Evaluation of Quadrature-based Moment Methods in turbulent premixed combustion
Martin Pollack, Federica Ferraro, Johannes Janicka, Christian Hasse
EEVALUATION OF QUADRATURE-BASED MOMENT METHODS INTURBULENT PREMIXED COMBUSTION
MARTIN POLLACK ∗ , , FEDERICA FERRARO , JOHANNES JANICKA ,CHRISTIANHASSE Institute for Simulation of Reactive Thermo-Fluid Systems (STFS), Technische Universit¨atDarmstadt, Otto-Berndt-Straße 2, Darmstadt 64287, Germany Institute of Energy and Power Plant Technology (EKT), Technische Universit¨at Darmstadt,Otto-Berndt-Straße 3, Darmstadt 64287, Germany
Abstract.
Transported probability density function (PDF) methods are widely used to modelturbulent flames characterized by strong turbulence-chemistry interactions. Numerical methodsdirectly resolving the PDF are commonly used, such as the Lagrangian particle or the stochasticfields (SF) approach. However, especially for premixed combustion configurations, characterizedby high reaction rates and thin reaction zones, a fine PDF resolution is required, both in physicaland in composition space, leading to high numerical costs. An alternative approach to solve aPDF is the method of moments, which has shown to be numerically efficient in a wide rangeof applications. In this work, two Quadrature-based Moment closures are evaluated in thecontext of turbulent premixed combustion. The Quadrature-based Moment Methods (QMOM)and the recently developed Extended QMOM (EQMOM) are used in combination with a tabulatedchemistry approach to approximate the composition PDF. Both closures are first applied to anestablished benchmark case for PDF methods, a plug-flow reactor with imperfect mixing, andcompared to reference results obtained from Lagrangian particle and SF approaches.Second, a set of turbulent premixed methane-air flames are simulated, varying the Karlovitznumber and the turbulent length scale. The turbulent flame speeds obtained are compared with SFreference solutions. Further, spatial resolution requirements for simulating these premixed flamesusing QMOM are investigated and compared with the requirements of SF.The results demonstrate that both QMOM and EQMOM approaches are well suited to reproducethe turbulent flame properties. Additionally, it is shown that moment methods require lowerspatial resolution compared to SF method.
Keywords:
Transported PDF, Quadrature-based moment methods, Stochastic fieldss, Turbu-lent premixed flames 1.
Introduction
Different approaches have been developed for modeling turbulence-chemistry interaction. Amongthem, one very popular approach is the transported probability density function (PDF) method,first proposed in [1]. Here, the interaction between turbulence and chemistry is described by theevolution of a one-point joint PDF of selected flow variables [2]. In composition PDF methods,the PDF maps the likelihood of a specific state φ on the composition space ψ , usually containingthe species mass fractions and enthalpy, φ = ( Y , Y , · · · , Y N s , h ) T . The main advantage of PDFmethods is that the non-linear chemical source term appears in closed form. The PDF P ( x , t ; ψ )is characterized by high dimensionality and a finite-volume discretization is impractical due toexcessive numerical costs.Different Monte-Carlo (MC) approaches have been established for solving the PDF transportequation. The first was the Lagrangian approach [3], where the PDF is modeled by the evolutionof a large set of stochastic particles. An alternative PDF representation based on Eulerian MCstochastic fields (SF) was derived in [4]. In the SF method, a set of fields evolves to approximatethe transported PDF. Its Eulerian nature provides fields which are continuous and whose mean E-mail address : [email protected] . a r X i v : . [ phy s i c s . f l u - dyn ] F e b roperties are smooth and differentiable in space [4]. This method, relatively new compared to theparticle PDF approach, has been applied to premixed [5], partially premixed [6] and non-premixedsystems [7].The high computational effort, especially because of the chemical reactions, has limited theapplication of transported PDF methods to reduced mechanisms or less complex fuels. In [5] theEulerian SF method was combined with the tabulated chemistry approach, yielding a substantialreduction in the PDF dimensionality and the computational costs.An alternative and very efficient mathematical approach to solve PDF-based systems is themethod of moments (MOM). Similarly to SF method, MOM can be solved using conventionalEulerian discretization techniques. Here, however, only a set of integral PDF properties, i.e. itsmoments m k ( t, x ), are solved. Moment methods have been applied successfully to a wide rangeof problems, such as nano-particles and aerosols [8], sprays [9], and combustion-related problemssuch as soot formation [10, 11].Although the general idea of applying Quadrature-based Moment Methods (QbMMs) for trans-ported PDF systems was already proposed in [12], QbMMs still play a minor role. Only fewpublications, described in the following, are available in the literature. They already indicate thegreat potential in combustion PDF modeling. In the context of QbMM formulations, different clo-sures were derived, starting with the general univariate QMOM [8] and later the Direct QuadratureMethod of Moments (DQMOM) [13], which is considered the first multivariate approach. DQMOMwas applied to solve the moment transport equations for the composition PDF in [14–18]. However,the direct transport of nodes and weights does not satisfy the conservation rules and the approachcan also become numerically unstable [19]. Therefore, Donde et al. [19] extended this approachto Semi-discrete QMOM (SeQMOM), combining DQMOM with a direct transport of moments.After recent developments, such as Conditional QMOM (CQMOM) [20] for multivariate PDFs, orExtended QMOM (EQMOM), enabling continuous PDF reconstructions, PDF-QbMMs were nolonger restricted to closure by DQMOM. In [21] a local, pure-mixing problem was solved using twodifferent micro-mixing models: interaction by exchange with the mean (IEM) and Fokker-Planck.EQMOM was shown to provide an accurate PDF reconstruction. A first proof of concept that areacting system could be closed using CQMOM was provided in [22], while the underlying generalidea for the closure of different systems was additionally discussed in [23].Similarly to MC transported PDF methods, which have been primarily applied to turbulent non-premixed reacting flows, most QbMM studies have investigated non-premixed flames [14, 15, 17–19, 22].Following these recent developments, this work aims to evaluate the QbMM methods in turbulentpremixed flames using a tabulated chemistry approach. Two different closures, standard QMOMand EQMOM, will be employed. The QbMM framework will be applied to simulate (i) a plug-flowreactor with imperfect mixing proposed by Pope in [3] with both linear and non-linear chemistrysource term, and (ii) a set of freely-propagating turbulent flames for different Karlovitz numbersand turbulent length scales. The simulation results will be compared with Lagrangian [3] and SFmethod [4, 24] results taken from the literature.2. Moments transport equations
For variable density flows, the transport equation for the density-weighted joint compositionPDF (cid:101) P ( x , t ; ψ ) can be derived under the assumption of equal diffusivity of the species and unityLewis number [1] ∂ (cid:16) ¯ ρ (cid:101) P ( ψ ) (cid:17) ∂t (cid:124) (cid:123)(cid:122) (cid:125) I − accumulation + ∂∂x i (cid:16) ¯ ρ ˜ u i (cid:101) P ( ψ ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) II − macromixing + ∂∂ψ d (cid:16) ¯ ρ ˙ ω d ( ψ ) (cid:101) P ( ψ ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) III − chemical reactions − ∂∂x i (cid:104) ¯ ρ (cid:104) u (cid:48)(cid:48) i | φ = ψ (cid:105) (cid:101) P ( ψ ) (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) IV − meso − mixing + ∂∂x i (cid:32) D (cid:101) P ( ψ ) ∂x i (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) V − micro − mixing − ∂ ∂ψ d ∂ψ d (cid:48) (cid:20)(cid:28) D ∂φ d ∂x i ∂φ d (cid:48) ∂x i (cid:12)(cid:12)(cid:12) φ = ψ (cid:29) (cid:101) P ( ψ ) (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) V − micro − mixing , (1)where summation over repeated indexes is assumed. Here ρ is the density, u i is the velocity vectorcomponent, ˙ ω d is the chemical source term of the reactive scalar d , D is the laminar diffusivityand ψ is the sample space of the composition vector φ ; ˜ · denotes density-weighted Favre-averagedquantities and ¯ · time-averaged quantities. All the terms on the left-hand side are in closed form,while the two terms on the right-hand side need to be modeled. The meso-mixing term (IV), i.e.the composition-conditioned turbulent transport in physical space, is commonly modeled by thegradient-diffusion hypothesis (cid:104) u (cid:48)(cid:48) i | φ = ψ (cid:105) (cid:101) P ( ψ ) = D T ∇ (cid:101) P ( ψ ), with D T the turbulent diffusivity.The closure of the micro-mixing term (V), representing the molecular mixing in composition space,is obtained in this work with the IEM model [25], yielding (cid:68) D ∂φ d ∂x i ∂φ d (cid:48) ∂x i (cid:12)(cid:12)(cid:12) φ = ψ (cid:69) = − / τ − MM ( ψ − ˜ φ ),with the decay time scale τ MM of the scalar fluctuations. The IEM is widely used in combustionsimulations [12], also recently as SF closure in e.g. [5, 7]. However, IEM is known to have someshortcomings [12], i.e. it does not assure local mixing in composition space and it does not accountfor the dependency of the scalar dissipation rate on the chemical source term, which stronglycharacterizes turbulent premixed flames, as discussed in [26, 27]. Here IEM is chosen to facilitatethe comparison on benchmark calculations in the following, where the same model was applied [4,24]. However, the proposed approach is not limited to this specific micro-mixing model.Further, a combination of transported PDF and manifold approaches is promising to describethe thermochemical state of the reacting mixture. The main advantage is that the compositionmanifold can be parametrized with a reduced number of variables, instead of the complete setincluding species and enthalpy, and tabulated in pre-processing [5]. If a single progress variable Y c can be used to access the manifold, this approach leads to a univariate representation of the PDF,reducing the composition space vector to φ = ( Y c ).In [4] Vali˜no formulated the stochastic differential equations for an ensemble of N sf stochasticfields φ j ( x , t ) for j = 1 , · · · , N sf , whose evolution is equivalent to the evolution of the compositionPDF in Eq. (1) d( ¯ ρφ j ) = − ∂ (cid:0) ¯ ρ ˜ u i φ j (cid:1) ∂x i d t + ∂∂x i (cid:20) ¯ ρ ( D + D T ) ∂φ j ∂x i (cid:21) d t + (cid:0) ρ D T (cid:1) ∂φ j ∂x i d W ji − ¯ ρ τ MM (cid:16) φ j − ˜ φ (cid:17) d t + ¯ ρ ˙ ω ( φ j )d t. (2)Here the IEM micro-mixing closure is applied. The Wiener term dW i introduces the stochasticcharacter into the equation, which induces the different evolution of fields. Consequently, thisstochastic approach requires a minimum number of fields to ensure a smooth evolution of theintegral properties and limit the stochastic noise. In contrast to stochastic approaches, where onlythe ensemble describes the overall system, moment equations are deterministic. The transportequation for a moment of order k is obtained by applying the definition ˜ m k = (cid:82) (cid:60) ψ ψ k (cid:101) P ( ψ )d ψ tothe composition PDF Eq. (1): ∂ ( ¯ ρ ˜ m k ) ∂t + ∂∂x i ( ¯ ρ ˜ u i ˜ m k ) − ∂∂x i (cid:20) ¯ ρ ( D + D T ) ∂ ˜ m k ∂x i (cid:21) = k (cid:90) (cid:60) ψ ψ k − ¯ ρ ˙ ω ( ψ ) (cid:101) P d ψ − k ¯ ρ τ MM (cid:90) (cid:60) ψ ψ k − (cid:16) ψ − ˜ φ (cid:17) (cid:101) P d ψ. (3)Similarly to the SF formulation discussed above, the IEM model is employed. A micro-mixingformulation in Eq. (3) is also possible in terms of moments [21]. The chemical source term is, owever, unclosed, contrary to the Lagrangian and SF methods, since it directly depends on theunknown PDF. Two different moment closures employed will be described in the following.2.1. Closure with QMOM and EQMOM.
The first QbMM approach to close the momentequation was provided in [8], and this standard QMOM is discussed first. The recent EQMOMapproach, which provides more insight to the underlying PDF, is explained afterwards.QMOM. The basic idea of QMOM is to approximate integrals containing the PDF, such as thechemical reaction source term in Eq. (3), using a set of N α weighted quadrature nodes, φ α , asfollows: (cid:90) (cid:60) ψ q ( ψ ) (cid:101) P ( ψ )d ψ ≈ N α (cid:88) α =1 q ( φ α ) w α , (4)where w α are the weights and q ( ψ ) contains all terms except the PDF itself. Setting q = ψ k ,directly yields the moments definition:˜ m k = (cid:90) (cid:60) ψ ψ k (cid:101) P ( ψ )d ψ ≈ N α (cid:88) α =1 φ kα w α , (5)showing that the system of N mom moments (with m equal to unity per definition, m as the mean,etc.) is fully determined by N α = N mom / k (cid:80) N α α =1 φ k − α ¯ ρ ˙ ω ( φ α ) w α .Finally, the QMOM yields a discontinuous PDF representation as weighted sum of Dirac deltafunction (cid:101) P ( ψ ) ≈ N α (cid:88) α =1 w α δ ( ψ − φ α ) . (6)EQMOM. Contrary, the basic idea of EQMOM [29] is to provide a continuous PDF reconstruc-tion using a set of N α known Kernel Density Functions (KDFs), e.g. Gamma, Beta or Gaussiandistributions. The continuous reconstructions are essential for several applications such as sprayevaporation or soot burn-off, as shown in [11]. Here, Beta distributions are used, since they canrepresent a domain bounded on either side. This applies for the progress variable, which is lim-ited between the unburned state Y c,min and the burned state Y c,max . To satisfy the mapping onthe beta space ψ ∈ [0 , m [0 , k = ( Y c,max − Y c,min ) − k k (cid:88) n =0 k ! n !( k − n )! ˜ m n · ( − Y c,min ) ( k − n ) . (7)Further, it has been shown [21] that this formulation is well-suited to reconstruct a univariate,pure mixing system. The PDF reconstruction using Beta EQMOM is given as [29]: (cid:101) P ( ψ ) ≈ N α (cid:88) α =1 w α ψ λ α − (1 − ψ ) µ α − B ( λ α , µ α ) (8)with λ α = φ α /σ and µ α = (1 − φ α ) /σ , where σ is a shape parameter obtained iteratively and B isthe beta function. Introducing the additional parameter σ requires one more moment to be solved, N mom = 2 N α + 1. Since each of the N α KDFs is known, each can be represented by an arbitrarylarge number N α,β of secondary quadrature nodes φ α,β . This step allows the source term to beintegrated with a fine ψ -space resolution ( N α · N α,β secondary nodes compared to N α primarynodes in QMOM, Eq. (4)): (cid:90) (cid:60) ψ q ( ψ ) (cid:101) P ( ψ )d ψ ≈ N α (cid:88) α =1 w α N α,β (cid:88) β =1 q ( φ α,β ) w α,β , (9) hich yields the moments approximation as ˜ m k ≈ (cid:80) N α α =1 w α (cid:80) N α,β β =1 φ kα,β w α,β , similarly to Eq. (5)for QMOM.It is worth noting that EQMOM does not assume the shape of the PDF but it reconstructs ageneral PDF shape by a sum of weighted Beta-distributions. If EQMOM uses a single Beta-KDF,the closure is the same as the presumed Beta-PDF approach [23].The algorithm used here coupled with the CFD solver is [22, 23]:(1) solve mass and momentum transport equations(2) update D T and τ MM (3) solve only the advection and diffusion of the moments, left-hand side of Eq. (3)(4) invert the moments (determine the nodes and the weights) and calculate the source termson the nodes, reconstruct the updated moments(5) update the thermo-physical properties using the updated nodes(6) solve the pressure equation.While in this work flame extinction/ignition are not considered, it is worthwhile to note that forsuch cases the Strang splitting should be replaced [31]. Step 4 includes the solution of the systemfor each EQMOM node φ α,β , d φ α,β d t = ˙ ω ( φ α,β ) − τ MM (cid:16) φ α,β − ˜ φ (cid:17) , (10)from t to t + ∆ t . Analogously, replacing φ α,β with φ α Eq. (10) yields the QMOM formulation.As mentioned, the micro-mixing closure is possible both using nodes as in Eq. (10) or directlyusing moments as in [21], before updating the chemical source term. As discussed in [29] shiftingthe nodes is more consistent. Finally, replacing q ( φ α,β ) in Eq. (9) with Φ ( φ α,β ) yields the majorquantities, such as composition, temperature, density and the thermo-physical properties Φ ( φ ) =( Y , T, ρ, c p , µ, λ ) T , in step 5. Each requested state is interpolated from the tabulated manifold,using φ = Y c as a parameter. 3. Results
In the following, two test cases of increasing complexity are simulated with QMOM and EQ-MOM closure and compared with reference results obtained using the Lagrangian particle andSF methods. The QbMM results are analyzed in terms of the closures applied, the number ofmoments solved and the grid resolution.3.1.
Plug-flow reactor with imperfect mixing.
The test case proposed in [3] has become anestablished benchmark for PDF methods. It describes the evolution of a reaction progress variable φ in a non-dimensionalized sub-space t ∗ , x ∗ . Uniform non-dimensional velocity u ∗ , density ρ ∗ ,diffusivity D ∗ , chemical source term ˙ ω ∗ and micro-mixing time scale τ ∗ MM are considered. Theprogress variable is set to zero at the reactor inlet, φ = 0 at x ∗ = 0 (unburned gas), and every-where initialized with φ = 1 (burned gas). For the chemical reaction source term, two differentformulations are considered: a linear expression with ˙ ω ∗ = 3(1 − φ ) and an Arrhenius-type ex-pression with ˙ ω ∗ = 21830 φ (1 − φ ) exp( − / (1 + 3 φ )). Reference solutions were obtained using theLagrangian method in [3] with 400 particles per cell and using SF in [4] with 800 fields. Figure 1shows the steady-state profiles for the mean and standard deviation of the progress variable overreactor length. For the linear source, an analytic solution [3] is available for comparison. Resultsare shown using the standard QMOM approach with two ( Q
2) and three ( Q
3) quadrature nodesas well as with EQMOM with two KDFs and N α,β = 10 secondary nodes ( EQ − Q
2) are sufficientto provide very good results. According to the basic theory [32], the closure in Eq. (4) is exact,if q is a polynomial of degree ≤ N α −
1, as in this case for the linear chemical source terms andmicro-mixing terms. Consequently, in the calculations ( Q
3) or ( EQ −
10) no further significantimprovement could be obtained.Next, the results of the Arrhenius-type formulation are shown. In Fig. 2 the mean values us-ing Lagrangian particles from Pope [3] and SF from Vali˜no [4] are plotted as references for two Φ m e a n , s t d d e v i a t i o n x * Ref.-meanRef.-StdDevEQ2-10Q2Q3
Fig. 1.
Mean and standard deviation profiles of the progress variable for the plug-flow reactor [3] with linear chemical source term, compared to analytic solution(symbols) for all applied setups (lines). Φ m e a n , s t d d e v i a t i o n x * Q1Q2Q3mean-sf [4]mean-Lagr. [3]StdDev.-Lagr. [3]
Fig. 2.
Mean and standard deviation profiles of the progress variable for the plug-flow reactor at different times with Arrhenius chemical source term. References areLagrangian [3] and SF [4] solutions. QbMM solutions are shown for an increasingnumber of quadrature nodes ( Q Q t ∗ = 0 . t ∗ = 1 .
5, the variance curves (crosses) for t ∗ = 1.time instances, t ∗ = 0 . t ∗ = 1 . t ∗ = 1 istaken from [3]. The QbMM results were generated with the same numerical setup, i.e. the timestep (∆ t ∗ = 5 · − ) and the same ∆ x resolution (50 cells) as in [4] in order to ensure compa-rability. Simulations considering only the mean of the distribution ( Q
1) up to higher resolvedsetups ( Q
3) are shown in Fig. 2. Obviously is it not possible, to reproduce this configuration with( Q Q
2) setup already yieldsvery similar results to the SF reference, but requires only 3 moments to be solved. An increaseup to ( Q
3) yields even better results than SF. They are comparable to the Lagrangian particlemethod results, indicating a clear convergence towards the reference data by increasing the num-ber of moments. The standard deviation is slightly underestimated in the downstream area forall setups. Additionally, it has been observed that further increasing to 4 quadrature nodes (notshown here) yields no substantial improvements in the QbMM results. These results demonstratethat 3 quadrature nodes are a well-suited QMOM-configuration to guarantee sufficient accuracy.It is thus feasible to employ higher-order moments for closing this system.Additionally, EQMOM with 3 KDFs was applied for this system with 3 ( EQ −
3) and 20 ( EQ − EQ −
3) and ( EQ − Q
3) results from Fig. 2, included here forcomparison. Thus, EQMOM, which has the advantage of reconstructing a continuous PDF dis-tribution of the progress variable at a slightly higher computational cost, as it is discussed below,yields a very similar evolution to QMOM in this configuration. This indicates that the system isalready sufficiently well resolved with standard QMOM. Instead, the numerical treatment of themicro-mixing closure has a greater effect on the results. Figure 3 shows this for both closures ( Q Q
3) using either the node-shifting representation (MMN) following Eq. (10) or the direct Φ m e a n , s t d d e v i a t i o n x * Q2,MMNQ2,MMMQ3,MMNQ3,MMMEQ3-3EQ3-20
Fig. 3.
Mean and standard deviation profiles of the progress variable for the plug-flow reactor with Arrhenius chemical source term. Q2 and Q3 are the same asin Fig. 2. EQMOM results are shown for N α = 3, N α,β = 20. Different treatment ofthe micro-mixing IEM closure are compared for QMOM: closure applying directlythe moments (MMM) and closure by shifting the nodes (MMN), as in Eq. (10).Symbol legend: see Fig. 2.moment-based closure ˙ m k = 1 / τ − MM ( m k − m k − m ) (MMM), see [21]. ( Q
2) is only slightly in-fluenced by this choice, since mostly lower order moments ( m · · · m ) are used. For ( Q m · · · m ), the closure by shifting the nodes (MMN) shows a betteragreement with the Lagrangian reference than the direct moment-based closure (MMM).The numerical costs of QbMM methods can be analyzed in terms of the CPU time required forthe solution of all moment transport eqs. ( t cmomT r ) and for the moment inversion ( t cinv ). The lattercan be seen as a computational overhead compared to other Eulerian methods (SF or DQMOM).It, however, depends on the test case investigated. For the local absence of a variance (e.g. a fullyburned- or unburned mixture), which can be directly determined by the moments ( ˜ m = ˜ m ), theQbMM solution is simply a single quadrature node with φ = m , requiring a negligible compu-tational time. Consequently this test case, with a reaction zone spanning over the whole domain,must be considered as upper costs-limit, i.e. as the setup with greatest inversion effort. Comparingthe normalized CPU time t c, ∗ inv = t cinv /t cmomT r for QMOM ( t c, ∗ inv = 1 .
18 for Q
2, 1 .
22 for Q
3, 1 .
30 for Q . Q
3, which provides very good results,would correspond to the solution of 11 transport equations. While there is no predefined minimumof stochastic fields, Vali˜no [4] reported for the same case a number of 50 SFs to achieve statisticalconvergence. Consequently, it can be summarized that especially QMOM offers benefits in termsof computational efficiency, since (i) only a few moment transport equations are solved and (ii)the inversion time is low. Since the PDF-moment equations applicability has been demonstrated,the more complex turbulent methane flame is investigated next.3.2.
Turbulent premixed methane flame.
A freely propagating turbulent methane-air flamein stoichiometric conditions, based on the work of Picciani et al. [24], is investigated in the fol-lowing. In a homogeneous isotropic turbulence, a parameter variation of characteristic turbulenceproperties was calculated. For different specific turbulent length scales L T , the turbulent veloc-ity fluctuation u (cid:48) was increased yielding different turbulent flame velocities and thicknesses. Thissetup is based on DNS calculations [33] investigating the bending effect in turbulent flames, i.e. thereduced acceleration of the turbulent flame at higher turbulence intensities. Unlike the DNS, theSF-reference [24] was solved on a 1D grid resolving the PDF with 512 SFs. A similar setup is usedin this work. A schematic representation is shown in Fig. 4. Two different values are consideredfor the ratio of the turbulent length scale and laminar flame thickness L T /δ L , namely 1 and 2 . u (cid:48) /s L is varied between 1and 20. Analogously to [24], the turbulent diffusivity is modeled as D T = C µ u (cid:48) L T with C µ = 0 . τ − MM = ( C φ u (cid:48) ) / (2 L T ), where C φ = 2( D L /D T + 1). Thetabulated manifold is built solving a freely propagating flame with the same mechanism as in [24],also assuming Le = 1 and Sc = 0 .
7. This leads a laminar flame speed s L = 0 .
385 m/s and a flame Fig. 4.
A sketch of a freely-propagating turbulent premixed flame, showing aninstantaneous flame front, the mean progress variable profile ˜ φ and the turbulentflame thickness δ T . The progress variable PDF is also shown at three differentlocations in the flame: unburned gas (left), reaction zone (center), burned gas(right). s T / s L u'/s L Ref.-sf [24], L T / δ L =1Ref.-sf [24], L T / δ L =2.5L T / δ L =1L T / δ L =2.5 Fig. 5.
QMOM results of the non-dimensionalized turbulent flame speed for thecase investigated in [24]. The QbMM results using Q L T /δ L = 1 and L T /δ L = 2 . δ L = 0 .
408 mm. All thermo-physical properties Φ and ˙ ω are mapped on the progressvariable Y c = Y CO . The results for the turbulent flame speed are plotted in Fig. 5 togetherwith the reference SF results. The turbulent flame speed and the velocity fluctuations are non-dimensionalized with the laminar flame speed. In Fig. 5 the QbMM results are based on threequadrature nodes ( Q Q L T /δ L = 1 and L T /δ L = 2 .
5, where the flame speed evolution is almost identical.A point further studied for SFs in [24] is the flame thickness and the corresponding resolutionrequirements. The QMOM simulations in Fig. 5 predict (as expected) an increase in the flamethickness δ T over the range of u (cid:48) /s L (not shown here for brevity). A direct comparison with thedata reported by Picciani et al. [24] is, however, not possible since only the averaged thermalthickness of the individual stochastic fields is provided in their work. This definition based on in-dividual fields and the flame thickness based on the progress variable mean, as provided in QMOM,can vary greatly. However, as discussed in [24], the relevant flame thickness to be resolved in thecontext of SFs is the average thickness of all individual fields, which is lower than δ T . Conse-quently, resolving the structure of premixed flames demands very fine numerical grids. The griddependency of QMOM will be analyzed and compared to SFs in the following. Six different flames,indicated with symbols in Fig. 5, three for L T /δ L = 1 and three for L T /δ L = 2 . s T / s L δ L / Δ x L T / δ L =1, u ' /s L =3L T / δ L =1, u ' /s L =5L T / δ L =1, u ' /s L =7.5L T / δ L =2.5, u ' /s L =3L T / δ L =2.5, u ' /s L =7.5L T / δ L =2.5, u ' /s L =15 Fig. 6.
The turbulent flame speed dependence on the grid resolution. The left-hand side of the diagram corresponds to a coarse resolution. δ L / ∆ x = 1 representsa grid resolution equal the laminar flame thickness. For each of the cases shown,see also Fig. 5, the minimum grid resolution required for SF [24] is marked by asymbol.by large numerical diffusivity. It was shown in [24] that in order to properly resolve the turbu-lent flames, a minimum number of 16 grid nodes are required within the average stochastic fieldthickness. In Fig. 6 the SF resolution is marked with a symbol for each setup and it is generallymuch lower than the laminar flame thickness. As expected, L T /δ L = 1 shows that a fine gridresolution is required for both methods. Since the flame thicknesses are comparable for this setup,the resolution requirements are also similar. For the flames with L T /δ L = 2 .
5, however, it is seenthat with QMOM s T /s L converges with a resolution almost one order of magnitude smaller thanwith SFs, obviously due to the increased turbulent flame thickness. This resolution robustness isa further advantage worth noting, also with respect to the numerical costs. For real applications,commonly employing coarser grids, the applicability of this approach requires further study.4. Summary
Two QbMM closures, QMOM and (for the first time) EQMOM, were used to close the momentsequation formulation of the composition PDF, which, differently from classical TPDF methods,requires an additional closure for the chemical source term. A tabulated manifold approach hasbeen employed to account for chemical reactions. This configuration was first applied to abenchmark case, a reactive plug-flow reactor, and compared with SF and Lagrangian particlereference results from the literature. The influence of different numbers of moments was tested,with up to 4 QMOM nodes (7 moment transport equations) and 3 EQMOM KDFs (6 moments).It was observed that 3 quadrature nodes or KDFs, respectively, yield very good results for allsetups. The second application, a turbulent premixed methane-air flame, shows results whichcompare well with the SF results from the literature. As with many other composition TPDFmethods, a very high grid resolution is necessary for premixed flames, lower than the laminar flamethickness. Especially for higher levels of turbulence, QbMM shows that the flame speed can bepredicted with a substantially coarser grid. The results and the numerical benefits, in terms of lownumber of moment transport equations to be solved and the spatial resolution requirements, arevery promising for the future and represent a significant advance for the use of QbMM methodsfor reactive flows.
Acknowledgments
This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) - Projektnummer 237267381 - TRR 150 and by the Clean Sky 2 Joint Undertakingunder the European Union’s Horizon 2020 research and innovation programme under the ESTi-MatE project, grant agreement No 821418. We further thank Dr. M. Picciani for the fruitfuldiscussions. eferences [1] S. B. Pope. The statistical theory of turbulent flames. Philos. Trans. R. Soc. London. Ser.A, Math. Phys. Sci. , 291(1384):529–568, 1979. doi: 10.1098/rsta.1979.0041.[2] M. Muradoglu, P. Jenny, S. B. Pope, and D. A. Caughey. A Consistent Hybrid Finite-Volume/Particle Method for the PDF Equations of Turbulent Reactive Flows.
J. Comput.Phys. , 154:342–371, 1999. doi: 10.1006/jcph.1999.6316.[3] S. B. Pope. A Monte Carlo Method for the PDF Equations of Turbulent Reactive Flow.
Combust. Sci. Techn. , 25(5-6):159–174, jan 1981. doi: 10.1080/00102208108547500.[4] L. Vali˜no. Field Monte Carlo formulation for calculating the probability density functionof a single scalar in a turbulent flow.
Flow, Turbul. Combust. , 60(2):157–172, 1998. doi:10.1023/A:1009968902446.[5] A. Avdi´c, G. Kuenne, and J. Janicka. Flow Physics of a Bluff-Body Swirl Stabilized Flameand their Prediction by Means of a Joint Eulerian Stochastic Field and Tabulated ChemistryApproach.
Flow, Turbul. Combust. , 97(4):1185–1210, 2016. doi: 10.1007/s10494-016-9781-y.[6] W. P. Jones and S. Navarro-Martinez. Numerical Study of n-Heptane Auto-ignition Us-ing LES-PDF Methods.
Flow, Turbul. Combust. , 83(3):407–423, oct 2009. doi: 10.1007/s10494-009-9228-9.[7] W. P. Jones and V. N. Prasad. Large Eddy Simulation of the Sandia Flame Series (D-F)using the Eulerian stochastic field method.
Combust. Flame , 157(9):1621–1636, 2010. doi:10.1016/j.combustflame.2010.05.010.[8] R. McGraw. Description of aerosol dynamics by the quadrature method of moments.
AerosolSci. Techn. , 27(2):255–265, 1997. doi: 10.1080/02786829708965471.[9] M. Pollack, S. Salenbauch, D. L. Marchisio, and C. Hasse. Bivariate extensions of the ExtendedQuadrature Method of Moments ( EQMOM ) to describe coupled droplet evaporation andheat-up.
J. Aerosol Sci. , 92:53–69, 2016. doi: 10.1016/j.jaerosci.2015.10.008.[10] S. Salenbauch, C. Hasse, M. Vanni, and D. L. Marchisio. A numerically robust method ofmoments with number density function reconstruction and its application to soot formation ,growth and oxidation.
J. Aerosol Sci. , 128:34–49, 2019.[11] A. Wick, T.-T. Nguyen, F. Laurent, R. O. Fox, and H. Pitsch. Modeling soot oxidationwith the Extended Quadrature Method of Moments .
Proc. Comb. Inst. , 2016. doi: http://dx.doi.org/10.1016/j.proci.2016.08.004.[12] R. O. Fox.
Computational Models for Turbulent Reacting Flows , volume 51. CambridgeUniversity Press, 2003. ISBN 0521659078. doi: 10.2516/ogst:1996020.[13] D. L. Marchisio and R. O. Fox. Solution of population balance equations using the directquadrature method of moments.
J. Aerosol Sci. , 36(1):43–73, jan 2005. doi: 10.1016/j.jaerosci.2004.07.009.[14] V. Raman, H. Pitsch, and R. O. Fox. Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion.
Combust. Theory Model. , 10(3):439–458, 2006. doi: 10.1080/13647830500460474.[15] Q. Tang, W. Zhao, M. Bockelie, and R. O. Fox. Multi-environment probability density functionmethod for modelling turbulent combustion using realistic chemical kinetics.
Combust. TheoryModel. , 11(6):889–907, nov 2007. doi: 10.1080/13647830701268890.[16] J. Akroyd, A. J. Smith, L. R. McGlashan, and M. Kraft. Numerical investigation of DQMoM-IEM as a turbulent reaction closure.
Chem. Eng. Sci. , 65(6):1915–1924, 2010. doi: 10.1016/j.ces.2009.11.010.[17] H. Koo, P. Donde, and V. Raman. A quadrature-based LES/transported probability densityfunction approach for modeling supersonic combustion.
Proc. Combust. Inst. , 33(2):2203–2210, jan 2011. doi: 10.1016/j.proci.2010.07.058.[18] J. Jaishree and D. C. Haworth. Comparisons of Lagrangian and Eulerian PDF methods in sim-ulations of non-premixed turbulent jet flames with moderate-to-strong turbulence-chemistryinteractions.
Combust. Theory Model. , 16(3):435–463, 2012. doi: 10.1080/13647830.2011.633349.
19] P. Donde, H. Koo, and V. Raman. A multivariate quadrature based moment method for LESbased modeling of supersonic combustion.
J. Comput. Phys. , 231(17):5805–5821, jul 2012.doi: 10.1016/j.jcp.2012.04.031.[20] J. C. Cheng, R. Vigil, and R. Fox. A competitive aggregation model for Flash NanoPrecipi-tation.
J. Colloid Interface Sci. , 351(2):330–342, nov 2010. doi: 10.1016/j.jcis.2010.07.066.[21] E. Madadi-Kandjani, R. O. Fox, and A. Passalacqua. Application of the Fokker-Planck molec-ular mixing model to turbulent scalar mixing using moment methods.
Phys. Fluids , 29(6):065109, 2017. doi: 10.1063/1.4989421.[22] E. Madadi-Kandjani.
Quadrature-based models for multiphase and turbulent reacting flows .PhD thesis, Iowa St. Univ., 2017.[23] R. O. Fox. Quadrature-Based Moment Methods for Multiphase Chemically Reacting Flows.In
Adv. Chem. Eng. 52 , pages 1–50. Acad. Pr., jan 2018. ISBN 9780128150962. doi: 10.1016/bs.ache.2018.01.001.[24] M. A. Picciani, E. S. Richardson, and S. Navarro-Martinez. Resolution Requirements inStochastic Field Simulation of Turbulent Premixed Flames.
Flow, Turbul. Combust. , 101(4):1103–1118, 2018. doi: 10.1007/s10494-018-9953-z.[25] J. Villermaux and J. Devillon. Repr´esentation de la coalescence et de la redispersion desdomaines de s´egr´egation dans un fluide par un mod`ele d’interaction ph´enom´enologique. In
Proc. Second Int. Symp. Chem. React. Eng. , pages 1–13, New York, 1972.[26] Z. Ren, M. Kuron, X. Zhao, T. Lu, E. Hawkes, H. Kolla, and J. H. Chen. Micromixing Modelsfor PDF Simulations of Turbulent Premixed Flames.
Combust. Sci. Techn. , 191(8):1430–1455,aug 2019. doi: 10.1080/00102202.2018.1530667.[27] B. T. Zoller, M. L. Hack, and P. Jenny. A PDF combustion model for turbulent premixedflames.
Proc. Combust. Inst. , 34(1):1421–1428, 2013. doi: 10.1016/j.proci.2012.05.053.[28] J. C. Wheeler. Modified moments and Gaussian quadratures.
Rocky Mt. J. Math. , 4(2):287–296, jun 1974. doi: 10.1216/RMJ-1974-4-2-287.[29] C. Yuan, F. Laurent, and R. O. Fox. An extended quadrature method of moments for popu-lation balance equations.
J. Aerosol Sci. , 51:1–23, 2012.[30] M. Pollack, M. P¨utz, D. L. Marchisio, M. Oevermann, and C. Hasse. Zero-flux approximationsfor multivariate quadrature-based moment methods.
J. Comp. Physics , 398:108879, 2019. doi:https://doi.org/10.1016/j.jcp.2019.108879.[31] Z. Lu, H. Zhou, S. Li, Z. Ren, T. Lu, and C. K. Law. Analysis of operator splitting errors fornear-limit flame simulations.
J. Comp. Physics , 335:578 – 591, 2017. doi: https://doi.org/10.1016/j.jcp.2017.01.044.[32] W. Gautschi.
Orthogonal Polynomials Computation and Approximation . 2008.[33] G. Nivarti and S. Cant. Direct Numerical Simulation of the bending effect in turbulentpremixed flames.
Proc. Combust. Inst. , 36(2):1903–1910, 2017. doi: 10.1016/j.proci.2016.07.076., 36(2):1903–1910, 2017. doi: 10.1016/j.proci.2016.07.076.