering a simulatedindicator function appropriate for combustion is investigated in 2-d. In Section 3,Conditional Filtered versions of the Navier–Stokes Equations (NSE) are presented,  4 G. TABOR AND H.G. WELLER together with transport equations for properties of the surface, in particular areaper unit volume  and wrinkling  . Finally, in Section 4 possible closures of theequations are discussed. The evolution of a spherical flame is calculated using sucha model and its properties discussed. 2. Conditional and Surface Filtering The Navier–Stokes Equations (NSE) for a compressible fluid are ∂ρ∂t  +∇ .ρ U = 0 ,∂ρ U ∂t  +∇ .(ρ U ⊗ U ) =−∇ p +∇ . S ,∂ρe∂t  +∇ .ρe U = − p ∇ . U + S . D +∇ .κ ∇ e, (3)where S = λ ∇ . UI + 2 µ D , D = 12 ( ∇ U +∇ U T  ). (4)In Conditional Averaging in RANS, an indicator function is introduced [11] whichtakes the value 1 in the unburnt region phase and 0 in the burnt region. The NSEare multiplied by this function and then ensemble-averaged: the indicator functionprojects out one of the components, and so the resulting equation is for that com-ponent alone. This process introduces additional terms (in addition to the standardReynolds Stress term arising from the ensemble averaging process) which canbe written in terms of a surface average operation which represents the effect of the interface on the dynamics of the phase under consideration. These terms willcommonly require modelling. Transport equations can also be formulated in thisway for the ensemble averaged indicator function, which has the interpretation of the probability of finding the phase at that point, and for quantities relating to thesmall-scale geometry of the interface.InLESitisassumed that thedependent variables intheNSEcan bedecomposedinto GS and SGS components, i.e. ψ = ψ + ψ  . The GS component is obtainedby filtering ψ , which is a convolution between it and a filter function G with theproperties   D G( x ) d 3 x = 1, lim  → 0 G( x ,) = δ( x ) and G( x ,) ∈ C n ( R 3 ) with compact support. The decomposition into mean and fluctuating componentsis thus analogous to the decomposition in RANS, but with differing interpretationsof the resulting variables. We can adapt LES to include the concept of conditionalaveraging by introducing an indicator function I which is a generalised function(or distribution) such that I ( x ,t) =  1 if  ( x ,t) is in phase A (say the unburnt gas),0 otherwise.(5)
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