The relative alignment of the eigenvectors of pressure Hessian with reactive scalar gradient and strain rate eigenvectors in turbulent premixed flames have been analyzed for Karlovitz number values ranging from 0.75 to 126 using a detailed chemistry three-dimensional direct numerical simulations database of H2–air premixed flames. The reactive scalar gradient preferentially aligns with the most extensive strain rate eigendirection for large Damköhler number and small Karlovitz number values, whereas a preferential collinear alignment between the reactive scalar gradient with the most compressive strain rate eigendirection is observed in flames with small Damköhler number and large Karlovitz number. By contrast, the eigenvectors of pressure Hessian do not perfectly align with the reactive scalar gradient, and the net effect of the pressure Hessian on the evolution of the normal strain rate contribution to the scalar dissipation rate transport acts to reduce the scalar gradient in the zone of high dilatation rate. The eigenvectors of pressure Hessian and the strain rate are aligned in such a manner that the contribution of pressure Hessian to the evolution of principal strain rates tends to augment the most extensive principal strain rate for small and moderate values of Karlovitz numbers, whereas this contribution plays an important role for the evolution of the intermediate principal strain rate for large values of Karlovitz number. As the reactive scalar gradient does not align with the intermediate strain rate eigenvector, the influence of pressure Hessian contributions to the scalar–turbulence interaction remains weak for large values of Karlovitz number.

In premixed flames, the flow acceleration across the flame due to thermal expansion gives rise to a self-induced pressure gradient within the flame front.1–3 This pressure gradient p within the flame brush is known to have significant influences on turbulent scalar flux,2,3 turbulent kinetic energy,3–6 and enstrophy3,7–10 evolutions in premixed turbulent combustion. In comparison, limited attention has been given to the statistics of the pressure Hessian tensor Π (i.e., Πij=2p/xixj are the components of the pressure Hessian tensor) and its alignment with local principal strain rates in turbulent premixed flames although this aspect was analyzed in detail for non-reacting turbulent flows,11–14 especially for homogeneous isotropic turbulence11,13 and homogeneous axisymmetric turbulence.14 

To appreciate the importance of pressure Hessian, consider the transport equation of the strain rate tensor with components Sij=0.5ui/xj+uj/xi, where ui is the ith component of fluid velocity,

Sijt+ukSijxk=SikSkjωiωjδijωkωk4+12ρ2pxiρxj+pxjρxiΠijρ
12ρ2pxiτjkxk+pxjτikxk+12ρ2τikxjxk+2τikxixk.
(1)

Here, ρ is density, ωi=ϵijkuk/xj is the ith component of vorticity, and τik=2μSik2μδikSll/3 is the component of the viscous stress tensor with μ being the dynamic viscosity. Equation (1) shows that the pressure Hessian components Πij play a role in the evolution of strain rate components. In particular, Π̂=ETΠE was found to play a key role in the evolution of principal strain rates in turbulent premixed flames,15,16 where E=[êαêβêγ] with êα (êγ) being the eigenvector associated with the most extensive (compressive) principal strain rate and êβ is the eigenvector corresponding to the intermediate strain rate. Moreover, the dissipation rate of kinetic energy is given by ε=τikSik, and thus, it can be understood from Eq. (1) that the inner product between the pressure Hessian tensor and the strain rate tensor (i.e., ΠikSik) plays an important role in the evolution of ε.

Ahmed and Prosser17,18 demonstrated that the relative alignment of the eigenvectors of the pressure Hessian tensor Π with the flame normal vector N affects the evolution of the scalar–turbulence interaction term (i.e., the normal strain rate aN=NiNjSij contribution to the scalar dissipation rate transport) ρ¯Δ̃c=2ρDcNiNjSijc2¯=2ρDcaNc2¯, where ρ is the gas density, c is the reaction progress variable, Dc is the reaction progress variable diffusivity, and the overbar (tilde) suggests a Reynolds (Favre) averaging/filtering operation, as appropriate. The transport equation of Δ̃c takes the following form:18 

ρ¯Δ̃ct+ũjΔ̃cxj=xnρDcuncxiSijcxj¯+2cxiSijxjxkρDc2cxk¯+Dccxicxjxkτikxj¯+F11+F12+F2+F3+F4.
(2)

Here, F11 is the reaction rate contribution, F12=2DcΠijNiNjc2¯ is the contribution of pressure Hessian, and the terms F2,F3, and F4 arise due to turbulent transport, dilatation rate, and turbulent straining, respectively. A positive value of F12 acts to increase (decrease) the magnitude of the positive (negative) value of Δ̃c, which is indicative of the increased (decreased) extent of collinear alignment between N and êα (êγ). The remaining terms F11,F2,F3, and F4 are not important for the current analysis and, thus, are not provided here, but interested readers are referred to Ref. 18 for further information in this regard. Gonzalvez and Paranthoën19,20 concluded from their inviscid flow analysis that the anisotropy of vorticity and pressure Hessian is responsible for the modification of the scalar gradient alignments with strain rate eigenvectors from a preferential alignment of c with êγ in the case of passive scalar mixing to a preferential alignment of c with êα in turbulent premixed flames when the heat release effects overwhelm turbulent straining.21 Moreover, Keylock22 argued that the pressure Hessian is a major contributor to the non-local effects on the vorticity and scalar gradient alignments in turbulent flows.

All the aforementioned information indicates that the statistics of the alignment of pressure Hessian with strain rate eigenvectors and reactive scalar gradient are important for the analysis of turbulent premixed flame physics, but this aspect is yet to be analyzed in detail. This paper addresses this gap in the existing literature by analyzing the pressure Hessian statistics using detailed chemistry Direct Numerical Simulation (DNS) data of statistically planar H2air flames with an equivalence ratio of ϕ=0.7 for different turbulence intensities across different combustion regimes. The main objectives of the current analysis are (i) to demonstrate the effects of turbulence intensity and regime of combustion on the alignment of pressure Hessian with local principal strain rate eigenvectors, and c, and (ii) to explain the observed behavior based on physical principles and provide modeling implications.

The components of Π̂=ETΠE in the strain rate eigendirections (i.e., Π̂α,Π̂β,andΠ̂γ) are given as16,

(3)
Π̂α=Παcos2(π̂α,êα)+Πβcos2(π̂β,êα)+Πγcos2(π̂γ,êα),
(3a)
Π̂β=Παcos2(π̂α,êβ)+Πβcos2(π̂β,êβ)+Πγcos2(π̂γ,êβ),
(3b)
Π̂γ=Παcos2(π̂α,êγ)+Πβcos2(π̂β,êγ)+Πγcos2(π̂γ,êγ),
(3c)
where Πα,Πβ, and Πγ are the most extensive, intermediate, and the most compressive eigenvalues of the pressure Hessian and π̂α,π̂β, and π̂γ are the corresponding eigenvectors. Similarly, the contribution of F12=2DcΠijNiNjc2¯ in the transport equation of Δ̃c [see Eq. (2)] can be expressed as18 

F12=2DcΠijNiNjc2¯=2Dcc2[Παcos2(π̂α,N)+Πβcos2(π̂β,N)+Πγcos2(π̂γ,N)]¯.
(4)

The flame normal vector N is defined as N=c/|c|, where the reaction progress variable c is defined based on a suitable species mass fraction Yα as c=(Yα0Yα)/(Yα0Yα), where subscripts 0 and refer to the values in the unburned gas and fully burned products, respectively. For the present analysis, α=H2,O2, and H2O have been used, but the qualitative nature of the pressure Hessian statistics does not depend on the choice of the definition of c. It is also possible to define a non-dimensional temperature cT as cT=(TT0)/(TadT0), where T0 and Tad are the unburned gas temperature and adiabatic flame temperature, respectively. In the context of multi-step chemistry, c=cT is not necessarily maintained, and this assumption does not play any role in the analysis conducted in this paper.

It can be appreciated from Eqs. (3) and (4) that the relative alignments of pressure Hessian with strain rate eigenvectors and c are important for the analysis of fluid turbulence in premixed combustion. These statistics have been explored in this paper for H2–air premixed flames with an equivalence ratio ϕ of 0.7 (i.e., ϕ=0.7) using a three-dimensional DNS database,10,23 which uses a detailed chemical mechanism24 with 9 species and 19 chemical reactions. H2–air premixed flames with an equivalence ratio ϕ of 0.7 are thermo-diffusively neutral in terms of stretch rate effects.25,26 Therefore, the effects of differential diffusion are relatively weak for ϕ=0.7 in the case of H2–air premixed flames. For this database, the unburned gas temperature T0 is taken to be 300 K, which yields an unstrained laminar burning velocity SL=135.6cm/s under atmospheric pressure. Turbulent inflow and outflow boundaries are considered in the direction of mean flame propagation and are specified using the Navier–Stokes characteristic boundary conditions technique, and transverse boundaries are taken to be periodic. High order finite-difference (eighth order for the internal grid points) and third order explicit Runge–Kutta schemes are used for numerical differentiation and explicit time advancement, respectively, and the interested readers are referred elsewhere10,23 for detailed information on numerical implementation. The mean inlet velocity has been gradually modified as the simulation progresses to match the turbulent burning velocity, so that a statistically stationary state can be obtained. The inflow values of normalized root mean square turbulent velocity fluctuation u/SL, the most energetic turbulent length scale to flame thickness ratio lT/δth, the Damköhler number Da=lTSL/uδth, the Karlovitz number Ka=ρ0SLδth/μ00.5u/SL1.5lT/δth0.5, and the turbulent Reynolds number Ret=ρ0ulT/μ0 for all cases are listed in Table I, where ρ0 is the unburned gas density, μ0 is the unburned gas viscosity, δth=(TadT0)/maxTL is the thermal flame thickness, and the subscript “L” is used to refer to the unstrained laminar flame quantities. The longitudinal integral scale L11 is a factor of 2.5 smaller than the most energetic scale lT, and thus, the values of Ka (Da) in Table I increase (decrease) by a factor of 1.6 (2.5) if L11 instead of lT is used for their definitions. For cases A and B (case C), the domain size is taken to be 20×10×10mm3 (8×2×2mm3), which has been discretized by a uniform Cartesian grid of dimension 512×256×256 (1280×320×320). The simulation time corresponds to {2.5,17,16.75}L11/u for cases A–C, respectively, and is comparable to several previous analyses.2,4,5,27,28 The cases investigated here belong to the corrugated flamelets (i.e., case A with Ka<1), thin reaction zones (i.e., case B with 1<Ka<100), and broken reaction zones (i.e., case C with Ka>100) regimes according to Peters.29 Whether the broken reaction zones combustion is realized in case C is not the focus of this analysis, and without doubt, cases A–C allow for the analysis of the effects of Ka on the pressure Hessian statistics. In this regard, it is worthwhile to note that most practical combustion devices operate within the corrugated and thin reaction zones combustion regimes,30 and the upper range of Ka in conventional engines remains about 13.0.31 However, in lean premixed prevaporized (LPP) gas turbine combustors32 and some laboratory-scale configurations33 using lean hydrocarbon flames, Ka values can be high especially under elevated pressure and can locally reach close to the Ka=100 boundary. The Ka>100 conditions are also likely in next generation combustors involving NH3 combustion.34 

TABLE I.

List of inflow turbulence parameters.

Caseu/SLlT/δthRetDaKa
0.7 14.0 227 20.0 0.75 
5.0 14.0 1623 2.8 14.4 
14.0 4.0 1298 0.29 126 
Caseu/SLlT/δthRetDaKa
0.7 14.0 227 20.0 0.75 
5.0 14.0 1623 2.8 14.4 
14.0 4.0 1298 0.29 126 

The distributions of vorticity, flow topology, and scalar gradient within the flame-front for cases A–C are presented elsewhere3,10,23 and, thus, will not be repeated here. However, the distribution of the log(SijSij×δth/SL) at the central mid-plane is shown in Figs. 1(a)–1(c) for cases A–C, respectively, and the contours of cT=0.15,0.3,0.5,0.7, and 0.85 (left to right) are superimposed on top of it. The distributions of SijSij in cases A–C are markedly different. In case A, occasional local augmentations of SijSij from the unburned to the burned gas side of the flame front can be discerned, whereas the strain rate magnitude drops significantly from the unburned to the burned gas side of the flame front in cases B and C. Moreover, the strain rate magnitudes change significantly in a short span of space in the unburned gas region in cases B and C, whereas the length scale associated with the variation of SijSij in case A is much greater than in cases B and C.

FIG. 1.

Distributions of log(SijSij×δth/SL) in the central midplane for (a)–(c) cases A–C (note the different scale for case C). The white broken lines indicate c=0.15,0.3,0.5,0.7, and 0.85 from left to right.

FIG. 1.

Distributions of log(SijSij×δth/SL) in the central midplane for (a)–(c) cases A–C (note the different scale for case C). The white broken lines indicate c=0.15,0.3,0.5,0.7, and 0.85 from left to right.

Close modal

It is worthwhile to note that cases B and C have higher values of Ret than in case A, and thus, they exhibit a larger range of length scales than in case A. Note that lT is identical in cases A and B, but the Kolmogorov length scale ηlTRet3/4 is smaller in case B than in case A because of the higher Ret value in case B. Moreover, lT in case C is smaller than that in case A, and thus, the Kolmogorov length scale ηlTRet3/4 in case C is smaller than in case A due to the much higher Ret value in case C. As the Karlovitz number Ka scales as Kaδth2/η2,29 the flame thickness δth remains smaller than the Kolmogorov length scale η in case A (where Ka<1), but δth>η is obtained for cases B and C. Thus, the scale separation between δth and η is greater in case C than in case B due to the greater Ka value for case C. As a result, the inner flame structure in cases B and C gets affected by turbulent fluctuations, and this is reflected in the local flame thickening in these cases, which can, indeed, be verified from the lack of the parallel nature of the contours of cT in Figs. 1(b) and 1(c). By contrast, the contours of cT remain parallel to each other in case A as the inner flame structure is not affected by turbulent fluid motion in this case. From the foregoing discussion, it can be appreciated that the strain rate distribution and the flame–turbulence interaction are significantly affected by the Karlovitz number, which will be reflected in the alignment statistics of pressure Hessian eigenvectors with strain rate eigenvectors and reaction scalar gradient (or flame normal vector N).

The mean values of normalized principal strain rates {sα,sβ,sγ}×δth/SL (with sα>sβ>sγ) conditional upon cT are shown in Figs. 2(a)–2(c) for cases A–C, respectively, along with the mean normalized dilatation rate ·u×δth/SL={sα+sβ+sγ}×δth/SL. It is evident from Fig. 2 that the individual principal strain rates tend to increase from case A to case C, which is consistent with the observations made from Figs. 1(a)–1(c). However, the magnitudes of ·u remain comparable. Figures 2(a)–2(c) also show that the peak mean value of ·u is obtained for 0.2<cT<0.4 for all cases because the effects of heat release are strong in this region of the flame for H2air flames with ϕ=0.7, for which the maximum heat release rate occurs at cT0.3 under laminar conditions. Figures 2(a)–2(c) further reveal that the magnitude of the most extensive principal strain rate sα remains greater than the magnitudes of the intermediate and the most compressive principal strain rates (i.e., sβ and sγ) in case A, whereas the mean values of sα and sγ are of similar magnitude in case C, and the behavior in case B is somewhere in between cases A and C. This suggests that the preferential augmentation of the magnitude of sα weakens with increasing (decreasing) Ka (Da) (i.e., from cases A and C), and this behavior affects the statistics of the scalar–turbulence interaction term ρ¯Δ̃c, which can be expressed as18,21

ρ¯Δ̃c=2ρDcc2sαcos2(êα,N+sβcos2(êβ,N)+sγcos2(êγ,N¯)].
(5)

Equation (5) suggests that a preferential alignment between N and êα (êγ) yields a positive (negative) value of Δ̃c, which is indicative of destruction (generation) of the reactive scalar gradient by flame normal straining.18,21

FIG. 2.

Profiles of mean values of {sα,sβ,sγ, and ·u}×δth/SL (with sα>sβ>sγ) conditional upon cT for (a)–(c) cases A–C. Note that sα,sβ,andsγ are multiplied by 0.1 in case C.

FIG. 2.

Profiles of mean values of {sα,sβ,sγ, and ·u}×δth/SL (with sα>sβ>sγ) conditional upon cT for (a)–(c) cases A–C. Note that sα,sβ,andsγ are multiplied by 0.1 in case C.

Close modal

The relative alignment of N and êi (with i=α,β,andγ) can be quantified by Γα=|cos(N,êα)|, Γβ=|cos(N,êβ)|, and Γγ=|cos(N,êγ)|, so that a unity value of these quantities (i.e., Γi=cos0°=1 for i=α,β,andγ) signifies a perfect collinear alignment, whereas a zero value (i.e., Γi=cos90°=0 for i=α,β,andγ) represents a perpendicular alignment. The mean values of Γα, Γβ, and Γγ conditional upon cT are shown in Figs. 3(a)–3(c) for cases A–C, respectively, for c based on H2 mass fraction, but the same qualitative behavior has been observed for reaction progress variables using O2 and H2O mass fractions. It is seen from Figs. 3(a)–3(c) that the flame normal vector N preferentially aligns collinearly with êα in case A for most of the flame front except for the burned gas side, where an increase in the collinear alignment between N and êγ is observed. In case C, N exhibits a predominant collinear alignment with êγ for a major part of the flame, but an increase in the collinear alignment between N and êα is observed in the burned gas region. In case B, N also collinearly aligns with êα, but the extent of this alignment is weaker (stronger) than in case A (case C). In contrast, the extent of collinear alignment of N with êγ in case B is stronger (weaker) than in case A (case C). It has been shown elsewhere3,21 that N (or c) preferentially aligns with êα for TadT0Dal/T01 (where Dal=L11SL/δthu is the Damköhler number based on the longitudinal integral length scale L11), which signifies that the strain rate induced by flame normal acceleration arising from thermal expansion overwhelms turbulent straining. On the contrary, N (or c) preferentially aligns with êγ similar to non-reacting flows, for TadT0Dal/T01 when turbulent straining dominates over the strain rate due to thermal expansion. The values of TadT0Dal/T0 are 45.68, 6.40, and 0.66 in cases A, B, and C, respectively, and thus, N (or c) predominantly aligns with êα (êγ) in cases A and B (case C).

FIG. 3.

Profiles of mean values of Γα (blue solid), Γβ (red broken), and Γγ (magenta chain dotted) conditional upon cT for (a)–(c) cases A–C. N is calculated based on c definition based on H2 mass fraction in Figs. 3 and 4. Horizontal green, orange, and purple lines indicate cos30°,cos45°, and cos60°, respectively, in Figs. 3–7.

FIG. 3.

Profiles of mean values of Γα (blue solid), Γβ (red broken), and Γγ (magenta chain dotted) conditional upon cT for (a)–(c) cases A–C. N is calculated based on c definition based on H2 mass fraction in Figs. 3 and 4. Horizontal green, orange, and purple lines indicate cos30°,cos45°, and cos60°, respectively, in Figs. 3–7.

Close modal

Based on the above information, it will be useful to consider the relative alignment between N and π̂i (for i=α,β,andγ) in order to understand how the pressure Hessian contribution (i.e., F12) affects the evolution of ρ¯Δ̃c within the flame front [see Eqs. (2) and (4)]. Based on the close relation of the pressure Hessian Π with the evolutions of Δ̃c and si (for i=α,β,andγ), the alignments of π̂i (for i=α,β,andγ) with N and with êi (for i=α,β,andγ) will be analyzed next.

The mean values of Ψα=|cos(N,π̂α)|, Ψβ=|cos(N,π̂β)|, and Ψγ=|cos(N,π̂γ)| conditional upon cT are shown in Figs. 4(a)–4(c) for cases A–C, respectively, where N is evaluated based on c using H2 mass fraction, but the same qualitative behavior has been observed for c definitions using O2 and H2O mass fractions (not shown here). Figures 4(a) and 4(b) suggest that the mean values of Ψα and Ψγ conditional upon cT remain comparable, and both remain greater than the mean value of Ψβ in cases A and B. In contrast, the mean value of Ψγ assumes greater values than that of Ψα and Ψβ toward the unburned gas side (cT<0.4) in case C. Figures 4(a)–4(c) indicate that the mean values of Ψα,Ψβ, and Ψγ are sufficiently different from unity, suggesting that π̂α and π̂γ remain in imperfect alignment with N (and c), and π̂β does not collinearly align with N (and c) in all cases considered here.

FIG. 4.

Profiles of mean values of Ψα (blue solid), Ψβ (red broken), and Ψγ (magenta chain dotted) conditional upon cT for (a)–(c) cases A–C.

FIG. 4.

Profiles of mean values of Ψα (blue solid), Ψβ (red broken), and Ψγ (magenta chain dotted) conditional upon cT for (a)–(c) cases A–C.

Close modal

The pressure gradient p remains imperfectly aligned with N for high values of u/SL such as in cases B and C (not shown here), whereas p aligns with N for small values of u/SL (e.g., case A) similar to that in laminar flames. As shown in Fig. 3, N preferentially collinearly aligns with êα in case A, and also in the region of intense heat release in case B, whereas a predominant alignment between N and êγ is obtained in case C. Therefore, it is of interest to consider the relative alignments between π̂i (for i=α,β,andγ) and êk (for k=α,β,andγ).

The mean values of Φij=|cos(π̂i,êj)| (for i=α,β,andγ and j=α,β,andγ) conditional upon cT are shown in Fig. 5 for case A. The corresponding variations for cases B and C are shown in Figs. 6 and 7, respectively. Figure 5(a) shows that the mean value of Φαα (Φαβ) assumes the highest (smallest) value among the mean values of Φαα,Φαβ, and Φαγ in case A. However, all of these mean values remain comparable and smaller than cos45°= 0.707, suggesting an imperfect alignment of π̂α for all strain rate eigenvectors êα,êβ, and êγ. A comparison between Figs. 5(a) and 5(c) reveals that the profiles of the mean values of Φγα,Φγβ, and Φγγ remain qualitatively similar to that of Φαα,Φαβ, and Φαγ, respectively, which also suggests an imperfect alignment of π̂γ with êα,êβ, and êγ. In contrast, the mean values of Φββ and Φβγ remain greater than that of Φβα and assume values close to 0.6 throughout the flame front, suggesting an imperfect alignment of π̂β with êβ and êγ, and a very little alignment between π̂β with êα. Under homogeneous isotropic non-reacting turbulent flows, π̂β has a tendency to align collinearly with êβ,11,13 whereas in homogeneous axisymmetric non-reacting flows, π̂γ preferentially aligns, collinearly with êβ.14Figure 5 suggests that the presence of heat release in case A alters the collinear alignment of êβ with either π̂β or π̂γ.

FIG. 5.

Profiles of mean values of (a) Φαα (blue solid), Φαβ (red broken), and Φαγ (magenta chain dotted); (b) Φβα (blue solid), Φββ (red broken), and Φβγ (magenta chain dotted); and (c) Φγα (blue solid), Φγβ (red broken), and Φγγ (magenta chain dotted) conditional upon cT for case A.

FIG. 5.

Profiles of mean values of (a) Φαα (blue solid), Φαβ (red broken), and Φαγ (magenta chain dotted); (b) Φβα (blue solid), Φββ (red broken), and Φβγ (magenta chain dotted); and (c) Φγα (blue solid), Φγβ (red broken), and Φγγ (magenta chain dotted) conditional upon cT for case A.

Close modal
FIG. 6.

Profiles of mean values of (a) Φαα (blue solid), Φαβ (red broken), and Φαγ (magenta chain dotted); (b) Φβα (blue solid), Φββ (red broken), and Φβγ (magenta chain dotted); and (c) Φγα (blue solid), Φγβ (red broken), and Φγγ (magenta chain dotted) conditional upon cT for case B.

FIG. 6.

Profiles of mean values of (a) Φαα (blue solid), Φαβ (red broken), and Φαγ (magenta chain dotted); (b) Φβα (blue solid), Φββ (red broken), and Φβγ (magenta chain dotted); and (c) Φγα (blue solid), Φγβ (red broken), and Φγγ (magenta chain dotted) conditional upon cT for case B.

Close modal
FIG. 7.

Profiles of mean values of (a) Φαα (blue solid), Φαβ (red broken), and Φαγ (magenta chain dotted); (b) Φβα (blue solid), Φββ (red broken), and Φβγ (magenta chain dotted); and (c) Φγα (blue solid), Φγβ (red broken), and Φγγ (magenta chain dotted) conditional upon cT for case C.

FIG. 7.

Profiles of mean values of (a) Φαα (blue solid), Φαβ (red broken), and Φαγ (magenta chain dotted); (b) Φβα (blue solid), Φββ (red broken), and Φβγ (magenta chain dotted); and (c) Φγα (blue solid), Φγβ (red broken), and Φγγ (magenta chain dotted) conditional upon cT for case C.

Close modal

It was demonstrated earlier3,21 that the strength of the strain rate induced by thermal expansion in comparison to turbulent straining processes weakens with increasing (decreasing) Ka (Da), and the same is also valid for this database (i.e., from case A to case C).3,21 Thus, the alignments between π̂i (for i=α,β,andγ) and êj (for j=α,β,andγ) change from case A to case C. In case B, the mean values of Φαα,Φαβ, and Φαγ assume comparable values [see Fig. 6(a)], and the mean values of Φβα and Φγα remain the highest among Φβα,Φββ, and Φβγ and Φγα,Φγβ, and Φγγ, respectively [see Figs. 6(b) and 6(c)]. The mean value of Φβα is the smallest among Φβα,Φββ, and Φβγ in case B, which is qualitatively similar to case A. This indicates that heat release effects in case B induce a misalignment between pressure Hessian (i.e., π̂i for i=α,β,andγ) and strain rate eigenvectors (i.e., êj for j=α,β,andγ). The alignment statistics between pressure Hessian eigenvectors (i.e., π̂i for i=α,β,andγ) and strain rate eigenvectors (i.e., êj for j=α,β,andγ) in cases A and B is consistent with previous findings based on simple chemistry DNS data.16 

A comparison among Figs. 5–7 reveals that the alignment statistics between π̂α and strain rate eigenvectors (i.e., êj for j=α,β,andγ) in case C is significantly different from that in cases A and B [see Figs. 5(a), 6(a), and 7(a)]. In case C, the mean values of Φαβ and Φαγ remain greater than the mean value of Φαα for a major part of the flame front except for the burned gas side of the flame. The mean value of Φββ remains the cosine magnitude with the highest value for a major part of the flame front except for the burned gas side among Φβα,Φββ, and Φβγ in case C, but their values remain comparable. The mean value of Φββ in case C is smaller than that obtained in cases A and B. The mean value of Φγα remains the highest among Φγα,Φγβ, and Φγγ in case C, which is qualitatively similar to the behavior observed for cases A and B [see Figs. 5(c), 6(c), and 7(c)]. The observations from Fig. 7 also indicate that the heat release effects within the high Karlovitz number flames remain strong enough to exhibit differences from the predominant alignment between π̂β (π̂γ) and êβ in homogeneous isotropic (axisymmetric) non-reacting turbulence.11,13,14 The imperfect alignments between π̂i (i.e., π̂i for i=α,β,andγ) and strain rate eigenvectors (i.e., êj for j=α,β,andγ) and comparable values of Φαα,Φβα, and Φγα (Φαβ,Φββ, and Φγβ) [Φαγ,Φβγ, and Φγγ] reveal that Π̂α(Π̂β)[Π̂γ] is determined by the eigenvalues of the pressure Hessian (i.e., Πα,Πβ, and Πγ) according to Eq. (3). Similarly, comparable values of Ψα and Ψγ indicate that the behavior of F12 is driven by the relative magnitudes of Πα and Πγ [see Eq. (4)]. It has been found that the mean values of Ψα remain comparable to that of Ψγ for a major part of the flame front in cases A–C [see Figs. 4(a)–4(c)], and the mean value of the trace of pressure Hessian (Πα+Πβ+Πγ) is determined by the mean value of (Πα+Πγ) (not shown here). The mean values of {Π̂N=(Παcos2Ψα+Πβcos2Ψβ+Πγcos2Ψγ), Π̂α, Π̂β, Π̂γ)}×δth2/ρ0SL2 conditional upon cT for cases A–C are shown in Figs. 8(a)–8(c), respectively.

FIG. 8.

Profiles of the mean values of {Π̂N,Π̂α, Π̂β, Π̂γ}×δth2/ρ0SL2 conditional upon cT for (a)–(c) cases A–C.

FIG. 8.

Profiles of the mean values of {Π̂N,Π̂α, Π̂β, Π̂γ}×δth2/ρ0SL2 conditional upon cT for (a)–(c) cases A–C.

Close modal

Figures 8(a)–8(c) show that the mean value of Π̂N assumes negative values for 0.1<cT<0.4 where the effects of the dilatation rate are the strongest (see Fig. 2). Equation (4) suggests that a negative value of Π̂N tends to induce positive values of F12, which acts to increase the positive value of Δ̃c [see Eq. (2)], indicating an increased (a reduced) extent of collinear alignment between êα (êγ) and N. Figure 8 also demonstrates that Π̂α assumes the highest magnitude among the quantities Π̂α, Π̂β, and Π̂γ in cases A and B, indicating that Π influences particularly the evolution of sα in these cases. Furthermore, Eqs. (1) and (3) suggest that the predominantly negative values of Π̂α act to increase sα in cases A and B, but both sβ and sγ in these cases are not severely affected by the pressure Hessian due to small magnitudes of Π̂β and Π̂γ in comparison to Π̂α. However, Π̂β assumes the highest magnitude among the quantities Π̂α, Π̂β, and Π̂γ in case C, and thus, the pressure Hessian does not significantly influence the evolutions of sα and sγ. However, the mean value of Π̂α remains predominantly negative in case C and, therefore, acts to increase sα, but the influences of Π on sα and sγ weaken with increasing Ka (i.e., from case A to case C). The above findings suggest that the influence of pressure Hessian could be important for flames with small (large) values of Ka (Da), which is consistent with recent findings by Kasten et al.16 in the context of simple chemistry DNS and experimental findings by Steinberg et al.15 The current analysis suggests that Π may play a key role in the alteration of c (or N) alignment with strain rate eigenvectors in high Da and low Ka turbulent premixed flames in comparison to that in the case of passive scalar mixing, as previously conjectured by Gonzalez and Paranthoën.19,20

The alignment statistics of the eigenvectors of pressure Hessian Π with c and strain rate eigenvectors in turbulent premixed flames has been analyzed for Ka (Da) values ranging from 0.75 to 126 (0.29 to 20.0) using a detailed chemistry three-dimensional DNS database of fuel-lean H2–air premixed flames with ϕ=0.7. It was found that c preferentially aligns with the eigenvector corresponding to the most extensive principal strain rate with large Da and small Ka, but the reactive scalar gradient in flames with small Da and large Ka exhibits a preferential collinear alignment with the eigenvector corresponding to the most compressive principal strain rate. It was also found that the eigenvectors of Π exhibit imperfect alignment with c. However, the net contribution of the pressure Hessian to the evolution of the scalar–turbulence interaction term (i.e., normal strain rate contribution to the scalar dissipation rate transport) in the zone of high dilatation rate acts to increase the normal strain rate and, thus, to reduce |c|. Moreover, the relative collinear alignments of Π and strain rate eigenvectors have been found to be different from those in the non-reacting turbulent flows. Furthermore, the relative alignments of the eigenvectors of Π and the strain rate are such that the contribution of Π to the evolution of principal strain rates tends to augment the most extensive principal strain rate for small and moderate values of Ka, but this behavior weakens for Ka100 where the contribution of Π plays an important role for the evolution of the intermediate principal strain rate.

The findings of the current analysis suggest that the contributions of Π to the evolution of scalar–turbulence interaction and principal strain rates weaken with increasing the Karlovitz number, but these effects might play significant roles for small and moderate values of Ka. The results further indicate that the contributions of Π might be pivotal to the alteration of c alignment with strain rate eigenvectors in comparison to that in the case of passive scalar mixing in turbulent premixed flames with high Da and low Ka, as conjectured previously based on analytical studies19,20 on inviscid flows.

N.C. and U.A. are grateful to EPSRC and ARCHER (No. EP/R029369/1) for computational support. H.G.I. is grateful to KAUST for research funding and computational support.

The authors have no conflicts to disclose.

Nilanjan Chakraborty: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Project administration (lead); Software (lead); Supervision (lead); Writing - original draft (lead); Writing - review & editing (lead). Umair Ahmed: Conceptualization (supporting); Formal analysis (equal); Writing - review & editing (supporting). Markus Klein: Formal analysis (supporting); Writing - review & editing (supporting). Hong G. Im: Funding acquisition (lead); Project administration (supporting); Resources (lead);Writing - review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
R.
Borghi
and
D.
Escudie
, “
Assessment of a theoretical model of turbulent combustion by comparison with a simple experiment
,”
Combust. Flame
56
,
149
164
(
1984
).
2.
D.
Veynante
and
T.
Poinsot
, “
Effects of pressure gradient in turbulent premixed flames
,”
J. Fluid Mech.
353
,
83
114
(
1997
).
3.
N.
Chakraborty
, “
Influence of thermal expansion on fluid dynamics of turbulent premixed combustion and its modelling implications
,”
Flow, Turbul. Combust.
106
,
753
848
(
2021
).
4.
S.
Zhang
and
C. J.
Rutland
, “
Premixed flame effects on turbulence and pressure-related terms
,”
Combust. Flame
102
,
447
461
(
1995
).
5.
S.
Nishiki
,
T.
Hasegawa
,
R.
Borghi
, and
R.
Himeno
, “
Modeling of flame-generated turbulence based on direct numerical simulation databases
,”
Proc. Combust. Inst.
29
,
2017
2022
(
2002
).
6.
N.
Chakraborty
,
M.
Katragadda
, and
R. S.
Cant
, “
Statistics and modelling of turbulent kinetic energy transport in different regimes of premixed combustion
,”
Flow, Turbul. Combust.
87
,
205
235
(
2011
).
7.
A. N.
Lipatnikov
,
S.
Nishiki
, and
T.
Hasegawa
, “
A direct numerical study of vorticity transformation in weakly turbulent premixed flames
,”
Phys. Fluids
26
,
105104
(
2014
).
8.
N.
Chakraborty
,
I.
Konstantinou
, and
A.
Lipatnikov
, “
Effects of Lewis number on vorticity and enstrophy transport in turbulent premixed flames
,”
Phys. Fluids
28
,
015109
(
2016
).
9.
C.
Dopazo
,
L.
Cifuentes
, and
N.
Chakraborty
, “
Vorticity budgets in premixed combusting turbulent flows at different Lewis numbers
,”
Phys. Fluids
29
,
045106
(
2017
).
10.
V.
Papapostolou
,
D. H.
Wacks
,
M.
Klein
,
N.
Chakraborty
, and
H. G.
Im
, “
Enstrophy transport conditional on local flow topologies in different regimes of premixed turbulent combustion
,”
Sci. Rep.
7
,
11545
(
2017
).
11.
C.
Kalelkar
, “
Statistics of pressure fluctuations in decaying isotropic turbulence
,”
Phys. Rev. E
73
,
046301
(
2006
).
12.
B.
Lüthi
,
M.
Holzner
, and
A.
Tsinober
, “
Expanding the Q-R space to three dimensions
,”
J. Fluid Mech.
641
,
497
507
(
2009
).
13.
S.
Suman
and
S. S.
Girimaji
, “
Velocity gradient dynamics in compressible turbulence: Characterization of pressure-Hessian tensor
,”
Phys. Fluids
25
(
12
),
125103
(
2013
).
14.
J. M.
Lawson
and
J. R.
Dawson
, “
On velocity gradient dynamics and turbulent structure
,”
J. Fluid Mech.
780
,
60
98
(
2015
).
15.
A. M.
Steinberg
,
B.
Coriton
, and
J. H.
Frank
, “
Influence of combustion on principal strain-rate transport in turbulent premixed flames
,”
Proc. Combust. Inst.
35
,
1287
1296
(
2015
).
16.
C.
Kasten
,
U.
Ahmed
,
M.
Klein
, and
N.
Chakraborty
, “
Principal strain rate evolution within turbulent premixed flames for different combustion regimes
,”
Phys. Fluids
33
,
015111
(
2021
).
17.
U.
Ahmed
,
R.
Prosser
, and
A. J.
Revell
, “
Towards the development of an evolution equation for flame turbulence interaction in premixed turbulent combustion
,”
Flow, Turbul. Combust.
93
,
637
663
(
2014
).
18.
U.
Ahmed
and
R.
Prosser
, “
Modelling flame turbulence interaction in RANS simulation of premixed turbulent combustion
,”
Combust. Theory Modell.
20
,
34
57
(
2016
).
19.
M.
Gonzalez
and
P.
Paranthoën
, “
Effect of density step on stirring properties of a strain flow
,”
Fluid Dyn. Res.
41
,
035508
(
2009
).
20.
M.
Gonzalez
and
P.
Paranthoën
, “
Effect of variable mass density on the kinematics of scalar gradient
,”
Phys. Fluids
23
,
075107
(
2011
).
21.
N.
Chakraborty
and
N.
Swaminathan
, “
Influence of Damköhler number on turbulence-scalar interaction in premixed flames. Part I: Physical insight
,”
Phys. Fluids
19
,
045103
(
2007
).
22.
C.
Keylock
, “
The Schur decomposition of the velocity gradient tensor for turbulent flows
,”
J. Fluid Mech.
848
,
876
905
(
2018
).
23.
H. G.
Im
,
P. G.
Arias
,
S.
Chaudhuri
, and
H. A.
Uranakara
, “
Direct numerical simulations of statistically stationary turbulent premixed flames
,”
Combust. Sci. Technol.
188
,
1182
1198
(
2016
).
24.
M. P.
Burke
,
M.
Chaos
,
Y.
Ju
,
F. L.
Dryer
, and
S. J.
Klippenstein
, “
Comprehensive H2-O2 kinetic model for high-pressure combustion
,”
Int. J. Chem. Kinet.
44
,
444
474
(
2012
).
25.
J. H.
Chen
and
H. G.
Im
, “
Stretch effects on the burning velocity of turbulent premixed hydrogen/air flames
,”
Proc. Combust. Inst.
28
,
211
218
(
2000
).
26.
H. G.
Im
and
J. H.
Chen
, “
Preferential diffusion effects on the burning rate of interacting turbulent premixed hydrogen-air flames
,”
Combust. Flame
131
,
246
258
(
2002
).
27.
I.
Han
and
K. Y.
Huh
, “
Effects of the Karlovitz number on the evolution of the flame surface density in turbulent premixed flames
,”
Combust. Flame
152
,
194
205
(
2008
).
28.
C.
Dopazo
,
L.
Cifuentes
,
J.
Martin
, and
C.
Jimenez
, “
Strain rates normal to approaching iso-scalar surfaces in a turbulent premixed flame
,”
Combust. Flame
162
,
1729
1736
(
2015
).
29.
N.
Peters
,
Turbulent Combustion
(
Cambridge University Press
,
2000
).
30.
T.
Poinsot
and
D.
Veynante
,
Theoretical and Numerical Combustion
(
R.T. Edwards Inc
.,
Philadelphia
,
2001
).
31.
T.
Falkenstein
,
S.
Kang
,
L.
Cai
,
M.
Bode
, and
H.
Pitsch
, “
DNS study of the global heat release rate during early flame kernel development under engine conditions
,”
Combust. Flame
213
,
455
466
(
2020
).
32.
X.
Wang
, “
Direct numerical simulation of lean premixed turbulent flames at high Karlovitz numbers under elevated pressures
,” Ph.D. thesis (
University College London
,
London
,
2019
).
33.
E.
Inanc
,
A. M.
Kempf
, and
N.
Chakraborty
, “
Scalar gradient and flame propagation statistics of a flame-resolved laboratory-scale turbulent stratified burner simulation
,”
Combust. Flame
238
,
111917
(
2022
).
34.
M.
Reith
,
A.
Gruber
,
F. A.
Williams
, and
J. H.
Chen
, “
The effect of pressure on ammonia/hydrogen/nitrogen premixed flames in intense sheared turbulence, vol. 66, paper no: 17
,” in
74th Annual Meeting of the APS Division of Fluid Dynamics
,
2021
.