Noname manuscript No.
(will be inserted by the editor)
Onset of Convection in a Triangular Porous Prism
with Robin-Type Thermal Wall Condition
Peder A. Tyvand · Jonas Kristiansen
Nøland
Received: date / Accepted: date
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 fea-
sible. The present eigenvalue problem results in a non-normal mode behaviour
in the horizontal cross-sectional plane. Further, it is identified that the stagna-
tion points for the horizontal flow are displaced from the extremal points of the
temperature perturbation, for both symmetric and antisymmetric eigenfunc-
tions. 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 ra-
tios. 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
confirms 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.
Keywords Convection · Cylinder · Porous medium · Vertical cylinder
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 (Corresponding Author)
Faculty of Information Technology and Electrical Engineering
Norwegian University of Science and Technology
E-mail: Jonas.k.Noland@ntnu.no
2 Peder A. Tyvand, Jonas Kristiansen Nøland
1 Introduction
The Horton-Rogers-Lapwood (HRL) problem for the onset of convection in a
horizontal porous layer has a two-dimensional (2D) structure for the mathe-
matical solution, even though the physical problem is 3D. The general solution
is a superposition of individual 2D Fourier modes (Horton and Rogers [1], Lap-
wood [2]). Any Fourier mode of convection onset is 2D in a vertical (x, z) plane
aligned with its wave number vector. This is a trivial degeneracy in dimension
due to the lack of boundaries in the horizontal plane.
The HRL problem has two types of degeneracy. (i) The solution in the
vertical direction is a normal mode. (ii) A Fourier element of the solution in
the horizontal direction is by definition a normal mode. Since normal modes
are governed by a second-order equation, these degeneracies are essentially
reductions from 4th order to 2nd order for the full eigenvalue problem.
The challenges of bringing more generality and physical realism into the
HRL solution are two-fold: (1) To provide the need for a 3D solution, by con-
sidering a finite 3D porous medium. A vertical cylinder is a natural case to
consider (Wooding [3]). (2) To provide mathematical solutions that are not of
the normal-mode type. Any solution that is not of normal-mode type confirms
that the eigenvalue problem is a genuine fourth-order problem. As soon as a
normal-mode type of solution applies, the problem reduces to an essentially
second-order problem. A number of papers have been written on porous cylin-
ders, with normal modes as the natural starting point. Beck [4] and Zebib [5]
carried out the details of the theory by Wooding [3], for a rectangular box
and a circular cylinder, respectively. Wooding [3] had pointed out the neces-
sity of degeneration in the boundary condition, where thermal and mechanical
conditions coincide mathematically.
Tyvand and Storesletten [6] developed from first principles the restrictions
for the normal-mode class of solutions for vertical cylinders. They solved the
problem of a vertical cylinder with a triangular cross-section. Only the simple
case of a right-angle isosceles cylinder was considered. The equilateral triangle
is another cross-section for which exact analytical eigenfunctions can be found
of the Helmholtz equation. This more complicated solution is known from the
theory of vibrating membranes in elasticity. Barletta and Storesletten [7] have
written the only paper where a cylinder with an elliptical cross-section has
been studied. For mathematical convenience, they chose the Dirichlet condi-
tions of open/conducting cylinder walls instead of the classical case of imper-
meable/insulating cylinder walls.
There exists only a handful of papers that treat the onset problem in
a 3D porous medium where the normal-mode type of spatial dependence is
challenged. Haugen and Tyvand [8] wrote the first paper on a vertical porous
cylinder full-normal dependency in the radial direction of the circular cross-
section. This model was generalized by Nyg˚ard and Tyvand [9] to account
for partial conduction and partial penetration at the cylinder walls. In the
present paper, we study the same type of model for a triangular cylinder, with
the Robin-type condition of partially conducting cylinder walls.
Title Suppressed Due to Excessive Length 3
Our problem contrasts the circular cross-sections, where the azimuthal sep-
arability of the eigenfunctions restricts the non-normal mode dependency to
the 1D radial direction only. Non-normal modes in 1D can be studied analyti-
cally. Our triangular cross-section results in 2D non-normal modes that cannot
be separated in space, where no analytical methods are known.
The study of 1D non-normal modes for the HRL problem started with Nield
[10] who solved the onset problem with all possible Dirichlet and/or Neumann
conditions for the temperature perturbation and the vertical velocity at the
lower and upper boundaries. Barletta et al. [11] extended Nield’s analysis for
the HRL problem by allowing general Robin conditions at the lower and upper
boundaries. 2D non-normal modes for the HRL problem is a new challenging
topic. Tyvand et al. [12] has solved a 2D problem of non-normal modes in a
vertical rectangle. In the present paper, we carry this type of analysis further
by considering a fully 3D problem of a vertical cylinder, with normal-mode
dependency in the vertical direction. By spatial separation, we will have a
non-normal modes dependency the horizontal cross-section plane.
The significance of the present paper is that it is the first theoretical study
of a three-dimensional Darcy-B´enard eigenvalue problem with full non-normal
dependency over the horizontal cross-section of a cylinder. The present type of
modeling has the disadvantage that no analytical solution methods are known.
The advantage is that more physical realism can be included in the boundary
conditions, compared with the implicit degeneracy of the existing models based
on normal modes.
2 Mathematical formulation
A three-dimensional porous medium is bounded by horizontal planes z = −h/2
and z = h/2. The porous medium is homogeneous and isotropic. Cartesian
coordinates (x, y, z) are introduced. The z axis is directed vertically upwards.
We will consider a vertical cylinder, noting that the linear theory has been
established both for impermeable insulating walls (Wooding [3]), and for open
conducting walls (Barletta and Storesletten [7]). We will here develop the
general linear theory for vertical cylinders with impermeable and thermally
conducting walls, and we will perform calculations for the case of an isosceles
triangular cylinder. We will allow a thermal Robin condition for the cylinder
wall, whereby the standard case of impermeable adiabatic walls (Wooding [3])
arises as a limit case.
The velocity vector v has Cartesian components (u, v, w). The temperature
field is represented as T (x, y, z, t) with t denoting time. In the undisturbed
state, the lower plane z = −h/2 is kept at a constant temperature T = T
0
,
and the upper plane z = h/2 is kept at a constant temperature T = T
0
− ∆T .
Here ∆T is a positive temperature difference. The gravitational acceleration
g is written in vector form as g.
4 Peder A. Tyvand, Jonas Kristiansen Nøland
The standard Darcy-Boussinesq equations for free thermal convection in a
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.
The lower and upper boundaries support a given temperature difference
∆T across the porous layer. The undisturbed basic state of pure conduction
has the boundary temperatures
T = T
0
+ ∆T, at z = 0 − h/2, (4)
T = T
0
, at z = h/2. (5)
T
0
is a reference temperature. These boundary temperatures will be main-
tained also when the basic state is disturbed with infinitesimal perturbations.
The kinematic conditions for the impermeable lower and upper boundaries
are
w = 0, at z = −h/2, (6)
w = 0, at z = h/2. (7)
Figure 1 shows definition sketches for a vertical enclosure with the trian-
gular cross-section which will be our calculated example.
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, v, w) → (u, v, 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,
(8)
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.
Title Suppressed Due to Excessive Length 5
Fig. 1 Definition sketch of a vertical porous enclosure with triangular cross-section. The
investigated mid plane is indicated.
The dimensionless governing equations can then be written
v + ∇P − RT k = 0. (9)
∇ · v = 0 (10)
∂T
∂t
+ v · ∇T = ∇
2
T, (11)
with the boundary conditions of impermeable and conducting lower and upper
horizontal planes
w = T − 1 = 0, z = −1/2, (12)
w = T = 0, z = 1/2. (13)
Here the Rayleigh number R is defined as
R =
ρ
0
gβK∆T h
µκ
m
. (14)
