3.1. Route R1

Once the isoflux boundary condition is imposed on the pipe, a thermal boundary layer forms adjacent to the inner surface of the pipe. When $t$ is small at the very early stage, the flow is weak so that the heat transfer by convection is neglected. The energy equation (2.4) can be simplified as the balance of unsteady and conduction terms, which is written in the integration form as (Nie & Xu Reference Nie and Xu2022)

(3.1) \begin{equation} \int _{R -\delta _{T,1}}^R \frac {\partial T}{\partial t}r{\rm d}r \sim \int _{R -\delta _{T,1}}^R \kappa \frac {\partial }{\partial r}\left (r\frac {\partial T}{\partial r} \right ){\rm d}r. \end{equation}

The left-hand side term can be estimated by $\Delta T (R^2-(R-\delta _{T,1})^2)/(2t)$ and the right-hand side term by $\kappa R \Delta T / \delta _{T,1}$ . We obtain a relation of $\delta _{T,1}$ as

(3.2) \begin{equation} \delta _{T,1} \sim \kappa ^{1/2} t^{1/2} \boldsymbol{\cdot }\left (1 + \frac {\delta _{T,1}^3}{2 \kappa t R}\right )^{1/2} \sim \kappa ^{1/2} t^{1/2} \eta _{T,1} , \end{equation}

where $\kappa ^{1/2} t^{1/2}$ is the thickness of the thermal boundary layer on a flat wall (Lin & Armfield Reference Lin and Armfield2013; Ma et al. Reference Ma, Nie and Xu2018a ). Hence, $\eta _{T,1}$ represents the non-dimensional curvature effect on $\delta _{T,1}$ , which is always larger than unity. Here $\eta _{T,1}$ increases with $\delta _{T,1}$ , implying that the curvature effect becomes increasingly prominent with the increase of $\delta _{T,1}$ .

When the buoyancy-driven convection becomes non-negligible, a two-layer structure can be observed, which is a thermal boundary layer and a velocity boundary layer. As the Prandtl number of water is larger than unity, the thickness of the velocity boundary layer is always larger than that of the thermal boundary layer. It is noteworthy that the velocity boundary layer has also been divided into an inner and an outer viscous layer in the study of transient natural convection adjacent to a flat wall (Lin et al. Reference Lin, Armfield, Patterson and Lei2009). Instead of the three-layer model in Lin et al. (2009), we follow the two-layer model as in Patterson & Imberger (Reference Patterson and Imberger1980) for the sake of simplicity of the scaling laws.

In the thermal boundary layer where $\delta _{T,1} \lt r \lt R$ , (2.3) can be simplified as the balance between the viscous and buoyancy terms, as the inertial term is much smaller than the viscous term for ${\textit{Pr}} \gt 1$ (Patterson & Imberger Reference Patterson and Imberger1980), i.e.

(3.3) \begin{equation} - \nu \int _{R -\delta _{T,1}}^R \frac {\partial }{\partial r}\left ( {r\frac {{\partial {u_z}}}{{\partial r}}} \right ) {\rm d}r \sim \int _{R - \delta _{T,1}}^R {g\beta \Delta T r} {\rm d}r. \end{equation}

The left-hand side term may be estimated by $\nu u_{v,1} R/\delta _{T,1} + \nu u_{v,1} (R-\delta _{T,1})/(\delta _{v,1} - \delta _{T,1})$ , where $u_{v,1}$ is the typical velocity of the velocity boundary layer. The right-hand side term by $g \beta q \delta _{T,1}(R^2-(R-\delta _{T,1})^2)/(2 \lambda )$ considering $q \sim \lambda \Delta T/\delta _{T,1}$ . Using (3.2) and considering that $\delta _{v,1}$ is much larger than $\delta _{T,1}$ for water, (3.3) gives a scaling law of the velocity as

(3.4) \begin{equation} u_{v,1}\sim \frac {{\textit{Ra}}_q}{S^{4}} \frac {\kappa ^{2}t}{R^{3}} \boldsymbol{\cdot }{\frac {1-\delta _{T,1}/\delta _{v,1}}{(\delta _{T,1}/R)^{-1}-\delta _{T,1}/\delta _{v,1}}} \sim \frac {{\textit{Ra}}_q}{S^{4}}\frac {\kappa ^{2} t}{R^{3}}{\frac {\delta _{T,1}}{R}} \sim \frac {{\textit{Ra}}_q}{S^{4}} \frac {\kappa ^{5/2} t^{3/2}}{R^4}\eta _{T,1}. \end{equation}

Notably, for natural convection adjacent to a flat vertical wall, the typical vertical velocity scales as $t$ under isothermal conditions (Patterson & Imberger Reference Patterson and Imberger1980) and as $t^{3/2}$ under isoflux conditions (Lin & Armfield Reference Lin and Armfield2013). Equation (3.4) shows that the curvature effect on the velocity is captured by $\eta _{T,1}$ , similarly to its effect on $\delta _{T,1}$ .

In the layer $ \delta _{T,1} \lt r \lt \delta _{v,1}$ , the dominant balance in the momentum equation (2.3) is between the viscous and inertial terms, which can be written as

(3.5) \begin{equation} \int _{R-\delta _{v,1}}^{R-\delta _{T,1}}r\frac {\partial u_{z}}{\partial t}{\rm d}r\sim \nu \int _{R-\delta _{v,1}}^{R-\delta _{T,1}}\frac {\partial }{\partial r}(r\frac {\partial u_{z}}{\partial r}){\rm d}r. \end{equation}

The left-hand side term can be estimated by $u_{v,1}((R-\delta _{T,1})^2-(R-\delta _{v,1})^{2})/(2t)$ and the right-hand side term by $\nu (R-\delta _{T,1})u_{v,1}/(\delta _{v,1}-\delta _{T,1})$ as $\partial u_z / \partial r = 0$ at $r = R - \delta _{v,1}$ . We can obtain

(3.6) \begin{equation} \delta _{v,1} \sim \nu ^{1/2} t^{1/2} {\left [\frac {\delta _{T,1}}{\nu ^{1/2} t^{1/2}}+ \left (\frac {R-\delta _{T,1}}{2R-\delta _{T,1}-\delta _{v,1}}\right )^{1/2}\right ]} \sim \nu ^{1/2} t^{1/2} {\eta _{v,1}} .\end{equation}

Here, $\nu ^{1/2} t^{1/2}$ is the thickness of the velocity boundary layer for natural convection adjacent to a flat vertical wall, as obtained previously (Patterson & Imberger Reference Patterson and Imberger1980); $\eta _{v,1}$ represents the non-dimensional curvature effect on the thickness of the velocity boundary layer.

Additionally, the flow rate $Q_{o,1}$ at the outlet of the pipe can be estimated by $ 2 \pi R \delta _{v,1} u_{v,1}$ , which reads

(3.7) \begin{equation} Q_{o,1} \sim \frac {{\textit{Ra}}_q}{S^4} \boldsymbol{\cdot }\frac {\kappa ^3 t^2}{R^3}\eta _{T,1} \eta _{v,1}. \end{equation}

The subscript $o$ represents quantities measured at the outlet of the pipe hereafter.

With the increase of the flow rate, the heat convected away increases and balances that conducted in, and the flow and heat transfer become fully developed. The balance between the advection and diffusion terms in the energy equation (2.4) gives

(3.8) \begin{equation} \int _{R-\delta _{Ts,1}}^{R}u_{z}\frac {\partial T}{\partial z}r{\rm d}r\sim \int _{R-\delta _{Ts,1}}^{R}\kappa \frac {\partial }{\partial r}\left (r\frac {\partial T}{\partial r}\right ){\rm d}r. \end{equation}

The left-hand side term can be estimated by $u_{v\textit{s},1} q \delta _{Ts,1} (R^2-(R-\delta _{Ts,1})^2)/(2\lambda H)$ and the right-hand side term by $\kappa R q /\lambda$ . The variables with subscript $s$ represent the corresponding quantities in the steady stage here and hereafter. Thus, we have

(3.9) \begin{equation} u_{v\textit{s},1} \sim \frac {\kappa \textit{HR}}{\left ( 2R{\delta _{Ts,1}} - {\delta _{Ts,1}}^2 \right ){\delta _{Ts,1}}}. \end{equation}

It is noteworthy that the velocity $u_{v\textit{s},1}$ depends on $Raq$ and $S$ implicitly.

Assuming heat transfer and flow in the pipe enter the steady stage at time $t_{s,1}$ and inserting $t_{s,1}$ into (3.2), (3.4) and (3.6), we have

(3.10) \begin{align} \delta _{Ts,1} &\sim \kappa ^{1/2} t_{s,1}^{1/2} \left (1 + \frac {\delta _{Ts,1}^3}{2 \kappa t_{s,1} R}\right )^{1/2}, \end{align}

(3.11) \begin{align} u_{v\textit{s},1} &\sim \frac {{\textit{Ra}}_q}{S^4}\frac {\kappa ^2 t_{s,1}} {R^3} \frac {\delta _{Ts,1}}{R} \end{align}

and

(3.12) \begin{equation} \delta _{v\textit{s},1} \sim \nu ^{1/2} t_{s,1}^{1/2} \left [\frac {\delta _{Ts,1}}{\nu ^{1/2} t_{s,1}^{1/2}}+ \left (\frac {R-\delta _{Ts,1}}{2R-\delta _{Ts,1}-\delta _{v\textit{s},1}}\right )^{1/2}\right ] \sim \nu ^{1/2} t_{s,1}^{1/2}\eta '_{v\textit{s},1}, \end{equation}

respectively.

Equation (3.10) can be written as $(2R\delta _{Ts,1}-\delta _{Ts,1}^2)\delta _{Ts,1} \sim 2R \kappa t_{s,1}$ , and substituting this into (3.9) yields $u_{v\textit{s},1} \sim L/t_{s,1}$ . Physically, (3.9) states the balance between heat convected away and conducted in, while (3.10) is based on the assumption of self-similar growth of the thermal boundary layer. The resulting scale $u_{v\textit{s},1} \sim L/t_{s,1}$ reflects a simple yet fundamental constraint: for a steady convection to be established from start-up, fluid parcels must travel a distance ( $u_{v\textit{s},1}t_{s,1}$ ) comparable to several times the heated length $L$ . Inserting $u_{v\textit{s},1} \sim L/t_{s,1}$ into (3.11), we can obtain

(3.13) \begin{equation} t_{s,1} \sim \left (\frac {S^5}{{\textit{Ra}}_q} \frac {R}{\delta _{Ts,1}}\right )^{1/2} \frac {R^2}{\kappa }. \end{equation}

Using (3.10) and (3.13), $\delta _{Ts,1}$ and $t_{s,1}$ can be finally expressed as

(3.14) \begin{align} \delta _{Ts,1} &\sim \frac {\textit{SR}}{{\textit{Ra}}_q^{1/5}}{\left [1+\frac {1}{2} \frac {{\textit{Ra}}_q^{1/2}}{S^{5/2}} \left (\frac {\delta _{Ts,1}}{R}\right )^{7/2} \right ]^{2/5} } \sim \frac {\textit{SR}}{{\textit{Ra}}_q^{1/5}}{\eta _{Ts,1}}, \end{align}

(3.15) \begin{align} t_{s,1}&\sim \frac {S^2}{{\textit{Ra}}_q^{2/5}} \frac {R^2}{\kappa }\eta _{Ts,1}^{-1/2}. \end{align}

Furthermore, inserting (3.14) and (3.15) into (3.11) and (3.12) gives

(3.16) \begin{align} u_{v\textit{s},1}&\sim \frac {{\textit{Ra}}_q^{2/5}\kappa }{\textit{SR}}\eta _{Ts,1}^{1/2}, \end{align}

(3.17) \begin{align} \delta _{v\textit{s},1} &\sim { \frac {\textit{SR}}{{\textit{Ra}}_q^{1/5}} \eta _{Ts,1}^{-1/4} Pr^{1/2} \eta '_{v\textit{s},1}} \sim \frac {\textit{SR}}{{\textit{Ra}}_q^{1/5}} {\eta _{v\textit{s},1}}. \end{align}

Based on $2 \pi R \delta _{v\textit{s},1} u_{v\textit{s},1}$ , the flow rate at the outlet of the pipe can be expressed as

(3.18) \begin{equation} Q_{os,1} \sim {\textit{Ra}}_q^{1/5} R\kappa \eta _{Ts,1}^{1/2}\eta _{v\textit{s},1}. \end{equation}

In the fully developed stage the heat conducted from the boundary, $2\pi R H q$ , balances that convected away, $\rho c_p \pi u_s \Delta T _{s,1}$ , where $\Delta T _{s,1}$ is the temperature difference between the inlet and outlet. Using (3.14) and (3.16), a scaling law about $\Delta T _{os,1}$ can be written as

(3.19) \begin{equation} \Delta T _{os,1} \sim \frac {S}{{\textit{Ra}}_q^{1/5}}\frac {qR}{\lambda }\eta _{Ts,1}. \end{equation}

Comparing (3.14)−(3.19) with the corresponding scaling relations for a type I fully developed boundary layer flow at ${\textit{Pr}} \lt 1$ (Nie & Xu Reference Nie and Xu2022) reveals that the expressions differ only in the curvature coefficients.

3.2. Route R2

When $\delta _{v,1}$ increases further and merges at the centre of the pipe, the flow transitions to the scenario of type II. The time scale for the merge of the viscous boundary layer, $t_v$ , satisfies $R \sim \delta _{v,1}|_{t=t_v}$ . According to (3.6), which can be simplified as $\nu ^{1/2}t_v^{1/2} \eta _{v,1 } /R \sim 1$ considering the fact that the velocity boundary layer is much thicker than the thermal boundary layer for water, we have

(3.20) \begin{equation} t_v \sim \frac {R^2}{\nu \eta _{v,1 }^2 }. \end{equation}

Assuming the merging of the viscous boundary layer has a negligible impact on the development of the thermal boundary layer, as the viscous boundary layer is much thicker than the thermal boundary layer and the velocity has become much weaker at the tail, this means that $\delta _{T,2}$ still follows the same scaling law as (3.2) for $\delta _{T,1}$ , that is,

(3.21) \begin{equation} \delta _{T,2} \sim \kappa ^{1/2} t^{1/2} \eta _{T,2}, \end{equation}

with $\eta _{T,2} \sim \eta _{T,1}$ .

The balance of the momentum equation (2.3) is still between the viscous and buoyancy terms for the thermal boundary layer ( $R - \delta _{T,2} \lt r \lt R$ ). As the velocity boundary layer merges at the centreline, integrated along the axial direction to obtain a global relation,

(3.22) \begin{equation} - \nu \int _{0}^{L} \int _{R -\delta _{T,2}}^R \frac {\partial }{\partial r}\left ( {r\frac {{\partial {u_z}}}{{\partial r}}} \right ) {\rm d}r {\rm d}z \sim \int _{0}^{L} \int _{R - \delta _{T,2}}^R {g\beta \Delta \textit{Tr}} {\rm d}r {\rm d}z. \end{equation}

Although the velocity boundary layer has merged at the centreline, the buoyancy force is confined within the still-distinct thermal boundary layer and drives the dominant velocity variation near the wall. The characteristic wall-normal velocity gradient $ \partial u_{z}/ \partial r$ at $r = R$ is taken as $u_{v,2}/\delta _{T,2}$ . Therefore, (3.22) can be estimated as

(3.23) \begin{equation} \nu R \frac {u_{v,2}}{\delta _{T,2}} L+ \nu (R-\delta _{T,2})L\frac {\partial u_{z}}{\partial r}\mid _{R-\delta _{T,2}} \sim g \beta q L \delta _{T,2}(R^2-(R-\delta _{T,2})^2)/(2 \lambda ). \end{equation}

In the layer $0 \lt r \lt R- \delta _{T,2}$ , the balance between the viscous and inertial terms of (2.3) gives

(3.24) \begin{equation} \int _{0}^{L} \int _{0}^{R-\delta _{T}}r\frac {\partial u_z}{\partial t}{\rm d}r{\rm d}z \sim \nu \int _{0}^{L}\int _{0}^{R-\delta _{T}}\frac {\partial }{\partial r} \left (r\frac {\partial u_{z}}{\partial r}\right ){\rm d}r{\rm d}z, \end{equation}

As $\partial u_z/\partial r = 0$ at $r=0$ , it can be simplified as

(3.25) \begin{equation} \frac { u_{v,2}}{ t}\frac {(R-\delta _{T,2})^{2}}{2}L \sim \nu (R-\delta _{T,2})L \frac {\partial u_{z}}{\partial r}\mid _{R-\delta _{T,2}}. \end{equation}

Using (3.23) and (3.25), we have

(3.26) \begin{equation} u_{v,2}\sim \frac {{\textit{Ra}}_q}{S^{4}} \frac {\kappa ^{2}t}{R^{3}} \frac {\delta _{T,2}}{R} \boldsymbol{\cdot }\left [2+\frac {\delta _{T,2}}{R}\boldsymbol{\cdot }\frac {(1-\delta _{T,2}/R)^{2}}{\nu t/R^2 }\right ]^{-1} \sim \frac {{\textit{Ra}}_q}{S^{4}}\frac {\kappa ^{5/2} t^{3/2}}{R^4}\eta _{T,2}. \end{equation}

The flow rate $Q_{o,2}$ can be estimated by $ \pi R^2 u_T$ . Using (3.26) we obtain

(3.27) \begin{equation} Q_{o,2} \sim \frac {{\textit{Ra}}_q}{S^4}\frac {\kappa ^3 t^2}{R}\eta _{T,2}. \end{equation}

Comparing $\delta _{T,2}$ with $\delta _{T,1}$ and $u_{v,2}$ with $u_{v,1}$ , it shows that the thickness of the thermal boundary layer and the typical velocity in the scenario of unsteady type II develop as the same laws as that of the scenario of unsteady type I. This implies that the emergence of the velocity boundary layer introduces only a secondary effect on the development of the temperature boundary layer and the subsequent buoyancy-driven forces.

The flow and heat transfer become fully developed once the heat convected away balances that conducted in. As the velocity boundary layer merges at the centre, $\delta _{v\textit{s},2}$ remains at $R$ . Repeating the discussion of (3.8)–(3.15), the velocity and time scales in the fully developed stage read

(3.28) \begin{equation} \frac {\delta _{Ts,2}}{R} \sim \frac {S}{{\textit{Ra}}_q^{1/5}} \left [1+\frac {1}{2}\frac {{\textit{Ra}}_q^{1/2}}{S^{5/2}}\left (\frac {\delta _{Ts,2}}{R}\right )^{7/2}\right ]^{2/5} \sim \frac {S}{{\textit{Ra}}_q^{1/5}}\eta _{Ts,2} \end{equation}

and

(3.29) \begin{equation} t_{s,2}\sim \frac {S^{2}}{{\textit{Ra}}_q^{2/5}} \frac {R^2}{\kappa } \eta _{Ts,2}^{-1/2}. \end{equation}

respectively.

Repeating the discussion about $u_{v\textit{s},1} \sim L/t_{s,1}$ based on (3.9) and (3.10), we have $u_{v\textit{s},2} \sim L/t_{s,2}$ . The similarity arises because only the thicker velocity boundary layer merges, with merely a secondary effect on thermal growth and buoyancy. Then, the scale for the velocity reads

(3.30) \begin{equation} u_{v\textit{s},2}\sim \frac {{\textit{Ra}}_q^{2/5}}{S} \frac {\kappa }{R} \eta _{Ts,2}^{1/2} . \end{equation}

Based on $ \pi R^2 \delta _{vs} u_{qs}$ , the flow rate at the outlet of the pipe can be expressed as

(3.31) \begin{equation} Q_{os,2} \sim \frac {{\textit{Ra}}_q^{2/5}}{S} \kappa R \eta _{Ts,2}^{1/2} . \end{equation}

Repeating the discussion of (3.19) gives $\rho c_p \pi (R^2-(R-\delta _{Ts,2})^2)u_{v\textit{s},2}\Delta T _{os,2} \sim 2\pi RHq$ . Accordingly, the scale of the temperature difference between the inlet and outlet can be written as

(3.32) \begin{equation} \Delta T _{os,2} \sim \frac {S}{{\textit{Ra}}_q^{1/5}}\frac {qR}{\lambda }\eta _{Ts,2}. \end{equation}

3.3. Route R3

The merged thermal boundary layer and pipe flow develop from the unsteady stage of type II. When the thermal boundary layer increases and merges at the centre, $\delta _{T,3}$ is limited at $R$ . The time scale for the merge of the thermal boundary layer, $t_s$ , satisfies $R \sim \delta _{T,1}|_{t=t_s}$ as $\delta _{T,1}$ and $\delta _{T,2}$ follow the same relation. As (3.2) can be rewritten as $(\delta _T/R)^2 \sim \kappa t_T/R^2 + 1/2 (\delta _T/R)^3$ and is further simplified to $\kappa t_T/R^2 \sim 0.5$ , we have

(3.33) \begin{equation} t_T \sim \frac {R^2}{\kappa }. \end{equation}

For the pipe flow, the balance of the momentum equation (2.3) is still between the viscous and buoyancy terms:

(3.34) \begin{equation} - \nu \int _{0}^{L}\int _{0}^{R} \frac {\partial }{\partial r}\left ( {r\frac {{\partial {u_z}}}{{\partial r}}} \right ) {\rm d}r{\rm d}z \sim \int _{0}^{L}\int _{0}^{R} {g\beta \Delta T r} {\rm d}r{\rm d}z. \end{equation}

As the $\partial u_z/\partial r = 0$ at $r = 0$ , the left-hand side term can be estimated by $1/2 \nu u_{v,3} L$ . The right-hand side term can be estimated by $1/2 g \beta \Delta T R^2 L$ . Considering that most of the heat conducted in, $2\pi RLq t$ , is converted into the internal energy of the water, $\rho c_p \pi R^2 L \Delta T$ , the right-hand side term can be further simplified as ${\textit{Ra}}_q \kappa ^2 \nu R t/L^4$ . Therefore, (3.34) can be simplified as

(3.35) \begin{equation} u_{v,3}\sim \frac { {\textit{Ra}}_q}{S^{4}} \frac {\kappa ^2 }{ R^3}t. \end{equation}

Accordingly, the flow rate estimated by $\pi R^2 u_{v,3}$ reads

(3.36) \begin{equation} Q_{o,3}\sim \frac {{\textit{Ra}}_q}{S^4} \frac {\kappa ^2 }{R}t. \end{equation}

In the fully developed stage the heat conducted in will be convected away, that is, $\rho c_p \pi R^2 u_{v\textit{s},3} \Delta T_{os,3} \sim 2\pi RHq$ . We have

(3.37) \begin{equation} \Delta T_{os,3} \sim \frac { \kappa Hq}{\lambda R u_{v\textit{s},3}}. \end{equation}

Equation (3.34) is still valid for the fully developed stage:

(3.38) \begin{equation} \nu u_{v\textit{s},3} \sim g\beta \Delta T_{os,3} R^{2}. \end{equation}

Using (3.37) and (3.38), the scaling law of velocity and temperature difference can be written as

(3.39) \begin{equation} u_{v\textit{s},3}\sim \frac {{\textit{Ra}}_q^{1/2}}{S^{3/2}} \frac {\kappa }{R} \end{equation}

and

(3.40) \begin{equation} \Delta T_{os,3}\sim \frac {S^{5/2}}{{\textit{Ra}}_q^{1/2}} \frac {q R}{\lambda } , \end{equation}

respectively.

Based on (3.39), the flow rate can also be estimated as

(3.41) \begin{equation} Q_{os,3}\sim \frac {{\textit{Ra}}_q^{1/2}}{S^{3/2}}\kappa R. \end{equation}

Using (3.35) and (3.39), the time scale for the flow to develop into the steady type III regime can be written as

(3.42) \begin{equation} t_{s,3}\sim \frac {S^{5/2}}{{\textit{Ra}}_q^{1/2}} \frac {R^2}{\kappa }. \end{equation}

3.4. Regimes

The criterion for route R1 is $t_{s,1} \leqslant t_v$ , for route R2 is $t_v \lt t_{s,2} \leqslant t_T$ and for route R3 is $t_{s,3} \gt t_T$ . According to (3.15), (3.29) and (3.42), $t_{s,1}/(R^2/\kappa )$ , $t_{s,2}/(R^2/\kappa )$ and $t_{s,3}/(R^2/\kappa )$ are functions of $S^5/{\textit{Ra}}_q$ . Considering (3.20) and (3.33), the criteria distinguishing the three routes are

(3.43) \begin{equation} \mbox{Routes} \left \{ \begin{array}{l} \displaystyle \mbox{R1}: {\textit{Ra}}_q \geqslant I_{c,12} S^5,\\[3pt] \displaystyle \mbox{R2}: I_{c,12}S^5 \lt {\textit{Ra}}_q \leqslant I_{c,23}S^5,\\[3pt] \displaystyle \mbox{R3}: {\textit{Ra}}_q \gt I_{c,23} S^5. \end{array} \right . \end{equation}

Here, $I_{c,12}$ and $I_{c,23}$ are some critical values and can be determined by numerical or physical experiments.

The scales obtained in §§ 3.1– 3.3 for the three regimes are summarised in table 1.

Table 1.

Summary of the scaling laws for natural convection in a pipe with a fluid of ${\textit{Pr}} \gt 1$ .

Table 1. Long description

The table presents the scaling laws for natural convection in a pipe with a fluid of varying properties across three regimes. It is divided into two main stages: Steady and Transition time, with each stage further divided into two subcategories: Steady and Unsteady. The table has four rows and five columns. The columns are labeled Stages, Quantities, and Regimes with subcategories R1, R2, and R3. The rows list different quantities such as Thermal layer thickness, Velocity layer thickness, Velocity, Flow rate, and Temperature. Each cell under the Regimes columns contains specific scaling laws or references to equations. For example, under the Steady stage, Thermal layer thickness for regime R1 is referenced as (3.2), for regime R2 as (3.21), and for regime R3 as approximately R. Similar references are provided for other quantities and regimes. The Transition time stage follows a similar structure with references to specific equations for each quantity and regime.

3.5. Curvature effect

In our previous paper (Nie & Xu Reference Nie and Xu2022), the curvature effects for a non-merged thermal boundary layer adjacent to a vertical pipe wall were discussed and compared with those on a flat vertical wall; however, that analysis was limited to fluids with ${\textit{Pr}} \lt 1$ , for which the thermal and velocity boundary layers are of equal thickness. In the present study, we focus on the case of ${\textit{Pr}} \gt 1$ and have derived the corresponding curvature coefficients quantifying the wall curvature effect on the flow and heat transfer in § 3.1.

The main findings regarding wall curvature effects from § 3.1 are summarised as follows. (i) In the unsteady stage, the curvature effects are quantified by the coefficients $\eta _{T,1}$ for the thermal boundary layer thickness (3.2), $\eta _{v,1}$ for the velocity boundary layer thickness (3.6), $\eta _{T,1}$ for the velocity (3.4) and $\eta _{T,1}\eta _{v,1}$ for the flow rate (3.7). (ii) In the fully developed stage, the corresponding effects are quantified by $\eta _{Ts,1}$ for the thermal boundary layer thickness (3.14), $\eta _{v\textit{s},1}$ for the velocity boundary layer thickness (3.17), $\eta _{Ts,1}^{1/2}$ for the velocity (3.16) and $\eta _{Ts,1}^{1/2}\eta _{v\textit{s},1}$ for the flow rate (3.18).

A comparison with our previous paper (Nie & Xu Reference Nie and Xu2022) shows that the curvature effect on the thermal boundary layer thickness and the velocity remains the same in the unsteady stage for both ${\textit{Pr}} \lt 1$ and ${\textit{Pr}} \gt 1$ . However, the curvature effect has no influence on the flow rate for ${\textit{Pr}} \lt 1$ but enhances it by a factor of $\eta _{T,1}\eta _{v,1}$ for ${\textit{Pr}} \gt 1$ . This is because the flow rate is determined by the velocity boundary layer thickness. While the velocity and thermal boundary layer thicknesses are equal for ${\textit{Pr}} \lt 1$ , this is not the case for ${\textit{Pr}} \gt 1$ .

In contrast to flat vertical walls, the type II and type III scenarios for ${\textit{Pr}} \gt 1$ discussed in §§ 3.2 and 3.3, respectively, arise entirely due to the curvature effect of the wall. They also differ from the ${\textit{Pr}} \lt 1$ case, in which the thermal and velocity boundary layers merge simultaneously. For the type II scenario, the curvature effect in the unsteady stage is quantified by $\eta _{T,2}$ for the thermal boundary layer thickness (3.21), velocity (3.26) and flow rate (3.27). In the fully developed stage, the corresponding effects are quantified by $\eta _{Ts,2}$ for the thermal boundary layer thickness (3.28), $\eta _{Ts,2}^{1/2}$ for the velocity (3.30) and $\eta _{Ts,2}^{1/2}$ for the flow rate (3.31).

We now examine the curvature coefficients: $\eta _{T,1}$ , $\eta _{v,1}$ and $\eta _{T,1}\eta _{v,1}$ . Since $\eta _{T,1}$ and $\eta _{v,1}$ are functions of $\kappa ^{1/2} t^{1/2} /R$ according to (3.2) and (3.6), these coefficients are plotted against this dimensionless time in figure 4. The curvature coefficient $\eta _{T,2}$ for the unsteady type II scenario, which has the same functional form as $\eta _{T,1}$ , is also presented. Similarly, we examine the coefficients for the fully developed stage: $\eta _{Ts,1}$ , $\eta _{v\textit{s},1}$ , $\eta _{Ts,1}^{1/2}$ and $\eta _{Ts,1}^{1/2}\eta _{v\textit{s},1}$ . As $\eta _{Ts,1}$ and $\eta _{v\textit{s},1}$ are functions of $S {\textit{Ra}}_q^{-1/5}$ according to (3.14) and (3.17), these coefficients are plotted against this parameter in figure 5. The curvature coefficient $\eta _{Ts,2}$ for the scenario of steady type II is also shown. Figures 4 and 5 show that the curvature effect is comparable in magnitude to the leading-order term and increases with the thickness of the boundary layers. Accordingly, its impact is significant and can be quantitatively assessed.

Figure 4.

Dependence on $\kappa ^{1/2} t^{1/2} /R$ of the curvature coefficients for the scenarios of unsteady type I and type II.

Figure 4. Long description

A line graph displays the dependence of curvature coefficients for unsteady type I and type II scenarios. The x-axis represents the variable kappa to the power of one half times t to the power of one half divided by R, ranging from 0.01 to 1. The y-axis represents the variables eta subscript T 1 or eta subscript T 2, and eta subscript nu 1, ranging from 0.8 to 1.6. The solid red line represents eta subscript T 1 or eta subscript T 2, which starts at 1 and increases sharply as the x-axis value increases. The dashed red line represents eta subscript nu 1, which remains relatively constant around 1. All values are approximated.

Figure 5.

Curvature coefficients for the scenarios of fully developed type I and type II.

Figure 5. Long description

A line graph with two data lines representing the relationship between S R a q inverse 1 over 5 on the x axis and efficiency parameters eta T S 1, eta T S 2, and eta v s 1 on the y axis. The solid red line represents eta T S 1 or eta T S 2, while the dashed red line represents eta v s 1. The x axis ranges from 10 to the power of negative 2 to 10 to the power of 0, and the y axis ranges from 0.8 to 1.4. The solid red line starts at 1.0 and increases steadily, reaching approximately 1.4 at the right end of the graph. The dashed red line starts at 1.0, dips slightly below 1.0, and then increases again, ending slightly above 1.0. All values are approximated.

5.1. Impinging

The volume of fluid entrained in the impinging region is much less than $Q_{\textit{ps},3}$ . Therefore, the flow rate at the exit of the impinging region scales with $Q_{\textit{ps},3}$ , which gives

(5.1) \begin{equation} 2\pi r_{\textit{pg}} \delta _{Is0,3} u_{Is0,3} \sim Q_{\textit{ps},3}. \end{equation}

Here, $\delta _{Is0,3}$ and $u_{Is0,3}$ are the thickness and velocity of the thin-layer intrusion flow at $r = r_{\textit{pg}}$ , respectively.

The pressure increase in the vertical direction, $\Delta p$ , for buoyancy-driven flow can be estimated as $\rho g \beta \Delta T_{\textit{ps},3} \delta _{Is0,3}$ . In the radial momentum equation, the pressure term ( $-1/\rho \partial P/\partial r$ ) is balanced by the convection term ( $u_{r} \partial u_{r}/\partial r$ ) due to the intense acceleration. The radial flow driven by the pressure difference can thus be written as

(5.2) \begin{equation} \frac { u^{2}_{Is0,3} }{r_{\textit{pg}}} \sim \frac {1}{\rho }\frac {\Delta P}{ r_{\textit{pg}}} \sim \frac {1}{\rho }\frac {\rho g \beta \Delta T_{\textit{ps},3} \delta _{Is0,3}}{r_{\textit{pg}}}. \end{equation}

By combining the flow rate constraint (5.1) and the momentum balance (5.2), and substituting the scalings for the plume temperature (4.13) and flow rate (4.15), we derive the following scaling laws for the intrusion flow at $r = r_{\textit{pg}}$ :

(5.3) \begin{align} \delta _{Is0,3} &\sim \frac { D^{41/48} S^{25/32} } { {\textit{Ra}}^{25/96} } \frac {R^{5/3}}{r_{\textit{pg}}^{2/3}}, \end{align}

(5.4) \begin{align} u_{Is0,3} &\sim \frac { D^{1/12} {\textit{Ra}}^{7/24}}{S^{7/8}} \frac { \kappa }{ r_{\textit{pg}}^{1/3} R^{2/3}}. \end{align}

Consequently, the axisymmetric intrusion flows with a thickness of $\delta _{Is0,3}$ and a velocity of $u_{{\kern-1pt}Is0,3}$ are formed. Considering the mixing in the impinging region, the temperature of the intrusion flow is assumed to be quasi-uniform and scales with $\Delta T_{\textit{ps},3}$ .

5.2. Spreading

The driving force of a spreading intrusion flow varies. For subcritical flows (discharge Froude number ${\textit{Fr}} \sim 1$ or less), such as oil spilling at sea (Hoult Reference Hoult1972) or spreading of lighter current along the free surface (Didden & Maxworthy Reference Didden and Maxworthy1982), discharging heavier current at the bottom (Huppert Reference Huppert1982; Bonnecaze, Huppert & Lister Reference Bonnecaze, Huppert and Lister1993), the flow is typically buoyancy-driven (or gravity driven). In contrast, supercritical horizontal buoyant jets are primarily inertia-driven (Didden & Maxworthy Reference Didden and Maxworthy1982). In our scenario, the Froude number of the spreading flow is estimated as ${\textit{Fr}} \sim u_{Is0,3}( g\beta \Delta T _{\textit{ps},3} \delta _{Is0,3})^{-1/2}$ . Inserting (5.3), (5.4) and (4.13), ${\textit{Fr}}$ is only a function of the Prandtl number and independent of ${\textit{Ra}}_q$ , $S$ or $D$ . Considering the thermal plume and flow in the impinging region fall in the buoyancy force-driven flow regime, it confirms that the buoyancy force drives the spreading flow.

The opposing force that balances buoyancy can be either viscous or convection terms, depending on the local Reynolds number ( $Re$ ). It is noteworthy that the role of the convection term evolves with the flow process. In the impinging region the convection term balances the driving force, buoyancy-driven pressure gradient, resulting in flow acceleration. In the spreading region the flow decelerates as the viscous term balances the convection term, in which now convection acts as the driving mechanism. Accordingly, the transition from a convection-dominated impinging region to a viscous-dominated spreading region occurs at $r_{\textit{pg}}$ . Using (5.3) and (5.4), and applying the criterion $Re \sim 1$ , i.e. $u_{Is0,3} \delta _{Is0,3}/ \nu$ is unity, gives

(5.5) \begin{equation} r_{\textit{pg}} \sim \frac {{\textit{Ra}}^{1/32} D^{15/16}}{ S^{3/32}} R. \end{equation}

We adopt a one-layer model for the spreading flow, as buoyancy confines heat and momentum transport in the vertical direction. Let $\delta _{Is,3}$ and $u_{Is,3}$ denote the thickness and velocity of the spreading current, respectively, as shown in figure 7. For simplicity, a local coordinate downward with $z' = 0$ at the free surface is used for the discussion of the spreading flow.

Figure 7.

Typical temperature and velocity profiles for the intrusion flow. The profiles shown are the numerical results of the case ${\textit{Ra}}_q = 10^6$ , $S = 6.0$ and $D = 4.00$ at $r = 2.0R$ .

Figure 7. Long description

A line graph showing temperature and velocity profiles for intrusion flow. The x-axis is labeled as delta T over delta T max and u over u max, ranging from 0 to 1.2. The y-axis is labeled as z over R, ranging from 6.0 to 10.0. The graph features two lines: a red line representing the temperature profile (delta T over delta T max) and a blue line representing the velocity profile (u over u max). The red line starts at the bottom left and curves upward, indicating the temperature distribution. The blue line also starts at the bottom left and curves upward, representing the velocity distribution. The graph highlights a region labeled ‘Spreading layer,’ indicating the area where the temperature and velocity profiles interact. All values are approximated.

The balance of the buoyancy-driven pressure difference and viscous force suggests that

(5.6) \begin{equation} \int _{0}^{\delta _{Is,3}} \frac {g\beta \Delta T_{\textit{ps},3} \delta _{Is,3}}{r} {\rm d}z' \sim \nu \int _{0}^{\delta _{Is,3}}\frac {\partial ^{2}u_{r}}{\partial z'^{2}} {\rm d}z'. \end{equation}

Substituting the scaling for $\Delta T_{\textit{ps},3}$ from (4.13) and using the definition of ${\textit{Ra}}_q$ to simplify, (5.6) can be expressed as

(5.7) \begin{equation} \frac {{\textit{Ra}}^{27/32}}{ D^{11/16} S^{81/32}} \frac {\nu \kappa }{R^3}\frac {\delta _{Is,3}^2}{r} \sim \nu \frac {u_{Is,3}}{\delta _{Is,3}}. \end{equation}

Conservation of the flow rate gives

(5.8) \begin{equation} u_{Is,3}2\pi r \delta _{Is,3} \sim Q_{\textit{ps},3}, \end{equation}

in which $Q_{\textit{ps},3}$ has already been given in (4.15).

With (5.7) and (5.8), we can have the thickness and velocity of the spreading flow at $r$ as

(5.9) \begin{align} u_{Is,3} &\sim \frac { D^{17/32} {\textit{Ra}}^{15/64}} {S^{45/64}} \frac {\kappa }{r}, \end{align}

(5.10) \begin{align} \delta _{Is,3} &\sim \frac {S^{39/64}D^{13/32}} { {\textit{Ra}}^{13/64}}{R}.\\[10pt] \nonumber \end{align}

The radial position $r_{tl}$ , marking the transition to a conduction-dominated tail region, is found by balancing advective and conductive heat transfer. The advective heat flux scales as $\rho c_p 2 \pi r \delta _{Is,3} u_{Is,3} \Delta T_{\textit{ps},3}$ , while the conductive rate is $2 \pi r \delta _{Is,3} \boldsymbol{\cdot }\lambda \Delta T_{\textit{ps},3}/\delta _{Is,3}$ . Using (5.9) and (5.10), the balance of conduction and convection gives

(5.11) \begin{equation} r_{tl} \sim \frac {{\textit{Ra}}^{1/32} D^{15/16}}{ S^{3/32}} R. \end{equation}

In summary, the intrusion flow transitions between distinct dynamical regimes. The impinging region ( $r \leqslant r_{\textit{pg}}$ ) is characterised by a buoyancy-inertia balance, while the spreading region ( $r_{\textit{pg}} \lt r \leqslant r_{tl}$ ) is governed by a buoyancy-viscosity balance. Heat transfer remains advection-dominated in both the impinging and spreading regions, transitioning to conduction dominance only in the tail region ( $r \gt r_{tl}$ ). Equations (5.5) and (5.11) also indicate the non-dimensional length scale (normalised by $R$ ) of the impinging region and spreading region is proportional to ${\textit{Ra}}^{1/32} D^{15/16} S^{-3/32}$ .

6.1. Flow in the pipe

The parameter space of ${\textit{Ra}}_q$ - $S$ - $D$ can be divided into three regimes corresponding to the routes R1, R2 and R3 according to the flow within the pipe. There exists a critical value $D_c$ , beyond which a further increase in $D$ ceases to have a significant influence on the pipe flow. Figure 8 shows that the boundaries between regimes R2 and R3 are nearly identical for both $D = 1.3$ and $D = 4.0$ . This suggests that the critical value $D_c$ is likely less than 1.3. According to the criteria in (3.43), the flow transitions from route R3 to R2 when ${\textit{Ra}}_q$ increases to approximately $2000S^5$ .

Figure 8.

Regimes of the flow in the pipe in the parameter space ${\textit{Ra}}_q$ - $S$ - $D$ . Cases following route R2 are shown by the red cycle, whereas blue squares represent the cases undergoing route R3. The dashed line represents the regime boundary according to (3.43) with $I_{c,23} = 2000$ . Note that this regime map is valid for the case runs 10–20 and 20–40, it implies that variation of $D$ from 1.3 to 4.0 does not change the division of the regimes.

Figure 8. Long description

A scatter plot illustrates the flow regimes in a pipe within a parameter space. The x-axis represents the variable S on a logarithmic scale ranging from 0.1 to 10, while the y-axis represents the variable Raq on a logarithmic scale ranging from 10⁴ to 10⁷. The plot features several red circles and blue squares, indicating different flow routes. Red circles denote cases following route R2, while blue squares represent cases undergoing route R3. A dashed line marks the regime boundary according to the equation (3.43) with specific parameters. The plot highlights that the regime map is valid for case runs 1020 and 2040, implying that variations in a certain parameter from 1.3 to 4.0 do not alter the division of the regimes. The data points are distributed across the plot, showing distinct clusters for each route. The dashed line serves as a reference for distinguishing between the two regimes. All values are approximated.

Figure 9.

Variation of the thicknesses of the temperature boundary layer in the pipe over time in the start-up stage. The blue, black and red scatters are numerical results at $z/L$ = 0.25, 0.5 and 0.75, respectively. The thickness of the thermal boundary layer with the solid line corresponds to the scaling law (3.2).

Figure 9. Long description

A line graph showing the variation of the thicknesses of the temperature boundary layer in the pipe over time in the start-up stage. The x-axis represents the dimensionless time, while the y-axis represents the dimensionless thickness of the temperature boundary layer. The blue, black, and red scatters represent numerical results at different radial positions, specifically at R equals 0.25, 0.5, and 0.75, respectively. The solid line corresponds to the scaling law (3.2). The graph illustrates how the thickness of the temperature boundary layer changes over time for different radial positions within the pipe. All values are approximated.

The scaling laws for the transient boundary layer flow during the start-up stage are validated in figure 9–11. As the transient boundary layer development is negligibly affected by variations in $D$ from 1.3 to 8, we used data from numerical runs with $D = 1.3$ for this analysis.

The temperature profile at $z/L =$ 0.25, 0.50 and 0.75 are monitored in each case. The thermal boundary layer thickness $\delta _{T,1}$ for representative cases ( $S = 0.1, 2.0, 6.0$ ; ${\textit{Ra}}_q = 10^4, 10^7$ ) is presented in figure 9. This thickness is quantified using an integral measure to avoid discrepancies from arbitrary truncation of the temperature profile tail, which is defined as

(6.1) \begin{equation} \delta _{T,1} = \frac {\int _0^R (T(r,t)-T_0) {\rm d}r}{T_w -T_0}. \end{equation}

Figure 9 shows that the numerical results for the equivalent thicknesses of the temperature boundary layer collapse onto the line predicted by (3.2), which has a slope of 1.03. The variation in thickness between different axial locations ( $z/L =$ 0.25, 0.50 and 0.75) is negligible. This means that the thickness along the pipe is almost uniform in the start-up stage, which is consistent with the fact that heat conduction dominates over convection initially.

Figure 10.

Variation of the maximal velocity of flow in the pipe over time in the start-up stage. The blue, black and red scatters are numerical results at $z/L$ = 0.25, 0.5 and 0.75, respectively. The solid line represents the scaling law (3.4) with a proportional coefficient of 0.42.

Figure 10. Long description

The line graph illustrates the variation of the maximal velocity of flow in the pipe over time during the start-up stage. The blue, black, and red scatters represent numerical results at different values, specifically at 0.25, 0.5, and 0.75, respectively. The solid line on the graph represents the scaling law with a proportional coefficient of 0.42. The x-axis is labeled with the dimensionless parameter Ra sub q k to the power of 3 divided by 2 t to the power of 3 divided by 2 η sub T, 1 S to the power of negative 4 divided by R cubed, ranging from 10 to the power of negative 8 to 10 to the power of 2. The y-axis is labeled with the dimensionless parameter u sub v, 1 divided by (k divided by R), ranging from 10 to the power of negative 8 to 10 to the power of 2. All values are approximated.

Figure 11.

Variation of flow rate at the outlet of the pipe over time in the start-up stage. The solid line represents the scaling law (3.7).

Figure 11. Long description

A line graph showing the variation of flow rate at the outlet of the pipe over time in the start-up stage. The x-axis represents the dimensionless parameter RaqS⁻⁴ηT,1ηv,1κ^2t^2/R^4, ranging from 10⁻⁸ to 10⁴. The y-axis represents the dimensionless flow rate Q1/(κR), ranging from 10⁻⁸ to 10⁴. The graph includes multiple data series represented by different symbols: squares, circles, down triangles, up triangles, pentagons, and stars. Each symbol corresponds to different values of A and Raq. The solid line represents the scaling law (3.7). The data points show a general trend where the flow rate increases with the dimensionless parameter. All values are approximated.

Figure 12.

Maximal velocity of flow in the pipe in the fully developed stage. The scatters are numerical results monitored at the $ z/L =0.75$ . (a) Cases in regime R2 with the solid line corresponding to the scaling law (3.30). (b) Cases in regime R3 with the solid line corresponding to the scaling law (3.39).

Figure 12. Long description

The image contains two scatter plots side by side. The left plot (a) shows cases in regime R2 with numerical results represented by various symbols: triangles, diamonds, stars, and circles. The solid line corresponds to the scaling law (3.30). The x-axis is labeled with a complex expression involving Raq, S, and other variables, while the y-axis is labeled with a velocity term. The right plot (b) shows cases in regime R3 with symbols: stars, circles, and squares. The solid line corresponds to the scaling law (3.39). The x-axis is labeled with a different complex expression involving Raq, S, and other variables, while the y-axis is labeled with a different velocity term. Both plots illustrate the relationship between the scaled variables and the maximal velocity of flow in the pipe in the fully developed stage.

Figure 10 shows the numerical results of the maximal velocity in the velocity boundary layer. The data follow the linear relation predicted by the scaling law (3.4). The velocities at all three monitored heights ( $z/L = 0.25, 0.50, 0.75$ ) collapse onto the same line, again reflecting the weak convective effect and the resulting axial uniformity in the start-up stage.

The flow rates at the outlet of the pipe are shown in figure 11. The numerical results show that there is a linear relation between $Q_1/(\kappa R)$ and ${\textit{Ra}}_q S ^{-4} \eta _{T,1} \eta _{v,1} \kappa ^2 t^2 /R^4$ . Discrepancies between the data and the scaling law arise from two sources. In the very early stages, the grid resolution is limited compared with the extreme thin velocity boundary layer. At later times, the deviation occurs as the flow begins its transition toward the fully developed regime. The observed scaling coefficient is larger than unity because the flow rate estimate $Q_{o,1} \sim 2\pi R \delta _{v,1} u_{v,1}$ used in the derivation omits the factor of $2\pi$ , which is subsequently absorbed into the final fitted coefficient.

In general, the thickness, maximal velocity and flow rate during the start-up stage show excellent agreement with the numerical results, as evidenced by the close match in figures 9–11. The near-uniform profiles along the pipe axis and the dominance of heat conduction over convection are consistent with the physical assumptions underlying these scaling laws.

We now present the validation of the scaling laws for the fully developed flow inside the pipe. As the velocity boundary layer in regime R2 and the temperature and velocity boundary layers in regime R3 have merged (with their thicknesses of the order of the pipe radius $R$ ), the validation focuses on the global flow parameters: the outlet velocity, flow rate and temperature difference, as shown in figure 12–14.

The maximal velocities of the numerical cases in regime R2 and regime R3 are collected separately when the heat transfer and flow are fully developed. Figure 12(a) shows the maximal velocities for regime R2, in which the velocity boundary layer has merged at the centre but the thermal boundary layer has not. The non-dimensional velocities collapse onto the line predicted by the scaling law (3.30). The numerical results are fitted by the relation

(6.2) \begin{equation} u_{v\textit{s},2}/(\kappa / R) = 0.62 {\textit{Ra}}_q^{2/5} S^{-1} \eta _{Ts,2}^{1/2}+ 27.75, \end{equation}

as indicated by the solid line. As for regime R3, the numerical results in figure 12(b) also collapse onto the linear solid line, described by

(6.3) \begin{equation} u_{v\textit{s},3}/(\kappa /R) = 0.36 {\textit{Ra}}_q^{2/5} S^{-1} \eta _{Ts,2}^{1/2}+6.67. \end{equation}

Figure 13.

Flow rate in the pipe in the fully developed stage. The open and solid scatters are numerical results from cases of $D = 1.3$ and 4.0, respectively. (a) Cases in regime R2 with the solid line corresponding to the scaling law (3.31). (b) Cases in regime R3 with the solid line corresponding to the scaling law (3.41).

Figure 13. Long description

The image contains two scatter plots labeled (a) and (b), each depicting the flow rate in a pipe at the fully developed stage. In plot (a), open and solid scatters represent numerical results from cases with different parameters, and a solid line corresponds to the scaling law (3.31) for regime R2. Plot (b) similarly shows numerical results for regime R3, with a solid line representing the scaling law (3.41). The x-axis in both plots is labeled with different scaling parameters, while the y-axis measures the flow rate. The symbols used include triangles, diamonds, stars, and circles, each representing different values of the parameter S. The plots illustrate how the flow rate scales differently under varying conditions and regimes within the pipe.

Figure 14.

Validation of the temperature scales of flow at the outlet of the pipe in the fully developed stage. The open and solid scatters are numerical flow rates for cases in regimes R2 and R3, respectively. (a) Temperature scale for regime R2 (3.32). (b) Temperature scale for regime R3 (3.40).

Figure 14. Long description

Two scatter plots validate temperature scales of flow at the outlet of a pipe in the fully developed stage. The open and solid scatters represent numerical flow rates for cases in regimes R2 and R3, respectively. The first scatter plot (a) shows the temperature scale for regime R2, while the second scatter plot (b) shows the temperature scale for regime R3. Each plot includes a line of best fit and an inset zooming in on a specific range of values. The x-axes and y-axes are labeled with specific variables related to the temperature scale and flow rate. The symbols used in the plots include triangles, diamonds, stars, and circles, each representing different values of a parameter S.

Figure 13(a) shows the numerical flow rate from the cases of $D$ = 1.3 (open scatters) and $D$ = 4.0 (solid scatters). The data for $D = 1.3$ and $D = 4.0$ fall on distinct linear relations. The dashed line and the solid line correspond to

(6.4) \begin{equation} Q_{os,2}/(\kappa R) = 0.36 {\textit{Ra}}_q^{2/5}S^{-1} \eta _{Ts,2}^{1/2}+173 \end{equation}

and

(6.5) \begin{equation} Q_{os,2}/(\kappa R) = 0.89 {\textit{Ra}}_q^{2/5}S^{-1} \eta _{Ts,2}^{1/2}+19, \end{equation}

respectively. For the same ${\textit{Ra}}_q^{2/5} S^{-1} \eta _{Ts,2}^{1/2}$ , the flow rate for $D = 1.3$ is slightly lower than for $D = 4.0$ . It suggests that the free surface introduces a resistance to the vertical motion of the flow. In all, the scale ${\textit{Ra}}_q^{2/5} S^{-1} \eta _{Ts,2}^{1/2}$ effectively predicts the fully developed flow rate at the pipe outlet in regime R2.

In regime R3, when the thermal boundary layer merges, the numerical results for both $D = 1.3$ and $D = 4.0$ collapse onto a single solid line, as shown in figure 13(b). Consequently, the outlet flow rate in regime R3 follows a ${\textit{Ra}}_q^{1/2} S^{3/2}$ scaling and becomes insensitive to $D$ for $D \gt 1.3$ . This is because the upward flow velocity and flow rate in regime R3 are lower than in regime R2, as can be observed by comparing figures 12 and 13.

Figure 14 compares the numerical outlet temperatures with the scaling laws, showing excellent agreement in both regimes R2 and R3. The data collapse onto lines with slopes and intercepts of (0.787, $-0.001$ ) for regime R2 and (2.061, 0.169) for regime R3, in panels (a) and (b) respectively. This close agreement confirms that the scales $S {\textit{Ra}}_q^{-1/5} \eta _{Ts,2}$ and $S^{5/2} {\textit{Ra}}_q^{-1/2}$ accurately predict the outlet temperature for regimes R2 and R3, respectively.

Figure 15.

Validation of the velocity scale (4.14) of the thermal plume in the fully developed stage. The scatters are numerical results of cases in regime R3, and the maximal velocities at $z'= 0.5 H$ are used.

Figure 15. Long description

A scatter plot with approximately 20 data points representing the relationship between two variables. The x-axis represents Ra sub q raised to the power of 15/32, multiplied by S raised to the power of negative 45/32, and D raised to the power of 1/16. The y-axis represents u sub ps,3 divided by (k/R). The data points are color-coded and shaped differently: stars for S equals 2.0, circles for S equals 4.0, and squares for S equals 6.0. Open scatters represent D equals 1.3, solid scatters represent D equals 4.0, and half-up scatters represent D equals 8.0. Two trend lines are present: a black line and a pink line. The data points show a positive correlation along these lines. All values are approximated.

6.2. Thermal plume and intrusion flow

Figure 15 presents the maximum upward velocity in the vertical profile of the thermal plume, obtained at a location midway between the outlet and the free surface (i.e. at $z' = 0.5H$ ). The numerical results for $D = 4.0$ collapse onto the solid line defined by

(6.6) \begin{equation} u_{\textit{ps},3}/(\kappa / R) = 0.57 {\textit{Ra}}_q^{15/32} S^{-45/32} D^{1/16} +5.64. \end{equation}

In contrast, the data for $D = 1.3$ follow a separate, lower trend described by

(6.7) \begin{equation} u_{\textit{ps},3}/(\kappa / R) = 0.46 {\textit{Ra}}_q^{15/32} S^{-45/32} D^{1/16} -0.72. \end{equation}

This indicates that for the same value of the scale ${\textit{Ra}}_q^{15/32} S^{-45/32} D^{1/16}$ , the plume velocity for $D=1.3$ is lower than that for $D=4.0$ . This is because a smaller $D$ brings the free surface closer, which impedes the vertical flow by constraining fluid motion and modifying the local pressure distribution. Compared with the significant change between $D = 1.3$ and $D = 4.0$ , the difference between the data for $D = 4.0$ and $D = 8.0$ is marginal. This trend suggests that further increases in $D$ no longer significantly affect the measured maximum velocity, and the plume at the measurement location gradually approaches a state where the influence of the pipe exit and upper boundary becomes negligible.

Figure 16.

Validation of the velocity scale (5.9) of the spreading flow. The scatters are the maximal numerical velocity of cases in regime R3.

Figure 16. Long description

A scatter plot with data points representing maximal numerical velocity in regime R3. The x-axis represents the dimensionless parameter Raq^15/64 D^17/32 S⁻⁴⁵/64 R r⁻¹, while the y-axis represents the dimensionless velocity uIs,3/(kappa/R). The plot includes data points in various colors (red, black, and blue) and shapes (stars, circles, and squares), indicating different values of S and D. Open and solid scatters differentiate between r = 2.0R and r = 3.0R. A regression line is present, showing a positive correlation. All values are approximated.

Figure 17.

Validation of the thickness scale (5.10) of the spreading flow. The scatters are the numerical thickness of cases in regime R3.

Figure 17. Long description

A scatter plot illustrates the relationship between two variables, with approximately 30 data points. The x-axis represents the dimensionless parameter Raq⁻¹³/64 * S^39/64 * D^13/32, while the y-axis represents the dimensionless parameter δIs,3/R. The data points are color-coded: red for S = 2.0, black for S = 4.0, and blue for S = 6.0. They are also shaped differently: stars for D = 1.3, circles for D = 4.0, and squares for D = 8.0. Open scatters indicate r = 2.0R, and solid scatters indicate r = 3.0R. A regression line is present, showing a positive correlation between the variables. The plot includes a legend explaining the color and shape coding. All values are approximated.

The velocity and thickness of the spreading flow were monitored at two streamwise locations, $r = 2.0R$ and $r = 3.0R$ , and are compared with the scales in (5.9) and (5.10), respectively. The comparisons are presented in figure 16 for velocity and figure 17 for thickness.

Figure 16 shows that the scaling law tends to underestimate the velocity at $r = 3.0R$ but overestimate it at $r = 2.0R$ . A similar deviation is observed for the flow thickness in figure 17. This discrepancy arises because the scaling analysis assumes a balance between buoyancy and viscosity. However, at $r = 2.0R$ , which is closer to the impinging region, inertial forces still play a certain driving role, resulting in a higher velocity and a thicker boundary layer due to more intense mixing than predicted. Furthermore, as shown in figure 17, the scaling law predicts the data for $D=4$ and $D=8$ better than for $D=1.3$ . This is consistent with the notion that the closer proximity to the free surface at smaller $D$ introduces more significant mixing.

Despite these local discrepancies, the overall trends of the spreading flow are well captured by the scaling law. Based on the numerical data, the velocity and thickness can be estimated by

(6.8) \begin{equation} u_{Is,3}/(\kappa / R) = 10.69 {\textit{Ra}}_q^{15/64} D^{17/32} S^{-45/64} (R/r) -20.57, R_{\textit{adj}}^2 = 0.90 \end{equation}

and

(6.9) \begin{equation} \delta _{Is,3}/ R = 1.38 {\textit{Ra}}_q^{-13/64} S^{39/64} D^{13/32} + 0.41, R_{\textit{adj}}^2 = 0.79, \end{equation}

respectively. Here, $R_{\textit{adj}}^2$ is the adjusted $R$ -squared value. The $R_{\textit{adj}}^2$ of the scaling law for spreading flow is lower than that for the pipe and plume flows. This is due to the accumulation of errors caused by the sequential assumptions in the scaling analysis from the pipe to the plume and finally to the spreading flow.