high wavenumber convection in tall porous...

HIGH WAVENUMBER CONVECTION INTALL POROUS CONTAINERS HEATED

FROM BELOW

By SAMANTHA LEWIS, D. A. S. REES

(School of Mechanical Engineering, University of Bath,Claverton Down, Bath BA2 7AY)

and ANDREW P. BASSOM

(Department of Mathematics, University of Exeter,North Park Road, Exeter, Devon EX4 4QE)

[Received 16 April 19%. Revise 20 January 1997]

SUMMARYIn this paper we consider convection in a high-aspect-ratio porous container

which is heated from below and whose sidewalls are insulated. This paper presentsasymptotic analyses of weakly nonlinear and highly nonlinear convection in the limitof large wavenumber, a. When a is 0(1), strongly nonlinear convection must beanalysed using fully numerical methods, though, when a is large, some progress canbe made using asymptotic methods without recourse to a direct simulation of thefull governing equations.

The onset of convection is given by the usual linear stability analysis for theDarcy-Be'nard problem, and the critical Rayleigh number is given by Rc ~ a2 atleading order. As the Rayleigh number increases there are three nonlinear asymptoticregimes which may be considered, the first of which yields the familiar weaklynonlinear flow. In this regime, for which R — Rc is asymptotically small, the weaklynonlinear evolution is given by the solution of a cubic Landau equation for theamplitude. When R- Rc = 0(1), the second regime, the flow is characterized bythe fact that the mean correction to the temperature profile and the first harmonicappear at the same order in the asymptotic expansion. The amplitude of convectionis now found to be given by an integro-differential equation and numerical solutionsindicate that when R- Rc is large boundary layers develop at the top and thebottom of the container.

The final regime emerges when R is so large that it is of size Rc + 0(a). The flowthen separates into three distinct domains: two boundary layers form, one at eitherend of the container, and a core flow exists in the interior. These flows are given bya singular perturbation analysis and the equations describing the boundary layer flowconstitute a nonlinear eigenvalue problem which requires numerical solution.

1. Introduction

THE~stJBJECT^of~ihennai convection in~porons media "has been studiedextensively in recent years and the growing volume of work devoted to thisand other topics in porous medium convection has been documented in thereviews by Tien and Vafai (1) and Nield and Bejan (2). There are manyimportant reasons for the development of this subject On the practical side,the need to model geothermal systems, packed-bed catalytic reactors and

[Q. Jl MfdL appL MattL, VoL SO, PL 4, 1997] © Oxford Untrcntty Prtn 1997

546 S. LEWIS ETAL

heat storage beds continues to motivate further research towards a betterunderstanding of transport phenomena in porous media. On the other hand,there is also a need to comprehend fully the implications of recent modelsof fluid flows within a porous matrix. An example of the latter is the recentpaper of Rees (3) which details how the presence of inertia modifies stabilityboundaries in the Darcy-Bdnard problem and hence determines the rangesof heat transfer rates which might be found in practice.

The motivation for the present work is the desire to understand themechanisms which may drive nonlinear thermal convection in porous mediaflow in a long, thin xectangular container. We demonstrate below that highlynonlinear modes can be obtained using formal asymptotic methods as longas the Rayleigh number is not too large. In a typical weakly nonlinearanalysis (see Stuart (4) or Watson (5)) a small-amplitude disturbance in theform of a cellular motion is imposed on the basic flow. If this amplitude isof size O(e), say, then the interaction of the cell with itself forces a secondharmonic and a mean state correction both of size O(e2). These in turndrive an O(e3) correction to the fundamental component of the mode. Asolvability (or orthogonality) criterion applied to the O(e3) equations yieldsan equation for the scaled amplitude of the original disturbance. This type ofanalysis has been applied to a wide variety of situations but Palm et al. (6)were the first authors to apply the technique to porous medium convectionin a layer heated from below.

This weakly nonlinear approach remains valid only if the amplitudeof the convection cell is asymptotically small, and the deviation of theRayleigh number from its critical value is O(e2). One significant feature ofthis methodology is that the correction to the mean state induced by theinteraction of the fundamental component with itself is asymptotically smallerthan the original basic flow. But when the amplitude of the disturbancebecomes too large, such as would be the case when R is too far abovecritical, then this is no longer true. Instead, the mean state is then alteredby the convective motions and the mathematical nature of the problem ischanged dramatically. In most stability problems whenever the leading ordermean flow and the highly nonlinear disturbance are inextricably linked inthis way then analytical progress is virtually impossible and one has to resortto a numerical solution of the full governing equations. However, recentlyit has been shown that this scenario does not occur in all cases. The firstdemonstration of an alternative was presented by Hall and Lakin (7) whostudied nonlinear small wavelength Gtirtler vortices which can evolve inan isothermal boundary layer above a suitably concave plate. Such modescan be described by an asymptotic flow structure which shows that thevortices are confined to a well-defined region within the boundary layer.The vortices are sufficiently large so as to interact and drive a correctionto the mean boundary layer profile which is of size comparable with themagnitude of the basic flow that would exist in the absence of vortices.

CONVECTION IN TALL POROUS CONTAINERS 547

The overall mean flow is totally changed by the vortex modes and, inturn, the resulting form of the mean flow dictates the quantitative detailsof the vortex structure. Thus the mean flow-vortex interaction is fullycoupled but, rather than necessitating a computational solution of the fullNavier-Stokes equations, the problem is amenable to analytical treatment. Inthis case the problem is reduced to one in which the solution of a singlefree-boundary problem yields the location of the edges of the main activityzone of the vortex as the boundary layer evolves downstream.

The essential key for the analytical description of the nonlinear Gflrtlermodes is the observation that a marked disparity exists between the twoprincipal lengthscales involved; the spanwise vortex wavelength is muchshorter than the characteristic distance over which the the boundary layerdevelops. Similar features have been subsequently exploited in order toderive strongly nonlinear modes in a number of other situations; by 'stronglynonlinear' here we mean vortex amplitudes which are far larger than thelimit of the validity of standard weakly nonlinear theories. Blennerhassettand Bassom (8) illustrated that steady, highly nonlinear rolls are mathemat-ically acceptable solutions of the classical B6nard convection problem. Theirresults were extended by Bassom and Zhang (9) who investigated nonlinearconvection rolls in a rapidly rotating fluid layer. Such modes are believed tobe important in a variety of geophysical applications and it was demonstratedhow strongly nonlinear cells can affect significantly the characteristics of theflow and lead to greatly enhanced rates of heat transfer.

In the present article we apply the same asymptotic strongly nonlinearvortex theory to the problem of convection cells generated in a saturatedporous medium which is confined within a large aspect ratio, tall container.The fact that the flow domain has a large aspect ratio makes the developmentof an appropriate strongly nonlinear theory possible. We restrict our attentionto two-dimensional flows—this is a reasonable assumption for if the heightof the (cuboidal) cavity is substantially greater than either its depth or itsbreadth, then the onset problem yields two-dimensional flow aligned suchthat fluid motion takes place in planes parallel with the vertical sides ofgreatest width. It is possible that three-dimensionality could arise if thehorizontal cross-section is nearly square, and therefore we assume that thecross-section has a sufficiently large aspect ratio to preclude this. Given thatmodes in such a tall cavity are identical to those of the same wavelengthin a horizontally unbounded layer (though presumably highly unstable inthe latter case) our present work has more physical realism than those

raises—treated^ » ^8^-or—(9)r^ind—isr^ndeedr ^irlrjr realistic subject to theabove geometrical constraint and the assumed absence of time-dependentinstabilities.

The three increasingly nonlinear regimes we describe here can best beestablished sequentially by taking appropriate limits of the results arisingfrom each regime and thereby inferring the scalings for the next. Indeed, the

548 S. LEWIS ETAL

details of the structure governing strongly nonlinear modes are sensitivelydependent on the particular problem being considered. For example, theGtirtler vortex work of Hall and Lakin (7) shows that the mode occupies awell-defined zone contained well within the boundary layer, whilst the theoryof (8) shows that nonlinear modes in a Benard layer fill the whole gapbetween the bounding horizontal surfaces. Thus it is not easy to anticipatethe appropriate strongly nonlinear structure and this is best deduced viaa gradual increase in vortex amplitude starting from the familiar weaklynonlinear case.

In view of this fact, the remainder of the paper proceeds as follows. Inthe next section we formulate the governing equations and review brieflythe results of a standard weakly nonlinear analysis. As the vortex strengthis increased a new stage of convection is reached in which the disturbancestructure throughout the container is significantly altered. We call this state'intermediate strength convection' for we then examine one further limit inwhich fully nonlinear boundary layers are set up at the top and bottom ofthe container. A simple eigenvalue problem can be obtained for intermediatestrength convection and this is formulated and solved in section 3. Thestrongly nonlinear analysis is discussed in section 4 and we draw someconclusions in section 5.

2. Equations of motion and weakly nonlinear solutions

Consider the problem of free convection in an infinite layer of saturatedporous medium bounded by two horizontal impermeable surfaces a distanced apart The temperatures of the upper and lower surfaces bounding themedium are taken to be uniform and equal to Tu and 7), respectively, with7) > Tu. Cartesian coordinates (x, y, z) are oriented such that the y-axis isnormal to the bounding surfaces of the fluid and gravity acts in the negativey-direction. The fluid is assumed to be such that Darcy's law holds andthe Oberbeck-Boussinesq approximation is valid. The coefficient of thermalexpansion, a, the thermal diffusivity, K, and the viscosity, ft, are therefore alltaken to be independent of the temperature T, whilst the density, p, dependslinearly on variations in T. A scaled temperature 0 is defined according toT = TU + 6(T, - Tu) and thence p = pu[\ - a(Tt - Tu)0] where a(T, - Tu) isassumed small. The resulting dimensionless Darcy-Boussinesq equations are

ux + vy = 0, (2.1a)

u = -Px, (2.1b)

v = -py + R6, (2.1c)

u9x +vOy = exx + eyy, (2.1 d)


if steady two-dimensional velocity fields are considered. Here R denotes theDarcy-Rayleigh number of the flow

2hw (2,e)

and the specification of the problem is completed by the boundary conditionsu = 0, 9 - l - y on y = 0 and y = l. (2.2)

It is well known that the conduction state 9=1— y, u = v = 0is unstableto small temperature perturbations proportional to cos ax, say, when theDarcy-Rayleigh number is sufficiently high, and, in particular, there is aneutral stability curve R = RN(") relating R and the wavenumber, a, at theonset of convection. When R > RN(O) then small-amplitude disturbancesgrow in time, but if e2 = R — RN(O) > 0 is small then a nonlinear analysisshows that equilibrium solutions with O(e) amplitude are possible; see Palm,Weber and Kvernvold (6) and Rees and Riley (10, 11). A full stabilityanalysis of weakly nonlinear solutions in a layer of infinite horizontal extentmay be gleaned from (10, 11), but as the present paper is ultimatelyconcerned with convection in a large-aspect ratio container we shall notdwell on their results here.

For the purpose of the weakly nonlinear analysis of the basic conductionstate we introduce a streamfunction ty^x, y) such that

u = -fy, v = \/fx,

so that the governing equations (2.1a to d) may be recast as

V V = ROx, (2-3 a)

f*0y - fy0x = V20, (2.3b)

subject to the boundary conditions

f = 0, 6=1 on y = 0 and V = °> 6> = 0 o n > = ! • (2-3 c)

In (2.3b), V2 denotes the two-dimensional Laplacian operator in the (x, v)-plane.

For Darcy-Rayleigh numbers close to the neutral value RN(<I) thestandard weakly nonlinear analysis for vortices of wavenumber a starts withexpansions

^ ] . . . . (2.4a)

6 = 1 - y + ei4ri1Cy)cosojt-|-f2i42[r2oCy) + 7'22Cy)cos2ajc]

+ £3A3[r3 1(y)cosax + r330')cos3aj:]-|-..., (2.4b)

R = RN + e2R2 +..., (2.4c)

550 S. LEWIS ETAL

where R2 = 0(1) as E -*• 0. The equations for the </>y and 7"y termsform pairs of second order coupled linear equations which must be solvedsequentially. The leading order equations for <pn and Tu constitute a lineareigenvalue problem for RN whose solution is

RN = 5 , Tu = sin Try, <pu = sin^y. (2.5a, b, c)az a

At second order in e we obtain

(7r + a ^ (2.5d,e,f)^ 0 , 7 ^ 0, 72o o

whilst the equations for 031 and T31 in expansions (2.4) are in-homogeneous versions of those for <pn and Tu with forcing termsgiven in terms of the linearized eigensolutions. These equations onlyhave a solution if a solvability criterion is satisfied which fixes thevalue of R2. This criterion is , derived by the usual procedure ofexamining adjoint systems, although in the present case the linearizedsystem is self-adjoint. This procedure is omitted for brevity but it leads toa Landau equation for the amplitude, A, whose steady supercritical value isgiven by

The onset of convection is via a pitchfork bifurcation from the stable zerostate when R2 < 0 to a weakly convecting state when R2 > 0.

In this paper we are interested in convection in a high aspect ratiocontainer, and this corresponds to a large vortex wavenumber, a » 1.Further, it is necessary to impose boundary conditions on the sidewalls,which will be taken to be at x = 0 and x — n/a. For simplicity we shallassume that the boundaries are insulated and impermeable; cases where thesidewalls have a fixed, linearly decreasing temperature have recently beenconsidered by Rees and Lage (12). Unlike the case of the B6nard problemin which no-slip conditions must be satisfied, the infinite layer analysis, asfar as we have already expounded it, is immediately applicable to convectionin a container with rectangular cross-section. Equation (2.6) shows that withR2 = 0(1) then A ~ O(a~2) as a -*• 00 and in this large wavenumber limitexpansions (2.4) attain the forms

\J/ = ea~1i4sin;rysinax + e2a~4i42^22sin2ax-h.., (2.7a)

0= 1 - 2 2 2 2 1 f

where we have written A = a2A, (fa = <^<hi and T-n = a*T-n with thecircumflex notation denoting 0(1) quantities. It is clear that because Abecomes small in the a » 1 limit (see (2.6)), it is possible to relax theassumption that e is small. A little experimentation reveals that in fact e


can be as large as 0(1) without changing the essential structure of theanalysis. With this size of e, the perturbations to the temperature fieldremain asymptotically smaller than the basic conduction profile, but nowthe corrections to the mean temperature field and the first harmonic arethe same size, namely 0(a~2) (notice that in (2.7b) f̂ = 0 from (2.5e)).Furthermore, the vertical velocity of the fluid is now 0(1)—this can beseen by forming v = \frx and allowing a —• oo with e = 0(1). This regimetherefore heralds a much stronger convection field which is examined indetail presently. For convenience, much of the notation introduced in thissection will be reused in sections 3, 4, sometimes with altered meanings.Whilst this may seem somewhat perverse, each of the coming sections ismore-or-less self-contained so no confusion should arise through the repeateduse of notation and, in compensation, we are not forced to introduce aplethora of different symbols.

3. Intermediate-strength large wavenumber convection

The results described in section 2 indicate that for large wavenumbersthere is a solution of (2.1), (2.2) with

6 = 1 - y + a~262 + a~2(0io + a~2612 +...) cosa*

+ a~802ocos2ax+..., (3.1a)

u = a"1(MiO + a~2«i2+-)sincu + a~7U2osin2ar+..., (3.1b)

v = (v\0 + a~2v\2+-)cosax + a~6v2ocos2ax+..., (3.1c)

where

p = a2~p0 + ~p~2 + a 2 ( P i o + a2 P n

= a2 + R2+... as a -> oo

and where />y. fy and vtj are functions of y alone. It is worth remarkingon the very small size of the second harmonic terms. The relevant scalingsarise as a result of the findings of section 2 that in expansions (2.4) theharmonic terms #22 and T22 vanish. Calculation of O(e4) terms in (2.4)then reveals that in our intermediate-strength expansions (3.1) the secondhanriQnic_quantities_vanish_down_to_the orders-of-a-indicated. It-is the~verysmall size of these terms which allows the subsequent analytical progress.

On substitution of expansions (3.1) into the governing equations (2.1) wefind that the leading order vortex quantities satisfy

,1«io = O, t>io = 0io ^ d "io = Pio- (3.2a, b,c)dy

552 S. LEWIS ETAL.

The mean pressure field is determined by the leading order v-momentumequation (2.1c) to give

(3.3a, b)

whilst the leading order mean relations arising from the energy and^-momentum equations are automatically satisfied. The second order tem-perature term is derived from the mean flow components of the energyequation

^ = \0\0~~A2, (3.4)dy 2

where A 2 is the constant of integration which, on applying the boundaryconditions fy = 0 on y = 0 and y = 1, is given by

<(3.5)

At next order the equations for the second order vortex terms v\2 and u\2

derived from (3.1) and (2.1) lead to a consistent solution only if

- f = 0

subject to

on y = 0,1.

(3.6a)

(3.6b)

We note that (3.6a) may be rescaled to obtain the corresponding equationfor the Bdnard problem found in (8). This second order nonlinear equationshows that the correction to the mean flow 62 and the fundamental term0io are determined simultaneously rather than sequentially (cf. usual weaklynonlinear theory). Analytical solution of (3.6) is possible and if E(m) andK(m) denote the elliptic integrals of the first and second kind respectivelythen we can write 6\Q parametrically as

s:

K{m) = I

m

+m m (3.7c)

(3.7d)


Equations of the form (3.6) have been previously derived by Knoblochand Proctor (13) and Proctor and Holyer (14) in the context of convectioncells in double diffusive systems. Proctor and Holyer examined the planformselection mechanisms for tall thin modes and used two-scale methodsto derive nonlinear evolution equations for interacting roll solutions. Theparametric form of (3.7) admits multiple solutions and, for small amplitudes,these match directly onto the weakly nonlinear solutions of section 2. (Thephysically most important weakly nonlinear mode, in the sense that itoccurs at the least value of RN, is as given by (2.5). However, there is asequence of higher modes in which T\\ and <p\\ are proportional to sinnny,n = 2,3,....) When R2 is large, the parameter m is very close to unity andthe sn function in (3.7c) is then approximately constant away from theboundaries. Multiple solutions continue to exist in this limit but they differfrom each other by only exponentially small quantities.

Although (3.7) is the formal solution of (3.6), it is not particularlyilluminating and, for the range of values 2n2 < R2 < oo, system (3.6) can beeasily solved using a standard fourth order Runge-Kutta numerical techniquecoupled to a multiple shooting code. The_dependence of A2 on R2 is shownin Fig. la and the structures of 6\Q and 02 are illustrated in Figs lb, c. Forvalues of R2 close to 2JT2 an expansion of the form

R2 = In2 + r2S +..., ~A2 = a2S +..., (3.8a, b)

0,o = S* (Bio + SBl2 +...) sinny +..., 62 = -(B2O5 sin27ry)/87r +...,

(3.8c, d)gives B2

0 = 4a2 = 8r2, which matches with the structure of the appropriateweakly nonlinear solution described in section 2. For large values of R2 thenumerical solutions show the emergence of the distinctive asymptotic stateand it may be shown, using (3.4) to (3.6), that as R2 —• oo,

A2 ~ j^R2 - R2 + exponentially small terms, (3.9a)

and

#io = —^[tanh^R2y + tanh£/?2(l — y) — 1] + exponentially small terms2V2

(3.9b)is the appropriate composite solution of (3.6a) for 0 < y < 1. Furthermore itis straightforward to show that

&2(y) ~ Riiy — i ) (3.9c)

across the majority of 0 < y < 1. However, as this solution does not satisfy therequisite conditions on the endwalls, boundary layers need to be introducedthere. The asymptotic prediction (3.9a) is shown in Fig. la and is seen tobe in excellent agreement with the numerical solution of (3.6a) once R2 isgreater than approximately 40.

55470-,

60-

50-

40-

30-

20-

10-

0

S. LEWIS ETAL

(a)

Asymptotic

15 20—i—25

— i —30

— i —

35—i

40 45

30-1

-200 0 OS ! 0

Fio. 1. (a) The dependence of A 2 on the Darcy-Rayleigh number R2- Thedotted line denotes the large R2 asymptotic prediction (3.9a). (b) Thestructure of the eigenfunction #io(y) corresponding to R2 = 20,25,30,40,50,70. (c) The form of the mean flow temperature term ^ (y ) for the same

choices of Ri as used in (b)


The large R2 results (3.9) and the numerical solutions confirm that thestrength of the cellular motion increases with R2. This numerical solutionalso demonstrates the formation of the expected boundary layers attachedto y = 0 and y = 1. These layers are of thicknesses 0(/?^') and so, whenR2~ a, they are of depth comparable to the wavelength of the convectioncells. Then we can expect that all the harmonics of the vortex disturbancewill be of equal amplitudes inside the boundary zones which will therebybecome fully nonlinear and only amenable to a numerical solution. It is thisnonlinear structure which we describe in section 4.

4. Strongly nonlinear convection

The results described above indicate that when the Rayleigh number isO(a) above the linear neutrally stable value the flow field develops a three-layer structure which comprises a central core supplemented by boundarylayers at each endwall. Based on the analysis detailed above we anticipateexpansions in the core of the forms

9 = 1 - y i ( l o + l i + )

+ a~462ocos2ax+..., (4.1a)

u = (u\o + a~xu\\ +...)sinax + a~3«2osin2aa:+..., (4.1b)

v = a(v\o + a~]vu +...)cosax + a~2v2ocos2ax+..., (4.1c)

p = a2~p0 + a~p~2 + a~x (p\o + a~XP\\ +...)cosaa:

with

and where /?,, %. "y etc. are functions of v only. The leading order vortexterms derived by substitution of (4.1) into (2.1) are given by

io = 0, uio = 0io, 1dy

On considering the leading order mean terms in the energy equation weobtain an equationfor #f(pwhicti can be integrated easily to give

0io = 2y, (4.3)

where y is a constant of integration which measures the core cellularstrength. The corresponding terms in the ^-momentum equation lead toexpressions for the mean flow pressure distribution but this is not required

556 S. LEWIS ETAL

here. The equations for the second order cellular terms, as in section 3, leadto the consistency condition

f = R,, <4.4,dy

thereby giving the mean temperature distribution. It is obviously impossibleto both solve (4.4) and simultaneously satisfy both temperature boundaryconditions at the endwalls. We appeal to the numerical results of Fig. lcand to the large Ri asymptotic solution of (3.6), which together suggest thatin the present strongly nonlinear convection the mean temperature correctionis a function which is odd about y = \- Therefore we take

h = R\iy-{) (4.5)

and introduce boundary layers in order to fulfill boundary conditions (2.2).As in the nonlinear B6nard convection work of (8) and the nonlinear Gortlervortex study by Bassom and Blennerhassett (15) all the harmonics of thecellular motion are of comparable size within these O(a~]) depth zones.

In the layer adjacent to v = 0 we define the 0(1) coordinates X, Y by

X = ax, Y = ay (4.6)

and the flow quantities are expanded as

u = aU(X, Y)+..., v = aV(X, Y)+..., (4.7a, b)

p = constant + aY +P(X,Y)+..., 9 = 1 +a~1T(X, Y) +.... (4.7c, d)

Substitution of (4.7) into (2.1) yields boundary layer equations of the form

UX+VY = O, (4.8a)

U = -Px, (4.8b)

V = -Pr + T + Ri, (4.8c)

UTX + VTY = Txx + TYY, (4.8d)

which must be solved subject to the conditions

V = 0, T = 0 on Y = 0 (4.8e,f)

along with the matching conditions

V-+2ycosX, U-+0, T -> -Y - ±rt, +2ycosX as Y -> oo.(4.8g.h,i)

The system of equations (4.8) constitutes a nonlinear eigenvalue problem fory as a function of the scaled Rayleigh number parameter R\.


4.1 Numerical solution of the boundary layer equationsIn order to obtain numerical solutions of (4.8) it is convenient to use a

streamfunction formulation for the equations. Thus we define

V = ¥*, U = - * K , (4.9)

and hence system (4.8) can be rewritten as

TX, (4.10a)

xx + TrY. (4.10b)

The symmetries inherent to the equations and boundary conditions implythat T and * may be expressed as Fourier cosine and sine series in Xrespectively. Therefore we look for solutions of the form

T = Y^Tj(Y) cos jX, * = ^VjiY) sin jX, (4.1 la, b)

where the Fourier series have necessarily been truncated after a finitenumber of modes N. The expansions (4.11) reduce equations (4.10) to aboundary value problem consisting of 2N + 1 coupled, nonlinear secondorder differential equations with y as an eigenvalue. The specification of theproblem is completed by the constraints

To = 0 , Tj = Vj = 0 l < j < N , (4.12a)

on the endwall Y = 0 together with matching conditions with the coresolution (4.5)

7b-* -Y-\RU 7, - • y, V i -»x as Y -> oo. (4.12b)

It should be noted at this stage that the appropriate boundary layer equationsin the thin zone attached to the other endwall at y = 1 can be easily scaledto take the form of (4.9) to (4.12). Thus the symmetry in the configurationassures us that we only need to solve one set of boundary layer equations.

The results presented here were obtained using a numerical procedureclosely allied to the Keller-box method. This method is very frequently usedto solve non-similar boundary layer flows in which the flow profiles varywith the streamwise coordinate. In such marching problems the governingequations are usually reduced to first order form in the cross-streamcoordinate and then approximated using central differences^ withjjig resultingdifferenee-equations-solved^using'Newton-Kaphson iteration. The Keller-boxmethod is one which has been applied to a very wide variety of problemsand the particular implementation which is used as the basic solver is onewhere a numerical differentiation procedure is used to generate die Jacobianiteration matrix rather than having to specify it explicitly (Rees and Pop(16), Rees and Bassom (17)). The numerical implementation used for the

558 S. LEWIS ETAL

present problem involved two fundamental adaptations of the basic Keller-box philosophy. First, the equations solved were not reduced to first orderform, but were left in second order form; this has the advantage of leadingto an increased numerical conditioning of the Jacobian matrix. Second, thecode was extended to include a procedure for computing eigenvalues. Theresulting Jacobian matrix is no longer block-tridiagonal for it is supplementedby an extra column and row of matrix blocks. The block tridiagonal matrixalgorithm was then adapted to allow for the modified configuration andthe entries were themselves computed numerically in preference to explicitspecification within the code. For the present calculation the 'streamwisecoordinate' used in the basic Keller-box method was replaced by theparameter R\, and therefore the structure of the program easily enabled usto perform a parameter sweep through a range of values of R\. In practicewe found it sufficient to truncate the computational domain at Y = Ya, = 40and to take N = 11 as the upper limit in the Fourier sums (4.11). Othervalues of N and Yx were also used in order to verify convergence ofresults. Taking N = l\ was found to be adequate for values of the Rayleighparameter R) up to about 200; beyond this value more modes are needed togive the required resolution in the X-direction. Our computations describedhere were based on /?] =£ 200 although the adjustments required in order toderive reliable results for higher Ri are quite straightforward to implement.

Figure 2 shows streamfunction plots corresponding to progressively largervalues of R\. In each subfigure the fluid is moving down the X = 0 sidewalland turning at the Y = 0 endwall before returning to the core. When R\ issufficiently small the solution is virtually symmetric about X = ^n and thenumerical solution is predominantly confined to the zeroth and first termsin (4.11); thus the solution of the last section is recovered in this R\ -*• 0limit. As R\ increases the solution loses its symmetry and the region withinwhich the fluid turns near Y = 0 becomes smaller. The fluid motion becomesparticularly intense near the Y = 0, X = n corner.

Figure 3 shows the mean temperature profile, TQ(Y), for a variety ofvalues of R\. All slopes shown in Fig. 3 asymptote to —1 when Y issufficiently large, but the solutions near Y = 0 are of interest. Near Y = 0the slope of the mean temperature profile becomes increasingly negative asR\ increases. When R] is sufficiently large the profile has a minimum valuenear to Y = 1. We note that this behaviour is qualitatively different from theresults presented by Blennerhassett and Bassom (8) who found that the meanflow component TQ(Y) in strongly nonlinear Bdnard convection appears todevelop a large region of near constant temperature before attaining thelarge- Y linear part of the solution. No evidence of a corresponding featureis observed here.

Figure 4 shows the variation of the parameter y with R\. Also shownare the large R} solution with N = 1 obtained in section 4.2 below and thenumerical solution of system (4.10) when N = 1 in the Fourier sums (4.11).

CONVECTION IN TALL POROUS CONTAINERS 559R, = 10 R, =200

FIG. 2. The lines of constant * for the solution of equations (4.8) when theFourier sums (4.11) are truncated with N = 11. In each case the domainshown i s O < X < ^ , 0 < y < 4 0 and the four plots correspond to R\ given by(a) 1, (b) 10, (c) 100, (d) 200. The interval A * between successive con-tours and the maximum values * „ , in each of the four cases is(a) A * = 0-032, *„» =0-321; (b) A * = 0-278, * m = 2-774; (c) A * =

1-303, *„» = 13-032; (d) A * = 1-949, *„» = 19-486

For this single harmonic truncation of (4.8) there is an offset between thenumerical and asymptotic results. The reason for this is that the asymptoticprediction derived in section 4.2 only considers the leading order behaviourand the ratio between successive terms in the expression for /?] wheny » 1 decays only very slowly. Thus we can only expect a good agreementbetween numerical and asymptotic predictions at extremely large values ofY (far greater than those considered here). However, the fact that the twomethods give qualitatively similar results based on only one term in theasymptotic expansion means that we can have confidence in results arisingfrom our numerical code.

4.2 Asymptotic solution of the boundary lower equations as R\ -* oo

We have seen that an asymptotic analysis shows that only the mean andfundamental modes of the vortex play a role in determining the core flow. Asan approximation this structure is imposed in the boundary layer as Y g e t s

large. This can be regarded as the single harmonic version of the equationssolved numerically in section 4.1 and provides some analytical results against

560 S. LEWIS ETAL

/?, = 10

= 125

Rt = 200

-1400 2 4 6 8 10 12 14 16 18 20

FIG. 3. The mean temperature profiles corresponding to (a) R\ = 10;(b) R{ = 50; (c) Rx = 125; (d) Rt = 200

which the numerical results described above may be compared. The requisiteequations are

V,"- V, = -7", and T'{ - (2V,2 + 1)7", = - ( 2 y 2 + 1)V, (4.13a,b)

with the mean temperature term 7b given by the integration of

1 - y 2 ) - l . (4.13c)

30-0

r1006-0

30-

1-00-6

0-3-

0-1

J V =

Asymptotic

N= 11

7 10 20 40 70 100 200

FIG. 4. The variation of y with R\. The thin solid line denotes the resultsobtained using truncation at 11 harmonics whilst the dotted line de-notes the results using only one harmonic. Finally, the thick solid line showsthe form of the one term asymptotic prediction (4.23) which is valid for


Equations (4.13) must be solved subject to the boundary conditions

7b = 7i = V i = 0 at y = 0 (4.14a)

along with the matching conditions

VuTx^y, TO-+-Y-\RX as Y -» oo. (4.14b)

In the region where Y = 0(1) we start from the hypothesis that

T\ = yT]0(Y) + O(y~x) +..., V, = yVw(Y) +... (4.15a, b)

which, on substitution into (4.13a,b), yield

7io Vio = 1, V"o - V10 = -7T- (4.16a, b)HO

After multiplication by V[Q equation (4.16b) can be integrated to give

= V12

0-21nV10- 1, (4.17)

where the constant of integration has been determined by insisting thatV10 -*• 1 as Y -* oo.

An analysis of (4.17) in the limit / —• 0 demonstrates that

V2YV-\nY+... (4.18)Ho

and the expression for Ti0 given by (4.16a) indicates the existence of aviscous region of depth S = O(y~i(lny)~*). Here we define

Y (4.19)

with Y ~ 0(1), and we write

, V, = yS(-\n8)l V +.... (4.20a,b)

Conditions (4.14a) and (4.14b) require that V = f = 0 at Y = 0 andV -»• y/2Y, f ->• (V2Y)'1 as Y - • oo. On substitution of (4.20) into (4.13)it is then straightforward to show that

74Yf 2y/2Y, —£ = 2(VT-l). (4.21a,b,c)dY2 dY

If we now define f = 2~^T* and Y = 2~^Y* then equation (4.21b) can bewritten

2 ( r ) r = _ 2 n ( 4 2 2 )

and is subject to T* = 0 on Y* = 0 and T' -* (\/Y*) as Y* -*• oo.

562 S. LEWIS ETAL

It follows from the boundary conditions (4.14b) together with (4.13c) and(4.20) to (4.22) that

Rl = -21/ASy1 [ ( r r - l ) d r + . . . = 3-00y3/2Gnyr1/4+- (4-23)J

and the comparison between this asymptote and the corresponding numericalsolution using only one harmonic is shown in Fig. 4.

5. Discussion

In this work we have examined the structure of steady strongly nonlinearvortices in a saturated porous medium within a tall, large aspect ratio verticalcontainer. The main result we have is that the previously found stronglynonlinear modes in Gtirtler flow and Be'nard convection have a directcounterpart in this porous medium configuration. However, there is a crucialdifference between our work here and some of the other studies which maywell have implications concerning the possible physical importance of themodes. In the GOrtler analysis, it is found that for large vortex wavenumbera the linear neutral GCrtler number Gs is O(aA) and fully nonlinearmodes can only arise once the GOrtler number (which plays a role akin tothat of the Rayleigh number here) satisfies (G - GN)/GN = 0(1). For dieBdnard problem discussed in (8), strongly nonlinear vortices appear once(R — RN)/RN = O(a~l), where RN is the linear neutral Rayleigh number.As RN — O(a*) the nonlinear domain is approached at parameter valuesO(a3) above the linear neutral stability curve. This should be contrasted withthe present formulation in section 4. Equation (4.1e) indicates that stronglynonlinear motions are possible at distances only O(a) above linear neutrality.Thus the nonlinear vortices discussed in this paper occur quite close to theneutral curve and it is therefore plausible that they are of significant practicalimportance. It must be recalled, however, dial we are strictly only entitled todraw this conclusion for modes within a box whose horizontal dimensionsare much shorter than its height. In cases where this length disparity isabsent we cannot legitimately appeal to our strongly nonlinear analysis forthe preferred modes will not be of short wavelength and a proper descriptionof the flow requires direct computations using the full equations.

Fully nonlinear vortices in the tall container acquire the distinctive core-boundary layer structure. The vortices in die central core can be deducedby solving simple consistency conditions which reflect the fact diat themean flow and first harmonic component are closely coupled together. Theboundary layers are necessary in order to ensure that die core solutionsare modified appropriately so as to satisfy the imposed conditions on theendwalls of the container. These boundary layers necessitate a numericalsolution which suggests that a well-defined vortex configuration must evolveas die intensity of the modes increases yet further. This next stage ofnonlineariry has yet to be identified in either the present porous medium


problem or the study of (8). In order to describe this stage it is necessaryto ascertain the formal asymptotic solution of (4.10) to (4.12) in the large-)'limit and preliminary studies suggest that this is a formidable task.

Lastly, it must be remembered that here we have concentrated on steadyvortices. It might be argued that time-dependent modes cannot be ignoredand, of course, for a complete description of the problem this should beconsidered. However, we note that the standard weakly nonlinear analysissummarized in section 2 shows that our flow is supercritically stable: thatis, according to usual Stuart-Landau theory it evolves to a stable finiteamplitude steady state. Naturally, we cannot therefore definitely concludethat our highly nonlinear steady motions must constitute the preferred modebut it would not be surprising if this turned out to be the case. A completelynonlinear unsteady computation is beyond the scope of the present workbut a natural extension of our study would be an investigation into thepossibility of any time-dependent modes.

Acknowledgements

We thank the referees for their useful comments on this paper. The firstauthor is grateful to the EPSRC for financial support whilst this work wascarried out.

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1992).3. D. A. S. REES, J. theor. appl. Mech. to appear.4. J. T. STUART, / Fluid Mech. 9 (1960) 353-370.5. J. WATSON, ibid. 9 (1960) 371-389.6. E. PALM, J. E. WEBER and O. KVERNVOLD, ibid. 54 (1972) 153-161.7. P. HALL and W. D. LAKIN, PTOC. R. SOC. A 415 (1988) 421-444.8. P. J. BLENNERHASSETT and A. P. BASSOM, IMA J. appl Maths 52 (1994)

51-77.9. A. P. BASSOM and K. ZHANG, Geophys. Astwphys. Fluid Dynamics 76 (1994)

223-238.10. D. A. S. REES and D. S. RiLEY, Proc. R. Soc. A 421 (1989) 303-339.11. and , J. Fluid Mech. 199 (1989) 133-154.12. and J. L. LAGE, Int. J. Heat Mass Transfer 40 (1997) 111-121.13. E. KNOBLOCH and M. R. E. PROCTOR, J. Fluid Mech. 108 (1981) 291-316.14. M. R. E. PROCTOR and J. Y. HOLYER. ibid 168 (1986) 241-253.15. A. P. BASSOM and P. J. BLENNERHASSETT, Proc. R. Soc. A 439 (1992)

317-336. _ _ _16. ©. ArSrREES-and-l: POT, Trans, porous Media 20 (1995) 223-234.17. and A. P. BASSOM, Int. J. Engng. Set. 34 (1996) 113-124.

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