4.1. Symmetry

First, we examine whether NOB-II effects disrupt temperature symmetry in the same way as NOB-I effects. Figure 8(a) shows the horizontal-averaged normalised temperature, $\langle \widetilde {T} \rangle _x$ , as a function of $y/H$ for various temperature differences. The results indicate that the overall symmetry of the temperature profile remains intact. All curves pass through the point (0.5, 0) without significant deviation as the temperature difference increases. We further present the variation of the normalised mean temperature of central plane $\widetilde {T}_{{c}}$ with $\alpha$ in figure 8(b). The results show that $\widetilde {T}_{{c}}$ fluctuates within a narrow range of $\pm 0.01$ . Previous studies on the NOB-I effects have reported deviations in $\widetilde {T}_{{c}}$ as follows: $+0.05{-}+0.17$ for glycerol (Zhang et al. Reference Zhang, Childress and Libchaber1997), $0{-} +0.08$ for water (Ahlers et al. Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006; Sugiyama et al. Reference Sugiyama, Calzavarini, Grossmann and Lohse2009; Demou & Grigoriadis Reference Demou and Grigoriadis2019), $0{-} +0.05$ for air (Wan et al. Reference Wan, Wang, Wang, Xia, Zhou and Sun2020), $-0.22{-} +0.08$ for supercritical SF $_6$ (Yik et al. Reference Yik, Valori and Weiss2020) and $-0.035{-} +0.02$ for supercritical ethane (Ahlers et al. Reference Ahlers, Araujo, Funfschilling, Grossmann and Lohse2007, Reference Ahlers, Calzavarini, Araujo, Funfschilling, Grossmann, Lohse and Sugiyama2008). Additionally, these studies consistently show that the deviation increases with the intensity of NOB-I effects. In contrast, the deviations observed in figure 8(b) are minimal (<0.01) and show no clear trend. Therefore, we conclude that the NOB-II effects have no impact on $\widetilde {T}_{{c}}$ , meaning that NOB-II effects do not affect $\widetilde {T}_{{c}}$ and thus do not break the temperature symmetry. The slight fluctuations in $\widetilde {T}_{{c}}$ are most likely caused by minor variations in $c_p$ and $\beta$ with temperature (see figure 1), which are attributable to the NOB-I effects.

Figure 8.

(a) Horizontal-averaged normalised temperature $\langle \widetilde {T} \rangle _x$ as a function of $y/H$ for various $\Delta T$ at $H=100$ . (b) Variation of the normalised temperature of centre plane $\widetilde {T}_{{c}}$ with $\alpha$ for various $H$ . (c) Horizontal-averaged normalised density $\langle \widetilde {\rho } \rangle _x$ as a function of $y/H$ for various $\Delta T$ at $H=100$ . Dashed lines represent the result of (4.2) $\widetilde {\rho } = -\widetilde {T}$ . (d) Variation of the normalised centre plane density $\widetilde {\rho }_{{c}}$ with $\alpha$ for various $H$ . Ten data points for each $H$ represent varying $\Delta T$ values, specifically from 1.1 to $2.0\Delta T_{ {on}}$ in increments.

Under OB and NOB-I conditions, the density in the buoyancy term decreases as temperature increases, maintaining a one-to-one correspondence. However, under NOB-II conditions, density is determined by both temperature and pressure, making the study of the density field particularly important (Amiroudine et al. Reference Amiroudine, Bontoux, LarroudÉ, Gilly and Zappoli2001; Carlès & El Khouri Reference Carlès and El Khouri2001). To examine the density symmetry, we define the normalised density as follows:

(4.1) \begin{align} \widetilde {\rho }=\frac {\rho -\rho _{{m}}}{\rho _{{m}} \beta \Delta T}, \end{align}

Under OB conditions, $\widetilde {\rho }$ and $\widetilde {T}$ are related by the following equation:

(4.2) \begin{align} \widetilde {\rho }=\frac {\rho -\rho _{{m}}}{\rho _{{m}} \beta \Delta T} =\frac {-\rho _{{m}}\beta (T-T_{{m}})}{\rho _{{m}}\beta (T_{ {b}}-T_{ {t}})} =-\widetilde {T} .\end{align}

This implies that $\widetilde {\rho }$ and $\widetilde {T}$ have opposite signs under OB conditions. Figure 8(c) shows the variation of the horizontal-averaged normalised density distribution $\langle \widetilde {\rho } \rangle _x$ with $y/H$ for various $\Delta T$ . The results indicate that the overall symmetry of the density profile remains intact. Figure 8(d) presents the variation of the normalised mean density at the centre plane $\widetilde {\rho }_{{c}}$ with $\alpha$ . The results show that $\widetilde {\rho }_{{c}}$ fluctuates within a narrow range of $\pm$ 0.01, with deviations of less than 0.01. This suggests that NOB-II effects do not break the density symmetry either. However, the profiles in figure 8(c) deviate significantly from the relationship described by (4.2). Therefore, it is necessary to further investigate the distribution of the density field, which is discussed in detail in § 4.2.

While studying the symmetry of temperature and density, values at the centre plane are of primary interest. In contrast, when analysing the symmetry of velocity under NOB conditions, the focus shifts to the velocity difference between the upper and lower regions (Horn et al. Reference Horn, Shishkina and Wagner2013). Under OB conditions, the velocity field is symmetric along the centre plane. However, under NOB conditions, this symmetry is disrupted. Experimental studies on NOB-I effects have revealed asymmetries in the upper and lower velocity fields (Valori et al. Reference Valori, Elsinga, Rohde, Tummers, Westerweel and van der2017). Therefore, it is essential to investigate the impact of NOB-II effects on the velocity field.

Figure 9.

(a) Variation of the horizontal-averaged absolute value of $u_y$ with $|y/H-0.5|$ for various $\Delta T$ at $H=100$ . Dashed lines represent the results for the lower region ( $y/H\lt 0.5$ ). Points represent the results for the upper region ( $y/H\gt 0.5$ ). (b) Variation of the velocity deviation $\varepsilon _u$ in the upper and lower regions with $|y/H-0.5|$ for various $\Delta T$ at $H=100$ . (c) Variation of the horizontal-averaged absolute value of $\rho u_y$ with $|y/H-0.5|$ for various $\Delta T$ at $H=100$ . (d) Variation of the momentum deviation $\varepsilon _{\rho u}$ in the upper and lower regions with $|y/H-0.5|$ for various $\Delta T$ at $H=100$ .

Figure 9(a) shows the variation of the horizontal-averaged absolute value of $u_y$ with $|y/H - 0.5|$ . Here, the horizontal coordinate is taken as $|y/H - 0.5|$ rather than $y/H$ to emphasise the asymmetry between the upper and lower regions. The results indicate that the velocity field is asymmetric along the central horizontal plane, with $\langle |u_y| \rangle _x$ in the lower region consistently smaller than in the upper region. The relative deviation $\varepsilon _u$ between the two regions is shown in figure 9(b), where $\varepsilon _u$ is defined as follows:

(4.3) \begin{align} \varepsilon _u = \frac {\langle |u_y| \rangle _{x,{upper}} - \langle |u_y| \rangle _{x,{lower}}} { \langle |u_y| \rangle _{x,y}}, \end{align}

with the subscripts ‘upper’ and ‘lower’ representing the upper and lower regions, respectively. Figure 9(b) shows that the deviation between the two regions increases and then decreases with $|y/H-0.5|$ and the maximum deviation increases with $\Delta T$ .

To further analyse the cause of this asymmetry, figure 9(c) presents the variation of the horizontal-averaged absolute value of $\rho u_y$ with $y$ . This figure shows that momentum symmetry between the upper and lower regions is maintained, with minimal differences observed between the two regions. Figure 9(d) displays the relative deviation $\varepsilon _{\rho u}$ , which is defined as follows:

(4.4) \begin{align} \varepsilon _{\rho u} = \frac {\langle |\rho u_y| \rangle _{x,{upper}} - \langle |\rho u_y| \rangle _{x,{lower}}} { \langle |\rho u_y| \rangle _{x,y}} .\end{align}

The results indicate that the momentum deviation remains within 0.02. In contrast, the maximum deviation in velocity is close to 0.1, as shown in figure 9(b). Thus, we conclude that under NOB-II conditions, the symmetry of momentum is preserved, while the asymmetry observed in the velocity field is attributed to variations in the density field.

4.2. Density inversion

Figure 10(a) presents the normalised density $\widetilde {\rho }$ contours at $\Delta T = 1.1\Delta T_{ {on}}$ for various heights. In OB-RBC, it is widely accepted that the fluid at the bottom is less dense than at the top. Due to buoyancy, the lighter fluid rises, driving thermal convection. However, as $H$ increases, leading to stronger NOB-II effects, the density field exhibits an intriguing inversion. As shown in figure 10(a), when $\alpha \leq 0.7$ , the density remains higher at the top and lower at the bottom, consistent with OB-RBC behaviour. At this point, the contours for $\widetilde {\rho } = 0$ show a distribution of shapes that are convex upwards at the hot plume and concave downwards at the cold plume. However, when $\alpha \geqslant 0.81$ , an inversion in the density field occurs, with the density becoming lower at the top and higher at the bottom. At this point, the contours for $\widetilde {\rho } = 0$ display shapes that are concave downwards in the hot plume and convex upwards in the cold plume.

We now examine the mechanism of density inversion, using hot plumes as a representative case (a similar analysis applies to cold plumes). Generally, density decreases with increasing temperature and increases with rising pressure. Due to fluid compressibility, the effect of pressure cannot be neglected. Although the fluid at the bottom is hotter, it is also subjected to higher pressure. As the hot plume ascends, its temperature decreases, which would typically increase its density. Simultaneously, the pressure also decreases, which tends to lower its density. When both the pressure gradient and fluid compressibility are significant, the effect of pressure on density outweighs the effect of temperature. Consequently, the density gradually decreases as the hot plume rises, leading to the phenomenon of density inversion.

Although the mechanism of density inversion is understood, it remains unclear how convection can persist when density inversion occurs. This is particularly intriguing, as it is generally accepted that a fluid is stable when its density increases with height. A typical example of this is the stable conduction stratification of penetrative RBC (Toppaladoddi & Wettlaufer Reference Toppaladoddi and Wettlaufer2018; Wang et al. Reference Wang, Zhou, Wan and Sun2019). To elucidate the mechanism of convection, figure 10(b) shows the normalised density distribution along the flow circulation lines (indicated by the solid lines in figure 10 a). As shown in figure 10(b), the fluid density decreases with increasing height when $H$ is large, further confirming the occurrence of density inversion. From figure 10(b), we observe a common feature in the density circulation lines: for all values of $H$ , the fluid exhibits counterclockwise circulation on the $\rho$ – $y/H$ plot. This implies that, for a given $y$ , the density of the rising hot plume is always lower than that of the sinking cold plume.

Figure 10.

(a) Normalised density $\widetilde {\rho }$ contours at $\Delta T = 1.1\Delta T_{ {on}}$ for various heights. Black arrows represent velocity vectors. The green dashed lines represent $\widetilde {\rho }=0$ . (b) Normalised density $\widetilde {\rho }$ distribution on the circulation lines (solid lines in panel a) for various heights.The arrows represent the direction of flow.

To better explain the flow mechanism when density inversion occurs, we schematically depict the flow structure of RBC in figure 11. As shown in figure 11(a) and 11(b), when density inversion is not present, the densities of both the hot and cold plumes increase as $y$ increases. For a given $y$ , the density of the hot plume is lower than that of the cold plume. When density inversion occurs, the densities of both the hot and cold plumes decrease as $y$ increases. Meanwhile, for a given $y$ , the density of the hot plume remains lower than that of the cold plume. We elaborate on the example of one cycle consisting of half a hot plume and half a cold plume, as is illustrated in figure 11(c). It can be seen in figure 10(a) that the flow velocity at the cycle centre is very low. Therefore, we can abstract the cycle as a natural circulation loop (Misale et al. Reference Misale, Garibaldi, Passos and de Bitencourt2007), with the dashed line at the centre of the loop representing the virtual boundary. For the natural circulation loop depicted in figure 10(c), the gravitational pressure drop on the left side is $\Delta p_{\mathrm {1}}=-\bar {\rho }_{{h}}gH$ and the gravitational pressure drop on the right side is $\Delta p_2=\bar {\rho }_{{c}}gH$ , where $\bar {\rho }_{{h}}$ and $\bar {\rho }_{{c}}$ represent the average densities of the hot and cold plumes, respectively. In addition, wall friction introduces a frictional pressure drop $\Delta p_{\mathrm {3}}=-\Delta p_{ {f}}$ , where $\Delta p_{ {f}}$ is positive. The total pressure drops is given by $\Delta p= (\bar {\rho }_{{c}}-\bar {\rho }_{ {h}})gH-\Delta p_{ {f}}$ . It is well known that, for a steady-state natural circulation loop, the net pressure drop must be zero. From the above analysis, we observe that $\bar {\rho }_{h}$ is lower than $\bar {\rho }_{{c}}$ . In this case, $\Delta p$ can be equal to 0, allowing the circulation to persist. A similar analysis of the stable stratification in penetrative RBC reveals that, for a given $y$ , $\bar {\rho }_{ {h}}$ is always greater than $\bar {\rho }_{{c}}$ . In this case, the total pressure drop is necessarily negative, and convection cannot persist.

To briefly summarise, under NOB-II conditions, the density of the hot plume is always less than that of the cold plume for a given $y$ , regardless of whether density inversion occurs. The buoyancy difference due to the density difference between the two drives the convection to persist. The flow mechanism is very similar to a natural circulation loop.

Figure 11.

(a) Schematic of the flow structure and density distribution when the density field is not inverted. (b) Schematic of the flow structure and density distribution when the density field is inverted. (c) Schematic of one cycle consisting of half a hot plume and half a cold plume. $\rho _{ {h}}$ represents the density of the hot plume, $\rho _{{c}}$ represents the density of the cold plume, and $\Delta p_1$ and $\Delta p_2$ represent the gravitational pressure drops on the left and right sides of the cycle, respectively.

4.3. Condition for density inversion

Building on our understanding of the convective mechanism, we now seek to identify the conditions under which density inversion occurs. To this end, we define the normalised density difference:

(4.5) \begin{align} \Delta \widetilde {\rho } = \widetilde {\rho }_{ {t}}-\widetilde {\rho }_{ {b}} = \frac {\rho _{ {t}}-\rho _{ {b}}}{\rho _{{m}}\beta \Delta T}. \end{align}

where $\rho _{{b}}$ and $\rho _{{t}}$ represent the mean densities of the bottom and top plates, respectively. Apparently, $\Delta \widetilde {\rho }$ is equal to 1 under OB conditions and $\Delta \widetilde {\rho } \lt 0$ indicates the occurrence of density inversion. Density is determined by both temperature and pressure. Thus, the density difference $\Delta \rho$ arises from the effects of temperature and pressure. The density difference from temperature is given by

(4.6) \begin{align} \Delta \rho _T = (T_{ {t}}-T_{ {b}})\left (\frac {\partial \rho }{\partial T}\right )_p = \rho _{{m}}\Delta T \frac {1}{\rho _{{m}}}\left (\frac {\partial \rho }{\partial T}\right )_p = \rho _{{m}}\beta \Delta T. \end{align}

The density difference from pressure can be expressed as

(4.7) \begin{align} \Delta \rho _p = (p_{ {t}}-p_{ {b}}) \left (\frac {\partial \rho }{\partial p}\right )_T =\rho _{{m}} gH \left (\frac {\partial \rho }{\partial p}\right )_T. \end{align}

Since the $\rho$ , $p$ and $T$ satisfy equation of state (2.14), triple product rule exists (see Elliott & Lira Reference Elliott and Lira2012, (6.15)):

(4.8) \begin{align} \left (\frac {\partial \rho }{\partial p}\right )_T \left (\frac {\partial p}{\partial T}\right )_\rho \left (\frac {\partial T}{\partial \rho }\right )_p = -1. \end{align}

Thus, $\Delta \rho _p$ can be further expressed as

(4.9) \begin{align} \Delta \rho _p = -\rho _{{m}} gH \left (\frac {\partial \rho }{\partial T}\right )_p \left (\frac {\partial T}{\partial p}\right )_\rho = -\rho ^2_{{m}} gH \beta \left (\frac {\partial T}{\partial p}\right )_\rho. \end{align}

Considering (2.19), the adiabatic temperature difference can be written in the following form (Kogan et al. Reference Kogan, Murphy and Meyer1999; Meyer & Kogan Reference Meyer and Kogan2002):

(4.10) \begin{align} \Delta T_{ {ad}} = \frac {gT\beta H}{c_p} = \rho _{{m}} g H\frac {T}{\rho _{{m}}^2 c_p}\left (\frac {\partial \rho }{\partial T}\right )_p \left (\frac {\partial p}{\partial T}\right )_\rho \left (\frac {\partial T}{\partial p}\right )_\rho = \rho _{{m}} g H \frac {c_p-c_v}{c_p}\left (\frac {\partial T}{\partial p}\right )_\rho. \end{align}

Thus, $\Delta \rho _p$ can also be expressed as

(4.11) \begin{align} \Delta \rho _p = -\frac {c_p}{c_p-c_v}\rho _{{m}}\beta \Delta T_{ {ad}} = -\frac {c_p}{c_p-c_v} \Delta \rho _{ {ad}}, \end{align}

where $\Delta \rho _{ {ad}}$ is the density difference from ATG. We assume that the effects of temperature and pressure on the density difference are independent of each other. At this point, the normalised density difference can be expressed as

(4.12) \begin{align} \Delta \widetilde {\rho } = \frac {\Delta \rho _T + \Delta \rho _p}{\rho _{{m}}\beta \Delta T} = 1 - \frac {c_p}{c_p - c_v}\alpha. \end{align}

We present both our theoretical results (4.12) and the calculated data in figure 12. Notably, the calculated data closely align with the predictions of (4.12). They both show that the condition for density inversion is $\alpha \gt (c_p - c_v)/{c_p}$ . Furthermore, (4.10) indicates that the precondition for density inversion is the inability of the density difference from ATG to fully counterbalance the density difference from pressure. From (4.6) and (4.11), the following equation can also be derived:

(4.13) \begin{align} \alpha = \frac {\Delta T_{ {ad}}}{\Delta T_{ {on}}} = \frac {\Delta \rho _{ {ad}}}{\Delta \rho _T} = -\frac {c_p-c_v}{c_p}\frac {\Delta \rho _p}{\Delta \rho _T}. \end{align}

This indicates that $\alpha$ is proportional to the ratio of the density difference from pressure to that from temperature. The above analyses suggest that $\alpha$ does indicate the intensity of NOB-II effects well.

4.4. Nusselt number

Next, we analyse the impact of NOB-II effects on heat transfer. Figure 13(a) illustrates the variation of the Nusselt number $ Nu$ with Rayleigh number $ Ra$ . The results clearly show that NOB-II effects deteriorate the heat transfer, as $ {Nu}$ is consistently lower than that of OB-RBC for a given $ Ra$ . Additionally, for a given $ Ra$ , the larger the $H$ , the smaller the $ Nu$ . We also calculated $Nu$ of OB-RBC across various $Ra$ using the thermal lattice Boltzmann method presented by Gao et al. (Reference Gao, Yu, Yang, Qin and Hou2021). The results show that the calculated results closely align with the results reported by Clever & Busse (Reference Clever and Busse1974). The slight deviation observed in figure 13(a) arises from the difference in $Pr$ . Clever & Busse (Reference Clever and Busse1974)set $Pr$ to 7.0, while in our calculations, we used a $Pr$ value of 8.25, which corresponds to supercritical fluid at $T_{{m}} = 1.05T_{ {cr}}$ and $p_{{m}} = 2p_{ {cr}}$ . Previous studies by Clever & Busse (Reference Clever and Busse1974) have already indicated that $Nu$ decreases slightly with increasing $Pr$ when $Pr \gt 2$ .

Figure 12.

Variation of the normalised density difference between the top and bottom plates $\Delta \widetilde {\rho }$ with $\alpha$ for various heights. The dash-dotted line represents the theoretical prediction of (4.12). Ten data points for each $H$ represent varying $\Delta T$ values, specifically from 1.1 to $2.0\Delta T_{ {on}}$ in increments.

Figure 13.

(a) Variation of Nusselt number $Nu$ with Rayleigh number $Ra$ for various $H$ . Black square points represent the results of OB-RBC from Clever & Busse (Reference Clever and Busse1974). Black cross points represent the calculated results of OB-RBC using the thermal lattice Boltzmann method presented by Gao et al. (Reference Gao, Yu, Yang, Qin and Hou2021). Dashed line is the best fitting of black cross points. (b) Variation of normalised Nusselt number $\widetilde {Nu}$ with $\alpha$ for various $H$ . Ten data points for each $H$ represent varying $\Delta T$ values, specifically from 1.1 to $2.0\Delta T_{ {on}}$ in increments.

To further quantify the heat transfer deterioration, we define the normalised Nusselt number as follows:

(4.14) \begin{align} \widetilde {Nu} = \frac {Nu-1}{Nu_{ {ob}}-1}, \end{align}

where $ Nu_{ {ob}}$ is the Nusselt number for the same $ Ra$ under OB conditions, obtained by fitting the black cross points in figure 13(a). The variation of $ \widetilde {Nu}$ with $ \alpha$ for various heights is shown in figure 13(b). The results indicate that the points fall nicely on the line $ \widetilde {Nu} = 1 - \alpha$ . This demonstrates that as $ \alpha$ increases from 0 to 1, $Nu$ decreases linearly from $Nu_{ {ob}}$ to 1, indicating significant heat transfer deterioration. Additionally, this further confirms that $\alpha$ is a reliable indicator of NOB-II intensity. Notably, several blue outliers are observed in figure 13(b), which primarily arise from the proximity of their $Ra$ to $Ra_{{c}}$ . This near-critical condition results in $Nu_{ {ob}}$ approaching unity, thereby causing the denominator in (4.14) to approach zero and consequently amplifying numerical uncertainties.

4.5. Reynolds number

Figure 14.

(a) Variation of Reynolds number $Re$ with Rayleigh number $Ra$ for various height $H$ . (b) Variation of Reynolds number $Re$ with reduced Rayleigh number $Ra^{\ast}$ for various height $H$ . Black asterisk points represent the calculated results of OB-RBC using the thermal lattice Boltzmann method presented by Gao et al. (Reference Gao, Yu, Yang, Qin and Hou2021). The solid line $Re=1.042(Ra^*)^{0.541}$ is the best power-law fit to asterisk points. Ten data points for each $H$ represent varying $\Delta T$ values, specifically from 1.1 to $2.0\Delta T_{ {on}}$ in increments.

We use the Reynolds number $Re$ to indicate the convective intensity and $Re$ is defined as follows (Shishkina & Wagner Reference Shishkina and Wagner2016):

(4.15a,b) \begin{align} Re = \frac {UH}{\nu }, \quad U = \sqrt {\left \lt \boldsymbol {u\cdot u}\right \gt _{x,y}}, \end{align}

where $\left \lt \cdot \right \gt _{x,y}$ denotes the spatial average over the $x$ and $y$ directions, $U$ represents the characteristic velocity. Figure 14(a) shows the variation of $Re$ with $Ra$ for various $H$ . The results indicate that $Re$ increases with $Ra$ . Meanwhile, $H$ has no impact on the relation between $Re$ and $Ra$ , as all points under both OB and NOB-II conditions fall on the same curve. Figure 14(b) shows the variation of $Re$ with $Ra^*$ . Under OB and NOB-II conditions, all points follow the same power-law relationship $Re=1.042 (Ra^* )^{0.541}$ , further confirming the above conclusion.

Combined with the discussion in § 4.4, we observe a unique phenomenon: under dominant NOB-II effects, despite the persistence of vigorous convection, the heat transfer efficiency decreases to a level comparable to pure heat conduction. In other words, as $\alpha$ approaches 1, convection ceases to enhance heat transfer relative to pure conduction. The mechanism underlying this significant HTD urgently needs to be explored.

4.6. Mechanism behind heat transfer deterioration

To understand the mechanism behind HTD resulting from NOB-II effects, we first analyse the temperature field. As shown in figure 15(a), as $\alpha$ increases, the convective intensity remains essentially unchanged, as reflected by the arrows. However, the temperature distribution becomes progressively flatter as $\alpha$ increases. When $\alpha = 0.84$ , the temperature distribution closely resembles that of pure heat conduction, characterised by a zero gradient in the $x$ -direction and a linear decrease in the $y$ -direction. At this point, HTD is observed, as shown in figure 15(b). This suggests that an unknown mechanism is absorbing the heat carried by the hot plume, thereby reducing its ability to transport heat to the top plate. We further investigate this mechanism by examining the energy conservation equation, which is given by

(4.16) \begin{align} \frac {\partial T}{\partial t} = \underbrace {-\boldsymbol {u\cdot \nabla } T}_{\displaystyle \phi _{\textit{conv}}} + \underbrace {\frac {1}{\rho c_p}\boldsymbol {\nabla \cdot }(\lambda \boldsymbol {\nabla } T)}_{\displaystyle \phi _{\textit{diff}}} + \underbrace {\frac {1}{\rho c_p}\left (\frac {\partial \mathrm {ln}\rho }{\partial \mathrm {ln}T}\right )_p\frac {\mathrm {d} p}{\mathrm {d} t}} _{\displaystyle \phi _{\textit{comp}}} ,\end{align}

where $\mathrm {d} p/\mathrm {d} t = \partial p/\partial t + \boldsymbol {u \cdot \nabla } p$ and (4.16) is equivalent to (2.18) (Moukalled et al. Reference Moukalled, Mangani and Darwish2016). Here, the first term on the right-hand side represents convection term ( $\phi _{\textit{conv}}$ ), the second term represents diffusion term ( $\phi _{\textit{diff}}$ ) and the third term represents compression work term ( $\phi _{\textit{comp}}$ ), which is ignored under OB conditions (Johnston & Doering Reference Johnston and Doering2009).

Figure 15.

(a) Normalised temperature $\widetilde {T}$ , convection item $\phi _{\textit{conv}}$ , diffusion item $\phi _{\textit{diff}}$ and compression work item $\phi _{\textit{comp}}$ (from top to bottom panel) contours at $H=100$ and $Ra=10^4$ for various $g$ . The green dashed lines represent $\widetilde {T}=0$ . (b) Variation of normalised Nusselt number $\widetilde {Nu}$ with $\alpha$ . The corresponding $g$ is indicated in the figure. (c) Variation of the spatial-averaged of the absolute value of source term $\langle |\phi |\rangle _{x,y}$ with $\alpha$ . Black circle points represent the results of $\phi _{\textit{conv}}$ . Blue pentagon points represent the results of $\phi _{\textit{diff}}$ . Red square points represent the results of $\phi _{\textit{comp}}$ . Connect points with lines for visual enhancement.

Figure 15(c) shows the spatial-averaged of the absolute value of $\phi$ , $ \left \lt |\phi |\right \gt _{x,y}$ , as a function of $ \alpha$ . The results indicate that $ \left \lt |\phi _{\textit{diff}}|\right \gt _{x,y}$ decreases with increasing $ \alpha$ , while $ \left \lt |\phi _{\textit{comp}}|\right \gt _{x,y}$ increases with increasing $ \alpha$ . The contours of $\phi$ in figure 15(a) reveal that when NOB-II effects are weak, the diffusion and convection terms dominate and the compression term is negligible. This observation is consistent with figures 15(c) and 16(a). Since the diffusion term is related to the temperature gradient, it primarily acts near the plates, where the temperature gradient is the largest (Demou & Grigoriadis Reference Demou and Grigoriadis2019; Wan et al. Reference Wan, Wang, Wang, Xia, Zhou and Sun2020), as shown in figure 8(a). Therefore, $\langle |\phi _{\textit{conv}}|\rangle _{x}$ and $\langle |\phi _{\textit{diff}}|\rangle _{y}$ present two peaks at the region near the plates, as shown in figure 16(a). At this point, the heat transfer mechanism is that the thermal plume carries the hot fluid near the bottom plate to the top, and then the heat of thermal plume is transfered to the top plate by thermal diffusion. This heat transfer mechanism enables the transfer of heat from the top plate to the bottom plate.

Figure 16.

Variation of the horizontal-averaged of the absolute value of source term $\langle |\phi |\rangle _{x}$ with $y/H$ for (a) $g=3\times 10^{-5}$ and (b) $g=2\times 10^{-4}$ at $H=100$ and $Ra=10^4$ . Black circle points represent the results of $\phi _{\textit{conv}}$ . Blue pentagon points represent the results of $\phi _{\textit{diff}}$ . Red square points represent the results of $\phi _{\textit{comp}}$ . Connect points with lines for visual enhancement.

In contrast, when NOB-II effects are strong, the compression and convection terms dominate, while the diffusion term is negligible, as shown in figures 15(a), 15(c) and 16(b). According to (4.16), the compression work term is related to $\boldsymbol {u \cdot \nabla } p$ . Although the pressure gradient is uniform throughout the domain ( $\boldsymbol {\nabla } p \approx \rho \boldsymbol {g}$ ), the velocity is higher in the central region and lower near the plates. Additionally, the velocity of the plume in the central region is parallel to the pressure gradient, whereas near the plates, it is perpendicular to the pressure gradient, as shown by the velocity vectors in figure 15(a). As a result, the compression term is larger in the central region and smaller near the plates, as illustrated in figures 15(a) and 16(b). At this point, the heat transfer mechanism changes: when thermal plumes transport hot fluid from the bottom plate to the top plate, the heat of thermal plume is absorbed in the central region by the compression work term. By the time the heat plume reaches the top, its temperature is significantly reduced, greatly reduceing its ability to transfer heat to the top plate. Consequently, the overall heat transfer from the bottom plate to the top plate is significantly impaired.

It is important to clarify that the previous mention of ‘the compressible work term absorbing heat’ is a metaphorical expression. In reality, the flow of fluid along in the opposite direction of the pressure gradient will lead to expansion, and the decrease in temperature during this process is a fundamental characteristic of compressible fluids.

4.7. Perfect gas

The above analysis and discussion pertain to supercritical fluids. However, our analysis is applicable to any compressible fluid. In this section, we perform simulations for a perfect gas to demonstrate the generalisability of our conclusions. The equation of state for a perfect gas is $ p = \rho RT$ . To align with the physical properties of air at room temperature, we take the following values for the parameters: $ R = 1.6$ , $ \nu = 0.05$ , $ c_v = 4$ , $ \lambda = 1.643$ , $ T_{{m}} = 0.15$ and $ p_{{m}} = 1$ . With these parameter choices, $ c_p/c_v = 1.4$ and $ Pr = 0.71$ , which correspond to the typical physical properties of air at room temperature. The height of the computational domain in this section is set to 100. The intensity of NOB-II effects is modulated by adjusting the gravitational acceleration.

Figure 17 illustrates the variation of $ \widetilde {Nu}$ and $ \Delta \widetilde {\rho }$ with $ \alpha$ . The results show that density inversion and HTD also occur in a perfect gas. The simulation results are consistent with the conclusions obtained for supercritical fluids. The density difference $ \Delta \widetilde {\rho }$ satisfies (4.12) and $ \widetilde {Nu}$ decreases from 1 to 0 with increasing $\alpha$ , which fully validates the universality of our conclusions for compressible fluids.

Figure 17.

(a) Variation of the normalised density difference $ \Delta \widetilde {\rho }$ with $ \alpha$ for $ Ra = 5000$ . (b) Variation of the normalised Nusselt number $ \widetilde {Nu}$ with $ \alpha$ for $ Ra = 5000$ . The four points in each panel, from left to right, correspond to $ g$ : $ 2 \times 10^{-4}$ , $ 3.2 \times 10^{-4}$ , $ 4.8 \times 10^{-4}$ and $ 7.5 \times 10^{-4}$ .