3.1 Review of linear stability results

The linear stability analysis of the present problem has been studied by a number of authors (Bera & Khalili Reference Bera and Khalili2002; Chen Reference Chen2004; Kumar et al. Reference Kumar, Bera and Khalili2010) in the literature. In these studies, the rigorous numerical study of different controlling parameters, as well as the impact of drag forces and inertia on the stability of the PMCF, was carried out. The problem contains eight important parameters such as Reynolds number $(Re)$ , Rayleigh number $(Ra)$ , Darcy number $(Da)$ , Prandtl number $(Pr)$ , Forchheimer number ( $F$ ), porosity $(\unicode[STIX]{x1D700})$ , viscosity ratio $(\unicode[STIX]{x1D706})$ and heat capacity ratio $(\unicode[STIX]{x1D70E})$ . The objective of the present study is to investigate the influence of the nonlinear interaction of different harmonic modes on the instability of flow through a highly permeable porous medium. For this, two fluids, air ( $Pr=0.7$ ) and water ( $Pr=7$ ), are chosen. Apart from this, to avoid too many parametric studies, we have fixed certain parameters for the present study. The porosity, viscosity ratio and heat capacity ratio are kept constant at 0.9, 1 and 1, respectively. Here, three different values ( $10^{-2},10^{-3}$ and $10^{-4}$ ) of $Da$ and three different values (0.0006, 0.006 and 0.06) of $c_{F}$ are taken to examine the present problem. There is a lack of experimental data on the value of $c_{F}$ related to a non-isothermal system, so the consideration of the above three values of $c_{F}$ is based on the information mentioned in Calmidi & Mahajan (Reference Calmidi and Mahajan2000) that the value of $c_{F}$ can be as small as or less than 0.1 (for metallic foam).

Figure 2.

Linear stability boundaries in the ( $Re,Ra$ )-plane: (a) $c_{F}=6\times 10^{-4}$ , (b) $c_{F}$   $=$ $6\times 10^{-3}$ and (c) $c_{F}=6\times 10^{-2}$ , where the solid line is $Pr=0.7$ and the dashed line is $Pr=7$ .

We present the linear stability results for air ( $Pr=0.7$ ) and water ( $Pr=7$ ) in the $(Re,Ra)$ -plane. The linear stability boundaries for three values, 0.0006, 0.006 and 0.06, of $c_{F}$ are plotted in figures 2(a), 2(b) and 2(c), respectively. Here, in general, based on the value of $c_{F}$ , $Pr$ and $Da$ the instability boundary curve first decreases on increasing the value of $Re$ and may reach a minimum value of $Re$ , above which the curve will have an increasing characteristic. This may continue further to a certain range of $Re$ and beyond such that it may asymptotically converge to a constant. The typical decreasing characteristic of the instability curve up to a minimum value of $Re$ is a consequence of the following fact. In general, the shear force induced by increasing $Re$ destabilizes the flow. Also, in a porous medium, increasing $Re$ increases the form drag induced in the flow which has a stabilizing characteristic. As mentioned in Bera & Khalili (Reference Bera and Khalili2002), under the present heating condition, a slow flow can carry denser fluid upward into the region of lighter fluid. While doing so, in addition to the energy due to the shear force existing in a viscous fluid, for a porous medium, two new disturbance energies with stabilizing characteristics, namely the one caused by surface drag associated with the Darcy term and the other caused by the form drag associated with the Forchheimer term, appear in the region of high velocity. However, the contribution of form drag may be very small for such types of flow. As a consequence, a point of inflection develops, leading to a local high shear layer. In this situation, a small enhancement of $Re$ makes the convection in the direction of the main flow stronger, and more dense fluid can be transported upwards to destabilize the flow, and hence the critical heating condition (i.e. critical $Ra$ ) falls drastically. As $Re$ is increased further, the induced form drag may take a dominant role and this results in an increased critical heating condition.

Furthermore, when $c_{F}=6\times 10^{-3}$ is replaced by $c_{F}=6\times 10^{-2}$ , figure 2(c) shows that, in an air-saturated porous medium, based on the value of $Da$ , there exists a threshold value of $Re$ (e.g. $Re=2000$ for $Da=10^{-2}$ and $Re=700$ for $Da=10^{-3}$ ) beyond which the instability curve decreases smoothly as a function of $Re$ . However, in the range $0<Re\leqslant Re$ , the (threshold) instability boundary curve first decreases up to a minimum value of $Re$ and then increases, which may be the consequence of the sizeable constant value of $c_{F}$ even in the Darcy regime. As compared to the PMCF of air, the same for water is much more unstable. The instability boundaries show a more stable flow with decreasing medium permeability. Apart from these, the induced form drag in the system delays the instability of the flow.

The disturbance kinetic energy (KE) balance corresponding to a neutral stability curve also provides some insight into the transport mechanisms during the flow instability. The mathematical details of the balance of KE can be seen in the paper by Bera & Khalili (Reference Bera and Khalili2006). The different energy terms of the KE balance for $c_{F}=6\times 10^{-4}$ are shown in figures 3(a) and 3(b) for air and water, respectively. Due to the similarity, the KE spectra for all other considered values of $c_{F}$ are not shown here. As can be seen from figure 3(a,b), based on the permeability of the medium considered, the type of instability may be buoyant (or thermal–buoyant), mixed (or interactive) and shear (or thermal–shear). Here, the type of instability is defined on the basis of the contribution to energy production (or, destruction) of the shear term ( $E_{s}$ ) and the buoyant term ( $E_{b}$ ). If, in the energy balance, the contribution of $E_{s}$ ( $E_{b}$ ) is more than or equal to 75 % then the type of instability is defined as shear (buoyant), otherwise it is defined as mixed. Thus, the mixed mode of instability is defined as 25 %–75 % contribution to the total KE production by both buoyant and convective terms. The kinetic energy production for $Da=10^{-2}$ , when the fluid is air (water), shows three different types of instability: buoyant for $Re\leqslant 500$ ( $Re\leqslant 500$ ), mixed for $500<Re\leqslant 3150$ ( $500<Re\leqslant 3460$ ) and shear for $Re>3150$ ( $Re>3460$ ). Similarly, for $Da=10^{-3}$ , it shows two types of instability: buoyant for $Re\leqslant 8000$ and mixed for $Re>8000$ in the case of air, whereas for water only the buoyant type occurs. For air as well as water, when $Da=10^{-4}$ , KE production due to the buoyant term remains prominent in the entire range of $Re$ considered in this study, consequently, buoyant instability is the only type of instability. Note that for $Da$ equal to $10^{-3}$ or $10^{-4}$ KE production or destruction is mainly balanced by dissipation of KE due to surface drag ( $E_{D}$ ). However, for $Da=10^{-2}$ it is different. Here, dissipation of KE is made through $E_{D}$ as well as $E_{F}$ .

Figure 3.

Linear disturbance kinetic energy (KE) balance at the critical level: (a) $Pr=0.7$ and (b) $Pr=7$ when $c_{F}=6\times 10^{-4}$ .

It has been also found that when $c_{F}=6\times 10^{-4}$ is replaced by $c_{F}=6\times 10^{-3}$ , in general, whether the porous medium is saturated by air or water, the type of instability is buoyant and energy production due to the buoyant term is balanced by dissipation of KE through the surface as well as form drag (figure is not shown here). For $Re>1100$ , $Da=10^{-2}$ and $Pr=0.7$ the mode of instability is of mixed type, i.e. both $E_{s}$ and $E_{b}$ are positive and take active roles in the KE balance. It is also worth mentioning here that, in an air-saturated porous medium for $Da=10^{-3}$ or $Da=10^{-4}$ , $E_{s}$ is negative (i.e. energy is lost to basic flow, and is in favour of stabilizing the flow).

The occurrence of instability in a flow may lead to the replacement of the original laminar flow by a new laminar flow due to the superimposition of finite disturbances. This flow may be expected to persist for a certain range of Rayleigh numbers beyond the critical value and then become unstable at some new Rayleigh number against a new (second) type of disturbance. A new equilibrium flow, consisting of a mean flow with two superimposed harmonics, is then conceivable for a range of Rayleigh numbers above the second critical value. In a purely fluid domain, as the Rayleigh number is increased further, additional modes of disturbance may appear successively until turbulence is achieved. In this way, a sequence of instabilities may exist before it leads finally to turbulence (Hof et al. Reference Hof, van Doorne, Westerweel, Nieuwstadt, Faisst, Eckhardt, Wedin, Kerswell and Waleffe2004). Since turbulent flow in a wall bounded porous medium is possible (Jin & Kuznetsov Reference Jin and Kuznetsov2017), therefore, the above similar phenomena may be expected in fluid flow through highly permeable porous media. Thus, a nonlinear stability analysis is necessary to study the structure of the flow field that results from linear stability analysis.

4.1 Derivation of cubic Landau equation

On substituting equation (4.1) into the governing equations (2.2)–(2.4) and separating the harmonic components, the equations for the harmonic $E^{0}$ are

(4.3) $$\begin{eqnarray}\displaystyle & & \displaystyle {\mathcal{L}}_{y}(V_{0},\unicode[STIX]{x1D6E9}_{0},Ra,Da,F,Re)-Re\frac{\text{d}P_{0}}{\text{d}y}+c_{i}|B|^{2}\left\{\unicode[STIX]{x1D706}\frac{\text{d}^{2}V_{1}}{\text{d}x^{2}}-\frac{1}{Da}V_{1}+Ra\unicode[STIX]{x1D6E9}_{1}-2F^{\prime }|V_{0}|V_{1}\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.Re\frac{\text{d}P_{1}}{\text{d}y}-\frac{Re}{\unicode[STIX]{x1D700}^{2}}\left[\frac{\text{d}}{\text{d}x}(\tilde{v}_{10}u_{10}+v_{10}\tilde{u} _{10})\right]\!\right\}=O((c_{i})^{2}),\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle {\mathcal{L}}(V_{0},\unicode[STIX]{x1D6E9}_{0})+c_{i}|B|^{2}\left\{{\mathcal{L}}(V_{1},\unicode[STIX]{x1D6E9}_{1})-RePr\left[\frac{\text{d}}{\text{d}x}(\tilde{u} _{10}\unicode[STIX]{x1D703}_{10}+u_{10}\tilde{\unicode[STIX]{x1D703}}_{10})\right]\!\right\}=O((c_{i})^{2}). & & \displaystyle\end{eqnarray}$$

Similarly, the equations for the harmonic $E^{1}$ are

(4.5) $$\begin{eqnarray}\displaystyle (c_{i})^{1/2}B\{\mathscr{L}_{o}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u_{10},v_{10},w_{10})\}+(c_{i})^{3/2}B|B|^{2}\{\mathscr{L}_{o}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u_{11},v_{11},w_{11})\}=O(c_{i}^{5/2}), & & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & & \displaystyle (c_{i})^{1/2}B\{\mathscr{L}_{x}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,u_{10},p_{10},V_{0},Da,F,Re)\}\nonumber\\ \displaystyle & & \displaystyle \quad +\,(c_{i})^{3/2}\left\{B|B|^{2}\mathscr{L}_{x}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,u_{11},p_{11},V_{0},Da,F,Re)\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.B|B|^{2}{\mathcal{G}}_{x}+\frac{Re}{\unicode[STIX]{x1D700}}\frac{\text{d}B}{\text{d}\unicode[STIX]{x1D70F}}u_{10}-\frac{\unicode[STIX]{x1D6FC}BRe}{\unicode[STIX]{x1D700}}u_{10}\right\}=O(c_{i}^{5/2}),\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle & & \displaystyle (c_{i})^{1/2}B\{\mathscr{L}_{y}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,u_{10},v_{10},p_{10},\unicode[STIX]{x1D703}_{10},V_{0},Ra,Da,F,Re)\}\nonumber\\ \displaystyle & & \displaystyle \quad +\,(c_{i})^{3/2}\left\{B|B|^{2}\mathscr{L}_{y}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,u_{11},v_{11},p_{11},\unicode[STIX]{x1D703}_{11},V_{0},Ra,Da,F,Re)\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.B|B|^{2}{\mathcal{G}}_{y}+\frac{Re}{\unicode[STIX]{x1D700}}\frac{\text{d}B}{\text{d}\unicode[STIX]{x1D70F}}v_{10}-\frac{\unicode[STIX]{x1D6FC}BRe}{\unicode[STIX]{x1D700}}v_{10}\right\}=O((c_{i})^{5/2}),\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & & \displaystyle (c_{i})^{1/2}B\{\mathscr{L}_{z}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,w_{10},p_{10},V_{0},Da,F,Re)\}\nonumber\\ \displaystyle & & \displaystyle \quad +\,(c_{i})^{3/2}\left\{B|B|^{2}\mathscr{L}_{z}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,w_{11},p_{11},V_{0},Da,F,Re)\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.B|B|^{2}{\mathcal{G}}_{z}+\frac{Re}{\unicode[STIX]{x1D700}}\frac{\text{d}B}{\text{d}\unicode[STIX]{x1D70F}}w_{10}-\frac{\unicode[STIX]{x1D6FC}BRe}{\unicode[STIX]{x1D700}}w_{10}\right\}=O((c_{i})^{5/2}),\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & & \displaystyle (c_{i})^{1/2}B\{\mathscr{L}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,u_{10},v_{10},\unicode[STIX]{x1D703}_{10},V_{0},\unicode[STIX]{x1D6E9}_{0})\}\nonumber\\ \displaystyle & & \displaystyle \quad +\,(c_{i})^{3/2}\left\{B|B|^{2}\mathscr{L}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c,u_{11},v_{11},\unicode[STIX]{x1D703}_{11},V_{0},\unicode[STIX]{x1D6E9}_{0})\right.\nonumber\\ \displaystyle & & \displaystyle \quad -\left.B|B|^{2}{\mathcal{G}}-RePr\left(\frac{\text{d}B}{\text{d}\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D703}_{10}-\unicode[STIX]{x1D6FC}B\unicode[STIX]{x1D703}_{10}\right)\!\right\}=O((c_{i})^{5/2}),\end{eqnarray}$$

where the ${\mathcal{G}}$ values are defined in appendix C.2.

The equations for the harmonic $E^{2}$ are

(4.10) $$\begin{eqnarray}\displaystyle c_{i}B^{2}\{\mathscr{L}_{o}(2\unicode[STIX]{x1D6FC},2\unicode[STIX]{x1D6FD},u_{20},v_{20},w_{20})\}=O((c_{i})^{2}), & & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & & \displaystyle c_{i}B^{2}\left\{\!\mathscr{L}_{x}(2\unicode[STIX]{x1D6FC},2\unicode[STIX]{x1D6FD},c,u_{20},p_{20},V_{0},Da,F,Re)+\frac{Re}{\unicode[STIX]{x1D700}^{2}}\left(\text{i}\unicode[STIX]{x1D6FC}u_{10}v_{10}+u_{10}\frac{\text{d}u_{10}}{\text{d}x}+\text{i}\unicode[STIX]{x1D6FD}u_{10}w_{10}\right)\!\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =O((c_{i})^{2}),\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & & \displaystyle c_{i}B^{2}\left\{\mathscr{L}_{y}(2\unicode[STIX]{x1D6FC},2\unicode[STIX]{x1D6FD},c,u_{20},v_{20},p_{20},\unicode[STIX]{x1D703}_{20},V_{0},Ra,Da,F,Re)\right.\nonumber\\ \displaystyle & & \displaystyle \qquad +\left.\frac{Re}{\unicode[STIX]{x1D700}^{2}}\left(\text{i}\unicode[STIX]{x1D6FC}v_{10}^{2}+u_{10}\frac{\text{d}v_{10}}{\text{d}x}+\text{i}\unicode[STIX]{x1D6FD}v_{10}w_{10}\right)\!\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =O((c_{i})^{2}),\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\displaystyle & & \displaystyle c_{i}B^{2}\left\{\mathscr{L}_{z}(2\unicode[STIX]{x1D6FC},2\unicode[STIX]{x1D6FD},c,w_{20},p_{20},V_{0},Da,F,Re)+\frac{Re}{\unicode[STIX]{x1D700}^{2}}\left(\text{i}\unicode[STIX]{x1D6FC}v_{10}w_{10}+u_{10}\frac{\text{d}w_{10}}{\text{d}x}+\text{i}\unicode[STIX]{x1D6FD}w_{10}^{2}\right)\!\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =O((c_{i})^{2}),\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & & \displaystyle c_{i}B^{2}\left\{\mathscr{L}(2\unicode[STIX]{x1D6FC},2\unicode[STIX]{x1D6FD},c,u_{20},v_{20},\unicode[STIX]{x1D703}_{20},V_{0},\unicode[STIX]{x1D6E9}_{0})-RePr\left(\text{i}\unicode[STIX]{x1D6FC}v_{10}\unicode[STIX]{x1D703}_{10}+u_{10}\frac{\text{d}\unicode[STIX]{x1D703}_{10}}{\text{d}x}+\text{i}\unicode[STIX]{x1D6FD}w_{10}\unicode[STIX]{x1D703}_{10}\right)\!\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =O((c_{i})^{2}).\end{eqnarray}$$

Here ${\sim}$ denotes the complex conjugate. Consideration of higher-order harmonics ( $E^{3},E^{4}$ , etc.) in (4.1) is not necessary to obtain the first Landau coefficient. Therefore, these terms are not included in the series.

The system of harmonic equations (4.3)–(4.14) can be solved sequentially in increasing power of $c_{i}$ . At order $(c_{i})^{0}$ , the harmonic $E^{0}$ contains exactly the same basic state equations (3.1)–(3.2). The unknown functions $V_{0}$ and $\unicode[STIX]{x1D6E9}_{0}$ are obtained from (3.1), (3.2). At order $(c_{i})^{1/2}$ , the harmonic $E^{1}$ contains linear stability equations and the contribution of other harmonics ( $E^{0}$ and $E^{2}$ ) is zero. The functions $u_{10}$ , $v_{10}$ , $w_{10}$ and $\unicode[STIX]{x1D703}_{10}$ are given by the eigenfunctions of the linear stability equations at a particular wavenumber as well as $Ra$ . At order $(c_{i})^{1}$ , the harmonics $E^{0}$ and $E^{2}$ produce the non-homogeneous equations for the basic flow distortion functions $V_{1}$ and $\unicode[STIX]{x1D6E9}_{1}$ as well as for the functions $u_{20}$ , $v_{20}$ , $w_{20}$ and $\unicode[STIX]{x1D703}_{20}$ , respectively. The non-homogeneous part of these equations contains the known variables $u_{10}$ , $v_{10}$ , $w_{10}$ , $\unicode[STIX]{x1D703}_{10}$ and their derivatives, which are calculated from a lower-order analysis. At order $(c_{i})^{3/2}$ , the equations of harmonic $E^{1}$ become non-homogeneous. The left-hand sides of these equations contain linear stability operators operating on $u_{11}$ , $v_{11}$ , $w_{11}$ , $\unicode[STIX]{x1D703}_{11}$ and $p_{11}$ , whereas the right-hand sides of these equations contain the terms proportional to $\text{d}B/\text{d}\unicode[STIX]{x1D70F}$ , $B$ and $B|B|^{2}$ . The coefficients of these terms on the right-hand sides are known from the lower-order analysis. The homogeneous form of the equations of $E^{1}$ is the same as for the linear stability equations. For a non-trivial solution of the $E^{1}$ equations, the integrability condition can be used to determine the unknown amplitude function $B$ . In order to formulate the integrability condition, the solution of a homogeneous adjoint system corresponding to the linear stability problem (see appendix C) is required. Therefore, the right-hand sides of the non-homogeneous equations of $E^{1}$ at order $(c_{i})^{3/2}$ must be orthogonal to the adjoint field variables. In this manner, multiplying the above right-hand side terms by the adjoint field variables $p^{\ast }$ , $u^{\ast }$ , $v^{\ast }$ , $w^{\ast }$ , $\unicode[STIX]{x1D703}^{\ast }$ and enforcing the condition (C 18), the following Landau equation results,

(4.15) $$\begin{eqnarray}\displaystyle \frac{\text{d}B}{\text{d}\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FC}B+a_{1}B|B|^{2}, & & \displaystyle\end{eqnarray}$$

where

(4.16) $$\begin{eqnarray}\displaystyle a_{1}=\frac{1}{Re}\int _{-1}^{1}({\mathcal{G}}_{x}u^{\ast }+{\mathcal{G}}_{y}v^{\ast }+{\mathcal{G}}_{z}w^{\ast }+{\mathcal{G}}\unicode[STIX]{x1D703}^{\ast })\,\text{d}x, & & \displaystyle\end{eqnarray}$$

and is known as the Landau constant, which is the first correction to the linear growth rate. The Landau equation (4.15) represents a modification to the exponential growth or decay of a disturbance predicted by linear theory.

In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system moves from one fixed point to three fixed points. This is closely tied to the concept of equilibrium amplitude (i.e. the finite amplitude equilibrium solution) and the real part of the Landau coefficient, as shown below.

On substituting $A=c_{i}^{1/2}B$ and $\unicode[STIX]{x1D70F}=c_{i}t$ into the Landau equation (4.15), the form of the modified Landau equation becomes

(4.17) $$\begin{eqnarray}\displaystyle \frac{\text{d}A}{\text{d}t}=\unicode[STIX]{x1D6FC}c_{i}A+a_{1}A^{2}\tilde{A}, & & \displaystyle\end{eqnarray}$$

and its corresponding complex conjugate equation is

(4.18) $$\begin{eqnarray}\displaystyle \frac{\text{d}\tilde{A}}{\text{d}t}=\unicode[STIX]{x1D6FC}c_{i}\tilde{A}+\tilde{a_{1}}\tilde{A}^{2}A, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}c_{i}$ is the growth rate from the linear stability analysis and $A$ is the physical amplitude of the wave. If we multiply (4.17) by $\tilde{A}$ , equation (4.18) by $A$ and add, we obtain

(4.19) $$\begin{eqnarray}\displaystyle \frac{\text{d}|A|^{2}}{\text{d}t}=2\unicode[STIX]{x1D6FC}c_{i}|A|^{2}+2(a_{1})_{r}|A|^{4}, & & \displaystyle\end{eqnarray}$$

where $(a_{1})_{r}$ is the real part of $a_{1}$ . The solution, $|A|$ , of (4.19) is said to be an equilibrium amplitude ( $A_{e}$ ) provided it satisfies the stationary condition

(4.20) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}|A|}{\text{d}t}}=0. & & \displaystyle\end{eqnarray}$$

Accordingly, there are three possible equilibrium amplitudes which are given by

(4.21a,b ) $$\begin{eqnarray}\displaystyle A_{e}=0\quad \text{and}\quad A_{e}=\pm \sqrt{-\unicode[STIX]{x1D6FC}c_{i}/(a_{1})_{r}}. & & \displaystyle\end{eqnarray}$$

Note that $A_{e}=0$ , which corresponds to steady basic flow, is stable for $Ra<Ra$ (critical) and unstable for $Ra>Ra$ (critical). As has been discussed in Drazin & Reid (Reference Drazin and Reid2004) and Shukla & Alam (Reference Shukla and Alam2011), when $\unicode[STIX]{x1D6FC}c_{i}$ and $(a_{1})_{r}$ are of the opposite sign, the existence of a finite amplitude solution is guaranteed. Thus, there are two such possible situations: first, the growth rate is positive and the real part of the Landau constant is negative; second, the growth rate is negative and the real part of the Landau constant is positive. The first situation leads to a supercritical pitchfork bifurcation (hereafter, supercritical bifurcation), whereas the second one leads to a subcritical pitchfork bifurcation (hereafter, subcritical bifurcation). For positive growth rates, the nonlinear stationary solution of (4.19) is called the equilibrium amplitude. However, for negative growth rates the nonlinear stationary solution is referred to as the threshold amplitude, since the latter finite amplitude solution is in fact unstable.

4.2 Nonlinear kinetic energy spectrum

To gain further insight into the mechanism of supercritical/subcritical bifurcation, an investigation of the energy transfer in parallel mixed convective flow is made. The Reynolds stress has a significant impact on the basic flow in a weakly nonlinear stability analysis. This distortion of the basic flow modifies the rate of transfer of energy from the basic flow to the disturbance. The balance of kinetic energy for the fundamental disturbance (here onward KE) is given as (Rogers, Moulic & Yao Reference Rogers, Moulic and Yao1993; Khandelwal & Bera Reference Khandelwal and Bera2015)

(4.22) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left\langle \frac{1}{2}\frac{1}{\unicode[STIX]{x1D700}}[\overline{u_{1}^{2}}+\overline{v_{1}^{2}}+\overline{w_{1}^{2}}]\right\rangle & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D700}^{2}}\left\langle \overline{u_{1}v_{1}}\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}x}\right\rangle +\frac{Ra}{Re}\langle \overline{v_{1}\unicode[STIX]{x1D703}_{1}}\rangle -\frac{1}{DaRe}\langle \overline{u_{1}^{2}}+\overline{v_{1}^{2}}+\overline{w_{1}^{2}}\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{Re}\left\langle \overline{\left(\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}y}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}z}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}v_{1}}{\unicode[STIX]{x2202}x}\right)^{2}}\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\overline{\left(\frac{\unicode[STIX]{x2202}v_{1}}{\unicode[STIX]{x2202}y}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}v_{1}}{\unicode[STIX]{x2202}z}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}w_{1}}{\unicode[STIX]{x2202}x}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}w_{1}}{\unicode[STIX]{x2202}y}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}w_{1}}{\unicode[STIX]{x2202}z}\right)^{2}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{\unicode[STIX]{x1D700}^{2}}\left\langle \overline{u_{1}^{2}\frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}x}}+\overline{u_{1}v_{1}\frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}y}}+\overline{u_{1}w_{1}\frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}z}}+\overline{u_{1}v_{1}\frac{\unicode[STIX]{x2202}v_{2}}{\unicode[STIX]{x2202}x}}+\overline{v_{1}^{2}\frac{\unicode[STIX]{x2202}v_{2}}{\unicode[STIX]{x2202}y}}\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\overline{v_{1}w_{1}\frac{\unicode[STIX]{x2202}v_{2}}{\unicode[STIX]{x2202}z}}+\overline{u_{1}w_{1}\frac{\unicode[STIX]{x2202}w_{2}}{\unicode[STIX]{x2202}x}}+\overline{v_{1}w_{1}\frac{\unicode[STIX]{x2202}w_{2}}{\unicode[STIX]{x2202}y}}+\overline{w_{1}^{2}\frac{\unicode[STIX]{x2202}w_{2}}{\unicode[STIX]{x2202}z}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle -\,F\langle |V_{0}|(\overline{u_{1}^{2}}+2\overline{v_{1}^{2}}+\overline{w_{1}^{2}})\rangle .\end{eqnarray}$$

Here, the bracket $\langle ~\rangle$ implies integration over the volume of the disturbance wave and $\overline{GH}$ for some field variables $G$ and $H$ is defined as $G\tilde{H}+\tilde{G}H$ . With the help of (4.15), the balance of kinetic energy leads to an amplitude equation (Rogers et al. Reference Rogers, Moulic and Yao1993)

(4.23) $$\begin{eqnarray}\displaystyle \frac{\text{d}|B|^{2}}{\text{d}\unicode[STIX]{x1D70F}}=2\unicode[STIX]{x1D6FC}|B|^{2}+(P_{101}+E_{12}+P_{110}+T_{11}+D_{11}+K_{11}+F_{11})|B|^{4}. & & \displaystyle\end{eqnarray}$$

The comparison of (4.15) and (4.23) shows that

(4.24) $$\begin{eqnarray}\displaystyle 2(a_{1})_{r}=P_{101}+E_{12}+P_{110}+T_{11}+D_{11}+K_{11}+F_{11}, & & \displaystyle\end{eqnarray}$$

where the right-hand side quantities are defined as

(4.25) $$\begin{eqnarray}\displaystyle P_{101}=-\frac{1}{e_{0}\unicode[STIX]{x1D700}^{2}}\left\langle \overline{u_{10}v_{10}}\frac{\text{d}V_{1}}{\text{d}x}\right\rangle , & & \displaystyle\end{eqnarray}$$

(4.26) $$\begin{eqnarray}\displaystyle E_{12} & = & \displaystyle -\frac{1}{e_{0}\unicode[STIX]{x1D700}^{2}}\left\langle \overline{u_{10}^{2}\frac{\unicode[STIX]{x2202}u_{20}}{\unicode[STIX]{x2202}x}}+\overline{u_{10}v_{10}\frac{\unicode[STIX]{x2202}u_{20}}{\unicode[STIX]{x2202}y}}+\overline{u_{10}w_{10}\frac{\unicode[STIX]{x2202}u_{20}}{\unicode[STIX]{x2202}z}}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\overline{u_{10}v_{10}\frac{\unicode[STIX]{x2202}v_{20}}{\unicode[STIX]{x2202}x}}+\overline{v_{10}^{2}\frac{\unicode[STIX]{x2202}v_{20}}{\unicode[STIX]{x2202}y}}+\overline{v_{10}w_{10}\frac{\unicode[STIX]{x2202}v_{20}}{\unicode[STIX]{x2202}z}}\nonumber\\ \displaystyle & & \displaystyle +\left.\overline{u_{10}w_{10}\frac{\unicode[STIX]{x2202}w_{20}}{\unicode[STIX]{x2202}x}}+\overline{v_{10}w_{10}\frac{\unicode[STIX]{x2202}w_{20}}{\unicode[STIX]{x2202}y}}+\overline{w_{10}^{2}\frac{\unicode[STIX]{x2202}w_{20}}{\unicode[STIX]{x2202}z}}\right\rangle ,\end{eqnarray}$$

(4.27) $$\begin{eqnarray}\displaystyle & \displaystyle P_{110}=-\frac{1}{e_{0}\unicode[STIX]{x1D700}^{2}}\left\langle (\overline{u_{10}v_{11}}+\overline{v_{10}u_{11}})\frac{\text{d}V_{0}}{\text{d}x}\right\rangle , & \displaystyle\end{eqnarray}$$

(4.28) $$\begin{eqnarray}\displaystyle & \displaystyle T_{11}=\frac{1}{e_{0}}\frac{Ra}{Re}\langle \overline{v_{10}\unicode[STIX]{x1D703}_{11}}+\overline{\unicode[STIX]{x1D703}_{10}v_{11}}\rangle , & \displaystyle\end{eqnarray}$$

(4.29) $$\begin{eqnarray}\displaystyle D_{11} & = & \displaystyle -\frac{1}{e_{0}}\frac{1}{Re}\left\langle \overline{\frac{\unicode[STIX]{x2202}u_{10}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}u_{11}}{\unicode[STIX]{x2202}x}}+\overline{\frac{\unicode[STIX]{x2202}u_{10}}{\unicode[STIX]{x2202}y}\frac{\unicode[STIX]{x2202}u_{11}}{\unicode[STIX]{x2202}y}}+\overline{\frac{\unicode[STIX]{x2202}u_{10}}{\unicode[STIX]{x2202}z}\frac{\unicode[STIX]{x2202}u_{11}}{\unicode[STIX]{x2202}z}}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\overline{\frac{\unicode[STIX]{x2202}v_{10}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}v_{11}}{\unicode[STIX]{x2202}x}}+\overline{\frac{\unicode[STIX]{x2202}v_{10}}{\unicode[STIX]{x2202}y}\frac{\unicode[STIX]{x2202}v_{11}}{\unicode[STIX]{x2202}y}}+\overline{\frac{\unicode[STIX]{x2202}v_{10}}{\unicode[STIX]{x2202}z}\frac{\unicode[STIX]{x2202}v_{11}}{\unicode[STIX]{x2202}z}}\nonumber\\ \displaystyle & & \displaystyle +\left.\overline{\frac{\unicode[STIX]{x2202}w_{10}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}w_{11}}{\unicode[STIX]{x2202}x}}+\overline{\frac{\unicode[STIX]{x2202}w_{10}}{\unicode[STIX]{x2202}y}\frac{\unicode[STIX]{x2202}w_{11}}{\unicode[STIX]{x2202}y}}+\overline{\frac{\unicode[STIX]{x2202}w_{10}}{\unicode[STIX]{x2202}z}\frac{\unicode[STIX]{x2202}w_{11}}{\unicode[STIX]{x2202}z}}\right\rangle ,\end{eqnarray}$$

(4.30) $$\begin{eqnarray}\displaystyle & \displaystyle K_{11}=-\frac{2}{e_{0}DaRe}\langle \overline{u_{10}u_{11}}+\overline{v_{10}v_{11}}+\overline{w_{10}w_{11}}\rangle , & \displaystyle\end{eqnarray}$$

(4.31) $$\begin{eqnarray}\displaystyle & \displaystyle F_{11}=-\frac{2}{e_{0}}F\langle |V_{0}|(\overline{u_{10}u_{11}}+2\overline{v_{10}v_{11}}+\overline{w_{10}w_{11}})\rangle , & \displaystyle\end{eqnarray}$$

and

(4.32) $$\begin{eqnarray}\displaystyle e_{0}=\frac{1}{2\unicode[STIX]{x1D700}}\langle \overline{u_{10}^{2}}+\overline{v_{10}^{2}}+\overline{w_{10}^{2}}\rangle . & & \displaystyle\end{eqnarray}$$

The physical interpretation of different terms in the balance of KE, given in (4.24), is as follows. The term $P_{101}$ , which is an integral of the product of the Reynolds stress and the mean velocity gradient, denotes the gradient production of KE due to the interaction between the fundamental mode (quantities assigned subscript 10) and the distorted mean flow strain rate. The energy needed for the distortion of the mean flow is obtained from the fundamental disturbance, consequently, this term will be negative. Hence, this term will reduce the growth rate of the disturbance. The second term $E_{12}$ represents the transfer of the KE from the fundamental to the second harmonic wave (quantities assigned subscript 20). The other five terms $P_{110}$ , $T_{11}$ , $D_{11}$ , $K_{11}$ and $F_{11}$ account for the energy exchange due to the modification of the shape of the fundamental disturbance wave. The term $P_{110}$ arises because of the modification to the gradient production of KE, and may have a positive or negative sign. The positive sign of $P_{110}$ indicates that the change in the shape of the fundamental disturbance will be more favourable for shear production. The term $T_{11}$ represents the modification to the buoyant production of KE. The terms $D_{11}$ , $K_{11}$ and $F_{11}$ represent dissipation of KE through modification to the viscous dissipation rate, modification to the surface drag and modification to the form drag, respectively.

6.1 Landau constant in the neighbourhood of the bifurcation point

The Landau constant is calculated at $\unicode[STIX]{x1D6FF}_{Ra}=\pm 0.01$ (i.e. in the vicinity of the critical Rayleigh number) with respect to the most unstable linear wave (with critical wavenumber $(\unicode[STIX]{x1D6FC}_{c})$ ). Note that $\unicode[STIX]{x1D6FF}_{Ra}=-0.01(+0.01)$ is a point from a linearly stable (unstable) region, and the corresponding growth rate is negative (positive). To identify the nature of the bifurcation of the flow, the real part of the Landau constant ( $(a_{1})_{r}$ ) as a function of $Re$ is plotted. The variation of $(a_{1})_{r}$ at $\unicode[STIX]{x1D6FF}_{Ra}=-0.01(+0.01)$ , when the porous medium is saturated by air, is shown in figures 4(a,b), 4(c,d) and 4(e,f) for $Da$ equal to $10^{-2}$ , $10^{-3}$ and $10^{-4}$ , respectively. As can be seen from figure 4(a,b), the sign of $(a_{1})_{r}$ is negative for $Da=10^{-2}$ with $c_{F}=0.0006$ as well as 0.06, indicating the supercritical bifurcation (see figure 4 b). However, on replacing the value 0.0006 of $c_{F}$ by 0.006 the same figures show supercritical bifurcation for $Re<1900$ and subcritical bifurcation for $Re>1900$ . Let $Re_{sb}$ denote the minimum value of $Re$ at which the shifting of bifurcation from supercritical to subcritical takes place. Accordingly, in this case, the value of $Re_{sb}$ is 1900. Note that the value of $(a_{1})_{r}$ is zero at $Re=Re_{sb}$ . In general, the value of $Re_{sb}$ increases on increasing the value of $c_{F}$ . For example, when the value 0.006 of $c_{F}$ is replaced by 0.03, the value of $Re_{sb}$ becomes 5000 (see figure 4 b). Furthermore, for each value of $c_{F}$ , figures 4(c) and 4(d) as well as the inset therein, show, in general, the existence of a $Re_{sb}$ . The values of $Re_{sb}$ for $c_{F}$ equal to 0.0006 and 0.006 are 4000 and 5340, respectively. For $Da=10^{-4}$ , figure 4(e,f) shows mainly a supercritical bifurcation except for $c_{F}=0.0006$ where the shifting of the bifurcation also takes place at $Re_{sb}=14\,000$ . Interestingly, $Re_{sb}$ increases on increasing the value of $c_{F}$ as well as on decreasing the value of $Da$ . This may be a consequence of the following fact. Since a decrease in the permeability of the medium or an increase in the form drag, in general, stabilizes the flow, therefore to achieve a change of bifurcation in PMCF in a porous medium in which the permeability is relatively low or induced form drag is relatively high, a relatively higher disturbance shear production is required.

Figure 4.

Variation of Landau constant ( $(a_{1})_{r}$ ) with respect to $Re$ : (a,b) $Da=10^{-2}$ , (c,d) $Da=10^{-3}$ and (e,f) $Da=10^{-4}$ when $Pr=0.7$ ; (a,c,e) $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ , and (b,d,f) $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ .

Figure 5.

Variation of threshold amplitude in the subcritical region (solid line) as well as equilibrium amplitude in the supercritical region with respect to $Re$ : (a) $Da=10^{-2}$ , (b) $Da=10^{-3}$ and (c) $Da=10^{-4}$ when $Pr=0.7$ .

The corresponding threshold amplitude in the subcritical region as well as the equilibrium amplitude in the supercritical region for $Da$ equal to $10^{-2}$ , $10^{-3}$ and $10^{-4}$ , are displayed in figures 5(a), 5(b) and 5(c), respectively. The solid line in each graph denotes the amplitude profile in the subcritical region. It is important to mention here that for $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ (i.e. positive growth rate) if the sign of $(a_{1})_{r}$ is positive then $A_{e}$ does not exist. Similarly, for $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ if the sign of $(a_{1})_{r}$ is negative then $A_{e}$ does not exist. Consequently, the amplitude in the above figure is drawn in the subcritical (i.e. $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ ) as well as in supercritical (i.e. $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ ) regimes only. Each of the above figures manifests an important feature. The notable feature is the sudden jump in the amplitude profile in the vicinity of $Re_{sb}$ , where the supercritical bifurcation changes to a subcritical one or vice versa. Note that at $Re_{sb}$ the function $A_{e}$ is not defined and the variation of growth rate in the vicinity of $Re_{sb}$ is almost constant (see figure 19 a,b in appendix B.2). Therefore, the above-mentioned sudden jump of the amplitude in the vicinity of $Re_{sb}$ may be a consequence of the very small value of $(a_{1})_{r}$ , and $A_{e}$ being inversely proportional to the square root of $(a_{1})_{r}$ . The jump in the amplitude profile is expected to lead to complex phenomena in the flow instability. More about its impact on the instability mechanism will be discussed later in this section.

To show the variation of $(a_{1})_{r}$ and the equilibrium amplitude as a function of $Re$ , when the porous medium is saturated by water, respectively figures 6(a) and 6(b) are drawn for $Da$ equal to $10^{-3}$ in the linearly unstable regime at $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ . As can be seen from figure 6(a), the sign of $(a_{1})_{r}$ is negative, for all considered values of $c_{F}$ , i.e. only supercritical bifurcations exist. The corresponding amplitude profiles shown in figure 6(b) vary smoothly. Note that the calculated values of $(a_{1})_{r}$ for both values (0.01 and $-$ 0.01) of $\unicode[STIX]{x1D6FF}_{Ra}$ are almost identical for air, which is also true for water. Therefore, variation of $(a_{1})_{r}$ at $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ is not shown here. Apart from this, it has been also checked that in a water-saturated porous medium the sign of $(a_{1})_{r}$ is negative for all other considered values of $Da$ . So, the variation of $(a_{1})_{r}$ for all other values of $Da$ is also not shown here. Since we are interested in the situation where a change of bifurcation takes place, therefore, in the rest of this subsection we shall be restricted to only an air-saturated porous medium.

Figure 6.

Variation of Landau constant (a) and equilibrium amplitude (b) with respect to $Re$ when $Da=10^{-3}$ and $Pr=7$ .

The above observations make us curious to know about the variation of physical quantities such as the heat transfer rate in terms of the Nusselt number and the friction coefficient as a function of $Re$ , especially in the neighbourhood of the point where a change of bifurcation takes place. As we have seen, in general $c_{F}$ delays the shifting whereas $Da$ accelerates it. Therefore, in the rest of the analysis in this subsection, we have fixed the value of $Da$ at $10^{-3}$ , $c_{F}$ at 0.006 and $Pr$ at 0.7.

6.1.1 Nusselt number and friction coefficient as a function of $Re$

The heat transfer rate in terms of the Nusselt number is defined as

(6.1) $$\begin{eqnarray}\displaystyle Nu=-2H_{1}/H_{2}, & & \displaystyle\end{eqnarray}$$

where $H_{1}$ and $H_{2}$ are given by $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}/\unicode[STIX]{x2202}x|_{x=1}$ and $\int _{-1}^{1}V\unicode[STIX]{x1D6E9}\,\text{d}x/\int _{-1}^{1}V\text{d}x$ , respectively. The functions $V$ and $\unicode[STIX]{x1D6E9}$ , calculated using the threshold or equilibrium amplitude, are given as $V_{0}+{A_{e}}^{2}V_{1}$ and $\unicode[STIX]{x1D6E9}_{0}+{A_{e}}^{2}\unicode[STIX]{x1D6E9}_{1}$ , respectively. The functions $V_{1}$ and $\unicode[STIX]{x1D6E9}_{1}$ are the basic flow distortion functions. Similarly, the friction coefficient at the wall is given as (Schlichting & Gersten Reference Schlichting and Gersten2004)

(6.2) $$\begin{eqnarray}\displaystyle C_{f}=-\frac{2}{Re}\left.\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}x}\right|_{x=1}. & & \displaystyle\end{eqnarray}$$

The friction coefficient at the other wall of the channel ( $x=-1$ ) is also the same as the wall $x=1$ .

The impact of disturbance growth/decay on the heat transfer rate in the region of subcritical as well as supercritical bifurcation is examined in figure 7(a). The dash-dotted line is curved in the subcritical region. Let $Nu_{bs}$ and $Nu_{ds}$ be defined as the Nusselt numbers predicted for the basic state (solid line) and the distorted state (dashed line as well as dash-dotted line), respectively (see figure 7 a), then the results show that the Nusselt number estimated with the help of weakly nonlinear stability analysis is more or less the same as the one calculated for the basic state, except in the vicinity where the type of bifurcation changes. In this vicinity, $Nu_{ds}$ is not defined at $Re=Re_{sb}$ . The impact of nonlinear interaction between different harmonic modes on the friction coefficient is displayed in figure 7(b). Similar to $Nu_{ds}$ , the difference between the friction coefficient for the distorted state ( $C_{fds}$ ) and basic state ( $C_{fbs}$ ) is also negligible, but the magnitude of $C_{fds}$ decreases significantly in the vicinity where the change in bifurcation takes place. The distorted friction coefficient is also not defined at $Re=Re_{sb}$ . It should be noted that the above phenomenon (rapid change of $Nu_{ds}$ as well as $C_{fds}$ in the vicinity of $Re_{sb}$ ) was also reported for purely fluid medium (Suslov & Paolucci Reference Suslov and Paolucci1999b ) while studying nonlinear stability of a mixed convection flow under a non-Boussinesq condition in a differentially heated channel.

Figure 7.

Variation of Nusselt number and friction coefficient with respect to $Re$ when $Da=10^{-3}$ , $Pr=0.7$ and $c_{F}=6\times 10^{-3}$ . The solid line represents the basic state, whereas the dash-dotted and dashed lines represent the distorted state.

6.1.2 Nonlinear energy spectrum as a function of $Re$

To understand the driving mechanisms of flow instability and change of bifurcation at a neighbouring point of the least linearly stable point, the spectrum of nonlinear kinetic energy balance of the fundamental disturbance (KE) in the supercritical regime (at $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ ) and the subcritical regime (at $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ ) is shown in figures 8(a) and 8(b), respectively. Note that the mathematical analysis of the nonlinear kinetic energy spectrum shows that the sum of the different energy terms is twice the real part of the Landau constant (see (4.24)). Therefore, the type of bifurcation predicted by the Landau constant can be reconfirmed through the sum of the different energy terms. Consequently, the sum is also shown in both figures 8(a) and 8(b). The following points can be noted from figure 8(a,b). The term $P_{101}$ is negative in both (super- and subcritical) regimes which implies that the gradient production of the KE for supercritical flow always tends to stabilize. Its magnitude decreases on increasing $Re$ and the contribution to the energy spectrum becomes almost negligible. The magnitude of the term $E_{12}$ or $D_{11}$ is very small, i.e. the transfer of KE from fundamental to the harmonic waves or modification to the dissipation of KE due to a change in the disturbance shape is also negligible. In both regimes, the modified disturbance shape is favourable to buoyant production of the KE (i.e. $T_{11}$ is positive throughout the range of $Re$ ) and acts as the main destabilizing factor. In the subcritical regime, $T_{11}$ is balanced mainly by dissipation of KE through induced surface drag ( $K_{11}$ ) and form drag ( $F_{11}$ ). However, in the supercritical regime, apart from $K_{11}$ , $F_{11}$ , another term $P_{110}$ also takes a significant role in the KE balance, especially for relatively low values of $Re$ . In the supercritical regime, the term $P_{110}$ is negative and its magnitude decreases on increasing $Re$ , i.e. the stabilizing impact decreases on increasing $Re$ , whereas in the subcritical regime it is positive and its contribution to the energy spectrum is almost negligible. A negative sign of $P_{110}$ indicates that the change in the shape of the fundamental disturbance is not favourable to shear production. It is important to mention here that for the present parametric set values of $Da$ and $c_{F}$ the contribution of the shear term, $E_{s}$ , in linear theory is negative and the type of instability is buoyant (see discussion § 3.1); therefore, the dominance of $T_{11}$ in both regimes is a consequence of the buoyant instability of the flow. We are also curious to know whether any relation between the sign of $E_{s}$ in linear theory and the sign of $P_{110}$ in weakly nonlinear analysis exists or not. This issue will be discussed further in § 6.2. Finally, figure 8(a,b) shows the negative (positive) value of the sum of all the terms (i.e. $(2a_{1})_{r}$ ) of the KE spectrum. So, the analysis of the nonlinear kinetic energy spectrum also predicts the supercritical (subcritical) bifurcation in figure 8(a,b).

Figure 8.

Variation of KE with respect to $Re$ for $Da=10^{-3}$ , $Pr=0.7$ and $c_{F}=6\times 10^{-3}$ : (a) supercritical regime ( $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ ) of $Re$ , (b) subcritical regime ( $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ ) of $Re$ .

6.2 Bifurcation as a function of $Ra$

Weakly nonlinear analysis is only valid near the linear stability boundary point (Drazin & Reid Reference Drazin and Reid2004) and it addresses the question of what happens to an unstable flow. Rogers et al. (Reference Rogers, Moulic and Yao1993) have shown, by comparison with the results from the direct numerical simulation as well as high-order weakly nonlinear results, that the cubic Landau equation gives the correct results when the magnitude of $c_{i}$ is small. They have also pointed out that the weakly nonlinear analysis is asymptotically correct for larger values of the Rayleigh number beyond the bifurcation point in the limit of $c_{i}$ approaching zero. Based on this fact, for a given $Re$ , we have studied the behaviour of the Landau constant as well as the equilibrium amplitude away from the critical $Ra$ (i.e. in the linearly stable and unstable regimes), as a function of $\unicode[STIX]{x1D6FF}_{Ra}$ . For this we have chosen four pairs of ( $Da$ , $Re$ ), two each from the supercritical and subcritical regimes. Here, $c_{F}$ is fixed at 0.006. Accordingly, for two values $10^{-2}$ and $10^{-3}$ of $Da$ , two different respective values, 1000 and 4000, of $Re$ from the supercritical regime and two respective values, 3000 and 7000, of $Re$ from the subcritical regime are chosen. The corresponding critical values of $Ra$ are 290, 12 269, 425, 12 366, respectively. We have checked from our numerical experiments by calculating the growth rate as a function of $\unicode[STIX]{x1D6FF}_{Ra}$ that the magnitude of $c_{i}$ for all of the above values of $Re$ is of the order of $10^{-2}$ in the considered range of $\unicode[STIX]{x1D6FF}_{Ra}$ (see figure 20 in appendix B.2). Therefore, we also analyse the limiting value of growth of the instabilities under nonlinear effects (i.e. nonlinear saturation) for larger (lower) values of the Rayleigh number beyond (before) the bifurcation point. However, a complete picture of nonlinear saturation can only be confirmed by use of direct numerical simulation, which is beyond the scope of the present work. Hence, the following analysis will give a qualitative characteristic of the nonlinear interaction of different harmonic modes for larger (lower) values of $Ra$ beyond (before) the bifurcation point.

Figure 9.

Variation of $(a_{1})_{r}$ and amplitude with respect to $\unicode[STIX]{x1D6FF}_{Ra}$ when $Pr=0.7$ and $c_{F}=6\times 10^{-3}$ : (a,c) subcritical regime of $Re$ , (b,d) supercritical regime of $Re$ .

The variation of the real part of the Landau constant, for pairs of ( $Da$ , $Re$ ) chosen from the subcritical and supercritical regimes, as a function of $\unicode[STIX]{x1D6FF}_{Ra}$ is shown in figures 9(a) and 9(b), respectively. As can be seen from figure 9(a), when $Da=10^{-3}$ , $Re=7000$ and the corresponding critical value of $Ra$ is 12 366, the sign of $(a_{1})_{r}$ in the linearly stable regime (i.e. $\unicode[STIX]{x1D6FF}_{Ra}<0$ ) remains positive for $\unicode[STIX]{x1D6FF}_{Ra}\geqslant -0.5$ , i.e. for $Ra\geqslant 6183$ . This indicates that $Ra=6183$ acts as a lower critical value, below which the bifurcated solution does not exist (Drazin & Reid Reference Drazin and Reid2004), and $-0.5\leqslant \unicode[STIX]{x1D6FF}_{Ra}\leqslant 0$ is a regime of subcritical bifurcation. Similarly, when $Da=10^{-2}$ , $Re=3000$ and the corresponding critical value of $Ra$ is 425, the sign of $(a_{1})_{r}$ in the linearly stable regime remains positive for $\unicode[STIX]{x1D6FF}_{Ra}\geqslant -0.1$ , i.e. for $Ra\geqslant 382.5$ . So, in this case, $Ra=382.5$ acts as a lower critical value, below which the bifurcated solution does not exist. Therefore, we can conclude that $6183\leqslant Ra\leqslant 12\,366$ ( $382.5\leqslant Ra\leqslant 425$ ) is a regime of subcritical bifurcation when $Da=10^{-3}$ and $Re=7000$ ( $Da=10^{-2}$ and $Re=3000$ ). In the linearly unstable regime, the same figure also shows the positive sign of $(a_{1})_{r}$ when $Da=10^{-2}$ , $Re=3000$ , However, in the case of $Da=10^{-3}$ , $Re=7000$ the sign of $(a_{1})_{r}$ remains positive up to $\unicode[STIX]{x1D6FF}_{Ra}=0.17$ , beyond which it becomes negative. This leads to supercritical bifurcation for $\unicode[STIX]{x1D6FF}_{Ra}>0.17$ . So, in this case the subcritical bifurcation changes to supercritical bifurcation. Note that the above-mentioned lower critical values obtained through the cubic Landau equation are only probable values. To get the exact value of $Ra$ in the subcritical region where the unstable threshold branch turns and becomes stable, a fifth-order Landau equation is required, which is beyond the scope of the present study.

Furthermore, the positive sign of $(a_{1})_{r}$ in the linearly unstable regime (i.e. positive growth rate) implies that both terms on the right-hand side of the Landau equation (4.19) are positive and therefore $|A|$ increases exponentially and the solution breaks down after a finite time. Of course this kind of breakdown does not occur in practice; rather, the breakdown points to the need for terms in higher powers of $|A|$ (Drazin & Reid Reference Drazin and Reid2004), which is beyond the scope of the present work.

For the other two pairs $(10^{-2},1000)$ and $(10^{-3},4000)$ of ( $Da$ , $Re$ ) chosen from the supercritical regime, figure 9(b) shows that the sign of $(a_{1})_{r}$ remains negative in the linearly stable as well as the unstable regime for $(Da,Re)=(10^{-2},1000)$ . This implies that the supercritical bifurcation of the flow remains supercritical even when $Ra$ is increased to 1.75 times its critical value. However, for $(Da,Re)=(10^{-3},4000)$ it is different. In this case, the sign of $(a_{1})_{r}$ in the linearly stable regime is positive for $-0.6\leqslant \unicode[STIX]{x1D6FF}_{Ra}\leqslant -0.15$ , consequently a regime of subcritical bifurcation appears and gives a lower critical value of $Ra=4907.6$ .

The variation of the amplitude as a function of $\unicode[STIX]{x1D6FF}_{Ra}$ for ( $Da$ , $Re$ ) chosen from the subcritical and supercritical regimes are shown in figures 9(c) and 9(d), respectively. As can be seen from figure 9(c), in the subcritical regime $A_{e}$ increases on increasing the magnitude of $\unicode[STIX]{x1D6FF}_{Ra}$ for $Re=3000$ , but for $Re=7000$ the same increases up to $\unicode[STIX]{x1D6FF}_{Ra}=-0.17$ and it decreases when $-0.17>\unicode[STIX]{x1D6FF}_{Ra}>-0.5$ . The increasing characteristic of $A_{e}$ away from the bifurcation point is a consequence of the fact that the magnitude of the growth rate increases (see figure 20 in appendix B.2) and $(a_{1})_{r}$ remains almost constant. In the supercritical regime, figure 9(d) shows that $A_{e}$ increases on increasing $\unicode[STIX]{x1D6FF}_{Ra}$ which is mainly due to increasing characteristic of the growth rate as a function of $\unicode[STIX]{x1D6FF}_{Ra}$ .

A similar study is also made when the medium is saturated with water (a figure is not shown here). However, the variation of the Landau constant as a function of $Ra$ predicts only the supercritical bifurcation. Therefore, in the rest of the section our analysis will be restricted to the impact of nonlinear interaction on the PMCF of air only.

To understand the impact of nonlinear effects on heat transfer rate and friction coefficient away from the bifurcation point, for the same set of $(Da,Ra)$ , figure 10(a,c) is drawn for $Da=10^{-2}$ and figure 10(b,d) is drawn for $Da=10^{-3}$ . As can be seen from figures 10(a) and 10(b), the Nusselt number estimated in the linearly unstable regime with the help of a weakly nonlinear stability analysis is more than the one calculated from the basic state. The substantial increase in $Nu_{ds}$ beyond the bifurcation point can be predicted due to the buoyant instability of the flow. Quantitatively, we have observed that the increase in $Nu$ due to nonlinear interaction is around 35 % for $(Da,Re)=(10^{-2},1000)$ (5.6 % for $(Da,Re)=(10^{-3},4000)$ ) at $\unicode[STIX]{x1D6FF}_{Ra}=0.5$ . However, for $Re$ in the subcritical regime, the change in $Nu$ under the basic state and the distorted state can be seen up to $\unicode[STIX]{x1D6FF}_{Ra}=-0.1$ ( $\unicode[STIX]{x1D6FF}_{Ra}=-0.17$ ) for $Re=3000$ ( $Re=7000$ ). The friction coefficient, $C_{f}$ , for the above four values of $Re$ is shown in figure 10(c,d). The friction coefficient due to nonlinear interaction in the linearly unstable regime is less compared to the same calculated in the basic state. It decreases as a function of $Ra$ . Quantitatively, the decrease in friction coefficient due to nonlinear interaction at $\unicode[STIX]{x1D6FF}_{Ra}=0.5$ in the supercritical regime is around 12.42 % for $(Da,Re)=(10^{-2},1000)$ (3 % for $(Da,Re)=(10^{-3},4000)$ ). However, it is almost zero in the linearly stable regime.

Figure 10.

Variation of (a,b) Nusselt number and (c,d) friction coefficient with respect to $\unicode[STIX]{x1D6FF}_{Ra}$ in the subcritical ( $Re=3000$ , $Re=7000$ ) and supercritical regime ( $Re=1000$ , $Re=4000$ ) for $Pr=0.7$ and $c_{F}=6\times 10^{-3}$ . The solid line represents the basic state, whereas the dash-dotted as well as dashed lines represent the distorted state. Here, (a,c) $Da=10^{-2}$ , (b,d) $Da=10^{-3}$ .

Figure 11.

Variation of KE with respect to $\unicode[STIX]{x1D6FF}_{Ra}$ for $Pr=0.7$ and $c_{F}=6\times 10^{-3}$ : (a) $Re=1000$ , $Da=10^{-2}$ , (b) $Re=3000$ , $Da=10^{-2}$ , (c) $Re=4000$ , $Da=10^{-3}$ and (d) $Re=7000$ , $Da=10^{-3}$ .

The nonlinear spectrum of KE away from the critical point for $Da=10^{-2}$ ( $Da=10^{-3}$ ) is shown in figure 11(a) (figure 11 c) for the supercritical regime and in figure 11(b) (figure 11 d) for the subcritical regime. The spectrum of KE in the supercritical as well as the subcritical regime for $Da=10^{-2}$ reveals that the three terms $P_{110}$ , $T_{11}$ and $D_{11}$ are positive and act as destabilizing factors. However, in the supercritical (subcritical) regime $T_{11}$ ( $P_{110}$ ) acts as a dominant destabilizing factor up to a certain distance from the bifurcation point. Beyond that distance, $P_{110}$ ( $T_{11}$ ) in the supercritical (subcritical) regime starts to play an equally important role in the KE spectrum and also becomes dominant. A notable contribution of $P_{110}$ in both the regimes is a consequence of the following fact. Note that in the supercritical (subcritical) regime $Re$ is fixed at 1000 (3000) and the basic flow has a buoyant (mixed) instability. Also, the contribution of the shear term is positive and not negligible (see discussion § 3.1). Therefore, modification to the gradient production due to a change in the shape of the fundamental wave will be more favourable for shear production. In the subcritical regime, figure 11(b) shows that the value of $P_{110}$ decreases as one moves down from the critical value. Since for this value of $Re$ the basic flow has a mixed type of instability and in the linearly stable regime the value of $Ra$ is less than the critical value, as a result, a relatively lower shear production is expected. Definitely the pattern of secondary flow will be influenced by this, which will be discussed in § 6.4. Furthermore, in contrast to the supercritical regime where $K_{11}$ , $F_{11}$ , $P_{101}$ of the disturbance KE increase on increasing $Ra$ , in the subcritical regime all these remain almost constant. For $Da=10^{-3}$ , figure 11(c,d) shows that the term $T_{11}$ is positive and it is the main destabilizing factor in the KE balance. The magnitude of it increases when it varies as a function of $Ra$ away from the critical point. Similar to figure 8(a,b) here also the term $P_{110}$ is negative (positive) in the supercritical (subcritical) regime. In the supercritical regime its contribution to the KE spectrum increases as it moves away from the critical point. The term $T_{11}$ is balanced by $P_{110}$ , $K_{11}$ and $F_{11}$ in the supercritical regime, whereas the same in the subcritical regime is balanced by $K_{11}$ and $F_{11}$ . Apart from these, figures 11(a,b) and 11(c,d) show that the sign of $2(a_{1})_{r}$ is negative (positive) in the entire supercritical regime (subcritical regime). Thus, the balance of KE supports the results obtained from the Landau constant.

6.3 Bifurcation as a function of $\unicode[STIX]{x1D6FC}$

The variation of the neutral stability curve with wavenumber is also important because an instability that is supercritical for some wavenumber may be subcritical or supercritical at other nearby wavenumber (Rogers et al. Reference Rogers, Moulic and Yao1993). In this situation, any of the potential unstable waves may grow and interact with other modes. Therefore, for the present problem we have chosen two different values of $Re$ , one from the supercritical regime and the other from the subcritical regime, for a given $Da$ . Accordingly, for three values $10^{-2}$ , $10^{-3}$ and $10^{-4}$ of $Da$ three different respective values 1000, 4000 and 10 000 of $Re$ from the supercritical regime are chosen. From the subcritical regime two values 3000 and 7000 of $Re$ are chosen for respectively two different values $10^{-2}$ and $10^{-3}$ of $Da$ .

Note that the corresponding critical points ( $\unicode[STIX]{x1D6FC},Ra$ ) for the above chosen values of $Re$ in the supercritical regime are (1.94, 290), (1.24, 12269) and (15.1, 137 491) and in the subcritical regime they are (1.95, 425), (1.22, 12 366), respectively. The type of instability is buoyant at all these points except for $Re=3000$ for $Da=10^{-2}$ where it is mixed or interactive. The neutral stability curves $(c_{i}=0)$ for different values of $Da$ are plotted in figure 12. Here, solid and dashed lines represent variation of $Ra$ with $\unicode[STIX]{x1D6FC}$ for a given $Re$ from the supercritical and subcritical regime, respectively. It can be seen from the above figure that for $Da=10^{-3}$ and $Re=7000$ ( $Da=10^{-4}$ and $Re=10\,000$ ), a large band of wavenumber exists in which subcritical (supercritical) bifurcation remains subcritical (supercritical). However, for $Da=10^{-2}$ and $Re=1000$ , subcritical bifurcation can be seen for $\unicode[STIX]{x1D6FC}<0.8$ and supercritical for $\unicode[STIX]{x1D6FC}>0.8$ . Similarly, for $Re=3000$ , supercritical bifurcation can be seen for $\unicode[STIX]{x1D6FC}<1.8$ and subcritical for $\unicode[STIX]{x1D6FC}>1.8$ . In the case of $Da=10^{-3}$ and $Re=4000$ , subcritical bifurcation can be seen for $\unicode[STIX]{x1D6FC}<1.12$ and supercritical for $\unicode[STIX]{x1D6FC}>1.12$ .

Figure 12.

Neutral stability curve ( $c_{i}=0$ ) in the ( $\unicode[STIX]{x1D6FC},Ra$ )-plane: (a) $Da=10^{-2}$ , (b)  $Da=10^{-3}$ and (c) $Da=10^{-4}$ when $Pr=0.7$ and $c_{F}=6\times 10^{-3}$ .

Note that for $Re=4000$ the basic flow at the least linearly stable point (1.24, 12 269) is supercritically stable whereas the same at the nearby neutral point (1.1, 14 519) is subcritically unstable. Similarly, for $Da=10^{-2}$ the basic flow which is subcritically unstable at the least linearly stable point (1.95, 425) is supercritically stable at the nearby neutral point (1.7, 436). So, from the above analysis we can point out that with the buoyant and mixed instabilities, both subcritical and supercritical branches appear on the neutral curves.

6.4 Pattern of secondary flow

The overview of the above discussion suggests the existence of secondary flow with finite amplitude beyond the bifurcation point. We have seen that depending on the value of input parameter the bifurcation may be supercritical or subcritical. Apart from this, the balance of kinetic energy depicts a different result relative to a purely fluid domain in the supercritical regime. Therefore, the nonlinear impact on secondary flow from the linearly least stable wave is analysed in the supercritical as well as the subcritical regime. The secondary flow under linear stability analysis is obtained from the eigenfunction associated with the linearly least stable eigenvalue and the same from the weakly nonlinear analysis is calculated as a superposition of different harmonics (here, $E^{1}$ and $E^{2}$ in (4.1)). Following Khandelwal & Bera (Reference Khandelwal and Bera2015), the functions $\hat{u} _{1}$ , $\hat{u} _{2}$ , etc., for the secondary flow are calculated. The eigenfunction includes the disturbance $u$ and $v$ velocities as well as disturbance temperature ( $\unicode[STIX]{x1D703}$ ). However, to avoid numerous figures the pattern of disturbance of the $v$ velocity is not shown in the following. We are interested in studying the pattern of secondary flow very close to the bifurcation point as well as away from the bifurcation point under linear and weakly nonlinear analyses.

We restrict our analysis to the same four pairs ( $10^{-2},1000$ ), ( $10^{-3},4000$ ), ( $10^{-2},3000$ ) and ( $10^{-3},7000$ ) of ( $Da,Re$ ) which are also used in § 6.2. As discussed therein, $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ and $\unicode[STIX]{x1D6FF}_{Ra}=0.5$ are points from the supercritical regime for the first two pairs of ( $Da,Re$ ), whereas $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ and $\unicode[STIX]{x1D6FF}_{Ra}=-0.5$ are points from the subcritical regime for the last pair of ( $Da,Re$ ) only. Since the subcritical regime of $(Da,Re)=(10^{-2},3000)$ is $0\geqslant \unicode[STIX]{x1D6FF}_{Ra}\geqslant -0.1$ , for this pair of $(Da,Re)$ , $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ and $\unicode[STIX]{x1D6FF}_{Ra}=-0.1$ are chosen as respective points from very close to the critical point and away from the critical point. In the following the solid and dashed lines in the contours are associated with respective positive and negative values of a field variable.

Figure 13.

Pattern of secondary flow by (a) linear stability analysis and (b) nonlinear stability analysis when $Da=10^{-2}$ , $Pr=0.7$ , $Re=1000$ and $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ as well as $\unicode[STIX]{x1D6FF}_{Ra}=0.5$ : (i) and (iii) disturbance $u$ -velocity; (ii) and (iv) disturbance temperature.

Figure 14.

Pattern of secondary flow by (a) linear stability analysis and (b) nonlinear stability analysis when $Da=10^{-3}$ , $Pr=0.7$ , $Re=4000$ and $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ as well as $\unicode[STIX]{x1D6FF}_{Ra}=0.5$ : (i) and (iii) disturbance $u$ -velocity; (ii) and (iv) disturbance temperature.

Figure 15.

Pattern of secondary flow by (a) linear stability analysis and (b) nonlinear stability analysis when $Da=10^{-2}$ , $Pr=0.7$ , $Re=3000$ and $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ as well as $\unicode[STIX]{x1D6FF}_{Ra}=-0.1$ : (i) and (iii) disturbance $u$ -velocity; (ii) and (iv) disturbance temperature.

Figure 16.

Pattern of secondary flow by (a) linear stability analysis and (b) nonlinear stability analysis when $Da=10^{-2}$ , $Pr=0.7$ , $Re=2000$ and $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ as well as $\unicode[STIX]{x1D6FF}_{Ra}=-0.1$ : (i) and (iii) disturbance $u$ -velocity; (ii) and (iv) disturbance temperature.

Figure 17.

Pattern of secondary flow by (a) linear stability analysis and (b) nonlinear stability analysis when $Da=10^{-3}$ , $Pr=0.7$ , $Re=7000$ and $\unicode[STIX]{x1D6FF}_{Ra}=-0.01$ as well as $\unicode[STIX]{x1D6FF}_{Ra}=-0.5$ : (i) and (iii) disturbance $u$ -velocity; (ii) and (iv) disturbance temperature.

The patterns of secondary flow in the supercritical regime, when $Re=1000$ and $Da=10^{-2}$ , from linear stability theory and weakly nonlinear stability theory are shown in figures 13(a) and 13(b), respectively. As can be seen from the above figures, the difference between the patterns of secondary flow using linear and weakly nonlinear theories at $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ is negligible as compared to the same at $\unicode[STIX]{x1D6FF}_{Ra}=0.5$ . However, magnitude-wise it differs significantly for both values of $\unicode[STIX]{x1D6FF}_{Ra}$ . Apart from these, although the shape of the contours of the disturbance temperature and the $u$ -velocity in the linear stability analysis are deformed significantly under the nonlinear analysis, the deformation in the disturbance temperature is much higher than the same in the disturbance velocity. It increases as one moves away from the bifurcation point. This type of modification in the fundamental disturbance is also predicted in the energy analysis in § 6.2. To check the above result for other values of $Da$ we have plotted the same for $Da=10^{-3}$ when $Re=4000$ in figures 14(a) and 14(b), respectively. Although these also give a similar impression, compared to $(Da,Re)=(10^{-2},3000)$ the change of shape in the disturbance velocity as well as the temperature under the nonlinear analysis for the considered pair of $(Da,Re)$ is less. Here, at $\unicode[STIX]{x1D6FF}_{Ra}=0.01$ the deformation of the contours is mainly in terms of shifting.

In the subcritical regime for $Da=10^{-2}$ , figure 15(a,b) shows that the deformation of the fundamental disturbance, in the vicinity of the bifurcation point, is much higher than the same under the supercritical regime. Here, the deformation of the cells is in the form of shifting as well as stretching. The deformation of the fundamental disturbance under nonlinear interaction in the neighbourhood of the $Re_{sb}$ is expected to be much more than the same far away from $Re_{sb}$ . Therefore, figures 16(a) and 16 b) are drawn for $Re=2000$ and $Da=10^{-2}$ . Note that for $Re=2000$ the calculated subcritical regime is $-0.12\leqslant \unicode[STIX]{x1D6FF}_{Ra}\leqslant 0$ . As can be seen from figure 16(a,b), the above expectation is correct. The shape of the uni-cellular structure of the $u$ velocity from linear stability theory shifts to a deformed multicellular structure for $\unicode[STIX]{x1D6FF}_{Ra}=-0.1$ . Deformation of the temperature cells under subcritical bifurcation is accelerated as one moves away from the bifurcation point. In order to cross-check the pattern variation of secondary flow in the subcritical regime when $Da$ is changed to $10^{-3}$ , figure 18 is drawn for $Re=7000$ . Here, deformation of the fundamental disturbance under nonlinear interaction is much less as compared to $Da=10^{-2}$ at both values $-0.01$ and $-0.5$ of $\unicode[STIX]{x1D6FF}_{Ra}$ , which is also expected from the nonlinear energy spectrum.

From the above analysis, we would like to mention that a jump in the amplitude profile of a physical problem, which was observed in §§ 6.1 and 6.2, leads to a transition from supercritical bifurcation to subcritical bifurcation or vice versa and deforms the pattern of secondary flow under linear stability analysis significantly (see figures 15 and 16).