Stability of plane Poiseuille Couette flow in a fluid layer overlying a porous layer

Transcription

1 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at J. Fluid Mech. (27), vol. 826, pp c Cambridge University Press 27 doi:.7/jfm Stability of plane Poiseuille Couette flow in a fluid layer overlying a porous layer Ting-Yueh Chang, Falin Chen and Min-Hsing Chang 2, Institute of Applied Mechanics, National Taiwan University, Taipei, 6, Taiwan 2 Department of Mechanical Engineering, Tatung University, Taipei, 4, Taiwan (Received February 27; revised 3 May 27; accepted 9 June 27; first published online 3 August 27) This paper performs a linear stability analysis to investigate the stability of plane Poiseuille Couette flow in a fluid layer overlying a porous medium saturated with the same fluid. The effect of superimposed Couette flow on the associated Poiseuille flow in such a two-layer system is explored carefully. The result shows that the presence of Couette flow may destabilize the Poiseuille flow at small depth ratio ˆd, defined by the ratio of the depth of the fluid layer to the depth of the porous layer, and induce a trimodal structure to the neutral curves. At moderate ˆd, the Couette component generally produces a stabilization effect on the flow. When the velocity of the upper moving wall is large enough, a bi-modal behaviour of the neutral curves appears and a shift of instability mode occurs from the long-wave fluid-layer mode to the porous-layer mode with higher wavenumber. These stability characteristics are remarkably different from those of the plane Poiseuille Couette flow in a single fluid layer in that the flow becomes absolutely stable when the wall velocity is over 7 % of the maximum velocity of the Poiseuille component of flow. The stability of pure Couette flow in such a two-layer system is also studied. It is found that the flow is still absolutely stable with respect to infinitesimal disturbances, which is the same as the stability characteristic of a single-layer plane Couette flow. Key words: absolute/convective instability, instability control, porous media. Introduction The stability of plane Poiseuille Couette (PPC) flow has been studied for many years. The early works on this subject for a single fluid layer can be found in Potter (966), Hains (967, 97), Reynolds & Potter (967) and Cowley & Smith (985). Potter (966) first derived the asymptotic solutions for the related Orr Sommerfeld equation and found that the moving upper plate exerts a strong stabilizing effect on the Poiseuille flow. Particularly, the flow is found to be stable for all finite Reynolds numbers when the plate velocity, U, exceeds 7 % of the maximum velocity of the Poiseuille component of flow denoted by W. His results also showed that the flow may become more unstable when the ratio of U/W is within a small range between address for correspondence: mhchang@ttu.edu.tw

2 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer and.4 with increased Couette flow. He observed that this effect occurs when U is approximately equal to the travelling wave speed. Hains (967) investigated this flow numerically and found that the critical Reynolds number is increased when the upper plate is moved in either the same or reverse direction with respect to the Poiseuille flow, and approaches infinity eventually with increasing the velocity of the upper plate. A weakly nonlinear analysis performed by Reynolds & Potter (967) shows that the PPC flow is stable to infinitesimal disturbances at all Reynolds numbers when the Couette component is sufficiently large, but remains unstable to finite-amplitude disturbances. Cowley & Smith (985) also considered the linear and nonlinear stability of the PPC flow with the upper and lower boundaries moving at the same speed but in opposite directions. They discussed the variation of neutral curves with increasing wall speeds and determined the cutoff velocity at which all the neutral curves disappear. The time evolution of disturbances superimposed on various mean flow profiles were investigated by Bergström (25). He found that the disturbance peak amplification depends on how the Poiseuille and Couette components are combined. Savenkov (28) applied the asymptotic triple-deck theory to explore the development of the most unstable waves in the PPC flow and showed how their growth leads to the formation of wave packets. Regarding the heat transfer effect, Özgen, Dursunkaya & Ebrinc (27) considered the stability of PPC flow of air and found that the flow is stabilized by the wall heating. Recently, Guha & Frigaard (2) presented a linear stability analysis of the PPC flow together with the presence of a uniform cross-flow. They used the cross-flow Reynolds number and the dimensionless wall velocity to characterize the stability behaviours. A detailed review for related studies in literature about the stability of PPC flow in a single fluid layer was given in this work. The stability of PPC flow is also an important subject for non-newtonian fluids due to its wide engineering applications (Nouar & Frigaard 29; Tran & Suslov 29; Moyers-Gonzalez & Frigaard 2). Nouar & Frigaard (29) conducted a linear stability analysis of the PPC flow for shear-thinning fluid. A parametric study was performed and the results demonstrate that the effect of shear thinning decreases the phase velocity of the travelling waves and causes an increase in stability. The later work of Moyers-Gonzalez & Frigaard (2) further considered the viscoelastic models of Oldroyd-B and modified finitely extensible nonlinear elastic (FENE) Chilcott Rallison fluids. The stability characteristics were found to be similar to the results for a Newtonian fluid and the cutoff wall velocity was still approximately.7 times the maximum velocity of the Poiseuille component. Tran & Suslov (29) examined the stability of incompressible fluids with a pressure-dependent viscosity, named piezo-viscous fluids. The results show that the piezo-viscous effect is generally a stabilizing factor for a base flow with increasing applied pressure gradient. The stability of plane parallel flow in a two-layer system is also an important subject in fluid mechanics due to its wide range of applications in industry and geophysical problems. The system comprises a fluid layer overlying a porous layer saturated with the same fluid. The instability of Poiseuille flow in such a two-layer system was first studied by Chang, Chen & Straughan (26) in which Darcy s law was used to model the porous layer. They found that there are three instability modes, with different stability characteristics, which are triggered by the shear stress of the Poiseuille flow in the fluid layer. A similar study was performed later by Liu, Liu & Zhao (28) in which Brinkman s model was employed to simulate the porous layer. Bars & Worster (26) also investigated the Poiseuille flow in a fluid overlying a porous layer. They derived the Darcy Brinkman equation using the volume-averaging

3 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at T.-Y. Chang, F. Chen and M.-H. Chang method and focused on the interaction between flow and solidification within the mushy layer during binary alloy solidification in a corner flow. They proposed a new interfacial condition at the mushy layer liquid interface and made a comparison with the Beavers & Joseph (967) boundary condition. Hill & Straughan (28) further considered a three-layer configuration composed of a fluid layer overlying a Brinkman porous transition layer and a Darcy-type porous layer. Two instability modes corresponding to the fluid and porous layers, respectively, were found in their work. Their results showed that the depth ratio between the fluid and porous layers and the depth of the transition layer are the key parameters which dominate the stability characteristics of the system. They also performed a study on the instability of Poiseuille flow in a two-layer system, in which a highly porous material is considered and simulated by the Darcy Brinkman equation (Hill & Straughan 29). The effects of plane Couette flow and Poiseuille flow on the instability of thermal convection in such a two-layer system has been investigated by Chang (25, 26), respectively. The interaction of instability mechanisms of the unstable stratification and the shear arising from the plane Couette flow or Poiseuille flow was explored in details for both longitudinal rolls and transverse rolls. Recently, Kumar et al. (23) performed a linear stability analysis for a pressure-driven two-layer plane Couette flow confined between an upper moving wall and a Darcy Brinkman porous layer at the bottom. Two liquid layers are considered in the system in which one fluid layer is overlying the other one which saturates the porous layer simultaneously. They found the moving boundary wall together with the slippage at the porous/liquid interface may stimulate a pair of finite-wavenumber shear modes in addition to the long-wave interfacial mode of the instability. The influence of the anisotropic and inhomogeneous permeability of the porous layer on the stability of plane Poiseuille flow in the two-layer system was studied in the work of Deepu, Anand & Basu (25). The generalized Darcy model was used to describe the flow in the porous medium. They suggested that a modulation of the permeability characteristic in the porous layer could be an effective method to control the flow stability. The stability of the two-layer system with a free and undeformable upper boundary was analysed by Lyubimova et al. (26) in order to simulate the instability of water flow in a river over aquatic plants. The system is inclined at a small angle to the horizon to model the riverbed slope. They made a detailed calculation for the neutral curves at various porous-layer permeabilities and the relative thicknesses of the porous layer. In this study we consider the instability of PPC flow in a two-layer system confined between two horizontal parallel plates. The system comprises a fluid layer under which lies a porous layer saturated with the same fluid. The object is to investigate the effects of an upper moving boundary on the instability behaviours of plane Poiseuille flow. Darcy s law is employed in the porous layer and the derived eigenvalue problem is solved by a D 2 Chebyshev tau method. Various depth ratios are considered and the influences of Couette component of the base flow on the onset and transition of the instability mode are examined. The results exhibit quite different stability characteristics in comparison with the case of PPC flow in a single fluid layer and give significant findings for the stability behaviours in such a two-layer system. 2. Problem formulation Consider a fluid layer within the domain z (, d) overlying a layer of porous medium occupying z ( d m, ) saturated with the same fluid. The interface between

4 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 379 the fluid and porous layers is naturally at z =. The continuity and Navier Stokes equations in the fluid layer are u i =, x i (2.) u i t + u u i j = p + ν u i, x j ρ x i (2.2) where u i and p are the velocity and pressure, ρ and ν are the density and kinematic viscosity. Standard indicial notation is employed throughout, with being the Laplace operator. The Darcy law is used to characterize the flow in the porous medium with the governing equations u m i χ t u m i =, (2.3) x i = p m ν ρ x i K um i, (2.4) where u m i and p m are the averaged pore velocity and interstitial pressure, χ and K are the porosity and permeability. The superscript m denotes porous medium where necessary. Here we first derive the governing equations of PPC flow and then reduce the system to the simplified plane Couette flow. 2.. The plane Poiseuille Couette flow in a fluid layer overlying a porous layer To model the Poiseuille Couette flow in such a two-layer system, appropriate boundary conditions should be defined first. At the upper surface of the fluid layer z = d, we have u = W, v = w =, (2.5a,b) where W is the horizontally constant speed of the upper moving plate. At the lower surface of porous layer z = d m, the fixed plate with non-penetrative boundary induces the condition w m =. (2.6) At the interface z =, we assume the normal velocity and the pressure are continuous and the experimentally suggested condition proposed by Beavers & Joseph (967) for the horizontal velocity is employed. Accordingly, we have w = w m, p = p m, u j z = α (u j u m j ) ( j =, 2), (2.7a c) K where α is the Beavers Joseph constant depending on the property of the porous layer and velocity components are indicated by (u, u 2, u 3 ) = (u, v, w) The basic flow For Poiseuille Couette flow we assume a constant pressure gradient in the x-direction and the upper plate moving with a constant speed W in the same direction. Accordingly, the basic solution to (2.2) and (2.4) with the boundary conditions specified above can be determined. In the fluid layer z d, we have u(z) = 2 A z 2 + A 2 z + A 3 + B z + B 2, (2.8)

5 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at 38 T.-Y. Chang, F. Chen and M.-H. Chang and in the porous layer d m z, we obtain u m = A K. (2.9) In (2.8) and (2.9), the coefficients are A = dp µ dx, A αa d 2 + 2α 2 A d K 2 = αa K 2(αd +, A 3 = A d 2 K + 2αA Kd K) 2(αd +, K) B = αw αd + K, B 2 = W K αd + K. (2.) Obviously, the basic flow in the fluid layer consists of two parts, u(z) = u p + u c, where u p is the Poiseuille flow due to the imposed horizontal pressure gradient defined by u p = (/2)A z 2 + A 2 z + A 3 and u c is the component of Couette flow defined by u c = B z + B 2. Note that the basic flow in the porous medium is driven by the imposed pressure gradient only. In order to interpret the relative magnitude of velocity in both the fluid and porous layers, the length scales in the fluid layer and porous layer are normalized by dividing by d and d m, respectively. The basic velocities in both layers are normalized by V which is the maximum velocity of Poiseuille flow u p. Then we have the following velocity functions for the fluid layer, and U(z) = u V = z2 + 2C z + 2C 2 C 2 2C 2 + W V (D z + D 2 ), z (, ) (2.) for the porous layer, where U m = u m V = 2δ 2 (C 2 2C 2 )ˆd 2, z m (, ) (2.2) C = αδ ˆd α ˆd + 2α 2 δ 2(α ˆd + δ), C 2 = δ ˆd + 2αδ 2 2(α ˆd 2 + δ ˆd), D = α ˆd α ˆd + δ, D 2 = δ α ˆd + δ. (2.3a d) Here the dimensionless parameter ˆd is the depth ratio defined by ˆd = d/d m and δ is the Darcy number defined by δ = K/d m. The speed ratio W/V indicates the relative magnitude of Couette component to that of Poiseuille flow. As will be shown in the present study, the presence of the porous layer is a crucial factor which may affect the system stability characteristics significantly Perturbation equations The governing equations are non-dimensionalized and perturbed by small disturbances to derive the perturbation equations. The scales of length, velocity, time and pressure in the fluid layer are d, V, d/v and µv/d; while in the porous layer they are d m, V m, d m /V m and µv m /d m, in which V m = u m. Based on the basic state, the linearized perturbation equations in the fluid layer z (, ) take the forms u i x i =, (2.4)

6 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 38 ( u Re i t + U u i j + u j x j U i x j ) = p x i + u i, (2.5) where the small perturbations are denoted by the prime symbol and Re is the Reynolds number defined by Re = Vd/ν. In the porous layer z m (, ) they are Re m χ u mi t m u mi x m i =, (2.6) = p m x m i u mi δ 2, (2.7) where Re m satisfy is the Reynolds number given by Re m = V m d m /ν. Note that Re and Re m Re m = 2δ 2 ˆd 3 ( C 2 2C 2 )Re. (2.8) It is assumed that the disturbances are two-dimensional and can be decomposed by the normal modes of the following forms (u i, p ) = [û i (z), ˆp(z)] exp[ia(x ct)], (2.9a) (u mi, p m ) = [û mi(z m ), ˆp m (z m )] exp[ia m (x m c m t m )]. (2.9b) Accordingly, we can obtain the equations of the amplitudes as follows iaû + Dŵ =, Re(U c)iaû + ReU ŵ = iaˆp + (D 2 a 2 )û, Re(U c)iaŵ = Dˆp + (D 2 a 2 )ŵ, (2.2a) (2.2b) (2.2c) for z (, ), and ia m û m + D p ŵ m =, ( ) δ Re ia m c m 2 m û m + ia m ˆp m =, χ ( ) δ Re ia m c m 2 m ŵ m + D p ˆp m =, χ (2.2a) (2.2b) (2.2c) for z m (, ), where D = d/dz, D p = d/dz m and U = du/dz. By introducing the streamfunction ψ which satisfies u = ψ/ z, w = ψ/ x and with a similar definition for ψ m in the porous layer, we can further give the eigenfunctions φ and φ m by ψ = φ(z) exp[ia(x ct)], ψ m = φ m (z m ) exp[ia m (x m c m t m )]. (2.22a,b) The set of (2.2) and (2.2) can be reduced to the following two equations, (D 2 a 2 ) 2 φ = iare(u c)(d 2 a 2 )φ iareu φ, z (, ), (2.23) ( ) δ Re ia m c m 2 m (D 2 p χ a2 m )φ m =, z m (, ). (2.24)

7 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at T.-Y. Chang, F. Chen and M.-H. Chang Note that the wavenumbers a, a m and eigenvalues c, c m are related by a = ˆda m, cre = ˆdc m Re m. (2.25a,b) Equations (2.23) and (2.24) comprise a sixth-order system and the boundary conditions on the upper moving plate z = are On the bottom of the porous layer z m =, and on the interface z = z m =, φ = Dφ =. (2.26) φ m =, (2.27) Reφ = Re m φ m, (2.28) (D 2 a 2 )Dφ + iareu φ iare(u c)dφ = ˆd ( 3 Re m ia m c m Re m ) D Re χ δ 2 p φ m, (2.29) D 2 φ α ˆd δ Dφ + α ˆd 2 Re m D p φ m =. (2.3) δre Hence, the equations (2.23) and (2.24) can be solved together with the boundary conditions (2.26) (2.3) which forms an eigenvalue problem. The D 2 Chebyshev tau method is employed to solve the eigenvalue problem and the details of the numerical procedures could be found in the work of Dongarra, Straughan & Walker (996) The plane Couette flow in a fluid layer overlying a porous layer In the absence of horizontal pressure gradient, the PPC flow reduces to the plane Couette flow. In this two-layer system, the dimensional basic flows in both the fluid and porous layers are u(z) = αw αd + K z + W K αd +, z (, d), (2.3) K u m =, z ( d m, ). (2.32) Taking the speed of the moving boundary W as the velocity scale, the dimensionless forms of base flows can be expressed by U(z) = α ˆd α ˆd + δ z + δ, z (, ), (2.33) α ˆd + δ U m =, z m (, ). (2.34) To non-dimensionalize the governing equations and derive the small perturbation equations, the following scales of length, velocity, time and pressure are employed. In the fluid layer, they are respectively d, W, d/w and µw/d; and in the porous layer they are d m, ν/d m, dm 2 /ν and ρν2 /dm 2, respectively. Accordingly, the linearized perturbation equations in both the fluid and porous layers can be written as follows u i x i =, (2.35)

8 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 383 ( u Re i t + U u i j + u j x j U i x j ) = p x i + u i, (2.36) χ u mi t m u mi x m i = p m x m i =, (2.37) u mi δ 2, (2.38) where the Reynolds number is defined by Re = Wd/ν. Following the same procedures as in the previous subsection, we can obtain the governing equations (D 2 a 2 ) 2 φ = iare(u c)(d 2 a 2 )φ, z (, ), (2.39) ( δ ia ) mc m (D 2 2 p χ a2 m )φ m =, z m (, ), (2.4) where c and c m satisfy cre = ˆdc m. Equations (2.39) and (2.4) also comprise a sixthorder system and the boundary conditions at z = and z m = are the same as (2.26) and (2.27). On the interface z = z m =, the boundary conditions reduce to the following forms Reφ = φ m, (2.4) (D 2 a 2 )Dφ + iareu φ iare(u c)dφ = ˆd ( 3 iam c m ) D Re χ δ 2 p φ m, (2.42) D 2 φ α ˆd δ Dφ + α ˆd 2 δre D pφ m =. (2.43) The eigenvalue problem consisting of (2.39) (2.43) is solved by the same numerical procedures. 3. Results and discussion In the following analyses, we first focus on the effect of the moving upper boundary indicated by the dimensionless speed ratio U = W/V on the stability behaviours of the PPC flow. And then the results for the plane Couette flow will be discussed. The results provide an overview for the stability characteristics of PPC flow in this twolayer system. 3.. The stability of plane Poiseuille Couette flow The depth ratio ˆd is the dominant parameter for the stability problem of a fluid layer overlying a porous layer. Several typical values of depth ratio ˆd are taken into consideration and the other parameters used in the calculations are given by δ = 3, χ =.3 and α =., which are practical properties of many porous materials (Straughan 22). Before proceeding to the stability analysis, we first pay attention to the variation of basic flows with depth ratio as illustrated in figure (a d) for the four assigned values of ˆd with three typical speed ratios of U. It is obvious that the flow velocity U m in the porous layer is independent of U and significantly less than the magnitude of U in the fluid layer. The velocity jump at the interface due to the Beavers Joseph boundary condition is significant in the case of ˆd =., as

9 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at T.-Y. Chang, F. Chen and M.-H. Chang (a). U (b). U (c). (d) FIGURE. The basic states of dimensionless velocities U and U m in the fluid and porous layers for four assigned depth ratios with three typical values of U : (a) ˆd =., (b) ˆd =.4, (c) ˆd =. and (d) ˆd =. shown in figure (a), in which U m = jumps to U =.36,.397 and.485 for U =, and 2, respectively. This result also indicates a large interfacial shear stress at the interface according to (2.7c). However, as the depth ratio increases, the fluid-layer velocity U at the interface decreases quickly and the porous-layer velocity U m diminishes simultaneously. For example, in figure (c) for the case of ˆd =., the velocity at the interface jumps from U m = to U = , and for U =, and 2, respectively. As a result, the fluid-layer velocity at the interface approaches zero with increasing ˆd and the effect of interfacial shear stress at the interface decays gradually. For the case of ˆd =, as shown in figure (d), the velocity U m reduces to a negligible value of and the base flow in the fluid layer is almost the same as the PPC flow in a single fluid layer. For the flow instability behaviours, we first consider the case of ˆd =.. It has been found in the work of Chang et al. (26) that the neutral curves display a bi-modal structure as shown in figure 2(a) for the case of U = in which the component of Couette flow is absent. The dominant mode of instability is determined by the left minimum at lower wavenumber, which is the so-called porous-layer mode at which

10 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 385 (a) (b) (c) Re (d) 6 (e) 6 ( f ) Re a a a FIGURE 2. The variations of neutral curve with velocity ratio U at several assigned values of depth ratio: (a) ˆd =., (b) ˆd =.2, (c) ˆd =.3, (d) ˆd =.4, (e) ˆd =. and ( f ) ˆd = the onset of instability occurs within both the fluid and porous layers and that is primarily dominated by the porous medium. The right minimum at higher wavenumber is named the odd-fluid-layer mode due to the odd-symmetry profile of streamfunction φ with respect to the centreline of fluid layer (Chang et al. 26). Once the Couette flow begins to exert its effect on the system, it is found that both the minima are without any significant variation initially while a small loop appears abruptly below the neutral curve of the two minima when the component of Couette flow is large enough, as shown in the case of U =.3. The onset of instability is then dominated by this mode and the loop grows rapidly with increasing the speed ratio and eventually merges with the upper neutral curve, forming a tri-modal structure as shown in the case of U =.5. Obviously, the third instability mode results from the enhancement of shear in the fluid layer due to the imposed Couette flow, which is named the evenfluid-layer mode because of its even-symmetry eigenfunction of φ in the fluid layer. The shear mode dips lower gradually with increasing the speed ratio U while the other two modes rise, recede and disappear eventually. The corresponding variations of the critical Reynolds number Re c, wavenumber a c and phase velocity c c r are shown in figures 3(a) 3(c), respectively. As illustrated in figure 3(a) for the case of ˆd =., the variation of neutral curves causes the critical Reynolds number to decrease suddenly at U =.3 due to the shift of the instability mode, and then to reduce gradually with increasing U. Such stability behaviours are quite different from those of pure Couette and PPC flows in a single fluid layer. It is well known that the Couette flow in a single fluid layer is unconditionally stable for small disturbances (cf. Drazin &

11 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at T.-Y. Chang, F. Chen and M.-H. Chang (a) (b) (c) FIGURE 3. The variations of (a) critical Reynolds number Re c, (b) critical wavenumber a c and (c) critical phase velocity c c r ; with U for several assigned values of depth ratio ˆd. Reid 985). For the PPC flow in a fluid layer, the work of Potter (966) had found that the presence of Couette flow produces a strong stabilizing effect on the Poiseuille flow. However, the present results show that the imposed Couette flow may destabilize the Poiseuille flow significantly under small depth ratio conditions in such a two-layer system. The Couette flow tends to enhance the shear effect of the Poiseuille flow in the fluid layer initially and thus destabilize the base flow. As the speed of upper plate increases further, the flow instability is gradually dominated by the Couette flow.

12 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 387 Hence, the critical Reynolds number reaches a minimum near U =.79 and then increases monotonically with increasing U, indicating that the base flow is stabilized by the imposed Couette flow when its magnitude is high enough. The corresponding critical wavenumber a c jumps to a higher value at U =.3 and the critical phase velocity c c r jumps to a lower value simultaneously due to the shift of critical mode. The value of a c increases to a maximum near U =.27 and then decreases gradually, while c c r increases monotonically after the shift of instability mode. Similar stability characteristics also can be observed in the typical cases of ˆd =.2 and.3, as shown in 2(b) and 2(c), respectively. In figure 2(b), the right minimum on the bi-modal neutral curve for the case of U = initially determines the critical mode. The imposed Couette flow triggers the onset of a third mode quickly and forms the tri-modal neutral curve, as seen in the case of U =.5. The third mode extends and dips down to dominate the onset of instability with increasing U while the other two modes recede gradually. In figure 2(c), the neutral curve is already a tri-modal structure in the absence of Couette flow at ˆd =.3 and the middle third mode gives the minimum on the neutral curve. It is apparent that an increase of Couette component enhances the growth of the third mode and destabilizes the flow. Note that the critical wavenumber corresponding to the location of the minimum of Re at first moves to the right on the (a, Re)-plane and then moves to the left gradually to form the so-called long-wave instability mode. The variations of Re c and a c with U are illustrated respectively in 3(a) and 3(b) for the typical case of ˆd =.3. It is noted that the results are similar to those of the case ˆd =. when the third shear mode determines the critical mode and the critical Reynolds number exhibits a minimum near U =.48. The critical phase velocity c c r increases monotonically with U, as shown in figure 3(c), since the onset of instability is always dominated by the shear mode in this case. The stability of the present two-layer flow is generally enhanced by increasing the depth ratio, as demonstrated in figure 3(a). Particularly, it is found that the destabilizing effect caused by the component of Couette flow decays gradually as the depth ratio increases. As shown in figure 2(d) for the case ˆd =.4, the lobes of the neutral curve corresponding to both the left porous-layer mode and the right odd-fluid-layer mode disappear and only the even-fluid-layer mode induced by the shear flow in the fluid layer controls the onset of instability. Once the speed ratio U increases, the critical Reynolds number first reduces slightly, as shown in the case of U =.5, and then the minimum on the neutral curve moves to the left horizontally within a small range of Re until U., which also can be observed in figure 3(a). The critical Reynolds number begins to rise significantly as U increases further and an instability mode switch occurs within the narrow range of.8 < U <.9 as indicated in figure 3(a), together with a jump of a c and a discontinuity in the slope of c c r, as shown in 3(b) and 3(c), respectively. The shift of instability mode can be observed clearly by the typical neutral curves displayed in figure 4. It is obvious that the neutral curve becomes bi-modal again and the left minimum determines the critical mode at U = However, a slight increase in U to.879 makes the neutral curve rise and the right minimum becomes lower than the left one to dominate the instability. The switch of instability mode causes the a c jump from.48 to.57. The bi-modal neutral curve occurs only within a limited range of U and the left mode recedes quickly after the occurrence of the mode switch. The characteristics of the right critical mode will be discussed later by examining the magnitude of the eigenfunction and the pattern of the streamfunction of the small disturbance.

13 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at T.-Y. Chang, F. Chen and M.-H. Chang Re a FIGURE 4. The neutral curves of two assigned values of U shift of the instability mode. at ˆd =.4, illustrating the The destabilizing effect of the Couette flow on the PPC flow as the Couette component increases from zero totally vanishes when the depth ratio is large enough. On the contrary, the presence of Couette flow begins to exhibit a stabilizing effect as shown in figure 2(e) for the case ˆd =.. The neutral curves rise and recede to the left with increasing U. Especially, the phenomenon regarding the shift of the instability mode when the velocity ratio U exceeds a certain critical value is more pronounced as ˆd increases. One can see that a local minimum gradually appears on the right branch to form a bi-modal structure on the neutral curve. Once the velocity ratio U is greater than.87 in this typical case, the critical mode shifts abruptly from the left long-wave mode to the right one at higher wavenumber. The long-wave instability mode disappears eventually as illustrated in the neutral curve of U = 2. and the right mode totally controls the onset of instability. The corresponding variation of a c, as displayed in figure 3(b), shows a more significant discontinuity due to the shift of the instability mode at U.87. Simultaneously, the critical phase velocity c c r also jumps to a higher value as illustrated in figure 3(c). The typical case ˆd = is taken into consideration and the results indicate that the influence of porous layer could be ignored when the thickness of the fluid layer is much greater than that of porous layer. The variations of the neutral curves in figure 2( f ) and the critical Reynolds number with U in figure 3(a) are exactly the same as the results found by Potter (966) that concluded that the superimposed Couette flow has a strong stabilizing effect on the Poiseuille flow. Besides, the critical Reynolds number and wavenumber for U = are respectively and 2.376, which are also in good agreement with the results of the Orr Sommerfeld equation in the common Poiseuille channel flow with Re c = and a c = 2.4 (Drazin & Reid 985) based on the channel height as the characteristic length. The comparison is more excellent for higher ˆd which verifies the accuracy of the present numerical algorithm. As illustrated in figure 3(a), the critical Reynolds number approaches infinity near U.7 which is the so-called cutoff velocity and the critical wavenumber approaches zero. Once the speed of the moving boundary is greater than the cutoff velocity, the PPC flow becomes unconditional stable for all infinitesimal disturbances. It is noted that in the range of.2 < U <.4, there exists

14 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 389 (a) (b) z (c) (d) z x x FIGURE 5. The normalized eigenfunctions φ, φ m and the corresponding patterns of the streamfunction ψ for the four typical values of U at ˆd =.: (a) U =, (b) U =.3, (c) U =.4 and (d) U = 2.. a local minimum which indicates the flow becomes more unstable with increased Couette flow. Potter observed that this effect occurs when U is approximately equal to the travelling phase velocity c r. A more detailed explanation for this phenomenon was given in the study of Özgen et al. (27), in which they proposed that this stability behaviour is related to the evolution of U with the phase velocity and the exchange of energy between the perturbation and the base flow due to the change in the location of the critical layer. The critical layer is defined as the location where both the phase velocity and the base flow velocity are equal. It plays an important role in transferring the energy of the base flow to the disturbances. They observed that the critical layer is very close to the moving wall in the range of.2 < U <.25 and thus produces a destabilizing effect. The present results show that the critical velocity ratio of U corresponding to the shift of instability mode from the long-wave mode to the mode at higher wavenumber decreases gradually with increasing ˆd. Once ˆd is large enough, the phenomenon of mode switch vanishes and only the neutral curve corresponding to the long-wave instability mode may appear on the (a, Re) plane. The instability behaviours are in good agreement with those studies for PPC flow in a single fluid layer. The characteristics of the instability modes can be explored further by observing the profiles of the normalized eigenfunctions φ and φ m, and the corresponding patterns of streamfunctions ψ and ψ m at the critical states, as displayed in figure 5(a d) for the four typical cases of U at ˆd =.. It is noted that both the vertical z-axis and horizontal x-axis are scaled by the thickness of the fluid layer. Thus, the fluid layer is always indicated in the range of z (, ) and the range of porous layer depends on the depth ratio in the following similar figures. In figure 5(a), the magnitude of

15 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at 39 T.-Y. Chang, F. Chen and M.-H. Chang streamfunction penetrates into the porous layer for the porous-layer mode and the corresponding flow pattern occupies both the fluid and porous layers at U =. Above the interface, the magnitude of φ first decreases slightly with increasing z, reaches a local minimum and then increases gradually to the maximum. Such a variation causes the occurrence of a small vortex motion on the streamfunction pattern of the small disturbances which occupies the thin region across the interface. It has been pointed out by Lyubimova et al. (26) that the rapid reduction of tangential velocity at the interface may result in flow instability similar to the Kelvin Helmholtz instability. Obviously, the small vortex motion at the interface is induced by the significant velocity jump of the basic state, as illustrated in figure (a). The onset of instability shifts to the fluid-layer mode at U =.3, in which the streamfunction pattern concentrates mainly within the fluid layer with shorter wavelength. The magnitude of φ then gradually penetrates into the porous layer again with increasing U as shown in figure 5(b d). Similar variations of the streamfunction pattern can also be observed for the cases of ˆd =.2 and.3. For the case of ˆd =.4, the fluid-layer mode is predominant at U = and the pattern is within the fluid layer only, as illustrated in figure 6(a). The pattern of the streamfunction gradually penetrates into the porous layer and the wavelength increases simultaneously, as shown in figures 6(b) and 6(c) for U =. and.5, respectively. Once the speed ratio U increases to.879, the Couette flow gradually prevails over the Poiseuille flow and induces the shift of the instability mode as shown in figure 4. Obviously, the left critical mode corresponds to the fluid-layer mode such that the magnitude of the streamfunction is mainly within the fluid layer, while the right one corresponds to the porous-layer mode because the unstable wave penetrates into the porous layer again, as shown in figure 6(d) for the case of U = 2.. A small and weak cell on the interface similar to that in figure 5(a) is also observed. As shown in figure (a d), the effect of the horizontal velocity jump reduces rapidly with increasing ˆd. Hence, the vortex motion at the interface induced by the velocity jump decays quickly in the porous-layer mode. The critical wavelength is shortened and the porous-layer mode becomes dominant after the shift of the instability mode. This phenomenon for the instability mode switch is more apparent for the case of ˆd =. as U near.87. The results are shown in figure 7(a,b) for the typical cases of U =.8 and.85. One can see that the magnitude and pattern of φ penetrate into the porous layer after the occurrence of mode switch as the effect of Couette flow becomes pronounced in the PPC flow. Note that the small vortex motion in the porous-layer mode at the interface totally vanishes in figure 7(b). However, once the influence of porous layer is negligible at higher depth ratio, for example, ˆd =, the behaviour of the instability mode switch from the fluid-layer mode to the porous-layer mode at high velocity ratio vanishes eventually. Only the fluid-layer modes are observed as shown in figure 8(a,b) for the two typical cases of U =.3 and.65 at ˆd = The stability of plane Couette flow It is well known that the Couette flow in a horizontal fluid layer is unconditionally stable with respect to small disturbances. Since the superimposed Couette flow may destabilize the Poiseuille flow in the present two-layer system, it is interesting to explore whether the pure Couette flow may become conditionally stable due to the existence of the porous layer. It is noted that there is no base flow in the porous layer, as indicated by (2.34). A velocity jump at the interface z = due to the Beavers Joseph boundary condition may possibly induce the occurrence of flow

16 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 39 (a).. (b).. z (c).. (d).. z x x FIGURE 6. The normalized eigenfunctions φ, φ m and the corresponding patterns of the streamfunction ψ for the four typical values of U at ˆd =.4: (a) U =, (b) U =., (c) U =.5 and (d) U = 2.. (a).. (b) z x x FIGURE 7. The normalized eigenfunctions φ, φ m and the corresponding patterns of streamfunction ψ for the two typical values of U at ˆd =.: (a) U =.8, (b) U =.85. instability. From (2.33), it is obvious that the fluid velocity at the interface depends on the Darcy number δ, Beavers Joseph constant α and the depth ratio ˆd. For the present assigned values of δ and α, the fluid velocity at the interface is only 9. % of the upper plate velocity for the case ˆd =.. This velocity ratio decreases with increasing ˆd which indicates the significance of the velocity jump at the interface decreases gradually. Because the tangential velocity jump at the interface is small even though at low depth ratio, it is to be expected that the Couette flow is still always stable. The corresponding linear stability problem has been solved numerically and the results indeed reveal that the pure Couette flow in this two-layer system is still an unconditionally stable flow. Table lists the leading eigenvalues at different

17 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at T.-Y. Chang, F. Chen and M.-H. Chang (a).. (b) z x x FIGURE 8. The normalized eigenfunctions φ, φ m and the corresponding patterns of streamfunction ψ for the two typical values of U at ˆd = : (a) U =.3, (b) U =.65. ˆd Re c i c r TABLE. For pure Couette flow, the leading eigenvalues with largest imaginary parts with respect to Re for several assigned values of ˆd at a =.. Reynolds numbers for several assigned depth ratios. It is found that no matter how large the Reynolds number is, the growth rate represented by the imaginary part of the eigenvalue c, c i, is always negative. Figures 9(a) and 9(b) illustrate the spectrum of the typical case of ˆd =. at Re = 4 and 5, respectively. One can see that the value of c i is closer to the neutral state (i.e. c i = ) with increasing Re. However, it never changes sign to become positive in any the cases considered here. 4. Conclusions The linear stability problem for the PPC flow in a fluid layer overlying a porous layer saturated with the same viscous fluid has been investigated in this work. The results reveal that the stability characteristics in such a two-layer system depend heavily on the depth ratio ˆd. For a smaller ˆd, for example, ˆd., the imposed Couette flow on the Poiseuille flow is found to enhance the shear in the fluid layer and exhibits a significant destabilizing effect initially on the base flow. The neutral curve is shifted from a bi-modal to a tri-modal form and the third mode induced by the Couette flow is the fluid-layer mode such that the streamfunction pattern of the critical disturbance is mainly within the fluid layer. However, once the magnitude

18 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at Stability of Poiseuille Couette flow in a fluid overlying a porous layer 393 (a) (b) FIGURE 9. The spectra of the eigenvalues at a =. with ˆd =.. (a) Re = 4 (b) Re = 5. and of Couette flow is large enough, the critical Reynolds number begins to increase with the speed ratio U after reaching a minimum. This result indicates the Couette flow begins to prevail over the Poiseuille flow to control the stability and eventually exhibits a stabilizing effect. By observing the variation of the streamfunction patterns, it is also found that the critical instability mode gradually penetrates into the porous layer with increasing magnitude of the Couette flow. That is, the onset of instability appears to shift gradually from the fluid-layer mode to the porous-layer mode with increasing U. At larger values of ˆd, such a switch of critical mode will occur abruptly and accompany a significant jump in the critical wavenumber. If ˆd increases further, the influence of the porous layer becomes negligible and finally the phenomenon of instability mode switch vanishes. Under such circumstances, the imposed Couette flow exerts a strong stabilizing effect and the stability characteristics are exactly the same as those of the PPC flow in a single fluid layer. The present results reveal that the presence of a porous layer significantly changes the stability characteristics of the PPC flow. At smaller ˆd, the upper moving boundary may significantly destabilize the base flow in contrast with the result for a single fluid layer where the imposed Couette flow produces a strong stabilizing effect. At moderate ˆd, the Couette flow enhances the flow stability greatly at high speed ratio U, while the onset of instability is dominated by the porous-layer mode eventually and the flow is still conditionally stable with the existence of a critical Reynolds number within the range of speed ratio U considered. Nevertheless, a cutoff speed appears at high ˆd and the flow becomes unconditionally stable once the magnitude of the moving boundary velocity exceeds the cutoff speed. The case of pure Couette flow is also studied and the results show that it is still an unconditional stable flow in such a two-layer system. It is noted that the instability may be affected by the Beaver Joseph constant α and the Darcy number δ. Several studies have taken the effects of α or δ into consideration for the instability of plane Poiseuille flow in this fluid porous system (cf. Chang et al. 26; Liu et al. 28; Lyubimova et al. 26). As discussed in the work of Chang et al. (26), a larger α tends to trigger the onset of instability in the fluid layer, leading to a more unstable system. A lower δ indicates a smaller permeability

19 Downloaded from National Taiwan University Library, on 9 Sep 27 at 9:53:45, subject to the Cambridge Core terms of use, available at T.-Y. Chang, F. Chen and M.-H. Chang in the porous layer and the fluid layer becomes dominant in the system. Instead of Darcy s law, different models also can be employed to simulate the flow in the porous layer such as Brinkman s model (Liu et al. 28) and the Darcy Forchheimer model (Lyubimova et al. 26). The boundary condition at the fluid porous interface also plays an important role affecting the instability behaviours as in the work of Bars & Worster (26). A future study considering these factors would be helpful to provide more complete instability characteristics for the PPC flow in this fluid porous system. REFERENCES BARS, M. L. & WORSTER, M. G. 26 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 55, BEAVERS, G. S. & JOSEPH, D. D. 967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 3, BERGSTRÖM, L. B. 25 Nonmodal growth of three-dimensional disturbances on plane Couette Poiseuille flows. Phys. Fluids 7, 45. CHANG, M.-H. 25 Thermal convection in superposed fluid and porous layers subjected to a horizontal plane Couette flow. Phys. Fluids 7, 646. CHANG, M.-H. 26 Thermal convection in superposed fluid and porous layers subjected to a plane Poiseuille flow. Phys. Fluids 8, 354. CHANG, M.-H., CHEN, F. & STRAUGHAN, B. 26 Instability of Poiseuille flow in a fluid overlying a porous layer. J. Fluid Mech. 564, COWLEY, S. J. & SMITH, F. T. 985 On the stability of Poiseuille Couette flow: a bifurcation from infinity. J. Fluid Mech. 56, 83. DEEPU, P., ANAND, P. & BASU, S. 25 Stability of Poiseuille flow in a fluid overlying an anisotropic and inhomogeneous porous layer. Phys. Rev. E 92, 239. DONGARRA, J. J., STRAUGHAN, B. & WALKER, D. W. 996 Chebyshev tau-qz algorithm methods for calculating spectra of hydrodynamic stability problems. Appl. Numer. Maths 22, DRAZIN, P. G. & REID, W. H. 98 Hydrodynamic Stability. Cambridge University Press. GUHA, A. & FRIGAARD, I. A. 2 On the stability of plane Couette Poiseuille flow with uniform crossflow. J. Fluid Mech. 656, HAINS, F. D. 967 Stability of plane Couette Poiseuille flow. Phys. Fluids, HAINS, F. D. 97 Stability of plane Couette Poiseuille flow with uniform crossflow. Phys. Fluids 4, HILL, A. A. & STRAUGHAN, B. 28 Poiseuille flow in a fluid overlying a porous medium. J. Fluid Mech. 63, HILL, A. A. & STRAUGHAN, B. 29 Poiseuille flow in a fluid overlying a highly porous material. Adv. Water Resour. 32, KUMAR, A. A. P., GOYAL, H., BANERJEE, T. & BANDYOPADHYAY, D. 23 Instability modes of a two-layer Newtonian plane Couette flow past a porous medium. Phys. Rev. E 87, 633. LIU, R., LIU, Q. S. & ZHAO, S. C. 28 Instability of plane Poiseuille flow in a fluid-porous system. Phys. Fluids 2, 45. LYUBIMOVA, T. P., LYUBIMOV, D. V., BAYDINA, D. T., KOLCHANOVA, E. A. & TSIBERKIN, K. B. 26 Instability of plane-parallel flow of incompressible liquid over a saturated porous medium. Phys. Rev. E 94, 34. MOYERS-GONZALEZ, M. & FRIGAARD, I. 2 The critical wall velocity for stabilization of plane Couette Poiseuille flow of viscoelastic fluids. J. Non-Newtonian Fluid Mech. 65, NOUAR, C. & FRIGAARD, I. 29 Stability of Couette Poiseuille flow of shear-thinning fluid. Phys. Fluids 2, 644. ÖZGEN, S., DURSUNKAYA, Z. & EBRINC, A. A. 27 Heat transfer effects on the stability of low velocity plane Couette Poiseuille flow. Heat Mass Transfer 43, POTTER, M. C. 966 Stability of plane Couette Poiseuille flow. J. Fluid Mech. 24,