3.1. Instantaneous flame image and velocity field

Figure 4 is the instantaneous flow field at the vertical plane passing through the burner centreline overlaid on synchronised colour flame images of colour CMOS for $\dot{Q}^{*}=$ 0.50. In figure 4(a–f), the burner rim was level with the floor, while the floor was absent in figure 4(A–F). Since the gas density in the flame is much lower than the ambient air density, the flame near the burner rim rolls up to form toroidal vortices (see figure 4 a–c and figure 4 A–C) due to the Rayleigh–Taylor instability, which entrains a large amount of ambient air into the flame. Under the action of strong buoyancy, the toroidal vortices accelerate downstream with increasing scale (see figure 4 c–f and figure 4 C–F), and the flame below the toroidal vortices contracts towards the centre axis. Then the toroidal vortices gradually burn out, and a new cycle starts near the burner surface. Notably, in figure 4(e–f), the toroidal vortices break at the flame tip, and the flame is coherent along the burner centreline. However, in figure 4(E–F), as the scale of the toroidal vortices increases steadily, the toroidal vortices could develop to the central axis of the burner, resulting in significant flame shedding. The shedding flame continues to rise and quickly burns out. This could result in stronger axial flame pulsation in the configuration without a floor.

Figure 4.

Instantaneous flow fields at the vertical plane passing through the burner centreline overlaid on synchronised colour flame images of colour CMOS for $\dot{Q}^{*}=$ 0.50. The time interval between adjacent images is 1/15 s. (a)–( f) With a flush floor; (A)–(F) without a floor.

The difference in the development process of the toroidal vortices can be revealed by the following vorticity equation:

(3.1) \begin{equation} {\rm D}\boldsymbol{\omega}/{\rm D}t=\big(\boldsymbol{\omega}\boldsymbol{\cdot }\boldsymbol{\nabla }\big)\boldsymbol{V}-\boldsymbol{\omega}\big(\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{V}\big)+\frac{1}{\rho ^{2}}\left(\boldsymbol{\nabla }\rho \times \boldsymbol{\nabla }p\right)+\frac{\rho _{\infty }}{\rho ^{2}}\big(\boldsymbol{\nabla }\rho \times \boldsymbol{g}\big)+\nu {\nabla} ^{2}\boldsymbol{\omega} ,\end{equation}

where $\rho$ , p and g are respectively the gas density, pressure and the gravitational acceleration. The right-hand side terms of (3.1) are the vortex stretching term, dilation term, baroclinic torque, gravitational term and viscous term, respectively. Based on the direct numerical simulation works by Jiang & Luo (Reference Jiang and Luo2000, Reference Jiang and Luo2001), the gravitational term is critical for the creation of flow vorticity in global and local senses and essential to the formation of the large vortical structures in turbulent buoyant diffusion flames.

Figure 5 shows the time-averaged velocity vectors in the central vertical plane and the schematic diagram of $(\boldsymbol{\nabla }\rho \times \boldsymbol{g})=| \boldsymbol{\nabla }\rho | | \boldsymbol{g}| \sin (\delta )$ for $\dot{Q}^{*}=$ 0.50. Here, $\delta$ is the angle between $\boldsymbol{\nabla }\rho$ and $\boldsymbol{g}$ . As shown in figures 5(a) and 5(b), due to the presence of the flush floor, the radial inflow of air at the bottom pushes the flame sheet to be nearly parallel to the burner exit. In figures 5(A) and 5(B), the ambient air flows radially upwards near the burner rim, and the angle ( $\delta _{wo}$ ) between the flame sheet at the burner rim and the burner exit decreases. Then we have $\sin (\delta _{w})\lt \sin (\delta _{wo})$ . The subscripts ‘ $w$ ’ and ‘ $wo$ ’ denote with and without the floor, respectively. Consequently, the gravitational term is lower when the floor is present, resulting in a slower increase in tangential vorticity of the toroidal vortices at the bottom of the flame. Compared with the configuration with a flush floor, the scale and vorticity of the toroidal vortices increase faster, and the turbulent mixing rate of fuel and air is greater at the same height for the configuration without a floor (see figure 4 a–f) (Moreno-Boza et al. Reference Moreno-Boza, Coenen, Carpio, Sánchez and Williams2018).

Figure 5.

Time-averaged velocity vectors in the central vertical plane for $\dot{Q}^{*}=$ 0.50 (a, A) and schematic diagram of $\boldsymbol{\nabla }\rho \times \boldsymbol{g}$ (b, B): (a) and (b) with a flush floor; (A) and (B) without a floor.

3.2. Mean flame shape

Figures 6(a) and 6(b) show the mean flame shapes of the turbulent buoyant flames with and without the floor for $\dot{Q}^{*}=$ 0.50–0.90, respectively. All the flames are anchored at the burner rim and exhibit the necking-in effect near the burner exit. With the increase of z, the flame width increases slowly and begins to decrease gradually above a certain height. As $\dot{Q}^{*}$ increases, the mean flame shape expands outward steadily. However, the mean flame shapes under the two boundary conditions differ greatly under equal $\dot{Q}^{*}$ . Compared with the configuration with a flush floor, the air entrainment rate and the turbulent mixing rate of fuel gas and air both increase in the configuration without a floor, leading to a decrease in the flame height and an increase in the maximum flame width.

Figure 6.

Mean flame shapes with an intermittency of 0.50 for the free buoyant diffusion flames. (a) With a flush floor, (b) without a floor.

3.3. Time-averaged velocity field

Figure 7 shows the time-averaged radial velocity ( $\overline{u}$ ) contours in the central vertical plane for $\dot{Q}^{*}=$ 0.50–0.90 under the two burner boundary conditions. Note that the 0 m s⁻¹ contour is roughly located near the central axis of the burner, which indicates that the flow field exhibits good symmetry. Similar to the experimental results of Weckman & Strong (Reference Weckman and Strong1996) and Tieszen et al. (Reference Tieszen, O’.Hern, Schefer, Weckman and Blanchat2002, Reference Tieszen, O.’Hern, Weckman and Schefer2004), a relatively strong air inflow is formed at the bottom of the flame (z/d $\leq$ 0.33) due to the presence of the floor. Notably, the contours of $0.15\lt \overline{u}\lt 0.30$ m s⁻¹ are tilted upward due to the strong buoyancy of the hot gas in the reactive zone. In the upper part of the flame, the ambient air is mainly engulfed into the flame by the toroidal vortices, named as radial air entrainment. As z increases, $\overline{u}$ decreases rapidly. With the increase of $\dot{Q}^{*}$ , the profile of $\overline{u}$ changes insignificantly. The comparison among figure 7(a–e) and figure 7(A–E) shows that the dimensionless radial locations ( $\xi =r/b$ ) with relatively greater $\overline{u}$ (the 0.25 ms⁻¹ contour) are 0.33 $\leq \xi \leq$ 0.67 and 0.67 $\leq \xi \leq$ 1.00 for the configurations with and without a flush floor, respectively. This is consistent with the experimental results of Tieszen et al. (Reference Tieszen, O’.Hern, Schefer, Weckman and Blanchat2002, Reference Tieszen, O.’Hern, Weckman and Schefer2004) (0.30 $\leq \xi \leq$ 0.70, with a flush floor, $\dot{Q}^{*}=$ 1.73–2.98) and Weckman & Strong (Reference Weckman and Strong1996) (0.67 $\leq \xi \leq$ 1.00, without a floor, $\dot{Q}^{*}=$ 0.39). Compared with the configuration with a flush floor, the peak value of $\overline{u}$ moves towards the burner rim in the configuration without a floor.

Figure 7.

Time-averaged radial velocity ( $\overline{u}$ ). (a)–(e) With a flush floor, (A)–(E) without a floor. From left to right in each row, $\dot{Q}^{*}=$ 0.50, 0.60, 0.70, 0.80 and 0.90, respectively; r $=$ 0 cm denotes the burner centreline.

Figure 8 presents the time-averaged axial velocity ( $\overline{v}$ ) in the central vertical plane under different $\dot{Q}^{*}$ and burner boundary conditions. Near the surface of the burner, $\overline{v}$ is only a few cm s^–1. The centreline $\overline{v}$ increases rapidly to around 2.60 m s⁻¹ due to the strong buoyancy of hot gas in the flame. As z increases, the flow changes from the laminar state to the turbulent state, and the radial profile of $\overline{v}$ gradually changes from a ‘top-hat’ profile to a Gaussian profile. Similar to the experimental results of Tieszen et al. (Reference Tieszen, O.’Hern, Weckman and Schefer2004), as $\dot{Q}^{*}$ increases, the radial profile of $\overline{v}$ varies from a unimodal to a bimodal type due to the increase of the fuel-rich core size. As shown in the discussion of figure 5, compared with the configuration with a flush floor (see figure 8 a–e), the mixing rate of fuel gas and air increases (see figure 8 A–E), and the fuel-rich core scale would be smaller in the configuration without a floor under equal $\dot{Q}^{*}$ . The transition of the radial profile of $\overline{v}$ from unimodal to bimodal is more pronounced in the configuration with a flush floor.

Figure 8.

Time-averaged axial velocity ( $\overline{v}$ ). (a)–(e) With a flush floor, (A)–(E) without a floor. From left to right in each row, $\dot{Q}^{*}=$ 0.50, 0.60, 0.70, 0.80 and 0.90, respectively; r $=$ 0 cm denotes the burner centreline.

3.4. Time-averaged turbulent statistics

Figure 9 presents the contour maps of the time-averaged radial velocity fluctuations squared ( $\overline{u'u'}$ ) in the central vertical plane under different $\dot{Q}^{*}$ and burner boundary conditions. It is found that $\overline{u'u'}$ is relatively small near the burner surface (z/d < 0.33), but increases rapidly with the increase of z. In the radial direction, $\overline{u'u'}$ first increases and then decreases, and its radial profile is bimodal. As $\dot{Q}^{*}$ increases, the contour area of $\overline{u'u'}=$ 0.18 m² s⁻² increases slightly, indicating that the radial flame pulsations are enhanced gradually. In figure 9(a–e) and the experiments of Tieszen et al. (Reference Tieszen, O’.Hern, Schefer, Weckman and Blanchat2002, Reference Tieszen, O.’Hern, Weckman and Schefer2004) ( $\dot{Q}^{*}=$ 1.73–2.98, with a flush floor), the maximum $\overline{u'u'}$ values in the radial direction are located at approximately $\xi =0\mathit{.}47$ and $\xi =0\mathit{.}40$ , respectively. However, the maximum value of $\overline{u'u'}$ is found near $\xi =0\mathit{.}67$ in figure 9(A–E) (without a floor). Consistent with figure 5, the gravitational term at the flame bottom is larger when the floor is absent, and the toroidal vortices grow quickly. This also results in the differences among figure 9(a–e) and figure 9(A–E). Compared with the configuration with a flush floor, the area of the $\overline{u'u'}=$ 0.18 m² s⁻² contour is much greater in the configuration without a floor under the same $\dot{Q}^{*}$ , indicating that the radial pulsation of turbulent buoyant flames is stronger in the configuration without a floor.

Figure 9.

Time-averaged radial velocity fluctuations squared ( $\overline{u'u'}$ ). (a)–(e) With a flush floor, (A)–(E) without a floor. From left to right in each row, $\dot{Q}^{*}=$ 0.50, 0.60, 0.70, 0.80 and 0.90, respectively; r $=$ 0 cm denotes the burner centreline.

Figure 10 presents the time-averaged axial velocity fluctuations squared ( $\overline{v'v'}$ ) in the central vertical plane under different $\dot{Q}^{*}$ and burner boundary conditions. Similar to the time-averaged axial velocity ( $\overline{v}$ ), the maximum $\overline{v'v'}$ occurs around the central axis and $\overline{v'v'}$ increases rapidly when z increases. With the increase of $\dot{Q}^{*}$ , the radial profile of $\overline{v'v'}$ near the burner surface changes from unimodal to bimodal. Compared with the configuration with a flush floor, the axial location where the flame shedding occurs is lower in the configuration without a floor (see figure 4). Consequently, at the same height, $\overline{v'v'}$ in the configuration without a floor (see figure 10(A)–10(E)) is greater than that in the configuration with a flush floor (see figure 10 a–e). The comparison among figure 10(a–e) and figure 10(A–E) indicates that $\overline{v'v'}$ is relatively lower in the configuration with a flush floor, i.e. the presence of the floor suppresses the axial flame pulsation. As shown in figures 9 and 10, $\overline{v'v'}$ is approximately 0.10 m² s⁻² −1.00 m² s⁻², and $\overline{u'u'}$ ranges from 0.02 m² s⁻² to 0.18 m² s⁻². Therefore, the axial pulsation is more significant than the radial pulsation, and the turbulent flow is anisotropic in turbulent buoyant diffusion flames (Kundu, Cohen & Dowling Reference Kundu, Cohen and Dowling2016).

Figure 10.

Time-averaged axial velocity fluctuations squared ( $\overline{v'v'}$ ). (a)–(e) With a flush floor, (A)–(E) without a floor. From left to right in each row, $\dot{Q}^{*}=$ 0.50, 0.60, 0.70, 0.80 and 0.90, respectively; r $=$ 0 cm denotes the burner centreline.

Figure 11 shows the time-averaged axial and radial fluctuation product ( $\overline{u'v'}$ ) in the central vertical plane of flames with and without a floor for 0.50 $\leq \dot{Q}^{*} \leq$ 0.90. Near the burner surface, $\overline{u'v'}$ is relatively small. As z increases, $\overline{u'v'}$ gradually increases. When z reaches a certain height, $\overline{u'v'}$ remains unchanged with the increase of z. In the radial direction, $\overline{u'v'}$ first increases and then decreases, and its radial profile is the bimodal type. With the increase in $\dot{Q}^{*}$ , the lowest height (z) of the contour $| \overline{u'v'}|=$ 0.06 m² s⁻² increases. The peaks of $\overline{u'v'}$ are located in the normalised radial locations of $\xi \approx 0\mathit{.}33$ and $\xi \approx 0\mathit{.}67$ in the configurations with and without a floor, respectively. Compared with the configuration with a flush floor, the radial location of the peak $\overline{u'v'}$ moves closer to the burner rim in the configuration without a floor.

Figure 11.

Time-averaged axial and radial fluctuation product ( $\overline{u'v'}$ ). (a)–(e) With a flush floor, (A)–(E) without a floor. From left to right in each row, $\dot{Q}^{*}=$ 0.50, 0.60, 0.70, 0.80 and 0.90, respectively; r $=$ 0 cm denotes the burner centreline.

Similar to the experimental results of Yuan et al. (Reference Yuan, Lei and Liu2024) for the configuration without a floor with $0.2\lt \dot{Q}^{*}\lt 0.5$ , the peak of time-averaged tangential velocity fluctuations squared ( $\overline{w'w'}$ ) is located on the central axis of the burner. In the configuration with a flush floor, $\overline{w'w'}$ is 0.05–0.30 m² s⁻². In the configuration without a floor, it becomes 0.05–0.40 m² s⁻². Figure 12 shows the turbulent kinetic energy ( $k=(\overline{u'u'}+\overline{v'v'}+\overline{w'w'})/2$ ) in the central vertical plane. The spatial distributions of $k$ and $\overline{v'v'}$ are similar, and their magnitudes are close (10⁻¹ m² s⁻²). The maximum $k$ and $\overline{v'v'}$ are both located on the burner centreline. As the height (z) increases, $k$ gradually increases. Since $\overline{v'v'}$ is approximately one order of magnitude greater than $\overline{u'u'} , k$ is determined by $\overline{v'v'}$ in the free turbulent buoyant diffusion flame. As $\dot{Q}^{*}$ increases, the radial profile of k near the burner surface changes from the unimodal to the bimodal type. Compared with the configuration with a flush floor (see figure 12 a–e), the flame pulsation is stronger and k increases slightly in the configuration without a floor under equal $\dot{Q}^{*}$ .

Figure 12.

Turbulent kinetic energy (k). (a)–(e) With a flush floor, (A)–(E) without a floor. From left to right in each row, $\dot{Q}^{*}=$ 0.50, 0.60, 0.70, 0.80 and 0.90, respectively; r $=$ 0 cm denotes the burner centreline.

Figure 13 shows the time-averaged tangential vorticity ( $\overline{\omega }_{\theta }$ ) in the vertical plane passing through the burner centreline. As shown by figure 13(a–e) and figure 13(A–E), although the peak value of $\overline{\omega }_{\theta }$ varies slightly in the two configurations with and without a floor, the area with $| \overline{\omega }_{\theta }| \gt$ 3.00 s⁻¹ in the configuration without a floor is much larger than that in the configuration with a flush floor under equal $\dot{Q}^{*}$ . This indicates that the growth of toroidal vortices is faster when the floor is absent, which is consistent with the discussion in § 3.1. As $\dot{Q}^{*}$ increases, the initial fuel velocity at the burner exit increases, and the lowest height of $\overline{\omega }_{\theta }=$ 12.00 s⁻¹ gradually increases.

Figure 13.

Time-averaged tangential vorticity ( $\overline{\omega }_{\theta }$ ). (a)–(e) With a flush floor, (A)–(E) without a floor. From left to right in each row, $\dot{Q}^{*}=$ 0.50, 0.60, 0.70, 0.80 and 0.90, respectively; r $=$ 0 cm denotes the burner centreline.

3.5. The mean turbulent viscosity

The turbulent viscosity ( $\nu _{t}$ ) is a local function in the flow (Katopodes Reference Katopodes2019). Based on the gradient transport approximation, $\nu _{t}$ could be determined by $\nu _{t}=-\overline{u'v'}/(\partial \overline{v}/\partial r)$ . For turbulent buoyant flames, $\overline{u'v'}$ and $\overline{v}$ are assumed to have self-similarity and can be written as

(3.2) \begin{align} &\overline{v}=\overline{v}_{m}\exp \big(-C_{1}\xi ^{2}\big),\end{align}

(3.3) \begin{align} \overline{u'v'}&=\textit{fcn}\left(\dot{Q},d,\rho _{\infty },c_{p},g,T_{\infty }\right)\phi \left(\xi \right)\!.\end{align}

Here, the radial profile of $\overline{u'v'}$ ( $\phi$ ) is taken as $\phi (\xi )=\xi (C_{2}+C_{3}\xi ^{3})(1+C_{4}\xi ^{3})^{-6}$ , which is suggested by Shabbir & George (Reference Shabbir and George2006) for a round buoyant plume and satisfies the boundary conditions of $\phi (0)=\phi (\infty )=0$ . Also, C ₁, C ₂, C ₃ and C ₄ are dimensionless parameters. Since the fire buoyancy flux $F=g\dot{Q}/(\rho _{\infty }c_{p}T_{\infty })$ , (3.3) can be written as $\overline{u'v'}=fcn(F,d,\overline{v}_{m})\phi (\xi )$ . Based on the dimensional analysis, we obtain

(3.4) \begin{equation} \overline{u'v'}=\overline{v}_{m}F^{1\mathit{/}3}d^{-1\mathit{/}3}\phi \left(\xi \right)\!.\end{equation}

Figure 14 shows the radial profile of $\overline{v}$ and $\overline{u'v'}$ under the two burner configurations. Near the surface of the burner, the initial laminar flow becomes turbulent quickly. To reduce the impact of the bottom transition zone, the data above the height where $| \overline{u'v'}|=$ 0.02 are used for analysis. All data sets of $\overline{v}$ and $\overline{u'v'}$ could be correlated well by (3.2) and (3.4), respectively. The radial profiles of $\overline{v}$ and $\overline{u'v'}$ are unimodal and bimodal, respectively. As shown in figures 14(a) and 14(A), the time-averaged axial velocity ( $\overline{v}$ ) decreases quicker in the radial direction in the configuration with a flush floor. In figures 14(b) and 14(B), the maximum $\overline{u'v'}$ values are located around $\xi =0\mathit{.}40$ and $\xi =0\mathit{.}58$ for the configurations with and without a floor, respectively.

Figure 14.

Radial profiles of $\overline{v}$ and $\overline{u'v'}$ : (a) and (b) with a flush floor; (A) and (B) without a floor.

Substituting (3.2) and (3.4) into $\nu _{t}=-\overline{u'v'}/(\partial \overline{v}/\partial r)$ , the local turbulent viscosity ( $\nu _{t}$ ) is given by

(3.5) \begin{equation} \nu _{t}=F^{1/3}d^{2/3}{\Phi} \left(\xi \right)\!,\end{equation}

where ${\Phi} (\xi )=(4C_{1}\xi )^{-1}\phi (\xi )\exp (C_{1}\xi ^{2})$ . Assuming that (3.5) is applicable within the mean flame height (z $\leq$ H), then the mean $\nu _{t}$ within the flame height is written as

(3.6) \begin{equation} \nu _{t}^{=}=\lambda F^{1/3}d^{2/3} ,\end{equation}

where the turbulent viscosity coefficient $\lambda =\xi _{1}^{-1}\int _{0}^{\xi _{1}}{\Phi} (\xi )\,\rm d\xi$ . The integration upper limit $\xi _{1}$ is taken as the maximum radial range of measurement at $\xi _{1}=$ 0.98. Note that previous works have shown that $\nu _{t}^{=}=\lambda _{1}F^{1/2}H^{3/2}$ for $\dot{Q}^{*}\gt 1$ (Lei & Liu Reference Lei and Liu2017) and $\nu _{t}^{=}=\lambda _{2}g^{1/2}d^{3/2}$ for $0.2\lt \dot{Q}^{*}\lt 0.5$ (Yuan et al. Reference Yuan, Lei and Liu2024). In this work, $\nu _{t}^{=}$ is scaled by the fire buoyancy flux (F) and fire source diameter (d) for $0.5\leq \dot{Q}^{*}\lt 1$ , regardless of the flame height (H).

Table 4 presents the data processing results under different $\dot{Q}^{*}$ and burner boundary conditions. The turbulent viscosity coefficients are $\lambda _{w}=(0\mathit{.}77\pm 0\mathit{.}06)\times 10^{-2}$ and $\lambda _{wo}=(1\mathit{.}03\pm 0\mathit{.}16)\times 10^{-2}$ for the configurations with and without a floor, respectively. Substituting $\lambda _{w}$ and $\lambda _{wo}$ into (3.6), the mean turbulent viscosities are respectively $\nu^{=} _{t\,w}=$ 0.0032∼0.0038 m² s⁻¹ and $\nu^{=} _{t\,w}=$ 0.0042∼0.0051 m² s⁻¹ for the configurations with and without a floor under different $\dot{Q}^{*}$ . Note that the mean turbulent viscosity is one order of magnitude larger than the air viscosity ( $\nu=$ 1.63 × 10⁻⁴ m² s⁻¹) in the flame (1200 K). Moreover, under equal $\dot{Q}^{*} , \nu^{=} _{t\,w}$ is significantly greater than $\nu^{=} _{t\,w}$ . Therefore, it is inferred that the presence of the floor around the burner could reduce the turbulent mixing rate of fuel and air in turbulent buoyant flames.

Table 4.

Data processing results.

3.6. Flame shape formulations based on turbulent transport properties

3.6.1. Flame height correlation

For the lightly sooting flames, the theoretical flame height should be close to the visible flame height determined by the DV images. The theoretical flame height (H) could be defined as the distance between the burner exit and the axial position where the mixture fraction $Z=Z_{st}$ at the flame central axis. The mixture fraction $Z$ is defined by $Z=(sY_{F}-Y_{O}+Y_{O,\infty })/(s+Y_{O,\infty }) , Y_{F}$ and $Y_{O}$ are the mass fractions of fuel and oxygen, $s=(a_{O}M_{O})/(a_{F}M_{F})$ is the mass stoichiometric ratio of oxygen to fuel, $a_{O}$ are $a_{F}$ the coefficients of fuel and oxygen in an overall unique reaction and $M_{O}$ and $M_{F}$ are respectively the molecular weights of fuel and oxygen. Here, $Z_{st}=Y_{O,\infty }/(s+Y_{O,\infty })$ is the stoichiometric value. Lei & Liu (Reference Lei and Liu2017) derived the flame height correlation by solving the controlling equation of mixture fraction through coordinate transformation

(3.7) \begin{equation} \frac{H}{d}=\frac{v_{F0}d}{4\left(D^{=}+D_{t}^{=}\right)}\frac{1}{4\ln \left[1/\left(1-Z_{st}\right)\right]} .\end{equation}

In the derivation of (3.7), the boundary conditions at the bottom apply to the turbulent diffusion flames for the configurations with and without a floor. Therefore, (3.7) should be able to provide accurate estimations of the flame heights under the two burner boundary conditions in this work. In the fully turbulent flow, the mean molecular diffusivity ( $D^{=}$ ) can be ignored compared with the mean turbulent diffusivity within the mean flame height ( $D_{t}^{=}$ ). $D_{t}^{=}$ could be related to the mean turbulent viscosity ( $v_{t}^{=}$ ) through the turbulent Schmidt number ( $Sc_{t}=v_{t}^{=}/D_{t}^{=}$ )

(3.8) \begin{equation} D_{t}^{=}=\lambda {Sc}^{=-1}_{t}F^{1/3}d^{2/3} .\end{equation}

Substituting (3.8) into (3.7) yields

(3.9) \begin{equation} H^{*}=\left\{\frac{Sc_{t}^{=}}{4\pi \lambda }\frac{\rho _{\infty }c_{p}T_{\infty }}{{\unicode[Arial]{x0394}} H_{c}\rho _{0}}\frac{1}{\ln \left[1/\left(1-Z_{st}\right)\right]}\right\}\dot{Q}^{*2/3}=\kappa \dot{Q}^{*2/3} ,\end{equation}

or

(3.10) \begin{equation} H=\kappa \dot{Q}^{*2/3}d=\kappa L_{0}^{5/3}d^{-2/3} ,\end{equation}

where $L_{0}=[\dot{Q}/(\rho _{\infty }c_{p}T_{\infty }g^{1/2})]^{2/5}$ . Equation (3.9) reproduced the classical ‘2/3’ exponential dependence of $H^{*}$ on $\dot{Q}^{*}$ for 0.5 $\leq \dot{Q}^{*}$ < 1 in Drysdale (Reference Drysdale2011). However, the coefficient $\kappa$ is a deterministic function of the turbulent transport characteristics, fuel and ambient parameters in (3.10). For the two configurations with and without a flush floor under equal $\dot{Q}^{*} , \kappa$ is only affected by the turbulent viscosity coefficient ( $\lambda$ ), and we have $H_{w}/H_{wo}=\lambda _{\textit{ratio}}^{-1}=(\lambda _{w}/\lambda _{wo})^{-1}$ .

3.6.2. Flame width correlation

For vertical turbulent diffusion flames, the axial convection time of fuel from the burner exit to the flame tip $(t_{conv}=\int _{0}^{H}\overline{v}_{c}^{-1}{d}z)$ is approximately equal to the radial diffusion time of air from the flame edge to the flame centreline $(t_{\textit{diff}}=d_{f}^{2}/(8D_{t}))$ (Jost Reference Jost1946; Saito, Williams & Gordon Reference Saito, Williams and Gordon1986; Sunderland et al. Reference Sunderland, Mendelson, Yuan and Urban1999; Lei et al. Reference Lei, Huang, Liu and Zhang2022b ). Then we have

(3.11) \begin{equation} d_{f}^{2}=8D_{t}\int _{0}^{H}\overline{v}_{c}^{-1}{d}z=16D_{t}k^{-1}H^{1/2} ,\end{equation}

where $\overline{v}_{c}=k_{0}z^{1/2}$ (McCaffrey Reference McCaffrey1979). The subscript ‘c’ denotes the centreline. It is found that $k_{0}$ varies slightly under different $\dot{Q}^{*}$ and burner boundary conditions in this work. For simplicity, $k_{0}$ is taken as 5.00 m² s⁻¹. Using (3.8) and (3.9) in (3.11) leads to

(3.12) \begin{equation} d_{f}=KL=K\dot{Q}^{*1/3}d=KL_{0}^{5/6}d^{1/6} .\end{equation}

The characteristic length L and the dimensionless constant K are respectively

(3.13) \begin{align}& L=\left(\frac{\dot{Q}d^{1/2}}{\rho _{\infty }c_{p}T_{\infty }g^{1/2}}\right)^{1/3}=\dot{Q}^{*1/3}d=L_{0}^{5/6}d^{1/6} , \\[-28pt] \nonumber \end{align}

(3.14) \begin{align}&\quad K=4Sc_{t}^{=-1/2}g^{1/4}\kappa ^{1/4}k_{0}^{-1/2}\lambda ^{1/2} .\end{align}

Note that $L_{0}$ is the sole characteristic length in the correlations of $H$ and $d_{f}$ for $\dot{Q}^{*}$ > 1 (Quintiere Reference Quintiere2006; Lei et al. Reference Lei, Huang, Liu and Zhang2022b ). However, for 0.5 $\leq \dot{Q}^{*}$ < 1, the characteristic length scales are $L_{0}^{5/3}d^{-2/3}$ and $L_{0}^{5/6}d^{1/6}$ in (3.10) and (3.12), respectively. Then, the dimension of the fire source (d) becomes an important length scale affecting the combustion behaviour of free buoyant flames (Zukoski, Kubota & Cetegen Reference Zukoski, Kubota and Cetegen1981; Zukoski, Cetegen & Kubota Reference Zukoski, Cetegen and Kubota1985). In addition, (3.13) indicates that $d_{f}$ is proportional to d for turbulent buoyant flames under equal $\dot{Q}^{*}$ . For the two configurations with and without a flush floor, it could be concluded that $d_{fw}/d_{fwo}=\lambda _{\textit{ratio}}^{1/4}$ from (3.14).

3.6.3. Verification of the flame height and width correlations

By substituting $\lambda _{w}=0\mathit{.}77\times 10^{-2}$ and $\lambda _{wo}=1\mathit{.}03\times 10^{-2}$ into (3.9) and (3.14), we have $\kappa _{w}=$ 3.99, $\kappa _{wo}=$ 2.99, $K_{w}=$ 0.45 and $K_{wo}=$ 0.49 for the methane buoyant flames with and without a floor in this work, respectively. In the calculation of $\kappa$ and $K , Z_{st}=0.055 , Sc_{t}^{=}=0.75 , {\unicode[Arial]{x0394}} H_{c}=$ 50 × 10³ kJ kg⁻¹, $\rho _{\infty }=$ 1.1614 kg m⁻³, $c_{p}=$ 1.007 kJ kg⁻¹ K⁻¹, $T_{\infty }=$ 300 K and $\rho _{0}=$ 0.2403 kg m⁻³ is the geometric mean of the methane density at the burner exit of 600 K and air density at the average flame temperature of 1500 K (Chuah et al. Reference Chuah, Kuwana, Saito and Williams2011). In addition, the coefficients $\kappa _{w}=$ 3.99 and $\kappa _{wo}=$ 2.99 are comparable to the fitted values of $\kappa=$ 2.6 and 3.30 by Zukoski et al. (Reference Zukoski, Cetegen and Kubota1985) and Delichatsios (Reference Delichatsios1987), respectively.

In the derivation of (3.10) and (3.12), there are some deviations between the assumptions and the actual situation. (i) The influence of the axial diffusion term on flame height was neglected in the derivation of (3.7). This may result in a relatively lower prediction of the mean flame height (Lee & Chung Reference Lee and Chung1991; Mahalingam Reference Mahalingam1993; Ban, Venkatesh & Saito Reference Ban, Venkatesh and Saito1994; Knott & Brewster Reference Knott and Brewster2000). (ii) The model ignored the transition of flames near the burner surface from laminar flame to turbulent flame. (iii) Although the methane flames of this work involve relatively less visible soot, the differences between the theoretical and visual flame heights and widths are inevitable. To include the potential effects of these factors, we added the correction factors on the right-hand sides in (3.9) and (3.12). Figure 15 presents the comparisons between the experimental and the predicted mean flame heights and widths for free buoyant flames with and without a floor of this work. The predicted results are in good agreement with the experimental data within an error range of ±10 %. The four correction coefficients of 0.93, 0.92, 0.95 and 0.99 are all close to unity, indicating that the models have good predictive ability and the influence of the above three factors is not significant. However, future works are demanded to study the influence of burner boundary conditions under larger burner scales.

Figure 15.

Demonstration of the flame height and width models according to (3.9) and (3.12) with experimental data in this work. The X-axis shows measured values from the flame images and the Y-axis shows predicted values from the models.