The effects of turbulence and pressure gradients on vorticity transport in premixed bluff-body flames

Vorticity transport describes the key fluid mechanical processes that govern the dynamics of turbulent reacting flows.^1–3 Turbulent reacting flows are experienced in propulsion and energy production technologies, where flame stabilization processes are dependent on the generation of vorticity and/or turbulent mixing between the reactants and products. Once reactions are stabilized, the turbulent combustion process is dominated by the continuous distortion, production, extension, and dissipation of the flame surface by vortices and turbulence at different scales.⁴ For premixed flames, large-scale turbulent eddies will wrinkle the surface of the flame and accelerate its propagation velocity,^5,6 while small-scale turbulence promotes preheat region broadening and increases the mean burning rate.⁷ Additionally, the presence of the flame results in combustion-induced thermal expansion, which drives local pressure perturbations and changes the behavior of the turbulent flow.⁸ Thus, the dynamics between turbulent flows and premixed flames are tightly coupled and remain a topic of significant interest.

A common approach to quantify turbulent flame-flow dynamics is to analyze the transport of vorticity across the premixed flame. For a constant dynamic viscosity (μ), the vorticity transport equation can be written as^7,9–11

The first term on the right-hand side of the equation accounts for the stretching and tilting of vorticity due to flow nonuniformities. The second term accounts for the annihilation or generation of vorticity by volumetric dilatation. The third and fourth terms represent the torques generated by pressure and viscous forces, respectively. The fifth and final term is the diffusion of vorticity from viscosity. For constant density non-reacting flows, only the vortex stretching/tilting and viscous diffusion terms exist on the right-hand side of the equation. For reacting flows, combustion heat release induces gaseous expansion and a non-uniform flow density, which induces the vorticity terms of dilatation, baroclinic torque, and viscous torque. For premixed bluff-body flame configurations, dilatation and viscous torque both act as a vorticity sink in the reacting domain, while baroclinic torque is a flame-generated vorticity component that induces flame-scale turbulence production at scales O(l_f), where l_f is the laminar flame thermal thickness.^12,13 This subsequently influences the volumetric burning rate and turbulence downstream of the flame.^8,14 As such, each term in the vorticity transport equation has a specific contribution to the behavior of the turbulent reacting flow. Thus, decomposing the individual vorticity terms provides better physical insights into the behavior of turbulent reacting flows and offers additional knowledge for predicting and modeling efforts.^4,15,16

Previous researchers have evaluated the influence of individual vorticity terms on reacting flow dynamics, with particular emphasis on the combustion-induced terms of baroclinic torque and dilatation.^17,18 Baroclinic torque arises due to the misalignment of pressure and density gradients along the flame front and acts as a source of vorticity generation for premixed bluff-body flames.^19–21 The influence of baroclinic torque on a reacting flow field has been examined numerically and experimentally^18,22,23 and will augment or suppress the existing vorticity in the domain through spatial superposition. The vorticity generated from baroclinic torque subsequently affects the flame topology, wrinkling characteristics, velocity fields, turbulence, and combustion heat release.^22,24 The dilatation vorticity term has similarly been analyzed for reacting bluff-body flows. Dilatation results from combustion heat release, which causes flow expansion as reactants are consumed.¹⁹ The magnitude of dilatation is directly dependent on the density ratio between the reactants and products.²¹ The dilatation term has also been shown to influence the shear layer dynamics in the bluff-body wake region and helps stabilize the flow field by suppressing the periodic asymmetric vortex shedding instability [the Bénard–von Kármán (BVK) instability] that occurs in nonreacting flows around a bluff-body.^18,25

Recent numerical studies have focused on quantifying the relative balance of the vorticity mechanisms under various turbulence conditions.^15,17,26,27 Chakraborty²⁷ conducted a study using a direct numerical simulation (DNS) database to analyze the vorticity vector alignment of planar premixed flames at various turbulence conditions, which were quantified with the Karlovitz number,

where τ_chem is the chemical timescale and τ_flow is the Kolmogorov time micro-scale. For experimental measurements, the Karlovitz number is typically quantified using integral timescales and flame scales. The formulation is included in Eq. (2) where l_f is the laminar flame thickness, L₁₁ are the integral length scales, $urms′$ are the turbulent velocity fluctuations, and S_L is the laminar flame speed. For Karlovitz numbers of Ka = 0.54 and 13.17, the study reported that the mean vortex stretching contribution remained positive under reacting conditions (i.e., the vortex stretching increases the magnitude of vorticity in the reacting domain). In a similar DNS study, Lipatnikov et al.⁷ characterized the relative balance of the vortex stretching, baroclinic torque, and dilatation mechanisms for flames with Ka ≪ 1. The data indicated that baroclinicity and dilatation are significant in the conservation of vorticity and enstrophy under low turbulence (or Ka) conditions. Increasing the turbulence intensity will impact the relative balance of the vorticity mechanisms, also known as the vorticity budget.^15,17 Specifically, the contributions of baroclinic torque and dilatation are diminished at high turbulence/Ka conditions, while vortex stretching increases significantly. This suggests that there is a fundamental change in the relative balance between the vorticity mechanisms across the combustion regimes.¹⁷ Additional work from Bobbit et al.¹⁵ considered turbulent flames with Ka > 20, with the inclusion of the viscous diffusion term. The study demonstrated that vortex stretching and viscous diffusion are the dominant vorticity terms at high turbulence conditions.

The existing literature has provided valuable insights regarding the influence of vorticity on turbulent premixed flames. However, most of these studies are numerical, specifically DNS studies with flames embedded in a box of homogeneous isotropic turbulence, and there is a lack of experimental investigations to validate the computational results. Thus, the current research seeks to experimentally quantify the vorticity transport terms and understand their respective impact on a reacting bluff-body flow field. To accomplish this, a high-speed combustion facility is paired with high-speed particle image velocimetry (PIV) and CH^* chemiluminescence diagnostics to experimentally analyze vorticity transport. The novelty of the research is that the turbulence conditions and stream-wise pressure gradient can be modified, thereby allowing for the investigation of vorticity and turbulence transport across premixed flames within an environment similar to propulsion and energy production technologies. The analysis is conducted using a mean, conditionally filtered, fluid element tracking method²³ to quantify vorticity transport across the premixed flame. The results demonstrate that vorticity transport is augmented as the magnitude of the pressure gradient is increased; however, the vorticity balance is more sensitive to turbulence effects.

The turbulent reacting flow is analyzed in the high-speed combustion facility shown in Fig. 1(a). The premixed fuel/air mixture flows through the nozzle and enters the optical test section. The flow from the test section is exhausted to atmospheric conditions. The test section is rectangular with a cross-sectional area of 127 × 45 mm² (depth × height). The test section includes a ballistic bluff-body, mounted in the channel centerline, to stabilize flames. The bluff-body and dimensions are provided in Fig. 1(b). The height of the bluff-body is H = 16 mm, resulting in a blockage ratio of 35%. The bluff-body geometry shown in Fig. 1(b) also extends across the full depth of the test section. Downstream of the bluff-body is the optical viewing region. The side and bottom walls of the viewing region are recessed and fitted with fused quartz windows; the windows provide 95% transmissibility for wavelengths between 250 nm and 700 nm.

To modify the combustor pressure gradient, the upper and lower walls of the viewing region can be modulated using ±3° angled wedges. This allows for three geometric configurations to be explored: a diffuser, nominal, and nozzle configuration, as shown in Fig. 2. The ±3° wall angles provide a maximum change in the mean axial pressure gradient without choking the flow for the nozzle or inducing flow separation along the walls for the diffuser. To quantify the stream-wise pressure gradient for each configuration, pressure taps are installed along the axial length of the top wall.²³ The static pressure is measured at each location using a Scanivalve (PDCR23D) pressure transducer system. At each location, 3000 pressure samples are recorded at a frequency of 2500 Hz, and the data are averaged to obtain a single pressure measurement at each location. The nonreacting (constant density) and reacting stoichiometric pressure profiles are plotted in Fig. 2 as C_P = P/(0.5ρU_o²). For the nonreacting conditions, the nozzle accelerates the flow in the axial (x) direction and results in a favorable pressure gradient, whereas the diffuser decelerates the flow and results in an adverse pressure gradient. Under reacting conditions, all pressure profiles depict a favorable axial pressure gradient, where the magnitude is driven from the combination of the variable geometry of the walls and the effects of combustion, which decreases the fluid density and accelerates the flow. Within the region of interest (5.5 ≤ x/H ≤ 8), the mean axial pressure gradient $(dP∕dx¯)$ for the diffuser, nominal, and nozzle configurations are −1.8 kPa/m, −4.0 kPa/m, and −7.6 kPa/m, respectively. The uncertainty in the pressure measurements is ±8 Pa based on equipment specifications, and the corresponding uncertainty in the axial pressure gradient is ±0.46 Pa/m.

Each geometric configuration in Fig. 2 is subjected to multiple turbulence conditions using a custom-designed turbulence generator, which can be installed 64 mm upstream from the bluff-body leading edge; a schematic of the turbulence generator is shown in Fig. 3(a). Turbulence is induced using a combination of a passive grid and premixed impinging jets, which allows the turbulence to be dynamically controlled. The design consists of 12 stainless-steel tubes, each with an outer and inner diameter of 4.4 mm and 4.0 mm, respectively. The tubes are oriented in a rectangular grid with a mesh size of 15.9 × 11.3 mm², yielding a blockage ratio of 59.6%. To allow for fluid jet impingement, injection orifices are drilled along the length of each tube with a diameter of 0.8 mm. The tubes are supplied with a premixed fuel/air mixture described in Sec. II B. The orifices are oriented orthogonal to the bulk flow and evenly spaced across the horizontal and vertical directions. In the current study, three turbulence conditions are considered using three configurations: first, without the turbulence generator installed, second, with the turbulence generator installed passively, and third, with the turbulence generator installed and the impinging jets choked at 35 psi. The choked jets provide under-expanded impinging sonic jet plumes, which are visualized with schlieren imaging in Fig. 3(b). Additional details of the turbulent test conditions are quantified and characterized in Sec. II D.

Flow into the test section [Fig. 1(a)] is supplied with 300 K air, which is regulated from 300 psi to 120 psi using a Fisher 546 Electro-Pneumatic valve. A Preso CV-300-65 Venturi flowmeter and Dwyer Series 626 pressure transducer are used to measure the main air flowrate, which is controlled using a DM4500 series JFlow valve operated via the pneumatic SR actuator with a rotary positioner. The flow control system is operated using a feedback loop, integrated with the LabVIEW software. This allows for desired flowrates to be maintained within the combustion test section.

The fuel and PIV seed particles are independently injected far upstream from the combustor (∼2.5 m upstream of the test section). Gaseous propane is regulated in-line and measured using a Dwyer VFC-121 rotameter. Similarly, the PIV seed particles (1 μm diameter Al₂O₃) are injected using a swirl seeder supplied by an external air source. The supply air is regulated in-line, and flow into the seeder is measured with a Dwyer RMC-122 rotameter. The total uncertainty in the propane/air flow supplied into the combustor test section is 5 × 10⁻⁴ kg/s.

For high turbulence studies, the turbulence generator tubes are supplied with a propane/air mixture equivalent to that of the main flow. Supply air is delivered from a constant 300 psi source, the flowrate of which is measured with a Dwyer VFC-123 rotameter. Propane is regulated in-line and injected into the air supply line, the flowrate of which is measured using a Key Instruments (FR4A67SVVT) rotameter.

Simultaneous high-speed particle imaging velocimetry (PIV) and CH^* chemiluminescence are used to obtain flow field measurements and evaluate the location of the flame front. A BNC model 575 pulse/delay generator is used to synchronize the components of the optical diagnostic system. The 1 μm aluminum oxide seed particles (Al₂O₃) are illuminated with a 532 nm, dual-cavity, diode pumped, solid state ND:YAG laser. The laser is pulsed in a frame-straddling mode at a frequency of 10 000 Hz with a time differential of 15 µs between the PIV images. PIV particle motion is captured using a high-speed CMOS camera (Photron Fastcam SA-Z 2100 K) pulsed in synchrony with the PIV lasers. The PIV camera is fitted with a Nikon1 50 mm f^# 1.2 focal length lens. The field-of-view resulting from this setup is 45 × 45 mm captured on a camera sensor composed of 1024 × 1024 pixel². The images are acquired using the Photron Fastcam computer software, and velocity vector fields are obtained using the commercial software (LaVision DaVis 10). The images are pre-processed using a contrast-limited adaptive histogram equalization (CLAHE) filter to mitigate image noise. A multi-pass cross-correlation method is used to obtain vector fields, with a final interrogation region of 16 × 16 pixel² and a 50% region overlap for each pass. The final interrogation box size is 703 μm, and the PIV vector spacing is λ_m = 359 μm, which provides well-resolved measurements with respect to the laminar flame thickness²⁸ and Kolmogorov scales,^14,29,30 as shown in Table I. Among all test conditions, the maximum Stokes number is St = (ωd_p/ν)^1/2 = 0.02, the particle Reynolds number is Re_p = 0.05, and the Stokes drag coefficient is C_D = 24/Re_p = 480.³¹ The bias error based on the particle image diameter is d_τ ∼ 1 pixel,³² and the correlation-based uncertainty in velocity fields is nominally 0.3 m/s.

Additionally, independent, high-resolution stereoscopic PIV measurements (SPIV) are used to validate velocity fields, ensure spatial velocity gradients are accurate, and provide information about the velocity component in the third dimension. The SPIV system included two high-speed CMOS cameras (Photron SA-Z) on opposite sides of the facility.³³ The cameras are oriented at 22.5° (measured from the perpendicular to the laser plane) and are equipped with 180 mm focal length lenses and scheimpflug adaptors to mitigate blurring from off-axis imaging. The field-of-view for the SPIV is small and nested within the upper half of the PIV domain, covering a spatial domain of 25 × 25 mm² between 6 ≤ x/H ≤ 7.5. Images are collected at 20 000 Hz onto a sensor comprised of 1024 × 1024 pixel², giving a pixel resolution of 24.4 μm/pixel. The SPIV data are processed in commercial software (LaVision DaVis 10) with a multi-pass cross-correlation algorithm and a 50% overlap for all passes. The final interrogation window is 16 × 16 pixel², resulting in a 390 μm interrogation box and a vector spacing of 195 μm, or λ_m/l_f = 0.59. The statistical correlation uncertainty in the SPIV velocity fields is nominally 0.05 m/s.

The instantaneous location of the flame front was determined from the PIV images and verified with the CH^* data.^33,34 The PIV flame boundary is extracted using the seed-particle density jump from the Mie scatter. The step-change in seed density approximates the interface between the reactants and products from the flames heat release and has been used across a variety of applications to obtain turbulent flame contours.^35–38 The advantage of using the PIV images is that the flame front coordinates are extracted from the same plane as the velocity measurements. In contrast, the CH^* chemiluminescence is a line-of-sight technique where the emitted light from the chemical reaction is path integrated across 3D space. Hence, the PIV images provide a better local representation of the flame front, while the CH^* images provide a global perspective of heat release oscillations.

The flame front extraction process is outlined in Fig. 4. An example of a PIV image is shown in Fig. 4(a). The flame boundary is obtained by first applying a median filter (10 × 10 window size) to eliminate background noise and blur individual particles within the image [Fig. 4(b)]. Next, the intensity of the image is inverted to enhance the contrast between the reactant and product regions. The image is then binarized using Otsu’s method [Fig. 4(c)], which applies automated cluster based thresholding.³⁹ Canny edge detection is then used to trace the binarized image and obtain instantaneous flame coordinates.⁴⁰ 

The instantaneous location of the flame front is verified using the CH^* images.³³ The CH^* images are recorded at 10 000 Hz using a Photron Fastcam SA1.1 675K (CMOS image sensor) fitted with a Nikon 50 mm f^# 1.2 lens. The resulting field-of-view is 45 × 90 mm² collected on a sensor comprised of 576 × 1024 pixel². The data result in a CH^* pixel resolution of 78-μm/pixel. When comparing flame front contours obtained from the CH^* images and PIV images, the root mean square difference of the flame location is nominally 52 μm, with a maximum difference between 1 and 2 laminar flame thickness. The agreement between the measurement methods provides confidence in the location of the PIV flame boundary.

For the current study, each geometric configuration in Fig. 2 is subjected to the turbulence conditions (A–F) provided in Table II. The turbulent fluctuating velocity (u_turb) is calculated using the stream-wise and cross-stream velocity fluctuations from the freestream portions of the domain (outside of the flame boundary). For this experimental configuration, the span-wise velocity and its fluctuations are not included in the analysis. Both quantities were assessed from the SPIV data, and the relative contributions, with respect to the dominant stream-wise component, are provided in Table III for each turbulence condition. At maximum turbulence conditions, the span-wise (w) velocity is less than 6% of the stream-wise velocity magnitude, and the span-wise turbulence fluctuations are less than 7% of the stream-wise fluctuations. This is due to the nature of the experiment; the bluff-body extends the full depth of the test section, and the test section is confined with only one downstream outlet. Thus, variations in the velocity and vorticity fields are minimal in the span-wise direction. To provide additional confidence, tomographic PIV measurements were acquired to assess three-dimensional velocity fields and estimate 3D spatial gradients and other vorticity components. Although the experiment is optically accessible, it is not an optimal design for 3D measurements and full volumetric flow reconstructions. Thus, the tomographic imaging domain used for validation does not perfectly align with the measurement plane and is not used for analysis. Nonetheless, the tomographic velocity data acquired from the freestream are included in Table III to provide general insights into 3D effects. For the turbulence conditions in Table II, spectral analysis has confirmed that the u_turb spectra closely follows Kolmogorov’s (−5/3) decay law. An example of turbulence spectra taken from the high turbulence condition is provided in Fig. 5. The turbulent velocity fluctuations in Table II are normalized by the laminar flame speed to obtain the turbulence intensity (u_turb/S_L). The integral length scales (L₁₁) and Kolmogorov length scales (η) are also included in Table II. The integral length scale is computed using the two-point velocity autocorrelation, and the Re and Ka numbers are calculated using formulations provided by Driscoll¹⁴ and Skiba et al.²⁹ It is important to note that the Reynolds number (Re) is calculated as (u_turbL₁₁)/(l_FS_L), which captures the ratio of turbulence scales to flame scales. The turbulent Reynolds number Re_T = u_turbL₁₁/ν is 685–790 under the current conditions. For all conditions explored here, the Lewis number is nominally constant: Le ∼ 1.7 and 1.8 for Φ = 1 and 0.7, respectively.⁴¹ The test conditions for this study are also plotted on the premixed combustion regime diagram in Fig. 6.^30,42 It is noteworthy that there are two different Karlovitz boundaries that separate the premixed combustion regimes: Ka = 1 and Ka_δ = 1; the former is calculated with respect to the laminar flame thickness, l_f²/η² (which includes the preheat region and reaction zone), and the latter is calculated with respect to the reaction zone thickness, l_δ²/η².

The current study seeks to evaluate flame-vortex dynamics for turbulent premixed flames. The reacting vortex dynamics are investigated by determining the relative magnitudes of the vorticity transport terms. The flow field vorticity is generally computed across the Eulerian PIV grid (which is presented in Sec. IV). However, the spatial gradients of PIV tend to be noisy, and an Eulerian-based calculation of the vorticity transport terms would not likely yield accurate results.^43–45 Therefore, a mean, conditionally filtered fluid element tracking method is employed to quantify the vorticity terms.²³ The method involves tracking theoretical fluid elements (TFEs) in space and time; however, the data are conditioned to only include the information from fluid elements that reside within the 2D laser plane and also filtered to only include elements that interact with the flame. This method is advantageous as it provides grid-free derivatives and sufficient spatial gradients within the reacting flow.

The tracking technique is initiated by inserting theoretical fluid elements (TFEs) onto a computational grid equivalent to the PIV vector grid. The tracking technique is based on a Lagrangian approach, where fluid elements are tracked using

where X(t) is the instantaneous position of a TFE and V(X(t), t) is the corresponding velocity at time t. This same tracking technique has been implemented in previous research; however, fluid elements were tracked backward in time to increase robustness and preserve desired quantities.^43,44 The current study employs the same technique, where fluid elements are inserted into the computational grid and tracked backward in time using Eq. (3).

To improve the accuracy of the TFE tracking, four additional time steps are added between the PIV velocity fields using a cubic-spline interpolation method, which is optimal for turbulent flows.⁴⁶ The resulting time resolution for fluid element tracking is 20 µs and is used to document the particles position and velocity; this time resolution is slightly smaller than the Kolmogorov timescales at the highest turbulence condition (≈21 µs)^29,30 and is adequate for particle sampling. Additionally, TFE displacement is documented on a scale of ≈36 μm, which is one-tenth of the PIV vector spacing. If a TFE falls between PIV grid spaces, the local velocity is determined from a cubic-spline interpolation from neighboring vectors. To ensure that the particle trajectories are accurate, the same backward-tracking technique is conducted using the raw PIV data without the added time steps. The deviation of the TFE position and velocity were less than 0.5% between the two methods, which ensures that the interpolation technique provides sufficient accuracy. The position and velocity data obtained from the fluid element tracking are used to calculate the vorticity terms in Eq. (1), and all spatial gradients (∂V_i/∂X_j and $∂2Vi/∂Xj2$⁠) are calculated using Lagrangian derivatives along the TFE trajectories. The detailed formulations used in this study are provided in Secs. III B and III C.

Dilatation is induced from combustion heat release, which occurs as reactants are burned.^18,25 This causes flow expansion, which results in a density gradient across the flame.²¹ The computational formulation for dilatation is given by

where x, y, u, and v are the stream-wise and cross-stream position and velocity vector components and ω_z is the span-wise vorticity component. Conditioning the TFEs within the 2D plane filters-out all spatial gradients in the third dimension. Thus, contributions from ω_x, ω_y, and ∂w/∂z are not included in the dilatation calculation and have minor contributions as quantified in Table III. It is also noted that any additional vorticity, induced, for example, by spatial variations of the local flame displacement velocity vector, is not included in the calculations and is usually not captured or quantified from PIV measurements.

Baroclinic torque is a flame-generated vorticity component, which arises from density gradients across the flame thickness and pressure gradients within the combustor; the misalignment of these gradients generates vorticity.^47,48 The formulation for baroclinic torque is

where the pressure gradient is replaced with the material acceleration, DV/Dt. The material acceleration includes Du/Dt and Dv/Dt terms to account for the local TFE pressure gradient in the x- and y-directions, respectively. Note that this pressure gradient includes both the induced pressure gradient from flow confinement and the local pressure gradient induced by the flame. To verify that DV/Dt ∼ ∇P, the Du/Dt term is compared to $dP∕dx¯$ in Fig. 7(a). Here, $dP∕dx¯$ comes from the static pressure measurements shown in Fig. 2(e), while Du/Dt is calculated along the fluid element trajectories. As shown, the material acceleration closely matches the pressure gradient within the root mean square bands, providing confidence in the substitution. The density gradient in Eq. (5) is computed by applying conditional-mean density profiles provided by Bobbitt et al.¹⁵ The conditional-mean density profiles are applicable for all turbulence conditions explored here and are used to calculate the density gradient across the flame. An example of the density profile, as well as the corresponding gradient in the progress variable space (⁠$ĉ$⁠, within the flame thickness), is provided in Fig. 7(b). The exact gradient is dependent on the density ratio between the reactants and products, as well as the turbulent flame thickness (δ_T). In this study, the density ratio is acquired from the work of Tang et al.²⁸ and Vagelopoulos et al.,⁴⁹ and the turbulent flame thickness is computed from the laminar flame thickness, l_F, using correlations provided by Driscoll.¹⁴ The density gradient is calculated normal to the flame front and then decomposed into its respective x and y components, as depicted in Fig. 7(c).

Viscosity effects are captured by the terms in Eq. (6). The first term on the right-hand side is the viscous diffusion, which accounts for dissipation of vortical structures from viscosity. For the current study, the kinematic viscosity across the flame is assessed using the conditional mean viscosity profiles provided by Bobbitt et al.¹⁵ The second term on the right-hand side accounts for the torque induced by viscous forces,

The final term of the vorticity transport equation is the vortex stretching/tilting, which acts to redistribute vorticity across the three dimensions of the domain.¹⁷ This vorticity mechanism transfers energy between large and small-scale structures in turbulent flows and is inherently three-dimensional.^17,50,51 For this study, the vortex stretching/tilting term is not explicitly determined from the data; it is computed by inverting the vorticity equation, as presented in Eq. (7). For a single component of vorticity, the stretching and tilting components can be isolated, as shown in Eq. (8). Using the maximum ∂w/∂z term from Table III, the vortex stretching term is calculated and compared to the total term calculated from Eq. (7). The results indicate that the vortex stretching contributes a maximum of 0.026% to the total S · ω; therefore, the S · ω term is dominated by vortex tilting. It is also noted that the uncertainty in the vortex stretching/tilting term will include error from all other terms; however, the maximum uncertainty in the vortex stretching/tilting term was 15.1% among all cases considered,

The accuracy of any computed quantity from a Lagrangian-based tracking scheme is sensitive to the number of fluid elements considered.⁴⁵ In order to minimize random errors from single element tracks and ensure that all calculated quantities are accurate, a mean convergence sensitivity analysis is conducted by varying the total number of fluid elements (N) conditioned in the plane. To do so, all fluid elements are temporally aligned first such that t = 0 is where each element first interacts with the flame. An example is provided in Fig. 8, where five fluid elements are tracked and aligned to evaluate the average dilatation for the set of elements. In Fig. 8, the time axis is normalized by the PIV acquisition time, t₀ = 100 µs. The effect of adding fluid elements to mean quantities are presented in Fig. 9(a). Here, the velocity, vorticity, and baroclinic torque terms are evaluated from a bundled average of N elements. Maximum convergence is observed for test cases with N ≥ 100, with all properties shown in Fig. 9(a) differing by less than 4% when N = 100. Increasing the number of fluid elements beyond 100 results in increased computational time with minimal improvements to property convergence. Averaging at least 100 elements minimizes random errors associated with out of plane motion and provides a better statistical representation of relevant quantities. As such, all the results are presented as a bundled average of at least 100 elements, which cross the planar flame front. An example of the conditionally filtered fluid element tracking is provided in Fig. 9(b), where 100 elements evolve from reactants to products.

The vorticity transport across the turbulent flame is analyzed for the various pressure gradient and turbulence conditions. The analysis focuses on the axial domain between 5.5 ≤ x/H ≤ 8, which is downstream of the closeout region^24,36 and free from the effects of flow recirculation.²³ Additionally, the analysis is carefully conducted to only include flame segments away from the walls, so boundary layer effects are not considered. Each of the geometric configurations in Fig. 2 is analyzed for the different turbulence conditions outlined in Table II. The results show the impacts of turbulence (u_turb) and pressure gradients on vorticity transport in premixed flames, as well as the subsequent impact on the evolution of the flame front. Throughout the results, the terms “augmented” or “increasing” pressure gradient refers to the increasing magnitude of $dP∕dx¯$ (the true value decreases and becomes more negative). Likewise, the turbulence conditions are commonly presented as a turbulence intensity of u_turb/S_L.

Instantaneous velocity and vorticity fields are shown in Figs. 10 and 11 for stoichiometric combustion. For all cases considered, the heat release from the flame will induce a volumetric expansion of the flow and drive an increase in flow velocity. However, the extent to which the flow is accelerated is dependent on both the inflowing turbulence and the induced pressure gradient. For instance, increasing the inflowing freestream turbulence intensity will augment the turbulent flame speed. This allows the flame brush to extend farther in the transverse (y) direction, occupy a greater portion of the domain, and entrain more reactants. Consequently, there is a greater portion of low-density products in the domain; thus, the augmented flow velocity is observed with elevated turbulence. Notably, the velocity also increases with dP/dx; however, this is purely driven from flow confinement.

Accelerating the flow velocity will lead to enhanced spatial gradients in the flow field. Higher spatial gradients will enhance the turbulent velocity fluctuations and lead to higher vorticity magnitudes in the flow field, as depicted in Fig. 11. At low turbulence conditions, spatial gradients are minimal, and the vorticity field is tightly wrapped around the flame brush. An increase in either the turbulence or the pressure gradients leads to larger spatial gradients and contributes to a broad distribution of high-magnitude vorticity across the domain. To explore this further, the evolution of turbulence across the flame is assessed using the TFE tracking method as fluid elements are convected through the flame, and the results are provided in Fig. 12. In essence, the profiles capture the effect of turbulent eddies convecting through a premixed flame. For all conditions, the turbulent eddy velocity scales are elevated as they propagate through the flame due to thermochemically induced volumetric expansion and resulting stream-wise flow acceleration. It is also noted that this trend is different than those observed in the DNS literature or piloted flame experiments. The primary difference between these data and previous literature is the configuration specific flow field; for DNS studies, there is typically no bulk fluid motion in one direction. For piloted flames, it is likely that the piloting mechanism induces a broad turbulent shear region to stabilize the flame, thereby biasing the spatial distribution of turbulence from reactants to products. For this configuration, the region of interest lies downstream of the bluff-body recirculation zone and shear layers to quantify the behavior of a freely propagating flame in a bulk flow. Within this region, the bulk flow expansion and acceleration drive an increase in velocity and turbulence fluctuations. However, the turbulence transport dynamics are a function of both the incoming freestream velocity scale (the u_turb that convects into the flame) and the pressure gradient. The turbulence transport through the flame results in the largest velocity fluctuations when u_turb and $dP∕dx¯$ are maximal. The increased turbulence fluctuations lead to large spatial velocity gradients and highly distributed vorticity in the reacting domain.

Instantaneous flame fronts are also overlaid onto the contours in Figs. 10 and 11. For all cases, the topology of the flame front is highly tailored to the vortex dynamics. For the low turbulent flames, the large-scale coherent vortex structures lead to smooth, large-scale flame wrinkles. As the turbulence increases, the distribution of high-magnitude vorticity leads to highly corrugated flame fronts, with small-scale wrinkled structures. Hence, the topology of the flame and the behavior of the flow field are both influenced by turbulence and pressure gradient tailoring effects. Additional analysis of the flame structure and its convecting dynamics are discussed in Sec. IV.

To further evaluate the influence of u_turb and $dP∕dx¯$ on the reacting vorticity field, the instantaneous and time-averaged vorticity field is evaluated for each test condition. An example of time-averaged velocity and vorticity fields, taken from a sequence of 2727 image pairs, is provided in Fig. 13(a). Using these data, as well as the instantaneous data presented in Fig. 11, the vorticity along the upper half of the domain (y/H ≥ 0) is presented as histograms in Figs. 13(b)–13(d). The histograms show similar trends as the contours in Fig. 11; the vorticity magnitudes increase with both $dP∕dx¯$ and u_turb. The histograms also encompass a wider range of values with increasing $dP∕dx¯$⁠. The increased magnitude and distribution of vorticity are expected to be coupled with a shift in the relative balance of the individual vorticity mechanisms.^15,17 For this reason, it is beneficial to decompose the individual terms of the vorticity transport equation and discern the physical mechanisms governing the behavior of the reacting flow field.

The mean conditionally filtered tracking is first applied to the time-averaged velocity and vorticity fields to demonstrate how the individual vorticity terms are affected by $dP∕dx¯$ and u_turb. An example of the time-averaged velocity and vorticity fields at stoichiometric conditions (Φ = 1) is provided in Fig. 13. TFE’s are inserted into the time-averaged flow fields and allowed to propagate through the average flame location to quantify the vorticity transport terms. Note that the time-averaged tracking analysis is presented as a function of time; however, the time axis is with respect to the TFE’s evolution and not the flow field (the average flow field does not change with time). Since the present experiment is a confined channel with only one downstream outlet, time-averaging the data further minimizes velocity fluctuations and spatial gradients in the third dimension and ultimately minimizes any bias errors based on the conditional-filtering approach and formulations described in Sec. III.

The TFE tracking and vorticity decomposition are then repeated for the time-resolved data using the instantaneous flow field and flame front, as depicted in Fig. 9(b). This approach accounts for the temporal evolution of the flow field and flame front. The time-resolved analysis is also conducted for two equivalence ratios (Φ = 1 and Φ = 0.7) to evaluate the consequent impacts on the vorticity budget. The time-resolved vorticity results are then used to characterize the coupling between vorticity dynamics and the evolution of the flame front.

Uncertainty in the time-resolved vorticity analysis arises from random errors as well as bias errors from the conditional-filtering approach and formulations in Sec. III. To address random errors, all time-resolved tracking is repeated multiple times with different sets of at least 100 TFEs and presented as an average of the multiple runs. A comparison is then performed between the instantaneous and time-averaged vorticity budgets as validation; since time-averaging provides a predominately 2D flow, the impacts of instantaneous fluctuations and gradients can be identified. When analyzing the contributions of individual terms to the overall vorticity budget, the maximum error is in the vortex stretching term, which varies by 4.7% with respect to the time-averaged Dω/Dt. Similarly, the baroclinic torque term varied by 3.8%, and the remaining terms all varied by <1%. The minimal variation between the time-averaged and time-resolved vorticity budgets confirms that the instantaneous fluctuations and gradients from three-dimensional effects have a minimal impact on the results. Bias errors were addressed using a propagation of error approach, which accounts for the uncertainty in the velocity fields, spatial gradients, as well as neglected terms that are provided in Table III. The maximum uncertainty in the vortex stretching term is 7% at the highest turbulence condition. This provides confidence in the time-resolved vorticity budget and ensures that the mean, conditionally filtered technique and associated assumptions are adequate for analysis.

To implement the average tracking, TFEs are tagged and tracked along the time-averaged data, and Lagrangian derivatives are used to calculate the individual vorticity terms; the results are shown in Fig. 14. The time axis is normalized by t₀ = 100 µs and begins at t/t₀ = 0, which is when the particles first interact with the flame boundary. All vorticity terms are normalized using (H/U_oS_T), where S_T is the turbulent flame speed provided in Table II for each turbulence condition. It is noted that since the time-averaged flow field is used for analysis, spatial gradients and fluctuations are dampened, resulting in minimal changes in the magnitude of the vorticity terms as the TFE’s propagate through the flame.

The combined effects of turbulence and $dP∕dx¯$ on the dilatation mechanism are first shown in Fig. 14. The dilatation mechanism is associated with the expansion of the TFEs as they cross the flame. The dilatation increases with $dP∕dx¯$ and decreases with u_turb. The increased dilatation with $dP∕dx¯$ is due to augmented stream-wise velocity and vorticity magnitudes as a result of flow confinement. This corresponds with increased velocity gradients and fluctuations experienced as the TFEs progress through the flame previously depicted in Figs. 10–12. Alternately, the scaled dilatation decreases with u_turb, which suggests that the dilatation has a decreasing impact on vorticity evolution at high turbulence conditions.¹⁷ Although there is an increase in velocity gradients and turbulence fluctuations, the dilatation experienced when normalized by the turbulent flame speed decreases. This dilatation is primarily responsible for attenuating vorticity, but at high turbulence conditions, it becomes overwhelmed by the strong vorticity production that occurs at the flame front. As $dP∕dx¯$ and u_turb are changed simultaneously, the aforementioned trends are preserved.

The baroclinic torque shown in Fig. 14 has similar behavior to the dilatation, showing an increase in magnitude with $dP∕dx¯$ and a decrease with u_turb. The increase in $dP∕dx¯$ directly enhances the flame-generated vorticity, as visualized in Fig. 11, which shows increasing vorticity magnitude. The increased flame-generated vorticity results in increased turbulence generated at the flame front, which was previously depicted in Fig. 12. In contrast, the scaled baroclinic torque decreases as u_turb increases, although it should be noted that the true magnitude of baroclinic torque increases due to increased flame corrugation, which exacerbates the misalignment between density and pressure gradients. This trend is consistent with previous DNS studies¹⁷ and indicates that baroclinic torque has the most substantial impact on vorticity evolution at low turbulence conditions and becomes less significant with increasing turbulence.

The remaining vorticity terms are the vortex stretching/tilting, viscous diffusion, and viscous torque. The vortex stretching and tilting contribute to the formation of small-scale structures in turbulent flows.⁵¹ As $dP∕dx¯$ increases, the vortex stretching is augmented. This corresponds to the increased flame corrugation with $dP∕dx¯$⁠, as shown in Figs. 10 and 11. Similarly, as u_turb increases, the vortex stretching/tilting also increases. This behavior is opposite of the trends exhibited by the baroclinic torque and dilatation, which decreased with u_turb. This indicates that as u_turb increases, the reacting vorticity field becomes dominated by the deformation of vortex structures, which is indicative of turbulence energy exchange.¹⁷ As $dP∕dx¯$ and u_turb are simultaneously increased, their effects superimpose onto one another and the vortex stretching reaches a maximum. The viscous diffusion and viscous torque follow the same trends as the vortex stretching, although their magnitude is notably smaller. The viscous diffusion experiences an increase in magnitude as a result of the increased velocity gradients and small-scale vortices, as shown in Figs. 10 and 11. The viscous torque undergoes an increase with both $dP∕dx¯$ and u_turb due to the increased flame corrugation experienced as both quantities are augmented, which creates further misalignment between the density gradient and divergence of the viscous stress tensor.

The results in Fig. 14 can be expressed quantitatively by integrating the data and obtaining an average value for each curve; these results are shown in Fig. 15. Specifically, Fig. 15(a) depicts the overall vorticity budget for each pressure gradient, while Fig. 15(b) shows the impacts of turbulence and $dP∕dx¯$ on the individual vorticity terms.

For all pressure gradient conditions, the dominant vorticity terms in Fig. 15(a) are the baroclinicity and the vortex stretching, while dilatation, viscous torque, and viscous diffusion are orders of magnitude weaker. The primary difference in the vorticity budget is that the magnitude of the vortex stretching term overtakes the baroclinic term at high $dP∕dx¯$ conditions. Increasing the baroclinic torque will enhance flame-scale turbulence production of order O(δ_T) along the flame front. Augmenting the flame-scale turbulence will enhance spatial gradients of velocity and vorticity in the reacting domain and thus invoke stronger magnitudes of vortex stretching/tilting, viscous torque, and viscous diffusion.

Increasing the turbulence intensity has a much more prominent impact on the vorticity budget, as seen in Fig. 15(a). At low turbulence conditions, the baroclinic torque and vortex stretching terms have the most substantial influence on vorticity transport, with the baroclinic term being most dominant low turbulence conditions. As the turbulence intensity is increased, Figs. 15(a) and 15(b) both demonstrate that the scaled magnitude of baroclinicity and dilatation decreases, while the magnitudes of vortex stretching/tilting, viscous torque, and viscous diffusion increase. This is a similar trend described in the DNS literature,^15,17 where baroclinic torque and dilatation are suppressed at high turbulence conditions, and the physical behavior of the reacting flow field approaches non-reacting turbulence conditions. These trends are upheld for all pressure-gradient conditions, meaning that the physical behavior of the reacting flow is more sensitive to turbulence effects.

The TFE tracking and vorticity decomposition are repeated using the time-resolved flow fields. This is advantageous as it captures the temporal dynamics and evolution of the flow field and flame front. Additionally, the analysis considers two equivalence ratios: a stoichiometric condition of Φ = 1 and a lean condition of Φ = 0.7. Decreasing the equivalence ratio results in decreased turbulent flame speed (S_T), as well as an increase in the turbulent flame thickness (δ_T). Thus, it is possible to assess how these relevant flame scales impact the vorticity budget and contribution of each term. The time-resolved vorticity decomposition results are first plotted in Fig. 16 for Φ = 1; Fig. 17 shows similar results for Φ = 0.7.

For the dilatation term, the trends for the stoichiometric and lean data presented in Figs. 16 and 17 are consistent with one another. The dilatation increases from zero to a maximum value when the TFEs first interact with the flame. The dilatation then decreases back to zero after the TFEs are burned. For the lean condition in Fig. 17, the dilatation magnitude is less than the stoichiometric case due to the decreased density ratio between the reactants and products.⁴⁹ For both Φ conditions, increasing $dP∕dx¯$ augments the dilatation; maximum dilatation is obtained with the nozzle and minimum with the diffuser. Alternately, the dilatation decreases with increasing u_turb. Although both Φ conditions experience the same u_turb value, the normalized turbulence intensity (u_turb/S_L) is larger for Φ = 0.7; this is caused by the reduced flame speed at lean conditions.^49,52,53 In this manner, the magnitude of dilatation is decreased with increasing turbulence in the flow or decreasing equivalence ratio. Altering the pressure gradient and turbulence conditions simultaneously results in coupled effects where the maximum dilatation occurs for the low-turbulent nozzle configuration.

The temporal evolution of the baroclinic torque is similarly shown in Figs. 16 and 17. The baroclinic torque arises from the density gradient across the flame thickness and spikes to a maximum value when the TFEs first interact with the flame boundary at t/t₀ = 0. The magnitude then decreases after passing through the flame thickness. Similar to dilatation, Figs. 16 and 17 show that the baroclinic torque production decreases as Φ is reduced; this is a result of the decreased density ratio across the flame at lean conditions.⁴⁹ Augmenting $dP∕dx¯$ yields higher baroclinic torque; however, increasing u_turb decreases the magnitude of baroclinic torque. Altering u_turb and $dP∕dx¯$ in tandem has similar effects as dilatation, where the maximum baroclinic torque occurs within the low-turbulent nozzle configuration.

The vortex stretching/tilting, viscous diffusion, and viscous torque terms are also provided in Figs. 16 and 17. The vortex stretching/tilting similarly reaches a maximum value when t/t₀ = 0 as a result of the formulation in Eq. (7). The stretching and tilting of vorticity increase as a result of the augmented u_turb and $dP∕dx¯$ but also increases as Φ is lowered to lean conditions. This trend occurs due to the thermochemically induced turbulence evolution across the premixed flame, as depicted in Fig. 12. Increasing u_turb and $dP∕dx¯$ augments turbulence velocity fluctuations and spatial gradients; thus, the stretching and tilting of vorticity are magnified. Additionally, as Φ is decreased, the influence of combustion is diminished and non-reacting turbulence drives the behavior of the flow field. In a similar manner, the magnitudes of viscous diffusion and viscous torque are also amplified with u_turb and $dP∕dx¯$⁠, but at several orders of magnitude lower than the stretching and tilting term.

To summarize the data in Figs. 16 and 17, the curves are integrated to obtain an average value, and the results are shown in Fig. 18. Figure 18(a) depicts the overall vorticity budget, while Fig. 18(b) shows the impacts of $dP∕dx¯$⁠, u_turb, and Φ on the individual vorticity terms. To quantify the effects of Φ, it is useful to consider the vorticity budget in the context of the Reynolds (Re) and Karlovitz (Ka) numbers. The Reynolds number is defined using the definition provided by Skiba et al.²⁹ 

which captures the balance between the turbulence and flame scales. When computed for the current test conditions (Table II), Re is nominally constant for a given Φ. Between the Φ = 1.0 and Φ = 0.7 cases, Re is larger for the lean flame because of the reduced flame speed. The stoichiometric and lean Reynolds numbers are also included in Fig. 18(a). The Karlovitz number describes the ratio of chemical to fluidic timescales, and the formulation is provided in Eq. (2). The Ka number will increase with the turbulence intensity as documented in Table II. Thus, the curves in Fig. 18(a) also capture the effects of increasing Ka on the vorticity budget.

As Re and Ka numbers increase, the vortex stretching, viscous torque, and viscous diffusion terms increase in magnitude, while the baroclinic torque and dilatation decrease. When Ka ≫ 1, the fluid mechanical timescales are much faster than the chemical timescales, and turbulent energy exchange governs the vorticity dynamics in the flow field. This is captured in Fig. 18(a), where the vortex stretching dominates the vorticity budget at high Ka numbers. Decreasing the equivalence ratio (and increasing Ka) exaggerates this trend, and the vortex stretching is an order of magnitude larger than the baroclinic torque. This demonstrates that the vorticity budget is indeed a function of the equivalence ratio. Specifically, the transition to a non-reacting turbulence condition is exaggerated at lean equivalence ratios where the chemical timescales are overshadowed by the fast fluid mechanical timescales.

Figures 10 and 11 demonstrate that altering the turbulence and pressure gradient influences the flow field as well as the structure of the flame front. Additionally, increasing the pressure gradient was shown to increase the magnitude of all terms in the vorticity transport equation, while increasing the turbulence intensity led to a vortex stretching dominated flow with minor influence from the exothermic terms of baroclinic torque and dilatation. Based on the literature,^4,21,54–56 vorticity and turbulence dynamics are strongly coupled with the structure and evolution of the flame front. For this reason, it is of interest to investigate how altering the pressure gradient and turbulence intensities influences the topology of the flame front. For each test condition, flame segments are isolated and depicted within Fig. 19, where each subsequent snapshot is separated by 100 µs. The flame segments are obtained by first truncating a portion of the PIV flame trace that interacted with TFEs in the first half of the domain, near 5.5 ≤ x/H ≤ 6. The initial flame segment is tagged with theoretical flame elements and propagated forward in time using the Lagrangian equations of motion and the turbulent flame speed normal to the flame front. This allows for the subsequent flame segments to be determined and extracted from the PIV trace.

For the low turbulence condition, the flame in Fig. 19(a) (top left corner) is nominally smooth with minimal wrinkling. Augmenting the pressure gradient results in a global counterclockwise (ccw) rotation, and a distinct wrinkle is formed. The time duration for the wrinkle to form decreases with $dP∕dx¯$ and is likely a result of the increased magnitude (and decreased timescales) of baroclinic torque,^18,24 which is pertinent at low turbulence conditions. These distinct wrinkles vanish as the turbulence intensity increases. Increasing the turbulence intensity leads to greater magnitudes of vortex stretching and tilting, which transfers energy from large-scale to small-scale structures in turbulent flows. This effect is visualized in Fig. 19, where small-scale flame corrugations develop with increased turbulence. When comparing the flame structures for the highly turbulent flames (test case E), the amplitude of the turbulent corrugations appears to increase with $dP∕dx¯$⁠. To quantify this, the length of the flame segments is computed for each time step shown in Fig. 19 and presented in Fig. 20. For all test cases, the data in Fig. 20 demonstrate that the flame wrinkles increase in length as they evolve through time. Furthermore, the length of the flame segments increase with both $dP∕dx¯$ and turbulence intensity. Previous research investigations have indicated that there is an inherent link between the local flame front topology and the turbulent flame speed.^57,58 The growth of the flame segments corresponds to an increase in the flame curvature, resulting in an augmented flame consumption speed (i.e., the turbulent flame speed), which results in amplified streamwise velocities, as depicted in Fig. 10. This corresponds to a more efficient mode of combustion, where the conversion from chemical potential to kinetic energy is enhanced by the imposed pressure gradient and turbulence–flame interactions. Based on the vorticity transport analysis above, flow fields can be tailored toward conditions, which would promote greater turbulent flame speeds. Optimal reactions can be achieved within flow fields dominated by the stretching and tilting of vorticity accompanied with high-magnitudes of flame-generated baroclinic torque production.

This study explored the effects of turbulence and pressure gradients on vorticity transport in premixed bluff-body flames. The goal was to elucidate their effects on the reacting flow field and subsequently assess if the flow behavior was driven by combustion-induced or non-reacting vorticity terms. Additionally, lean and stoichiometric conditions were analyzed to evaluate the effects of chemical/turbulence timescales on reacting vorticity transport. The analysis was conducted using a mean, conditionally filtered fluid element tracking method to decompose the vorticity transport terms. The flame topology was then examined to determine the impacts the vorticity dynamics in the flow field has on the evolution of the flame front.

The results confirmed that the fundamental behavior of the reacting flow field is dependent on the pressure gradient and turbulence conditions as the stream-wise velocity and span-wise vorticity magnitudes both increased with $dP∕dx¯$ and turbulence. To assess the source of increased vorticity, the vorticity transport terms were decomposed across the flame front to determine the vorticity budget for each test case. For all test conditions, the dominant vorticity terms were the baroclinic torque and vortex stretching terms, while the dilatation, viscous torque, and viscous diffusion terms were orders of magnitude smaller.

At low turbulence conditions, baroclinic torque was the dominant vorticity term in the reacting domain. As the turbulence intensity was increased, the baroclinic and dilatation terms decreased in magnitude, while the vortex stretching/tilting, viscous torque, and viscous diffusion terms increased. This demonstrates that the combustion-induced terms become overshadowed by the non-reacting terms, primarily the vortex stretching, at high turbulence conditions. This finding is comparable to DNS studies and indicates that the behavior of the reacting flow approaches non-reacting behavior at high turbulence conditions. To further explore the effects of turbulence, the vorticity transport analysis was conducted for different equivalence ratios, which alters the ratio of chemical to turbulence timescales. At lean equivalence ratios (high Ka conditions), the fast timescales of turbulence overshadow the chemical timescales and combustion effects. This led to a diminished impact of dilatation and baroclinic torque due to a decreased density ratio between the reactants and products at lean conditions. This exaggerates the overall transition to a nonreacting turbulence behavior, where the flow field becomes increasingly dominated by vortex stretching/tilting, and viscous diffusion has a more substantial influence on the vorticity budget.

Increasing the favorable pressure gradient was expected to shift the vorticity budget to favor the flame-generated baroclinic term; however, this was not found to be true. The dilatation and baroclinic torque were both augmented with $dP∕dx¯$⁠, which enhanced flame-generated turbulence and spatial gradients of velocity and vorticity in the domain. Thus, by increasing the dilatation and baroclinicity terms, there was a coupled increase in the viscous diffusion and vortex stretching/tilting with the vorticity budget remaining nominally unchanged. This is contrary to turbulence effects, where the vorticity budget was dominated by vortex stretching, and reveals that the vorticity dynamics within the reacting flow field are more sensitive to turbulence rather than pressure gradient effects.

The flame structure and evolution were also shown to be heavily dependent on the vorticity dynamics within the domain. For low turbulence flames, augmenting the pressure gradient led to large-scaled wrinkles forming along the flame front. These features were attributed to baroclinic torque, which significantly contributes to the vorticity budget. As the turbulence intensity was increased, the flame structure became dominated by vortex stretching, which induced corrugations and the formation of small-scale structures. For all turbulence conditions, the flame corrugations increased in amplitude with $dP∕dx¯$⁠; this was verified by evaluating the length of the flame segments through time. However, large-scale features were linked with augmented $dP∕dx¯$ and baroclinic effects, while small-scale corrugations were primarily a function of the turbulence and vortex stretching.

The work was sponsored by the Air Force Office of Scientific Research (Grant Nos. 16RT0673/FA9550-16-1-0441 and 19RT0258/FA9550-19-0322, Program Manager: Dr. Chipping Li). Furthermore, the authors would like to acknowledge the Science, Mathematics & Research for Transformation (SMART) Graduate Fellowship funded by the National Defense Education Program for the graduate student support.

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