In the present work, direct numerical simulation (DNS) of a laboratory premixed turbulent jet flame was performed to study turbulence-flame interactions. The turbulent flame features moderate Reynolds number and high Karlovitz number (Ka). The orientations of the flame normal vector **n**, the vorticity vector **ω** and the principal strain rate eigenvectors **e**_{i} are examined. The in-plane and out-of-plane angles are introduced to quantify the vector orientations, which also measure the flame geometry and the vortical structures. A general observation is that the distributions of these angles are more isotropic downstream as the flame and the flow become more developed. The out-of-plane angle of the flame normal vector, *β*, is a key parameter in developing the correction of 2D measurements to estimate the corresponding 3D quantities. The DNS results show that the correction factor is unity at the inlet and approaches its theoretical value of an isotropic distribution downstream. The alignment characteristics of **n**, **ω** and **e**_{i}, which reflect the interactions of turbulence and flame, are also studied. Similar to a passive scalar gradient in non-reacting flows, the flame normal has a tendency to align with the most compressive strain rate, **e**_{3}, in the flame, indicating that turbulence contributes to the production of scalar gradient. The vorticity dynamics are examined via the vortex stretching term, which was found to be the predominant source of vorticity generation balanced by dissipation, in the enstrophy transport equation. It is found that although the vorticity preferentially aligns with the intermediate strain rate, **e**_{2}, the contribution of the most extensive strain rate, **e**_{1}, to vortex stretching is comparable with that of the intermediate strain rate, **e**_{2}. This is because the eigenvalue of the most extensive strain rate, *λ*_{1}, is always large and positive. It is confirmed that the vorticity vector is preferentially positioned along the flame tangential plane, contributing to the dominance of cylindrical curvature of the flame front. Finally, the effect of heat release on the turbulence-flame interactions is examined. It is found that heat release has only limited impact on the statistics due to the minor role played by the strain rate induced by heat release rate in the current high Ka flame.

## I. INTRODUCTION

The development of future premixed combustion technologies having higher efficiencies and lower NO_{x}, CO_{2}, and other emissions would be facilitated by the availability of more accurate predictive models. In turn, the development of these models may be aided by an improved fundamental understanding of flame-turbulence interactions. Among these interactions, in premixed combustion, the most fundamental effect of turbulence is to change the burning rate relative to a laminar flame. For example, the turbulent flame speed, *S _{T}*, may be expressed as

*S*=

_{T}*S*

_{L}I_{0}∫Σ

*dξ*,

^{1}where

*S*is the laminar flame speed,

_{L}*I*

_{0}is the Bray stretch factor, i.e., the ratio of the burning rate of the turbulent flame per unit surface area to that in the laminar flame,

^{2}Σ is the flame surface density, i.e., flame surface area per unit volume,

^{3}and

*ξ*is the normal direction of the mean flame brush. Here, ∫Σ

*dξ*may be interpreted as the ratio of the wrinkled flame surface area to the unwrinkled one, and this ratio can be much larger than unity. Both ∫Σ

*dξ*and

*I*

_{0}are closely related to the relative orientations of the flame front and the turbulent strain rate fields. Turbulence acts on the flame front to realign scalar gradients, resulting in wrinkling and stretching that cause the significant flame surface area to increase. The process by which this stretching occurs depends on the relative alignment of the flame with straining and vortical motions. In addition, turbulent stretch is also known to affect

*I*

_{0}, and this effect may also become quite significant for high stretch rate conditions.

The significance of flame alignment has led to a number of studies being carried out. To identify the flame surface, a scalar iso-surface is usually employed. The scalar gradient can then be used to compute the flame normal direction, enabling studies of flame orientations. Early studies were connected with the BML (Bray-Moss-Libby) model,^{2,4} in which an algebraic expression for the flame surface density based on a geometrical description was proposed which explicitly involved the flame orientation,

In the above, *g* is a model constant of order unity, *L _{y}* is the scalar integral length scale measured along the mean progress variable $(c\xaf)$ contour, and

*σ*

_{y}is the mean of a direction cosine defining the flamelet orientation relative to the $c\xaf$ contour.

^{2}This expression has been investigated experimentally in various premixed flame configurations, where flame orientations were determined.

^{5–8}Flame and strain rate orientations in non-premixed flames were also measured.

^{9–11}These studies suggested that the mean strain field modifies the turbulence field and affects the flame orientations. Therefore, Σ is also influenced.

Both Σ and *I*_{0} are strongly dependent on the stretch rate *κ* = (*dA*/*dt*)/*A*,^{3,12,13} which quantifies the rate of change in the flame surface element area due to the turbulence-flame interactions. The stretch rate includes the effects of both strain rate and curvature on the flame surface via the expression,^{3}

where *a _{n}* is the normal strain rate, ∇ ⋅

**u**the dilatation,

*S*the displacement velocity, and ∇ ⋅

_{d}**n**the curvature. The normal strain rate

*a*, given by

_{n}*n*, is therefore a key quantity in turbulence-flame interactions, where

_{i}S_{ij}n_{j}*n*is the ith component of the flame normal,

_{i}**n**, and

*S*is the strain rate tensor. Early studies of passive scalar mixing showed that the production of scalar gradients occurs when the gradient vector preferentially aligns with the most compressive principal strain rate.

_{ij}^{14–17}Recent experimental and direct numerical simulation (DNS) studies have also shed some light on the alignment between the flame normal and the strain rates in turbulent reacting flows. It was found that the above alignment is not altered in the presence of passive chemical reactions,

^{15,18}or for flames having sufficiently high Karlovitz number (Ka).

^{19–21}However, the most extensive strain rate has an increased tendency to be aligned with the flame normal in low Ka flames, because the flame-induced dilatation, which is aligned in the normal direction, is dominant over turbulent strain rate.

^{20,22,23}It is noted, however, that this effect does not necessarily lead to destruction of scalar gradients, since it is balanced by normal gradients of displacement speed.

^{24,25}

Turbulence is very complex in that it consists of vortical structures of various sizes, shapes, and orientations. These structures are often described in terms of the velocity gradient tensor, strain rate tensor, and vorticity vector.^{26–29} The orientations of the principal strain rates and the vorticity vector have been investigated in homogeneously sheared turbulent flows.^{17,30} The alignment between the principal strain rates and the vorticity vector has been suggested to be correlated with vortex stretching, which is a source for production of vorticity.^{31} Both DNS^{14–16,32–34} and experimental^{35} studies have shown that the vorticity preferentially aligns with the intermediate strain rate, which is consistent with the fact that the intermediate strain rate is more often positive than negative, and thus contributes to vortex stretching.

The interactions between turbulence and flame may also be characterised by the alignment between the scalar gradient (flame normal) and the vorticity vector. Earlier studies focused on the configuration of a passive scalar in non-reacting flows. Kerr^{14} found that the scalar gradient in the vicinity of the largest vortex tube is orthogonal to the vorticity. Later, the alignment between the flame normal and the vorticity was studied in turbulent premixed flames in isotropic turbulence.^{21,29} The general scenario is that the flame normal misaligns with the vorticity, which indicates that the vortical structures are preferentially tangential to the scalar iso-surface, where cylindrical iso-surfaces are favoured.^{18}

The previous DNS results, reviewed above, concerning the turbulence-flame interactions were based on simple configurations of freely propagating flames in isotropic turbulence, in which a mean shear layer was absent. However, it is expected that the existence of mean shear and non-trivial geometry has a significant impact on the flame and flow dynamics. This motivates the present work to study turbulence-flame interactions in DNS of a jet flame in which the mean shear effects are significant.

The present work employs a recently reported DNS database, which modelled a laboratory high Ka round jet flame measured by Zhou *et al.*^{36} Detailed comparisons between the DNS results and the experimental measurement were reported in the work of Wang *et al.*,^{37} and overall good agreement was obtained. In the present work, turbulence-flame interactions are investigated; in particular, the orientation and alignment statistics of the flame normal vector, vorticity, and principal strain rates are reported. As in spatially developing flows the characteristics of both the turbulence and flame depend on spatial locations, the manner in which they evolve spatially is examined. The sensitivity of the results to the selected scalar iso-surface is also analysed. The paper is organised as follows. Section II describes the computational approach, key parameters, and post-processing methods. Statistical results of the orientations and alignments of the flame normal vector, vorticity and principal strain rates, their implications to the flame and flow dynamics, and the effect of heat release are presented in Section III before drawing conclusions in Section IV.

## II. DESCRIPTION OF THE DNS

### A. Configuration

The DNS is described in full in the work of Wang *et al.*,^{37} so only a brief description is recounted here. A round jet burner of Zhou *et al.*^{36} was employed for the DNS. The central jet operates at 1 atm and 300 K. The jet bulk velocity is U_{b} = 110 m/s and the equivalence ratio of the CH_{4}/air mixture is 0.7. At these conditions, the laminar flame velocity *S _{L}* = 0.193 m/s, the flame thermal thickness

*δ*= 0.663 mm, and the flame time scale

_{L}*τ*=

_{L}*δ*

_{L}/

*S*

_{L}= 3.44 ms. The jet diameter is D = 1.5 mm and the jet Reynolds number based on U

_{b}and D is Re

_{j}= 10 510. A pilot flame of CH

_{4}/air mixture at an equivalence ratio of 0.9 was established surrounding the burner to provide a hot coflow and aid in stabilising the flame. The temperature of the coflow is 1800 K and the velocity is 1.8 m/s.

The mean jet axial velocity and scalar profiles at the inlet approximate to those of the experiment using a power law velocity profile for an axisymmetric fully developed turbulent pipe flow.^{38} A fluctuation velocity obtained by generating an auxiliary isotropic field based on a prescribed Passot-Pouquet energy spectrum^{39} is then added to the mean inlet velocity using Taylor’s hypothesis. The turbulent velocity is 4 m/s and the integral length scale is 0.75 mm, consistent with the laser Doppler anemometry (LDA) measurement of the experiment.^{36} The mean and root-mean-square axial velocity of the DNS was compared with the LDA measurements in the near-field (*x*/D = 4) with good agreement,^{40} confirming the validity of the inlet velocity profile and inflow turbulence. Ka at the inlet is evaluated as Ka = *τ _{L}*/

*τ*

_{η}and is 253, where

*τ*

_{η}is the Kolmogorov time scale. The Damköhler number (Da) defined as Da =

*τ*/

_{f}*τ*is 0.0545, where

_{L}*τ*is the turbulent time scale.

_{f}### B. Numerical methods

The physical domain for the simulation is large owing to the long flame length: L_{x} × L_{y} × L_{z} = 48D × 36D × 36D in the streamwise *x* and lateral directions *y* and *z*, respectively. The resolution of the DNS is chosen to adequately resolve both the flame and turbulence structures. A uniform grid spacing of Δ*x* = 30 μm is used in the streamwise direction *x*. The grid in the *y*(*z*) direction is uniform with Δ*y*(Δ*z*) = 30 μm in the region between *y*/D (*z*/D) = − 5 and 5 and gradually stretched outside of this region. Approximately 22 grid points across *δ _{L}* are obtained with this spatial resolution. In the present case, the spatial resolution is limited by turbulence. The Kolmogorov length scale defined as

*η*= (

*ν*

^{3}/

*ϵ*)

^{1/4}exhibits a minimum of 10 μm in a narrow region near the potential core, where the flame does not overlap (the distribution of Kolmogorov length scale was shown in the supplementary material of Ref. 37). The criteria,

*η*/Δ

*x*> 0.5,

^{41}are satisfied elsewhere. Thus, the smallest scales of the turbulent flow are reasonably resolved. The resultant number of grids is N

_{x}× N

_{y}× N

_{z}= 2400 × 900 × 900, i.e., in total ∼2 × 10

^{9}.

To flush out initialisation artefacts, the simulation was first advanced for 10*τ _{j}* on a half-resolved grid, where

*τ*is the flow through time defined as

_{j}*τ*= L

_{j}_{x}/U

_{b}. The results were then mapped to the fine grid. The solution was advanced for another 10

*τ*to provide stationary statistics. It was found that statistical results of the half-resolution run and the production run are very similar, confirming that the resolution is adequate.

_{j}The DNS code “S3D” was employed to solve the compressible transport equations for continuity, momenta, species mass fractions, and total energy.^{42} The code uses a fourth-order Runge-Kutta method^{43} and a skew-symmetric, eighth-order explicit finite difference spatial scheme. A tenth-order filter was applied to damp high-wave number oscillations. A previously published reduced chemical mechanism for premixed CH_{4}/air flames with NO_{x} chemistry derived from GRI-Mech 3.0 was employed.^{44} The mechanism contains 268 elementary reactions and 44 species, of which 16 species were identified as quasi-steady state species. The remaining 28 species were transported on the DNS grid. The mechanism has been validated against the detailed mechanism comprehensively.^{44} Constant Lewis numbers, determined from a fit to mixture-averaged transport properties in a premixed flame, were employed.

### C. Post-processing methods

As mentioned above, the simulation is performed in a Cartesian coordinate system (*x*, *y*, *z*). However, this coordinate system must be transformed into a cylindrical coordinate system (*x*, *r*, *θ*) for collecting the statistics, where *r* is the radial direction and *θ* is the azimuthal direction. The coordinates (*x*, *r*, *θ*) are related to (*x*, *y*, *z*) by

where arctan2 is the quadrant-corrected arctangent function with the two arguments *y* and *x*. By definition, *θ* is in the interval [0°, 360°). A schematic of the coordinate transformation is delineated in Fig. 1, where **δ**_{x}, **δ**_{y}, and **δ**_{z} are three mutually perpendicular unit vectors in Cartesian coordinates, and **δ**_{x}, **δ**_{r}, and **δ**_{θ} are the three unit vectors in the cylindrical coordinate.

During the post-processing, the position vector and other vectors at each grid point are converted from Cartesian to cylindrical coordinates. It is clear that a vector **φ** can be represented in either coordinate system, i.e., **φ** = *φ _{x}*

**δ**

_{x}+

*φ*

_{y}**δ**

_{y}+

*φ*

_{z}**δ**

_{z}or

**φ**=

*φ*

_{x}**δ**

_{x}+

*φ*

_{r}**δ**

_{r}+ φ

_{θ}

**δ**

_{θ}. The components of

**φ**in cylindrical coordinates are related to those in Cartesian coordinates by

## III. RESULTS AND DISCUSSIONS

### A. Flame orientation

Due to the existence of the shear layer and turbulent fluctuations, the instantaneous flame front is wrinkled. The orientations of the flame normal are analysed to quantify the wrinkling structures. The flame normal vector **n** is defined as

where *c* is the progress variable defined based on the scalar, O_{2} mass fraction, as

Here, *Y*_{O2,u} is the O_{2} mass fraction of the reactant at the inlet and *Y*_{O2,b} is the O_{2} mass fraction in the coflow. By definition, the flame normal vector points towards the reactants. The instantaneous flame front is defined as *c* = 0.8, which corresponds to the location of the maximum heat release rate in the unstrained laminar flame.

An example of the instantaneous flame front is shown in Fig. 2. The flame is long and the instantaneous flame length is about 34D. In cylindrical coordinates, the flame normal could be written as **n** = **n**(*n _{x}*,

*n*,

_{r}*n*

_{θ}), where

*n*,

_{x}*n*, and

_{r}*n*

_{θ}are the flame normal components in the axial, radial, and azimuthal directions, respectively. The distributions of

*n*,

_{x}*n*, and

_{r}*n*

_{θ}along the flame front are examined to study the flame orientations. Note that as the flame evolves significantly in the axial direction, the statistics are expected to be dependent on the axial distance where they are evaluated. In the present paper, three locations,

*x*/D = 4, 16, and 28, are selected to report the statistics as delineated in Fig. 2, which represents the upstream, intermediate, and downstream regions of the flame, respectively.

In a laminar cylindrical flame, one may expect that the flame normal points towards the central line; the distributions of *n _{i}* in this case are delta functions at

*n*= − 1,

_{r}*n*= 0, and

_{x}*n*

_{θ}= 0. Fig. 3 shows the probability density functions (PDFs) of the flame normal components at the three axial locations of the turbulent flame. It is readily observed the most probable value for

*n*is −1 at all the locations. Close to the inlet, the flame is laminar-like, and distributions of

_{r}*n*centred on the expected values for a laminar flame are observed. As the flame develops downstream, the distributions of

_{i}*n*become broader and more complex due to turbulence-flame interactions.

_{i}Although the PDFs of *n _{i}* provide a wealth of information regarding the flame normal, it is unclear how these components are coupled. In the following, two angles are introduced to further quantify the flame orientation, a schematic of which is shown in Fig. 4(a). The first one is the in-plane angle,

*α*, which is defined as

By definition, *α* is 0° when the projection of the flame normal in the *x* − *r* plane aligns with the positive direction of the radial axis (*n _{x}* = 0 and

*n*≠0), and it increases counter-clockwise from 0° to 360°. Four different quadrants may be identified based on the value of

_{r}*α*(I: 0° ≤

*α*< 90°; II: 90° ≤

*α*< 180°; III: 180° ≤

*α*< 270°; and IV: 270° ≤

*α*< 360°), as illustrated in Fig. 4(b). It is obvious that in a laminar cylindrical flame, the distribution of

*α*is a delta function at

*α*= 180°. In a high Ka flame, the instantaneous flame front is so corrugated that all the four quadrants can play a role.

The second angle is the out-of-plane angle, *β*, which is written as

By definition, *β* is 0° when *n*_{θ} = 1, and it increases from 0° to 180° when *n*_{θ} decreased from 1 to −1. *β* is 90° when the flame normal locates in the *x* − *r* plane (*n*_{θ} = 0).

Figures 5(a) and 5(b) show the PDFs of *α* and *β* conditioned on the instantaneous flame front, respectively. Also shown in the figures are the isotropic distributions of *α* and *β*. Assuming an isotropic distribution of the flame normal results in the following relationship:^{45,46}

It is evident that the distribution of *α* is very different from isotropy in the present jet flame configuration, and the deviation from isotropy is more evident in the upstream region. At *x*/D = 4, the most probable angle of *α* is about 180°, consistent with the result of a laminar cylindrical flame. The angles are confined in the two quadrants II and III. The scenario is delineated in the left half of Fig. 4(b). At *x*/D = 16 and 28, there are two peaks in the PDFs of *α*; the first is at 45°, and the second is near 225°. The distributions of *α* could be understood with the help of the right half of Fig. 4(b). As will be shown later, in the downstream region, the flame normal preferentially aligns with the most compressive strain rate of the flow, which orients roughly 45°/225° to the radial direction. Therefore, the most probable angle for *α* is expected to be 45° (225°) when the flame propagates outwards (inwards) from (to) the central line. It is evident that it is also possible for *α* to be larger than 270° in the downstream region, though the probability is comparably low. The PDFs of *β* indicate that the distributions of the out-of-plane angle are, however, close to being isotropic in the downstream region. The isotropy increases with increasing axial distance. Note that at the inlet the distribution of *β* is a delta function at *β* = 90°.

The joint PDFs of *α* and *β* at the three axial locations are shown in Fig. 6. It is readily seen that the two angles are poorly correlated, which implies that *α* and *β* are basically independent. Consequently, the joint PDF of *α* and *β* could be modelled using the product of their marginal PDFs, i.e., *p*(*α*, *β*) = *p*(*α*) *p*(*β*). Similar results were reported by Veynante *et al.*^{45} and Chen and Bilger.^{8}

The mean and fluctuating values of *α* and *β* conditioned on the flame front as a function of the axial distance are revealed in Fig. 7(a). The mean of *α* equals to 180° at the inlet and decreases slightly downstream, while the mean of *β* remains 90° throughout the entire domain. On the other hand, the fluctuating *α* and *β* are both small near the inlet, confirming a laminar-like flame front near the inlet. The fluctuating *α* increases progressively with axial distance and reaches 90° at the flame tip. At *x*/D = 10, the fluctuating *β* reaches its maximum, which is close to the theoretical value of 39.20° calculated as

where $\beta \xaf=90\xb0$ and the isotropic assumption (Eq. (12)) is used as the PDF of *β*.

As most experimental measurements are performed in two spatial dimensions, it is necessary to develop a theoretical correction of the 2D measurements to estimate the corresponding 3D quantities. This correlation for the flame surface density Σ is examined. The key parameter is the out-of-plane angle, *β*. If the iso-surface *c* = *c*^{∗} defines the flame surface, the measured 2D flame surface density is written as

where 〈 〉 denotes the ensemble average, |∇*c*|_{2} is the 2D surface density function in the *x* − *r* plane, and δ is the delta function. Similarly, the 3D flame surface density is written as

It was previously shown^{45,46} that Σ_{3} and Σ_{2} are related by

where 〈 〉_{s} denotes the surface average. The surface average of a quantity *q* is calculated as

If we assume that the out-of-plane angle, *β*, has an isotropic distribution, it is evident that Σ_{3}/Σ_{2} = 4/*π* = 1.27. The results of Σ_{3}/Σ_{2} as a function of the axial distance are shown in Fig. 7(b). The evolution of Σ_{3}/Σ_{2} indicates how turbulence-flame interactions, and therefore flame wrinkling structures, develop in the axial direction. It is seen that Σ_{3}/Σ_{2} is unity at the inlet, indicating that the flame is unwrinkled. As the flame develops, it is progressively wrinkled by turbulence and Σ_{3}/Σ_{2} increases accordingly. Near the flame tip, Σ_{3}/Σ_{2} approaches its maximum value of 1.25, which is closer, but slightly smaller, than the theoretical value of 1.27 for the isotropic distribution of *β*. It is also obvious that the increase of Σ_{3}/Σ_{2} is more significant before *x*/D = 10, after which the change of Σ_{3}/Σ_{2} slows down.

### B. Vorticity orientation

The orientations of the vorticity vector are examined in a similar way as for those of the flame normal vector. The vorticity vector is defined as $\omega \u02c6=\u2207\xd7u$. In the present paper, only the direction of vorticity is of interest; therefore, the normalised vorticity vector $\omega =\omega \u02c6/|\omega \u02c6|$ is used. The normalised vorticity in cylindrical coordinates may be written as **ω** = **ω**(*ω _{x}*,

*ω*,

_{r}*ω*

_{θ}), where

*ω*,

_{x}*ω*, and

_{r}*ω*

_{θ}are the axial, radial, and azimuthal components of the normalised vorticity, respectively. The vorticity vectors are interpolated to the instantaneous flame front where their statistics are evaluated. Fig. 8 represents the PDFs of the normalised vorticity vector components at the three axial locations. It is clear that the dominant component is the azimuthal one. At all locations the most probable value for

*ω*

_{θ}is unity, though the probability of finding

*ω*

_{θ}= 1 decreases with increasing axial distance.

To visualise the vortical structures of the turbulent flow better, instantaneous snapshots of the vorticity components in a typical *x* − *r* plane at *x*/*D* = 4–8 are shown in Fig. 9. Note that in the upstream region, the vorticity dynamics is closely related to the Kelvin-Helmholtz instability in the shear layer. It is clear that in the inner jet region, the three components of the vorticity vector are comparable. However, in the shear region, the azimuthal component is the most prominent. As the flame front also locates in the shear region, it is suggested that the vortical structures lie in the flame tangential plane, which will be discussed in more detail in Section III D.

For the vorticity vector, two angles could also be defined, i.e., the in-plane angle (Eq. (9)), *γ*, and the out-of-plane angle (Eq. (10)), *δ*. The PDFs of *γ* are shown in Fig. 10(a). It is interesting to see that there are two maximums for the PDFs of *γ*, the first is near 135° and the second is near 315°. Actually, these two angles indicate the same vortex axis, and they are consistent with the direction of the most extensive strain rate of the mean flow, as will be discussed in Sec. III C. Similar results were reported by Nomura and Elghobashi^{17} and Rogers and Moin^{30} in homogeneously sheared non-reacting flows. It is suggested that the preferred alignment between the 2D vorticity vector, **ω**_{2D} = **ω**_{2D} (*ω _{x}*,

*ω*), and the most extensive strain rate of the mean flow is related to vortex stretching by the mean shear.

_{r}^{30}Fig. 10(b) shows the PDFs of the out-of-plane angle,

*δ*. As expected, the most probable angle is small in the upstream region and increases with increasing axial distance. The isotropic distribution (Eq. (12)) is also superimposed. Although the distribution of

*δ*is more isotropic downstream, it is still considerably different from isotropy. The joint PDFs of

*γ*and

*δ*were also studied, although not shown for brevity. Again, no evident correlation between these two angles was observed.

Figure 11 represents the mean and fluctuating values of *γ* and *δ* conditioned on the flame front as a function of the axial distance. It is seen that the mean and fluctuating values of the in-plane angle, *γ*, remain relatively constant along the flame. In contrast, the mean and fluctuating values of the out-of-plane angle, *δ*, increase with increasing axial distance, confirming that the vortical structures are more isotropic in the downstream region, where the fluctuating *δ* approaches the theoretical value of 39.20°.

### C. Strain rate orientation

In turbulent flows, small structures may be described in terms of the strain rate tensor *S _{ij}*, which is defined as

where *u _{i}* is the

*i*th component of the instantaneous velocity.

*S*can be characterised by its principal eigenvalues

_{ij}*λ*

_{1},

*λ*

_{2}, and

*λ*

_{3}designated by the convention

*λ*

_{1}≥

*λ*

_{2}≥

*λ*

_{3}, which are determined from the following characteristic equation of

*S*:

_{ij} where *P*, *Q*, and *R* are the three invariants of *S _{ij}*,

The eigenvectors of *λ*_{1}, *λ*_{2}, and *λ*_{3} are **e**_{1}, **e**_{2}, and **e**_{3}, respectively. *λ _{i}* and

**e**

_{i}are interpolated to the instantaneous flame front and their statistics are evaluated.

Similarly, the strain rate tensor of the mean flow $S\xafij$ is defined as

where *U _{i}* is the

*i*th component of the Reynolds averaged velocity. Note that in a round jet configuration, the only non-zero components of the mean velocity are

*U*and

_{x}*U*, and the non-zero components of $S\xafij$ include $S\xafxx$, $S\xafxr$ ($S\xafrx$), and $S\xafrr$. Due to the existence of the strong shear layer, $S\xafxr$ ($S\xafrx$), which is negative and large, is the dominant component of $S\xafij$. The three eigenvectors of the mean strain rate tensor, i.e., $e\xaf1$, $e\xaf2$, and $e\xaf3$, could then be estimated. The angle between the eigenvector of the most extensive strain rate of the mean flow $e\xaf1$ and the radial axis is roughly 135°/315°, while the angle between the eigenvector of the most compressive strain rate of the mean flow $e\xaf3$ and the radial axis is roughly 45°/225°. Here, the angle is defined as 0° in the positive direction of the radial axis, and it increases counter-clockwise from 0° to 360° in the

_{r}*x*-

*r*plane. The intermediate strain rate of the mean flow $e\xaf2$ is expected to be perpendicular to the

*x*-

*r*plane.

Figure 12 shows the PDFs of the principal eigenvalues of the strain rate tensor, *λ _{i}*. It is evident that

*λ*

_{1}(

*λ*

_{3}) is always positive (negative). The most probable value for

*λ*

_{2}is zero. However, the distribution of

*λ*

_{2}is positively skewed and a majority of the samples have a positive

*λ*

_{2}, which is consistent with the previous DNS results for passive scalars.

^{15,17}It is also seen that the magnitude of the principal strain rates decreases with increasing axial distance, due to the decay of the jet.

The in-plane angles of **e**_{1}, **e**_{2}, and **e**_{3} based on Eq. (9) are determined and denoted as *μ*_{1}, *μ*_{2}, and *μ*_{3}, respectively. Meanwhile, their out-of-plane angles computed with Eq. (10) are denoted as *σ*_{1}, *σ*_{2}, and *σ*_{3}, respectively. The distributions of *μ _{i}* are studied as shown in Fig. 13. It is clear that the most probable angles for

*μ*

_{1}are 135° and 315°, indicating that

**e**

_{1}has a similar orientation with $e\xaf1$. This is because the flame is located in the shear layer, where the strain rate of the mean flow plays an important role in the total strain rate. On the other hand, the distribution of

*μ*

_{3}(Fig. 13(c)) shows that the most probable angles for

*μ*

_{3}are 45° and 225°, indicating that

**e**

_{3}preferentially aligns with $e\xaf3$. The probability for

**e**

_{1}(

**e**

_{3}) to align with $e\xaf1$ ($e\xaf3$) is the highest at

*x*/D = 4. In the downstream regions,

**e**

_{1}(

**e**

_{3}) still preferentially aligns with $e\xaf1$ ($e\xaf3$). However, the distribution of

*μ*

_{1}(

*μ*

_{3}) is more isotropic.

Figure 14 shows the PDFs of *σ _{i}*. The distributions of

*σ*

_{1}and

*σ*

_{3}are very similar. Particularly, the most probable angle is 90°, implying that

**e**

_{1}and

**e**

_{3}, most probably, are located in the

*x*-

*r*plane. The distributions of

*σ*

_{2}are more complex. Recall that for the intermediate strain rate of the mean flow $e\xaf2$, the out-of-plane angle can only be 0° or 180°. The distributions of

*σ*

_{2}indicate that at

*x*/D = 4, the orientation of

**e**

_{2}is close to $e\xaf2$. In the downstream region, the orientation of

**e**

_{2}is nearly isotropic.

### D. Alignment statistics

From the above analyses, it is obvious that there are correlations between the orientation of the vorticity **ω**, the flame normal **n**, and the strain rate eigenvectors **e**_{i}, reflecting the complex interactions between the turbulence and the flame. In this section, the turbulence-flame interactions are quantified by examining the alignment between **ω**, **n**, and **e**_{i}. The orientations of the flame normal/vorticity relative to the principal strain rate directions are characterised by the absolute value of the cosine of the angle between the flame normal/vorticity vector and each of the strain rate eigenvectors at each grid point in the domain, i.e., |**n** ⋅ **e**_{i}| and |**ω** ⋅ **e**_{i}|. The alignment between the vorticity and the flame normal is denoted as |**n** ⋅ **ω**|. Once again, the results are interpolated to the instantaneous flame front to report the statistics.

Figure 15 shows the PDFs of |**n** ⋅ **e**_{i}| at the three axial locations. It can be seen that **n** preferentially aligns with the most compressive strain rate, **e**_{3}, in the downstream region. Accordingly, there is a tendency for **n** to point away from **e**_{1} and **e**_{2}. The results are consistent with previous DNS results of turbulent flows with passive scalars or low Damköhler number flames^{14–21} and experimental measurements.^{22,47}

Notably, it is found that in the upstream region, e.g., *x*/D = 4, the PDFs of |**n** ⋅ **e**_{1}| and |**n** ⋅ **e**_{3}| are very similar, and the most probable angle between **n** and **e**_{1}/**e**_{3} is approximately 45°. This is because the upstream flame front is rarely influenced by the turbulent eddies and remains relatively laminar (Fig. 2). Based on the statistics of the orientations of the flame front and the principal strain rates, it is expected that the flame normal, **n**, lies in the middle of the two principal strain rate eigenvectors, **e**_{1} and **e**_{3}, resulting in the similar distributions of |**n** ⋅ **e**_{1}| and |**n** ⋅ **e**_{3}|.

The axial evolutions of the means of |**n** ⋅ **e**_{i}| conditioned on the flame front, 〈|**n** ⋅ **e**_{i}|〉, are further studied. Fig. 16 shows 〈|**n** ⋅ **e**_{i}|〉 as a function of the axial distance. It is confirmed that in the near-field, *x*/D < 6, 〈|**n** ⋅ **e**_{1}|〉 and 〈|**n** ⋅ **e**_{3}|〉 are very close. Further downstream, 〈|**n** ⋅ **e**_{1}|〉 (〈|**n** ⋅ **e**_{3}|〉) decreases (increases) with increasing axial distance. 〈|**n** ⋅ **e**_{2}|〉 is shown to play a minor role along the entire flame, although it also increases downstream.

The alignment characteristics of **n** and **e**_{i} indicate that the flame and turbulence dynamics and their interactions depend significantly on the axial distance. In particular, in the upstream region, both **n** and **e**_{i} have their respective preferred orientations and there is little influence of turbulence on the flame geometry. Therefore, the flame front remains laminar-like. As the flame and flow develop, the flame front behaves like a passive scalar surface, and it is progressively wrinkled by the turbulent flow. Accordingly, the flame front (flame normal) increasingly aligns with the most extensive (compressive) strain rate of the flow. It is suggested that the orientations of the flame normal are more isotropic downstream due to the increasing isotropy of the flow.

The response of the flame to fluid dynamic strain rate influences also the strain rate on the flame front, i.e., the tangential strain rate, *a _{t}*, which is an important quantity determining the flame structure

^{48}and other flame properties such as flame speed and flame thickness. The correlations of the fluid dynamic strain rate and the flame tangential strain rate,

*a*, are therefore examined. The evolutions of

_{t}*a*and ∇ ⋅

_{t}**u**conditioned on the flame front along the axial direction are shown in Fig. 17. It is clear that

*a*is small near the inlet and increases downstream. The behaviour of

_{t}*a*can be explained as follows.

_{t}*a*is written as

_{t} Fig. 17 shows that ∇ ⋅ **u** is small compared with *a _{n}* or

*a*in the high Ka condition. This can be explained by the scaling analysis,

_{t}^{20}

where Δ*ρ* is the density difference between the reactants and the products, and *ρ _{b}* is the density of the products. The small-scale strain rate is of the order 1/

*τ*

_{η},

^{49}where

*τ*

_{η}is the Kolmogorov time scale. Therefore, the ratio of ∇ ⋅

**u**and small-scale strain rate is

^{19,50}

where ε (ε = Δ*ρ*/*ρ _{b}*) is the heat release parameter, which is about 5 in the present flame. It is seen that the ratio decreases with Ka. In high Ka flames (Ka > 100), ∇ ⋅

**u**can be neglected and Eq. (24) can be recast as

As shown in Fig. 16, the magnitude of |**n** ⋅ **e**_{1}| is comparable to the magnitude of |**n** ⋅ **e**_{3}| in the near-field. *λ*_{1} and *λ*_{3} are also comparable but with opposite signs. Therefore, in the near-field, −*λ*_{2}|**n** ⋅ **e**_{2}|^{2} is the dominant term for *a _{t}*, which is small (Fig. 17). Further downstream,

**n**preferentially aligns with

**e**

_{3}. Consequently, |

**n**⋅

**e**

_{3}| overtakes |

**n**⋅

**e**

_{1}|, and −

*λ*

_{3}|

**n**⋅

**e**

_{3}|

^{2}plays a more important role. Similar behaviours of the alignment between

**n**and

**e**

_{i}were also reported in previous numerical and experimental flames.

^{9,51}

In the context of turbulent combustion models, such as the BML approach,^{52} the scalar dissipation rate, *N _{c}*, which characterises turbulent mixing, is an important quantity to be modeled. It is also needed in presumed PDF methods where the progress variable equation is solved,

^{53}in transported probability density function methods

^{54,55}and in conditional moment closure methods.

^{56}The turbulence-scalar interaction, signified by −2

*ρN*(

_{c}*n*), appears as a term in the transport equation for

_{i}S_{ij}n_{j}*N*.

_{c}^{57}The normal strain rate,

*a*(also denoted as

_{n}*n*), therefore, plays an important role in the turbulence-scalar interaction. The modelling of

_{i}S_{ij}n_{j}*N*is beyond the scope of the present study. However, from the above analyses, it is clear that the alignment of the flame normal and the principal strain rates influences

_{c}*N*. It has been shown that for low Ka flames, the flame normal preferentially aligns with the most extensive strain rate,

_{c}^{22,23}acting as sinks for

*N*. In contrast, the flame normal aligns with the most compressive strain rate in high Ka flames,

_{c}^{19–21}acting as sources for

*N*, as is the case in the present flame.

_{c}The enstrophy transport equation may write

where *ω* is the vorticity magnitude, *ρ* is the density, *P* is the pressure, and *τ _{kl}* is the viscous tensor. The terms on the LHS are the transient term and the convection term (

*T*

_{0}). The terms on the RHS correspond to vortex stretching (

*T*

_{1}), dilatation (

*T*

_{2}), baroclinic torque (

*T*

_{3}), and dissipation (

*T*

_{4}). As the flow is statistically stationary, on average, the transient term is expected to be zero. The means of other terms in Eq. (28) across the flame brush at

*x*/D = 16 are plotted in Fig. 18. Note that the low $c\xaf$ part is not plotted as there is no DNS data for the low $c\xaf$ limit in the downstream region. It is seen that vortex stretching is the predominant source of vorticity generation, which is balanced by viscous dissipation. The dilatation and the baroclinic torque terms are small compared with the vortex stretching and viscous dissipation terms, as expected given the large Karlovitz number of the present flame. The peak of the vorticity production/dissipation occurs near $c\xaf=0.4$.

The orientation of **ω** relative to **e**_{i} is important in that it determines the vortex stretching term, *ω _{i}ω_{j}*∂

*u*/∂

_{i}*x*(or

_{j}*ω*), and thus influences the structure and dynamics of the turbulent flows.

_{i}S_{ij}ω_{j}*ω*can be written as

_{i}S_{ij}ω_{j}^{34}

Figure 19 shows the PDFs of |**ω** ⋅ **e**_{i}| at the three axial locations. It is evident that **ω** has a tendency to align with the intermediate strain rate and misalign with the other two strain rates, which is consistent with other DNS results of various configurations for non-reacting, incompressible turbulent flows^{15,17,34} and turbulent premixed flames in isotropic turbulence.^{21} It seems the compressibility effect does not change the alignment significantly, which might be due to the minor role played by dilatation as shown in Fig. 17. It has been shown that **ω** can be weakly aligned with **e**_{1} in low Ka flames where the effect of dilatation is more significant.^{21} The axial evolutions of 〈|**ω** ⋅ **e**_{i}|〉 are shown in Fig. 20(a). It is clear that 〈|**ω** ⋅ **e**_{2}|〉 decreases while 〈|**ω** ⋅ **e**_{1}|〉 and 〈|**ω** ⋅ **e**_{3}|〉 increase with increasing axial distance before *x*/D = 10. In the downstream region, 〈|**ω** ⋅ **e**_{i}|〉 remains relatively constant. 〈|**ω** ⋅ **e**_{2}|〉 is confirmed to be the dominant term throughout the entire flame.

The implication of the alignment between **ω** and **e**_{i} for the vorticity dynamics is now discussed. As shown in Fig. 12, most of the intermediate strain rates are positive. Therefore, the preferred alignment between **ω** and **e**_{2} contributes to the positive values of *ω _{i}S_{ij}ω_{j}*, consistent with the vortex stretching mechanism. It is also possible for

**ω**to align with

**e**

_{1}, which always contributes to a positive value of

*ω*as

_{i}S_{ij}ω_{j}*λ*

_{1}is large and positive. The roles of the three terms on the RHS of Eq. (29) are quantified. Fig. 20(b) shows that vortex stretching is observed to be the highest near

*x*/D = 7. As expected, both 〈

*ω*

^{2}

*λ*

_{1}|

**ω**⋅

**e**

_{1}|

^{2}〉 and 〈

*ω*

^{2}

*λ*

_{2}|

**ω**⋅

**e**

_{2}|

^{2}〉 are positive while 〈

*ω*

^{2}

*λ*

_{3}|

**ω**⋅

**e**

_{3}|

^{2}〉 is negative. It is seen that 〈

*ω*

^{2}

*λ*

_{1}|

**ω**⋅

**e**

_{1}|

^{2}〉 and 〈

*ω*

^{2}

*λ*

_{2}|

**ω**⋅

**e**

_{2}|

^{2}〉 are comparable throughout the entire flame. Between

*x*/D = 10 and 20, 〈

*ω*

^{2}

*λ*

_{1}|

**ω**⋅

**e**

_{1}|

^{2}〉 is even larger than 〈

*ω*

^{2}

*λ*

_{2}|

**ω**⋅

**e**

_{2}|

^{2}〉. Similar results were found in DNS of isotropic turbulence.

^{34}

Finally, the alignment between the flame normal and the vorticity is examined. Fig. 21 shows the PDFs of |**n** ⋅ **ω**| at the three axial locations. It is readily observed that **n** misaligns with **ω**, indicating that the vorticity vector preferential locates along the flame tangential plane, which is consistent with the observations of Fig. 9. The dominance of cylindrical curvature of the flame front (not shown) can be explained by considering the preferred alignment of the flame front with the vorticity vector.^{18} The misalignment between **n** and **ω** is expected, also, as **n** preferentially aligns with the most compressive strain rate, **e**_{3} (Fig. 15) while **ω** aligns with the intermediate strain rate, **e**_{2} (Fig. 19). The evolution of 〈|**n** ⋅ **ω**|〉 along the axial direction is shown in Fig. 22. The misalignment between **n** and **ω** is the strongest in the near-field. As the flame develops, 〈|**n** ⋅ **ω**|〉 increases and approaches a value of 0.18 near the flame tip.

### E. Effects of heat release

The influence of Ka in turbulent premixed flames on the alignment between the flame normal and the principal strain rates has been discussed in Refs. 20, 21, and 58. In general, the flame normal aligns with the most extensive (compressive) strain rate in low (high) Ka flames. It was concluded that the strain rate induced by heat release, which always acts in the flame normal direction, plays an important role in the alignment in low Ka flames. Ka can also have impact on the alignment between the vorticity and the principal strain rates.^{21,59} In particular, it was demonstrated that the alignment of the vorticity with the principal strain rates in low Ka flames exhibits deviation from passive turbulent flow conditions. In the present work, a parametric DNS study of Ka was not possible due to the limitation of the computational resource. However, the effect of heat release on the turbulence-flame interactions can be evaluated.

The results presented so far are based on the instantaneous flame front which corresponds to the location of the maximum heat release rate. It is of interest to investigate the sensitivity of the turbulence-flame interactions to the location of analysis. To this end, another scalar iso-surface, *c* = 0.5, is extracted and the statistics conditioned on this surface are studied. Note that heat release rate at *c* = 0.5 is small, roughly 35% of the maximum heat release rate. A smaller value of *c* than 0.5 is not preferred, as the number of samples available on the surface decreases significantly with decreasing *c* in the round jet configuration.

Figures S1-S14 in the supplementary material show the statistics of the flame normal, vorticity and principal strain rate orientations and the alignment characteristics of these vectors conditioned on *c* = 0.5. It is clear that the results are quantitatively similar to those conditioned on *c* = 0.8. No evident difference in the orientations and alignments is observed on the two scalar iso-surfaces. As discussed earlier, the strain rate induced by heat release can be scaled as ε*S _{L}*/

*δ*, which is negligible compared with the strain rate induced by small-scale turbulence of the order 1/

_{L}*τ*

_{η}in high Ka flames. As a result, it is not surprising that heat release has only limited impact on the statistics of flame and flow dynamics and their interactions.

## IV. CONCLUSIONS

Three-dimensional DNS of a laboratory high Ka premixed round jet flame was analysed to study turbulence-flame interactions. The orientation statistics of the flame normal vector **n**, the vorticity vector **ω**, and the principal strain rate eigenvectors **e**_{i} were investigated, where the vectors are represented in a cylindrical coordinate system. Two angles were introduced to quantify the vector orientations, i.e., the in-plane angle and the out-of-plane angle. For the flame normal, it is found that there are two peaks in the distributions of the in-plane angle; the first is at 45°, and the second is near 225°, which correspond to alignment with the most compressive direction of the mean shear, the scenarios when the flame locally propagates outwards and inwards with respect to the centre line, respectively. The out-of-plane angle is used to develop the correction of the 2D measurements to estimate the corresponding 3D quantities. The DNS results show that Σ_{3}/Σ_{2} is unity at the inlet and approaches its theoretical value of 1.27 in the downstream. For the vorticity vector, the most probable values of the in-plane angle are 135° and 315°, which are consistent with the direction of the most extensive strain rate of the mean flow. The distribution of the out-of-plane angle is considerably different from isotropy throughout the entire flame, due to the dominance of the azimuthal vorticity component in the shear region. As for the strain rate eigenvectors, it is seen that **e**_{1} (**e**_{3}) has a similar orientation with $e\xaf1$ ($e\xaf3$), especially in the upstream region.

The alignment characteristics of **n**, **ω**, and **e**_{i}, reflecting the complex interactions of turbulence and flame dynamics, are examined with the absolute value of the cosine of the angle between two vectors. It is found that there is a tendency for **n** to point towards the most compressive strain rate, **e**_{3}, and away from **e**_{1} and **e**_{2} in the fully developed region, while the PDFs of |**n** ⋅ **e**_{1}| and |**n** ⋅ **e**_{3}| are very similar near the inlet. This explains why the tangential strain rate, *a _{t}*, is small near the inlet and increases downstream. The preferred alignment between

**n**and

**e**

_{3}also indicates the production of scalar gradient by turbulence. The vorticity,

**ω**, is observed to align with the intermediate strain rate,

**e**

_{2}, which contributes to vortex stretching, the predominant source of vorticity production balanced by dissipation. Detailed analyses show that although

**ω**preferentially aligns

**e**

_{2}, the contribution of

*ω*

^{2}

*λ*

_{1}|

**ω**⋅

**e**

_{1}|

^{2}to vortex stretching is comparable with that of

*ω*

^{2}

*λ*

_{2}|

**ω**⋅

**e**

_{2}|

^{2}. It is observed that

**n**misaligns with

**ω**, which is expected as

**n**preferentially aligns with

**e**

_{3}, and

**ω**with

**e**

_{2}. The dominance of cylindrical curvature of the flame front can be explained by considering the preferred alignment of the flame front with the vorticity vector.

Overall the results demonstrate that the mean flow geometry can have a significant, and indeed leading order, effect on turbulence-flame interactions. However, the effect of heat release on the interactions is minor in the current high Ka flame. Future work will be exploring predictive models describing turbulence-flame interactions in high Ka flames.

## SUPPLEMENTARY MATERIAL

See supplementary material for the statistics conditioned on the scalar iso-surface, *c* = 0.5.

## Acknowledgments

This work was supported by the Australian Research Council. This research used resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. The research was also supported by computational resources at Pawsey awarded through the National Computational Merit Allocation Scheme. The work at Sandia National Laboratories was supported by the Division of Chemical Sciences, Geosciences and Biosciences, the Office of Basic Energy Sciences, the US Department of Energy (DOE). Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy under Contract No. De-AC04-94-AL85000.