1 Introduction

Laminar flame speed, also known as unstrained burning velocity, is an important fundamental property of flames, key to understanding fuel utilization and optimizing combustion efficiency. In the presence of turbulence, flame surfaces can be distorted, resulting in increased volumetric flame surface density as well as modifications to the local combustion chemistry. As a result, turbulent flame speed can be dramatically different than its laminar counterpart and hence serves a separate and distinct role in combustors. For internal combustion engines and turbines, high laminar flame speeds enable strong flame kernel development that is resistant to localized extinction, while high turbulent flame speeds allow the fuel to be consumed before end-gas conditions reach autoignition (i.e., knock). However, a significant capability gap exists for collecting and interpreting turbulent flame speed measurements. Such data are scant in the literature (Bradley et al. 2013; Bagdanavicius et al. 2015; Marshall et al. 2017; Turner et al. 2019) and lack sufficient generality to inform and validate predictive models (Farrell et al. 2004; Daniele et al. 2011) of turbulent flame phenomena across a broad range of device configurations.

Experimentally, turbulent flame speeds are typically measured using up to four canonical configurations: Bunsen flames, rod-stabilized flames, counterflow (stagnation) flames and spherically expanding flames (Law 2006). The first three of these feature stationary flames, while the final one requires dedicated devices capable of generating turbulence inside a spherical constant-volume combustion chamber (CVCC). CVCC facility surveys have been summarized elsewhere (Ravi et al. 2012). Generally, most turbulent CVCCs employ impellers (Ravi et al. 2012; Bradley et al. 2019; Mannaa et al. 2019; Morones et al. 2019) for relatively uniform turbulence generation, although recent reports suggest that traditional jets (i.e., nozzles) may also be used to create highly uniform turbulence (Davani and Ronney 2017; Davani et al. 2019). However, each experimental agitation method has unique limitations, such as vortical biases induced by impellers (Bonhomme et al. 2014; Bradley et al. 2019; Mannaa et al. 2019) or the pumping challenges (i.e., high pressure drop) required by the small openings of jet nozzles. Likewise, significant safety hazards are introduced by improper sealing and wear of high-rpm impeller parts, or, in the case of traditional jets, by the practical challenge of needing to premix the fuel and oxidizer in the absence of incidental ignition sources prior to entering the CVCC.

In this article, we explore the possibility of synthetic jets as an alternative and potentially advantaged turbulence generation device for turbulent combustion experiments. Unlike traditional jets, synthetic jets move fluid via periodically oscillating diaphragms that are contained within a wall-recessed cavity and eject fluid through a narrowed opening into the chamber (Smith and Swift 2003). Thus, synthetic jets result in high fluid ejection velocities with net-zero mass flux, driven by a significant momentum flux—attractive features for several applications encompassing power electronics cooling (Mahalingam and Glezer 2004; Arik et al. 2012; He et al. 2015), aerodynamics (Hassan and JanakiRam 1998; McCormick 2000), microfluidic devices (Mautner 2004) and granular flows (Han et al. 2020), among others. Turbulent premixing for combustion is one possible extension of synthetic jets, and we assess their technical viability as turbulence generation devices in a hypothetical 4,189-cm3 CVCC using computational fluid dynamics (CFD) modeling. This model framework serves as an early basis for synthetic jet design and engineering for next-generation flame speed measurement apparatuses.

2 Results and Discussion

To assess the agitation behavior of a premixed fuel in air, we first define the appropriate system size, geometry and conditions to establish a synthetic jet modeling framework (Table 1). A stoichiometric (φ = 1.0) mixture of methane (CH4) and air is assumed as a base-case fluid initially dispersed within a 200-mm-diameter spherical chamber (4,189-cm3 volume). The eight synthetic jets are placed at the corners of a cube inscribed inside of the chamber (Fig. 1), a basis of configuration selected from prior studies on synthetic (Hwang and Eaton 2004) and traditional (Davani and Ronney 2017) jets. Further, the 10-mm-wide synthetic jets are “cavityless” (Fig. 1a), such that the fluid in the chamber is subjected to a periodic velocity boundary condition at the jet opening (Eq. 1), accomplishing net-zero mass flux without needing to define an explicit cavity geometry (Kral et al. 1997; Matiz-Chicacausa and Lopez Mejia 2020). (Notably, cavity shape, depth and internal volume are expected to significantly influence the behavior of a real experimental device (Kotapati et al. 2007; Chaudhari et al. 2008, 2009; Jain et al. 2011; Capuano et al. 2019), but these considerations are beyond the scope of this study.)

Table 1 Baseline system parameters and conditions for synthetic jet models (parenthetical values denote the additional conditions evaluated in this study)
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Fig. 1
figure 1

a Illustration of three-dimensional spherical model control volume. Small circles at the surface represent 10-mm synthetic jet openings; four additional jets are symmetrically and oppositely located on the backside of the sphere (not shown). b Meshing of control volume, with refined finite volume sizing at and near jet openings. c Magnified illustration showing finer meshing at a jet opening

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Transient, finite-volume simulations are performed in ANSYS Fluent 2021 R2 to spatially discretize and solve the governing Reynolds-Averaged Navier–Stokes (RANS) equations using a shear stress transport (SST) k-omega (k–ω) turbulence model. We employ pressure–velocity coupling to compute the conservation equations via a segregated, guess-and-correct method and enable data sampling for time-based turbulence statistics (see Eqs. S1–S3 and discussion below). Full details of model development, sampling and sensitivity are included in the Supplementary Information (SI). To begin characterizing synthetic jet behavior predicted by this model framework, the root mean square (RMS) jet velocity vRMS is calculated from Eq. (1) to describe a nominal average of synthetic jet velocities v from periodic fluid pulses of frequency f and amplitude v0; this metric enables comparison to traditional jet behavior (Davani and Ronney 2017).

$$v={v}_{0}\mathrm{ sin}(2\pi ft)$$
(1)

First, a 2D circular cut plane with four symmetrically opposed synthetic jets is employed as a simplified, reduced-order test case for the eight-jet 3D spherical volume to verify reasonable velocity profiles and an appropriate turbulence model (i.e., SST k–ω). Briefly, the CFD model conditions and parameters shown in Table 1 yield physically meaningful velocity profiles and turbulence intensities within expected orders of magnitude (Fig. S1), suggesting that the assumed physics models are valid for this system and may be readily extended to the 3D system (Fig. 1). Full details and discussion of assumptions and sensitivities (Roache 1994) are included in SI.

Turbulence magnitudes and uniformity metrics are derived from time-averaged sampling calculations of the mean velocity \(\overline{{u }_{j,i}}\) (Eq. S1) and the RMS of fluctuating velocities \({u}_{j,i}^{^{\prime}}\), where j = x, y, z Cartesian coordinates at location i, allowing for calculation of time-averaged (Eq. S2) and volume-averaged (Eq. S3) turbulence intensities \({u}_{i}^{^{\prime}}\) and \(\overline{{u }_{V}^{^{\prime}}}\), respectively. Simulation times are sufficiently long to allow for at least 25 jet pulses. Following the volume-averaged quantitative analyses described by others (Davani and Ronney 2017), we similarly define the mean flow index (MFI), homogeneity index (HI) and isotropicity index (II) in Eqs. 2–4. Full definitions and discussion of index terminology are provided in SI.

$$MFI\equiv \frac{\sqrt{\frac{1}{V}\sum_{i=1}^{n}{\left(\overline{{u }_{i}}\right)}^{2}{v}_{i}}}{\overline{{u }_{V}^{^{\prime}}}}$$
(2)
$$HI\equiv \frac{\sqrt{\frac{1}{3V}\sum_{i=1}^{n}\left[{\left({u}_{x,i}^{^{\prime}}-\overline{{u }_{V}^{^{\prime}}}\right)}^{2}+{\left({u}_{y,i}^{^{\prime}}-\overline{{u }_{V}^{^{\prime}}}\right)}^{2}+{\left({u}_{z,i}^{^{\prime}}-\overline{{u }_{V}^{^{\prime}}}\right)}^{2}\right]{v}_{i}}}{\overline{{u }_{V}^{^{\prime}}}}$$
(3)
$$II\equiv \frac{\sqrt{\frac{1}{3V}\sum_{i=1}^{n}\left[{\left({u}_{x,i}^{^{\prime}}-{u}_{i}^{^{\prime}}\right)}^{2}+{\left({u}_{y,i}^{^{\prime}}-{u}_{i}^{^{\prime}}\right)}^{2}+{\left({u}_{z,i}^{^{\prime}}-{u}_{i}^{^{\prime}}\right)}^{2}\right]{v}_{i}}}{\overline{{u }_{V}^{^{\prime}}}}$$
(4)

Briefly, MFI represents a volume-averaged RMS flow normalized by the volume-averaged turbulence intensity. HI represents the extent to which turbulence intensity remains constant within the entire control volume, whereas II represents the spatial uniformity of the local intensity values. All three turbulence metrics, along with \(\overline{{u }_{i}}\) and \({u}_{i}^{^{\prime}}\), are used here to characterize synthetic jet agitation effectiveness and uniformity for various test cases.

Figure 2 depicts the 3D model velocity fields resulting from simulation at conditions described by Table 1. These heat maps of velocity magnitude illustrate spatially symmetric fluid velocity profiles at the middle section plane within the chamber, suggesting that synthetic jets are capable of highly uniform agitation. Time-averaged turbulence statistics are reported in Table 2 for this base case (No. 2) and for variations of the velocity constant v0 (Eq. 1), oscillation frequency f, methane/air equivalence ratio φ, synthetic jet slit size dsl, and chamber diameter dch.

Fig. 2
figure 2

Center-plane contours of 3D model results of a time-averaged mean velocity \(\overline{{u }_{i}}\) (m·s–1) and b time-averaged turbulence intensity \({u}_{i}^{^{\prime}}\) (m·s.–1). Color legend (right) illustrates magnitudes of each calculated velocity. See Table 1 for baseline conditions and Eqs. S1–S2 for definitions of \(\overline{{u }_{i}}\) and \({u}_{i}^{^{\prime}}\)

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Table 2 Summary of results of base case (No. 2, Table 1) and varied parameters
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Overall, \(\overline{{u }_{i}}\) and \({u}_{i}^{^{\prime}}\) monotonically increase with increasing v0, vRMS and dsl (Nos. 1–3, 11, 13), while f exhibits an inverse relationship with these two turbulence descriptors (Nos. 2, 4–8). When the total mass flowrate is kept constant with changing dsl (Nos. 2, 12, 14), similar \(\overline{{u }_{i}}\) and \({u}_{i}^{^{\prime}}\) are calculated for the 5-, 10- and 25-mm openings, with some loss in turbulence intensity noted for the 25-mm case; relatedly, as dch is increased two-fold (No. 15) and four-fold (No. 16), time-averaged turbulence statistics precipitously decrease for a given fixed timescale. However, across all these cases, maximum local Reynolds numbers Remax (Eq. S4) are identical within two significant figures for these three trials. Indeed, both dsl and dch appear to be important yet flexible design parameters, potentially allowing for tunable flow profiles and extents of turbulence in a prospective real device. Finally, base-case RANS results (No. 2) are compared to predictions of a large eddy simulation (LES; No. 17), an alternate turbulence model which yields internally agreeable values for \(\overline{{u }_{i}}\), \({u}_{i}^{^{\prime}}\) and Remax. Sensitivities of other model parameters are further assessed to verify the robustness of the 3D model relative to baseline model conditions and to a traditional jet analogue; full results are described in SI (Figs. S2–S4, Table S1).

Next, Fig. 3 illustrates calculated volume-averaged turbulence metrics as a function of dimensionless radial position r/R and oscillation frequency f for the base-case velocity constant v0 of 10 m·s–1. Above 50 Hz, turbulence indices MFI (Eq. 2, Fig. S5), HI (Eq. 3, Fig. 3a), II (Eq. 4, Fig. 3b) and time-averaged turbulence intensity \({u}_{i}^{^{\prime}}\) (Eq. S2, Fig. S6) are effectively constant for r/R < 0.8, suggesting that flame speed measurements taken in this field of view would feature highly homogeneous, spatially uniform turbulence and flow fields. For r/R > 0.8 (i.e., near the chamber wall), viscous dissipation likely damps the turbulent fluctuations and leads to observable deviations in index profiles; these wall effects are more pronounced for lower frequencies, which necessarily feature longer timescales for turbulent fluctuation settling. Overall, while turbulence descriptors here do not yet match or exceed the magnitudes of those predicted for traditional jets (Davani and Ronney 2017), the model framework instead serves to illustrate that synthetic jets may indeed be capable of generating spatially uniform turbulence (Figs. 2–3, S2) at sufficient intensities relevant to CVCC-based experiments (Fig. S6, Table 2).

Fig. 3
figure 3

Spatial variability of a HI and b II values along the dimensionless radial coordinate r/R calculated for various oscillation frequencies. See Table 1 for baseline conditions and Eqs. 3–4 for definitions

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Temperature and pressure also impact predicted turbulence profiles by changing effective fluid densities, and thus momentum flux. Simulation results of stoichiometric fluid mixtures initially at 600 K, 30 atm or both are summarized in Table S2. While elevated temperature increases the magnitude of calculated mean and RMS velocities, system pressure indeed has a pronounced retardation effect on turbulent fields. However, MFI, HI and II are generally lower for these cases, suggesting hypothetical CVCC experiments with denser fluids may feature highly uniform turbulence at elevated pressures. Additionally, future synthetic jet studies with well-defined cavity geometries and heated chamber walls may reveal heat transfer enhancements for pre-experiment thermal equilibration.

Finally, beyond the effects of slit diameter (Table 2, Nos. 2, 11–14), we assess the geometric impact of the total number of jets N. We construct new hypothetical chambers containing 4 and 20 symmetrically placed synthetic jets and compare them at Table 1 conditions, for cases of both constant flow velocity and constant mass flowrate (Table S3, Fig. S7). For the N = 20 case at constant flow velocity, values of \({u}^{^{\prime}}\) as high as 0.53 m·s–1 are achieved, although practical constraints such as placement of optical windows may prohibit such a large N in a real device. Nonetheless, these results illustrate the numerous important, flexible design handles available to researchers seeking to construct a synthetic jet-agitated CVCC in the future.

3 Conclusions

Synthetic jets are demonstrated to be a new, potentially promising turbulence generation device for CVCCs, and the model framework established here allows for initial testing of important fundamental and practical design hypotheses. These devices and their prospective applications warrant further investigation from the combustion community, particularly on fundamental mixing physics, cavity design, slit geometry, vibrational mechanisms and phasing, suitable materials, device scaling relationships and safety considerations.