2.1 Experimental set-up

Figure 2.

(a) Schematic of the rotating cylinder in the present configuration. (b) Experimental field of view (EFOV) and definition of the controlling parameters. The shaded area corresponds to the computed domain.

The experimental set-up consists of a rectangular cross-section combustion chamber ( $h=34$  mm by $l=94$  mm), operated at atmospheric pressure, in which a stainless steel cylinder with a diameter $d=8$  mm, serves as the flame holder (figure 2 a). The cylinder is connected to a brushless direct current electric motor with rotational speed ranging from 600 to 20 000 revolutions per minute (RPM). During experiments, the motor experienced low frequency rotation fluctuations so that the uncertainty on the rotation speed was $\pm$ 100 RPM. The flame shape is however weakly sensitive to these fluctuations so that it remains steady. The combustion chamber is equipped with three planar quartz windows allowing optical access. The lean air–methane mixture is injected in a plenum through six injectors. The flow is then laminarised by a bed of 1 mm glass balls and two honeycombs. The upper part of the plenum is water cooled to ensure a constant temperature for the fresh mixture ( $T_{in}=288.15$  K). The velocity field is measured with particle image velocimetry (PIV). A double cavity Nd:YAG laser (Quantel Big Sky), operating at 532 nm, fires two laser beams, with a delay of $200~\unicode[STIX]{x03BC}\text{s}$ . The laser beam is expanded through a set of fused silica lenses (spherical and diverging). Since the flow is not symmetric when the cylinder is rotating, a tilted mirror is inserted to lighten the cylinder shadow region with a reflection of the incoming laser sheet (figure 2 a). The laser sheet thickness is measured to $500~\unicode[STIX]{x03BC}\text{m}$ . Olive oil particles of $1~\unicode[STIX]{x03BC}\text{m}$ are seeded through two injection systems located just before the glass balls bed (venturi seeder). Mie scattering is collected on a Imager Intense camera (Lavision), operating at a repetition rate of 1 Hz and a resolution of 1376 $\times$ 1040 pixels. A f/16 182 mm telecentric lens (TC4M64, Opto-engineering) is used to reduce parallax displacements occurring with classical lenses and obtain more accurate velocity vector fields in the vicinity of the cylinder. A narrow band-pass optical filter, centred at $\unicode[STIX]{x1D706}_{c}=532\pm 10$  nm is inserted in front of the camera sensor to record only Mie scattering of olive oil particles. PIV images are processed with a cross-correlation multi-pass algorithm (Davis 8.2.3), resulting in a final window of 16 $\times$ 16 $\text{pix}^{2}$ and a 50 % overlap. Ninety images are collected for each operating condition, with a spatial vector spacing of 0.2 mm.

The flame is also visualised thanks to a second Imager Intense camera (Lavision), equipped with a similar telecentric lens as the PIV system. The exposure time is set to 500 ms and a band-pass optical filter, centred at $\unicode[STIX]{x1D706}_{c}=430\pm 10$  nm, is placed in front of the camera to record spontaneous $\text{CH}^{\ast }$ emission.

A $50/50$ beam splitter is inserted between the two cameras to simultaneously record PIV and $\text{CH}^{\ast }$ signals, with only one optical access for visualisation. The experimental field of view (EFOV) shown in figure 2(b) captures a part of the combustion chamber section (36 mm $\times$ 26 mm) and the origin is taken at the centre of the cylinder.

The inlet temperature of fresh gases $T_{in}$ and the ambient pressure $P_{0}$ are systematically checked prior to measurements, and images are recorded when thermal equilibrium of the combustion chamber is reached (measured with a K-thermocouple fixed on the burner structure). During experiments, the flame started to slightly flicker for certain operating conditions (pressure fluctuations less than 2 Pa). These effects contribute to the slight flame thickening observed in the experiments.

2.2 Direct numerical simulations

The reactive multi-species Navier–Stokes equations are solved using a fully compressible unstructured solver called AVBP (Schonfeld & Rudgyard Reference Schonfeld and Rudgyard1999; Moureau et al. Reference Moureau, Lartigue, Sommerer, angelberger, Colin and Poinsot2005). A major advantage of the configuration of figure 2 is that it is almost perfectly two-dimensional so that a two-dimensional DNS with detailed chemistry can be used. Chemical kinetics are described by the ‘Lu 19’ analytical mechanism of Lu & Law (Reference Lu and Law2008) for methane/air combustion. This mechanism involves $19$ transported species and $11$ quasi-steady state (QSS) species. The viscosity is modelled with a power law, each species has its own Lewis number and a constant Prandtl number is assumed. It was validated for a large range of equivalence ratio ( $\unicode[STIX]{x1D719}=0.5{-}1.5$ ), pressure ( $P_{0}=1{-}30$  atm) and correctly predicts auto-ignition times as well as CO levels. It was implemented in the DNS code and validated by comparison with GRI-MECH3.0 computations using the Cantera open source solver (Smith et al. Reference Smith, Golden, Frenklach, Moriarty, Eiteneer, Goldenberg, Bowman, Hanson, Song and Gardiner1999; Goodwin Reference Goodwin2002). Laminar flame speeds and adiabatic temperatures are correctly reproduced (figure 3 b). In addition, properties of flame/wall interactions were validated by running one-dimensional flames. For the canonical head-on quenching scenario, the maximum normalised wall heat flux is 0.35 and the corresponding Peclet number is 3, being in agreement with the literature (Huang, Vosen & Greif Reference Huang, Vosen and Greif1988; Jarosinski Reference Jarosinski1988; Wichman & Bruneaux Reference Wichman and Bruneaux1995). The laminar flame thickness for the $\unicode[STIX]{x1D719}=0.7$ equivalence ratio flame studied here is $0.680$  mm while the mesh resolution in the flame zone is $0.06$  mm so that at least $12$ points are used to resolve the flame front. Mesh sensitivity was verified by testing a finer mesh, with a cell resolution of $25~\unicode[STIX]{x03BC}\text{m}$ (instead of $60~\unicode[STIX]{x03BC}\text{m}$ in this study), leading to negligible modifications in flame position and velocity field.

The two-step Taylor–Galerkin convection scheme (Colin & Rudgyard Reference Colin and Rudgyard2000) used for DNS provides third-order accuracy in time and space. This scheme is characterised by a high spectral resolution, excellent dispersion and dissipation properties. Navier–Stokes characteristic boundary conditions (NSCBC) (Poinsot & Lele Reference Poinsot and Lele1992; Moureau et al. Reference Moureau, Lartigue, Sommerer, angelberger, Colin and Poinsot2005) are applied at inlet and outlet. Lateral walls were modelled as no-slip and isothermal ( $T_{w}=288$  K). In order to account for the rotation of the cylinder, the velocity of its wall is set at $u_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D714}/2$ in the frame of the burner, where $\unicode[STIX]{x1D714}$ is the angular rotation speed. The temperature $T_{c}$ is imposed at the cylinder external face (see § 2.2.1). The computational domain extends from $z=-35$ to $z=45$  mm, which corresponds to the shaded zone in figure 2(b). The velocity profile at the inlet (figure 3 a) is imposed by fitting experimental velocity data.

Figure 3.

(a) Experimental inlet velocity profile (hot-wire anemometry) measured at $z/d=-4.6$ and fit used in the simulation (figure 2 a). (b) Laminar burning velocities for air–methane mixtures. Comparison between different solvers, chemical schemes and the experiment of Varea et al. (Reference Varea, Modica, Vandel and Renou2012).

2.3 Operating conditions

A large range of operating conditions have been tested in order to determine the stability map of the burner. Based on these observations, the bulk velocity taken in the plenum was set to $u_{b}=1.07~\text{m}~\text{s}^{-1}$ and the equivalence ratio of the air–methane mixture to $\unicode[STIX]{x1D719}=0.70$ . The Reynolds number $Re$ of the incoming flow is $580$ (based on the cylinder diameter  $d$ ). To characterise the rate of rotation of the cylinder, the normalised parameter $\unicode[STIX]{x1D6FC}$ is introduced as:

This parameter compares the radial velocity at the cylinder to the bulk flow velocity $u_{b}$ , and is the control parameter for non-reactive flows around rotating cylinders. The fresh gas temperature is $T_{in}=288.15$  K and the outlet pressure $P_{0}=0.99$  bar. In this study, all parameters are kept constant except for $\unicode[STIX]{x1D6FC}$ whose variations are reported in table 1 together with the corresponding cylinder temperature.

Table 1.

Operating conditions investigated in the present study. The last column gives the temperature of the cylinder ( $T_{c}$ ) obtained with (2.2).

Figure 4.

Comparison between experimental and DNS flame fronts for increasing rotation rates $\unicode[STIX]{x1D6FC}$ . CH* traces from experiments are shown in grey scale and red dashed lines are an iso-contour of the heat release rate from the DNS (20 % of the maximum). (a) $\unicode[STIX]{x1D6FC}=0.00$ ; (b) $\unicode[STIX]{x1D6FC}=1.16$ ; (c) $\unicode[STIX]{x1D6FC}=2.30$ ; (d) $\unicode[STIX]{x1D6FC}=3.07$ ; (e) $\unicode[STIX]{x1D6FC}=4.10$ . The temperatures of the cylinder are given in table 1.

4.1 Excess or defect of flow enthalpy

As shown in figure 7, the flow is not adiabatic in the vicinity of the cylinder because of heat transfer with the flame holder. The proper quantity to measure the departure from adiabadicity is the reduced enthalpy defect/excess ${\mathcal{L}}$ given by:

where $h_{t}$ is the total chemical enthalpy taking into account sensible, chemical and kinetic energies. $h_{t}^{in}$ is a reference enthalpy, taken in the fresh gases. The normalizing term $QY_{F}^{in}$ in (4.1) is the heat of the reaction. Positive values of ${\mathcal{L}}$ correspond to sensible or chemical energy given to the fresh gases: the flame is therefore superadiabatic. Conversely, negative values of ${\mathcal{L}}$ correspond to an enthalpy loss and a subadiabatic flame. Figure 9 presents reduced enthalpy fields for different rotation rates $\unicode[STIX]{x1D6FC}$ . Far away from the cylinder and the flame fronts, the flow is adiabatic and ${\mathcal{L}}=0$ . When the cylinder is not rotating ( $\unicode[STIX]{x1D6FC}=0$ , figure 9 a), ${\mathcal{L}}$ is negative in the wake of the cylinder because hot gases loose energy to the cylinder through conductive heat transfer. Conversely, ${\mathcal{L}}$ is positive upstream of the cylinder because incoming gases are heated by the cylinder. The latter trend is still valid for moderate values of $\unicode[STIX]{x1D6FC}$ and even prominent as ${\mathcal{L}}$ goes up to 0.2. However, this region of reduced enthalpy gain narrows and eventually vanishes for larger rotation rates (figure 9 e). The distribution of ${\mathcal{L}}$ around the cylinder is shown in figure 10 ( $\unicode[STIX]{x1D703}$ is defined in figure 7). When the cylinder is at rest or has a low rotation rate (i.e. $\unicode[STIX]{x1D6FC}=0.00$ or 1.16), ${\mathcal{L}}$ is negative in the wake of the cylinder where burned gases are located, and it is positive on the upstream part where fresh gases impact the hot cylinder. Increasing the rotation rate reduces ${\mathcal{L}}$ all along the cylinder so that it is almost negative at all angles for the higher rotation rate ( $\unicode[STIX]{x1D6FC}=4.10$ ). Therefore, it is speculated that the flow induced by the rotation (figure 6) may explain this trend.

Figure 9.

Field of enthalpy loss ${\mathcal{L}}$ for different rotation rates $\unicode[STIX]{x1D6FC}$ . Iso-contour lines of ${\mathcal{L}}$ are shown in black. The red thick lines are the heat-release rate crest of each branch. (a) $\unicode[STIX]{x1D6FC}=0.00$ ; (b) $\unicode[STIX]{x1D6FC}=1.16$ ; (c) $\unicode[STIX]{x1D6FC}=2.30$ ; (d) $\unicode[STIX]{x1D6FC}=3.07$ ; (e) $\unicode[STIX]{x1D6FC}=4.10$ .

Figure 10.

Enthalpy loss ${\mathcal{L}}$ around the cylinder ( $\unicode[STIX]{x1D703}$ ).

4.2 Dilution effect at high rotation rates

Figure 11 displays the field of $\text{CO}_{2}$ mass fraction at $\unicode[STIX]{x1D6FC}=1.16$ (a) and $\unicode[STIX]{x1D6FC}=4.10$ (b). For $\unicode[STIX]{x1D6FC}=4.10$ , $\text{CO}_{2}$ is found all around the cylinder. This recirculation of burned gases, also observed in figure 6(b), changes the local composition by diluting incoming fresh gases and diminish ${\mathcal{L}}$ to negative values (figure 10).

Figure 11.

Reduced carbon dioxide mass fraction $c_{co_{2}}$ field for (a) $\unicode[STIX]{x1D6FC}=1.16$ and (b) $\unicode[STIX]{x1D6FC}=4.10$ . Black lines are iso-contour of $c_{co_{2}}$ .

To separate preheating and dilution effects, ${\mathcal{L}}$ can be written as:

where ${\mathcal{L}}_{s}$ measures gas preheating by the cylinder while ${\mathcal{L}}_{c}$ measures the effect of dilution and ${\mathcal{L}}_{k}$ accounts for the kinetic effect. Because the Mach number is 0.003, ${\mathcal{L}}_{k}$ can be neglected. Figure 12 plots the reduced sensible ${\mathcal{L}}_{s}$ (black line) and reduced chemical ${\mathcal{L}}_{c}$ enthalpies (dashed line) contributing to ${\mathcal{L}}$ , for two rotation rates $\unicode[STIX]{x1D6FC}=1.16$ (figure 12 a) and 4.10 (figure 12 b). The shape or the amplitude of ${\mathcal{L}}_{s}$ is similar in both cases because gases near the wall receive energy from the hot cylinder. However, the contribution of ${\mathcal{L}}_{c}$ differs from the two cases. At a moderate rotation rate (figure 12 a), the reduced chemical enthalpy parameter ${\mathcal{L}}_{c}$ is null on the upstream part of the cylinder, meaning that only fresh unburned gases are found at this location. In contrast, when the rotation rate is increased (figure 12 b), ${\mathcal{L}}_{c}$ is negative on the leading edge of the cylinder. Therefore, the dilution observed in figure 11(b) is the parameter that decreases ${\mathcal{L}}$ at high rotation rates.

In summary, the excess or defect of the reduced enthalpy ${\mathcal{L}}$ around the cylinder evolves due to the competition between two effects. First, the warmer cylinder heats up near-wall gases that results in positive values of ${\mathcal{L}}_{s}$ for all rotation rates investigated. Second, the induced flow created by the cylinder carries burned gases that diminish the reduced chemical enthalpy ${\mathcal{L}}_{c}$ . This latter effect is prominent when the rotation rate exceeds a certain threshold ( $\unicode[STIX]{x1D6FC}_{c}\approx 3$ ).

Figure 12.

Sensible ( ${\mathcal{L}}_{s}$ ) and chemical ( ${\mathcal{L}}_{c}$ ) enthalpy loss contributions versus position on the cylinder $\unicode[STIX]{x1D703}$ . (a) $\unicode[STIX]{x1D6FC}=1.16$ ; (b) $\unicode[STIX]{x1D6FC}=4.10$ .

4.3 Super- and subadiabatic flame structure

This section focuses on the differences between the two flame branches observed in figure 4 and their link with the reduced enthalpy field (figure 9). Figure 13 shows two flame profiles extracted from the DNS ( $\unicode[STIX]{x1D6FC}=1.16$ and $z/d=1.25$ ). The comparison between the overdriven lower branch (dashed lines with symbols) and an adiabatic unstrained flame (lines) in figure 13(a,b) shows a good agreement for the species profiles but the maximum heat-release rate is much higher for the flame of the DNS. This observation is also verified with the reduced enthalpy parameter as this branch is located in a superadiabatic flow ( ${\mathcal{L}}>0$ , figure 13 b). In contrast, the upper branch (figure 13 c,d) presents smoother species profiles and a heat-release rate profile quite low. Similarly, as the upper branch stabilises in a subadiabatic flow, no self-sustained or weak flame can exist at this location ( ${\mathcal{L}}<0$ , figure 13 d). Figure 14 presents the evolutions of the maximum heat release rate along the lower flame branch for different rotation rates. The location of maximum heat-release rate is detected with a crest sensor, for both flames and these maximum values are plotted along the curvilinear abscissa $s$ normalised by the flame thickness of a planar unstretched adiabatic flame $\unicode[STIX]{x1D6FF}_{L}^{0}=(T_{b}-T_{u})/\max (\text{d}T/\text{d}x)$ (0.680 mm for the $\unicode[STIX]{x1D719}=0.7$ equivalence ratio studied here).

Figure 13.

Flame profiles extracted from the DNS at $z/d=1.25$ for the case $\unicode[STIX]{x1D6FC}=1.16$ . $n$ is an axis normal to the flame front. Solid lines correspond to an adiabatic unstrained flame. Dashed lines with markers represent the overdriven and cooled branches. (a) Species profiles on the overdriven branch. (b) Reduced heat-release rate profiles on the overdriven branch. (c) Species profiles on the cooled branch. (d) Reduced heat-release rate profiles on the cooled branch.

Figure 14.

Normalised maximum heat-release rate along the lower flame front (overdriven) branch, for rotation rates $\unicode[STIX]{x1D6FC}$ from 0.00 to 4.10.

When the cylinder is not rotating (figure 14, solid line), the heat-release rate monotonically increases to reach the value obtained in an adiabatic flame. Close to the cylinder ( $s/\unicode[STIX]{x1D6FF}_{L}^{0}<10$ ) the reduced heat-release rate drops, due to the heat losses to the cylinder and the recirculation zones. The zone over which the cylinder inhibits chemical reactions is about $10\unicode[STIX]{x1D6FF}_{l}^{0}$ , being larger in comparison to canonical head-on quenching scenarios (Baum et al. Reference Baum, Poinsot, Haworth and Darabiha1994; Poinsot & Veynante Reference Poinsot and Veynante2011). When the cylinder is rotating at a moderate level ( $\unicode[STIX]{x1D6FC}=1.16$ , 2.30), the quenching zone narrows to about $\unicode[STIX]{x1D6FF}_{l}^{0}$ , and the heat-release rate increases near the wall, being locally twice as intense as in an adiabatic flame. When the rate of rotation is much higher ( $\unicode[STIX]{x1D6FC}=3.07$ , 4.10), the flame intensity starts to decrease, and it is even lower than in the non-rotating case when $\unicode[STIX]{x1D6FC}=4.10$ . This behaviour is an indicator of the dilution effect mentioned previously.

Figure 15 presents the reduced heat-release rate along the upper flame. In contrast with the lower superadiabatic flame branch, all cases exhibit heat-release rates lower than adiabatic conditions. The zone over which the flame is quenched becomes very large. For the case $\unicode[STIX]{x1D6FC}=2.30$ , the flame has not recovered its adiabatic structure after $40\unicode[STIX]{x1D6FF}_{l}^{0}$ . However, beyond a critical rotation rate ( $\unicode[STIX]{x1D6FC}_{c}\approx 3$ ), as dilution of burned gases shield the cylinder with burned gases, the flame starts to retrieve its adiabatic behaviour.

Figure 15.

Normalised maximum heat-release rate along the curvilinear abscissa of the upper flame (subadiabatic) branch, for various rotation rates $\unicode[STIX]{x1D6FC}$ .

At this point, ${\mathcal{L}}$ may be an adequate parameter explaining the flame topologies near the cylinder. In other words, could the two flame structures be explained by considering only local values of ${\mathcal{L}}$ ? It is therefore relevant to compute non-adiabatic stretched planar flame with a one-dimensional solver and to compare the local flame structures obtained with DNS flame profiles.

Figure 16.

Schematic illustration of two Cantera prototypes. (a) Overdriven lower branch with preheating; (b) cooled upper branch with cold burned gases. Dashed lines mark the heat-release rate crest used to define the curvilinear abscissa $s$ .

Two different counterflow flame prototypes are considered to mimic the two flame branches only with the reduced enthalpy parameter ${\mathcal{L}}$ . The lower branch corresponds to a superadiabatic flame where fresh gases have been preheated (figure 16 a) whereas the upper branch corresponds to a flame where fresh gases interact with burned gases that have lost energy to the cylinder (figure 16 b). However, the computation of a one-dimensional counterflow flame also introduces a strain effect. The range of strain rates to be used for the Cantera computations is evaluated by post-processing the DNS. Figure 17 shows the probability density functions of strain rate $\unicode[STIX]{x1D705}$ for the two branches for $\unicode[STIX]{x1D6FC}=0$ to 4.1. For all rotation rates, strain rate values are quite low and the possible mechanism of flame extinction due to an important strain rate is not supported here (Vagelopoulos & Egolfopoulous Reference Vagelopoulos and Egolfopoulous1994; Williams Reference Williams2000; Shanbhogue et al. Reference Shanbhogue, Sanusi, Taamallah, Habib, Mokheimer and Ghoniem2016). A small value $\unicode[STIX]{x1D705}=300~\text{s}^{-1}$ is therefore used in the Cantera calculations and only the reduced enthalpy parameter ${\mathcal{L}}$ is varied to reproduce the flame structures observed in figures 13–15.

Figure 17.

Probability density functions of the strain rate $\unicode[STIX]{x1D705}$ for different rotation rates. (a) Lower branch; (b) upper branch.

Figure 18.

Evolution of the reduced maximum heat release $\dot{\unicode[STIX]{x1D714}}/\dot{\unicode[STIX]{x1D714}}_{adia}$ with the enthalpy loss ${\mathcal{L}}$ , and for different rotation rates. The Cantera results (solid black line) are given with the methodology given in figure 16.

Figure 19.

Evolution of the reduced consumption speed $S_{c}/S_{c}^{adia}$ versus enthalpy loss ${\mathcal{L}}$ , for various rates of rotation. Planar unstretched flames computed with Cantera are plotted for comparison (solid black line).

Figures 18 and 19 present plots of the reduced heat-release rate and consumption speed versus the reduced enthalpy ${\mathcal{L}}$ . The consumption speed is evaluated locally by integrating the heat release rate on profiles normal to $s$ :

where $\dot{\unicode[STIX]{x1D714}}_{F}$ , $\unicode[STIX]{x1D70C}$ and $Y_{F}^{0}$ are the fuel source term, density and fuel mass fraction in the fresh gases, respectively. Data from the DNS have been obtained from profiles taken at $s/\unicode[STIX]{x1D6FF}_{L}^{0}=0.7$ , 4.7 and 40 for the overdriven branch, and $s/\unicode[STIX]{x1D6FF}_{L}^{0}=0.1$ , 5.0 and 40 for the cooled branch, and for different rotation rates. The corresponding Cantera results for the two counterflow flame prototypes are shown with the black line. The agreement between the one-dimensional and DNS flame properties is very good, showing that the flame structures observed locally in the DNS can be described to first order by the reduced enthalpy loss parameter ${\mathcal{L}}$ . However, some discrepancies are observed for the lower, superadiabatic branch where curvature effects could modify the flame structure (label 3). This comparison with one-dimensional flames proves that ${\mathcal{L}}$ is a robust and adequate parameter to describe the behaviour of the observed flames. Another approach to study this peculiar flame stabilisation is to look at the details of chemical kinetics, as for example done by Michaels & Ghoniem (Reference Michaels and Ghoniem2016). This analysis for the rotating cylinder is left to a further study.