Download Turbulent Combustion: The State of the Art

 Lecture 10 Turbulent Combus2on: The State of the Art 10.-­‐1 •  Engineering applica0ons are typically turbulent →  Turbulence models •  These models use systema0c mathema0cal deriva0ons based on the Navier-­‐
Stokes equa0ons •  They introduce closure hypotheses that rely on dimensional arguments and require empirical input → Semi-­‐empirical nature of turbulence models 10.-­‐2 •  The apparent success of turbulence models in solving engineering problems for non-­‐reac0ve flows has encouraged similar approaches for turbulent combus0on →  Turbulent combus0on models •  Problems: •  Combus0on requires that fuel and oxidizer are mixed at the molecular level à Depends on small-­‐scale mixing •  How this takes place in turbulent combus0on depends on the turbulent mixing process 10.-­‐3 •  The general view is that once a range of different size eddies has developed, strain and shear at the interface between the eddies enhance mixing •  During the eddy break-­‐up process and the forma0on of smaller eddies, strain and shear will increase and thereby steepen the concentra0on gradients at the interface between reactants, which in turn enhances their molecular inter-­‐
diffusion •  Molecular mixing of fuel and oxidizer, as a prerequisite of combus0on, therefore takes place at the interface between small eddies •  There remains, however, the ques0on how chemical and turbulent scales interact 10.-­‐4 Sta2s2cal Descrip2on of Turbulent Flows •  The aim of stochas0c methods in turbulence is the descrip0on of the fluctua0ng velocity and scalar fields in terms of their sta0s0cal distribu0ons •  For example, distribu0on func0on of a single variable of the velocity component u •  The distribu0on func0on Fu(U) of u is defined by the probability p of finding a value of u < U: •  U is the so-­‐called sample space variable associated with the random stochas0c variable u •  Sample space of the random stochas0c variable u consists of all possible realiza0ons of u 10.-­‐10 •  The probability of finding a value of u in a certain interval U-­‐ < u < U+ is given by •  Probability density func0on pdf of u is now defined as •  It follows that Pu(U)dU is the probability of finding u in the range U ≤ u ≤ U+dU •  If the possible realiza0ons of u range from -­‐∞ ≤ u ≤ +∞, it follows that which states that the probability of finding the value u in the range -­‐∞ ≤ u ≤ +∞ is certain, i.e. it has probability one •  This serves as a normalizing condi0on for Pu 10.-­‐11 •  In turbulent flows, the pdf of any stochas0c variable depends, in principle, on the posi0on x and on 0me t. •  These func0onal dependencies are expressed by the following nota0on •  Semicolon indicates that Pu is probability density in U-­‐space and a func0on of x and t •  In the following we will, for simplicity of nota0on, not dis0nguish between random stochas0c variable u and sample space variable U, drop the index and write the pdf as 10.-­‐12 •  Once the pdf of a variable is known one may define its moments by •  The overbar denotes the average or mean value, some0mes also called expecta0on, of un •  The first moment (n = 1) is called the mean of u •  Similarly, the mean value of a func0on g(u) can be calculated from 10.-­‐14 •  Reynolds decomposi0on u(x, t) = u(x, t) + u0 (x, t)
divides random variable in determinis0c mean (overline) and random fluctua0on (prime) •  For flows with large density changes as they occur in combus0on, it is convenient to introduce a density-­‐weighted average ũ, called the Favre average, by spli\ng u(x,t) •  This averaging procedure is defined by requiring that the average of the product of u'' with the density ρ (rather than u'' itself) vanishes •  The defini0on for ũ may then be derived by mul0plying by the density ρ and averaging 10.-­‐15 •  Favre averaging considerably simplifies the equa0ons for the averaged quan00es •  For instance, the average of the product of the density ρ with the velocity components u and v would lead with conven0onal averages to four terms •  Using Favre averages one writes •  When averaging, only the first two terms remain 00 v 00
⇢uv = ⇢ũṽ + ⇢u]
10.-­‐17 •  A Favre pdf of u can be derived from the joint pdf P(ρ,u) as •  Mul0plying both sides with u and integra0ng yields which is equivalent to •  The Favre mean value of u therefore is defined as 10.-­‐19 Navier-­‐Stokes Equa2ons and Turbulence Models •  The classical approach to model turbulent flows •  It is based on single point averages of the Navier-­‐Stokes equa0ons. •  These are commonly called Reynolds Averaged Navier-­‐Stokes Equa0ons (RANS) •  We will formally extend this formula0on to non-­‐constant density by introducing Favre averages 10.-­‐20 •  Even though it certainly is the best compromise for engineering design using RANS, the predic0ve power of the k-­‐ε model is, except for simple shear flows, oden found to be disappoin0ng. •  We will present it here, mainly to be able to define turbulent length and 0me scales •  For non-­‐constant density flows, the Navier-­‐Stokes equa0ons are wrieen in conserva0ve form (cf. Lecture 3) •  Con0nuity •  Momentum 10.-21
•  Using Favre averaging, one obtains •  This equa0on is similar to except for the third term on the l.h.s. containing the correla0on which is called the Reynolds stress tensor 10.-­‐23 •  An important simplifica0on is obtained by introducing the so called eddy viscosity νt in a model for the Reynolds stress tensor where I is the unit tensor •  This model for the Reynolds stress tensor then has the same form as the molecular stress •  The kinema0c eddy viscosity is related to the Favre average turbulent kine0c energy and its dissipa0on by 10.-­‐24 •  This requires that modeled equa0ons are available for . •  These equa0ons are given here in their most simple form •  Turbulent kine0c energy •  Turbulent kine0c energy dissipa0on rate •  In these equa0ons, the two terms on the l.h.s. represent the local rate of change and convec0on, respec0vely 10.-­‐25 •  The first term on the r.h.s. represents the turbulent transport, the second one turbulent produc0on and the third one turbulent dissipa0on. •  As in the standard k-­‐ε model, the constants σk=1.0, σε = 1.3, cε1= 1.44 and cε2= 1.92 are generally used •  A more detailed discussion concerning addi0onal terms in the Favre averaged turbulent kine0c equa0on may be found in Libby and Williams (1994) 10.-­‐26 Two-­‐Point Velocity Correla2ons and Turbulent Scales •  A characteris0c feature of turbulent flows is the occurrence of eddies of different length scales •  If a turbulent jet enters with a high velocity into ini0ally quiescent surroundings, the large velocity difference between the jet and the surroundings generate a shear layer instability which ader a transi0on, becomes turbulent further downstream from the nozzle exit. 10.-­‐27 •  The two shear layers merge into a fully developed turbulent jet •  In order to characterize the distribu0on of eddy length scales at any posi0on within the jet, one measures at point x and 0me t the axial velocity u(x,t), and simultaneously at a second point (x+r,t) with distance r apart from the first one, the velocity u(x+r,t) •  Then the correla0on between these two veloci0es is defined by the average 10.-­‐28 •  For homogeneous isotropic turbulence, the loca0on x is arbitrary and r may be replaced by its absolute value r=|r|. •  For this case the normalized correla0on is ploeed schema0cally here •  Kolmogorov's 1941 theory for homogeneous isotropic turbulence assumes that there is a steady transfer of kine0c energy from the large scales to the small scales and that this energy is being consumed at the small scales by viscous dissipa0on This is the eddy cascade hypothesis 10.-­‐29 •  Energy transfer rate is independent of the size of the eddies within the so called iner0al range and equal to the dissipa0on rate •  Then, the dissipa0on ε is the only dimensional quan0ty apart from the correla0on co-­‐ordinate r that is available for the scaling of f(r,t) •  Since ε has the dimension m2/s3, this leads to f(r,t) = where C is a universal constant called the Kolmogorov constant 10.-­‐30 Turbulent Time, Length, and Velocity Scales •  Large scale eddies contain most of the kine0c energy •  These eddies s0ll have a rela0vely large correla0on f(r,t) before it decays to zero •  The length scale of these eddies is called the integral length scale defined by 10.-­‐31 •  We denote the root-­‐mean-­‐square (r.m.s.) velocity fluctua0on by which represents the turnover velocity of integral scale eddies •  The turnover 0me of these eddies is then propor0onal to the integral 0me scale 10.-­‐32 •  The mo0on of very small eddies is influenced by viscosity which provides an addi0onal dimensional quan0ty for scaling •  Dimensional analysis then yields the Kolmogorov length scale 10.-­‐33 •  In addi0on to η, a Kolmogorov 0me and a velocity scale may be defined as •  According to Kolmogorov's 1941 theory, the energy transfer from the large eddies of the integral scale is equal to the dissipa0on of energy at the Kolmogorov scale •  Therefore, we will relate ε directly to the turnover velocity and the length scale of the integral scale eddies 10.-­‐34 Kolmogorov Scaling •  We now define a discrete sequence of eddies within the iner0al sub-­‐range by •  Since ε is constant within the iner0al sub-­‐range, dimensional analysis relates the turnover 0me and the velocity difference across the eddy size to ε in that range as •  This rela0on includes the integral scales and also holds for the Kolmogorov scales as 10.-­‐35 Energy Spectrum and k–5/3 Law •  A Fourier transform of the isotropic two-­‐point correla0on func0on leads to the kine0c energy spectrum E(k), which is the density of kine0c energy per unit wave number k. •  Here, rather than to present a formal deriva0on, we relate the wave number k to the inverse of the eddy size as •  The kine0c energy vn2 is then and its density in wave number space is propor0onal to •  This is the well-­‐known k-­‐5/3 law for the kine0c energy spectrum in the iner0al sub-­‐
range 10.-­‐36 Energy Spectrum •  For small wave numbers (corresponding to large scale eddies), the energy per unit wave number increases with a power law between k2 and k4 •  This range is not universal and is determined by large scale instabili0es which depend on the boundary condi0ons of the flow 10.-­‐37 •  The spectrum aeains a maximum at a wave number that corresponds to the integral scale, since eddies of that scale contain most of the kine0c energy •  k-­‐5/3 law in the iner0al sub-­‐range •  Cut-­‐off due to viscous effects at the Kolmogorov scale η. •  Beyond this cut-­‐off, in the range called the viscous subrange, the energy per unit wave number decreases exponen0ally due to viscous effects 10.-­‐38 Balance Equa2ons for Reac2ve Scalars •  For simplicity, we will assume that the specific heat capaci0es cp,i are all equal and constant, the pressure is constant and the heat transfer due to radia0on is neglected •  Then, the temperature equa0on for unity Lewis number becomes (cf. Lecture 3) •  The heat release due to chemical reac0ons is wrieen as •  This form of the temperature equa0on is similar to that for the mass frac0ons of species i (cf. Lecture 3), which becomes with the binary diffusion approxima0on 10.-­‐40 •  In the following we will use the term "reac0ve scalars" for the mass frac0on of all chemical species and temperature and introduce the vector •  The balance equa0on for the reac0ve scalar ψi will be wrieen where i=1,2,… ,k+1. The diffusivi0es Di (i=1,2,… ,k) are the mass diffusivi0es for the species and Dk+1=D denotes the thermal diffusivity •  Similarly, σi (i=1,2,… ,k) are the species source terms and σk+1 is ωT . •  The chemical source term will also be wrieen as σi=ρSi 10.-­‐42 Moment Methods for Reac2ve Scalars •  Favre averaged equa0ons for the mean of the reac0ve scalars can be derived as •  In high Reynolds number flows the molecular transport term containing the molecular diffusivi0es Di are small and can be neglected •  Two last terms on RHS need modeling 10.-­‐44 •  Test of simple model for non-­‐premixed combus0on in isotropic turbulence •  Mean transport for reac0ve scalars in decaying isotropic turbulence Product Mass Frac?on @ ei
= Sei
@t
DNS data Flamelet closure assump0on Evalua0on of chemical source term with mean quan00es •  Neglec0ng appropriate closure leads to leading order error! Dissipa2on and Scalar Transport of Non-­‐Reac2ng Scalars •  As an example for a nonreac0ve scalar we will use the mixture frac0on Z •  It is general prac0ce in turbulent combus0on to employ the gradient transport assump0on for non-­‐reac0ng scalars •  The scalar flux then takes the form •  Here, Dt is a turbulent diffusivity which is modeled by analogy to the eddy viscosity as where Sct is a turbulent Schmidt number 10.-­‐47 •  The equa0on for the mean mixture frac0on then reads where the molecular term has been neglected •  In order to derive an equa0on for the mixture frac0on variance, we first must derive an equa0on for the fluctua0on Z '' by subtrac0ng from the instantaneous equa0on which leads to 10.-­‐48 •  If deriva0ves of ρ and D and their mean values are neglected for simplicity, the first two terms on the r.h.s. of can be combined to obtain a term propor0onal to Di ΔZ ''. •  Introducing this and mul0plying the equa0on by 2ρZ '' one obtains an equa0on for Z ''2. •  Using the con0nuity equa0on and averaging one obtains 10.-­‐49 •  As before, the terms on the LHS describe the local change and convec0on •  The first term on the RHS is the turbulent transport term •  The second term on the RHS accounts for the produc0on of scalars fluctua0ons •  The mean molecular transport term has been neglected, but the molecular diffusivity s0ll appears in the dissipa0on term •  The Favre scalar dissipa0on rate is defined as 10.-­‐50 •  An integral scalar 0me scale can be defined by •  It is oden set propor0onal to the flow 0me scale •  The constant of propor0onality cχ is of order unity but varies •  cχ=2.0 is oden used •  Combining and leads to the model 10.-­‐51 The Eddy Break Up and the Eddy Dissipa2on Model •  An early aeempt to provide a closure for the chemical source term is due to Spalding (1971) •  Turbulent mixing may be viewed as a cascade process from the integral down to the molecular scales •  Cascade process also controls the chemical reac0ons as long as mixing rather than reac0on is the rate determining process •  This model was called the Eddy-­‐Break-­‐Up model (EBU) 10.-­‐52 •  The turbulent mean reac0on rate of products was expressed as is the variance of the product mass frac0on and CEBU is the Eddy-­‐Break-­‐Up constant •  Problems for lean and rich regions in non-­‐premixed combus0on •  This model has been modified by Magnussen and Hjertager (1977) who replaced by the mean mass frac0on of the deficient species (fuel for lean or oxygen for rich mixtures), which is then called the Eddy Dissipa0on Model (EDM) 10.-­‐53 •  The Eddy Dissipa0on Model takes the minimum of three rates, those defined with the mean fuel mass frac0on, with the mean oxidizer mass frac0on and with the product mass frac0on in order to calculate the mean chemical source term •  A and B are modeling constants and ν is the stoichiometric oxygen to fuel mass ra0o defined by 10.-­‐54 •  The Eddy Break-­‐Up model and its modifica0ons are based on intui0ve arguments •  The main idea is to replace the chemical 0me scale of an assumed one-­‐step reac0on by the turbulent 0me scale τ = k/ε. •  Thereby, the model eliminates the influence of chemical kine0cs, represen0ng the fast chemistry limit only •  When these models are used in CFD calcula0ons, it turns out that the constants CEBU or A and B must be "tuned" within a wide range in order to obtain reasonable results for a par0cular problem 10.-­‐55 The Pdf Transport Equa2on Model •  Similar to moment methods, models based on a pdf transport equa0on for the velocity and the reac0ve scalars are usually formulated for one-­‐point sta0s0cs •  Within that framework, however, they represent a general sta0s0cal descrip0on of turbulent reac0ng flows, in principle, independent of the combus0on regime •  A joint pdf transport equa0on for the velocity and the reac0ve scalars can be derived, Pope (1990) 10.-­‐56 •  For simplicity, we will consider here the transport equa0on for the joint pdf of velocity and reac0ve scalars only •  Deno0ng the set of reac0ve scalars, such as the temperature and the mass frac0on of reac0ng species by the vector ψ, is the probability of finding at point x and 0me t the velocity components and the reac0ve scalars within the intervals 10.-­‐57 •  There are several ways to derive a transport equa0on for the probability density P(v, ψ ; x, t) (cf. O'Brien, 1980) •  We refer here to the presenta0on in Pope (1985, 2000), but write the convec0ve terms in conserva0ve form 10.-­‐58 •  The symbol denotes the divergence operator with respect to the three components of velocity •  The angular brackets denote condi0onal averages, condi0oned with respect to fixed values of v and ψ. •  For simplicity of presenta0on we do not use different symbols for the random variables describing the stochas0c fields and the corresponding sample space variables which are the independent variables in the pdf equa0on 10.-­‐59 •  The first two terms on the l.h.s. of are the local change and convec0on of the probability density func0on in physical space •  The third term represents transport in velocity space by gravity and the mean pressure gradient •  The last term on the l.h.s. contains the chemical source terms •  All these terms are in closed form, since they are local in physical space 10.-­‐60 •  Note that the mean pressure gradient does not present a closure problem, since the pressure is calculated independently of the pdf equa0on using the mean velocity field •  For chemical reac0ng flows, it is of par0cular interest that the chemical source terms can be treated exactly •  It has oden been argued that in this respect the transported pdf formula0on has a considerable advantage compared to other formula0ons 10.-­‐61 •  However, on the r.h.s. of the transport equa0on there are two terms that contain gradients of quan00es condi0oned on the values of velocity and composi0on •  Therefore, if gradients are not included as sample space variables in the pdf equa0on, these terms occur in unclosed form and have to be modeled 10.-­‐62 •  The first unclosed term on the r.h.s. describes transport of the probability density func0on in velocity space induced by the viscous stresses and the fluctua0ng pressure gradient •  The second term represents transport in reac0ve scalar space by molecular fluxes This term represents molecular mixing 10.-­‐63 •  When chemistry is fast, mixing and reac0on take place in thin layers where molecular transport and the chemical source term balance each other •  Therefore, the closed chemical source term and the unclosed molecular mixing term, being leading order terms in a asympto0c descrip0on of the flame structure, are closely linked to each other •  Pope and Anand (1984) have illustrated this for the case of premixed turbulent combus0on by comparing a standard pdf closure for the molecular mixing term with a formula0on, where the molecular diffusion term was combined with the chemical source term to define a modified reac0on rate •  They call the former distributed combus0on and the laeer flamelet combus0on and find considerable differences in the Damköhler number dependence of the turbulent burning velocity normalized with the turbulent intensity 10.-­‐64 •  From a numerical point of view, the most apparent property of the pdf transport equa0on is its high dimensionality •  Finite-­‐volume and finite-­‐difference techniques are not very aerac0ve for this type of problem, as memory requirements increase roughly exponen0ally with dimensionality •  Therefore, virtually all numerical implementa0ons of pdf methods for turbulent reac0ve flows employ Monte-­‐Carlo simula0on techniques (cf. Pope, 1981, 1985) •  The advantage of Monte-­‐Carlo methods is that their memory requirements depend only linearly on the dimensionality of the problem 10.-­‐65 •  Monte-­‐Carlo methods employ a large number, N, of so called no0onal par0cles •  In the Lagrangian algorithm (Pope, 1985), the par0cles are not bound to grid nodes, but each par0cle has its own posi0on and moves through the computa0onal domain with its own velocity •  Par0cles should be considered as different realiza0ons of the turbulent reac0ve flow problem under inves0ga0on •  The state of the par0cle is described by its posi0on and velocity, and by the values of the reac0ve scalar that it represents as a func0on of 0me •  These par0cles should not be confused with real fluid elements, which behave similarly in a number of respects 10.-­‐66 The Laminar Flamelet Concept •  The view of a turbulent diffusion flame as an ensemble of stretched laminar flamelets is due to Williams (1975) •  Flamelet equa0ons based on the mixture frac0on as independent variable, using the scalar dissipa0on rate for the mixing process, were independently derived by Peters (1980) and Kuznetsov (1982) •  A first review of diffusion flamelet models was given by Peters (1984) •  For premixed and diffusion flames, the flamelet concept was reviewed by Peters (1986) and Bray and Peters (1994) 10.-­‐67 •  Flamelets are thin reac0ve-­‐diffusive layers embedded within an otherwise non-­‐reac0ng turbulent flow field •  Once igni0on has taken place, chemistry accelerates as the temperature increases due to heat release •  When the temperature reaches values that are of the order of magnitude of those of the close-­‐to-­‐equilibrium branch, the reac0ons that determine fuel consump0on become very fast •  For methane combus0on, for example, the 0me scale of the rate determining reac0on in the fuel consump0on layer was es0mated in Lecture 1 10.-­‐68 •  Since the chemical 0me scale of this reac0on is short, chemistry is ac0ve only within a thin layer, namely the fuel consump0on or inner layer •  If this layer is thin compared to the size of a Kolmogorov eddy, it is embedded within the quasi-­‐laminar flow field of such an eddy and the assump0on of a laminar flamelet structure is jus0fied •  If, on the contrary, turbulence is so intense, that small eddies of a certain size become smaller than the inner layer and can penetrate into it, they are able to destroy its structure •  Under these condi0ons the en0re flame is likely to ex0nguish 10.-­‐69 •  The loca0on of the inner layer defines the flame surface •  Differently from moment methods or methods based on a pdf transport equa0on, sta0s0cal considera0ons in the flamelet concept focus on the loca0on of the flame surface and not on the reac0ve scalars themselves •  That loca0on is defined as an iso-­‐surface of a non-­‐reac0ng scalar quan0ty, for which a suitable field equa0on is derived •  For non-­‐premixed combus0on, the mixture frac0on Z is that scalar quan0ty, for premixed combus0on the level set scalar G was introduced •  Once equa0ons describing the sta0s0cal distribu0ons of Z and G are solved, the profiles of the reac0ve scalars normal to the surface are calculated using flamelet equa0ons 10.-­‐70 •  These profiles are assumed to be aeached to the flame surface and are convected with it in the turbulent flow field •  Therefore the sta0s0cal moments of the reac0ve scalars can be obtained from the sta0s0cal distribu0on of the scalar quan00es Z and G •  Details of this procedure will be discussed later 10.-­‐71 The BML-­‐Model and the Coherent Flamelet Model •  For premixed combus0on, flamelet models are typically based on the progress variable c •  The progress variable c is defined as a normalized temperature or normalized product mass frac0on which implies a one-­‐step reac0on and a corresponding heat release raising the temperature from Tu to Tb •  In flamelet models based on the progress variable, the flame structure is assumed to be infinitely thin and no intermediate values of temperature are resolved •  This corresponds to the fast chemistry limit 10.-­‐72 •  The progress variable therefore is a step func0on that separates unburnt mixture and burnt gas in a given flow field •  The classical model for premixed turbulent combus0on, the Bray-­‐Moss-­‐Libby (BML) model (Bray & Moss, 1977) by assuming the pdf of the progress variable c to be a double delta func0on distribu0on •  This assump0on only allows for states at c = 0 and c = 1 in a turbulent premixed flame, but is able to illustrate important features, such as counter-­‐
gradient diffusion of the progress variable •  This appears in the equa0on for the Favre mean progress variable where the molecular diffusion term has been neglected 10.-­‐73 •  This equa0on requires the modeling of the turbulent transport term and the mean reac0on term •  Libby and Bray (1981) and Bray et al. (1981) have shown that a gradient transport assump0on like is not applicable to . •  This is due to gas expansion effects at the flame surface and is called counter-­‐
gradient diffusion 10.-­‐74 •  A model for the chemical source term by Bray and Libby (1994) leads to the expression where sL is the laminar burning velocity, I0 is a stretch factor and Σ is the flame surface density (flame surface per unit volume) •  A model for Σ has been proposed by Candel et al. (1990), which is called Coherent Flame Model (CFM) 10.-­‐76 •  Modeling based on DNS data has led Trouve and Poinsot (1994) to the following equa0on for the flame surface density Σ: •  The terms on the l.h.s. represent the local change and convec0on, the first term on the r.h.s. represents turbulent diffusion, the second term produc0on by flame stretch and the last term flame surface annihila0on •  The stretch term is propor0onal to the inverse of the integral 0me scale τ = k/ε, which is to be evaluated in the unburnt gas •  Mul0plying this equa0on with sL leads to an equa0on for sL Σ, which is independent of sL •  Then also in this model is independent of sL 10.-­‐77 Combus2on Models used in Large Eddy Simula2on •  Turbulence models based on Reynolds Averaged Navier-­‐Stokes Equa0ons (RANS) employ turbulent transport approxima0ons with an effec0ve turbulent viscosity that is by orders of magnitude larger than the molecular viscosity •  In par0cular, if steady state versions of these equa0ons are used, this tends to suppress large scale instabili0es which occur in flows with combus0on even more frequently than in non-­‐reac0ng flows •  If those instabili0es are to be resolved in numerical simula0ons, it is necessary to recur to more advanced, but computa0onally more expensive methods such as Large Eddy Simula0on (LES) 10.-­‐78 •  Large Eddy Simula0on does not intend to numerically resolve all turbulent length scales, but only a frac0on of the larger energy containing scales within the iner0al subrange •  Modeling is then applied to represent the smaller unresolved scales, which contain only a small frac0on of the turbulent kine0c energy •  Therefore, the computed flows are usually less sensi0ve to modeling assump0ons •  The dis0nc0on between the resolved large scales and the modeled small scales is made by the grid resolu0on that can be afforded 10.-­‐79 Concept of LES E(k)
•  Spa0al filtering rather than ensemble average u=
∫ G(x, x ')u(x ')dx '
-5/3
resolved
modeled
u
u
1/!
k
Representa0on taken from Pope (2000) Computa0onal Grid 10.-­‐79a •  The model for the smaller scales is called the sub-­‐filter or subgrid model •  In deriving the basic LES equa0ons, the Navier-­‐Stokes equa0ons are spa0ally filtered with a filter of size Δ, which is of the size of the grid cell (or a mul0ple thereof) in order to remove the direct effect of the small scale fluctua0ons (cf. Ghosal and Moin, 1995). •  These show up indirectly through nonlinear terms in the subgrid-­‐scale stress tensor as subgrid-­‐scale Reynolds stresses, Leonard stresses, and subgrid-­‐scale cross stresses •  The laeer two contribu0ons result from the fact that, unlike with the tradi0onal Reynolds averages, a second filtering changes an already filtered field 10.-­‐80 •  In a similar way, ader filtering the equa0ons for non-­‐reac0ng scalars like the mixture frac0on, one has to model the filtered scalar flux vectors which contain subgrid scalar fluxes, Leonard fluxes, and subgrid-­‐scale cross fluxes •  The reason why LES s0ll provides substan0al advantages for modeling turbulent combus0on is that the scalar mixing process is of paramount importance in chemical conversion •  Nonreac0ve and reac0ve system studies show that LES predicts the scalar mixing process and dissipa0on rates with considerably improved accuracy compared to RANS, especially in complex flows (Pitsch, 2006) 10.-­‐81 •  For example, to study the importance of turbulent scalar dissipa0on rate fluctua0ons on the combus0on process and to highlight the differences between RANS and LES, Pitsch (2002), compared the results of two different LES simula0ons using unsteady flamelet models in which the scalar dissipa0on rate appears as a parameter •  The only difference between the simula0ons was that only the Reynolds-­‐
averaged dissipa0on rate was used in one simula0on, Pitsch (2000), whereas the other considered the resolved fluctua0ons of the filtered scalar dissipa0on rate predicted by LES •  The results show substan0ally improved predic0ons, especially for minor species, when fluctua0ons are considered 10.-­‐82 •  Results from large-­‐eddy simula0on of Sandia flame D (Pitsch, 2002,2000), using the Eulerian flamelet model (solid lines) and the Lagrangian flamelet model (dashed lines) compared with experimental data of Barlow (1998). Temperature distribu0on (led), scalar dissipa0on rate distribu0on (center), and comparison of mixture frac0on–condi0oned averages of temperature and mass frac0ons of NO, CO, and H2 at x/D = 30 10.-­‐102 •  Another such example is the simula0on of a bluff-­‐body stabilized flame, Raman (2005), where a simple steady-­‐state diffusion flamelet model in the context of an LES with a recursive filter refinement method led to excellent results •  Such accuracy has not been achieved with RANS simula0ons of the same configura0on, e.g. by Kim (2002) or Muradoglu (2003) •  Similar arguments can be made for premixed turbulent combus0on LES 10.-­‐83 •  Results from large-­‐eddy simula0on of the Sydney bluff-­‐body flame (Raman, 2005) •  Flame representa0on from simula0on results (led) and 0me-­‐averaged radial profiles of temperature and CO mass frac0on •  The led figure shows computed chemiluminescence emissions of CH collected in an observa0on plane with a ray tracing technique 10.-­‐96 •  Because the probability density func0on (PDF) plays a central role in most models for nonpremixed combus0on, it is necessary to emphasize the special meaning of the FPDF in LES •  Here, the example of the marginal FPDF of the mixture frac0on is discussed, but similar arguments can be made for the joint composi0on FPDF •  In Reynolds-­‐averaged methods, a one-­‐point PDF can be determined by repea0ng an experiment many 0mes and recording the mixture frac0on at a given 0me and posi0on in space •  For a sufficiently large number of samples, the PDF of the ensemble can be determined with good accuracy 10.-­‐87 •  In LES, assuming a simple box filter, the data of interest is a one-­‐0me, one-­‐
point probability distribu0on in a volume corresponding to the filter size surrounding the point of interest •  If an experimentally observed spa0al mixture frac0on distribu0on is considered at a given 0me, the FPDF cannot simply be evaluated from these data, because the observed distribu0on is characteris0c of this par0cular realiza0on and it is not a sta0s0cal property •  As a sta0s0cal property, the FPDF must be defined by an ensemble that can poten0ally have an arbitrary large number of samples. In the context of transported PDF model formula0ons for LES, which are discussed below, Pope (1990) introduced the no0on of the filtered density func0on (FDF), which describes the local subfilter state of the considered experiment 10.-­‐88 •  The FDF is not an FPDF, because it describes a single realiza0on •  The FPDF is defined only as the average of the FDF of many realiza0ons given the same resolved field, Fox (2003) •  It is important to dis0nguish between the FDF and the FPDF, especially in using direct numerical simula0on (DNS) data to evaluate models, and in the transported FDF models discussed below •  Only the FDF can be evaluated from typical DNS data, whereas the FPDF is required for subfilter modeling 10.-­‐89 Modeling the Scalar Dissipa2on Rate •  Although different conceptual ideas and assump0ons are used in the combus0on models discussed here, most of them need a model for the scalar dissipa0on rate •  The dissipa0on rate of the mixture frac0on is a fundamental parameter in non-­‐premixed combus0on, which determines the filtered reac0on rates, if combus0on is mixing controlled •  High rates of dissipa0on can also lead to local or global flame ex0nc0on •  Models based on presumed FPDFs also require a model for the sub-­‐filter scalar variance 10.-­‐93 •  In RANS models, typically a transport equa0on is solved for the scalar variance, in which the Reynolds-­‐averaged scalar dissipa0on rate χ appears as an unclosed sink term that requires modeling •  The addi0onal assump0on of a constant ra0o of the integral 0mescale of the velocity τt and the scalar fields leads to the expression where cφ is the so-­‐called 0mescale ra0o 10.-­‐97 •  In the models most commonly used in LES, Girimaji (1996), Pierce (1998), the scalar variance transport equa0on and the 0mescale ra0o assump0on are actually used in the opposite sense •  Instead of solving the sub-­‐filter variance equa0on, the assump0on that the scalar variance produc0on appearing in that equa0on equals the dissipa0on term leads to an algebraic model for the dissipa0on rate of the form where an eddy diffusivity model was used for the sub-­‐filter scalar flux in the produc0on term •  The eddy diffusivity is where cZ can be determined using a dynamic procedure and is the characteris0c Favre-­‐filtered rate of strain 10.-­‐98 •  Wri0ng for the sub-­‐filter scales and combining this with then leads to the model for the scalar variance •  A new coefficient cV is introduced, which can be determined dynamically following Pierce (1998) •  From , and the dynamically determined coefficients of the eddy diffusivity and the scalar variance, the 0mescale ra0o cφ can be determined as 10.-­‐99 LES of real Combus2on Devices •  Several inves0gators have reported simula0ons of real combus0on devices with LES •  Many of these use either structured or block-­‐structured curvi-­‐linear meshes, which cannot deal with very complex geometries •  Simula0ons of gas turbines, for instance, typically require unstructured meshing strategies, for which the formula0on of energy conserving and accurate numerical algorithms, of par0cular importance for combus0on 10.-­‐104 •  LES, proves to be even more difficult •  Among the few fully unstructured mul0physics LES codes are the AVBP code of CERFACS, which has been applied in many studies on combus0on instabili0es and flashback in premixed gas turbines, Selle (2004), Sommerer (2004), and the Stanford CDP code •  CDP solves both low-­‐Ma number variable-­‐density and fully compressible LES equa0ons using the unstructured collocated finite volume discre0za0on of Mahesh (2004) and its subsequent improvements by Ham (2004). •  It applies Lagrangian par0cle tracking with adequate models for breakup, par0cle drag, and evapora0on for liquid fuel sprays 10.-­‐105 •  Closure for subfilter transport terms and other turbulence sta0s0cs is accomplished using dynamic models •  The Flamelet/Progress Variable combus0on model is applied to model turbulence/chemistry interac0ons •  The code is parallelized with advanced load balancing procedures for both gas and par0cle phases •  Computa0ons have been conducted with over two billion cells using several thousand processors •  A state-­‐of-­‐the-­‐art simula0on of a sec0on of a modern Prae & Whitney gas turbine combustor that uses all these capabili0es has been performed Mahesh (2005), Apte (2005), Mueller (2012) 10.-­‐106 LES of Prae & Whitney Aircrad Engine •  The figure shows the spray and temperature distribu0on and demonstrates the complexity of the geometry and the associated flow physics. Engine out Soot Emissions (Mueller & Pisch, 2012) 10.-­‐107