Transp Porous Med DOI 10.1007/s11242-015-0489-6

Onset of Convection in an Anisotropic Porous Layer with Vertical Principal Axes Peder A. Tyvand1 · Leiv Storesletten2

Received: 9 February 2015 / Accepted: 11 March 2015 © Springer Science+Business Media Dordrecht 2015

Abstract A Horton–Rogers–Lapwood problem for onset of thermal convection in a horizontal porous layer with anisotropic permeability and diffusivity is solved analytically. There is full 3D anisotropy, with the restriction that one of each set of principal axes points in the vertical direction. The porous layer has infinite horizontal extent. The upper and lower boundaries are taken to be impermeable and kept at constant temperatures. The critical Rayleigh number for the onset is calculated as a function of the five parameters governing the anisotropy: two permeability ratios, two diffusivity ratios and the angle (in the horizontal plane) between the principal axes of permeability and diffusivity. Keywords

Anisotropic porous media · Convection · Stability analysis · Rayleigh number

1 Introduction The onset of Rayleigh–Bénard convection in a porous layer heated from below was studied by Horton and Rogers (1945) and later by Lapwood (1948). Their results were generalized by Beck (1972) to rectangular porous boxes with thermally insulating and impermeable sidewalls. Castinel and Combarnous (1974) solved the onset problem for a porous layer with anisotropic permeability. Epherre (1975) extended the analysis to porous media with anisotropic thermal conductivity. These early papers assumed that one principal axis was aligned in the vertical direction, and there was horizontal isotropy. Later papers by McKibbin (1986) and Rosenberg and Spera (1990) also assumed the principal axes of anisotropy to coincide with the coordinate axes for the porous layer.

B

Peder A. Tyvand [email protected]

1

Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, 1432 Ås, Norway

2

Department of Mathematics, University of Agder, 4604 Kristiansand, Norway

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Tyvand and Storesletten (1991) took into account oblique anisotropy, where one principal axis of the permeability tensor makes an arbitrary angle with the vertical direction. The permeability was assumed transversely isotropic, which means that two of the principal axes have equal permeabilities. Storesletten (1993) solved the analogous onset problem for a porous layer with anisotropy in the thermal conductivity. Mahidjiba et al. (2000) studied the effects of vertical sidewalls on the onset of two-dimensional convection with oblique anisotropy. The review articles by Storesletten (1998, 2004) gave an overview with additional references on convection in anisotropic porous media, see also Nield and Bejan (2013). In the present paper, we will solve the onset problem for a homogeneous porous layer with three-dimensional anisotropy in permeability as well as thermal conductivity. The only restrictions for this anisotropy are that one principal axis of permeability is vertical, and one principal axis of thermal conductivity (or diffusivity) is vertical. In this paper, we want to find out how an angle between the horizontal principal axes of permeability and conductivity affects the onset of Darcy–Bénard convection. Kvernvold and Tyvand (1979) studied linear and nonlinear convection in porous media where this angle is zero, which means that the permeability and conductivity have coinciding principal axes. The present idealized class of porous media is not found in nature, but it can be designed and man-made. In fact, it is straightforward to construct porous media with different principal axes of permeability and conductivity. A set of evenly distributed parallel metal threads introduced into an otherwise weakly conducting medium can generate a desired anisotropy in thermal conductivity (and diffusivity), without significantly affecting the anisotropic permeability of the medium.

2 Mathematical Formulation We consider an unlimited horizontal porous layer with thickness h. The gravitational acceleration is g. The permeability and thermal conductivity are anisotropic, with one of their principal axes in the vertical direction. The vertical permeability is denoted by K 3 . The saturated porous medium has effective thermal conductivity λ3 in the vertical direction. We introduce the Cartesian coordinates x, y and z, where the z axis points vertically upward. The x and y axes are directed along the principal axes of permeability. Thus, K 1 and K 2 denote the permeabilities in the x and y directions, respectively. Without loss of generality, we choose K1 ≥ K2, (1) which means that the x axis is directed along the maximal horizontal permeability. The unit  respectively. vectors in the x, y and z directions are i, j and k, The principal values of effective horizontal conductivity are λ1 and λ2 , and their respective principal axes have the unit vectors i  and j  . The principal axes of horizontal conductivity x  and y  are rotated by an angle φ with respect to the coordinate axes, see the definition sketch in Fig. 1. The relationships between the unit vectors are i  = i cos φ + j sin φ, j  = −i sin φ + j cos φ.

(2) (3)

The porous layer is confined by the planes z = 0 and z = h, which are impermeable and perfectly heat-conducting. The velocity vector is v, with Cartesian components (u, v, w). The temperature field is T (x, z, t), where t denotes time. The lower plane (z = 0) is kept

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λ3

z

K3

λ1

K1 K2

x

φ λ2

z’

y φ y’ Fig. 1 Definition sketch of the anisotropic porous medium. The principal axes of permeability (green) and of thermal conductivity (red) are indicated. The coordinate axes (x, y, z) are aligned along the principal axes of permeability. The rotated coordinate axes (x  , y  , z  ) are the principal axes of thermal conductivity. The z  axis coincides with the z axis because one of each principal axes points in the vertical z direction. The angle between the x  and the x axes is denoted by ϕ, and it is equal to the angle between the y  and y axes

at the constant temperature T = T0 , and its upper plane (z = h) is kept at the constant temperature T = T0 − T , where T is a positive temperature difference. The standard Darcy–Boussinesq equations for convection in a homogeneous and anisotropic porous medium can be written ∇ p + μ M · v + ρ0 β (T − T0 ) g = 0,

(4)

∇ · v = 0, ∂T (ρc p )m + (ρc p ) f v · ∇T = ∇ · ( · ∇T ). ∂t The boundary conditions are

(5) (6)

w = 0, T − T0 = 0, at z = 0,

(7)

w = 0, T − T0 + T = 0, at z = 1.

(8)

In these equations, p is the dynamic pressure, v is the velocity vector with Cartesian components (u, v, w), g = | g | is the gravitational acceleration, β is the coefficient of thermal expansion, ρ = ρ0 is the fluid density at the reference temperature T0 , μ is the dynamic viscosity of the saturating fluid, c p is the specific heat at constant pressure, and  is the tensor of thermal conductivity. The subscript m refers to an average over the mixture of solid and fluid, while the subscript f refers to the saturating fluid alone. We have introduced the inverse permeability tensor M.

2.1 Dimensionless Equations From now on, we will work with dimensionless variables. We will reformulate the equations and boundary conditions in dimensionless form by means of the following transformations 1 h (x, y, z) → (x, y, z), (u, v, w) → (u, v, w), h∇ → ∇, K 3 M → M, h κ3 (ρc p ) f κ3 1 1 K3 (T − T0 ) → T,  → D, ( p − p0 ) → p, t → t, λ3 T μκ3 (ρc p )m h 2

(9)

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where κ3 = λ3 /(ρc p ) f is the thermal diffusivity in the vertical direction. The dimensionless governing equations can then be written M · v + ∇ p − R T k = 0.

(10)

∇ · v = 0 ∂T + v · ∇T = ∇ · (D · ∇T ), ∂t

(11) (12)

with the boundary conditions w = T = 0, z = 0,

(13)

w = T + 1 = 0, z = 1.

(14)

Here the Rayleigh number R is defined as R=

ρ0 gβ K 3 T h . μκ3

(15)

The dimensionless version of the inverse permeability tensor M is written as  M = ξ1−1i i + ξ2−1 j j + kk,

(16)

where we introduce the permeability ratios ξ1 =

K1 K2 , ξ2 = , K3 K3

(17)

noting that ξ1 ≥ ξ2 by the convention (1) above, choosing the x axis in the direction of maximal horizontal permeability. The dimensionless diffusivity tensor is written as    D = η1i i  + η2 j  j  + kk = D11i i + D22 j j + D12 i j + j i + kk, (18) where we have introduced the diffusivity ratios η1 =

λ1 λ2 , η2 = , λ3 λ3

(19)

being identical to the ratios of effective conductivity. We have now defined the four anisotropy parameters (ξ1 , ξ2 , η1 , η2 ). The first two of these four anisotropy parameters are subject to a constraint (1) implying ξ1 ≥ ξ2 . Adding the angle φ for principal axes of conductivity, we have a total set of five dimensionless parameters that specify the anisotropy. Without loss of generality, we take the constraint 0 ≤ φ < π/2. The components of the diffusivity tensor are given by D11 = η1 cos2 φ + η2 sin2 φ, D12 = (η1 − η2 ) sin φ cos φ, D22 = η1 sin2 φ + η2 cos2 φ. (20)

2.2 Basic Solution There exists a stationary basic solution of Eqs. (10)–(14). It is denoted by the subscript “b” and given by  z , (21) vb = 0, Tb = 1 − z, pb = Rz 1 − 2 The basic state is a state where the fluid is at rest in hydrostatic equilibrium, with a linear temperature profile of pure conduction.

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2.3 Linearized Perturbation Equations Stability is analyzed by perturbing the basic state given by Eq. (21). The perturbed field is represented by the variables v = vb + v, T = Tb (z) + θ, p = pb (z) + p  .

(22)

where the perturbations v, θ, p  are functions of x, y, z and t. The governing Eqs. (10)–(12) are linearized with respect to the perturbations. Eliminating the pressure by taking the curl of Eq. (10) gives the relationship    (23) ∇ × ξ1−1 ui + ξ2−1 v j + w k = R∇ × (θ k), which we combine with the continuity Eq. (11) to obtain the equation   ξ1 wx x + ξ2 w yy + wzz = R ξ1 θx x + ξ2 θ yy .

(24)

Here the subscripts x, y, z denote partial derivatives. The linearized heat Eq. (12) reduces to ∂θ − w = D11 θx x + 2D12 θx y + D22 θ yy + θzz . (25) ∂t These two coupled second-order equations combined with the homogeneous boundary conditions w = θ = 0 at z = 0 and z = 1. (26) represent the eigenvalue problem of convection onset.

3 Stability Analysis We assume that the principle of exchange of stabilities holds. The analysis of neutral stability can then be carried out for time-independent normal modes, without loss of generality. The preferred solution of the eigenvalue problem (24)–(26) can be written as (w, θ ) = (W, )ei(k x x+k y y) sin(π z),

(27)

where the real part represents the physical disturbance. W and  are perturbation amplitudes, while k x and k y are wavenumbers. The Rayleigh number at neutral stability is     π2 2 2 2 R = D11 k x + 2D12 k x k y + D22 k y + π , (28) 1+ ξ1 k x2 + ξ2 k 2y representing the onset of convection for a given wavenumber vector. We introduce polar coordinates (k, α) in Fourier space, defined by 

ky (k, α) = k x2 + k 2y , arctan . (29) kx It is advantageous to introduce a function F of four given physical parameters, being defined as ζ1 − ζ2 ζ1 + ζ2 + cos 2(φ − α). (30) F(ζ1 , ζ2 , φ, α) = 2 2 Here (ζ1 , ζ2 ) is chosen as a common notation to represent the anisotropy parameters for permeability (ξ1 , ξ2 ) and for diffusivity (η1 , η2 ).

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With this notation, the dispersion relation for the onset of convection gets a relatively simple form

   π2 . (31) R(k 2 , α) = π 2 + F(η1 , η2 , φ, α)k 2 1 + F(ξ1 , ξ2 , 0, α)k 2 We are going to minimize the Rayleigh number in two steps, first searching for the value of k = k x2 + k 2y that gives a minimum with α kept fixed. The resulting critical absolute value of the wavenumber vector is given by π kc (α) = . (32) F(ξ1 , ξ2 , 0, α)1/4 F(η1 , η2 , φ, α)1/4 The critical Rayleigh number as a function of the direction of the wavenumber vector is then given by   2 Rc (α) F(η1 , η2 , φ, α) = 1+ . (33) π2 F(ξ1 , ξ2 , 0, α) We determine the global minimum for Rc with the corresponding value of α. The minimizing value of α is determined by the condition 

d F(η1 , η2 , φ, α) = 0, (34) dα F(ξ1 , ξ2 , 0, α) leading to the equation (ξ1 +ξ2 )(η1 −η2 ) sin 2(αc −φ) = (ξ1 −ξ2 )(η1 +η2 ) sin 2αc +(ξ1 −ξ2 )(η1 −η2 ) sin 2φ (35) which determines the critical value α = αc . We simplify the notation by introducing two anisotropy quotients P and Q 1 + ξ2 /ξ1 K1 + K2 = , 1 − ξ2 /ξ1 K1 − K2 λ1 + λ2 1 + η2 /η1 = . Q= 1 − η2 /η1 λ1 − λ2 P=

(36) (37)

We recall the convention (1), which implies P ≥ 1. The parameter Q can have any value, positive or negative. Equation (35) is compacted as follows by this new notation P sin 2(αc − φ) − Q sin 2αc = sin 2φ = r sin(2αc − ψ).

(38)

Here we have taken the opportunity to introduce a polar coordinate type of variables (r, ψ) in parameter space. Their introduction into Eq. (38) gives the transformations r cos ψ = P cos 2φ − Q, r sin ψ = P sin 2φ, implying r=



P 2 + Q 2 − 2P Q cos 2φ.

(39) (40)

Above, we have made the conventions 0 ≤ φ < π/2 and P ≥ 1. From Eq. (39), it follows that sin ψ ≥ 0, or equivalently 0 ≤ ψ < π. Then ψ is given uniquely by the formula   P cos 2φ − Q . (41) ψ = arccos P 2 + Q 2 − 2P Q cos 2φ

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It follows from Eq. (38) sin(2αc − ψ) =

sin 2φ ≥ 0, r

(42)

but this equation has an ambiguity with respect to αc . Therefore, we have two solution branches for αc 



sin 2φ sin 2φ 1 1 ψ + arcsin , αc2 = ψ + π − arcsin . (43) αc1 = 2 r 2 r Both these values for the wavenumber angle represent legal candidates for the critical Rayleigh number at the onset of convection, given by Eq. (33). We therefore need to evaluate both Rc (αc1 ) and Rc (αc2 ) in order to select the smallest of these as the preferred Rayleigh number for the onset of convection. We have now found a double-valued functional relationship αc = αc (ξ2 /ξ1 , η2 /η1 , φ),

(44)

which reduces the number of independent physical parameters affecting the preferred wavenumber angle from five to three. The critical Rayleigh number for the onset of convection still depends on all the five parameters that govern the anisotropy Rc = Rc (ξ1 , ξ2 , η1 , η2 , φ).

(45)

4 Discussion of Results In the present paper, we study the onset of convection in a porous layer with a general threedimensional anisotropy in the permeability as well as conductivity. The only restriction is that one of each set of principal axes points in the vertical direction. We have illustrated the configuration of the principal axes in Fig. 1. There is an angle φ between the horizontal principal axes of permeability and conductivity. The cases φ = 0 and φ = π/2 have been solved by Kvernvold and Tyvand (1979). Our aim was to investigate how the critical Rayleigh number Rc depends on the angle φ where 0 < φ < π/2. Here Rc is given by the formula (33), where the wavenumber angle α is replaced with the critical value αc given by Eq. (43). Since Eq. (42) has an ambiguity with respect to αc , we have two candidates for the critical Rayleigh number Rc , and the smallest of these is the valid critical Rayleigh number for the onset of convection. The other candidate for a critical Rayleigh number does not possess physical significance, provided the porous layer is of infinite horizontal extent. Figure 2 displays the marginal Rayleigh number Rc /π 2 as a function of φ and α where ξ1 = 10, ξ2 = 5, η1 = 10 and η2 = 1. This plot illustrates a normal situation where we find two extremal values for Rc in the open interval 0 < α < π for each choice of φ. The chosen anisotropy parameters give a clear exposition of these two extremal points. The figure shows how the minimum point moves continuously to the upper endpoint of the interval (α = π) when φ → π/2. Note that the other minimum point α = 0 is identical to α = π, because a physical Fourier component (27) is invariant under the transformation (k x , k y ) → (−k x , −k y ), or equivalently α → α + π. Figure 3 represents the same anisotropy parameters as Fig. 2 and shows in detail how the marginal Rayleigh number at onset varies with the Cartesian components k x and k y of the wavenumber vector, for a chosen direction φ = π/8 of the thermal principal axes. αc2 is greater than π/2, and it gives the critical Rayleigh number for the onset of convection.

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Fig. 2 Marginal Rayleigh number Rc /π 2 as a function of the angle ϕ of thermal principal axis and the wavenumber angle α. The intervals are 0 < ϕ < π/2 and 0 < α < π. We have chosen ξ1 = 10, ξ2 = 5, η1 = 10 and η2 = 1. This plot illustrates the two extremal values that exist for the Rayleigh number within the open interval 0 < α < π . The figure shows how the minimum point for a given angle ϕ moves toward the upper bound α = π as ϕ tends to π/2

Fig. 3 Marginal Rayleigh number Rc /π 2 as a function of the wavenumber components k x and k y for an angle ϕ = π/8 of the thermal principal axis. We have chosen ξ1 = 10, ξ2 = 5, η1 = 10 and η2 = 1 (same as in Fig. 2). The plotting intervals are −2 < k x < 1.5 and 0.5 < k y < 3.5

We cannot identify αc1 from this plot, but it is not important physically since it is not preferred. In Fig. 4, we plot the two preferred wavenumber angles αc1 and αc2 as functions of the ratios ξ2 /ξ1 and η2 /η1 , where 0 < ξ2 /ξ1 < 1 and 0 < η2 /η1 < 3. We have here chosen φ = π/4. We subtract −π/2 from the value of αc2 , in order to reveal how much the two directions for extremal Rayleigh number differ from being perpendicular to one another. These two plots coincide at the degeneracy value η2 /η1 = 1, a case that is included in the linear stability analysis of Kvernvold and Tyvand (1979).

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Fig. 4 The preferred wavenumber angles αc1 and αc2 plotted as functions of ξ2 /ξ1 and η2 /η1 where 0 < ξ2 /ξ1 < 1 and 0 < η2 /η1 < 3. The angle of the thermal principal axes is ϕ = π/4. a The lower value αc1 . b The higher value αc2 , represented by αc2 − π/2

Figure 5 displays the critical Rayleigh number Rc /π 2 as a function of the mixed anisotropy parameters ξ1 and η1 with ξ2 = η2 given and the fixed orientation φ = π/4 for the principal axes. Figure 5a represents 1 < ξ1 < 5 and 0 < η1 < 3 where ξ2 = η2 = 1. Figure 5b represents 2 < ξ1 < 5 and 0 < η1 < 3 where ξ2 = η2 = 2. Figure 6 displays the critical Rayleigh number Rc /π 2 as a function of the mechanical anisotropy parameters ξ1 and ξ2 . We have chosen η1 = 5, η2 = 1 and φ = π/4. We display the intervals 1 < ξ1 < 4 and 0 < ξ2 < ξ1 . We note that the critical Rayleigh number decreases monotonically with both anisotropy parameters. Thus, increasing each of the horizontal permeabilities will promote instability, which is plausible. Figure 7 displays the critical Rayleigh number Rc /π 2 as a function of the thermal anisotropy parameters 0 < η1 < 3, 0 < η2 < 4. Here we have chosen ξ1 = 5, ξ2 = 1

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Fig. 5 a The critical Rayleigh number Rc /π 2 as a function of two mixed anisotropy parameters: 1 < ξ1 < 5, 0 < η1 < 3. Here ξ2 = η2 = 1 and ϕ = π/4. b The critical Rayleigh number Rc /π 2 as a function of two mixed anisotropy parameters: 2 < ξ1 < 5, 0 < η1 < 3. Here ξ2 = η2 = 2 and ϕ = π /4

and φ = π/4. We note that the critical Rayleigh number increases monotonically with both anisotropy parameters. Thus, increasing each of the horizontal conductivities will stabilize the state, because buoyancy is lost by diffusion in each horizontal direction. Figures 8 and 9 demonstrate how the critical Rayleigh number depends on φ, the angle between the horizontal principal axes of permeability and conductivity. Figure 8 displays the critical Rayleigh number Rc /π 2 as a function of the mechanical anisotropy parameter ξ2 and the angle φ where 0 < ξ2 < 5 and 0 < φ < π/2. Here we have chosen ξ1 = 5, η1 = 5 and η2 = 1. For each given value of φ, we note that the critical Rayleigh number decreases monotonically with ξ2 , which is the same trend as we commented in Fig. 6. Figure 9 displays the critical Rayleigh number Rc /π 2 as a function of the thermal anisotropy parameter η1 and φ, with the choices ξ1 = 2, ξ2 = 1 and η2 = 1. For each given value of φ, we note that the critical Rayleigh number increases monotonically with η2 , which is the same trend as we found in Fig. 7. The limiting values φ = 0 and φ = π/2 in Figs. 8 and 9 are included in the analysis by Kvernvold and Tyvand (1979). We note the discontinuous slopes in these plots, which

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Fig. 6 The critical Rayleigh number Rc /π 2 as function of mechanical anisotropy parameters: 1 < ξ1 < 4, 0 < ξ2 < 1. Here η1 = 5, η2 = 1 and ϕ = π/4

Fig. 7 The critical Rayleigh number Rc /π 2 as a function of thermal anisotropy parameters: 0 < η1 < 3, 0 < η2 < 4. Here ξ1 = 5, ξ2 = 1 and ϕ = π/4

correspond to anisotropy values where the direction of the preferred roll of disturbance switches by a finite angle.

5 Concluding Remarks Onset of convection in a horizontal porous layer with anisotropy in the permeability as well as conductivity (or diffusivity) is studied. There is a three-dimensional anisotropy, with the only restriction that one of each set of principal axes points in the vertical direction. The

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Fig. 8 The critical Rayleigh number Rc /π 2 as a function a mechanical anisotropy parameter ξ2 and the angle ϕ for the thermal principal axes: 0 < ξ2 < 5, 0 < ϕ < π/2. Here ξ1 = 5, η1 = 5 and η2 = 1

Fig. 9 The critical Rayleigh number Rc /π 2 as function of a thermal anisotropy parameter η1 and the angle ϕ of a thermal principal axis: 0 < η1 < 3, 0 < ϕ < π/2. Here ξ1 = 2, ξ2 = 1, η2 = 1

porous layer is confined by two horizontal planes, where the upper and lower boundaries are impermeable and kept at constant temperatures. There is an angle φ between the horizontal principal axes of the permeability and the conductivity. This is a generalization of the model discussed in Kvernvold and Tyvand (1979), who made a full analysis of the cases φ = nπ/2, where n is an integer. On account of this fact, it is especially important to find out how the onset of convection varies with the angle φ. The critical Rayleigh number Rc for the onset of convection depends on five parameters governing the anisotropy: two permeability ratios, two diffusivity ratios and the angle φ. Exact formulas for the critical Rayleigh number and the corresponding critical wavenumber vector are found analytically. Since there is an ambiguity with respect to the critical wavenumber

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angle αc , there exist two candidates for Rc , where the smallest of these is the preferred Rayleigh number for the onset of convection. Figures displaying the critical Rayleigh number and the critical wavenumber are plotted for various values of the anisotropy parameters. The two last figures show explicitly how Rc depends on the angle φ that specifies the direction of the thermal principal axes. In general, the critical Rayleigh number varies quite smoothly with the five anisotropy parameters. There are exceptional cases of discontinuous slopes in the Rayleigh number plots. These exceptions occur when we have degeneracies where the anisotropy is no longer fully three-dimensional in both permeability and diffusivity.

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