1. Introduction
The recirculation zone, a region where flow reverses due to factors like swirl, bluff bodies or abrupt expansions in the flow area, is crucial for flame stabilization (Chen et al. Reference Chen, Driscoll, Kelly, Namazian and Schefer1990; Nair & Lieuwen Reference Nair and Lieuwen2007; Chaudhuri et al. Reference Chaudhuri, Kostka, Renfro and Cetegen2010; Klusmeyer et al. Reference Klusmeyer, Cross, Lubarsky, Bibik, Shcherbik and Zinn2013; Candel et al. Reference Candel, Durox, Schuller, Bourgouin and Moeck2014). It meets two primary criteria: it produces adequate heat to maintain the ignition of the fuel–air mixture and allows sufficient time for the incoming mixture of fresh reactants to interact with the recirculation zone, facilitating effective heat transfer and raising the mixture’s temperature to its ignition point (Lieuwen Reference Lieuwen2021). These criteria are inherently tied to the mixing conditions within the recirculation zone. The level of reactants mixing significantly affects combustion efficiency, flame length, stability of the combustion process and the emission of pollutants (Hawthorne, Weddell & Hottel Reference Hawthorne, Weddell and Hottel1948; Chen & Driscoll Reference Chen and Driscoll1989; Lacarelle et al. Reference Lacarelle, Moeck, Konle, Vey, Nayeri and Paschereit2007). While combustion processes involving recirculation zones have been extensively studied, the detailed understanding of the mixing dynamics within these zones remains less evolved. As a result, several notable questions in this area remain to be thoroughly investigated.
One question that arises is how to characterize the mixing state within the recirculation zone. Traditionally, this state is quantified using global parameters. For example, Chen & Driscoll (Reference Chen and Driscoll1989) measured flame length as an indicator of mixing within the recirculation zone of swirl flames. Coats, Richardson & Wang (Reference Coats, Richardson and Wang2000) used the equivalence ratio to describe the mixing state in the mixed layer, which forms as a result of stream separation at a rearward-facing step. However, relying on flame length or global equivalence ratio to represent mixing within the recirculation zone inherently assumes a homogeneously mixed environment, which may not reflect the true nature of the mixing state. Klusmeyer et al. (Reference Klusmeyer, Cross, Lubarsky, Bibik, Shcherbik and Zinn2013) demonstrated that the global equivalence ratio alone is insufficient to reliably predict the combustion limits of stable flames in the presence of a recirculation zone. Masri et al. (Reference Masri, Dally, Barlow and Carter1994) investigated the recirculation zone in bluff body flames and the non-uniform nature of the mixing state within the recirculation zone was observed through the use of the mixture fraction. These findings underscore the need for more nuanced approaches to characterize the mixing dynamics within recirculation zones, beyond the traditional use of global parameters.
Another question concerns the variations in the mixing state within the recirculation zone as inflow parameters change. The interplay between inflow conditions and flame characteristics in the presence of a recirculation zone has been a subject of extensive research. Chen & Driscoll (Reference Chen and Driscoll1989) demonstrated a scaling relationship between flame length and swirl number, indicating a direct influence of inflow parameters on the mixing state. Additionally, Everson, Manin & Sirovich (Reference Everson, Manin, Sirovich and Winter1998) introduced an entropy-based approach to describe the degree of mixing, highlighting the impact of the jet orifice shape on mixing efficiency, with slotted orifices shown to enhance mixing. In the context of jet flows, the momentum flux ratio emerges as a critical factor shaping the mixing state. Dally, Fletcher & Masri (Reference Dally, Fletcher and Masri1998) found that the momentum flux ratio of the jet to the coflow significantly affects the structure of the recirculation zone. Wei et al. (Reference Wei, Xie, Zhang and Ren2021) further explored the profound influence of the fuel jet velocity on flow, mixing and flame stabilization in bluff body flames.
To address the questions raised regarding the mixing state within recirculation zones and its response to changes in inflow parameters, we have chosen to investigate the Sydney burner. The Sydney bluff-body flame, a classical flame stabilized by a recirculation zone, offers a wealth of detailed and authoritative experimental data (Masri & Bilger Reference Masri and Bilger1985; Dibble, Masri & Bilger Reference Dibble, Masri and Bilger1987; Masri et al. Reference Masri, Dibble and Bilger1988, Reference Masri, Dibble and Barlow1992, Reference Masri, Dally, Barlow and Carter1994). This makes it an ideal candidate for validating numerical simulation models and methods (Kempf, Lindstedt & Janicka Reference Kempf, Lindstedt and Janicka2006; Navarro-Martinez & Kronenburg Reference Navarro-Martinez and Kronenburg2007; Popov & Pope Reference Popov and Pope2014; Miranda, Coelho & Ströhle Reference Miranda, Coelho, Ströhle and Janicka2020). The combustion process in the Sydney burner involves both jet flow and separated flow, with the fuel jet and air coflow creating the recirculation zone behind the bluff body. Previous studies have demonstrated that altering the inflow velocity ratio significantly affects the recirculation zone (Huang & Lin Reference Huang and Lin1994; Chen et al. Reference Chen, Chang, Pan and Yang1998; Esquiva-Dano, Nguyen & Escudie Reference Esquiva-Dano, Nguyen and Escudie2001). Consequently, the Sydney bluff-body burner is a suitable choice for our objective of studying the mixing state in the recirculation zone, as it allows for a comprehensive analysis of the interplay between inflow parameters and flame dynamics.
Computational fluid dynamics is one of the powerful tools for studying fluid problems. Concurrently, the refinement of numerical simulation methods has led to their extensive application in the field of combustion research. Large-eddy simulations (LES) have shown promising results in capturing combustion dynamics (Fureby & Lofstrom Reference Fureby and Lofstrom1994; Klimenko & Bilger Reference Klimenko and Bilger1999; Raman & Pitsch Reference Raman and Pitsch2005; Pitsch Reference Pitsch2006; El-Asrag & Menon Reference El-Asrag and Menon2007), with the flamelet/progress-variable approach (FPV) proving effective for predicting unsteady-state combustion processes (Pierce Reference Pierce2001; Pierce & Moin Reference Pierce and Moin2004; Ihme & Pitsch Reference Ihme and Pitsch2008a ,Reference Ihme and Pitsch b ,Reference Ihme and Pitsch c ; Ihme, Shunn & Zhang Reference Ihme, Shunn and Zhang2012). To gain a deeper understanding of the mixing state within the recirculation zone, we utilize numerical simulation to obtain more detailed data.
In this study, we utilize the Sydney bluff-body burner as our research subject and employ the LES-FPV approach to conduct a comprehensive numerical simulation. We introduce a novel parameter, the characteristic mixture fraction, to quantify the mixing state within the recirculation zone. Furthermore, we investigate the effects of varying momentum flux ratios on the statistical similarity of mixing conditions within this zone.
The remainder of the paper is organized as follows: § 2 provides a detailed introduction to the numerical simulation method employed (LES-FPV method); § 3 presents the Sydney bluff-body burner, describes the numerical simulation set-up used in this study and specifies the test cases configured for analysis. The core of our findings is presented in § 4. This section validates our numerical approach, probes a characteristic mixture fraction within the recirculation zone, rigorously examines whether the power-law holds true and explores its intriguing implications for reacting flows. Finally, § 5 synthesizes the key discoveries and draws the overarching conclusions.
2. Numerical framework and mathematical formulation
Building upon the established research context, this section systematically establishes the numerical computational framework, with focused elaboration on the theoretical foundation and implementation methodology of the LES-FPV method. In LES of turbulent reactive flows combined with the FPV approach, the mass and momentum equations after Favre-filtering can be expressed as (Chen, Zhou & Ren Reference Chen, Zhou and Ren2019)
\begin{align} \frac {\partial \overline {\rho }}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\overline {\rho }\widetilde {\boldsymbol{u}})=0, \\[-28pt] \nonumber \end{align}
\begin{align} \frac {\partial \overline {\rho }\widetilde {\boldsymbol{u}}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\overline {\rho }\widetilde {\boldsymbol{u}}\widetilde {\boldsymbol{u}}) =-\boldsymbol{\nabla }\overline {p}+\boldsymbol{\nabla }\!\left [\overline {\rho }\widetilde {\nu }\left (\boldsymbol{\nabla }\widetilde {\boldsymbol{u}}+\left (\boldsymbol{\nabla }\widetilde {\boldsymbol{u}}\right )^T-\frac 23\boldsymbol{I}\boldsymbol{\nabla }\boldsymbol{\cdot }\widetilde {\boldsymbol{u}}\right )\right ]+\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau }_u^{\textit{res}}, \end{align}
where the bar and tilde denote Reynolds filtering and Favre filtering, respectively. Here
$\overline {\rho }, \widetilde {\boldsymbol{u}}, \overline {p}, \widetilde {\nu }$
represent the filtered density, velocity, pressure and kinematic viscosity, respectively. Here
$\boldsymbol{I}$
and
$\boldsymbol{\nabla}$
are the unit vector and del operator, respectively. Here
$\boldsymbol{\tau }_u^{\textit{res}}$
is the subgrid stress, which is modelled through the turbulent viscosity hypothesis as
\begin{equation} \boldsymbol{\tau }_u^{\textit{res}}=\overline {\rho }\widetilde {\nu }_t\bigg (\boldsymbol{\nabla }\widetilde {\boldsymbol{u}}+\left (\boldsymbol{\nabla }\widetilde {\boldsymbol{u}}\right )^T-\frac 23\boldsymbol{I}\boldsymbol{\nabla }\boldsymbol{\cdot }\widetilde {\boldsymbol{u}}\bigg ), \end{equation}
where
$\widetilde {\nu }_t$
is the subgrid turbulent viscosity, calculated using the dynamic Smagorinsky model (Germano et al. Reference Germano, Piomelli, Moin and William1991).
The combustion model selected for this study is the FPV approach, which stands as an enhanced version of the steady-flamelet model. For the steady-flamelet model, the temperature and the mass fraction of species
$i$
can be obtained by solving the equations in mixture fraction space,
\begin{equation} \rho \frac \chi 2\frac {\partial ^2Y_i}{\partial Z^2}+\dot {\omega }_{i}=0, \end{equation}
\begin{equation} \rho \frac \chi 2\frac {\partial ^2T}{\partial Z^2}+\sum _{i=1}^N\frac {h_i}{c_{\!p}}\dot {\omega }_i=0, \end{equation}
where
$\rho$
,
$T$
,
$c_{\!p}$
are density, temperature and specific heat at constant pressure, respectively. Here
$Y_i$
,
$\dot {\omega }_i$
,
$h_i$
are mass fraction, the chemical source term and specific enthalpy for species
$i$
, respectively. Here
$\chi$
is the scalar dissipation rate of mixture fraction. Then, the thermochemical quantities
$\phi$
, including species mass fractions, temperature and chemical source term, are parameterized by mixture fraction and scalar dissipation rate,
\begin{equation} \phi =F(Z,\chi _{\textit{st}}), \end{equation}
where
$\chi _{\textit{st}}$
is the scalar dissipation rate at stoichiometry.
However, the steady-state flamelet equations encounter the issue of multiple solutions when solving for unsteady flames (Zhang, Liu & Yang Reference Zhang, Liu and Yang2017). To address this problem, the FPV approach uses a progress variable
$C$
to replace the scalar dissipation rate for parameterizing the flamelet solution (Pierce Reference Pierce2001). In this paper, the progress variable
$C$
is defined as the sum of the main species mass fractions as (Ihme & Pitsch Reference Ihme and Pitsch2008b
)
\begin{equation} C=Y_{\mathrm{CO}}+Y_{\mathrm{CO_2}}+Y_{\mathrm{H_2}}+Y_{\mathrm{H_2O}}. \end{equation}
Filtered combustion variables are obtained by integrating chemical state relationships over the joint subgrid probability density function (PDF) of
$Z$
and
$C$
. The filtered thermal–chemical quantities
$\widetilde {\phi }$
are calculated by
\begin{equation} \widetilde {\phi }=\int \phi (Z,C)\widetilde {P}(Z,C)\,{\rm d}Z\,{\rm d}C, \end{equation}
where
$\phi (Z,C)$
are the chemical state relationships obtained from the steady-state flamelet library. Here
$\widetilde {P}(Z,C)$
is the density-weighted joint filtered density function of mixture fraction and progress variable. The joint subgrid PDF is modelled by first writing
\begin{equation} \widetilde {P}(Z, C) = \widetilde {P}(C|Z)\widetilde {P}(Z), \end{equation}
where
$\widetilde {P}(Z)$
is the
$\beta$
-PDF parameterized by the filtered mixture fraction
$\widetilde {Z}$
and its variance
$\widetilde {Z^{\prime \prime 2}}$
. The
$\widetilde {P}(C|Z)$
is modelled as a Dirac delta function by assuming that each subgrid chemical state is represented by a single flamelet solution,
\begin{equation} \widetilde {P}(C|Z) = \delta (C-\widetilde {C|Z}), \end{equation}
where
$\widetilde {C|Z}$
is given by the steady-flamelet solution,
\begin{equation} \widetilde {C|Z} = C(Z,\chi _{\textit{st}}). \end{equation}
Eventually, the filtered thermal–chemical quantities
$\widetilde {\phi }$
are obtained
$\widetilde {Z}$
,
$\widetilde {Z^{\prime \prime 2}}$
and
$\widetilde {C}$
as
\begin{equation} \widetilde {\phi }=F(\widetilde {Z},\widetilde {Z^{\prime \prime 2}},\widetilde {C}), \end{equation}
where
$\widetilde {Z}$
,
$\widetilde {Z^{\prime \prime 2}}$
and
$\widetilde {C}$
can be obtained by simultaneously solving the transport equations, mass conservation equation and momentum equation. In the FPV approach, the transport equations of the filtered mixture fraction
$\widetilde {Z}$
and its variance
$\widetilde {Z^{\prime \prime 2}}$
and the filtered progress variable
$\widetilde {C}$
are as follows (Chen et al. Reference Chen, Zhou and Ren2019):
\begin{equation} \frac {\partial \overline {\rho }\widetilde {Z}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\overline {\rho }\widetilde {\boldsymbol{u}}\widetilde {Z}\right )=\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\overline {\rho }\widetilde {D}\boldsymbol{\nabla }\widetilde {Z}\right )+\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau }_Z^{\textit{res}}, \end{equation}
\begin{equation} \frac {\partial \overline {\rho }\widetilde {Z^{\prime \prime 2}}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\big (\overline {\rho }\widetilde {\boldsymbol{u}}\widetilde {Z^{\prime \prime 2}}\big )=\boldsymbol{\nabla }\boldsymbol{\cdot }\big (\overline {\rho }\widetilde {D}\boldsymbol{\nabla }\widetilde {Z^{\prime \prime 2}}\big )+\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau }_{Z^{\prime \prime 2}}^{\textit{res}}\\-2\boldsymbol{\tau }_Z^{\textit{res}}\boldsymbol{\nabla }\widetilde {Z}-\overline {\rho }\widetilde {S}_\chi , \end{equation}
\begin{equation} \dfrac {\partial \overline {\rho }\widetilde {C}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\overline {\rho }\widetilde {\boldsymbol{u}}\widetilde {C}\right )=\boldsymbol{\nabla }\boldsymbol{\cdot }\left (\overline {\rho }\widetilde {D}\boldsymbol{\nabla }\widetilde {C}\right )+\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau }_C^{\textit{res}}+\overline {\rho }\widetilde {\dot {\omega }}_c, \end{equation}
where
$\boldsymbol{\tau }_Z^{\textit{res}}$
,
$\boldsymbol{\tau }_C^{\textit{res}}$
and
$\boldsymbol{\tau }_{Z^{\prime \prime 2}}^{\textit{res}}$
are the subgrid scalar flux terms, modelled based on the gradient diffusion hypothesis with the diffusion coefficient being calculated by dynamic Smagorinsky model. Here
$\widetilde {D}$
is the filtered molecular diffusion coefficient,
$\widetilde {\dot {\omega }}_c$
is the filtered the chemical source term of the progress variable and
$\widetilde {S}_\chi$
is the subgrid scalar dissipation rate, modelled as
\begin{equation} \widetilde {S_\chi }=C_\chi \frac {\widetilde {Z^{\prime \prime 2}}}{\tau _t}, \end{equation}
where
$\tau _t$
is the subgrid turbulent time scale and
$C_\chi$
is a constant with the typical value of 2.0 (Kemenov, Wang & Pope Reference Kemenov, Wang and Pope2012).
3. Experimental configuration and computational set-up
Having established the numerical foundation, this section describes the experimental configuration, commencing with description of the Sydney bluff-body burner and proceeding to delineate the computational implementation parameters.
The Sydney bluff-body burner consists of a cylindrical bluff body with a central orifice, located at the centre of a coflow air wind tunnel. The diameter of the cylinder
$D_b$
is 50 mm, and the diameter of the central hole
$D_{\!j}$
is 3.6 mm. The central jet is a fuel jet which is composed of CH
$_{4}$
and H
$_{2}$
(
$1:1$
in volume), with an inflow temperature
$T_{\!j}$
= 298 K. The flow around the exterior of the bluff body is air coflow at a velocity of
$U_{\!c}$
= 35 m s−1, and inflow temperature
$T_c$
= 298 K.
The simplified computational domain is shown in figure 1, with the origin of the coordinate system at the centre of the jet outlet. The geometric dimensions of the computational domain are
$x\times {r}\times \theta =80D_{\!j}\times 24D_{\!j}\times 2\pi$
. A structured mesh is used, with grid nodes along the axial (
$x$
), radial (
$r$
) and circumferential (
$\theta$
) directions as 256
$\times$
165
$\times$
64. Grid nodes are uniformly distributed in the circumferential direction, while they are non-uniformly distributed in the other two directions. Grid refinement is performed in the recirculation zone, inlet zone and shear layer region, with approximately
$2.7\times 10^6$
grid cells in total. According to the research by Kempf et al. (Reference Kempf, Lindstedt and Janicka2006), this grid resolution is sufficient to resolve an adequate amount of turbulent kinetic energy, meeting the demands for high-precision simulation (Wei et al. Reference Wei, Xie, Zhang and Ren2021).
Simplified computational domain.
The simulations are conducted in a three-dimensional cylindrical coordinate system based on a conservative, second-order finite-volumes method. The diffusion term is approached using a second-order central scheme, and the convection term uses a kinetic-energy-conserving second-order central scheme (Pierce & Moin Reference Pierce and Moin2004). Time advancement is achieved through a second-order semi-implicit method. The time step is controlled by the Courant–Friedrichs–Lewy number with a value of 0.8 to ensure stability and accuracy during computation. A detailed reaction mechanism GRI-Mech 2.11 which contains 277 elementary chemical reactions of 49 species, is used to generate a laminar steady state flamelet library consisted of all thermochemical quantities.
For boundary conditions, the fuel jet inlet is specified with the mean velocity profile of a fully developed turbulent pipe flow and turbulence intensity. The coflow inlet is set with a plug flow mean velocity profile with a turbulence intensity of 2 %. The outlet boundary is treated with a convective boundary condition. The outer boundary is considered a far-field boundary. For all walls, the velocity is bound by a no-slip Dirichlet condition, while Neumann conditions are applied for other boundary scalars.
Initially, the HM1E case is chosen to validate the accuracy and reliability of the numerical method. To investigate the influence of the momentum flux ratio (
$J$
) on the mixing within the recirculation zone, we manipulate
$J$
by adjusting the velocity of the central fuel jet. The fuel jet velocity is systematically reduced from 108 m s−1 in the HM1E case to 48 m s−1, encompassing six cases in total, including the HM1E case. The specific inflow conditions for these cases are detailed in table 1.
Inlet parameters.
4. Results and discussion
4.1. Validation of the numerical calculations
This section aims to validate the numerical simulation results by comparing them with experimental data. The scalar data (temperature and mixture fraction) are from the HM1 flame, while the velocity data are from the HM1E flame (Kempf et al. Reference Kempf, Lindstedt and Janicka2006). The HM1E condition, characterized by a lower coflow velocity (35 m s−1) and adapted jet velocity (108 m s−1), resulted from facility constraints at Sydney and reflects a slight operational adjustment from the original HM1 set-up. This minor difference in operating conditions does not affect the validity of the present comparative analysis. Figure 2 shows a side-by-side comparison of the simulation and experimental data for (figure 2
a) mean axial velocity
$U_m$
, (figure 2
b) mean temperature
$T_m$
, (figure 2
c) mean mixture fraction
$Z_m$
and (figure 2
d) mixture fraction variance
$Z_{rms}$
across various radial positions.
The radial profiles of (a) mean axial velocity, (b) mean temperature, (c) mean mixture fraction and (d) mixture fraction variance.
In figure 2(a), the simulation’s velocity field aligns closely with the experimental data, indicating the accuracy of the LES-FPV method in predicting velocities. Notably, regions with significant velocity gradients in the simulation correspond accurately to those identified in the experimental data, affirming the precise prediction of the shear layer’s position. Moreover, the simulation accurately captures the locations with negative velocities, indicative of the recirculation zone’s position and scale.
Turning to figure 2(b), the radial distribution of mean temperatures across different cross-sections shows a high degree of agreement between simulation and experiment, especially within the reaction zone. However, at the upstream location (
$x/D_{b}$
= 0.3), the peak temperature is slightly underestimated. Despite this, the overall consistency between the simulation results and experimental data is maintained.
Figure 2(c) illustrates the radial distribution of mean mixture fraction, with the simulation results exhibiting a strong correlation with the experimental data. A slight overestimation is observed at
$x/D_{b}$
= 0.3, which is consistent with the temperature distribution prediction at the same location. This suggests an interconnected relationship between temperature and mixture fraction predictions.
Figure 2(d) presents a comparison between the simulation results and experimental data for the mixture fraction variance. It can be observed that the LES-FPV results closely match the experimental data, thoroughly validating the accuracy of the simulation method in capturing turbulent mixing characteristics. Regions with higher mixture fraction variance reflect areas of stirring, such as the shear layers. In contrast, regions with lower variance indicate that the turbulent mixing is becoming more uniform. Notably, in the recirculation zone, the significant decrease in variance suggests that thorough and effective mixing occurs in this region.
These comparisons underscore the efficacy of the LES-FPV method and the numerical settings employed in this study to reliably predict the flow field and combustion characteristics of the Sydney bluff-body burner. The validation process not only supports the accuracy of our numerical approach but also provides a robust foundation for subsequent theoretical research.
4.2. A characteristic mixture fraction
$w$
ithin the recirculation zone
In a detailed analysis of the HM1E case, figures 3 and 4 illustrate the distribution of mean axial velocity and streamlines, as well as the mean mixture fraction. The black dashed line in these figures denotes the contour where the mean axial velocity
$U_m$
= 0 m s−1, clearly demarcating the recirculation zone.
The distribution of mean axial velocity and streamline (dashed line, isoline of
$U_m$
= 0).
The distribution of mean mixture fraction (dashed line, isoline of
$U_m$
= 0).
As depicted in figure 3, the interaction between the fuel jet and the air coflow creates the recirculation zone behind the bluff body. Within this zone, two distinct vortex structures emerge, one on each side of the fuel jet and air coflow, characterized by strong shear effects. The presence of these large-scale vortex structures significantly enhances the mixing of fuel and air.
Corroborating this observation, figure 4 illustrates that the gradients of mixture fraction changes are most pronounced near these vortex structures, confirming that they play a pivotal role in promoting fuel–air mixing.
Figure 4 also reveals that the mixture fraction within the recirculation zone is not entirely uniform, indicating a heterogeneous mixing state. However, localized regions within the recirculation zone exhibit a tendency towards mixture fraction consistency, suggesting the presence of localized uniformity.
To further substantiate this localized uniformity, figure 5 presents the radial distribution of mixture fractions at various positions. Notably, near the upstream position (at
$x = 15$
mm), away from the vortex centre, the mixture appears to be relatively uniform, with the gradient of the mixture fraction within the recirculation zone approaching zero.
The radial profiles of mean mixture fraction at
$x$
= 15 mm,
$x$
= 30 mm and
$x$
= 40 mm.
The localized uniformity observed within the recirculation zone suggests a statistical property that could be of significant importance for characterizing the mixing state in this region. To quantify this property and describe the mixing state, we introduce a new parameter: the characteristic mixture fraction, denoted as
$Z_{u}$
. The concept of characterizing mixing through the nearly uniform state within large-scale structures finds a strong analogy in the established research on planar shear layers. In such configurations, the mixture fraction within large-scale structures also tends towards uniformity (Pickett & Ghandhi Reference Pickett and Ghandhi2002). In this work, we extend this conceptual framework to the recirculation zone of bluff-body flows, which acts as the dominant large-scale structure governing mixing behind the bluff body. Consequently,
$Z_{u}$
is defined as the mixture fraction value at the position within the recirculation zone,
$\textit{Rc}$
, where the gradient of the mixture fraction approaches zero. Mathematically,
$Z_{u}$
is defined as
\begin{equation} Z_u=\lim _{\frac {\partial Z_m}{\partial r}{\operatorname *{\to }0}}Z_m,\ Z_m \quad \text{in} \quad \textit{Rc}; \end{equation}
where
$Z_m$
is within the region
$\textit{Rc}$
. While the introduction of a new definition or concept such as
$Z_{u}$
may not, in itself, hold intrinsic significance, its true value lies in the potential to uncover hidden physics, enhancing our understanding of the mixing state within the recirculation zone by revealing previously obscured insights.
Traditional approaches often tend to oversimplify the mixing process by assuming uniformity within the recirculation zone, which may not fully capture the complex and non-uniform nature of the mixing dynamics. By focusing on the characteristic mixture fraction
$Z_{u}$
, we can begin to move beyond these oversimplified assumptions and provide a more detailed and nuanced representation of the mixing state.
However, focusing on
$Z_{u}$
alone may not be sufficient. Should
$Z_{u}$
prove meaningful, it should shed light on the relationship between locally uniform mixing regions and key parameters such as inflow conditions and bluff body geometry, which is challenging to quantify accurately. Given this potential,
$Z_{u}$
could serve as a valuable metric to characterize the mixing state in a non-uniform environment, providing a statistical measure that captures the essence of the mixing dynamics. This approach may help us identify patterns and correlations that are otherwise difficult to discern through traditional methods alone.
The momentum flux ratio
$J$
, defined as the ratio of the coflow momentum flux to the fuel jet momentum flux, is a critical dimensionless parameter governing the flow and mixing characteristics within the recirculation zone of the Sydney bluff-body burner.
Previous studies (Dally et al. Reference Dally, Fletcher and Masri1998) have established that variations in the momentum flux ratio dominantly control the recirculation zone geometry (e.g. length, vortex scale) and turbulent mixing intensity. Dally et al. (Reference Dally, Fletcher and Masri1998) also demonstrated that in isothermal flows, the momentum flux ratio of the fuel jet to the coflow serves as the primary scaling parameter dictating the flow field structure.
Building on this foundation, we propose that the characteristic mixture fraction (
$Z_{u}$
) should be inherently correlate with
$J$
, if
$Z_{u}$
is meaningful. By definition,
$Z_{u}$
quantifies the localized uniformity of mixing within the recirculation zone, while the momentum flux ratio directly influences the mixing process by regulating the distribution of flow energy and the evolution of vortices. This connection would suggest a framework for understanding the link between flow dynamics and mixing in recirculating flows.
To simplify, we fixed the burner’s geometry. Under this condition, the momentum flux ratio becomes a key parameter governing the flow field structure. Based on dimensional analysis theory (Barenblatt Reference Barenblatt1996), we propose a relationship between the characteristic mixture fraction and the momentum flux ratio. This relationship is expressed as
\begin{equation} \frac {Z_u}{Z_{\textit{st}}}=f(J). \end{equation}
Here,
$Z_{\textit{st}}$
is the stoichiometric mixture fraction, which is species dependent and kept constant in the present study and
$f$
is a general unknown function. This step establishes the framework for scaling but does not specify the functional form of
$f$
.
To progress beyond this general relationship, we introduce a central hypothesis motivated by phenomena frequently observed in turbulent flows and recirculating systems, self-similarity. The power-law form is the mathematical form consistent with self-similarity. This power-law relationship represents as
\begin{equation} \frac {Z_u}{Z_{\textit{st}}}\propto {J^a}. \end{equation}
It remains invariant under scale transformations of the form
\begin{equation} \frac {Z_u}{Z_{\textit{st}}}\rightarrow {A^a\boldsymbol{\cdot }\frac {Z_u}{Z_{\textit{st}}}}, \quad {J}\rightarrow {A\boldsymbol{\cdot }{J}}, \end{equation}
revealing inherently self-similar mixing patterns within the recirculation zone. Should this relationship (4.3) hold true, it would greatly enhance our understanding of the mixing state within the recirculation zone by offering a new framework for interpreting the complexities of mixing dynamics.
4.3. Is the power-law true?
The emergence of such an elegant scaling law might seem counterintuitive when considering the inherent complexity of flow structures within recirculation zones, particularly the multiscale vortical interactions and three-dimensional separation phenomena. This seeming incongruity becomes even more pronounced in reacting flow systems, where exothermic chemical processes introduce additional nonlinear couplings between fluid dynamics and thermochemical evolution.
Given these added complexities in reacting flows, we first establish the relationship’s validity in non-reacting flows. Here, the absence of chemical energy coupling allows us to isolate and study the purely fluid dynamics scaling behaviour. For consistency with the reacting cases, the fuel stream in these non-reacting simulations uses a mixture of CH
$_{4}$
and H
$_{2}$
(
$1:1$
in volume), and its corresponding density and viscosity are employed.
To assess the power-law (4.3),
$J$
is adjusted by systematically varying the fuel jet velocity in the burner, while we maintain the velocity of coflow constant at
$U_{\!c}$
= 35 m s−1. Specifically, the fuel jet velocity is gradually decreased from 108 m s−1 to 48 m s−1, resulting in a corresponding increase in
$J$
from 0.33 to 1.66. Detailed inflow conditions are provided table 1.
There are six cases in total. First, we will check whether localized mixing uniformity exists across all cases. As we previously demonstrated in the HM1E reacting flow case (
$U_{\!j}$
= 108 m s−1), the presence of localized uniformity enabled us to define the characteristic mixture fraction
$Z_{u}$
. This approach will be consistently applied to all non-reacting flow cases to ensure a thorough analysis.
As the flow passing the bluff body undergoes an abrupt expansion, a recirculation zone is anticipated to form in each of these cases. Conventionally, one might expect the mixture within these recirculation zones to be uniform, attributed to the effects of large-scale convective mixing. However, the actual mixing within the recirculation zone is more complex and not as uniformly distributed as initially presumed.
In figure 6(b), we present radial profiles of the time-averaged mixture fraction
$Z_m$
, obtained from our numerical simulations at the axial location
$x$
= 15 mm. The recirculation zone in this plot is delineated in figure 6(a), which shows the radial profiles of axial velocity
$U_m$
. In figure 6(a), the recirculation zone corresponds to the region where
$U_m \lt 0$
, situated between the inner shear layer and the outer shear layer. Referring back to figure 6(b), it is evident that the mixture fraction within this region is not uniformly distributed.
The radial profiles of mean velocity and mixture fraction at different jet velocities.
In contrast, when examining a farther upstream location at
$x$
= 5 mm (as shown in figures 6
c and 6
d), the gradient of the mixture fraction is found to be close to zero within the recirculation zone. This suggests the presence of localized uniformity, which aligns with our expectations. Although the mixture is not uniform throughout the entire recirculation zone, localized uniformity is observed within certain region of the recirculation zone.
In light of this observation, we can define the characteristic mixture fraction
$Z_{u}$
in the localized uniform mixing region according to the definition provided in (4.1) for all six cases. To verify the power-law (4.3), we plotted
$Z_{u}$
versus
$J$
for each case in figure 7 using a log–log coordinate system. In this coordinate system, the relationship between
$Z_{u}$
and
$J$
would appear as a straight line with its slope
$a$
, if the power-law (4.3) holds true. As shown in figure 7, the data points align closely along a straight line, providing strong evidence in support of the proposed power-law relationship. This alignment not only confirms the validity of our hypothesis but also underscores the robustness of the power-law model in describing the mixing behaviour within the recirculation zone under varying
$J$
.
The characteristic mixture fraction with
$J$
in non-reacting flows.
The distribution of mean mixture fraction under non-reacting condition at
$U_{\!j}$
= 108 m s−1.
It is worth noting that the power-law exhibited here has a negative slope, indicating that as
$J$
increases, the characteristic mixture fraction decreases. This trend reflects a flow dominated by the coflow. Figure 8 illustrates the mean mixture fraction distribution for the case where
$U_{\!j}$
= 108 m s−1, which corresponds to the lowest
$J$
among the six cases examined. The black dashed line in the figure denotes the isoline for
$U_m$
= 0 m s−1, delineating the recirculation zone. Even in this lowest
$J$
scenario, the recirculation zone fully envelops the central fuel jet. As
$J$
increases, relatively more of the coflow is entrained into the recirculation zone and mixes with the fuel jet. This increased coflow content leads to a more diluted mixture, resulting in a lower characteristic mixture fraction
$Z_{u}$
.
However, this power-law also presents some confusing aspects. The mixture fraction, as well as the characteristic mixture fraction
$Z_{u}$
, should be bounded, with values ranging between 0 and 1. Nevertheless, according to the power-law depicted in figure 8, if
$J$
continues to decrease, the characteristic mixture fraction would keep increasing as per the power-law trend. This is not supposed to happen. Once
$J$
decreases beyond a certain point, there must be some hidden physics that we have not found yet. For now, we will leave it here as an open question and proceed to the next section to see what new insights we could discover.
4.4. Does power-law scaling persist in reacting flows?
The exploration of non-reacting flows has unveiled intriguing dynamics, particularly the power-law relationship and its implications. These insights naturally lead us to a critical question: Does power-law scaling persist in reacting flows?
In the transition from non-reacting to reacting flows, several new factors come into play that can significantly alter the dynamics observed in the non-reacting case. In reacting flows, chemical reactions introduce additional complexity through the release of heat and changes in fluid properties such as density and viscosity. These factors can modify the flow field and potentially influence the existence and form of a power-law relationship. The heat release from exothermic reactions can lead to thermal expansion, which affects the velocity field and mixing processes. Additionally, the progress of the chemical reactions can be influenced by the local mixture fraction, creating a coupling between the chemical and fluid dynamical processes that is not present in non-reacting flows. Our investigations into reacting flows therefore account for these additional complexities to fully understand the role of the power-law in such systems.
To explore this question, we will apply the methodology previously used for non-reacting flows to reacting flows. This includes systematically varying
$J$
while accounting for the additional complexities introduced by chemical reactions, observing the localized mixing uniformity and defining the characteristic mixture fraction
$Z_{u}$
. The inflow conditions are maintained identical to those of the non-reacting flows in table 1. Through this approach, we aim to assess if the power-law relationship remains valid and to identify any significant deviations or new patterns that may arise due to the reactive nature of the flow.
Similar to the non-reacting cases, there are also six reacting flow cases where
$J$
varies from 0.33 to 1.66. In figure 9, the contours of mixture fraction are plotted for all the six cases, along with the recirculation zones delineated by the isolines for
$U_m$
= 0 m s−1 (black dashed lines). Upon examining each recirculation zone, it is evident that the mixture fraction distribution is generally non-uniform. However, localized uniformity can still be observed within these zones. Figure 10 further illustrates the radial profiles of the time-averaged mixture fraction
$Z_m$
at the axial location
$x$
= 15 mm, showing near-zero gradients of
$Z_m$
within the recirculation zone.
The distributions of mean mixture fraction at different jet velocities (dashed line, isoline of
$U_m$
= 0 m s−1).
The radial profiles of mean mixture fraction at different jet velocities in reacting flows.
We can therefore define the characteristic mixture fraction
$Z_{u}$
within the localized uniform mixing regions using the same criteria outlined in (4.1). To test the endurance of the power-law relationship (4.3) in reacting flows, we again employ a log–log coordinate system, plotting
$Z_{u}$
against
$J$
for each reacting case as shown in figure 11(a). As expected, figure 11(a) was anticipated to display a straight line indicative of the power-law relationship. However, the actual results significantly diverged from this expectation. Contrary to non-reacting cases, figure 11(a) reveals two distinct straight lines intersecting at a point, rather than a single straight line.
The characteristic mixture fraction with
$J$
in reacting flows.
To verify whether the right segment in figure 11(a) forms a straight line, we added another case. The three data points from these three cases align well along a straight line in figure 11(b). Meanwhile, we also added an additional data point to the left-hand segment, and the left-hand segment continues to form a straight line as well.
Taking into account that
$Z_{u}$
must be between 0 and 1, the two intersecting lines in figure 11(b) make more sense than the single line in figure 8. At the intersection point,
$Z_{u}$
reaches a peak below 1, which fits the definition of the mixture fraction. This prompts a question: Do these two intersecting lines have physical meaning? As explained in the previous section, the negative slope of the right-hand straight line reflects coflow dominated flow conditions, where the recirculation zone fully envelops the central jet, as depicted in figures 9(e) and 9(f). The right-hand straight line with a negative slope in figure 11(b) is physically meaningful.
The left-hand straight line with a positive slope in figure 11(b) also has physical meaning. It indicates the central jet penetrating the recirculation zone, a scenario depicted in figure 9(a–d). As
$J$
increases, it becomes harder for the jet to fully penetrate the recirculation zone. Consequently, relatively more of the fuel jet is entrained into the recirculation zone to mix with coflow air, increasing the characteristic mixture fraction and producing a positive slope in the power-law.
The two power-law lines in figure 11(b) represent two distinct flow regimes: one where the central jet penetrates the recirculation zone and another where the zone fully envelops the jet. These two scenarios are physically meaningful and not due to differences between reacting and non-reacting flows. In non-reacting flows, only a single power-law was observed because
$J$
tested were too high to reveal jet penetration. Reducing
$J$
in non-reacting flows should also result in two intersecting power-law lines.
The intersection point of these two lines also has a special meaning. It marks the transition between flow regimes, from the recirculation zone fully enveloping the central jet to the central jet penetrating the recirculation zone, and vice versa. This distinctive transition point was also noted in Dally et al. (Reference Dally, Fletcher and Masri1998) work, though it was not yet correlated with physical parameters at that time. Our current research bridges this gap by identifying the characteristic mixture fraction
$Z_{u}$
as a key parameter, suggesting a potential opportunity for understanding this transition point more comprehensively.
The transition point corresponds to a flow state where the recirculation zone just envelops the central jet, such as when the zone’s boundary is tangent to the jet centreline. This flow state is inherently unstable, representing a delicate balance unlikely to be sustained in reality. Figure 11(b) illustrates this instability, as the intersection of the two power-laws is a singularity – an unstable critical point. The existence of this transition point reveals the dynamic and complex nature of flow regime transitions.
4.5. Differences between reacting and non-reacting flows
Sections 4.3 and 4.4 have established the power-law scaling of the characteristic mixture fraction
$Z_u$
with the momentum flux ratio
$J$
in both non-reacting and reacting flows. The empirically derived exponents are not merely fitting parameters but reflect the physical response of mixing to the momentum balance and heat-release effects within the recirculation zone. In this section, we compare the reacting and non-reacting flows to examine how heat release modifies the mixing dynamics and alters the underlying scaling behaviour, through the distinct differences in the scaling exponents and flow regimes.
To further examine whether a dual-power-law relationship also exists under non-reacting conditions, additional non-reacting cases were simulated. Additional non-reacting cases were simulated by lowering the coflow velocity in order to reach lower momentum flux ratios. This approach allows us to systematically explore the governing parameter
$J$
while keeping the Reynolds number within a well-validated range for our LES framework, thereby ensuring computational accuracy and consistency with the established baseline cases.
The inclusion of these additional cases reveals a dual-power-law relationship in non-reacting flows, analogous to that established in reacting flows (§ 4.4), as shown in figure 12. This result confirms that the dual-power-law behaviour is a general feature of bluff-body flows, independent of chemical reactions. However, a clear distinction remains: compared with the reacting cases, the transition point in non-reacting flows occurs at a smaller
$J$
, and the absolute value of the scaling exponent is noticeably larger.
The characteristic mixture fraction with
$J$
under different conditions.
These quantitative differences in transition point and scaling exponent point to a fundamental modification of the flow and mixing dynamics by heat release. To unravel the underlying physics, we attribute these modifications to two synergistic mechanisms rooted in heat-release-induced thermophysical effects.
-
(i) Heat release induces significant thermal expansion, drastically reducing the gas density within the recirculation zone. This density reduction lowers the effective momentum resistance of the recirculation bubble, effectively making it a ‘lighter’ hydrodynamic obstacle. Consequently, the central jet can penetrate this weakened bubble at a higher critical momentum flux ratio compared with non-reacting flows. This mechanism, which alters the inertial balance between the jet and the recirculation zone, is consistent with the previous observation that heat release weakens entrainment in reacting shear layers through density reduction (Dahm Reference Dahm2005).
-
(ii) Heat release also markedly increases the molecular viscosity while simultaneously reducing the density, which significantly alters the local Reynolds number of the flow. The increased viscosity suppresses turbulent mixing, reducing mixing efficiency. This effect, as reported in studies of reacting shear layers where heat release lowers the local entrainment velocity (Jahanbakhshi & Madnia Reference Jahanbakhshi and Madnia2018), causes the characteristic mixture fraction
$Z_u$
to respond more gradually to changes in
$J$
. This results in a shallower power-law exponent (e.g. −0.15 for reacting versus −0.5 for non-reacting flows in the jet-envelopment regime).
These two mechanisms act synergistically to reshape the momentum and mixing balance within the recirculation zone. The first mechanism predominantly influences the macroscopic flow topology by weakening the recirculation bubble and shifting the regime transition towards higher
$J$
. The second mechanism modulates the microscopic mixing dynamics, suppressing turbulence and altering the rate at which
$Z_u$
varies with
$J$
. Together, they yield the modified scaling observed in reacting flows.
Overall, the comparison demonstrates that while the two-regime power-law behaviour is a fundamental feature of bluff-body recirculating flows, the presence of combustion profoundly modifies the scaling characteristics. The differences in transition location and scaling exponent arise from the nonlinear interplay among heat-release-driven expansion, viscosity variation and turbulent transport, revealing the coupled nature of combustion and fluid dynamics in determining the statistical structure of turbulent mixing. By delineating the distinct roles of inertial balance and mixing efficiency, this analysis provides a mechanistic framework for understanding the coupled nature of combustion and fluid dynamics in recirculating flows.
5. Conclusions
In this study, the mixing state within the recirculation zone was investigated using LES combined with the FPV approach. The accuracy and applicability of the LES-FPV method were validated, which has been shown to predict the flow field and combustion characteristics of the Sydney bluff-body burner accurately. Further detailed studies on the mixing state within the recirculation zone led to the following conclusions.
-
(i) Localized uniformity and characteristic mixture fraction (
$Z_u$
). Research on the HM1E case revealed that while the mixing state in the recirculation zone is not entirely uniform, it possesses localized uniformity. Specifically, at some positions within the zone, the mixture fraction is almost constant. Based on this observation, a novel characteristic quantity, termed the characteristic mixture fraction (
$Z_u$
), was defined. This parameter is subsequently correlated with the momentum flux ratio, assuming a power-law relationship, providing deeper mechanistic insights. -
(ii) Power-law relationship. Both non-reacting and reacting cases exhibit a clear power-law relationship between the characteristic mixture fraction and the momentum flux ratio. Two distinct power-law regimes are identified, corresponding to different flow structures: a jet-penetration regime (positive slope) and a jet-envelopment regime (negative slope). The intersection of the two power-law lines marks a critical transition point between flow regimes.
-
(iii) Effect of heat release. In comparison with non-reacting flows, the distinct scaling behaviour in reacting flows is attributed to thermophysical effects induced by heat release. This modification operates through two interconnected pathways:
-
(a) Macroscale hydrodynamic weakening. Thermal expansion reduces density within the recirculation zone, lowering its effective momentum resistance. This weakens the recirculation bubble, allowing the central jet to penetrate at a higher critical momentum flux ratio
$J$
, thereby shifting the transition point. -
(b) Microscale mixing suppression. The concomitant increase in viscosity suppresses turbulent mixing. This attenuation of mixing rates causes the characteristic mixture fraction
$Z_u$
to respond more gradually to changes in
$J$
, resulting in a shallower power-law exponent.
The confluence of these mechanisms, one altering the inertial balance and the other modulating the mixing rate, quantitatively explains the modified scaling law. This analysis demonstrates that the momentum flux ratio remains the governing parameter, but its functional influence is fundamentally rescaled by combustion-induced thermophysical effects.
These findings provide new insights into the complex mixing processes within recirculation zones and offer a novel perspective for understanding the dynamic evolution of turbulent flow structures. The derived scaling law further implies that changes in the stoichiometric mixture fraction – such as through varying fuel composition – may shift the scaling curve without altering the exponent, a hypothesis that warrants systematic validation. Future studies should focus on theoretically deriving the power-law exponents, extending the analysis across different fuels and reaction conditions, and exploring potential connections between scaling breakdown and flame blowoff dynamics.
Funding
This work was supported by National Natural Science Foundation of China (52025062).
Declaration of interests
The authors report no conflict of interest.
Author contributions
J.Z. conceived the ideals. X.L. performed the numerical simulations. X.L. and J.Z. analysed the data. All authors wrote the paper, discussed the physics and proofread the paper.
