3.1.1. Initial stage

As the flow domain and boundary and initial conditions are axisymmetric, the natural convection flow is axisymmetric before the transition occurs for high Rayleigh numbers. This implies that the heat conduction in the circumferential direction is negligible. In addition, t is small at the very early stage, and thus ∂T/∂z can be assumed to be negligible. Further, the flow is so weak initially that the heat transfer by convection is neglected (Patterson & Imberger Reference Patterson and Imberger1980). Accordingly, the energy equation (2.5) can be simplified as

(3.1)\begin{equation}\frac{{\partial T}}{{\partial t}}\sim \kappa \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial T}}{{\partial r}}} \right).\end{equation}

It is noteworthy that the analytical solution based on Bessel function has also been derived for the pure conduction in the fleeting early stage by, e.g. Bergman et al. (Reference Bergman, Lavine, Incropera and Dewitt2011).

Integrating (3.1) over the radial direction, we have

(3.2)\begin{equation}\int_{R - {\delta _T}}^R {\frac{{\partial T}}{{\partial t}}r\,\textrm{d}r} \sim \kappa \int_{R - {\delta _T}}^R {\frac{\partial }{{\partial r}}\left( {r\frac{{\partial T}}{{\partial r}}} \right)\,\textrm{d}r} ,\end{equation}

where δ_T is the thickness of the TBL and the subscript T represents the isothermal condition hereafter.

The left-hand side of (3.2) can be estimated by $\Delta T({R^2} - {(R - {d_T})^2})/(2t)$. Considering ∂T/∂r = 0 at r = R − δ_T, the right-hand side can be estimated by $\kappa R\Delta T/{\delta _T}$. Therefore, an implicit relation about δ_T can be given by

(3.3)\begin{equation}{\left( {\frac{{{\delta_T}}}{R}} \right)^2}\sim \frac{{\kappa t}}{{{R^2}}} + \frac{1}{2}{\left( {\frac{{{\delta_T}}}{R}} \right)^3},\end{equation}

which is dealt with in the following section.

The streamwise velocity of the boundary layer flow is always much stronger than the radial and circumferential velocity. Moreover, the order of the unsteady term in (2.4) is O(u_T/t), and those of the advection, viscous and buoyancy terms are $O(u_T^2/h),\;O(\nu {u_T}/\delta _T^2)$ and O(gβΔT), respectively (Xu et al. Reference Xu, Patterson and Lei2009; Lin & Armfield Reference Lin and Armfield2012). Therefore, the balance is mainly among the unsteady term, viscous term and buoyancy term owing to the weak advection for a small time, which gives

(3.4)\begin{equation}\frac{{\partial {u_T}}}{{\partial t}}\sim \nu \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial {u_T}}}{{\partial r}}} \right) + g\beta \mathrm{\Delta }T,\end{equation}

where u_T is the characteristic velocity of the TBL.

Integrating (3.4) over the radial direction, we have

(3.5)\begin{equation}\int_{R - {\delta _T}}^R {\frac{{\partial {u_T}}}{{\partial t}}r\,\textrm{d}r} \sim \nu \int_{R - {\delta _T}}^R {\frac{\partial }{{\partial r}}\left( {r\frac{{\partial {u_T}}}{{\partial r}}} \right)\,\textrm{d}r} + \int_{R - {\delta _T}}^R {g\beta \mathrm{\Delta }Tr\,\textrm{d}r} .\end{equation}

In (3.5), the first term may scale with ${u_T}({R^2} - {(R - {\delta _T})^2})/(2t)$, the second term with νu_TR/δ_T because ∂u_T/∂r = 0 at r = R − δ_T, and the third term with $g\beta \Delta T({R^2} - {(R - {\delta _T})^2})/2$. Further, because the fixed Prandtl number is assumed in this study, (3.5) may be simplified and, thus, a scaling law of a typical streamwise velocity may be expressed as

(3.6)\begin{equation}{u_T}\sim R{a_T}\frac{{{\kappa ^2}t}}{{{H^3}}}.\end{equation}

The fluid in the TBL heated by the pipe wall rises owing to the buoyancy effect. That is, the flow rate at the outlet of the pipe in the initial stage Q_To may be described by ${u_T}({R^2} - {(R - {\delta _T})^2})$ where ${R^2} - {(R - {\delta _T})^2}$ may be described by 2Rκt/δ_T based on the discussion of (3.3). Further, considering (3.6), Q_To in the initial stage can be expressed as

(3.7)\begin{equation}{Q_{To}}\sim R{a_T}\frac{{{\kappa ^3}{t^2}}}{{{H^3}}}{\left( {\frac{{{\delta_T}}}{R}} \right)^{ - 1}}.\end{equation}

Note that the subscript o represents the quantity at the outlet of the pipe hereafter.

3.1.2. Fully developed stage

With the passage of time, the heat convected away increases and reaches that conducted in from the pipe wall. As a result, the development of the TBL enters a fully developed stage. Considering the energy equation, we may obtain

(3.8)\begin{equation}{u_{Ts}}\frac{{\partial T}}{{\partial z}}\sim \kappa \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial T}}{{\partial r}}} \right),\end{equation}

where u_Ts is the velocity of the TBL in the fully developed stage and the subscript s represents the fully developed stage hereafter.

Integrating (3.8) over the radial direction, we have

(3.9)\begin{equation}\int_{R - {\delta _{Ts}}}^R {{u_{Ts}}\frac{{\partial T}}{{\partial z}}r\,\textrm{d}r} \sim \kappa \int_{R - {\delta _{Ts}}}^R {\frac{\partial }{{\partial r}}\left( {r\frac{{\partial T}}{{\partial r}}} \right)\,\textrm{d}r} ,\end{equation}

where, the left-hand side term can be estimated by ${u_{Ts}}\Delta T({R^2} - {(R - {\delta _{Ts}})^2})/(2z)$, the right-hand side term by κzRΔT/δ_Ts, and δ_Ts is the thickness of the TBL in the fully developed stage. Thus, we have

(3.10)\begin{equation}{u_{Ts}}\sim \frac{{\kappa zR}}{{(2R{\delta _{Ts}} - {\delta _{Ts}}^2){\delta _{Ts}}}}.\end{equation}

Assume that the TBL reaches the fully developed stage at the time t_Ts. Inserting t_Ts into (3.3) and (3.6), respectively, we have

(3.11)\begin{gather}{\left( {\frac{{{\delta_{Ts}}}}{R}} \right)^2}\sim \frac{{\kappa {t_{Ts}}}}{{{R^2}}} + \frac{1}{2}{\left( {\frac{{{\delta_{Ts}}}}{R}} \right)^3}.\end{gather}

(3.12)\begin{gather}{u_{Ts}}\sim R{a_T}\frac{{{\kappa ^2}{t_{Ts}}}}{{{H^3}}}.\end{gather}

In addition, we also have u_Ts~z/t_Ts based on (3.10) and (3.11). Thus, inserting u_Ts~z/t_Ts into (3.12), we can obtain

(3.13)\begin{equation}{t_{Ts}}\sim Ra_T^{ - 1/2}{\left( {\frac{z}{H}} \right)^{1/2}}\frac{{{H^2}}}{\kappa }.\end{equation}

Further, inserting (3.13) into (3.11) and (3.12), we can obtain the relations for the thickness and velocity of the TBL in the fully developed stage,

(3.14)\begin{equation}{\left( {\frac{{{\delta_{Ts}}}}{R}} \right)^2}\sim \frac{{{A^2}}}{{Ra_T^{1/2}}}{\left( {\frac{z}{H}} \right)^{1/2}} + \frac{1}{2}{\left( {\frac{{{\delta_{Ts}}}}{R}} \right)^3},\end{equation}

(3.15)\begin{equation}{u_{Ts}}\sim Ra_T^{1/2}\frac{\kappa }{H}{\left( {\frac{z}{H}} \right)^{1/2}}.\end{equation}

In addition, the flow rate Q_Tso at the outlet of the pipe in the fully developed stage can be described by ${u_{Tso}}({R^2} - {(R - {\delta _{Tso}})^2})$, expressed as

(3.16)\begin{equation}{Q_{Tso}}\sim \frac{{\kappa {H^2}}}{{{\delta _{Tso}}A}}.\end{equation}

In the previous discussion, the thickness of the TBL is assumed to be so thin (particularly initially) that the curved TBL does not merge at the centre of the pipe. However, the curved TBL may merge at the centre of the pipe when δ_T~R. Here, if merging occurs, the flow rate of the TBL at the outlet of the pipe in the fully developed stage $({Q_{Tsom}}\sim {u_{Ts}}{\rm \pi} {R^2})$ can be expressed as

(3.17)\begin{equation}{Q_{Tsom}}\sim Ra_T^{1/2}\frac{\kappa }{H}{R^2},\end{equation}

where, the subscript m represents the merging TBL hereafter.

3.1.3. Heat transfer

In the initial stage, the rate of the heat conducted into the fluid from the pipe wall can be estimated by $2{\rm \pi} RH \cdot \lambda \Delta T/{\delta _T}$, the part of which is stored by the fluid in the pipe and the other part is convected away by convection. The rate of the heat convected away by the boundary layer flow can be estimated by $\rho {c_p}{Q_{To}}\Delta T$. In the fully developed stage, the heat stored by the fluid in the pipe cannot increase; that is, the rate of the heat convected away $\rho {c_p}{Q_{Tso}}\Delta T$ (or $\rho {c_p}{Q_{Tsom}}\Delta T$ when merging occurs) balances that conducted in $2{\rm \pi} RH \cdot \lambda \Delta T/{\delta _{Ts}}$ from the pipe wall. Further, based on (3.16) without merging and (3.17) with merging of the TBL, respectively, we may obtain the Nusselt number of the whole pipe wall in the fully developed stage, which is the rate of the heat convected away $\rho {c_p}{Q_{Tso}}\Delta T$ (or $\rho {c_p}{Q_{Tsom}}\Delta T$) normalised by $2{\rm \pi} RH \cdot \lambda \Delta T/R$, expressed as

(3.18) \begin{gather} N{u_T}\sim \left\{\begin{array}{@{}ll} \dfrac{R}{\delta_{Ts}}&{\delta_{Ts}} < R\\ \dfrac{Ra_T^{1/2}}{A^2}&\delta_{Ts} = R \end{array}\right. .\end{gather}

3.2.1. Initial stage

Initially, the balance of the energy equation (2.5) is between the unsteady term and radial conduction term. Similar to the discussion of (3.3), the thickness of the TBL may be given by

(3.19) \begin{equation}{\left( {\frac{{{\delta_q}}}{R}} \right)^2}\sim \frac{{\kappa t}}{{{R^2}}} + \frac{1}{2}{\left( {\frac{{{\delta_q}}}{R}} \right)^3},\end{equation}

where δ_q is the thickness of the TBL in the initial stage and the subscript q represents the isoflux condition hereafter.

Further, we have a scaling relation between the heat flux q and temperature difference ΔT for the isoflux TBL (also see Ma, Nie & Xu Reference Ma, Nie and Xu2018),

(3.20)\begin{equation}q\sim \lambda \frac{{\mathrm{\Delta }T}}{{{\delta _q}}}.\end{equation}

When the buoyancy-driven convection becomes non-negligible, we need to consider the balance among the unsteady term, viscous term and buoyancy term in (2.4) (similar to the discussion of (3.4)). The unsteady term may be described by ${u_q}({R^2} - {(R - {\delta _q})^2})/(2t)$, the viscous term by νu_qR/δ_q, and the buoyancy term by $g\beta q{\delta _q}({R^2} - {(R - {\delta _q})^2})/(2\lambda )$ based on (3.20). The balance gives the scale of the velocity for the fixed Prandtl number based on (3.19),

(3.21)\begin{equation}{u_q}\sim \frac{{R{a_q}}}{{{A^4}}}\frac{{{\kappa ^2}t}}{{{R^3}}}\frac{{{\delta _q}}}{R}.\end{equation}

Clearly, u_q is proportional to δ_q, but u_T is not dependent on δ_T based on (3.6). This is because the buoyancy-driven flow is determined by the temperature difference between the fluids in and out of the TBL, which is a constant value for the isothermal condition but is influenced by δ_q for the isoflux condition.

In addition, the flow rate in the initial stage Q_qo can be described by ${u_q}({R^2} - {(R - {\delta _q})^2})$ in which ${R^2} - {(R - {\delta _q})^2}$ can be rewritten as 2Rκt/δ_T using (3.19). Further, based on (3.21), Q_qo can be expressed as

(3.22)\begin{equation}{Q_{qo}}\sim \frac{{R{a_q}}}{A}\frac{{{\kappa ^3}{t^2}}}{{{H^3}}}.\end{equation}

3.2.2. Fully developed stage

With the passage of time, the heat convected away increases and balances that conducted in, resulting in the fully developed stage. Considering the balance between advection and diffusion terms in (2.5) and repeating the discussion of (3.8)–(3.10), we may obtain the velocity of the isoflux TBL in the fully developed stage,

(3.23)\begin{equation}{u_{qs}}\sim \frac{{\kappa zR}}{{(2R{\delta _{qs}} - {\delta _{qs}}^2){\delta _{qs}}}}.\end{equation}

Here, δ_qs is the thickness of the TBL in the fully developed stage.

Assume that the development of the isoflux TBL enters the fully developed stage at time t_qs. Inserting t_qs into (3.19) and (3.21), respectively, we have the relations

(3.24)\begin{equation}{\left( {\frac{{{\delta_{qs}}}}{R}} \right)^2}\sim \frac{{\kappa {t_{qs}}}}{{{R^2}}} + \frac{1}{2}{\left( {\frac{{{\delta_{qs}}}}{R}} \right)^3},\end{equation}

(3.25)\begin{equation}{u_{qs}}\sim \frac{{R{a_{qs}}}}{{{A^4}}}\frac{{{\kappa ^2}{t_{qs}}}}{{{R^3}}}\frac{{{\delta _{qs}}}}{R}.\end{equation}

Similar to the discussion of (3.13), based on (3.23), (3.24) and (3.25), the time scale t_qs may be obtained, expressed as

(3.26)\begin{equation}{t_{qs}}\sim {\left( {\frac{A}{{R{a_q}}}} \right)^{1/2}}{\left( {\frac{z}{H}} \right)^{1/2}}{\left( {\frac{{{\delta_{qs}}}}{R}} \right)^{ - 1/2}}\frac{{{H^2}}}{\kappa }.\end{equation}

Inserting (3.26) into (3.24) and (3.25), respectively, the thickness of the isoflux TBL in the fully developed stage can be expressed as

(3.27)\begin{equation}{\left( {\frac{{{\delta_{qs}}}}{R}} \right)^{\textrm{5/2}}}\sim {A^{5/2}}Ra_q^{ - 1/2}{\left( {\frac{z}{H}} \right)^{1/2}} + \frac{1}{2}{\left( {\frac{{{\delta_{qs}}}}{R}} \right)^{\textrm{7/2}}},\end{equation}

and the velocity as

(3.28)\begin{equation}{u_{qs}}\sim \frac{{Ra_q^{1/2}}}{{{A^{1/2}}}}{\left( {\frac{z}{H}} \right)^{1/2}}{\left( {\frac{{{\delta_{qs}}}}{R}} \right)^{1/2}}\frac{\kappa }{H}.\end{equation}

In addition, the flow rate at the outlet of the pipe in the fully developed stage can be expressed as

(3.29)\begin{equation}{Q_{qso}}\sim \frac{{\kappa H}}{{{\delta _{qs}}\textrm{/}R}},\end{equation}

based on ${Q_{qso}}\sim {u_{qs}}({R^2} - {(R - {\delta _{qs}})^2})$, (3.27) and (3.28) for the scenario without merging, but

(3.30)\begin{equation}{Q_{qsom}}\sim \frac{{Ra_q^{1/2}}}{{{A^{1/2}}}}\frac{\kappa }{H}{R^2},\end{equation}

based on ${u_{qs}}{\rm \pi} {R^2}$ and (3.28) for the scenario with merging of the TBL.

3.2.3. Temperature difference

Clearly, the rate of the heat convected away balances that conducted from the pipe wall q once the isoflux TBL is fully developed; that is, $\rho {c_p}{Q_{qso}}\Delta T\sim \textrm{ }2{\rm \pi} RHq$ or $\rho {c_p}{Q_{qsom}}\Delta T\sim 2{\rm \pi} RHq$, which is dependent on whether the isoflux TBL merges. The temperature difference ($\varOmega$) between the fluids convected away from and into the pipe can be normalised by qR/λ, and $\varOmega$_o can be written as

(3.31)\begin{equation}{\varOmega _o}\sim \left\{ {\begin{array}{*{20}{l}} {\dfrac{{{\delta_{qs}}}}{R}}&{{\delta_{qs}} < R}\\ {\dfrac{{{A^{5/2}}}}{{Ra_q^{1/2}}}}&{{\delta_{qs}} = R} \end{array}} \right..\end{equation}

5.2.1. Regimes

Natural convection in the pipe is axisymmetric, and thus only the profiles of the temperature and velocity adjacent to the wall are presented in figure 5 for A = 0.5 and Ra_T = 10⁶. The fluid in the core of the pipe is not heated as shown in figure 5(a) and even is motionless as shown in figure 5(b). This implies that the flow rate of the pipe mainly originates from the thin thermal layer adjacent to the wall. Figure 5(a) also shows that the velocity does have the maximum in the TBL but may be negative outside the TBL at the downstream heights (Z = 0.75 and 0.1) owing to the existence of the backflow. The examination of numerical results shows that the axial symmetry may remain and the transition and turbulence do not occur in all isothermal and isoflux cases.

Figure 5.

Temperature and velocity in the fully developed stage for A = 0.5 and Ra_T = 10⁶. (a) Temperature distribution in the computational domain and (b) temperature (upper) and velocity (lower) distributions at slice A.

Figure 6 further plots half of the temperature and velocity for A = 5 and Ra_T = 10³ owing to symmetry. The examination of numerical results shows that the TBL is thick and merges at the centre of the pipe. Further, figure 6(a) shows that the temperature profiles are different at the upstream and downstream heights. The non-dimensional temperature of the fluid in the core is much larger than zero. In addition, figure 6(b) shows that the velocity profiles rarely vary at different heights with the maximum in the core, which are different from those in figure 5(c). This is because the fluid in the core at the upstream is also driven upward owing to the suction effect caused by the downstream flow once the TBL merges.

Figure 6.

(a) Temperature and (b) velocity in the fully developed stage for A = 5 and Ra_T = 10³.

Repeating the observations in figures 5 and 6 for the other numerical cases, numerical cases with and without merging of the TBL were distinguished and are shown in the parameter space of Ra_T and A in figure 7. That is, regime I_T is dominated by the boundary layer flow for which the TBL is so thin that the flow rate is contributed by the boundary layer flow; regime II_T is dominated by the duct flow in which the TBL merges in the core, resulting in the conservation of the flow rate at each cross-section.

Figure 7.

Numerical cases for the isothermal condition for different A and Ra_T. The boundary layer flow regime (I_T) is shown by the open squares: the TBL without merging; the duct flow regime (II_T) is shown by the solid squares: the TBL with merging. Here, the TBL is regarded as the layer with (T − T ₀)/(T_w − T ₀) > 1 %. Note that the white border stripe is only a schematic separating the open and solid squares.

5.2.2. Initial stage

Figure 8(a) shows the thicknesses identified by using the different temperature contours as the edge of the TBL. Significant discrepancy can be observed among δ_{T 99}, δ_{T 95} and δ_{T 90}, which correspond to the thicknesses identified by using the temperature contour (T − T ₀)/(T_w − T ₀) = 0.99, 0.95 and 0.90 as the edge of the TBL, respectively. To avoid the discrepancy by artificial truncation thresholds, an equivalent thickness of the TBL δ_TE, defined as

(5.1)\begin{equation}{\delta _{TE}} = \frac{{\int_0^R {(T - {T_0})\,\textrm{d}r} }}{{{T_w} - {T_0}}},\end{equation}

is introduced and plotted in figure 8(a). Further, δ_{T 99}, δ_{T 95}, δ_{T 90} and δ_TE versus the scale C_TRη_T are further plotted in figure 8(b). Linear relations between the thicknesses and C_TRη_T can be observed. Thus, the growth law of the TBL may be validated using the definition (5.1). As a result, the equivalent thickness of the TBL δ_TE was used, which is also abbreviated to δ_T hereafter.

Figure 8.

Thicknesses of the TBL at the height of Z = 0.5 for Ra_T = 10⁵ and A = 1.0 identified by different definitions. (a) Dependence of the thickness on time. (b) Thickness δ_T/R versus the scale C_TRη_T.

We choose the maximal velocity in the streamwise as the characteristic velocity of the TBL. The time histories of u_T/(κ/R) at three heights, i.e. Z = 0.25, 0.50 and 0.75, are shown in figures 9(a) and 9(b) for A = 0.5 and Ra_T = 10³ and 10⁶ under regime I_T, but in figures 9(c) and 9(d) for A = 5.0 and Ra_T = 10³ and 10⁶ under regime II_T.

Figure 9.

Maximal velocities of the TBL versus time under regimes I_T and II_T. (a) and (b) Two typical cases under regime I_T, and (c) and (d) two typical cases under regime II_T.

Figures 9(a) and 9(b) show that when the TBL is smaller than the radius of the pipe, the maximal velocities at heights Z = 0.25, 0.50 and 0.75 can grow freely, and overshoots can be observed if the Rayleigh number is sufficiently large. In figures 9(c) and 9(d), the TBL merges at the centre of the pipe. Figure 9(d) shows that the TBL grows synchronously at heights Z = 0.25, 0.50 and 0.75 when C_TR < 0.2. Then, the overshoot of the maximal velocity at Z = 0.25 appears, which means that the upstream flow has entered a fully developed stage. However, the TBL at downstream continues to grow and the fluid in the core area may be entrained into the TBL. Once the TBL merges at the centre, the upstream flow may suffer the suction effect from the downstream; that is, the fluid at the upstream accelerates again, and in turn a distinct secondary growth appears in the velocity curve of Z = 0.25, as shown in figure 9(d).

For validation of the scaling law of δ_T (table 1), the numerical results of the thicknesses of the TBL under regime I_T were obtained and are shown in figure 10(a). Here, six numerical cases under regime I_T were chosen, i.e. the three cases in the corners, the one at the centre and the two near the separatrix in figure 7. In each case, the thickness of the TBL is presented at heights Z = 0.25, 0.5 and 0.75. Figure 10(a) shows that the thicknesses of the TBL from the numerical results fall in a line with the slope equal to 1.1. This means that the thickness of the TBL grows as C_TRη_T. A good linear relation also suggests that the thickness along the pipe is almost uniform, i.e. the growth of the thicknesses is not dependent on the height. It is consistent with the fact that conduction is dominant but convection is weak in the initial stage.

Figure 10.

Thicknesses of the TBL under regimes I_T and II_T. (a) Under the boundary layer flow regime I_T, and (b) under the duct flow regime II_T (also see figure 7).

The thicknesses of the TBL for six numerical cases under regime II_T are also plotted in figure 10(b). It shows that in the early stage in which the TBL does not merge, the numerical results follow the same scaling relation under regime I_T. However, the deviation can occur when C_TRη_T > 0.2. This is because the boundary layer is too thick and, in turn, the assumption of the boundary layer is invalid. Further, the numerical results fall upon the solid line when C_TRη_T > 0.2. This means that the growth rate of the thickness of the TBL speeds up when the TBL merges at the centre.

The maximal velocities of the TBL at three heights Z = 0.25, 0.50 and 0.75 were monitored for six numerical cases under regime I_T. The velocities in the initial stage are plotted versus the scaling law of u_T (see table 1) in figure 11(a). It is clear that the velocity of the TBL can be described by the scaling law of u_T. The scaling coefficient can also be obtained using a linear fitting, which is 0.32. In addition, the maximal velocity for six numerical cases under regime II_T is shown in figure 11(b). It is seen from this figure that the small difference between the numerical results and the scaling law of u_T exists when the TBL merges at the centre. However, when merging becomes strong, the maximal velocities from the numerical results deviate slightly from the scaling law. A typical case is for A = 5.0 and Ra_T = 1 × 10³. That is, the maximal velocity is consistent with the scaling law of u_T at the early period in the initial stage. However, as time increases further for which the TBL merges, the maximal velocity exceeds the prediction by the scaling law.

Figure 11.

Maximal velocities under regimes I_T and II_T. (a) Under the boundary layer flow regime I_T (also see figure 7), and (b) under the duct flow regime II_T.

The scaling law of the flow rate at the outlet in the initial stage is also validated. All 23 cases in figure 7 are plotted in figure 12 under regimes I_T and II_T. Here, only the flow rate at the outlet was monitored because of mass conservation. Clearly, the numerical flow rates versus that predicted by the scaling law of Q_To (see table 1) follow the linear relation under regime I_T, and the scaling coefficient is equal to 3.8. In addition, the difference between the flow rates under regimes I_T and II_T is small.

Figure 12.

Flow rates at the outlet under regimes I_T and II_T.

In general, the thickness, maximal velocity and flow rate of the TBL in the initial stage can be predicted by their scaling laws with good precisions. Moreover, they can be used in the cases with the weak merging of the TBL. The dependence on Ra_T and A of the thickness, maximal velocity and flow rate have been validated.

5.2.3. Fully developed stage

Figure 13 shows the numerical results of δ_Ts under regimes I_T and II_T. Clearly, the numerical results fall on the linear relation predicted by the scaling law of δ_Ts, as listed in table 1, except for the largest three scatters for Ra_T = 10³ and A = 5.0. This discrepancy is because δ_Ts is close to the upper limit δ_Ts/R = 1.0 at which the assumption of the boundary layer may fail in the scaling analysis, which, in turn, results in the invalid prediction of the scaling law. Merging of the TBL makes the temperature of the fluid in the pipe uniform, which is close to the wall temperature. In general, the scaling law ${\delta _{Ts}}/R\sim Ra_T^{ - 1/4}{Z^{1/4}}A{\eta _{Ts}}$ is consistent with the numerical results. Further, figure 13 suggests that the criterion for which the TBL merges is $ARa_T^{ - 1/4}{h_{Ts}} > 0.1$.

Figure 13.

Thicknesses of the TBL in the fully developed stage under regimes I_T and II_T. The dashed line corresponds to the upper limit of δ_Ts.

Figure 14 shows u_Ts under regime I_T for the validation of the scale $Ra_T^{1/2}{Z^{1/2}}{A^{ - 1}}$. The numerical results under regime I_T illustrated by red marks in figure 14 are consistent with the scaling law of u_Ts with a scaling coefficient of 0.46. However, the scaling coefficient changes once the TBL merges. As shown in figure 14, the scaling coefficient increases to 0.9 for the numerical results at Z = 0.25 under regime II_T. Moreover, the examination of the numerical results indicates that the increase of the scaling coefficient with the decrease of Z is not linear. In addition, u_Ts for Ra_T = 10³ and A = 5.0 is also away from the scaling law because the thickness of the TBL is close to 1.0 with strong merging of the TBL under regime II_T (also see figure 13).

Figure 14.

Maximal velocities of the TBL in the fully developed stage under regimes I_T and II_T. The solid line corresponds to regime I_T, and the dashed line corresponds to the numerical results at Z = 0.25 under regime II_T.

Figure 15(a) shows the flow rates from the numerical results and the scaling law of Q_Tso (see table 1). A good linear relation is clear between the numerical results and the scaling prediction under regime I_T, as illustrated by red marks in figure 15(a). That is, the scaling law $Ra_T^{1/4}h_{Ts}^{ - 1}$ with a scaling coefficient of 6.4 can perfectly capture the flow rate in the fully developed stage under regime I_T. However, merging of the TBL may significantly affect the flow rate in the fully developed stage, as shown by some blue marks far from the solid line in figure 15(a). Further, we also show the flow rates under regime II_T versus the scale $Ra_T^{1/2}{A^{ - 1}}$ in figure 15(b). The approximately linear relation confirms that the scaling law of Q_Tsom in table 1 with a scaling coefficient of 0.83 is capable of predicting the flow rate in the fully developed stage under regime II_T for which the TBL merges.

Figure 15.

Flow rates at the outlet in the fully developed stage under regimes I_T and II_T. (a) Numerical results versus the scaling law of Q_Tso under regime I_T. (b) Numerical results versus the scaling law of Q_Tsom under regime II_T.

Figures 15 also shows that the scaling law of the flow rate is different under the regimes with and without merging of the TBL in the fully developed stage but the same with different scaling coefficients in the initial stage. In fact, the overshoot of the thickness and velocity in figures 8 and 9 can be responsible for the difference.