Abstract

The thermoconvective instability of the parallel vertical flow in a fluid saturated porous layer bounded by parallel open boundaries is studied. The open boundaries are assumed to be kept at constant uniform pressure while their temperatures are uniform and different, thus forcing a horizontal temperature gradient across the layer. The anisotropic permeability of the porous layer is accounted for by assuming the principal axes to be oriented along the directions perpendicular and parallel to the layer boundaries. A linear stability analysis based on the Fourier normal modes of perturbation is carried out by testing the effect of the inclination of the normal mode wavevector to the vertical. The neutral stability curves and the critical Rayleigh number for the onset of the instability are evaluated by solving numerically the stability eigenvalue problem.

1 Introduction

The emergence of the thermoconvective instability in a fluid system is a typical feature of a horizontal fluid layer or fluid saturated porous layer where conduction heat transfer changes into convection due to a vertical downward temperature gradient. This phenomenon is typical of the Rayleigh–Bénard system, where a horizontal fluid layer is bounded by isothermal planes kept at different temperatures yielding conditions of heating from below [1]. As is well known, a version of the Rayleigh–Bénard stability problem involving a layer of fluid saturated porous medium has been studied in the classical papers by Horton and Rogers [2] and by Lapwood [3]. Nowadays, the porous medium version of the Rayleigh–Bénard problem is either known as Darcy–Bénard problem or Horton–Rogers–Lapwood problem [46].

The onset of multicellular convection patterns is a typical phenomenon also for vertical fluid layers [79], where a side-heating configuration is induced by boundaries kept at different uniform temperatures. Interestingly enough, the onset of multicellular convection patterns destabilizing the conduction regime was ruled out in the case of a vertical porous slab. This conclusion is a consequence of a pioneering paper by Gill [10]. In fact, Gill proposed a rigorous proof of the absence of thermoconvective instability in a vertical porous slab saturated by a fluid and bounded by isothermal and impermeable sidewalls. Other authors further enhanced Gill's results by taking into account other effects or developing a nonlinear stability analysis [1115]. Only recently, evidence that thermoconvective instability can indeed arise in a vertical porous layer has been given [16]. The key argument is the observation that Gill's proof of stability relies on the assumption that the boundaries are impermeable. When this assumption is relaxed, then the thermoconvective instability is possible at sufficiently large Rayleigh numbers.

Most of the studies regarding the onset of the thermoconvective instability in a fluid saturated porous medium are based on the model of isotropic permeability. The permeability is considered as a scalar, while for anisotropic porous materials, it is modeled as a second-rank tensor. Anisotropy effects may be significant in several materials of engineering interest. Thus, some authors extended the knowledge on the thermoconvective instability in porous media by envisaging anisotropic materials [5,17,18]. Among these papers, we mention the studies by Kvernvold and Tyvand [19] and by Nilsen and Storesletten [20], where the Rayleigh–Bénard instability is reconsidered for an anisotropic porous medium, as well as the analysis carried out by Rees and Postelnicu [21], where the effect of the porous layer inclination to the horizontal is taken into account. Very recent developments on the onset of the thermoconvective instability in anisotropic porous media have been achieved by Straughan [22], by Storesletten and Rees [23] and by Naveen et al. [24]. The latter contribution is of special interest for the present paper as it provides an extension of Gill's analysis [10] to the case of an anisotropic porous medium.

The aim of this paper is to provide an extension of the study carried out by Barletta [16] by envisaging a situation where the porous material has an anisotropic permeability. More precisely, we will consider a vertical porous slab with open boundaries, modeled as isobaric planes, kept at unequal uniform temperatures. The permeability is considered as anisotropic, meaning that it is given by a second-rank tensor. The principal axes will be assumed to be transversal and longitudinal with respect to the plane slab, thus following the description adopted by Storesletten and Rees [23]. The stability of the basic parallel buoyant flow in the slab will be studied by adopting a modal analysis. The stability eigenvalue problem will be solved numerically by employing the shooting method. The effects of the anisotropy will be discussed for the neutral stability curves and for the critical values of the wave number and of the Rayleigh number.

2 Mathematical Model

We consider a vertical porous slab and a Cartesian coordinate system with the x-axis horizontal and perpendicular to the slab, the y-axis horizontal and parallel to the slab, and the z-axis vertical and parallel to the slab. The origin of the axes is in the slab vertical midplane. The slab is assumed to be infinitely wide in the y and z directions, while its thickness is L. A sketch of the slab cross section in the xz-plane is given in Fig. 1. The permeable side boundaries at x=±L/2 are kept at a uniform hydrostatic pressure and uniform temperatures T1 and T2, with T2>T1.

Fig. 1
A sketch of the vertical porous slab
Fig. 1
A sketch of the vertical porous slab

2.1 Governing Equations.

The fluid saturated porous medium is anisotropic with principal axes directed along the (x, y, z) axes. The local momentum balance equation is modeled according to Darcy's law including the buoyancy force, namely [5]
μu=K¯[P+ρ0β(TT0)g]
(1)
where the permeability tensor is expressed as
K¯=(KT000KL000KL)
(2)

Equation (2) is a special case of anisotropic permeability where one direction, the transverse x-axis, is selected out of the other two longitudinal y and z directions. This is typical of fibrous or layered porous media where the direction of the fibers or layers displays an hydraulic resistance different from that of the other directions [17,23].

According to the Oberbeck–Boussinesq approximation, the only effect of the temperature-dependent density is the buoyancy force so that the mass, momentum, and energy balance equations can be written as [5,23]
ux+vy+wz=0
(3a)
u=KTμPx
(3b)
v=KLμPy
(3c)
w=KLμ[Pzρ0βg(TT0)]
(3d)
σTt+uTx+vTy+wTz=α2T
(3e)
Here, σ is the ratio between the heat capacity of the saturated porous medium and that of the fluid. Following Barletta [16], the boundary conditions can be written as:
P=0,T=T1inx=L/2P=0,T=T2inx=L/2
(4)
In order to keep the complexity of the anisotropy model at its minimum, we assumed an isotropic thermal diffusivity, while the most general situation would imply, at the right-hand side of Eq. (3e), a term
·(α¯T)

instead. Here, α¯ stands for the second-rank diffusivity tensor accounting for the thermal anisotropy. Since the thermal diffusivity α in Eq. (3e) comes out from a volume averaging of the conductivities of the solid and the fluid, we consider this parameter as one including average over the transverse and longitudinal directions as well. This simplification, similar to that adopted by Storesletten and Rees [23], will reduce the number of dimensionless parameters governing the onset of the instability.

2.2 Dimensionless Quantities.

A nondimensional formulation of Eqs. (3) is achieved by defining
(x*,y*,z*)=(x,y,z)/L,t*=tα/(σL2)(u*,v*,w*)=(u,v,w)L/α,P*=PKL/(μα)T*=(TT0)/(T2T1)
(5)
where the asterisks denote the dimensionless quantities. Hereafter, such asterisks will be omitted for the sake of brevity, as we will only deal with dimensionless quantities. Thus, the nondimensional reformulation of Eqs. (3) and (4), based on the fields T and P, is given by
ξ2Px2+2Py2+2Pz2=RaTz
(6a)
TtξPxTxPyTy(PzRaT)Tz=2Tx2+2Ty2+2Tz2
(6b)
and
P=0,T=±1/2inx=±1/2
(7)
In Eq. (6), the anisotropy parameter ξ and the porous medium version of the Rayleigh number Ra are defined as
ξ=KT/KL,Ra=[ρ0gβ(T2T1)KLL]/(μα)
(8)

3 The Basic Flow

A basic flow solution driven by the buoyancy force is obtained in a stationary regime and with a purely vertical velocity field, namely
Pb=0,Tb=x,wb=Rax
(9)

The basic flow rate is zero. In fact, the integral of wb over the interval 1/2x1/2 is zero.

4 Stability Analysis

Perturbations of the basic solution can be defined as
P=Pb+P̃,T=Tb+T̃
(10)
We substitute Eq. (10) into Eqs. (6) and (7), we take into account Eq. (9), and we linearize the resulting equations by assuming that the strength of the perturbation is much smaller than that of the basic state. Thus, we obtain
ξ2P̃x2+2P̃y2+2P̃z2=RaT̃z
(11a)
T̃tξP̃x+RaxT̃z=2T̃x2+2T̃y2+2T̃z2
(11b)
with boundary conditions
P̃=0,T̃=0inx=±1/2
(12)

4.1 The Stability Eigenvalue Problem.

We assume normal mode perturbations given by
[P̃(x,y,z,t)T̃(x,y,z,t)]=[f(x)h(x)]ei(kyy+kzz)eλt
(13)

Here, (ky, kz) are the components of the wavevector whose modulus, k=(ky2+kz2)1/2, is the wavenumber. The growth rate of the perturbation is Re(λ), while Im(λ) yields the angular frequency of the normal mode.

By substituting Eq. (13) in Eqs. (11) and (12), we are lead to the stability eigenvalue problem
ξfk2fikSh=0
(14a)
h(λ+k2+ikSx)h+ξf=0
(14b)
f=0,h=0inx=±1/2
(14c)

where the primes denote derivatives with respect to x. In Eq. (14), we introduced the scaled Rayleigh number, S=kzRa/k.

A numerical solution of Eq. (14) yields the complex eigenvalue λ and the complex-valued eigenfunctions (f, h) for prescribed input values of (k,S,ξ). In this paper, the numerical technique adopted to achieve this task is the shooting method. Here, such method is exploited and coded along the lines described in chapter 10 of the book by Barletta [6]. The main purpose is determining the neutral stability condition where Re(λ)=0. Such a condition marks the threshold of the transition from linear stability, Re(λ)<0, to instability, Re(λ)>0.

4.2 The Limiting Case ξ1.

Let us introduce the scaled eigenfunction
hˇ=ξ1h
(15)
Then, by substituting Eq. (15) into Eq. (14), by simplifying wherever needed and by taking the limit ξ, we get
fikShˇ=0
(16a)
hˇ(λ+k2+ikSx)hˇ+f=0
(16b)
f=0,hˇ=0inx=±1/2
(16c)

Equation (16) are independent of ξ and, as such, they serve for capturing the neutral stability condition when ξ is infinitely large. Physically, this limit means a transverse permeability much larger than the longitudinal permeability, i.e., KTKL.

4.3 The Limiting Case ξ1.

We define the scaled parameter and eigenfunction
kˇ=ξ1/2k,Sˇ=ξ1/2S,hˇ=ξ1h
(17)
We substitute Eq. (17) into Eq. (14), simplify and take the limit ξ0. Thus, we obtain
fkˇ2fikˇSˇhˇ=0
(18a)
hˇ(λ+ikˇSˇx)hˇ+f=0
(18b)
f=0,hˇ=0inx=±1/2
(18c)

Once again, Eq. (18) define an eigenvalue problem independent of ξ which serves to describe the features of the neutral stability condition when ξ1, namely, in an anisotropic medium where KLKT.

5 Discussion of the Results

The first evident feature observed with the numerical solution of the eigenvalue problem (14) is that the eigenvalues are real or, equivalently, the angular frequency Im(λ) is zero. This means that the onset of the thermoconvective instability reflects the principle of exchange of stabilities [1]. Such a behavior has been already pointed out for the isotropic case [16]. We also note that the stability eigenvalue problem (14) coincides with that formulated by Barletta [16] in the isotropic case, ξ = 1 meaning KT = KL.

Figure 2 displays the neutral stability curves for different values of ξ. The isotropic case (ξ=1) is evidenced with a gray line. The limiting case ξ, obtained by solving Eq. (16), is also displayed. The instability arises within the region entrapped by the neutral stability curve. Hence, the minimum value of S along each neutral stability curve defines the critical values (kc, Sc) which depend on ξ. Instability is possible for S>Sc. Thus, Fig. 2 shows clearly that the anisotropy is destabilizing when ξ>1, while it is stabilizing when ξ<1. This is physically sound as ξ>1 means a larger hydraulic resistance for the longitudinal directions, y and z, than for the transverse x direction. This feature inevitably tends to favor the onset of cellular flow patterns over the basic vertical and, hence, longitudinal flow. Another feature typical of the teardrop-shaped neutral stability curves is that, for each ξ, there exists a maximum k position where k = kt and S = St. Here, the subscript t stands for “tip”. With k>kt, no instability is observed for a fixed ξ.

Fig. 2
Neutral stability curves in the (k, S) plane for different values of ξ
Fig. 2
Neutral stability curves in the (k, S) plane for different values of ξ

Figure 3 explores the regime of small ξ, by comparing the neutral stability curves for ξ0 and for ξ=1/10 displayed in the (kˇ,Sˇ) plane. We note that the two curves are almost overlapped suggesting that the regime ξ1 is almost attained when ξ=0.1. We also point out that the numerical data for the limiting case ξ0 have been obtained from the solution of the eigenvalue problem (18).

Fig. 3
Neutral stability curves in the (kˇ,Sˇ) plane for ξ→0
Fig. 3
Neutral stability curves in the (kˇ,Sˇ) plane for ξ→0
The critical and tip values of S are plotted versus ξ in Fig. 4. This figure illustrates also the asymptotic behavior for both ξ1 and ξ1 obtained by solving Eqs. (16) and (18), respectively. In particular, for the limiting case where ξ1, we have
kc=3.101992,Sc=82.32102kt=4.593909,St=121.3962
(19)
Fig. 4
Plots of the critical and tip values of S versus ξ. The dotted lines illustrate the asymptotic trends for small ξ and for large ξ.
Fig. 4
Plots of the critical and tip values of S versus ξ. The dotted lines illustrate the asymptotic trends for small ξ and for large ξ.
On the other hand, for ξ1, we obtain
kc=1.153106ξ1/2,Sc=175.9625/ξ1/2kt=1.358588ξ1/2,St=222.7531/ξ1/2
(20)

The asymptotic trends revealed by Eq. (20) show that the instability cannot arise when ξ0 as the neutral stability curve is compressed to vanishingly small wavenumbers and pushed to infinitely large values of S. This feature is also suggested by Fig. 2. Figure 5 is the dual part of Fig. 4 with the critical and tip values of the wavenumber k plotted versus ξ. Again, the dotted lines show the asymptotic solutions found in the limits ξ and ξ0. Samples of the same sets of data are also reported in Table 1. We report that the numerical data for the isotropic case ξ = 1 perfectly coincide with those found in Barletta [16].

Fig. 5
Plots of the critical and tip values of k versus ξ. The dotted lines illustrate the asymptotic trends for small ξ and for large ξ.
Fig. 5
Plots of the critical and tip values of k versus ξ. The dotted lines illustrate the asymptotic trends for small ξ and for large ξ.
Table 1

Critical and tip values of k and S

ξkcScktSt
0.10.3611961563.64790.4265219714.8381
0.50.7795100264.38920.9281404337.5210
11.059498197.08121.272911253.3400
1.51.252150168.63911.516234218.0511
21.399648152.34921.706706197.9887
51.900118116.86692.389042155.1100
102.262886101.74672.933712137.7329
1002.95845784.642714.230445121.8534
ξkcScktSt
0.10.3611961563.64790.4265219714.8381
0.50.7795100264.38920.9281404337.5210
11.059498197.08121.272911253.3400
1.51.252150168.63911.516234218.0511
21.399648152.34921.706706197.9887
51.900118116.86692.389042155.1100
102.262886101.74672.933712137.7329
1002.95845784.642714.230445121.8534

A final remark is for the role of the wavevector orientation for a given k, which means the interpretation of the data provided so far in terms of the Rayleigh number Ra, instead of its scaled counterpart S. By relying on its definition, S=kzRa/k, we immediately infer that the most unstable modes are those with kz = k or, equivalently, ky = 0. Such two-dimensional modes are usually termed transverse rolls. That the instability arises with these modes is a feature that has been already pointed out, for the isotropic case, in Barletta [16].

6 Conclusions

The instability of the stationary and parallel flow in a vertical porous layer caused by the buoyancy force has been studied. The effect of the anisotropic permeability of the medium has been taken into account. The linear response to small-amplitude perturbations in the form of normal modes has been tested. The stability eigenvalue problem has been solved numerically for a wide range of the governing parameters, ξ (the anisotropy parameter) and Ra (the Rayleigh number).

The most important results of this analysis can be outlined as follows:

  • The onset of the instability is due to the transverse rolls, namely the two-dimensional normal modes lying in the plane of the basic buoyant flow.

  • When ξ>1, which means a transverse permeability larger than the longitudinal permeability, the effect of the anisotropy is destabilizing. When ξ<1, the longitudinal permeability is larger than the transverse permeability and the effect of anisotropy is stabilizing.

  • In the limit of very large values of ξ, the critical Rayleigh number attains the asymptotic value 82.32102, which is the lowest possible value for the onset of the instability in the anisotropic porous layer. In the limit of a vanishingly small ξ, no instability arises according to the linear analysis.

There are several possible future developments of the analysis reported in this paper. For instance, the nonlinear behavior in the supercritical regime needs a specific study, focused on the evaluation of the heat transfer rate across the porous layer and on the planforms of the multicellular patterns resulting from the instability.

Funding Data

  • Ministero dell'Istruzione, dell'Università e della Ricerca (Grant No. PRIN 2017F7KZWS; Funder ID: 10.13039/501100003407).

Nomenclature

     
  • f(x),h(x) =

    eigenfunctions

  •  
  • g =

    gravitational acceleration

  •  
  • g =

    modulus of g

  •  
  • k =

    wavenumber

  •  
  • k y, kz =

    wavevector components

  •  
  • K¯ =

    permeability

  •  
  • KL, KT =

    permeability components

  •  
  • L =

    slab thickness

  •  
  • P =

    difference between the pressure and the hydrostatic pressure

  •  
  • Ra =

    Rayleigh number

  •  
  • S =

    scaled Rayleigh number, S=kzRa/k

  •  
  • t =

    time

  •  
  • T =

    temperature

  •  
  • T0 =

    reference temperature, T0=(T1+T2)/2

  •  
  • T1, T2 =

    boundary temperatures

  •  
  • u =

    seepage velocity, u=(u,v,w)

  •  
  • x, y, z =

    Cartesian coordinates

Greek Symbols

    Greek Symbols
     
  • α =

    average thermal diffusivity

  •  
  • β =

    coefficient of thermal expansion for the fluid

  •  
  • λ =

    complex growth rate

  •  
  • μ =

    dynamic viscosity

  •  
  • ξ =

    anisotropy parameter

  •  
  • ρ0 =

    reference fluid density

  •  
  • σ =

    heat capacity ratio

Superscripts, Subscripts

    Superscripts, Subscripts
     
  • b =

    basic solution

  •  
  • c =

    critical values

  •  
  • t =

    tip values

  •  
  • * =

    dimensionless quantities in Eq. (5)

  •  
  • ∼ =

    perturbations

  •  
  • =

    scaled quantities in Eqs. (15) and (17)

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