Transfer of mass and momentum at rough and porous surfaces

Transfer of mass and momentum at rough and porous surfaces

Uǧis Lācis1, Y. Sudhakar1,2 Email address for correspondence: ugis@mech.kth.se    Shervin Bagheri1
Abstract

The surface texture of materials plays a critical role in wettability, lubrication, turbulence and transport phenomena. In order to design surfaces for these fluid applications, it is desirable to characterise non-smooth and porous materials by their ability to exchange mass and momentum with flowing fluids. Whereas the physics underlying tangential (slip) velocity at a fluid-solid interface is well understood, the treatment and importance of normal (transpiration) velocity and normal stress is unclear. We show that, when slip velocity varies at an interface above the texture, a non-zero transpiration velocity arises from mass conservation. The ability of a given surface texture to accommodate for normal velocity of this kind is quantified by a transpiration length. We further demonstrate that normal momentum transfer gives rise to resistance coefficients. For a porous material, they measure how normal fluid stress outside the porous medium is partitioned between the interstitial flow and the solid skeleton. The introduced measures of slip, transpiration and resistance can be determined for any anisotropic non-smooth surface by solving five Stokes problems. Moreover, these measures appear as constant coefficients in effective interface conditions. We validate and demonstrate the physical significance of the effective conditions on two canonical problems – a lid-driven cavity and a turbulent channel flow, both with non-smooth bottom surfaces. Our formulation provides a systematic approach for detailed characterisation and design of surfaces.

1 Introduction

The physical behaviour of a number of fundamental fluid systems is dramatically modified by the presence of small-scale surface roughness. For example, in partial wetting (figure 1a) – that is when a liquid in contact with solid reaches balance of surface tensions – the resulting apparent contact angle is very sensitive to the details of the surface texture (Wenzel 1936; Quéré 2008). Similarly, when a solid particle sediments in a quiescent fluid very close to a wall (figure 1c), it will experience a drag force which is sensitive to the texture of the wall (Jeffrey & Onishi 1981; Kaynan & Yariv 2017). At higher Reynolds numbers, the pressure loss in turbulent pipes is a function of wall roughness (figure 1b) (Nikuradse 1950; Jiménez 2004). Yet another example is transport phenomena involving porous media, where the exchange of energy, mass and particles (nutritions, ions, proteins, etc.) between a free flowing fluid and a porous medium depends very much on the roughness at the interface between the two domains (figure 1d).

Figure 1: (Colour online) Examples of problems that are sensitive to surface properties. Droplet spreading over rough surface (a), friction drag of turbulent flow on rough surface (b), sedimentation of particle near a rough wall (c) and particle transport through the interface with rough porous material (d).

Engineers take advantage of the sensitivity to surface texture to control large-scale flow and transport phenomena in a wide range of applications. Efficiency of heat exchangers (Mehendale et al. 2000; Agyenim et al. 2010) often depends on small surface roughness. In scaffold design for bone regeneration, the cell growth on the implant (a porous biomaterial such as calcium phosphate cements) depends on the interaction between the surrounding liquid and the surface texture of the implant (Dalby et al. 2007; Perez & Mestres 2016). The performance of fuel cells depends on the ability of gas flow to efficiently transport water vapour away from a thin porous medium, which in turn depends on the design of the gas diffusion layer (Prat 2002; Haghighi & Kirchner 2017). Turbulent skin friction on wings or turbine blades can be reduced by careful design of complex textures such as riblets, which are able to push quasi-streamwise vortices away from the wall (Walsh & Lindemann 1984).

1.1 Scope of the present work

The design of surface texture in the examples mentioned above is based on a trial and error procedure, that may require tremendous amount of effort, time and expensive surface manufacturing equipment. We lack a general formalism to characterise surfaces as to their ability to interact with fluids; this constrains our ability to efficiently design surfaces in applications. The only wide-spread fluid-related measure of non-smooth surfaces is the slip length , which is related to the Navier slip condition (Navier 1823),

(1.0)

Here, is the slip velocity (along the streamwise coordinate ) at an interface above the textured surface and is the wall-normal coordinate. Geometrically – as shown in figure 2(a) – the slip length is the distance that the velocity profile has to be extrapolated to reach zero value. Physically, it quantifies the extent which the tangential friction force is modified due to the presence of texture. A large slip length is associated with smaller resistance imposed on a horizontal fluid layer just above the texture, and thus serves as a measure of exchange of tangential momentum. For a viscous-dominated flow (low Reynolds number in the immediate vicinity of the surface) and assuming small textures compared to system scale, the slip length is a property of surface texture only, and thus a valuable characteristic for surface design.

Figure 2: (Colour online) Schematics of slip length (a), transpiration length (b) and resistance coefficient (c).

However, a non-smooth surface and an overlying fluid also interact through the exchange of mass and normal momentum; the level of exchange depends on the details of the surface texture, which means that, in addition to the slip length, other surface measures exist. In this paper, we introduce a transpiration length , which quantifies how much a surface texture allows exchange of mass with its surrounding due to streamwise variation of slip velocity, i.e.

(1.0)

Here, is the wall-normal velocity. Using continuity – and 2D flow for simplicity – the above condition can be written as . Geometrically (figure 2b), the transpiration length is thus the distance below the interface for which a non-zero transpiration velocity – arising from mass conservation – can exist. This depth is obtained under assumption of linear decay of the velocity with a slope . In other words, a large transpiration length is associated with smaller resistance for exchange of mass flow between the free fluid and the non-smooth surface.

In the case where the textured surface constitutes the top-most layer of a thick porous material, one may define additional measures arising from the roughness at the interface with the free flow. A normal force balance at the interface plane of the texture due to interface normal velocity yields

(1.0)

where the left hand side is the normal stress of the outside free flow on the interface plane. Here, is free fluid pressure and is free fluid viscosity. The right hand side is composed of the Darcian fluid pressure of the porous material and the stress borne by the solid skeleton of the porous material. The latter contains a resistance coefficient that provides a quantitative measure of how fluid stress outside the porous medium is partitioned between the interstitial flow and solid skeleton of the porous material (figure 2c). Large force coefficients – again properties of the surface of the porous material only – mean that a large fraction of shear stress can be borne by the solid skeleton of the material. For a given texture, the knowledge of these force coefficients provides important information of the diffusive/advective transport into the material as well as the ability of the solid skeleton to resist and absorb externally imposed shear stress.

In this paper,

  • we propose a set of measures (slip, transpiration, resistance coefficients) that provide a complete and flow-independent characterisation of surface textures. The surface coefficients and discussed above are generalised for three-dimensional deformable anisotropic non-smooth surfaces and introduced based on physical arguments of force balance and mass conservation;

  • we describe how these measures are obtained for any surface texture which – under the assumptions of creeping flow and length scale separation – is based on the Stokes solutions of five fundamental problems;

  • we validate the proposed coefficients by comparing effective simulations with geometrically resolved direct numerical simulations and we provide a showcase – a turbulent channel flow at friction Reynolds number 180 – to demonstrate the physical significance of the proposed coefficients.

1.2 Relation to previous work

The Navier slip condition (1.1), the transpiration condition (1.1) and force balance (1.1) can be considered as effective boundary conditions imposed at a planar interface. The main motivation behind the development of effective conditions is to overcome the numerical limitations imposed by resolving the microscopic details of small-scale surface texture.

By exploiting separation of characteristic lengths scales (between surface texture and fluid system), these methods typically employ a smoothed representation of fine grained materials and a sharp interface (so called two-domain modelling) between the material and surrounding medium. Some recent examples of effective modelling applied to drying, cell growth, heat exchange and free fluid flow can be found in works by Mosthaf et al. (2014); Vaca-González et al. (2018); Laloui et al. (2006); Lācis et al. (2017); Wang et al. (2018).

In general however, the effective boundary conditions used can be very different depending on the application. For example, a number of different effective models for over rough or textured surfaces can be found both for turbulent (Adams et al. 2012; Gómez de Segura et al. 2018; Zampogna et al. 2019) and laminar flows (Luchini et al. 1991; Stroock et al. 2002; Kamrin et al. 2010; Kamrin & Stone 2011; Luchini 2013). In particular, the boundary conditions between free fluid and porous beds has been the subject of much debate. A range of empirical condition has been proposed (Beavers & Joseph 1967; Han et al. 2005; Le Bars & Grae Worster 2006; Rosti et al. 2015; Zampogna & Bottaro 2016) and rigorous derivations of these empirical boundary conditions have been carried out (Mikelić & Jäger 2000; Jäger & Mikelić 2009; Carraro et al. 2015, 2018; Lācis & Bagheri 2016; Zampogna et al. 2019). Yet, it is unclear for many applied scientists if a pressure jump condition arises at the interface between free fluid and porous bed, and if so, what jump condition should be imposed.

Moreover, different approaches and viewpoints have been adopted by communities working with porous walls and on rough/textured surfaces. Despite these disperse approaches, it has been observed that flow over patterned surfaces and flow over porous media exhibit several common characteristics. At low permeabilities, the main effect of porous media is the slip flow at fluid-porous interfaces (Hahn et al. 2002); this feature is the same for walls with smaller roughness scales (Luchini et al. 1991). This gives an idea of a unified model, which could be suitable for describing all the mentioned configurations.

1.3 Structure of the paper

In section 2, we provide a physical introduction of proposed boundary conditions and a general framework for characterizing textured and porous surfaces. The procedure to determine surface measures – coefficients appearing in proposed boundary conditions – is outlined in section 3. In section 4 we validate the newly proposed boundary conditions using laminar lid-driven cavity flow. In section 5, we illustrate the importance of modelling small surface normal velocity using turbulent channel flow. Finally, we provide conclusions in section 6. The interested reader can find more detailed derivations, notes on non-dimensionalizing fundamental problems as well as scale estimates in Appendices A to D.

Figure 3: (Colour online) Effective description of flow over a complex surface. The left frame shows the microscopic porous surface exposed to a vortical free flow. To the right, the same configuration is described using effective (smoothed) model. This involves an artificial interface at coordinate (denoted with ), at which effective boundary conditions are specified.

2 Effective boundary conditions

We assume that there exists two distinct length scales in the system. A small length scale related to surface structures (such as size of spheres within the porous material, figure 3, left) and a large length scale characterizing to relevant large scale features of the flow field (such as a fluid vortex, figure 3, left). As illustrated in the right frame of figure 3, an effective representation of a complex surface involves defining an artificial boundary – interface at coordinate (also denoted with ). Below the interface, flow features are either neglected (for thin surfaces) or described using effective approaches (for thick porous surfaces). The interaction between these two domains is modelled by a set of boundary conditions applied at this interface. The accuracy of such description is determined by how closely these boundary conditions describe the physical features of the flow field in the neighbourhood of the interface.

In this section, we will describe effective boundary conditions for surfaces with gradually increasing complexity. We will begin with smooth and rough/patterned surfaces, followed by elastic and poroelastic surfaces. A porous surface will be treated as a special case of a poroelastic one. Our approach is intuitive as well as physics-motivated and is based on principles of mass conservation and force balance. These basic laws have to be complemented with a force model stating in what way force depends on various system parameters. At the end, we gather all the boundary conditions from different surfaces in a single set, which can be viewed as a physical framework suitable for modelling interaction between free flow and wide variety of complex surfaces.

2.1 Flow over a rigid smooth wall

A standard choice of boundary conditions for viscous flows over a rigid smooth wall is the well-known no-slip condition (Leal 2007)

This condition arises due to a very large friction force between the wall and nearby fluid. From molecular viewpoint, this friction force is sufficiently strong to drive the fluid molecule velocity to a very small value, which can be approximated with zero. The no-slip boundary condition is thus an effective boundary condition at length scales larger than the molecular size . Consequently this condition is not valid at molecular scales, in which there is a slip (Thompson & Troian 1997; Benzi et al. 2006). The no-slip condition is complemented with the kinematic condition for the wall normal velocity

to ensure no leakage of fluid through the wall. In this paper, we use and coordinates as tangential () directions and coordinate as normal () direction interchangeably.

2.2 Flow over a textured surface

Next, we consider a rigid wall with some texture/roughness characterised by a length scale . For such a surface, we assume that the length scale is much larger than molecular size , leading to the no-slip condition (section 2.1) on the surface of the texture. We denote length as microscale.

2.2.1 Tangential interface velocity

The commonly used effective interface condition for tangential velocity is Navier-slip condition (Navier 1823; Miksis & Davis 1994; Sarkar & Prosperetti 1996; Luchini 2013; Bolanos & Vernescu 2017),

(2.0)

where is vector containing fluid shear stress in both tangential directions, and is the surface slip length tensor, closely related to mobility tensor (Stroock et al. 2002; Kamrin et al. 2010; Kamrin & Stone 2011). The existence of slip velocity have been shown analytically  (Saffman 1971; Mikelić & Jäger 2000; Jäger & Mikelić 2001, 2009), numerically (Sahraoui & Kaviany 1992) as well as experimentally (Gupte & Advani 1997). The slip boundary condition can also be rewritten in a friction coefficient form

(2.0)

where is a friction coefficient matrix, similar as used by Bocquet & Barrat (2007). This expression is a linear force model, describing the resulting friction force on the flow from the textured wall. The friction at the effective flat interface is much smaller than at rigid smooth wall due to the existence of fluid between the surface texture. This internal fluid is able to respond to applied stress from the free fluid and move along with the lower layer of the free fluid. Fluid movement within the textured surface thus induces a non-negligible slip velocity.

Figure 4: (Colour online) A control volume (CV) below the interface () is used to determine the interface normal velocity condition (2.0). The width of the CV is , whereas and are slip velocities at the left and right sides of the CV, respectively. The velocity profiles linearly decay to zero over slip length (i.e. component of slip length tensor ).

2.2.2 Wall-normal interface velocity

It is common to set the wall-normal velocity to zero (Mohammadi & Floryan 2013; Jiménez Bolaños & Vernescu 2017). This condition – although often appropriate – is only the leading order approximation (discussed further in section 3.3). To obtain a more accurate condition, let us consider a two-dimensional flow, in which there is a slip velocity variation . We define a control volume (CV) near the textured surface as shown in figure 4. The net mass flux in the -direction through the CV boundaries is

where is a proportionality constant linking the slip velocity at the interface with a mass flux below the interface. The mass flux in -direction has to be balanced by a flux in the -direction. Since there is a rigid wall at the bottom of the CV, this flux can occur only at the interface with the free fluid . Setting both fluxes equal yields a vertical velocity

which in the limit of infinitesimal CV in the -direction becomes . Here it is apparent that if there is no fluid flux below the interface – this happens for molecular slip, because there are no fluid molecules below the slipping free fluid layer – then there is no generation of wall-normal velocity and the condition is exact. The no-penetration condition is also exact if there is no spatial variation of slip velocity.

More generally, applying mass conservation to a three dimensional control volume below the interface results in the following expression for the wall-normal velocity component

(2.0)

where is slip velocity variation vector and is a transpiration length vector. Using this notation, one can observe that the first transpiration length component is the same as the coefficient introduced above, i.e. . A more detailed discussion will follow in section 3 and appendix A.

2.3 Flow over an elastic surface

The next complex surface considered is an elastic surface with a finite thickness. To solve the coupled fluid-surface interaction problem, one introduces a displacement field for the elastic surface. The displacement field gives rise to a solid stress tensor (governed by either linear or non-linear law). To compute the displacement field, a force balance condition is employed

(2.0)

where is the free fluid stress tensor and is the unit normal vector of the surface. Solving a time dependent linear elasticity problem subjected to time dependent free fluid forcing will give access to displacement and velocity solutions.

To complete the set of boundary conditions, the interface fluid velocity is determined in analogous way as discussed in sections 2.1 and 2.2,

(2.0)
(2.0)

where are interface tangential displacements and is the interface normal displacement. When the wetted surface is smooth, and , the velocity conditions (2.02.0) simplify to , which is the classical kinematic condition. This condition, together with stress balance (2.0), have been widely used in computational modelling of various coupled fluid-structure-interaction phenomena (Küttler & Wall 2008; Wall & Rabczuk 2008). If the material is very stiff, velocity conditions (2.02.0) reduce to those introduced for rigid smooth or textured wall (2.0,2.0) and stress condition (2.0) becomes redundant.

2.4 Flow over a poroelastic surface

Next, consider a porous and elastic surface with two constituents (one solid, and one liquid). To fully describe this surface, we need an effective displacement field as well as an effective fluid velocity , i.e. describing the interstitial fluid below the interface. This seepage flow through porous material is described using Darcy’s law , where is the pore pressure and is the tensorial version of the classical permeability. Note that is fluid velocity relative to the solid velocity ; in the absence of any driving forces imposed directly on the fluid, the fluid will move with the solid skeleton due to the no-slip condition at microscopic pore scale.

2.4.1 Velocity conditions at the interface

For a poroelastic surface the velocity conditions for the free fluid are

(2.0)
(2.0)

Note that the interface permeability is smaller compared to the interior permeability , because the flow around the interface region is exposed to the porous material only below the interface and therefore experiences smaller resistance. The interface normal permeability , however, is the same as the interior permeability due to mass conservation principle (Lācis & Bagheri 2016). Moreover, from mass conservation, the variation of the interface Darcy velocity ( in equation 2.0) will induce a transpiration length . The associated transpiration velocity is however negligible and not considered here, since the Darcy contribution is much smaller than the slip velocity contribution. The relative sizes of the terms appearing the boundary conditions above will be discussed in section 3.3.

2.4.2 Stress conditions at the interface

In order to solve the full problem, one has to obtain both displacement of the poroelastic surface and the relative flow velocity inside the porous surface. To this end, we define a total stress tensor of the composite poroelastic material, containing contribution from both the solid skeleton and the fluid component . A force balance at the interface yields

(2.0)

This condition, however, is not sufficient for solving for both displacement field and pore fluid velocity , as there are not enough boundary conditions for a given number of unknowns.

Since the pore flow is governed by the Darcy’s law, the pore fluid stress tensor is a diagonal tensor

It is well known that to solve for the pore pressure, a pressure condition is needed at the interface (Ene & Sanchez-Palencia 1975; Levy & Sanchez-Palencia 1975; Hou et al. 1989; Mikelić & Jäger 2000; Jäger & Mikelić 2009; Marciniak-Czochra & Mikelić 2012; Carraro et al. 2013; Lācis & Bagheri 2016; Lācis et al. 2017; Carraro et al. 2018).

The pressure condition can be obtained by considering the normal stress balance between the free fluid and the composite surface,

(2.0)

The terms on the right hand side correspond, respectively, to stress transferred to the pore pressure and a friction term induced by the presence of solid skeleton. The velocity appearing in the friction term is the fluid velocity relative to the solid velocity; if the fluid parcels follow the motion of the surface, there is no friction. The resistance (or friction) coefficient vector provides the force in interface normal direction due to velocity in any direction. Note that normal stress balance (2.0) is different from the general force balance at the interface (2.0) because the stress absorbed by solid skeleton is directly modelled using resistance coefficient .

The force balance equation (2.0) can be analysed as a pressure condition, but we propose an alternative interpretation. We rewrite the condition in terms of the interior fluid stress tensor

On the right hand side we factor out the total fluid stress

This expression can be viewed as a stress partition condition. We introduce a partition function , which is a function of flow, displacement variables and pore geometry. Using , the force balance expression can be rewritten as

which shows the fraction of the total free fluid stress transferred to the fluid within the complex surface. Consequently the fraction of the total stress absorbed by a solid skeleton is . We have only considered a partition function for the normal stress. This is a direct consequence of choosing Darcy’s law as a model for the pore fluid; in such approach, pore fluid can only absorb the normal stress. In a more general situation, the partition function could be a tensor, containing information about partitioning of all stress components. The idea of stress partitioning has been proposed by Minale (2014). However, to the best of authors’ knowledge this is the first time a connection between pressure condition and stress partitioning has been made.

As a final remark, note that a porous surface is a special case of the poroelastic surface. The porous limit corresponds to sufficiently stiff solid skeleton or sufficiently slow free fluid such that any displacement can be neglected. To model these configurations, the total stress continuity (2.0) becomes redundant.

2.5 General framework

Figure 5 summarises the conditions stemming from the transfer mass and momentum between free flowing fluid and all the surfaces considered. The boundary conditions are

(2.0)
(2.0)
(2.0)
(2.0)
(2.0)

where we have complemented the previously explained expressions with additional stress partitioning expressions. This is done based on a conjecture that the same approach would be applicable also for complex surfaces consisting of components (see figure 5).

Figure 5: Illustration of the boundary condition framework proposed in the current work. Interaction between free fluid and different complex surface types are noted with connection through arrows. The corresponding equations are marked below each surface type. The multicomponent surface is included as a conjecture.

To give an overview, we shortly summarise all conditions. First, equations (2.02.0) are the dynamic and kinematic conditions, respectively; they determine free fluid velocity near the complex surface from a linear friction model (2.0) and mass conservation (2.0). In these equations, is the slip length tensor, is the fluid shear stress vector, is the transpiration length due to slip variations and is the slip velocity variation vector. The next expression in the list, (2.0), is the total stress continuity, which stems from the force balance. This condition is necessary for a deformable complex surface. This holds true for both purely elastic (built from monolithic elastic material) and poroelastic (built from porous elastic material) surfaces. And finally, if surface contains also second component (as in poroelastic surface case), stress partitioning condition (2.0) is needed. This defines the fraction of the total stress that is transferred into the second component of the surface. We conjecture that for complex surfaces built from more than two components, there should be a corresponding number of stress partition functions (2.0). This unified approach is illustrated in figure 5 with references to specific boundary conditions that apply for the specific complex surface type.

It is interesting to note that velocity conditions (2.02.0) for rough surfaces can be written in a more compact form,

valid for incompressible flow regime over isotropic geometries or for incompressible two-dimensional flows. The upper left block corresponds to the slip length tensor introduced before, while . The equivalence with previous formulation can be seen through application of continuity expression, i.e., . This form can be useful in practice, for example, when boundary conditions are imposed weakly in finite element method. A similar boundary condition was proposed empirically by Gómez de Segura et al. (2018). In their work, all the coefficients for the slip lengths could take different values.

2.6 Simplifications

The physically explained boundary conditions are applicable for rough, porous and poroelastic surfaces. Moreover, no assumptions have been made about coefficients appearing in boundary conditions. In principle, the slip and transpiration lengths ( and ) and friction coefficient () may all be functions of spatial/temporal coordinates, surface geometry, and flow properties.

In this work, however, we consider only time invariant, spatially regular (consisting of periodic structures) surfaces as well as linear (creeping) flow within the surface. These assumptions simplify the boundary condition coefficients to spatially and temporally constant parameters that only depend on surface geometry. In addition, we focus only on rough and porous surfaces in order to illustrate the novel parts of this work – addition of transpiration length (2.0) and accurate modelling of stress partition condition (2.0). Within these assumptions, the introduced boundary conditions are fully non-empirical and defined by the surface geometry only.

In the next section, we show how to determine the coefficients of the proposed boundary conditions.

3 Determining slip, transpiration and resistance coefficients

In this section, we show that the slip, transpiration and resistance coefficients in the boundary conditions (2.0)-(2.0) can be determined by a set of Stokes problems. In order to accomplish this, two major assumptions are made. First assumption we make is that of a creeping flow. This allows us to describe the full flow field as a linear combination of particular fundamental flow fields, which are obtained using unit boundary conditions or unit forcing (Leal 2007). The second major assumption is that of a scale separation , which means that from a viewpoint of single surface structure, the overlying free flow is nearly uniform. In this linear approach, the slip, transpiration and resistance coefficients are fully decoupled from the flow properties, and thus constitute characteristic measures of the surface texture.

Note that in this section effective quantities (as discussed in previous section) appear concurrently with microscopic resolved quantities (not yet introduced). The latter are denoted with a “hat”, e.g. is an effective flow field containing only large scale variations, while is a microscale flow field containing all variations owing to small scale surface structures.

3.1 Textured surface

Figure 6: (Colour online.) The left frame shows a flow domain, in which a generic free flow has been illustrated. The wall-parallel interface above surface texture is depicted using a transparent plane. The red rectangular domain indicates an interface unit cell. To the right, we illustrate the linear flow field decomposition in the unit cell. The tangential shear stress is decomposed as unit forcing terms along the tangential axis.

Consider a patterned wall and a vortical flow over it, as illustrated in figure 6. The interface is located at (transparent plane in figure 6). Let us consider a small volume near the surface of the texture with cross section , which contains one representative (periodic) surface structure. This volume extends from the valley of the texture (bottom coordinate ) over the interface and slightly (few texture size lengths) into the free fluid (top coordinate ), see figure 6. We refer to this volume as ”interface cell”. Within that interface cell, we assume that the shear stress from the free fluid at the interface is spatially constant (scale separation assumption) but may have arbitrary tangential direction and magnitude.

The flow field induced by a constant shear stress on the interface is then governed by

(3.0)
(3.0)

where a one dimensional Dirac delta function has been used to denote a surface forcing in the domain. At the solid structure the no-slip and no-penetration conditions are imposed. Due to the choice of a regular, repeating surface texture, it is natural to impose periodic conditions at the vertical sides of the interface cell. At the top boundary of the cell, we impose zero-stress condition to keep the shear stress at the interface as the only driving force of the problem.

3.1.1 Slip length

We use the linearity of the Stokes problem and write the solution as a linear combination of the two tangential components of shear

(3.0)

To obtain the effective boundary condition (2.0) for slip velocity, one can form a surface average of the velocity at the interface,

(3.0)

The surface average at the interface is defined as

(3.0)

where is an arbitrary quantity. This operation gives an effective, smoothed variable by averaging out the microscale variations.

To determine slip length, we need to solve for flow fields and , which are velocities per unit shear stress. We insert the velocity decomposition (3.0) – and analogous pressure decomposition expression – into the governing equations for the interface cell (3.03.0). Since the resulting expression should hold for any arbitrary shear stress, we are left with two fundamental problems. The first one is

(3.0)
(3.0)

and the second problem for contains a surface forcing in direction. Note that and have dimensions of length. The boundary conditions for the fundamental problems are same as for (3.03.0). By comparing (2.0) with (3.0), we can identify the slip length tensor components as

(3.0)

Note that due to symmetry of the problem, we have (Kamrin & Stone 2011).

Similar fundamental problems can be seen in the pioneering work of Luchini et al. (1991), and later obtained using method of matched asymptotic expansions (for example, Carraro et al. 2013) as well as multi-scale expansion (Lācis & Bagheri 2016).

3.1.2 Transpiration length

The fundamental problems (3.03.0) can be also used to predict the transpiration length for wall normal velocity condition (2.0). From the decomposition (3.0) we see that the flow field in the interface cell is a linear combination of solutions from two fundamental problems multiplied by shear. Consequently the flux in tangential directions below the interface will always be proportional to the shear. As an intermediate step, let us introduce a boundary condition, similar to expression (2.0), which depends on shear variations, instead of slip velocity variations. The condition reads

where we have introduced new proportionality coefficient . As explained in section 2.2 using figure 4, in order to determine these coefficients we need to look for proportionality coefficients relating the flux below the interface with variation of the slip velocity (or variation of shear stress for coefficients ). Detailed derivation of auxiliary coefficients can be found in appendix A; here we state that they can be obtained as

(3.0)
(3.0)

These coefficients are the averaged linear flux density across vertical boundaries below the interface. Note that we have only three independent parameters due to symmetry, i.e. , which is a direct consequence of the symmetry in slip length tensor (). To obtain in (2.0), it is sufficient to multiply the auxiliary coefficient with the inverse of slip length matrix

(3.0)

This relationship is derived in appendix A, equations (A 0A 0). We note that it is more convenient to work with transpiration length arranged in a vector form

which is the form appearing in equation (2.0).

3.2 Porous surface

Next, we consider a porous structure with a flow over it, as illustrated in figure 7. As before, the interface is located at . In analogous way as for textured surface, we define an interface cell with cross section . This cell encompasses one characteristic structure in tangential directions. The cell has a bottom coordinate chosen such that all interface induced effects has sufficient length to decay; i.e., the cell should contain a few (typically around three to five) solid structures. The top coordinate () of the cell is few micro lengths above the interface.

Figure 7: (Colour online.) In the left frame, we illustrate a flow over a porous domain consisting of fixed spheres arranged in a square lattice. The transparent plane above the porous material shows the interface location. Interface cell is indicated with a solid red cuboid. To the right, we show interface cells corresponding to five fundamental problems forced either with a shear at interface or pore pressure gradient below the interface. The linear decomposition of the forcing is shown. The flow field solution within the cell follows exactly the same decomposition.

We assume that the interface cell for porous material is forced by a spatially constant shear stress, and by a spatially constant pore pressure gradient. Darcy’s law is valid only in the porous material, therefore the pressure gradient is imposed below the interface. The governing equations are

(3.0)
(3.0)

where a one dimensional Heaviside step function is used to distinguish between regions above and below the interface. Boundary conditions for the interface cell are the same as in equations (3.03.0), except at the bottom of the domain we have to impose the interior solution corresponding to Darcy flow due to the same pressure gradient (Whitaker 1998; Mei & Vernescu 2010; Lācis & Bagheri 2016).

The flow field in the interface cell is written as a linear combination (figure 7) of all driving forces

(3.0)

The pressure field is correspondingly written as

(3.0)

We insert expressions (3.03.0) into the governing equations (3.03.0) and require the resulting equations to hold for any arbitrary forcing.

First two fundamental problems arising are the same as (3.03.0). The first fundamental problem arising from pore pressure gradient takes the form

(3.0)
(3.0)

The fundamental problems for and have the same form, but with a volume force in and directions, respectively. Note that these fundamental problems are defined per unit pressure gradient over viscosity and fields , and have the dimension of length squared, which is the same as for permeability.

3.2.1 Velocity conditions

The velocity boundary conditions (2.0-2.0) at the interface of porous surface contain in addition to terms related slip () and transpiration () lengths, also terms related to permeability . The former two terms are obtained from the two first fundamental problems and expressions (3.0, 3.03.0). In order to obtain the Darcian terms, we form the surface average (3.0) of the solutions , and , i.e.

3.2.2 Stress partitioning condition

Now we turn our attention to stress partitioning condition (2.0,2.0). For convenience, we repeat the expression here

Recall that the velocity condition is different depending on whether one considers tangential (2.0) or normal (2.0) component. We expand the friction term and insert the expressions (2.0-2.0), for the velocities to obtain

(3.0)

Observe that there are only three independent resistance coefficients (all three components of ). However, the stress distribution around the upper layer of the material will be different depending on whether shear stress or pore pressure gradient drives the interface velocity. This gives rise to different friction forces and more that three coefficients are needed, a more detailed explanation is provided in appendix B.1. We thus rewrite the normal stress balance as

(3.0)

where , and are independent resistance coefficients.

To determine these resistance coefficients, we can investigate term by term due to linearity of the system. By setting the derivative of free flow vertical velocity, pore pressure gradient, and slip velocity variations to zero (, , ), the stress partition condition (3.0) simplifies to

(3.0)

where we have introduced an auxiliary friction coefficient vector . This expression is very similar to the linear decomposition (3.0) including only first two terms. To determine the friction coefficients, we need to relate the effective pressures in porous and free fluid regions with the linear decomposition of pressure in the interface cell. We thus define effective pressures in the interior and free fluid as

(3.0)

respectively, where corresponds to fluid volume within the integration region. For more compact notation, we have introduced brackets. To neglect any transition effects of pressure field near the interface – which effective model is not intended to describe – these volume averages are taken at the bottom and at the top of the interface cell. In this way, the averaging operation is sufficiently far away from the interface to obtain a representative pressure value for the interior and the free fluid. Recalling the assumption that , the linear decomposition of pressure (3.0) takes form

Inserting this expression into (3.0) and then in (3.0) gives

Requiring that this expression hold for any arbitrary shear value, we obtain that

(3.0)

Here, we emphasise that and are pressure fields in the interface cell generated due to shear stress forcing (figure 7). These are the same fundamental problems, which were used to determine the slip length tensor . Finally, the resistance factor appearing in front of the slip velocity in (3.0), is obtained from

This resistance term is similar to the pressure jump expression derived by Marciniak-Czochra & Mikelić (2012); Carraro et al. (2013).

Now we turn to the resistance coefficient appearing in front of the Darcy velocity in (3.0). Lets assume that , and to simplify equation (3.0) and introduce an auxiliary friction coefficient . The pressure jump condition (3.0) becomes

(3.0)

The effective pressure jump expressed using volume averages (3.0) of the linear decomposition (3.0) is

(3.0)

By comparing expressions (3.0) and (3.0) we can identify,

Here pressure fields in the interface cell are generated due to pore pressure gradient forcing below the interface (figure 7). From the auxiliary friction factor we get

This friction coefficient term – which to the best of authors’ knowledge is reported for the first time – is important for capturing correct pressure jump across interfaces for layered problems (discussed in section 4.2).

The final resistance coefficient in (3.0) is . We recall that there is a fundamental difference between the measures and obtained from transfer of momentum and obtained from mass conservation. We show in appendix B.2 that

(3.0)

Finally note that it is more convenient to consider the fundamental problems in a non-dimensional setting, because they only need to be solved once for geometrically similar surfaces with any microscale length value. The non-dimensional fundamental problems are given in appendix C.

3.3 Scale estimates and a reduced set of conditions for numerical examples

In this section, we provide estimates of the magnitude of the terms appearing in boundary conditions (2.02.0). We repeat the expressions here (for rigid surfaces),

(3.0)
(3.0)

Here, the Darcy velocity , the slip velocity and transpiration velocity due to variations in slip are defined as

respectively. Recall that the shear stress vector is

and velocity variation vector is

In appendix D, we carry out scaling estimates of all these terms when a scale separation exists and show that

(3.0)

The Darcy and the transpiration velocities are much smaller compared to the slip velocity. The Darcy velocity has similar magnitude as the transpiration velocity due to variations in slip velocity. Based on these observations, we introduce an approximation of shear stress tensor

(3.0)

The terms and are neglected since . In table 1 we group all the terms appearing in velocity condition and stress partitioning condition according to their magnitude. We see in the table that slip velocity alone is often a sufficiently good approximation.

Tangential velocity Normal velocity Normal stress