On three-dimensional stable long-wavelength convection in the presence of Dirichlet thermal boundary conditions

Abstract

It is a well-known fact that the onset of Rayleigh–Bénard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. The aim of this paper is to quantify the exact dimensions of a cylinder of rectangular cross-section wherein stable three-dimensional Rayleigh–Bénard convection sets in via a long-wavelength instability from the motionless state at the same value of the critical Rayleigh number as the corresponding horizontally unbounded problem when the bounding horizontal walls have infinite thermal conductance. Hence, we consider three-dimensional Rayleigh–Bénard convection in a cell of infinite extent in the x-direction, confined between two vertical walls located at $y= \pm H$ and horizontal boundaries located at $z=0$ and $z=d$. Our analysis predicts the existence of the sought stable state for experimental velocity boundary conditions at the vertical walls provided the aspect ratio $\delta = H/d$ takes a certain value. In the limit $H \rightarrow \infty$, we retrieve the stability characteristics of the horizontally unbounded problem. As expected, the analysis predicts two counter-rotating rolls aligned along the y-direction of period $2 \pi /\delta$ equal to the period of the roll in the y-direction of the corresponding unbounded problem. A long-scale asymptotic analysis leads to the derivation of an evolution partial differential equation (PDE) that is fourth order in space and contains a single bifurcation parameter. The PDE, valid for a specific value of $\delta$, is analyzed analytically and numerically as function of the bifurcation parameter and for a variety of velocity boundary conditions at the vertical walls to seek the stable steady-state solutions. The same analysis is also extended to the case of convection in a fluid-saturated porous medium.

Appendices

Appendix A

This appendix is devoted to the case of side walls that are rigid and thermally conducting with the set of boundary conditions, $\phi =\phi _y=0$ at $y=\pm \delta$. Upon considering the Fourier modes expansions

$$\begin{aligned} {{\hat{\phi }}}(x,y,z)= & {} {{\hat{\phi }}}(y)\,\mathrm{{e}}^{\imath \alpha x}\,\sin {(n \pi z)}, \end{aligned}$$

(95)

$$\begin{aligned} \theta (x,y,z)= & {} {{\hat{\theta }}}(y)\,\mathrm{{e}}^{\imath \alpha x}\,\sin {(n \pi z)}, \end{aligned}$$

(96)

in the linearized steady system of equations, Eqs. (4) and (5), we obtain the eigenvalue problem,

$$\begin{aligned} (D^2 -(\alpha ^2+\pi ^2))^2\,{{\hat{\phi }}}= & {} R\,{{\hat{\theta }}}, \end{aligned}$$

(97)

$$\begin{aligned} (D^2 -(\alpha ^2+\pi ^2))\,{{\hat{\theta }}}= & {} -(\alpha ^2 + \pi ^2)\,{{\hat{\phi }}} \end{aligned}$$

(98)

with the corresponding boundary conditions ${{\hat{\theta }}}={{\hat{\phi }}}=D{{\hat{\phi }}}=0$ at $y=\pm \delta$. The transformation ${{\hat{y}}}=y/2 \delta$ then leads to

$$\begin{aligned}&({{\hat{D}}}^2 -4 \delta ^2\,(\alpha ^2+\pi ^2))^2\,{{\hat{\phi }}} = 16 \delta ^4\,R\,{{\hat{\theta }}}, \end{aligned}$$

(99)

$$\begin{aligned}&({{\hat{D}}}^2 -4 \delta ^2\,(\alpha ^2+\pi ^2))\,{{\hat{\theta }}} = -4 \delta ^2\,(\alpha ^2 + \pi ^2)\,{{\hat{\phi }}} \end{aligned}$$

(100)

with corresponding boundary conditions ${{\hat{\theta }}}={{\hat{\phi }}}={{\hat{D}}}{{\hat{\phi }}}=0$ at ${{\hat{y}}}=\pm 1/2$ where the symbol ${{\hat{D}}}$ stands for $d/d{{\hat{y}}}$. Upon setting $\alpha =0$ in Eqs. (99), we obtain the eigenvalue problem that governs the onset of steady Rayleigh–Bénard convection between two rigid and thermally conducting plates provided the wavenumber is $2 \delta \pi \approx 3.117$ and $R \approx (2 \pi )^4(1708)/(16)(3.117)^4 \approx 1762.5$. Therefore, the critical mode for the onset of convection with vanishingly small wavenumber requires $\delta = 3.117/2 \pi \approx 0.4961$. This finding is supported by the numerical solution of the boundary value problem, Eq. (97), the results of which are shown in Fig. 4. The numerical solution of the eigenvalue problem was carried out using the MATLAB code bvp4c. A weakly non-linear analysis using long-scale asymptotics could be performed in this case and a long-wavelength PDE derived provided the state variables are expanded using the complete set of Chandrasekhar functions, Eq. (85). It will lead to the emergence of a PDE having different numerical coefficients but qualitatively the same as Eq. (79) upon scaling.

Fig. 5

Plot of the steady-state solution of Eq. (79) for 4 distinct values of ${{\hat{\mu }}}$ starting with the initial condition Eq. (94). The solution consists of a single roll that occupies the full horizontal extent of the box. We note that the magnitude of F increases with ${{\hat{\mu }}}$

Fig. 6

On the left is a plot of the steady-state solution of Eq. (79) for 3 distinct values of ${{\hat{\mu }}}$ starting with the initial condition Eq. (94). The solution consists of a single roll that occupies the full horizontal extent of the box when ${{\hat{\mu }}}=100$. We note that more than one roll occupies the cell when ${{\hat{\mu }}}$ exceeds 100. On the right is a plot of the steady-state solution of Eq. (79) for 2 distinct and large values of ${{\hat{\mu }}}$ starting with the initial condition Eq. (94), namely, ${{\hat{\mu }}}=40000$ (dash-line) and ${{\hat{\mu }}}=200000$ (continuous line). A large increase in ${{\hat{\mu }}}$ seems to be associated with the emergence of more rolls having larger amplitude and sharper peaks that start at the boundaries and invade the domain

Appendix B

In this appendix, we examine the numerical solution of Eq. (79) on the interval $[-\pi ,\pi ]$ using the method of lines (MOL) by considering Dirichlet and Neumann boundary conditions for $F({{\hat{x}}},{{\hat{\tau }}})$, namely $F = {\partial F}/{\partial {{\hat{x}}}}=0$ at ${{\hat{x}}}=\pm \pi$. Upon using a central finite difference scheme for the fourth-order derivative term, we have for $1 \le i \le N-1$, $\varDelta {{\hat{x}}}=2 \pi /N$ and the grid in ${{\hat{x}}}$ has $N-1$ interior points,

$$\begin{aligned} {\mathrm{{d}}u_i \over \mathrm{{d}}{{\hat{\tau }}}} = -{u_{i+2}-4 u_{i+1}+ 6 u_i - 4 u_{i-1}+u_{i-2} \over \varDelta {{\hat{x}}}^4} + {{\hat{\mu }}} u_i - u_i^3. \end{aligned}$$

(101)

The boundary conditions imply $u_0=u_N=0$ and $u_{-1}=u_1$, $u_{N+1}=u_{N-1}$ and lead to the following matrix Q,

$$\begin{aligned} Q= \begin{bmatrix} 0 &{} 1 &{} 0 &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} 0 \\ 0 &{}0 &{} 0 &{} \ddots &{} &{}&{} &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \ddots &{} &{} &{} \vdots \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} &{} \vdots \\ \vdots &{} &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} &{} &{} \ddots &{} 0 &{} 0 &{} 0 &{} 0\\ \vdots &{} &{}&{} &{} \ddots &{} 0 &{} 0 &{} 0\\ 0 &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} 0 &{} 1 &{} 0 \\ \end{bmatrix} \end{aligned}$$

(102)

and the vectors $\mathbf{u}=[u_1, \ldots , u_{N-1}]^\mathrm{{T}}$. The discretized set of $(N-1)$ ODEs for the $u_i$, $1 \le i \le N-1$, is then given by

$$\begin{aligned} {\mathrm{{d}} \mathbf{u} \over \mathrm{{d}}t} = {{\hat{\mu }}} \mathbb {1}+A\,\mathbf{u} - ({1 / {\varDelta {{\hat{x}}}}^2})\,Q\,\mathbf{u}- \varvec{\Pi }^3, \end{aligned}$$

(103)

where $\mathbb {1}$ stands here for the $(N-1) \times (N-1)$ identity matrix. The set of equations, Eq. (103), is integrated using the MATLAB code ODE45 with the initial condition, Eq. (94). The results of the numerical solutions are shown in Figs. 5 and 6 wherein the steady-state solutions to Eq. (79) are depicted as function of the parameter ${{\hat{\mu }}}$. The imposed boundary conditions consist of $F={\mathrm{{d}}F/\mathrm{{d}}x}=0$ at $x = \pm \pi$. Figure 5 is a plot of F for ${{\hat{\mu }}}=O(1)$ quantity, making these results within the range of validity of the derivation of Eq. (79). We note that for a set of four distinct values of ${{\hat{\mu }}}$ and in contrast with the case of periodic boundary conditions, the solution does not evolve towards a uniform solution. Moreover, we note that the solution, the magnitude of which increases with ${{\hat{\mu }}}$ resembles that of the periodic box case for relatively larger values of ${{\hat{\mu }}}$. This difference may be attributed to the constraining effect of the imposed boundary conditions. This is more apparent in the cases where large parameter values are considered.

Although outside the range of validity of the derivation of Eq. (79), we have also investigated the effect of larger values of ${{\hat{\mu }}}$ on the solution. Figure 6 depicts the plot of F for 3 values of ${{\hat{\mu }}}$ varying between 100 and 500. We note that there is a tendency towards an even greater break-up of the spatially non-uniform solution with increasing values of ${{\hat{\mu }}}$. This situation becomes even more apparent in Fig. 6 where large values of ${{\hat{\mu }}}$ are considered, namely ${{\hat{\mu }}} =\,$40,000 and 400,000. This plot seems to support the analytical prediction that very large values of ${{\hat{\mu }}}$ lead to the formation of singularities in F when the boundary conditions $F = \mathrm{{d}}F/\mathrm{{d}}{{\hat{x}}}=0$ at ${{\hat{x}}}=\pm \pi$ are applied.