Numerical study of the flow through an annular gap with filtration by a rotating porous cylinder

ABSTRACT A numerical simulation approach is substantiated and verified for predicting the Taylor-Couette flow with an impermeable outer stationary cylinder and porous rotating inner one. An imposed throughflow is considered, supplied via one end of the annular gap and leaving the domain through another gap end (retentate) and from inside of the rotating cylinder (permeate). The flow is typical for dynamic filtration by rotating cylindrical filters. In contrast to known publications, the rotating porous cylinder is included into the computational domain and the liquid flow inside of it is fully resolved. The influence of the permeate-retentate ratio and permeability of the rotating inner cylinder onto the centrifugal stability boundary and supercritical flow details is discussed. It is found that filtration velocity becomes distributed not uniformly along the porous cylinder when its hydraulic resistance is less than a definite value. After some critical threshold, the flow structure drastically changes, and spiral vortices appears, which crosses the porous rotating cylinder back and forth several times, providing multiple flow recirculation through the porous cylinder. The results obtained create the basis for the development of dynamic rotational filters for various applications.


Introduction
The flow in the annular gap between two independently rotating coaxial cylinders, named as Taylor-Couette flow (TCF), is a subject of permanent interest for many researchers. It is not only a convenient configuration for the theoretical study of interaction phenomena between coupled to each other pressure and centrifugal force fields in a viscous liquid flow, but also a fruitful optimization site for various possible practical applications. Some of them are the lubricating flow between rotating shafts and in the bearing housing, the flow between rotor and stator in electric machines, etc. The main attention in our study is devoted to such an engineering application as a rotational filtration for the separation of solid admixtures from liquids. This application has a specific of a porous or perforated inner rotating cylinder.
A large number of publications were devoted to the study of TCF since the pioneer work of (Taylor, 1923), providing deep insight into the physics of this kind of flow. Particularly, the most notable feature is centrifugal instability resulted in the appearance of large-scale coherent vortical structures in the gap between the cylinders. The vortices change their shape and behavior in a step-wise manner while the flow passes through the CONTACT Jiancheng Cai cai_jiancheng@foxmail.com row of secondary instabilities when growing the rotation rate (Andereck et al., 1986;Coles, 1965). In spite of the distinguish features, the vortex structures, in general, are referred to as Taylor Vortices (TV). The fundamental role of TV is in their high impact on all transfer processes in the gap between the cylinders: momentum transfer (hydrodynamics), heat transfer, and mass transfer (Fenot et al., 2011). The primary instability boundary (vortex onset boundary) is well studied for classical TCF with the closed gap (no superimposed throughflow). Particularly, in Andereck et al. (1986), the authors obtained experimentally the map for all flow modes arising between two rotating coaxial cylinders. Here we note that the case of a stationary outer cylinder is the most widespread in practice and hereinafter we will focus on it by default. The TCF configuration with superimposed pressure driven axial throughflow also is studied well (Chung & Astill, 1977;DiPrima, 1960;DiPrima et al., 1979;Fenot et al., 2011;Gravas & Martin, 1978;Lueptow et al., 1992;Takeuchi & Jankowski, 1981), and the axial flow is shown to be a factor shifting the vortices onset towards the greater rotation rates. Radial throughflow inward the rotating inner cylinder also stabilizes the flow in the gap with respect to the centrifugal instability (Chang & Sartory, 1967, 1969Min & Lueptow, 1994a;Mochalin & Khalatov, 2015;Mochalin, E, et al., 2020;Serre et al., 2008). While the pure superimposed radial crossflow is rather a theoretical model (it requires both the inner and outer cylinders to be permeable), the most interesting for practical applications is the configuration with combined axial and radial flow through the gap. It is the case when the outer cylinder is solid and rotating inner cylinder is permeable (porous, punched, or meshed). Fluid is supplied into the gap through the annular inlet section, then, part of it passes inside the rotating cylinder and discharges through the hole shaft, while the remaining part leaves the gap through the opposite end. Such TCF configuration particularly is used in dynamic rotational filters for the separation of particulate admixtures from liquids (Beaudoin & Jaffrin, 1989;Figueredo-Cardero et al., 2012;Frappart et al., 2011;Jaffrin, 2008;Jaffrin & Ding, 2015;Lee & Lueptow, 2004;Ji et al., 2016;Wereley & Lueptow, 1999;Wereley et al., 2002;Zheng et al., 2019). The dynamic similarity parameters, characterizing the TCF with radial and axial flow, are rotational, axial, and radial Reynolds numbers which we will consider in the following form: where R 1 is the inner cylinder radius (R 2 is the radius of the outer cylinder); d = R 2 − R 1 is the gap width; ν is the fluid kinematic viscosity; V 1 , U 1 , W 0 are the typical azimuthal, radial, and axial velocities defined by formulas (2).
In equalities (2) the following notation is used: -the angular velocity of the inner cylinder; Q -the volumetric flow rate at the inlet annular cross section; Q 1 -the flow rate through the total surface of the porous inner cylinder (filtered flow or permeate); S 1 = 2π R 1 L -the surface area of the porous rotating cylinder of length L; S g = π(R 2 2 − R 2 1 ) -the cross-section area of the annular gap. Thus, the typical azimuthal velocity is the velocity of the inner rotating cylinder outer surface, the typical radial velocity is the average normal fluid velocity at this surface, and the typical axial velocity is the average velocity at the annular inlet. The geometrical similarity parameters usually used in TCF study are the radius ratio (η = R 1 /R 2 ) and aspect ratio ( = d/L).
According to definition (1), the axial Reynolds number Re a is defined by the inlet axial velocity, but the average axial velocity changes along the gap (is reduced) due to the flow inward the rotating cylinder (filtration). So, Re a itself does not estimate the axial flow intensity over the whole gap, especially when filtered flow rate Q 1 is a notable part of the total supplied flow (Q). Another point to be mentioned is the fact that the local value of the normal velocity component at the porous cylinder surface (V r (R 1 , θ , z)) depends on the transmembrane pressure and is influenced by the flow field in the gap. It results in that one actually is unable to apply a prescribed velocity boundary condition at the surface of the inner cylinder. At the same time, all known results on the vortices onset boundary and vortical flow modes, with respect to the TCF with axial and radial flow, are defined for the case of a uniform filtration velocity distribution over the whole rotating cylinder surface (V r (R 1 , θ , z) = U 1 ). The axial Reynolds number (Re a ) is also considered as the axial flow intensity parameter for the whole gap (Bahl & Kapur, 1975;Johnson & Lueptow, 1997;Kolyshkin & Vaillancourt, 1997;Martinand et al., 2009;Min & Lueptow, 1994b). We note here that almost all known implementations of rotational filters provide for the use of a fine porous membrane having large transmembrane pressure resulted in relatively low filtration flux (Re r < 5). Consequently, the filtration velocity distribution is close to uniform when porous cylinder resistance to the flow is large. A low Re r value is usually accompanied by the prevalence of the retentate flow over the permeate flow and the local axial Reynolds number nearly the gap outlet slightly differs from Re a defined by (1). So, the assumptions mentioned above are quite adequate for the case of a fine porous membrane and large retentate flow, but the theoretical database available is insufficient to substantiate the applications with a small retentate flow and low membrane resistance. Tilton et al. (2010) connected in their study the wallnormal velocity at the surface of the rotating porous cylinder (V r (R 1 , θ , z)) to the transmembrane pressure difference through Darcy's law. They considered only a subcritical Couette-Poiseuille flow (before the vortices onset) and demonstrated, in particular, the crossflow reversal phenomenon, when the radial velocity through the membrane changes its direction from inflow to outflow. Tilton and Martinand (2018) performed the study of instability development in the Couette-Poiseuille flow with a porous inner rotating cylinder. They described the velocity at the porous cylinder surface by Darcy's law with an assumption on constant pressure inside of the cylinder and supplemented a global mode linear analysis with a dedicated DNS to compare the results. The porous resistance effects and flow within the porous cylinder were not captured by Tilton and Martinand (2018) and low filtration rates have been considered (Re r ≤ 5). Figueredo-Cardero et al. (2014) also accounted for the variation of the filtration velocity in terms of Darcy's law implemented for each individual phase of the filtered suspension. They demonstrated periodical change in the transmembrane flow direction in accordance with the vortical structures periodicity in the gap. They considered a porous membrane with less resistance to the flow than in majority of other studies and a low filtration rate (Re r = 3).
Summarizing the introductory discussion, we can conclude that a gap exists in the theoretical database on the TCF with axial and radial flow. The influence of the permeate-retentate ratio and permeability of the rotating inner cylinder onto the centrifugal stability boundary and supercritical flow details have not been studied as on our knowledge. Thus, the goal of the present study is to investigate the effect of the permeate flow specific share and hydraulic resistance of the rotating porous cylinder onto the TV onset boundary and onto special features significant for possible engineering applications, particularly, for the rotational filtration.
To achieve the goal, one needs to resolve the flow inside of the filtering cylinder and move the outlet boundary away from its inner surface. The next challenge is in that much greater radial Reynolds numbers (filtration rate) should be considered resulted in sufficient change in the local axial Reynolds number along the annular gap. The above features make it impossible the application of the global mode analysis well developed in the previous stability investigation (Tilton et al., 2010;Tilton & Martinand, 2018). Numerical simulation is the most promising way to consider real case engineering applications of fluid dynamics in sufficiently general formulation. We can refer to Ez Abadi et al. (2020) and Ghalandari et al. (2019) as to recent examples of practically oriented numerical studies.
So, direct numerical simulation will be the main research method in the present work and one needs to elaborate and validate an appropriate modeling approach and its numerical implementation.

Problem formulation and governing equations
We will consider viscous flow in the Taylor-Couette system with axial flow supply through the annular inlet and with a porous inner rotating cylinder. Part of the flow passes inward the rotating cylinder through its porous surface (permeate flow), while another part leaves the gap through the opposite end (retentate flow). The flow domain is sketched in Figure 1. The dimensions are chosen as follows: R 1 = 0.044m; R 2 = 0.052m; R 3 = 0.0364m; R 4 = 0.0334m; L = 0.305m. It corresponds to η ≈ 0.85 and ≈ 38, providing the ability to compare some results to available experimental data.
The flow is supposed to obey the general Navier-Stokes and continuity equations for an incompressible liquid, which can be written in a rotating with the inner cylinder reference frame as follows: where ρ, μ are the fluid density and dynamic viscosity; p is the pressure; V i = {V x , V y , V z } are the projections of the absolute velocity V onto axes of the rotating reference frame (x, y, z); k are the projections of the inner cylinder angular velocity ( 1 = 2 = 0, 3 = ); ε ikj is the Levi-Civita symbol; V rel j are the projections of the relative velocity (V rel = V − × r) onto the moving axes (x, y, z), and r is the position vector started at the rotational axis. We used tensor notation when writing Equations (3), (4), following the Einstein summation convention.
There is an additional momentum loss for the flow within the porous cylinder, mainly conditioned by viscous resistance. We address it following the Darcy's law in the form (Batchelor, 2000): where b ij are the components of the resistance tensor for the porous medium. We will define the resistance of the porous cylinder as follows: According to definition (6), the porous structure is isotropic in the cross sections z = const and 3 orders of magnitude less permeable in the axial direction. That practically excludes internal axial currents within the porous medium without numerical convergence issues. Such a choice of the porous cylinder structure is connected to the rotational filtration application. The ideal modeling representation of a filtering membrane assumes permeability only in the normal direction to the surface, as it is implemented in all studies mentioned above. However, it does not correspond the real situation in the case of a porous filtering cylinder, though the pure radial filtration flow is the most desirable variant. In our approach we practically exclude axial flow motion within the filtering cylinder, considering the same porous structure in the radial and tangential directions. The greater than three-order-of-magnitude increase in the axial porous resistance relative to the resistance in the radial and tangential directions can lead to poor convergence of the numerical procedure, as it follows from computational experience. That explains ratio (6) between b ii . Regarding the practical feasibility of an axially impermeable porous cylinder, one can provide it (if necessary), for example, using transversal baffles.
Taking into account equality (5), momentum Equation (3) within the porous cylinder becomes following: Continuity Equation (4) remains unchanged within the porous zone. We should also mention that velocity V within the porous medium is an effective velocity which is defined regardless the porosity degree, considering the entire porous volume is open for fluid. We formulate the flow boundary conditions in the following analytical form (designation of the boundaries corresponds to Figure 1): (8) Regarding the initial conditions, we are able to get a proper start distribution of the flow parameters from the solution of the steady-state problem formulated for a subcritical flow mode. In other situations, one can start a new solving process from the final state, achieved in previous calculations, after some modification of the boundary conditions. As it follows from expressions (6)-(8), one can vary the values of Q, Q 1 , , to control parameters Re r , Re d , Re a , and vary the value of b, to control the porous cylinder resistance. For more generality, we will use below the dimensionless resistance in the form b * = bhR 1 , where h = (R 1 − R 3 ) is the thickness of the porous cylinder.
The inlet and two outlet flow rates are connected as Q = Q 1 + Q 2 , and Q 1 completely defines Q 2 when the inlet flow rate (Q) is specified. To generalize the analysis, we introduce the permeate specific share in the supplied flow as a parameter in the following way: that allows defining the outlet flows as So, m = 1 means 100% permeate flow, while m < 1 corresponds to certain retentate flow.

Numerical procedure and validation
A structured computational mesh of 1,780,800 cells has been adopted with 56 cell layers in the radial direction, 120 layers in the azimuthal direction, and 265 layers in the axial direction. General view of the mesh is presented in Figure 2. The mesh was approved after solution independence check on the mesh size. Integral flow parameters characterizing the overall velocity distribution in the annular gap outside of the rotating porous cylinder were chosen for the check. The data on their sensitivity to the number of computational cells are presented in Table 1.
Further refinement of the computational mesh, which we have adopted, by about a million cells changes the solution less than by 1%.
Finite volume method is applied for spatial discretization of Equations (3), (4), and (7). Momentum equations for each velocity component are solved separately in an iteration cycle including also the pressurevelocity coupling procedure based on the momentum and continuity equations (SIMPLE (Patankar, 2018) family algorithm). Approximation of the convection terms in Equations (3) and (7) is accomplished according to  the QUICK scheme (Leonard & Mokhtari, 1990) providing high order accuracy and low numerical diffusion on a structured mesh. The second order centraldifference approximation, described in (Ferziger & Peric, 2002), is applied to the diffusion terms. The PRESTO! (Pressure Staggering Option) scheme (Patankar, 2018) is utilized for interpolating the pressure values at the control volume faces. Temporal discretization is based on a three-layer second order difference scheme with fully implicit treatment providing unconditional stability with respect to time step size. SIMPLEC (SIMPLE-Consistent) algorithm (Van Doormaal & Raithby, 1984) is used for pressure-velocity coupling in steady-state calculations. We utilize also the Pressure-Implicit with Splitting of Operators (PISO) scheme with the 'neighbor correction' procedure (Issa, 1986) in transient computations. The PISO algorithm can dramatically decrease the number of iterations required for convergence at a time step.
The stored values of all flow variables (velocity components and pressure) are computed at the cell centroids (co-locative computational template) that is more convenient but tends to unphysical checkerboarding of pressure (Patankar, 2018). To prevent from the checkerboarding, we use a correction procedure similar to the one proposed by Rhie and Chow (1983).
Discretization procedure results in a set of linear algebraic equations with a spars coefficient matrix which is solved at each iteration step by Gauss-Seidel method (Ferziger, 1988). The solution procedure is designed to support parallel computing.
To verify the numerical procedure, first we compared the experimental data on primary centrifugal instability, obtained by Min and Lueptow (1994b), to our numerical predictions. From that purpose, we chose radii R 1 , R 2 and length L (the definitions are given in Section 2) to be the same as in Min and Lueptow (1994b). For the porous cylinder resistance, we adopted b * = 4 · 10 8 , which is almost the same as in Min and Lueptow (1994b) as well. Numerical simulation allows monitoring the flow template in the gap between the inner rotating and outer stationary cylinders. As an example, Figure 3 demonstrates the spatial position of the azimuthal velocity iso-surface (V θ = 0.5 R 1 ) for a same combination of the axial and radial Reynolds numbers and two rotational Reynolds number values. It follows from the two flow templates, that the flow mode transition is between Re d = 110 and Re d = 120 for the given Re a , Re r values. We present in Figure 4 our simulation data regarding the TV onset boundary together with the experimental result, obtained by Min and Lueptow (1994b), for a particular Re r value (Re r = 0.154).   (Min & Lueptow, 1994b); 2 -subcritical flow (numerical computation); 3 -supercritical flow (numerical computation).
In Figure 5 we bring also the calculated azimuthal velocity profiles, averaged in the azimuthal and axial directions, for the case of classical TCF (no the throughflow). The profiles for Re d = 180 and Re d = 342 are compared to the measurement data of Wereley and Lueptow (1994).
The numerically predicted data in Figures 4 and 5 are in sufficient agreement with the known experimental results that ensures our numerical simulation approach is adequate to be used in the present study. 3 -measurement (Wereley & Lueptow, 1994) at Re d = 180; 4measurement (Wereley & Lueptow, 1994) at Re d = 342.

Influence of the permeate flow specific share onto the vortex onset threshold
Large flow relaxation time in TCF is a known fact (Czarny & Lueptow, 2007) encumbering the stability boundary detection in numerical simulation based on direct solving of the Navier-Stokes equations in 3D domain, especially when a large aspect ratio. In case of the enclosed gap and the impermeable cylinders (classical TCF), the radial or axial flow velocity components can be monitored to detect the TV onset, but it does not work clearly when there is an imposed axial and radial flow through the gap. One can reasonably assume that large scale vortices onset affects the vorticity vector in the gap flow field, but we also should keep in mind that the axial vorticity component (ω z = 0.5(∂V y /∂x − ∂V x /∂y)) to great extent is conditioned by the inner cylinder rotation independently on the flow mode. Really, one can see from Figure 6 that the averaged over the gap axial vorticity component, normalized by the inner cylinder angular velocity, is nearly a constant independently on Re r , Re a , and Re d . Here we note that according to definitions (1), (2), and (9) the radial and axial Reynolds numbers are connected through permeate specific share m as While the axial vorticity component is defined mainly by the rotation rate, one can expect that the vorticity vector components in the cross-section planes (z = const) are sensitive to the TV onset and development. The data presented in Figures 7-9 support such an assumption. The value of the vorticity vector projection onto a plane z = 0 we define as follows: where ω is the vorticity vector magnitude and ω z is its z-projection. Average value of ω xy over the gap volume between the inner rotating and outer fixed cylinder ω xy m is plotted in Figures 7-9 in the normalized form by the angular velocity corresponding to the TV onset threshold when no axial and radial throughflow (Re z = 0, Re r = 0). The critical angular velocity is defined as where (Re d ) c0 is the critical rotational Reynolds number value for the enclosed TCF at a given gap configuration; for η = 0.85, in particular, we have (Re d ) c0 = 108 (Min & Lueptow, 1994a;Chang & Sartory, 1969).
It follows from Figures 7-9 that there is a distinct change in the transversal vorticity projection behavior, which can be explained only by the large-scale vortices onset due to centrifugal instability. ω xy m is nonzero in the subcritical gap flow and there is a trend of its reducing until approaching the stability threshold, when m < 1. For m = 1, that vorticity component can slightly increase in subcritical flow, as one can see in Figure 8 (line 4). We change the boundary condition at the gap outlet, in the case of m = 1 (no retentate flow), with the no-slip condition, introducing an end wall rotating together with the inner cylinder. The end-wall Eckman vortex appears in that case long before the centrifugal instability (Czarny et al., 2004), influencing the vorticity value in the subcritical flow. It is one of the reasons for no minimum of ω xy m near the transition, when m = 1. However, vorticity projection ω xy m behaves the same for all m when  large enough Re a (Figure 9). Anyway, one can detect the transition by notable change in slope for the graphs of ω xy m . It should be recognized that the approach proposed cannot provide the exact definition of the TV onset boundary, giving however its reasonable assessment for practical use.
We can see also from Figures 7-9 that the rotational Reynolds number (Re d ), corresponding to the TV onset, reduces with an increase in the permeate proportion. This is due to a decrease in the axial throughflow velocity along the gap because of filtration and fully agrees with the fact that the vortices first appear in the outlet part of the gap where the axial velocity is less (Figure 3(b)). The last feature is also in agreement with the result of Tilton and Martinand (2018) on the retentate outlet as the most unstable cross-section in the gap when filtration inward the rotating cylinder. To understand deeper the influence of the combined axial and radial throughflow onto the TV onset boundary, we defined the dependences of the critical rotational Reynolds number on the permeate specific share. The most demonstrative is a comparative form of the result presentation, giving a vision on stabilization degree relative to some base cases. We designate here as (Re d ) c our estimation of the critical Re d value for a particular combination of Re a and m. Figure 10 presents (Re d ) c normalized by the critical azimuthal Reynolds number (Re d ) c0 defined for the enclosed TCF with the similar geometry (η = 0.85). The ratio is plotted against m-factor for three fixed values of Re a . It gives an assessment of stabilization effect with respect to the case of no both the axial and radial throughflow. In Figure 11 the critical Reynolds number is normalized by its value ((Re d ) cr ) in the case of an imposed radial flow through the permeable inner rotating and outer stationary cylinders (no axial flow). The ratio of (Re d ) c to ((Re d ) ca ) is presented in Figure 12 as a function of m. ((Re d ) ca ) is the critical rotational Reynolds number for the case of pure axial throughflow (no permeate outflow).
Overall throughflow intensity in our analysis is defined by the axial Reynolds number Re a , as the typical axial velocity (W 0 ) is directly connected to the total flow rate at the inlet (Equations (1) and (2)). So, it is naturally that the stabilization depends mainly on Re a and slightly varies with parameter m which defines the relative portion of the flow passing through the surface of the rotating cylinder ( Figure 10). Increase in the amount of permeate reduces the stabilization effect at fixed Re a (Figures 11  and 12) and it is more visible for Re a = 25 in comparison to the cases of Re a = 11 and Re a = 200. We should note that (Johnson & Lueptow, 1997) demonstrated the essential stability increase with increase in Re r at fixed Re a , that may seem to be in contradiction with our result, but the main feature here is in that the radial throughflow   rate is independent on the axial one in their study due to a permeable outer cylinder. Both Re r and Re a in Johnson and Lueptow (1997) are defined independently and sustained externally, while we have a relationship (10) between them and only one inlet flow which is divided into the permeate and retentate flows. That means that the local axial Reynolds number reduces along the gap, with respect to Re a , and the more is m (or Re r ) the greater the drop rate is. That explains the reduction in stability at fixed Re a when increase the permeate specific share, as it was already discussed above with regard to Figures 7-9.

Effects of the porous cylinder resistance
The porous cylinder resistance of b * = 4 · 10 8 is large enough to provide uniform distribution of the filtration rate along the cylinder, as it follows from Figures 13 and 14. We presented in the graphs the variation of the radial velocity component along two lines parallel to the cylinder axis: the line r = R 1 , θ = 0 and line r = 0.98R 1 , θ = 0. The first line is at the interface between the gap area and porous zone, and the second one is displaced 1 mm inward the porous cylinder. We note here that in our model there is no any boundary condition at the surfaces of the porous cylinder and the discretized equations are solved jointly for all the domain containing two subdomains opened for fluid and one porous zone which differs by only an additional resistance to flow as described above. Consequently, the velocity at the interface depends to some extent on the local normal mesh step and it is influenced by both gap flow and flow in the porous zone. From one side, it is more realistic than in the case of the boundary conditions imposed at the porous cylinder surface, but this approach is not justified for cases when solid fraction of the porous cylinders matters notably (turbulent gap flow, surface heat transfer, etc.). This approach, however, is quite adequate in the case of laminar gap flow. Due to the reason above, we include into our analysis not only the radial velocity distribution at the interface, but the distribution along the line placed 1 mm deeper in the porous cylinder, to estimate the axial distribution of the filtration velocity close to the cylinder surface. The radial velocity is normalized by its mean value along the correspondent line. At Re a = 25 there is a subcritical flow mode in the gap when Re d = 130 (Figure 15(a)) and supercritical flow when Re d = 220 (Figure 15(b)). One can see from Figure 14 that V r at the interface is influenced by the vortices, but the filtration velocity axial distribution becomes uniform just under the interface. For the subcritical flow (Figure 13), the interface radial velocity is also uniformly distributed besides of short end regions where it is influenced by the  boundary conditions; but porous medium removes the influence right away.
The first thing happens when reducing the porous cylinder resistance is the breakup of uniformity in the filtration velocity distribution along the cylinder. One can see that from Figure 16 where the variation of the filtration velocity along the porous cylinder is plotted for different b * values and for the case of no retentate flow (m = 1). We also presented in Figure 17 the tangential velocity iso-surfaces to visualize flow structure for the same cases as in Figure 16.
The filtration velocity reduces by absolute value at the initial part of the cylinder (from the inlet end) and increases at the downstream part (close to the outlet) with decrease in the resistance. When b * = 4 · 10 6 (two orders of magnitude reduced with respect to the base value), the redistribution in V r through the porous cylinder stabilizing to some extent the gap flow, partially damping the vortices (Figure 17(a,b)). This can be explained by the fact that the axial velocity within first 60% of the gap becomes greater (in comparison to the case of b * = 4 · 10 8 ), while near the end of the gap, which is closed when m = 1, the radial throughflow increases; both factors are stabilizing ones. In the case of b * = 4 · 10 4 , almost all flow is filtered through a short downstream portion of the porous cylinder and the trend mentioned above formally out to be preserved, but in fact we see destabilization in Figure 17(c). One can see also that filtration velocity is influenced by vortices and porous medium does not Figure 17. Iso-surfaces V θ /( R 1 ) = 0.5 (Re a = 200, Re d = 600, m = 1) for b * = 4 · 10 8 (a), b * = 4 · 10 6 (b), b * = 4 · 10 4 (c), b * = 4 · 10 2 (d).
damp the gap flow perturbations, and that is probably a reason of reducing the stabilization. That flow configuration is undesirable for filtration applications due to very high local filtration velocity leading to quick fouling of the porous structure, and only small part of the whole filtration surface is in use.
In the case of low porous medium resistance (b * = 4 · 10 2 ), vortical structures occupy not only the whole gap, but they penetrate through the porous cylinder and there is fluid motion back and forth through the rotating cylinder with a high radial velocity (an order of magnitude greater than the average filtration velocity). The filtration velocity distribution for this case is presented by line 4 in Figure 16 and some details of the vortex structure in a fragment of meridional plane θ = 0 one can see in Figure 18. Vortical recirculation motion encompasses both the outer and inner gaps as well as the porous cylinder between them, but there is no axial velocity component within the porous zone because of much  greater resistance in the axial direction, as we described in Sect. II.
It is notable that the inclination angle of the spiral vortex structures changes with the opposite one for the low-resistance case (Figure 17(d)). To better illustrate this feature, we put in Figure 19 velocity vector plots in the surfaces presented in Figure 17(c,d). It is exhibited that the flow direction in a surface of a constant tangential velocity is across the vortex structures whenb * = 4 · 10 4 , while it is along the vortices forb * = 4 · 10 2 . Figure 20 shows flow pathlines or trajectories of liquid particles crossing the inlet annular cross section, which are built basing on the instantaneous velocity fields for the same flow modes: Re a = 200, Re d = 600, m = 1, b * = {4 · 10 8 , 4 · 10 6 , 4 · 10 4 , 4 · 10 2 }. The main distinguish feature of the lowest resistance case (Figure 20(d)) is in that the incoming fluid crosses first time the porous cylinder in a proximity of the inlet and crosses then it again back and forth while progressing to the outlet along the vortices and rotating in them. We note right  away that such flow mode is fully unsuitable for filtration applications because of evident reasons.
The relative velocity vector plot in a transversal cross section gives more vision of the flow structure for the low-resistance case; it is presented in Figure 21 for z/L = 0.5. However, the flow template is periodical in the axial direction as one can see from Figures 17(d), 19(b), and 20(d). It follows from Figure 21 that the flow is also azimuthally periodical.
As one can see from Figure 18, there is no relative axial flow motion within the porous cylinder, while significant azimuthal and radial relative flow velocity components (Figures 21 and 22) can be significant in addition to filtration. We remind here that we consider axial permeability of the porous structure to be 3 orders of magnitude less than that in the radial and azimuthal directions.   From the azimuthal velocity profiles in the radial direction at the mid length of the domain (they are presented in Figure 22), one can see that the azimuthal flow slip relative to the rotating cylinder appears at the lowest value of b * among those we have examined. When looking attentively, we can also note slightly less sharp change in the azimuthal velocity at the gap-porous cylinder interface (r/R 1 = 1) at b * = 4 · 10 4 (line 3 in Figure 22) in comparison to the greater resistance cases.
It is interesting how the retentate flow affects the porous cylinder resistance effects described above. To illustrate that, we bring in Figures 23 and 24 the same data as in Figures 16 and 17, but for the case m = 0.5 (50% retentate flow). The comparison shows that the filtration velocity distribution along the cylinder and flow templates are very similar for both no retentate flow and 50% retentate flow cases. So, one can conclude that the porous cylinder resistance effect manifests itself in the same way independently on the permeate flow specific share (at least for m ≥ 0.5).
( Figure 24(d,e)) to reflect the possible influence of transient flow nature. They are captured at two instant times sufficiently distant from each other. Evidently, the flow structure does not change after reaching the stable in average flow mode, and only small local evolutions can be noticed as well as translational progress of the vortices toward the outlet.

Conclusions
We substantiated a numerical simulation approach for study the Taylor-Couette flow with combined imposed throughflow: axial through the annular gap between a fixed outer cylinder and rotating inner one and radial through the inner porous cylinder. The flow is typical for dynamic rotational filters which are promising for wide use in engineering when cleaning liquids from mechanical admixtures. An important distinguishing feature of our solution is no boundary condition at the rotating cylinder surface and resolving the flow through the porous wall. That allowed extending the existing database with the regularities of the permeate specific share influence on the centrifugal stability boundary and with the porous cylinder resistance effects. We derived that flow stabilization from Taylor vortices onset mainly depends on the general throughflow rate estimated in terms of the axial Reynolds number (Re a ) calculated at the inlet annular cross section. The permeate to retentate ratio can affect the vortex onset threshold (critical Re d value) within 20%.
The reduction of the porous cylinder resistance firstly expresses itself in nonuniformity in the filtration velocity distribution. Until a definite resistance value, flow perturbations in the outer gap are damped by the porous structure, but beyond of the threshold the perturbations penetrate into the porous cylinder. When low resistance (b * = 4 · 10 2 in our case), the flow structure drastically changes, and incoming flow passes to the outlet in spiral vortices which crosses the porous rotating cylinder in both directions (inward and outward) many times. The vortices drift axially and there is axial and azimuthal flow periodicity at any instant time.
The results obtained provide the basis of the rotational filters development for the various applications where less transmembrane pressure and larger permeate specific share are required than known implementations have. We showed, in particular, that the porous resistance should be sufficient for the local filtration velocity to be less than a giving limit. It is also demonstrated that too low resistance results in a specific flow mode when rotational filtration is completely senseless. The regularities discovered of the permeate proportion effect on the flow nature open the way to consider greater filtration rates for rotational filters. In particular, we showed that the specific permeate share limitedly alter the vortices onset boundary, while the flow details in the rear part of the gap, significant for the filtration conditions, can be estimated by further numerical simulations. The possibility of a full-flow filtration mode (when m = 1) can also be considered, but a sedimentation bunker should be added to the flow geometry in that case.
One should notice that the configuration of the flow domain inside the porous cylinder can alter to some extent the regularities discovered as well as it can add some new ones. It can be a topic of further research as well as extending the study onto higher rotation rates when turbulent flow motion must be considered. The last issue requires additional research substantiating the adequacy of turbulent modeling near a rotating porous wall, which is a complex task including a dedicated experiment.

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