Linear Stability of a Developing Thermal Front Induced by a Constant Heat Flux

Abstract

A developing thermal front is set up by suddenly imposing a constant heat flux on the lower horizontal boundary of a semi-infinite fluid-saturated porous domain. The critical time for the onset of convection is determined using two main forms of analysis. The first of these is an approximate method which is effectively a frozen-time model while the second implements a set of parabolic simulations of monochromatic disturbances placed in the boundary layer at an early time. Results from the two approaches are compared and it is found that instability only occurs when the nondimensional disturbance wavenumber, $k$, is less than unity. The neutral curve for the primary mode possesses a vertical asymptote at $k=1$ in wavenumber/time parameter space which is in contrast to the more usual teardrop shape which occurs when the surface is subject to a constant temperature. Asymptotic analyses are performed for the frozen-time model which yield excellent predictions for both branches of the neutral curve and the locus of the maximum growth rate curve at late times.

Appendices

Appendix 1: Small-$k$ Analysis for the Fastest Growing Mode

In this Appendix we solve Eqs. (41) and (42) for small values of $k$ in order to determine that value of $\tau$ below which $k=0$ corresponds to the maximal value of $\lambda$. We introduce the following expansions,

$$\begin{aligned} \Psi&= k\Psi _0(\eta ) + k^3\Psi _1(\eta ) +\cdots ,\end{aligned}$$

(48)

$$\begin{aligned} \Theta&= \Theta _0(\eta ) + k^2\Theta _1(\eta ) +\cdots , \end{aligned}$$

(49)

and

$$\begin{aligned} \lambda =\lambda _0 + k^2\lambda _1 +\cdots . \end{aligned}$$

(50)

At leading order $\Theta _0$ satisfies,

$$\begin{aligned} \Theta _0^{\prime \prime }+2\eta \Theta _0^{\prime } = 2\tau \lambda _0\Theta _0, \end{aligned}$$

(51)

for which a suitable solution is

$$\begin{aligned} \Theta _0=e^{-\eta ^2}, \end{aligned}$$

(52)

where $\lambda _0=-1/\tau$. This shows that the $k=0$ mode is always stable. The corresponding equation for $\Psi _0$ is

$$\begin{aligned} \Psi _0^{\prime \prime } = 4\tau ^2e^{-\eta ^2}, \end{aligned}$$

(53)

and its solution is,

$$\begin{aligned} \Psi _0 = 2\tau ^2\Bigl [ e^{-\eta ^2}-1 -\sqrt{\pi }\,\eta \, \hbox {erfc}(\eta ) \Bigr ]. \end{aligned}$$

(54)

Although this expression for $\Psi _0$ satisfies $\Psi _0=0$ at $\eta =0$, it does not vanish as $\eta \rightarrow \infty$. However, an outer $\eta =O(k^{-1})$ layer may be evoked which will cause $\Psi _0$ to decrease exponentially to zero over that lengthscale.

At the next order we have,

$$\begin{aligned} \Theta _1^{\prime \prime } + 2\eta \Theta _1^{\prime } + 2\Theta _1 = \Bigl [4\tau ^2+2\tau \lambda _1\Bigr ]e^{-\eta ^2} +8\tau ^4\Bigl [ e^{-\eta ^2}-1 -\sqrt{\pi }\,\eta \, \hbox {erfc}(\eta ) \Bigr ]\hbox {erfc}(\eta ).\quad \end{aligned}$$

(55)

A straightforward solution using a fourth order Runge–Kutta scheme shows that $\lambda _1=0$ when $\tau =1.253314$. When $\tau$ takes larger values then $\lambda _1$ is positive, showing that $k=0$ is a local minimum for the growth rate. When $\tau$ takes smaller values then $\lambda _1$ is negative and therefore $k=0$ is a local maximum for the growth rate, as shown in Fig. 1.

Appendix 2: Asymptotic Theory for the Right-Hand Branch

In this Appendix we develop an asymptotic expression for the neutral curve as it approaches $k=1$ with $\tau \gg 1$. We begin with Eqs. (36) and (37) which are repeated below for the sake of completeness,

$$\begin{aligned}&\Psi ^{\prime \prime } - 4\tau ^2k^2\Psi = 4\tau ^2k\Theta ,\end{aligned}$$

(56)

$$\begin{aligned}&\Theta ^{\prime \prime } +2\eta \Theta ^{\prime }-4\tau ^2k^2\Theta -4\tau ^2 k \,\hbox {erfc}{(\eta )}\,\Psi = 0, \end{aligned}$$

(57)

and it is recalled that this system needs to be solved subject to $\psi ,\theta \rightarrow 0$ as $\eta \rightarrow \infty$ and $\theta ^{\prime }=\psi =0$ at $\eta =0$. The detailed numerical results indicate that the right-hand branch approaches the value $k=1$ as $\tau$ increases and that the corresponding eigenfunctions occupy a thinning region in terms of $\eta$. Under the assumption that $\tau \gg 1$, it turns out that $k$ expands in inverse powers of $\tau$ with

$$\begin{aligned} k=1+k_0\tau ^{-2/3}+k_1\tau ^{-1}+k_2\tau ^{-4/3}+\cdots . \end{aligned}$$

(58)

In what follows we evaluate $k_0, k_1$ and $k_2$ and remark that we anticipate that $k_0<0$ since the neutral curve approaches the $k=1$ asymptote from the left.

1.1 The Main Disturbance Layer

The disturbance is confined to a relatively thin region wherein the appropriate vertical variable is

$$\begin{aligned} \zeta =\eta \tau ^{2/3}\,=\,O(1), \end{aligned}$$

(59)

and for which the error function which appears in Eq. (57) becomes

$$\begin{aligned} \hbox {erfc}(\eta )= 1-{2\zeta \over \sqrt{\pi }}\tau ^{-2/3}+O(\tau ^{-4/3}). \end{aligned}$$

(60)

Eigensolutions are sought of the form where

$$\begin{aligned} (\psi ,\theta )&= (\psi _0,\theta _0) +\tau ^{-1/3}(\psi _1,\theta _1) +\tau ^{-2/3}(\psi _2,\theta _2) +\tau ^{-1}(\psi _3,\theta _3)\nonumber \\&+\tau ^{-4/3}(\psi _4,\theta _4)+\cdots , \end{aligned}$$

(61)

and the substitution of Eqs. (58), (59) and (61) into Eqs. (56) and (57) yields a sequence of equations for $(\psi _j,\theta _j)$.

At both leading (zeroth) and first orders, the streamfunction and heat transport equations are consistent and simply give

$$\begin{aligned} \psi _0=-\theta _0\quad \mathrm{and} \quad \psi _1=-\theta _1. \end{aligned}$$

(62)

To tie down these functions, we proceed to higher orders and, at $O(\tau ^{4/3})$, Eqs. (56) and (57) give

$$\begin{aligned} 4(\theta _2+\psi _2)=\psi _0^{^{\prime \prime }}-8k_0\psi _0-4k_0\theta _0, \end{aligned}$$

(63)

and

$$\begin{aligned} 4(\psi _2+\theta _2)=\theta _0^{^{\prime \prime }}-8k_0 \theta _0-4k_0\psi _0+{8\over \sqrt{\pi }}\zeta \psi _0\,, \end{aligned}$$

(64)

where a prime denotes differentiation with respect to $\zeta$ here. This pair then yields,

$$\begin{aligned} {d^2\psi _0\over d\zeta ^2}-4k_0\psi _0-\frac{4\zeta }{\sqrt{\pi }}\psi _0=0, \end{aligned}$$

(65)

which is a scaled form of Airy’s equation with solution

$$\begin{aligned} \psi _0=-\theta _0=C\mathrm{Ai\,}\left[ {4^{1/3}\over \pi ^{1/6}} \left( \zeta +k_0\sqrt{\pi }\right) \right] , \end{aligned}$$

(66)

where $C$ is an arbitrary constant.

It is clear that Eq. (66) satisfies $\psi _0, \theta _0\rightarrow 0$ as $\zeta \rightarrow \infty$, but it does not fulfil the requirements at $\zeta =0$, namely that $\theta ^{\prime }=\psi =0$. This points to the existence of an inner sublayer within the disturbance layer, which we analyse below, but here we can impose one of the two wall conditions and make the usual choice to enforce the Dirichlet condition $\psi _0=0$. If we denote the first zero of the Airy function as $Z_0$ ($=-2.338107$ to six decimal places), then Eq. (66) implies that

$$\begin{aligned} (4\pi )^{1/3}k_0=Z_0 \,\Rightarrow \, k_0=Z_0/(4\pi )^{1/3}. \end{aligned}$$

(67)

In order to derive further terms in the asymptotic expansion Eq. (58) it is necessary to consider the governing equations for $\theta _3$ and $\psi _3$ and for $\theta _4$ and $\psi _4$. The consistency of the pair of equations at $O(\tau )$ for $\theta _3$ and $\psi _3$ is guaranteed as long as

$$\begin{aligned} \theta _1=k_1\sqrt{\pi }{d\theta _0\over d\zeta } \end{aligned}$$

(68)

with $\psi _1$ then given by Eq. (62). At $O(\tau ^{2/3})$ we find that

$$\begin{aligned} 4(\psi _4+\theta _4)=\psi _2^{^{\prime \prime }}-8k_0\psi _2-4k_0\theta _2-8k_1 \psi _1-4k_1\theta _1-4(2k_2+k_0^2)\psi _0-4k_2\theta _0 \end{aligned}$$

and

$$\begin{aligned} 4(\psi _4+\theta _4)&= \theta _2^{^{\prime \prime }}-8k_0\theta _2-4k_0\psi _2+ {8\over \sqrt{\pi }}\zeta \psi _2-8k_1\theta _1-4k_1\psi _1\nonumber \\&-4(2k_2+k_0^2)\theta _0- 4k_2\psi _0+{8k_0\zeta \over \sqrt{\pi }}\psi _0. \end{aligned}$$

The solvability of this pair of equations leads to the governing equation for $\psi _2$ in the form,

$$\begin{aligned} {d^2\psi _2\over d\zeta ^2}-4k_0\psi _2-{4\zeta \over \sqrt{\pi }}\psi _2={1\over 8}\psi _0^{\prime \prime \prime \prime }+4k_1^2\sqrt{\pi }\psi _0^{\prime }+2(k_0^2+2k_2)\psi _0, \end{aligned}$$

(69)

and this can be solved explicitly by making use of the fact that $\psi _0$ satisfies Eq. (65). It follows that

$$\begin{aligned} \psi _2={\sqrt{\pi }\over 160}\psi _0^V +{k_1^2\pi \over 2}\psi _0^{\prime \prime }+{1\over 2}\sqrt{\pi }(k_0^2+2k_2)\psi _0^{\prime }; \end{aligned}$$

if we denote $\psi _0^{\prime }(\zeta =0)=D$ then it can be shown that

$$\begin{aligned} \psi _2\rightarrow D\left[ {k_0^2\sqrt{\pi }\over 10}+{\sqrt{\pi }(k_0^2+2k_2)\over 2}\right] \end{aligned}$$

as $\zeta \rightarrow 0$. Then Eq. (64) implies that

$$\begin{aligned} \theta _2\rightarrow \left[ {k_0^2\sqrt{\pi }\over 10}+{\sqrt{\pi }(k_0^2+2k_2)\over 2}\right] {d\theta _0\over d\zeta }\equiv M{d\theta _0\over d\zeta } \end{aligned}$$

(70)

in this limit. Taken together, Eqs. (66), (68) and (70) show that

$$\begin{aligned} \theta =\mathrm{Ai\,}\left[ {4^{1/3}\over \pi ^{1/6}}\left( \zeta +k_0\sqrt{\pi }\right) \right] +k_1\sqrt{\pi }{d\theta _0\over d\zeta }\tau ^{1/3}+ M{d\theta _0\over d\zeta }\tau ^{2/3}+\dots \quad \hbox {as } \zeta \rightarrow 0, \end{aligned}$$

and it is therefore necessary to consider the sub-layer in order to satisfy the Neumann boundary condition on $\theta$ at $\eta =0$.

1.2 The Sublayer

Within the sublayer, where $\zeta =\tau ^{-1/3}\hat{\eta }$, the main layer solutions imply that,

$$\begin{aligned} \theta =\tau ^{-1/3}\hat{\theta }_0(\hat{\eta })+\tau ^{-2/3} \hat{\theta }_1(\hat{\eta })+\dots ,\quad \psi = \tau ^{-1/3}\hat{\psi }_0(\hat{\eta })+\tau ^{-2/3}\hat{\psi }_0(\hat{\eta })+\cdots . \end{aligned}$$

Substituting gives at the first two orders

$$\begin{aligned} \hat{\psi }_j^{^{\prime \prime }}-4\hat{\psi }_j=4\hat{\theta }_j\quad \mathrm{and}\quad \hat{\theta }_j^{^{\prime \prime }}-4\hat{\theta }_j=4\hat{\psi }_j\quad (j=0,1), \end{aligned}$$

whereupon

$$\begin{aligned} \hat{\theta }_0=\lambda \left[ e^{-\sqrt{8}\hat{\eta }}+1+\hat{\eta }\sqrt{8}\right] ,\quad \hat{\psi }_0=\lambda \left[ e^{-\sqrt{8}\hat{\eta }}-1-\hat{\eta }\sqrt{8}\right] , \end{aligned}$$

for some constant $\lambda$ that can be determined by matching but whose precise value is not required here. These leading order solutions are only compatible with the main layer structure if

$$\begin{aligned} k_1\sqrt{\pi }={1\over \sqrt{8}}. \end{aligned}$$

(71)

The equations for $\hat{\theta }_1$ and $\hat{\psi }_1$ need to be solved such that both $\hat{\theta }_1^{\prime }$ and $\hat{\psi }_1$ vanish on the wall and tend to constants as $\hat{\eta }\rightarrow \infty$. The only way this is possible is if the constant is zero and then these functions are both identically zero. We deduce that

$$\begin{aligned} M={k_0^2\sqrt{\pi }\over 10}+{\sqrt{\pi }(k_0^2+2k_2)\over 2}=0 \quad \Rightarrow \quad k_2=-{3k_0^2\over 5}. \end{aligned}$$

(72)

Combining results of Eqs. (67), (71) and (72) yields the shape of the neutral curve for large $\tau$ in the form,

$$\begin{aligned} k&= 1+{Z_0(4\pi )^{-1/3}\over \tau ^{2/3}}+{1\over \sqrt{8\pi }\tau }- {0.6Z_0^2(4\pi )^{-2/3}\over \tau ^{4/3}}+\dots \nonumber \\&\approx 1-1.006\tau ^{-2/3}+0.199\tau ^{-1}-0.607\tau ^{-4/3}+\dots , \end{aligned}$$

(73)

where $Z_0=-2.338\,107$ was used in the computations depicted in Fig. 2.

Appendix 3: Asymptotic Form of the Fastest Growing Mode When $\tau \gg 1$

Here we outline the structure of the fastest growing mode for large scaled times. For completeness, it is convenient to start with the relevant system,

$$\begin{aligned}&\Psi ^{\prime \prime } - 4\tau ^2k^2\Psi = 4\tau ^2k\Theta ,\end{aligned}$$

(74)

$$\begin{aligned}&\Theta ^{\prime \prime } +2\eta \Theta ^{\prime }-4\tau ^2k^2\Theta -4\tau ^2k \,\hbox {erfc}{(\eta )}\,\Psi = 2\tau \lambda \Theta , \end{aligned}$$

(75)

which one may recall is the eigenvalue problem for $\lambda$, but where $\lambda$ is to be maximised over $k$ for chosen values of $\tau$.

To infer the structure of the desired mode it is easiest to consider the form of solution of the problem for a general value of $k=O(1)$ and then allow $k\rightarrow 0$. Doing this leads to the conclusion that the most dangerous mode resides in the regime where $k=O(\tau ^{-1/4})$ and it is compressed into a thin zone attached to $\eta =0$ and of depth $O(\tau ^{-1/2})$. In brief, the spatial variable is then

$$\begin{aligned} \tilde{\zeta }=\eta \tau ^{1/2}\,=\,O(1), \end{aligned}$$

(76)

the growth rate

$$\begin{aligned} \lambda =2\tau +\lambda _0\tau ^{1/2}+\dots , \end{aligned}$$

(77)

the wavenumber of interest is

$$\begin{aligned} k={\mathcal{K}}\tau ^{-1/4}\,+\dots \end{aligned}$$

(78)

and eigensolutions are sought of the form

$$\begin{aligned} \psi =\tilde{\psi }_0+\tau ^{-1/4}\tilde{\psi }_1+\tau ^{-1/4} \tilde{\psi }_2+\cdots \quad \theta =\tau ^{-1/4}\tilde{\theta }_0+\tau ^{-1/2}\tilde{\theta }_1 +\tau ^{-1/2}\tilde{\theta }_2+\cdots . \end{aligned}$$

(79)

At both leading and first orders, the streamfunction and heat transport equations are consistent and simply give

$$\begin{aligned} {\mathcal{K}}\tilde{\psi }_0=-\tilde{\theta }_0\quad \mathrm{and} \quad {\mathcal{K}}\tilde{\psi }_1=-\tilde{\theta }_1. \end{aligned}$$

(80)

Consistency at next order is only possible if

$$\begin{aligned} \frac{d^2\tilde{\theta }_0}{d\zeta ^2}-\left[ \frac{8{\mathcal{K}}^2}{\sqrt{\pi }}\zeta +4{\mathcal{K}}^2 \left( {\mathcal{K}}^2+\frac{1}{2}\lambda _0\right) \right] \tilde{\theta }_0=0. \end{aligned}$$

(81)

Much as in the case of discussion of the problem Eq. (65), the requirement that this equation possesses a solution with decay both as $\tilde{\zeta }\rightarrow 0$ and as $\tilde{\zeta }\rightarrow \infty$ requires that

$$\begin{aligned} \lambda _0=\frac{2Z_0}{\pi ^{1/3}}{\mathcal{K}}^{-2/3}-2{\mathcal{K}}^2, \end{aligned}$$

(82)

where $Z_0$ is the first zero of the Airy function.

It is now evident how the analysis has captured the fastest growing mode. The expression Eq. (82) for $\lambda _0\rightarrow -\infty$ both as ${\mathcal{K}}\rightarrow 0$ and ${\mathcal{K}}\rightarrow \infty$; it is a simple matter to show that $\lambda _0$ is maximised when ${\mathcal{K}}\approx 0.789\,332$ and hence we deduce the quoted result Eq. (44). There is good agreement between this analytical prediction and the numerical simulations—this comparison could be improved by including higher order terms in our analysis although the algebraic manipulations very rapidly become increasingly laborious. But this Appendix illustrates how that the wavenumber of the fastest growing mode decreases slowly with $\tau$.