Application of Helmholtz-Hodge decomposition and conditioned structure functions to exploring influence of premixed combustion on turbulence upstream of the flame
Vladimir A. Sabelnikov, Andrei N. Lipatnikov, Nikolay Nikitin, Shinnosuke Nishiki, Tatsuya Hasegawa
Application of Helmholtz-Hodge decomposition and conditioned structure functions to exploring influence of premixed combustion on turbulence upstream of the flame
Vladimir A. Sabelnikov , Andrei N. Lipatnikov , Nikolay Nikitin , Shinnosuke Nishiki , Tatsuya Hasegawa ONERA โ The French Aerospace Laboratory, F-91761 Palaiseau, France Central Aerohydrodynamic Institute (TsAGI), 140180 Zhukovsky, Moscow Region, Russian Federation Department of Mechanics and Maritime Sciences, Chalmers University of Technology, Gothenburg, 41296 Sweden Moscow State University,
Moscow Region, Russian Federation Department of Information and Electronic Engineering, Teikyo University, Utsunomiya 320-8551, Japan, Institute of Materials and Systems for Sustainability, Nagoya University, Nagoya 464-8603, Japan
Corresponding author:
Prof. Vladimir Sabelnikov, ONERA โ The French Aerospace Laboratory, F-91761 Palaiseau, France, [email protected]
Abstract
In order to explore the influence of combustion-induced thermal expansion on turbulence, a new research method is introduced. The method consists in jointly applying Helmholtz-Hodge decomposition and conditioned structure functions to analyzing turbulent velocity fields. Opportunities offered by the method are demonstrated by using it to process Direct Numerical Simulation data obtained earlier from two statistically 1D, planar, fully-developed, weakly turbulent, single-step-chemistry, premixed flames characterized by two significantly different (7.52 and 2.50) density ratios, with all other things being approximately equal. To emphasize the influence of combustion-induced thermal expansion on turbulent flow of unburned mixture upstream of a premixed flame, the focus of analysis is placed on structure functions conditioned to the unburned mixture in both points. Two decomposition techniques, i.e. (i) a widely used orthogonal Helmholtz-Hodge decomposition and (ii) a recently introduced natural Helmholtz-Hodge decomposition, are probed, with results obtained using them being similar in the largest part of the computational domain with the exception of narrow zones near the inlet and outlet boundaries. Computed results indicate that combustion-induced thermal expansion can significantly change turbulent flow of unburned mixture upstream of a premixed flame by generating anisotropic potential velocity fluctuations whose spatial structure differ substantially from spatial structure of the incoming turbulence. The magnitude of such potential velocity fluctuations is greater than the magnitude of the solenoidal velocity fluctuations in the largest part of the mean flame brush in the case of the high density ratio. In the case of the low density ratio, the latter magnitude is larger everywhere, but the two magnitudes are comparable in the middle of the mean flame brush. Contrary to the potential velocity fluctuations, the influence of the thermal expansion on the solenoidal velocity field in the unburned mixture is of minor importance under conditions of the present study.
Keywords: flame-generated turbulence, Helmholtz-Hodge decompositions, structure function, conditional averaging, thermal expansion, DNS
1. Introduction
Since an increase in burning rate due to flame-generated turbulence was hypothesized by Karlovitz et al. [1] and Scurlock and Grover [2], the influence of combustion-induced density variations on turbulent flow within a premixed flame brush was studied in a number of papers reviewed elsewhere [3-6], see also recent Refs. [7-11]. The majority of such studies addressed the first and second moments of velocity field within mean flame brush, with the moments being averaged in the conventional way by allowing for contributions from unburned reactants, intermediate mixture states, and combustion products simultaneously. However, as far as eventual increase in burning rate due to flame-generated turbulence is concerned, combustion-induced perturbations of the flow upstream of the flame appear to be of the most importance. Indeed, since a flame propagates into unburned gas, velocity perturbations upstream of the flame are required to affect the flame p ropagation. To distinguish the influence of thermal expansion on turbulence in unburned gas, reacting mixture, or products, conditioned flow characteristics are commonly used, as reviewed elsewhere [4-6]. Recently, Structure Functions (SFs) of the velocity field, conditioned to different mixture states, were introduced for that purpose in Refs. [12,13] and [14] using two different frameworks. Results obtained from weakly turbulent flames [12,13] indicated that combustion could significantly affect turbulent flow of unburned reactants not only within, but also upstream of the mean flame brush. Such effects were attributed to potential flow fluctuations caused by combustion-induced pressure perturbations upstream of the mean flame brush. The goal of the present work is to examine this hypothesis by decomposing the flow field into potential (irrotational) and solenoidal (rotational) subfields and separately exploring characteristics of each subfield. Direct Numerical Simulation (DNS) data analyzed in the following are summarized in Section 2. Methods applied to process these data are discussed in Section 3. Results are presented in Section 4, followed by conclusions.
2. DNS Attributes
In the present study, DNS data computed by Nishiki et al. [15,16] almost 20 years ago and used in the aforementioned papers [12,13] are further analyzed. The choice of this database, which appears to be outdated when compared to recent DNS data [17-22] generated invoking complex combustion chemistry at a high ratio of the rms turbulent velocity ๐ข โฒ to the laminar flame speed ๐ ๐ฟ , requires comments. The point is that the focus of the present study is placed on the influence of combustion-induced thermal expansion on the turbulent flow upstream of a flame. Accordingly, modeling of intermediate states of the mixture (e.g., complex combustion chemistry) appears to be of secondary importance when compared to two other major requirements. First, to clearly reveal thermal expansion effects, data obtained at different density ratios ๐ = ๐ ๐ข ๐ ๐ โ are very useful and the DNS data analyzed here were obtained at ๐ = 2.5 or 7.53, with all other things being roughly equal. Second, a new research method (e.g. conditioned SFs extracted from potential and solenoidal velocity subfields) should initially be probed under conditions associated with the strongest manifestation of effects (e.g. combustion-induced perturbations of the potential velocity subfield upstream of a flame) the method aims at. To make such effects as strong as possible, the flamelet regime of premixed turbulent burning [23], associated with a low ratio of ๐ขโฒ ๐ ๐ฟ โ , is of the most interest. The DNS by Nishiki et al. [15,16] did deal with the flamelet regime, as discussed in detail elsewhere [24], whereas the majority of recent DNS studies explored other combustion regimes. Because the DNS data analyzed here were extensively discussed by various research groups [12,13,15,16,24-43], we will restrict ourselves to a brief summary of those compressible simulations. They dealt with statistically1D, planar, adiabatic flames modeled by unsteady 3D continuity, Navier-Stokes, and energy equations, supplemented with a transport equation for the mass fraction ๐ of a deficient reactant and the ideal gas state equation. The Lewis and Prandtl numbers were equal to 1.0 and 0.7, respectively, and combustion chemistry was reduced to a single reaction. Accordingly, the combustion progress variable ๐(๐ฑ, ๐ก) = 1 โ ๐(๐ฑ, ๐ก) ๐ ๐ข โ = (๐(๐ฑ, ๐ก) โ ๐ ๐ข ) (๐ ๐ โ ๐ ๐ข )โ . The computational domain was a rectangular box ฮ ๐ฅ ร ฮ ๐ฆ ร ฮ ๐ง with ฮ ๐ฅ = 8 mm, ฮ ๐ฆ = ฮ ๐ง = 4 mm, and was resolved using a uniform rectangular ( ) mesh of
512 ร 128 ร 128 points. The flow was periodic in ๐ฆ and ๐ง directions. Homogeneous isotropic turbulence ( ๐ข โฒ = 0.53 m/s, an integral length scale ๐ฟ = 3.5 mm, the turbulent Reynolds number ๐ ๐ ๐ก =96 [15,16]) was generated in a separate box and was injected into the computational domain through the left boundary ๐ฅ = 0 . In the domain, the turbulence decayed along the direction ๐ฅ of the mean flow. At ๐ก = 0 , a planar laminar flame was embedded into statistically the same turbulence assigned for the velocity field in the entire computational domain. Subsequently, the inflow velocity was increased twice, i.e., ๐(0 โค ๐ก < ๐ก ๐ผ ) = ๐ ๐ฟ <๐(๐ก ๐ผ โค ๐ก < ๐ก ๐ผ๐ผ ) < ๐(๐ก ๐ผ๐ผ โค ๐ก) = ๐ ๐ก , in order to keep the flame in the domain till the end ๐ก ๐ผ๐ผ๐ผ of the simulations. Here, ๐ ๐ก is the turbulent flame speed. Solely data obtained at ๐ก ๐ผ๐ผ โค ๐ก โค ๐ก ๐ผ๐ผ๐ผ are discussed in the following. Two cases H and L characterized by High and Low, respectively, density ratios will be investigated. In case H, ๐ = 7.53 , ๐ ๐ฟ = 0.6 m/s, ๐ฟ ๐ฟ = 0.217 mm, ๐ท๐ = 18 , ๐พ๐ = 0.21 , ๐ ๐ก = 1.15 m/s. In case L, ๐ = 2.5 , ๐ ๐ฟ = 0.416 m/s, ๐ฟ ๐ฟ = 0.158 mm, ๐ท๐ = 17.3 , ๐พ๐ = 0.30 , and ๐ ๐ก = 0.79 m/s. Here, ๐ฟ ๐ฟ = (๐ โ ๐ ) max{|โ๐|}โ is the laminar flame thickness, ๐ท๐ =(๐ฟ ๐ขโฒโ ) (๐ฟ ๐ฟ ๐ ๐ฟ โ )โ and ๐พ๐ = (๐ขโฒ ๐ ๐ฟ โ ) (๐ฟ ๐ฟ ๐ฟ โ ) โ1 2โ are the Damkรถhler and Karlovitz numbers, respectively, evaluated at the leading edges of the mean flame brushes. The two flames are well associated with the flamelet combustion regime [23], e.g., various Bray-Moss-Libby (BML) expressions [44] hold, see figures 1-4 in Ref. [24].
3. Methods
If velocity field ๐ฎ(๐ฑ, ๐ก) is decomposed into solenoidal and potential subfields ๐ฎ ๐ (๐ฑ, ๐ก) and ๐ฎ ๐ (๐ฑ, ๐ก) , respectively, then, โ ร ๐ฎ ๐ = โ ร ๐ฎ, ๐ฎ ๐ = โ๐, ๐ฎ = ๐ฎ ๐ + ๐ฎ ๐ . (1) However, such a decomposition is not unique, unless the function ๐(๐ฑ, ๐ก) is defined. To solve the problem, an extra constraint should be invoked and there are different methods for doing so. In the present work, two such methods, i.e. (i) widely-used โorthogonalโ [45,46] and (ii) recently introduced โnaturalโ [47,48] Helmholtz-Hodge decompositions, are used.
The orthogonal decomposition invokes the following bulk constraint of orthogonality โญ ๐ฎ ๐ โ ๐ฎ ๐ ๐๐ฑ ๐ = 0 (2) of the subfields ๐ฎ ๐ (๐ฑ, ๐ก) and ๐ฎ ๐ (๐ฑ, ๐ก) . This constraint results in the additivity of the bulk kinetic energies of the solenoidal and potential flow fields, i.e. โญ ๐ฎ โ ๐ฎ๐๐ฑ ๐ = โญ ๐ฎ ๐ โ ๐ฎ ๐ ๐๐ฑ ๐ + โญ ๐ฎ ๐ โ ๐ฎ ๐ ๐๐ฑ ๐ , (3) but does not require that mean local correlation โฉ๐ข ๐ ,๐ (๐ฑ, ๐ก)๐ข ๐,๐ (๐ฑ, ๐ก)โช = 0 . Here, integration is performed over the computational domain ๐ . Substitution of Eq. (1) into Eq. (2) yields โญ ๐ฎ ๐ โ โ๐๐๐ฑ ๐ = โญ โ(๐๐ฎ ๐ )๐๐ฑ ๐ โ โญ ๐โ โ ๐ฎ ๐ ๐๐ฑ ๐ = โฏ ๐๐ฎ ๐ โ ๐ง๐๐ ๐ โ โญ ๐โ โ ๐ฎ ๐ ๐๐ฑ ๐ , (4) where ๐ is the boundary of the domain ๐ and the unit vector ๐ง is normal to the boundary. If ๐ฎ ๐ โ ๐ง = 0 and โ โ ๐ฎ ๐ = 0 , then integrals in Eq. (4) vanish, โ๐ = โ โ ๐ฎ ๐ = โ โ ๐ฎ (5) in the entire domain ๐ , and ๐๐๐๐| ๐ = ๐ง โ โ๐| ๐ = ๐ง โ ๐ฎ| ๐ (6) on its boundary. The Neumann problem given by Eqs. (5) and (6) has a unique solution for ๐(๐ฑ, ๐ก) . The natural decomposition [47,48] introduces an extra vector-field ๐ฐ(๐ฑ, ๐ก) , which (i) coincides with the velocity field ๐ฎ(๐ฑ, ๐ก) in the domain ๐ , i.e., ๐ฐ(๐ฑ, ๐ก) = ๐ฎ(๐ฑ, ๐ก) for all ๐ฑ โ ๐ , but (ii) is extrapolated to the entire 3D space โ such that |๐ฐ(๐ฑ, ๐ก)| โ 0 for |๐ฑ| โ โ . Subsequently, the velocity field ๐ฐ(๐ฑ, ๐ก) is decomposed as follows ๐ฐ = โฮ + โ ร ๐, ๐ฑ โ โ . (7) Consequently, โฮ = โ โ ๐ฐ, ๐ฑ โ โ , (8) โ ร โ ร ๐ = โ ร ๐ฐ, ๐ฑ โ โ . (9) Solutions to Poisson Eqs. (8) and (9) are unique ฮ(๐ฑ , ๐ก) = โญ ๐บ โ (๐ฑ, ๐ฑ )โ โ ๐ฐ(๐ฑ, ๐ก)๐๐ฑ โ , ๐ฑ , ๐ฑ โ โ , (10) ๐(๐ฑ , ๐ก) = โโญ ๐บ โ (๐ฑ, ๐ฑ )โ ร ๐ฐ(๐ฑ, ๐ก)๐๐ฑ โ , ๐ฑ , ๐ฑ โ โ . (11) where ๐บ โ (๐ฑ, ๐ฑ ) = โ 1 (4๐|๐ฑ โ ๐ฑ |)โ is the free-space Greenโs function in โ . Finally, integration in Eqs. (10) and (11) is truncated outside the domain ๐ by interpreting the truncated integrals to be an external influence. Thus, ฮ โ (๐ฑ , ๐ก) = โญ ๐บ โ (๐ฑ, ๐ฑ )โ โ ๐ฎ(๐ฑ, ๐ก)๐๐ฑ ๐ , ๐ฑ , ๐ฑ โ ๐, (12) ๐ โ (๐ฑ , ๐ก) = โโญ ๐บ โ (๐ฑ, ๐ฑ )โ ร ๐ฎ(๐ฑ, ๐ก)๐๐ฑ ๐ , ๐ฑ , ๐ฑ โ ๐. (13) Obviously, Eq. (1) with ๐ฎ ๐ = โ ร ๐ โ and ๐ฎ ๐ = โฮ โ holds by virtue of Eqs. (7)-(9). Since the DNS data analyzed here were obtained from fully-developed, statistically 1D, planar flames that propagated from right to left along the ๐ฅ -direction, the computed flow fields are considered to be statistically homogeneous and isotropic in any transverse plane ๐ฅ = const. Accordingly, the following discussion will be restricted to SFs measured for two points ๐ฑ ๐ด ={๐ฅ ๐ด๐ต , ๐ฆ ๐ด , ๐ง ๐ด } and ๐ฑ ๐ต = {๐ฅ ๐ด๐ต , ๐ฆ ๐ด + ๐ ๐ฆ , ๐ง ๐ด + ๐ ๐ง } that belong to the same transverse plane ๐ฅ = ๐ฅ ๐ด๐ต , i.e., ๐ฑ ๐ต โ ๐ฑ ๐ด โก ๐ซ = {0, ๐ ๐ฆ , ๐ ๐ง } . Then, the conditioned second-order SFs of the velocity field are defined as follows [12,13] ๐ท ๐๐๐ผ๐ฝ (๐ฅ, ๐) โก โฉ(๐ข ๐ต,๐ โ ๐ข
๐ด,๐ )(๐ข
๐ต,๐ โ ๐ข
๐ด,๐ )๐ผ ๐ต๐ผ ๐ผ ๐ด๐ฝ โช ฮก ๐ผ๐ฝ โ , (14) where โฉโโช designates averaging over a transverse plane and time; ๐ข ๐ is ๐ -th component of the velocity vector ๐ฎ = {๐ข, ๐ฃ, ๐ค} ; subscripts ๐ and ๐ refer to spatial coordinates; superscripts ๐ผ and ๐ฝ refer to the mixture state; the indicator function ๐ผ ๐ด๐ข =๐ผ ๐ข (๐ฑ ๐ด , ๐ก) is equal to unity if unburned mixture is observed in point ๐ฑ ๐ด at instant ๐ก and vanishes otherwise; ๐ผ ๐ด๐ = 1 if combustion products are observed in point ๐ฑ ๐ด at instant ๐ก and vanishes otherwise; ๐ผ ๐ด๐ = 1 if ๐ผ ๐ด๐ข = ๐ผ ๐ด๐ = 0 and vanishes otherwise; and ฮก ๐ผ๐ฝ (๐ฅ, ๐) = โฉ๐ผ ๐ต๐ผ ๐ผ ๐ด๐ฝ โช are the probabilities that the mixture states ๐ผ and ๐ฝ are recorded in points ๐ฑ ๐ต and ๐ฑ ๐ด , respectively, at the same instant. Depending on ๐ผ and ๐ฝ , there are different conditioned SF tensors. The following discussion will be restricted to conditioned SFs ๐ท ๐๐๐ข๐ข (unburned mixture in both points). Henceforth, superscript ๐ข๐ข will be skipped for brevity. When processing the DNS data, mean quantities ๐ฬ (๐ฅ) are averaged over transverse ๐ฆ๐ง -planes and over time (220 and 200 snapshots in cases H and L, respectively, stored during a time interval of ๐ก ๐ผ๐ผ๐ผ โ ๐ก ๐ผ๐ผ โ 1.5 ๐ฟ ๐ขโฒโ โ 10 ms). Subsequently, ๐ฅ -dependencies are mapped to ๐ฬ -dependencies using the spatial profiles ๐ฬ (๐ฅ) of the Reynolds-averaged combustion progress variable. The probability ฮก ๐ข๐ข (๐ฅ, ๐) and the unburned-mixture-conditioned SF are sampled from points characterized by ๐(๐ฑ, ๐ก) < 0.05 , with Eq. (14) being separately applied to the potential or solenoidal velocity subfield to extract three pairs of conditioned SFs. More specifically, first, the transverse SFs ๐ท ๐ฅ๐ฅ,๐ (๐ฅ, ๐) for the axial solenoidal and potential velocities are obtained by averaging (๐ข ๐ต โ ๐ข ๐ด ) over time and two sets of points; (i) ๐ฑ ๐ด = {๐ฅ, ๐ฆ, ๐ง} , ๐ฑ ๐ต = {๐ฅ, ๐ฆ + ๐, ๐ง} and (ii) ๐ฑ ๐ด = {๐ฅ, ๐ฆ, ๐ง} , ๐ฑ ๐ต = {๐ฅ, ๐ฆ, ๐ง + ๐} . To do so, both transverse coordinates are independently varied in intervals of ๐ฆ and ๐ง , whereas the separation distance is varied in an interval of ๐ฆ
2โ = ฮ ๐ง . Second, the transverse SFs for the transverse velocities ๐ท ๐ฆ๐ง,๐ (๐ฅ, ๐) = 0.5(๐ท ๐ฆ๐ฆ,๐ + ๐ท ๐ง๐ง,๐ ) , where ๐ท ๐ฆ๐ฆ,๐ and ๐ท ๐ง๐ง,๐ are obtained by averaging (i) (๐ฃ ๐ต โ ๐ฃ ๐ด ) and (ii) (๐ค ๐ต โ ๐ค ๐ด ) , respectively, over time and (i) ๐ฑ ๐ด = {๐ฅ, ๐ฆ, ๐ง} , ๐ฑ ๐ต = {๐ฅ, ๐ฆ, ๐ง + ๐} and (ii) ๐ฑ ๐ด = {๐ฅ, ๐ฆ, ๐ง} , ๐ฑ ๐ต = {๐ฅ, ๐ฆ + ๐, ๐ง} , respectively. Third, the longitudinal SFs for the transverse velocities ๐ท ๐ฆ๐ง,๐ฟ (๐ฅ, ๐) = 0.5(๐ท ๐ฆ๐ฆ,๐ฟ + ๐ท ๐ง๐ง,๐ฟ ) , where ๐ท ๐ฆ๐ฆ,๐ฟ and ๐ท ๐ง๐ง,๐ฟ are obtained by averaging (i) (๐ฃ ๐ต โ ๐ฃ ๐ด ) and (ii) (๐ค ๐ต โ ๐ค ๐ด ) , respectively, over time and (i) ๐ฑ ๐ด = {๐ฅ, ๐ฆ, ๐ง} , ๐ฑ ๐ต = {๐ฅ, ๐ฆ + ๐, ๐ง} and (ii) ๐ฑ ๐ด = {๐ฅ, ๐ฆ, ๐ง} , ๐ฑ ๐ต = {๐ฅ, ๐ฆ, ๐ง + ๐} , respectively.
4. Results and Discussion
Figure 1a shows that, upstream of the mean flame brush ( ), the mean solenoidal kinetic energy (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ (curves 1 and 4) is significantly higher than the mean potential kinetic energy (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ (curves 2 and 5) and is close to the mean kinetic energy (๐ฎ โ ๐ฎฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ (curve 3). Both (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ and (๐ฎ โ ๐ฎฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ decay in the ๐ฅ -direction in the unburned mixture ( ๐ฅ < 1 mm ), but begin increasing within the mean flame brush. On the contrary, the potential (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ begins increasing already in the unburned mixture upstream of the mean flame brush ( ๐ฅ โ 0.5 mm ) and exceeds the solenoidal (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ in the largest part of the mean flame brush ( or ). In case L characterized by a low density ratio, both (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ and (๐ฎ โ ๐ฎฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ 2โ decrease with ๐ฅ , see Fig. 1b, but there is a local peak of the kinetic energy of the potential motion within the mean flame brush so that (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ and (๐ฎ ๐ โ ๐ฎ ๐ ฬ ฬ ฬ ) ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ are comparable there. These results show generation of potential velocity fluctuations not only within the mean flame brush, but also in the unburned mixture upstream of the mean flame brush (at least in case H), with the effect magnitude being significantly increased by the density ratio. Note that generation of potential velocity perturbations is well known in the theory of the hydrodynamic instability of a laminar premixed flame [49], but solenoidal velocity vanishes upstream of the flame in that case. On the contrary, solenoidal velocity fluctuations dominate upstream of mean flame brushes in the present study. Fig. 1.
Contributions of solenoidal (curves 1 and 4) and potential (curves 2 and 5) velocity fields to the mean kinetic energy (curve 3) of velocity fluctuations. Results obtained using orthogonal (natural) decomposition are shown in curves 1 and 2 (4 and 5, respectively). To further explore the influence of combustion-induced thermal expansion on turbulence in unburned mixture upstream of the flame, let us consider behavior of SFs conditioned to the unburned mixture. Since comparison of curves 1 and 2 in Fig. 1 with curves 4 and 5, respectively, indicates that the two adopted Helmholtz-Hodge decompositions yield close results in the largest part of the computational domain with the exception of regions near the inlet and outlet boundaries, we will restrict ourselves to reporting data obtained using the natural decomposition. It is worth stressing that all trends discussed in the following are also observed when using the orthogonal decomposition. Figure 2 shows that longitudinal SFs ๐ท ๐ฆ๐ง,๐ฟ for solenoidal transverse velocities, normalized using conditioned solenoidal transverse rms velocity (๐ฃโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ , decrease slowly with ๐ฅ , as expected for spatially decaying turbulence. Similar results were obtained when normalizing these SFs with the total (๐ฃโฒ ๐ข2 ฬ ฬ ฬ ฬ + ๐คโฒ ๐ข2 ฬ ฬ ฬ ฬ ฬ ) 2โ . On the contrary, for the potential velocity field, such SFs normalized with (๐ฃโฒ ๐ข2 ฬ ฬ ฬ ฬ + ๐คโฒ ๐ข2 ฬ ฬ ฬ ฬ ฬ ) 2โ increase significantly with ๐ฅ in both cases, see Figs. 3a and 3b, with the effect being observed well upstream of the mean flame brush, see curves 1-3. If the potential ๐ท ๐ฆ๐ง,๐ฟ is normalized using the potential (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ , the evolution of such SFs is weakly pronounced at ๐ฬ โค 0.1 , see curves 1-4 in Figs. 3c and 3d. Thus, Fig. 3 implies that potential perturbations of the transverse velocity field in unburned mixture are mainly controlled by an increase in the magnitude (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ of such perturbations, whereas changes in the spatial scales of the perturbations are weakly pronounced. However, it is worth noting that, at large ๐, the solenoidal ๐ฆ๐ง,๐ฟ (๐ฃโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐ฃโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ )โ โ 2 , see Fig. 2, but the potential ๐ฆ๐ง,๐ฟ (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ )โ โ 4 , see Figs. 3c and 3d, thus, indicating that, contrary to a typical turbulent flow, where ๐ท ๐๐ (๐ โ โ) โ 2๐ข ๐โฒ ๐ข ๐โฒ ฬ ฬ ฬ ฬ ฬ ฬ [50], the discussed potential velocity perturbations can be negatively correlated at large distances. Fig. 2.
Longitudinal SFs ๐ท ๐ฆ๐ง,๐ฟ for solenoidal transverse velocities, conditioned to unburned mixture and normalized using conditioned solenoidal transverse rms velocity (๐ฃโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ . ( a ) case H, ( b ) case L. 1 โ ๐ฬ = 0, ๐ฅ = 0.50 mm; 2 โ ๐ฬ =0, ๐ฅ = 0.75 mm; 3 โ ๐ฬ = 0.01,๐ฅ = 1.0 mm; 4 โ ๐ฬ = 0.1, ๐ฅ = 1.3 or 1.4 mm in case H or L, respectively; 5 โ ๐ฬ = 0.25, ๐ฅ =1.5 or 1.8 mm in case H or L, respectively. Results computed for transverse SFs ๐ท ๐ฆ๐ง,๐ and ๐ท ๐ฅ๐ฅ,๐ , see Figs. 4-5 and 6-7, respectively, are qualitatively similar to the already discussed results for the longitudinal ๐ท ๐ฆ๐ง,๐ฟ , but there are some differences. First, the magnitudes of the potential ๐ฆ๐ง,๐ (๐ฃ ๐ขโฒ 2 ฬ ฬ ฬ ฬ ฬ + ๐ค ๐ขโฒ2 ฬ ฬ ฬ ฬ ฬ )โ or ๐ท ๐ฅ๐ฅ,๐ ๐ข ๐ขโฒ 2 ฬ ฬ ฬ ฬ ฬ โ are less than the magnitudes of the potential ๐ฆ๐ง,๐ฟ (๐ฃ ๐ขโฒ2 ฬ ฬ ฬ ฬ ฬ + ๐ค ๐ขโฒ2 ฬ ฬ ฬ ฬ ฬ )โ , cf. Fig. 4 or 6a and Figs. 3a-3b, respectively, with the effect being strongly pronounced in case L, cf. Fig. 3b and 4b. Fig. 3.
Longitudinal SFs ๐ท ๐ฆ๐ง,๐ฟ for potential transverse velocities, conditioned to unburned mixture and normalized using conditioned total ( a and b ) or potential ( c and d ) transverse rms velocity (๐ฃ ๐ขโฒ 2 ฬ ฬ ฬ ฬ ฬ + ๐ค ๐ขโฒ2 ฬ ฬ ฬ ฬ ฬ ) 2โ or (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ , respectively. ( a ) and ( c ) case H, ( b ) and ( d ) case L. Legends are explained in caption to Fig. 2. Fig. 4.
Transverse SFs ๐ท ๐ฆ๐ง,๐ for potential transverse velocities, conditioned to unburned mixture and normalized using conditioned transverse rms velocity (๐ฃ ๐ขโฒ 2 ฬ ฬ ฬ ฬ ฬ + ๐ค ๐ขโฒ2 ฬ ฬ ฬ ฬ ฬ ) 2โ . ( a ) case H, ( b ) case L. Legends are explained in caption to Fig. 2. Fig. 5.
Transverse SFs ๐ท ๐ฆ๐ง,๐ for ( a ) potential and ( b ) solenoidal transverse velocities, conditioned to unburned mixture and normalized using conditioned ( a ) potential or ( b ) solenoidal transverse rms velocities (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ or (๐ฃโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ , respectively. Case H. Legends are explained in caption to Fig. 2. Fig. 6.
Transverse SFs ๐ท ๐ฅ๐ฅ,๐ for potential axial velocity, conditioned to unburned mixture and normalized using conditioned ( a ) total or ( b ) potential axial rms velocity ๐ขโฒ ๐ข2 ฬ ฬ ฬ ฬ or ๐ขโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ , respectively. Case H. Legends are explained in caption to Fig. 2. Fig. 7.
Transverse SFs ๐ท ๐ฅ๐ฅ,๐ for solenoidal axial velocity, conditioned to unburned mixture and normalized using conditioned solenoidal axial rms velocity ๐ขโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ . ( a ) case H, ( b ) case L. Legends are explained in caption to Fig. 2. Second, the potential ๐ท ๐ฆ๐ง,๐ normalized using the potential (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ฬ ) 2โ increases with ๐ฅ , but remains smaller than 2, see Fig. 5a, whereas the potential ๐ท ๐ฅ๐ฅ,๐ ๐ขโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ โ evolves slowly with ๐ฅ if ๐ฬ โค 0.1 , see curves 1-4 in Fig. 6b, similarly to ๐ฆ๐ง,๐ฟ (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ฬ )โ see Fig. 3c. Moreover, the magnitudes of the normalized potential transverse SFs ๐ฆ๐ง,๐ (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ฬ )โ and ๐ท ๐ฅ๐ฅ,๐ ๐ขโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ โ are significantly less than the magnitude of the normalized potential longitudinal SF ๐ฆ๐ง,๐ฟ (๐ฃโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ ฬ )โ , cf. Figs. 5a and 6b with Fig. 3c. Thus, the spatial structure of the potential velocity fluctuations generated in unburned mixture upstream of the studied flames is highly anisotropic. Third, a decrease in the solenoidal SF ๐ฆ๐ง,๐ (๐ฃโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ )โ or ๐ท ๐ฅ๐ฅ,๐ ๐ขโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ โ with ๐ฅ , see Fig. 5b or 7a, respectively, is better pronounced when compared to the evolution of ๐ฆ๐ง,๐ฟ (๐ฃโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ + ๐คโฒ ๐ ,๐ข2 ฬ ฬ ฬ ฬ ฬ ฬ )โ with ๐ฅ , see Fig. 2a.
5. Conclusions
Helmholtz-Hodge decomposition and conditioned second-order structure functions of turbulent velocity field were jointly (for the first time to the best of the present authorsโ knowledge) applied to processing DNS data obtained earlier from two weakly turbulent premixed flames characterized by different density ratios. Computed results show that combustion-induced thermal expansion can significantly change turbulent flow of unburned mixture upstream of a premixed flame by generating anisotropic potential velocity fluctuations whose spatial structure differ substantially from spatial structure of the incoming turbulence. On the contrary, the influence of the thermal expansion on the solenoidal-velocity-field structure functions in the unburned mixture is of minor importance under conditions of the present st udy. Assessment of the effect of an increase in ๐ขโฒ ๐ ๐ฟ โ or Karlovitz number on the magnitude of the revealed effects appears to be of great fundamental interest for future studies. References [1]
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