Journal ArticleDOI

# A Non-normal-Mode Marginal State of Convection in a Porous Rectangle

01 Jun 2019-Transport in Porous Media (Springer Netherlands)-Vol. 128, Iss: 2, pp 633-651

Abstract: The fourth-order Darcy–Benard eigenvalue problem for onset of thermal convection in a 2D rectangular porous box is investigated. The conventional type of solution has normal-mode dependency in at least one of the two spatial directions. The present eigenfunctions are of non-normal-mode type in both the horizontal and the vertical direction. A numerical solution is found by the finite element method, since no analytical method is known for this non-degenerate fourth-order eigenvalue problem. All four boundaries of the rectangle are impermeable. The thermal conditions are handpicked to be incompatible with normal modes: The lower boundary and the right-hand wall are heat conductors. The upper boundary has given heat flux. The left-hand wall is thermally insulating. The computed eigenfunctions have novel types of complicated cell structures, with intricate internal cell walls.
Topics: , Heat flux (54%), Convection (52%), Finite element method (51%), Normal mode (51%)

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Noname manuscript No.
(will be inserted by the editor)
A non-normal mode marginal state of convection in
a porous rectangle
Peder A. Tyvand · Jonas Kristiansen
Nøland · Leiv Storesletten
Abstract The 4th order Darcy-B´enard eigenvalue problem for onset of ther-
mal convection in a 2D rectangular porous box is investigated. The conven-
tional type of solution has normal-mode dependency in at least one of the
two spatial directions. The present eigenfunctions are of non-normal mode
type both in the horizontal and the vertical direction. A numerical solution is
found by the ﬁnite element method, since no analytical method is known for
this non-degenerate fourth-order eigenvalue problem. All four boundaries of
the rectangle are impermeable. The thermal conditions are hand-picked to be
incompatible with normal modes: The lower boundary and the right-hand wall
are heat conductors. The upper boundary has given heat ﬂux. The left-hand
wall is thermally insulating. The computed eigenfunctions have novel types of
complicated cell structures, with intricate internal cell walls.
1 Introduction
The onset of convection in a horizontal porous layer was ﬁrst studied by Horton
and Rogers (1945) and Lapwood (1948). The reputation of this HRL problem
P. A. Tyvand
Faculty of Mathematical Sciences and Technology
Norwegian University of Life Sciences
1432
˚
As, Norway
Tel.: +47-67231564
E-mail: Peder.Tyvand@nmbu.no
J. K. Nøland
Faculty of Information Technology and Electrical Engineering
Norwegian University of Science and Technology
E-mail: Jonas.K.Noland@ntnu.no
L. Storesletten
Department of Mathematics
University of Agder
E-mail: Leiv.Storesletten@uia.no

2 Peder A. Tyvand et al.
is that it is the simplest mathematical problem of thermomechanical insta-
bility. Still there are mathematical challenges, especially those related to the
fact that the eigenvalue problem is of fourth order in space. Since the bihar-
monic operator does not separate in orthogonal coordinate systems, there is no
general closed-form analytical eigenfunction available, even for a rectangular
geometry.
In the literature on the HRL eigenvalue problem, the boundary conditions
have tacitly been restricted to make analytical solutions possible. Wooding
(1959) circumvented the restrictions of the non-separability of the biharmonic
operator for vertical porous cylinders, by a normal-mode assumption that re-
stricted the cylinder wall conditions to be equivalent to internal cell walls.
Tyvand and Storesletten (2018) veriﬁed Wooding’s normal-mode assumption
from ﬁrst principles.
A normal mode is the spatial dependency of free vibrations according to
the linear wave equation. Inserting the time dependency of a single oscilla-
tory frequency changes the wave equation to a Helmholtz eigenvalue equation
for the free vibrations, with the wave number as the eigenvalue. This is the
equation valid for free vibrations of strings in 1D, of elastic membranes in
2D and for sound waves in 3D. Normal modes are thus linked to diﬀerential
equations that are of second order in space, where any choice of homogeneous
boundary conditions may ﬁt automatically. This is not the case for higher-
order systems, where the full set of independent boundary conditions prevents
the general validity of normal modes. Still the normal modes are in common
use for analytical eigenfunctions of higher-order systems. This can only be
justiﬁed by applying degenerate boundary conditions, as was demonstrated
by Wooding (1959) for convection in a vertical porous cylinder. The details
of Wooding’s theory were carried out by Beck (1972) and Zebib (1978), for
rectangular and circular cylinders, respectively.
With normal modes valid over the horizontal cylinder cross-section, there
is no need for normal modes in the third (vertical) direction in order to de-
rive analytical eigenfunctions. This fact was demonstrated by Nield (1968, Ap-
pendix), who solved all the Dirichlet and Neumann type fourth-order problems
onset of convection in a porous medium between two horizontal planes. Wang
(1998, 1999) exempliﬁed how the cases solved by Nield (1968) can be adjusted
to be valid for cylinders with with normal-mode compatible wall conditions.
The wave number in Nield’s solutions will then appear as the eigenvalue of
the Helmholtz equation that applies to the cross-section of the cylinder. There
are two classes of normal-mode compatible eigenvalue problems for cylinders:
The Neumann type of impermeable/insulating conditions (Wooding 1959) and
Dirichlet type of conducting walls that are open to inﬂow/outﬂow. Barletta
and Storesletten (2015) solved the cylinder problem with these Dirichlet con-
ditions, both for a circular cross-section and an elliptical cross-section.
Non-normal modes in the horizontal direction is an interesting possibility
that was ﬁrst explored by Nilsen and Storesletten (1990). Lyubimov (1975)
had suggested but not derived such solutions. To solve the problem analyt-
ically requires normal modes in the vertical direction, following Horton and

A non-normal mode marginal state of convection in a porous rectangle 3
Rogers (1945). A non-normal mode solution for a full 3D problem was given
by Haugen and Tyvand (2003), for a circular cylinder with conducting walls.
In the tangential direction, continuity and periodicity gives normal modes in
terms of the azimuthal angle coordinate. Again the analysis rested on normal
modes in the vertical direction.
What will happen to the HRL eigenvalue problem for a rectangular box
when the set of boundary conditions fails to be compatible with normal modes,
in the horizontal as well as in the vertical direction? This is the topic of the
present paper, and for simplicity we consider only a 2D rectangular box. We
hand-pick the boundary conditions so that two out of four boundaries have
conditions that are compatible with normal modes, only to be overruled by the
other boundaries that prevent normal modes from contributing to the solution.
The resulting eigenfunction is a fully non-normal mode solution in 2D, and it
can only be determined numerically, since no analytical methods are known.
When we study non-normal mode convection cells, we search for qualitative
features of the cells: The shapes of the internal cell walls, the recirculation
patterns and the relationships between streamlines and isotherms. We expect
more complicated cell patterns than the well-known HRL cell, which has a full
normal-mode type sinusoidal spatial dependency.
Earlier analytical work on the onset of convection in homogeneous and
isotropic porous rectangles has not been able to identify ﬂow cells that have
irregular curved shapes of the internal cell walls. A tacit assumption has been
a normal-mode type of behavior either in the horizontal or vertical direction.
Thereby the cells are forced to have vertical internal cell walls, as well as
horizontal internal cell walls when higher vertical modes are taken into con-
sideration. The literature on the topic tends to assume that internal cell walls
in a marginal linearized state of convection onset in 2D rectangular porous
rectangles will always have rectangular internal cell walls, parallel to the ex-
ternal rectangular boundaries. However, this assumption has has never been
proven, and in the present paper we will demonstrate that it is incorrect.
In linearized theory of convection onset, one looks for the critical Rayleigh
number and the corresponding preferred cell width. The present numerical
solutions will produce precise Rayleigh numbers, but there will be no precise
concept of cell width. The shapes of the internal ﬂow cells and thermal cells
will diﬀer from one another, and their internal cell walls may have complicated
curved shapes that defy the wave number concept.
2 Mathematical formulation
A porous medium is considered, with 2D ﬂow in the x, z plane. Physically this
corresponds to a 3D porous medium with impermeable and insulating walls
y = 0 and y = b, where the distance b is suﬃciently small compared with
the vertical length scale, implying that the preferred mode of free convection
will be 2D. The z axis is directed vertically upwards. The porous medium is
homogeneous and isotropic. The assumption of 2D ﬂow in the x, z plane is

4 Peder A. Tyvand et al.
valid also for a 3D porous medium with thickness b in the y direction, if two
constraints are met: (i) The thermomechanical wall conditions at y = 0 and
y = b are those of impermeable and thermally insulating walls. (ii) The cell
width in the x direction must be suﬃciently large compared with b.
From now on we assume these conditions for 2D ﬂow in the x, z plane to
be met without specifying them further. Storesletten and Tveitereid (1991)
investigated a problem where the range of validity of this assumption of 2D
ﬂow has been scrutinized.
The velocity vector v has Cartesian components (u, w). The temperature
ﬁeld is T (x, z, t), with t denoting time. The undisturbed motionless state has
uniform vertical temperature gradient. The gravitational acceleration g is writ-
ten in vector form as g.
The standard Darcy-Boussinesq equations for convection in a homogeneous
and isotropic porous medium can be written
p +
µ
K
v + ρ
0
β (T T
0
)g = 0, (1)
· v = 0, (2)
(ρc
p
)
m
T
t
+ (ρc
p
)
f
v · T = λ
m
2
T. (3)
In these equations, p is the dynamic pressure, β is the coeﬃcient of thermal
expansion, ρ = ρ
0
is the ﬂuid density at the reference temperature T
0
, µ
is the dynamic viscosity of the saturating ﬂuid, K is the permeability, c
p
is
the speciﬁc heat at constant pressure, and λ
m
is the thermal conductivity of
the saturated porous medium. The subscript m refers to an average over the
solid/ﬂuid mixture, while the subscript f refers to the saturating ﬂuid alone.
We consider a 2D porous medium with a rectangular shape, bounded by
the four sides
x = 0, x = l, z = 0, z = h. (4)
The lower boundary is taken to be impermeable and perfectly heat-conducting
w = 0, T T
0
= 0, at z = 0, (5)
while the upper boundary is taken to be impermeable and with constant uni-
form heat ﬂux
w = 0,
T
z
=
∆T
h
, at z = h, (6)
where ∆T is the temperature diﬀerence across the layer in its undisturbed
state. T
0
is a reference temperature The left-hand boundary is assumed to be
impermeable and thermally insulating
u = 0,
T
x
= 0, at x = 0, (7)
while the right-hand boundary is assumed to be impermeable and perfectly
conducting
u = 0, T T
0
=
∆T
h
z, at x = l. (8)

A non-normal mode marginal state of convection in a porous rectangle 5
These four boundary conditions are hand-picked to violate in a mild way
the requirements for normal mode solutions, both in the horizontal and the
vertical direction. These conditions are designed to be as simple as possible.
The lower and the left-hand boundary condition alone are in themselves com-
patible with normal modes. The violation of normal modes is due to the upper
boundary condition as far as the vertical dependency of the eigenfunction is
concerned. The violation of normal modes in due to the right-hand boundary
condition as far as the horizontal dependency is concerned.
The upper right-hand corner (l, h) is thus the troublesome corner where
two non-normal mode type of conditions meet. Not surprising, we will discover
small convergence problems near the upper-right hand corner, and we will ap-
ply a ﬁner numerical grid around this corner than elsewhere in the rectangular
ﬂuid domain.
2.1 Dimensionless equations
From now on we work with dimensionless variables. We reformulate the math-
ematical problem in dimensionless form by means of the transformations
1
h
(x, z) (x, z),
h
κ
m
(u, w) (u, w), h ,
1
∆T
(T T
0
) T,
K
µκ
m
(p p
0
) p,
(ρc
p
)
f
κ
m
(ρc
p
)
m
H
2
t t,
(9)
where κ
m
= λ
m
/(ρ
0
c
p
)
f
is the thermal diﬀusivity of the saturated porous
medium. We denote the vertical unit vector by k, directed upwards.
The dimensionless governing equations can then be written
v + p R T k = 0. (10)
· v = 0 (11)
T
t
+ v · T =
2
T, (12)
with the boundary conditions at the lower and upper boundaries
w = T = 0, z = 0, (13)
w =
T
z
+ 1 = 0, z = 1. (14)
The boundary conditions at the sidewalls are
T
x
= u = 0, x = 0, (15)
T = u = 0, x = l/h. (16)
We have introduced the Rayleigh number R deﬁned as
R =
ρ
0
gβK∆T h
µκ
m
. (17)

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Journal ArticleDOI
Abstract: The onset of Darcy–Benard convection in an unlimited horizontal porous layer is studied theoretically. The thermomechanical boundary conditions of Dirichlet or Neumann type at the lower and upper plane are switched from one type to another, at certain values of the horizontal x-coordinate. A semi-infinite portion of the lower boundary is defined as thermally conducting and impermeable, while the remaining boundary is open and with given heat flux. At the upper boundary, the same thermomechanical conditions are applied, but with a relative spatial displacement L and in the opposite spatial order. A domain of local destabilization around the origin is generated between the lines of discontinuity $$x = \pm \,L/2$$ . The marginal state of convection is triggered centrally, while it is penetrative in the domains exterior to the central domain. The onset problem is solved numerically, with a general 3D mode of disturbance, but 2D disturbances are preferred in most cases. The critical Rayleigh number is given as a function of the dimensionless gap width L and the wavenumber k in the y direction along the lines of discontinuity in the boundary conditions. An asymptotic formula for 2D penetrative eigenfunctions is shown to be in agreement with the numerical results.

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Abstract: A special case of the fourth-order Darcy–Benard problem in a two-dimensional (2D) rectangular porous box is investigated. The present eigenfunctions are of non-normal-mode type in the horizontal and vertical directions. They compose a time-periodic wave with one-way propagation out of the porous rectangle. Asymmetry in the horizontal direction generates an oscillatory time dependence of the marginal state, similar to Rees and Tyvand (Phys Fluids 16:3706–3714, 2004a). No analytical method is known for this non-degenerate eigenvalue problem. Therefore, the problem was solved numerically by the finite element method (FEM). Three boundaries of the rectangle are impermeable. The right-hand wall is fully penetrative. The lower boundary and the left-hand wall are heat conductors. The upper boundary has a given heat flux. The right-hand wall is thermally insulating. As a result, the computed eigenfunctions show complicated periodic time dependence. Finally, the critical Rayleigh number and the associated angular frequency are calculated as functions of the aspect ratio and compared against the case of normal modes in the vertical direction.

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### Cites background from "A Non-normal-Mode Marginal State of..."

• ...Tyvand et al. (2019) designed and solved numerically a two-dimensional (2D) problem in a rectangle of the HRL type which is fully non-degenerate, which means that the onset is governed by a non-normal-mode solution where the horizontal and vertical dependencies cannot be separated from one another....

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Journal ArticleDOI
Abstract: The analytical theory on Darcy–Benard convection is dominated by normal-mode approaches, which essentially reduce the spatial order from four to two. This paper goes beyond the normal-mode paradigm of convection onset in a porous rectangle. A handpicked case where all four corners of the rectangle are non-analytical is therefore investigated. The marginal state is oscillatory with one-way horizontal wave propagation. The time-periodic convection pattern has no spatial periodicity and requires heavy numerical computation by the finite element method. The critical Rayleigh number at convection onset is computed, with its associated frequency of oscillation. Snapshots of the 2D eigenfunctions for the flow field and temperature field are plotted. Detailed local gradient analyses near two corners indicate that they hide logarithmic singularities, where the displayed eigenfunctions may represent outer solutions in matched asymptotic expansions. The results are validated with respect to the asymptotic limit of Nield (Water Resour Res 11:553–560, 1968).

### Cites background from "A Non-normal-Mode Marginal State of..."

• ...Little is still known on non-separable eigenfunctions of non-normal mode type (Tyvand and Nøland 2019; Tyvand et al. 2019)....

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Abstract: The onset of thermal convection in two-dimensional porous cavities heated from below is studied theoretically. An open (constant-pressure) boundary is assumed, with zero perturbation temperature (thermally conducting). The resulting eigenvalue problem is a full fourth-order problem without degeneracies. Numerical results are presented for rectangular and elliptical cavities, with the circle as a special case. The analytical solution for an upright rectangle confirms the numerical results. Streamlines penetrating the open cavities are plotted, together with the isotherms for the associated closed thermal cells. Isobars forming pressure cells are depicted for the perturbation pressure. The critical Rayleigh number is calculated as a function of geometric parameters, including the tilt angle of the rectangle and ellipse. An improved physical scaling of the Darcy–Benard problem is suggested. Its significance is indicated by the ratio of maximal vertical velocity to maximal temperature perturbation.

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Abstract: This paper investigates a peculiar case of thermal convection in a vertical porous prism with impermeable and partially conducting walls. We facilitate the analysis in the numerical finite-element environment alongside with analytical considerations, in special cases where direct solutions are feasible. The present eigenvalue problem results in a non-normal-mode behaviour in the horizontal cross-sectional plane. Further, it is identified that the stagnation points for the horizontal flow are displaced from the extremal points of the temperature perturbation, for both symmetric and antisymmetric eigenfunctions. In addition, the corresponding normal-mode counterparts are provided from an analogy solution. We show that the critical Rayleigh number decreases with increasing Robin parameter values for all of the investigated aspect ratios. Finally, the influence of the aspect ratio on the critical Rayleigh number for the fully conducting wall case is identified. An asymptotic benchmark case of the Robin condition is validated from well-known analytical solutions which confirm the effectiveness of the predictions made in this paper. In fact, this is the first contribution that reports a three-dimensional geometry with a two-dimensional non-normal mode.

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H. K. Moffatt1Institutions (1)
Abstract: Some simple similarity solutions are presented for the flow of a viscous fluid near a sharp corner between two planes on which a variety of boundary conditions may be imposed. The general flow near a corner between plane boundaries at rest is then considered, and it is shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity. The ratios of dimensions and intensities of successive eddies are determined for the full range of angles for which the eddies exist. The limiting case of zero angle corresponds to the flow at some distance from a two-dimensional disturbance in a fluid between parallel boundaries. The general flow near a corner between two plane free surfaces is also determined; eddies do not appear in this case. The asymptotic flow at a large distance from a corner due to an arbitrary disturbance near the corner is mathematically similar to the above, and has comparable properties. When the fluid is electrically conducting, similarity solutions may be obtained when the only applied magnetic field is that due to a line current along the intersection of the two planes; it is shown that the effect of such a current is to widen the range of corner angles for which eddies must appear.

1,386 citations

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• ...A wellknown example of this type is the Moffatt vortices (Moffatt 1964), representing a corner singularity for a driven viscous cavity with no-slip condition along the walls....

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Journal ArticleDOI
01 Oct 1948
Abstract: It is shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law. The criterion for marginal stability is obtained for three sets of boundary conditions, and the motion described. If such convection occurs in a stratum through which a bore-hole passes, the usual method of calculation of the heat flow must be modified, but in general the correction will not be large.

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• ...The onset of convection in a horizontal porous layer was first studied by Horton and Rogers (1945) and Lapwood (1948)....

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Journal ArticleDOI
Abstract: The problem is considered of the convection of a fluid through a permeable medium as the result of a vertical temperature‐gradient, the medium being in the shape of a flat layer bounded above and below by perfectly conducting media. It appears that the minimum temperature‐gradient for which convection can occur is approximately 4π2h2μ/kgρ0α D2, where h2 is the thermal diffusivity, g is the acceleration of gravity, μ is the viscosity, k is the permeability, α is the coefficient of cubical expansion, ρ0 is the density at zero temperature, and D is the thickness of the layer; this exceeds the limiting gradient found by Rayleigh for a simple fluid by a factor of 16D2/27π2kρ0. A numerical computation of this gradient, based upon the data now available, indicates that convection currents should not occur in such a geological formation as the Woodbine sand of East Texas (west of the Mexia Fault zone); in view of the fact, however, that the distribution of NaCl in this formation seems to require the existence of ...

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• ...These numerical eigenfunctions contain several novel features that we hope will stimulate more research on this topic, which is a well-established one since the pioneering work by Horton and Rogers (1945)....

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• ...To solve the problem analytically requires normal modes in the vertical direction, following Horton and Rogers (1945)....

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• ...The onset of convection in a horizontal porous layer was first studied by Horton and Rogers (1945) and Lapwood (1948)....

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• ...Wang (1998, 1999) exemplified how the cases solved by Nield (1968) can be adjusted to be valid for cylinders with normal-mode-compatible wall conditions....

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##### Network Information
###### Related Papers (5)

C. W. Horton, F. T. Rogers

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01 Oct 1948

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