Flow turbulence topology in regular porous media: from macroscopic to microscopic scale with direct numerical simulation Xu Chu,1, a) Bernhard Weigand,1 and Visakh Vaikuntanathan1 Institute of Aerospace Thermodynamics, University of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany (Dated: 27 March 2018)
Microscopic analysis of turbulence topology in a regular porous media is presented with a series of direct numerical simulation (DNS). The regular porous media are comprised of square cylinders in a staggered array. Triply periodic boundary conditions enable efficient investigations in a representative elementary volume (REV). Three flow patterns channel with sudden contraction, impinging surface and wake are observed and studied quantitatively in contrast to the qualitative experimental studies reported in literature. Among these, shear layers in the channel show the highest turbulence intensity due to a favorable pressure gradient and shed due to an adverse pressure gradient downstream. Turbulent energy budget indicates a strong production rate after the flow contraction and a strong dissipation on both shear and impinging walls. Energy spectra and pre-multiplied spectra detect large scale energetic structures in the shear layer and a breakup of scales in the impinging layer. However, these large scale structures break into less energetic small structures at high Reynolds number conditions. This suggests an absence of coherent structures in densely-packed porous media at high Reynolds number. Anisotropy analysis with a barycentric map shows that the turbulence in porous media is highly isotropic in the macro-scale, which is not the case in the micro-scale. In the end, proper orthogonal decomposition (POD) is employed to distinguish the energy-conserving structures. The results support the pore scale prevalence hypothesis (PSPH). However, energetic coherent structures are observed in the case with sparsely-packed porous media. Keywords: Porous media, turbulence, direct numerical simulation, microscopic analysis
a)
Electronic mail:
[email protected]
1
I.
INTRODUCTION
Porous media can be found in nature with a broad range of geometrical characteristics and are also involved in industrial and scientific applications with simplified regular structures. Selective laser melting, transpiration cooling in a gas turbine and pebble-bed reactor are some of the industrial examples where porous media are commonly encountered. Darcy 1 defined that porous media are “system ensembles of a solid matrix with its void filled with fluids”. The interconnected voids allow fluid to flow through the porous medium. Dybbs and Edwards 2 investigated flow through a bed of rods in a regular, hexagonal packing. Using laser anemometry and visualization over a wide range of Reynolds numbers, they highlighted the existence of Darcy, steady-laminar inertial, unsteady-laminar and turbulent regimes. Turbulent flows were observed in experimental studies with a pore Reynolds number Re = uint d/ν > 3002,3 , where uint and d are defined as double-averaged velocity in the void (interstice) and diameter of sphere/rod, respectively. But the definition of the exact Reynolds number to classify different flow regimes remains quite controversial4–6 . Furthermore, compared with extensive researches in the low-velocity flow in the Darcy or the laminar range, the high-velocity turbulent flow is less understood. Compared with macroscopic volume-averaged studies, microscopic analysis has been rarely considered due to its topological complexity. Horton and Pokrajac 4 performed experiments of turbulent flows through a regular porous matrix of spheres packed in a cubic arrangement. The velocity measurement within a pore is achieved by the way of ultrasonic velocity profiler (UVP) and particle image velocimetry (PIV). In the range of Reynolds numbers Re = uint d/ν = 70 − 430, three different regimes were detected: unsteady laminar, transition to turbulence and turbulent. Various statistics are attempted to differentiate these flow regimes. In the high-Re turbulent flow cases, the shedding of coherent motions with a range of spatial and temporal scales are captured. Patil and Liburdy 7 showed turbulent flow characteristics in a randomly packed porous bed of solid spheres with an alternatively defined pore Reynolds number Repore = uint dH /ν = 418−3964. The bed hydraulic diameter dH is defined as dH = ϕd/(1 − ϕ), where ϕ is solid bed porosity and d is the diameter of the solid spheres. The average bed interstitial velocity uint is the same as in Dybbs and Edwards 2 and Horton and Pokrajac 4 . These results show that beyond Repore ≈ 2800, the scaled turbulence characteristics are remarkably similar from pore 2
to pore despite very different mean flow conditions. The scaling relationship is based on the hydraulic diameter dH and other geometric characteristics. In a subsequent research8 , they determined three different regimes in the turbulent flow containing (i) tortuous channel flow, (ii) recirculating flow and (iii) jet like flow through measuring velocity fields collected in distinct pore geometries. In addition, pore scale flow structure contribution to dispersion are presented, where recirculating regions are found to contribute most to the total longitudinal dispersion, whereas the tortuous channels contribute most to the transverse dispersion. In a recent work9 , they further worked on scale estimation using the data of production and dissipation rate from the experiments. Hlushkou and Tallarek 10 defined the turbulent regime based on the macroscopic dependence of pressure drop on flow rate, while the physical mechanism behind this is still not completely understood. A preliminary microscopic analysis shows that the nonlinear flow regime is characterized by the development of an inertial core in the pore-level profile. Investigating complex flow patterns in the micro-scale is key to understanding the flow physics and developing models. In the experiments, access to detailed flow field measurements is quite challenging due to the inherent space constraints of the porous media. Moreover, the porous media is usually bounded by the walls of the container in which it is placed. Separating the porous media induced flows from the wall-bounded shear flow needs additional attention. In this case, direct numerical simulation (DNS) becomes a valuable approach to understand this special flow mechanism despite its high demand for computational resources. DNS is characterized by its ability to resolve until the Kolmogorov length scale and therefore eliminate the uncertainties associated with macroscopic modeling. Although it has been widely considered in various canonical conditions such as channel, pipe and turbulent boundary layers, it is not prevalent in the porous media community. Taking advantage of geometric periodicity in a regular porous media, a carefully chosen representative elementary volume (REV) with periodic boundary conditions is sufficient for DNS instead of a large computational domain. Hill and Koch 11 studied the transition regime from steady to unsteady chaotic flow through Lattice-Boltzmann simulations. This set of simulations is based on a REV with close-packed face-centered cubic array of spheres. They showed that the transition process occurs through a supercritical Hopf bifurcation, where the streamwise velocity starts to fluctuate first. Recent DNS is able to meet the resolution requirement in the turbulent regime. Jin et al. 12 investigated the presence of macroscopic 3
turbulence in porous media at Rep = uDarcy d/ν = 500 − 1000 using DNS on a REV scale. The Darcy velocity uDarcy is defined as uDarcy = ϕuint . Two numerical approaches have been adopted for the same cases: the finite-volume method (FVM) code OpenFOAM using body-conforming grids and a Lattice-Boltzmann method (LBM) using D3Q19 grid modeling. The results revealed that the pore size restricts the size of turbulent structures, which is referred as pore scale prevalence hypothesis (PSPH). This hypothesis was confirmed through examining two-point correlations and energy spectrum of velocity fields in all the cases they have considered. Moreover, the computational domain in their study is significantly larger than the one used in Hill and Koch 11 to examine this hypothesis fairly. In a further study13 , they extended their porous topology from a homogeneous two-dimensional porous structure to a three-dimensional heterogeneous porous media and a wall-bounded porous structure. The results continue to confirm this hypothesis. In a recent study14 , Jin and Kuznetsov concluded that macroscopic shear Reynolds stress in all REVs is negligibly small in their wall-bounded porous media with regularly arranged spheres. Also, the production rate of turbulence kinetic energy is generally balanced by the dissipation rate in each REV. Penha et al. 15 studied conjugate heat transfer in porous media within a REV. The REV is comprised of square cylinders in both in-line and staggered arrangement. A computational method is provided to approximate the heat transfer coefficient of porous media where the heat generated in the solid varies slowly with respect to the space and time scales of the developing fluid. Besides the computationally demanding DNS, different level of modeling attempts have been made to describe the macro-scale flow characteristics.
Kuwahara, Yamane, and
Nakayama 16 conducted large-eddy simulations (LES) as well as Reynolds-averaged NavierStokes (RANS) simulations on a single REV unit domain with triply periodic boundary conditions. The REV is constructed with a periodic array of square cylinders. The results are comprised of preliminary first- and second-order turbulence statistics at one monitoring point. Instead of a comprehensive microscopic analysis, the LES is prioritized to validate the RANS prediction and skin friction correlations. It revealed that Ergun’s equation may fairly well describe the drag force for the turbulent flow in porous media. Suga 17 reported LES studies based on REVs of three types of porous geometry: square rod arrays, staggered cube arrays and body centered cube. The computational domain is resolved using a D3Q27 LBM with WALE (wall adaptive local eddy-viscosity) model. Double averaged (volumetric 4
and Reynolds) momentum transport equations are derived and used for the development of a four equation k − ε eddy viscosity model. Kundu, Kumar, and Mishra 18 used both LES and RANS on a REV scale constructed with periodic array of staggered square cylinders. A wide range of Reynolds number from 100 to 40,000 was considered in combination with porosity ranging from 0.3 to 0.84. The predictions of the pressure gradient were found to be in good agreement with the Forchheimer-extended Darcy law except at low Reynolds number of 100. Pedras and De Lemos 3 developed and validated a macroscopic two-equation turbulence model. Besides, they derived a new set of equations for the transport of the volume-averaged turbulent kinetic energy and its dissipation rate. Apart from these, other numerical works dealing with porous media flow coupled with free flow are reported in literature19,20 . In the present study, a DNS-based microscopic investigation of turbulent flow in regular porous media at moderate and high Reynolds numbers is reported. Turbulence characteristics are demonstrated to reveal a topological relationship in porous media. Production and dissipation are discussed using kinetic energy budget information. Modal analysis with proper orthogonal decomposition (POD) is employed to capture coherent structures and validate the pore scale prevalence hypothesis (PSPH).
II.
COMPUTATIONAL DETAILS
A.
Numerical solver For handling turbulent flow in porous media, three-dimensional incompressible Navier−Stokes
equations, given by Eqns. 1-2, are solved in non-dimensional form, where Π is the corresponding source term in the momentum equation to maintain constant mass flux. ∂uj =0 ∂xj
(1)
∂ui ∂ui uj ∂p 1 ∂ 2 ui + =− + + Πδi1 ∂t ∂xj ∂xi Re ∂xi ∂xj
(2)
The equations are discretized and solved using the open-source solver package OpenFOAM based on the finite-volume method. The spatial discretization is handled with a central differencing scheme. The temporal discretization is applied using a second-order 5
implicit Euler scheme. The Pressure-Implicit with Splitting of Operators (PISO) algorithm introduced by Issa 21 is employed for the pressure-velocity coupling in a semi-implicit way. The Poisson equation for the pressure is iteratively solved with a Geometric-Algebraic MultiGrid approach. The source term Π is adjusted dynamically in each time step. The algorithm is second-order accurate in space and time. The solver is parallelized with pure MPI on the HPC platform. All numerical simulations were run on the Cray XC40 cluster Hazel Hen at HLRS (National High Performance Computing Center) located in Stuttgart, Germany.
B.
Simulation conditions Porous media exhibit a broad range of geometrical variance with high level of complexity.
Regular porous structure frequently appears in industrial applications and is advantageous in microscopic observation compared with heterogeneous porous media. In the current study, a novel porous structure consisting of a periodic staggered array of square cylinders is considered. FIG. 1 illustrates the simulation domain of the DNS with an instantaneous velocity field in the background. All the simulation cases are imposed with an identical flow direction in x. In this study, the length scales are non-dimensionalized with the characteristic length ds of square cylinders. In a domain size of 12 × 8 × 4 or 9 × 6 × 4, the porous media domain consists of 24 identical porous units with a square cylinder in each. Table I summarizes the simulation conditions considered in this study. Cases are named with Reynolds number Re and porosity ϕ information; for example, Re1500P75 denotes the case with Re = 1500 and ϕ = 0.75. The square cylinders are arranged in a staggered way, which differs from the sparse in-line arrangement of Jin et al. 12 . A direct flow impingement is expected in the staggered orientation, which is quite common in densely-packed porous media, for example in the randomly packed spherical bed of Patil and Liburdy 7 . The reference case Re568P88 is a reproduction of case A in Jin et al. 12 . It is composed of 12 square cylinders forming long clear channels, which differs from the others. The size of the computational domain is apparently larger than that of Kuwahara et al .16 and Suga17 to permit the existence of large coherent scales. The chosen domain size is comparable to Jin et al. 12 . In Appendix A, the justification for this choice is shown through a comparison with a larger domain. Here, a Reynolds number range of Re = uint ds /ν = 500 − 1500 is investigated in the DNS cases, as described in Table I. Two values of porosity, ϕ = 0.75 and 6
FIG. 1: Simulation domain with triply periodic boundary conditions, ds and dc indicate characteristic length of square cylinder and void respectively, background surface shows instantaneous velocity field of case Re1500P75
ϕ = 0.56 corresponding respectively to high and low porosity conditions, are considered. For ϕ = 0.75, the channel width dc between the square cylinders is equal to the characteristic length of the square cylinder ds . For ϕ = 0.56, the channel width dc is half of ds . Six boundary conditions of the domain are defined as periodic boundaries in triple pairs to represent a REV element in a large porous media domain. The walls of the square cylinders are defined as no-slip boundaries. In all cases, orthogonal hexahedral meshes are applied with local refinement close to the walls. The total mesh resolution ranges from 50 × 106 up to 200 × 106 in the case Re1500P75. For DNS in porous media, the evaluation of grid resolution is not straightforward, as the non-dimensional form of grid size e.g. y + is difficult to define due to the lack of a proper p definition of uτ as in wall-bounded flow. Therefore, the local grid spacing ∆ = 3 ∆x ∆y ∆z ∂u0 ∂u0
is compared with the local Kolmogorov length scale η = (−ν 3 /εk )(1/4) , where εk = −ν ∂xji ∂xji is the dissipation rate of the turbulent kinetic energy. To ensure the quality of DNS, the local grid resolution (∆/η)max < 3.9 is found overall in the entire domain for all cases. In 7
TABLE I: Summary of simulation conditions, naming cases with the Reynolds number and porosity, e.g. Re500P75 denotes Re = 500 and porosity ϕ = 0.75 Case
Re
ds /dc
Mesh resolution xy × z (∆/η)avg (∆/η)max
ϕ
(in total) Re500P75
500
1
0.75
294912×150 (59 ∗ 106 )
1.3
2.3
Re500P56
500
2
0.56
353280×150 (49 ∗ 106 )
1.4
2.3
Re1000P75
1000
1
0.75
695520×200 (99 ∗ 106 )
1.9
3.1
Re1000P56
1000
2
0.56
418560×200 (83 ∗ 106 )
1.9
3.3
Re1500P75
1500
1
0.75
783360×240 (200 ∗ 106 )
2.1
3.4
Re1500P56
1500
2
0.56
529920×240 (127 ∗ 106 )
2.2
3.9
Re568P88 (Ref)
568
2
0.875
344064×150 (51 ∗ 106 )
1.4
2.3
order to refine the mesh at the position where strong dissipation occurs, an initial coarse DNS is performed to determine the necessity of local mesh refinement. TABLE I lists the grid resolution information with averaged and maximal value of ∆/η. In addition, a grid sensitivity study is performed as described in Appendix A.
III.
RESULTS AND DISCUSSION
In the following, both temporal and volume averaging are involved. The overbar operator ¯ denotes temporal averaging and the operator h i denotes volume averaging over pore units.
A.
Macroscopic flow statistics Macroscopic statistics is the first step in modeling turbulence in porous media. FIG. 2
depicts the volume averaged turbulent kinetic energy hk/u2int i and volume averaged Reynolds stress components hu0i u0j /u2int i of all six DNS cases. An increasing trend of turbulent kinetic energy with Reynolds number is observed for both high and low porosity cases with a higher slope in the latter case. Further, the turbulent kinetic energy is decomposed into Reynolds stress components as shown in FIG. 2(b). The difference between the components decreases with Reynolds number, indicating a higher isotropy at higher Re. The Reynolds shear stress 8
0.5
0.6
0.4
0.5
0.3 0.4
0.2 0.3
0.1
0.2 500
1000
0 500
1500
1000
1500
(b) Reynolds stress components hu0i u0j /u2int i
(a) Turbulent kinetic energy hk/u2int i
4
-0.25
3.9 -0.3
3.8 3.7
-0.35
3.6 -0.4 500
1000
3.5 500
1500
(c) Skewness of u0x
1000
1500
(d) Flatness of u0x
FIG. 2: Volume-averaged macroscopic turbulence statistics component hu0x u0y /u2int i is zero as a result of symmetry in the geometry. The third and fourth order moments of velocity fluctuation, referred respectively as skewness and flatness, are
S(ux ) = h
u03 x i 3/2 (u02 x)
(3)
u04 x i 2 (u02 x)
(4)
F (ux ) = h
Here, skewness and flatness respectively reveal information about the symmetry and intermittency of the probability density function of u0x . In FIG. 2(c), the skewness ranges 9
from −0.39 to −0.27, which is similar to those at the lowest and highest Reynolds number cases in Horton and Pokrajac4 . The flatness in FIG. 2(d) ranges approximately from 3.6 to 3.9, which is close to majority of the results in Horton and Pokrajac4 . As the Reynolds number increases, both skewness and flatness tend to be that of a Gaussian-like distribution (S = 0, F = 3).
B.
Microscopic flow statistics In this section, microscopic flow statistics extracted from DNS are presented and inter-
preted in detail. Taking advantage of the geometric periodicity of the porous structure, the microscopic statistics are presented by sampling a sub-domain. FIG. 3 illustrates the mean velocity profile u¯x /uint and u¯y /uint in a sampled 4 × 4 sub-domain for the cases Re1000P75 and Re1000P56. The highest mean streamwise velocities u¯x /uint are observed in the confined channel between two parallel square cylinders. The sudden contraction results in a ‘vena contracta’ between two recirculation areas. The lowest streamwise velocities are seen in the vicinity of solid walls including impingement wall (e.g. x = 6.5, y = 3.5 − 4.5), the wake region behind the square cylinder (e.g. x = 7.5, y = 3.5 − 4.5) and a small recirculation zone attached to the entrance of the shear walls (e.g. x = 6.5 − 7, y = 4.5). Both the porosity cases show qualitatively similar mean velocity maps but differ in detail as a result of different porous geometry. Narrower channel in the low porosity case (FIG. 3(c)) leads to a shortened wake area compared to the characteristic length of square cylinders and a higher momentum after the sudden contraction. In FIG. 3(b) and FIG. 3(d), u¯y /uint depicts a strong evidence of flow acceleration on both sharp corners of the square cylinders, which is followed by a high momentum u¯x /uint in the confined channel as just mentioned. The mean streamwise velocity u¯x /uint profiles at various positions for different Reynolds numbers (Re = 500, 1000 and 1500) are plotted in FIG 3(e) for ϕ = 0.75. The value of the Reynolds number has only a marginal effect on these averaged velocity profiles. A slightly higher acceleration occurs close to the walls at the unconfined location x = 7.5 for Re =1000 and 1500 compared with Re=500. FIG. 4(a)-(d) describe the distribution of turbulent kinetic energy k =
1 2
P
u0i u0i for the
cases Re500P75, Re1500P75, Re500P56 and Re1500P56, respectively. It should be noted that the scale of the color map is adjusted for higher k level in the low porosity cases (FIG. 10
(a) ux /uint , case Re1000P75
(b) ux /uint , case Re1000P75
(c) ux /uint , case Re1000P56
(d) uy /uint , case Re1000P56
5.5
5
4.5 0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
(e) Mean velocity profiles ux /uint for ϕ = 0.75, lines from x = 6 to x = 8 are superimposed for clarity
FIG. 3: Mean velocity profiles u¯x /uint and u¯y /uint in a 4 × 4 sub-domain 11
4(c) and (d)). The peak of k is located in the shear layer of the confined channels in all four cases. In the high-porosity cases, the high k region in the shear layer occupies about the same size as the characteristic length ds of the square cylinder (Figure 4(a) and (b)), whereas in the low porosity cases it is only around half ds (FIG. 4(c) and (d)). This more concentrated distribution of k suggests a fast decay in turbulence. The majority of this peak is contributed by the streamwise component u0x u0x . A distribution map of all three components u0x u0x , u0y u0y and u0z u0z is hidden for a clean layout. A second observed peak is a consequence of flow mixing near the channel center plane at the entry (e.g. x = 4.75, y = 3.75 in FIG. 4(d)). The wake region shows an obscure turbulent kinetic energy irrespective of Reynolds number and porosity. However, this wake area is penetrated by the high turbulence intensity from flow mixing in the low-porosity cases. Moreover, the cold hue in the low-porosity cases (ϕ = 0.56) is more dominantly observed in all regions except shear layer, which suggests a large standard deviation of spatial distribution of k. This indicates that the higher turbulence intensity in the macroscopic scale for low-porosity cases as shown in FIG. 2(a) is mainly contributed by the shear layer in the confined channel. FIG. 4(e) shows the profiles of k at different streamwise locations and Reynolds numbers for ϕ = 0.75. As Re increases, the turbulent kinetic energy increases especially in the shear layer in the confined channel (x = 7, between y = 5.2 and y = 5.5). It reveals the fact that the shear layer contributes significantly to the increasing trend of k with Re observed in the macro-scale (FIG. 2(a)). FIG.5 shows the distribution of Reynolds shear stress component u0x u0y distribution for the cases Re1500P75 and Re1500P56. The low porosity case shows a higher maximum, whereas the high porosity case has an extended boundary layer with a clear track of u0x u0y . This is in qualitative agreement with the tendency of turbulent kinetic energy k discussed previously with respect to FIG.4. The magnitude of the Reynolds shear stress is certainly not negligible in the microscopic sense. However, volume-averaged u0x u0y tends to be zero as a result of geometrical symmetry in these DNS cases, as observed in Jin and Kuznetsov14 . Evaluating the microscopic pressure field in porous media experimentally is even more challenging than the velocity field; hence it is worth to demonstrate this explicitly in the DNS. As porous media is characterized with highly tortuous flows between solid obstacles, dramatic spatial changes in pressure are common. The mean pressure field p¯ of case Re1500P75 is depicted in FIG.6. High pressure magnitude is observed on the impinging surface, which corresponds to a flow deceleration around the stagnation point. Sudden flow 12
(a) case Re500P75
(b) case Re1500P75
(c) case Re500P56
(d) case Re1500P56
5.5
y
Re500 Re1000 Re1500
x=6
5
x = 6.5
x=7
x = 7.5
x=8
4.5 0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
k (e) k profiles for ϕ = 0.75, lines from x = 6 to x = 8 are superimposed for clarity
FIG. 4: Mean turbulent kinetic energy k distribution for a 4 × 4 sub-domain, note the scale difference from (a) to (d)
contraction into confined channels indicates a large favorable pressure gradient (∂ p¯/∂x > 0) which intensifies turbulence in the channel. It is followed by an adverse pressure gradient after the local pressure minimum at x ≈ 6.8, which stabilizes the flow again. This dramatic change of pressure gradient explains the spot of high turbulent kinetic energy in the shear 13
FIG. 5: Reynolds shear stress component u0x u0y /u2int for the cases Re1500P75 and Re1500P56, note the scale difference
FIG. 6: Mean pressure field p for the case Re1500P75
layer (FIG.4(b)) and its decay afterwards. FIG.7 illustrates vortex structures identified by the λ2 criteria. λ2 is the second eigenvalue of the S 2 + Ω2 tensor, where Sij = (ui,j + uj,i )/2 and Ωij = (ui,j − uj,i )/2 represent the symmetric part and anti-symmetric part of the velocity gradient tensor ∂ui /∂xj respectively. The vortex structures in the case Re1000P75 are extracted from the instantaneous velocity field with a threshold λ2 = −200 for an optimal visualization and colored by the instantaneous pressure field p. The majority of vortex structures are observed close to the 14
FIG. 7: Visualization of flow structure in the case Re1000P75 with λ2 = −200 threshold, vortex colored by instantaneous pressure field p
shear walls while the remaining found on the impinging wall. The other regions show a low enrichment of vortex, which is consistent with the turbulent kinetic energy distribution shown in FIG.4. On the surface of the impinging wall, vortices are produced and stretched along the transverse direction y. The high pressure (red color) on the vortex center and low pressure (blue color) on the vortex antennas indicate a large pressure gradient causing the stretching and breakup. This directional pressure gradient also explains the consistent heading (transverse direction only) of most vortices on this surface. The vortex cluster in the confined channel entry is characterized with a significantly low pressure, while a progressive shedding is observed in the streamwise direction. In this shedding process, they are lifted up by an adverse pressure gradient (∂p/∂x > 0, indicated by a color change from cold to warm in FIG.7). This observation suggests that the favorable pressure gradient (∂p/∂x < 0) creates an instability to the flow and produces clusters of vortices, while the subsequent adverse pressure gradient in the streamwise direction stabilizes the flow and leads to the shedding of vortices. FIG.8 illustrates the instantaneous streamline field in the case Re1000P75 with cross15
sectional colorization. Four different fields including velocity magnitude |u| (upper left), vorticity magnitude |ω| (lower left), instantaneous turbulent kinetic energy (upper right) and pressure p (lower right) are used to color the streamlines in four separate cross-sections. In general, streamlines with less tortuosity are observed in the channels parallel to the flow direction, while a majority of vortex structures are captured in the wake area and also attached to the shear layers. As for the adjoint field with velocity magnitude |u| in upper left cross section, the main flow streamlines (yellow hue) inherit high flow velocity and bifurcate/merge periodically. Around the stagnation point of the impinging jet, the streamlines are greatly decelerated and spread into the transverse direction. The magnitude of vorticity |ω| = |∇×u| is given in the lower left domain. Obviously, the curved streamlines correspond to high magnitude of vorticity, specially with the flow contraction over sharp corners. Moreover, high vorticity is observed close to the impinging and shear wall surfaces, which corresponds to large kinetic energy dissipation at these positions. On the upper right corner, the vortex appearing near the shear-wall is found with high turbulent kinetic energy. The pressure field p in the lower right corner is observed to be strongly correlated to the streamline patterns. High positive pressure (red to yellow hue) is seen in regions of flow expansion and jet impingement. The contracted channels (from walls to channel centers) show a negative pressure field indicated by a cold hue. Vortex structures attached to the shear walls are generally characterized with a low pressure center.
C.
Budget of turbulent kinetic energy Turbulent kinetic energy budget is comprised of production, diffusion and dissipation of
turbulent kinetic energy as follows: ∂k + Ck = Pk + εk + Πk + Dk + Tk ∂t
(5)
Where, 1 k = u0i u0i , 2 Πk = −
Ck = uj ∂p0 u0j , ∂xj
∂k , ∂xj Dk =
Pk = −u0i u0j ∂ ∂k (ν ), ∂xj ∂xj 16
∂ui , ∂xj
εk = −ν
Tk = −
∂ui 0 ∂ui 0 ∂xj ∂xj
1 ∂u0i u0i u0j 2 ∂xj
FIG. 8: Instantaneous streamlines for the case Re1000P75, cross-sectional colored with upper left corner: velocity magnitude |u|, lower left corner: vorticity magnitude |ω|, upper right: instantaneous turbulent kinetic energy, lower right: pressure p
The terms k, Ck , Pk , εk , Πk , Dk and Tk indicate turbulent kinetic energy, convection, production, dissipation, pressure diffusion, viscous diffusion and turbulent diffusion respectively. Their distributions for case Re1500P75 are depicted in FIG.9. The production Pk (FIG.9(a)) and dissipation εk (FIG.9(b)) are directly responsible for the gain and loss of k in the transport equation, respectively. In FIG.9, the maximum of shear production Pk is identified at the entry of channels (e.g. after x = 6.5) where a sudden flow contraction occurs and a shear layer begins to develop. This location does not exactly coincide with the maximum of k in FIG.4(b). Rather, the maximum of Pk is located upstream of the maximum of k. The peak of dissipation εk in FIG.9(b) is identified both on the impinging wall (e.g. x = 6.5, y = 3.5 − 4.5) and shear walls (e.g. x = 6.5 − 7.5, y = 4.5). On the shear walls, dissipation increases rapidly after the small recirculation zone. Wake wall shows only marginal dissipation. Both pressure diffusion Πk (FIG.9(c)) and viscous diffusion Dk (FIG.9(d)) show strong inhomogeneities on the impinging wall and shear wall without any significant trace away from the walls. In contrast, turbulent diffusion Tk shows clear trace with superimposed positive and negative tails at the entry of channel. Convection with the 17
mean flow Ck shows its maximal contribution around the corner. The shear production Pk and dissipation rate εk at key locations as a function of Reynolds number for ϕ = 0.75 are summarized in FIG.10. The shear production Pk (FIG.10(a)) is sampled at three streamwise locations in the confined channel (x = 6.5, x = 6.7 and x = 7 for y = 4.5 − 5.5). The sensitivity of Pk to changes in Re is maximum at the entry position x = 6.5. Interestingly, a slight decrease of Pk with Re is observed at the downstream location (x = 7, y = 4.7). It suggests that the shear production tends to concentrate with increasing Reynolds numbers. FIG.10(b) shows the sampling of the dissipation rate εk on three wall surfaces: impinging wall (x = 6.5, y = 3.5 − 4.5), shear wall (x = 6.5 − 7.5, y = 4.5) and wake wall (x = 7.5, y = 3.5 − 4.5). The wake wall shows only a weak dissipation almost independent of Re. Maximal dissipation rates are observed at the corners of the impinging wall and at the center of the shear wall. The sensitivity of the results to changes in Re is more readily observed at lower Re (500 to 1000) than at higher Re (1000 to 1500). FIG.11(a) shows map of the Kolmogorov length scale η/ds for the case Re1500P75. Under the condition of constant kinematic viscosity, the Kolmogorov length scale η is inversely proportional to (−εk )0.25 . Its minimum is observed near the corner of the impinging wall, where the strongest dissipation occurs as shown in FIG.10(b). And the maximal η is found in the wake region. The ratio of ηmax /ηmin is about 4 in case Re1500P75. Obtaining the information of η is relevant for an accurate length scale estimation, which can be only achieved indirectly in experiments9 . In FIG.11(b), maximum ηmax /ds , minimum ηmin /ds and the volume-averaged hηi/ds are summarized. All three measures of Kolmogorov length scale tend to decrease with increasing Re. Here, the low porosity cases (ϕ = 0.56) show smaller Kolmogorov length scales, which corresponds to more significant turbulent kinetic energy dissipation in FIG.4. However, this difference is less significant as Re increases.
D.
One-dimensional energy spectrum In this paragraph, the one-dimensional energy spectra Exx , Eyy and Ezz and their pre-
multiplied spectra kz Exx , kz Eyy and kz Ezz of three velocity components are investigated in detail. The energy spectra are based on a fast Fourier transform in spanwise direction (wavenumber kz ) and a temporal averaging. Two cutting planes normal to the y-axis, one in the channel center (y = 5 in ϕ = 0.75 cases) and one close to the shear wall (y = 4.7 in 18
(a) Pk
(b) εk
(c) Πk
(d) Dk
(e) Tk
(f) Ck
FIG. 9: Distributions of turbulent kinetic energy budget terms: Pk , εk , Πk , Dk , Tk and Ck for the case Re1500P75 19
5.5
5
4.5 0
2
4
6
8
0
2
4
6
8
0
2
4
(a) Production rate at three streamwise-normal planes
7.5
4.5
7
4
6.5
3.5 -8
-6
-4
-2
0
(b) Dissipation rate εk on three surfaces: impinging wall, shear wall and wake wall
FIG. 10: Production Pk and dissipation εk profiles as a function of Re (ϕ = 0.75 cases)
ϕ = 0.75 cases) are of special interest, since shear layer, turbulence mixing and impinging are observed there. The log-log spectra (non-pre-multiplied) at various locations on these two cutting planes for the case Re500P75 are given in FIG.12. In the figures, DNS shows the capability to resolve small scales in a broad dissipation range beyond the inertial range indicated by the slope of −5/3. Five to seven decades of energy can be resolved, which is a proof of the good numerical quality of the DNS. In the channel center (FIG.12(a),(c) and (e)), the wake region x = 5.6 (red curve) indicates the lowest magnitude of Exx , Eyy and Ezz in the entire wavenumber range. In FIG.12(a), a bulge (brown curve) in magnitude of Exx is observed at a wavenumber of around kz = 101 at the location x = 8.3, which is 20
0.025 ϕ = 0.75 ϕ = 0.56
ηmax
η/ds
0.02 0.015
hηi
0.01 0.005
ηmin 0 500
1000
1500
Re (b)
(a)
FIG. 11: (a) Map of Kolmogorov length scale η/ds for the case Re1500P75, and (b) an overview of ηmax /ds , ηmin /ds and hηi/ds for all the cases
absent at the upstream location x = 7.8. This indicates large-scale ux breaking into smaller scales on the impinging surface. This tendency is also identified in the pre-multiplied energy spectrum kz Exx from FIG.13(a). The highest magnitude of Eyy (FIG.12(c)) is found at x = 6.8, which corresponds to a significant turbulence mixing at the channel entry. In contrary to the dominant Exx at kz = 101 (‘bulge’) in the impinging area (x = 8.3, brown curve in FIG.12(a)), Eyy shows a cave (brown curve in FIG.12(b)) at low wavenumber at this position. FIG.12(b),(d) and (e) depict the spectra of Exx , Eyy and Ezz in the shear layer positions. The most energetic motion in Exx is found at x = 6.8 and x = 7.3 at low wavenumbers, which corresponds to the high turbulent kinetic energy in the shear layer as observed in FIG.4(a). Exx at x = 8.3 shows a significantly lower magnitude at low wavenumbers compared to most of the other x locations, which is a consequence of the impinging jet. Selected pre-multiplied spectra are shown in FIG.13. Gray arrows connecting the spectral curves point to the streamwise direction corresponding to increasing values of x. The highest energy contribution kz Exx (FIG.13(a)) is found where strong shearing occurs at wavenumber of about kz = 5, which suggests a significantly larger scale than that of the impinging jet (maximum at kz = 10). Compared with the observed highest kz Eyy (FIG.13(b)) from the 21
flow mixing (kz = 9), the shear layer still has a larger length scale. This indicates that energetic coherent structures might be found close to the shear layer. The effect of Reynolds number on the energy spectrum is discussed using cases Re500P75 (solid lines) and Re1500P75 (dash lines) as shown in FIG.14. An enormous growth of energy with Reynolds number in the high wavenumbers is clearly identified in both Exx (FIG.14(a)) and Eyy (FIG.14(c)). On the other hand, large-scale turbulence structures in the shear layer (Exx at x = 6.8 and x = 7.8) lose their energy content with increasing in Reynolds numbers, which corresponds to a clear shift of peak in the pre-multiplied spectrum in FIG.14(b). However, this kind of shift is less obvious in the impinging area (green curves in FIG.14(b)). For kz Eyy in FIG.14(d), a similar but mild peak shift is observed in the flow mixing area. This spectral analysis leads to the conclusion that with increasing Reynolds numbers, large turbulent scales are attenuated and small scales are strengthened. In FIG.15, a comparison between the cases Re1500P75 (solid lines) and Re1500P56 (dash lines) indicates the effect of porosity. Choosing a scaled position in the streamwise direction of the channel enables an objective comparison in the pore unit. Spectra with the same color are considered at the same scaled position and therefore compared with each other. Recalling the prior analysis, low-porosity cases (ϕ = 0.56) show a higher macroscopic turbulent kinetic energy at the same Reynolds number. This is highlighted by the prominent peak in the shear position (red dotted line in FIG.15(a)) from the microscopic point of view. Similarly, two spectra representing flow mixing show a higher peak for the low porosity cases (blue and red dotted lines from FIG.15(b)) compared to the corresponding high porosity cases (blue and red solid lines) with a slight shift to higher wavenumbers. The differences at other positions are rather marginal.
E.
Anisotropy characteristics Anisotropy of the Reynolds stress tensor can be represented by using the traditional
anisotropy-invariant map (AIM) introduced by Lumley22 . The Reynolds stress anisotropy tensor (bij ) is written as:
bij =
u0i u0j u0k u0k
−
δij 3
(6)
Its invariants I, II and III are obtained to analyze the anisotropy of the Reynolds stress 22
10 0
10 0
10 -2
10 -2
10 -4
10 -4
10 -6
10 -6
10 -8
10 -8
10 -10 10
1
10
10 -10
2
(a) Exx at channel center (y = 5.0)
10 0
-2
10
-2
10 -4
10 -4
10 -6
10 -6
10 -8
10 -8
10 -10 10
1
10
10 -10
2
(c) Eyy at channel center (y = 5.0)
10 0
10 -2
10 -2
-4
10
10 -6
10 -8
10 -8
-10
10 1
10 2
-4
10 -6
10
10 1
(d) Eyy at shear layer (y = 4.7)
10 0
10
10 2
(b) Exx at shear layer (y = 4.7)
10 0 10
10 1
10
10 2
(e) Ezz at channel center (y = 5.0)
-10
10 1
10 2
(f) Ezz at shear layer (y = 4.7)
FIG. 12: Energy spectra Exx , Eyy and Ezz at various locations for the case Re500P75, black solid line: −5/3 slope 23
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0 10
1
10
2
10
(a) kz Exx at shear layer (y = 4.7)
1
10
2
(b) kz Eyy at channel center (y = 5.0)
FIG. 13: pre-multiplied energy spectra kz Exx and kz Eyy at various locations for the case Re500P75, grey arrow indicates the streamwise direction x
tensor. The first invariant I is characteristically zero whereas the other invariants are given by II = λ21 + λ1 λ2 + λ22 and III = −λ1 λ2 (λ1 + λ2 ). With the eigenvalues of the tensor λi , they can be directly used to depict the anisotropy in a two-dimensional visualization (Lumley triangle) or through a new η − ξ coordinate system (turbulence triangle) with
η 2 = II/3, ξ 3 = III/2
(7)
This turbulence triangle improves the visualization of isotropic state aiming to evaluate trajectories of the return to isotropy of homogeneous turbulence23 . However, both approaches have difficulties to represent large amount of data in a complex geometry. A recent visualization technique is proposed by Emory and Iaccarino24 based on the barycentric map construction developed by Banerjee et al.25 . This method makes use of red, green and blue (RGB) coloration to represent three states: one-component, two-component and isotropic, as the three vertices of a triangle. This coloration is also suitable for the traditional Lumley triangle and turbulence triangle24 , yet the linearity in the barycentric map utilizes equally weighted three states. The barycentric map is based on a reconstruction of the eigenvalues λi . Placing the boundary states (one component x1C , two component x2C and isotropy x3C ) √ in a coordinate system at x1C = (1, 0), x2C = (0, 0) and x3C = (1/2, 3/2), the coordinate 24
10 -1
0.25 0.2
10 -3 0.15 0.1
10 -5
0.05
10 -7
0
10 1
10 2
10 1
(a) Exx at shear layer (y = 4.7) 10
10 2
(b) kz Exx at Shear layer (y = 4.7)
-1
0.25 0.2
10 -3 0.15 0.1
10 -5
0.05
10 -7
0
10 1
10 2
10 1
(c) Eyy at channel center (y = 5.0)
10 2
(d) kz Eyy at channel center (y = 5.0)
FIG. 14: Reynolds number effect on energy spectra Exx and Eyy and pre-multiplied spectra kz Exx and kz Eyy for the cases Re500P75 and Re1500P75, black solid line: −5/3 slope system (xB , yB ) is defined as 1 xB = C1c x1c + C2c x2c + C3c x3c = C1c + C3c , 2
(8)
√ yB = C1c y1c + C2c y2c + C3c y3c = C3c
3 2
(9)
where the weight parameters Cic are:
C1c = λ1 − λ2 , 25
(10)
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
101
0
102
(a) kz Exx at shear layer (y = 4.7 and y = 3.60)
101
102
(b) kz Eyy at channel center (y = 5.0 and y = 3.75)
FIG. 15: Porosity ϕ dependency of pre-multiplied energy spectrum kz Exx and kz Eyy at various locations for the cases Re1500P75 and Re1500P56
C2c = 2(λ2 − λ3 ),
(11)
C3c = 3λ3 + 1
(12)
In addition, the coefficients Cic in the barycentric map obey the relation C1c +C2c +C3c = 1. Following one of the implementations by Emory and Iaccarino24 , the red-green-blue (RGB) color map is constructed through: R 1 0 0 G = C1c 0 + C2c 1 + C3c 0 B 0 0 1
(13)
This simple idea distributes RGB colors to componentality behaviors in the way that one-component turbulence is red, two-component is green, isotropic is blue, and all other states within the map are linear mixtures of these colors. With this approach, the barycentric map is colored and shown in FIG.16. The anisotropy of the Reynolds stress can be visualized and quickly determined in a complex physical domain such as the porous media structure. In FIG.16, the anisotropy of volume-averaged (in the pore scale) Reynolds stress 26
FIG. 16: RGB color map based on the barycentric anisotropy with the trajectories of simulation cases, ϕ = 0.75 cases in blue, ϕ = 0.56 cases in red, arrows point to increasing Reynolds number
anisotropy tensor of all cases is plotted. Regardless of the porosity and Reynolds number, the volume-averaged macroscopic Reynolds stresses are located close to the isotropy corner. This tendency is more significant with increasing Reynolds number and porosity. Microscopic visualization in the porous geometry with this RGB color map for the cases Re500P75, Re1500P75, Re500P56 and Re1500P56 is depicted in FIG.17. In these cases, the isotropic turbulence (blue hue) is generally dominant in the flow distant from the wall. High-porosity cases show larger blue regions which means a larger fraction of isotropic turbulence. This microscopic perspective explains the observed isotropic tendency (FIG.16). One-component turbulence (red hue) is detected adjacent to the impinging wall and in the convergent flow after the sharp corner, which becomes less significant in the high Reynolds number cases Re1500P75 and Re1500P56. This explains the directional shift to isotropic turbulence in macroscopic scale as shown in FIG.16. Adjacent to the shear and wake walls in all the four cases, the state of two-component turbulence (green) is observed, which is a consequence of suppressed velocity fluctuations in the wall-normal direction. This twocomponent turbulence is partially replaced by the isotropic turbulence in high Reynolds number cases, similar to the observation for one-component turbulence. 27
(a) case Re=500P75
(b) case Re1500P75
(c) case Re500P56
(d) case Re1500P56
FIG. 17: Visualized anisotropy of Reynolds stress tensor with the RGB colormap in FIG.16
F.
Modal analysis with proper orthogonal decomposition (POD) Jin et al. 12 and Uth et al. 13 quantified the size of large eddies through two point correla-
tions and integral length scales. The main conclusion from their study is that the scales of the turbulent structure is limited by the pore size (PSPH: pore scale prevalence hypothesis). An alternative method to extract large eddies is a modal analysis such as proper orthogonal decomposition (POD). POD decomposes the flow field in a descending rank of energy, which enables a projected visualization of the leading energetic modes. In turbulent flows, POD has been used to identify and analyze coherent structures in both experiments26 and DNS27,28 . 28
The extracted POD modes are orthogonal to each other and can be used for constructing reduced-order models29,30 . Compared with classical POD, the snapshot POD by Sirovich 31 has the advantage of significantly reducing the size of the high-dimensional spatial data sets to determine the eigenvalue and eigenvector. This makes the snapshot-based approach the most widely used POD method. Given m snapshots of spatial field of a scalar or vector q(ξ, ti ), over discrete spatial points ξ and at discrete times ti , the fluctuation x(t) reads x(t) = q(ξ, ti ) − q¯(ξ) ∈ Rn
(14)
where n indicates the total spatial resolution. The covariance matrix R of x(t) is given as follows.
R=
m X
xT (ti )x(ti ) = X T X
(15)
i=1
where the matrix X consists of the m snapshot data x(ti ) stacked into a matrix X = [x(t1 ) x(t2 ) . . . x(tm )] ∈ Rn×m
(16)
With the method of snapshot POD, R is successfully downsized to m×m from the original eigenvalue problem of size n×n. The objective of the POD analysis is to identify the optimal basis vectors that can best represent the given data. The solution to this problem can be determined by finding the eigenvectors ψj and the eigenvalues λj of the equation.
Rψj = λj ψj
(17)
The POD mode φj is then recovered through projection of X on the direction of eigenvectors ψj . Xψj φj = √ λj
(18)
The original field can be reconstructed with a linear combination of the modes φj and their P corresponding temporal coefficients, as q(ξ, ti ) = j aj (t)φj (ξ). Two-dimensional surface snapshots normal to the spanwise direction z are well suitable to the current two-dimensional porous structure. A total of m = 5000 snapshots sampling three velocity components ui at the cell center are collected to calculate the temporal correlation. The sampling interval is 29
∆t = 0.1T , where T is defined as the characteristic time T = ds /uint . The total sampling period covers 500 T or about 42 domain flow through time. Additional attention has been paid to ensure the convergence of POD analysis. FIG.18 illustrates the first two POD modes of case Re568P88, which shows strong similarity between each other with a spatial shift. Two types of dominant structures are observed with alternative appearance in the streamwise direction. One shows dual-lateral attachment to the shear walls, while the other has only single-lateral attachment. These coherent structures with dual attachment are approximately of the size of one pore unit or slightly beyond (2ds to 2.5ds ), depending on including or excluding the outstretched feet. This obtained modal structure size is about twice the integral length scale (L11 = 1.04ds ) from the same case calculated by Jin et al. 12 . FIG.20(a) shows the percentage energy fraction of the first 50 modes in descending order and the accumulated energy proportion. It shows that a large percentage of energy (approximate 15% each) is observed in the first two modes, whereas the rest contains lower than 3% each. The first 50 modes contain about 50% of the total energy. FIG.19 describes the first two POD modes in the case of Re500P75 with a more dense packing than Re568P88. Compared with FIG.18, the observed coherent structures are narrowed and shortened as a result of the dense porous medium. The structures are also not significant enough to be extracted from the snapshots. All these structures inherit the same appearance: unlike traveling in the channel center as shown in FIG.18, they are closely attached to the shear walls with breakups near the next impinging surface, which indicates a length between ds and 2ds . This is shorter than the observation in FIG.18. In FIG.20(b), the first two modes of the case Re500P75 do not possess significantly more energy than the rest of the modes. The mode 1 contains less than 1% of total energy, which explains the weak appearance of both leading modes in FIG.19. A direct comparison between cases Re568P88 and Re500P75 reveals the fact that the shrunk void and increased tortuosity in porous media impair the coherent structures. In a sparsely packed porous medium, coherent structures are able to take advantage of the straight channel and occupy larger space with a larger portion of turbulent kinetic energy. This modal analysis generally proves the pore scale prevalence hypothesis (PSPH). However, large-scale energetic motion can not be ignored in a turbulent flow within high-porosity porous media with low tortuosity. Since the moderate Reynolds number case with larger void (Re500P75) is characterized with only obscure coherent structures, further attempts on cases with high-Re or low porosity is not 30
(a) Mode 1
(b) Mode 2
FIG. 18: First two POD modes for the case Re568P88
(a) Mode 1
(b) Mode 2
FIG. 19: First two POD modes for the case Re500P75 quite necessary.
IV.
CONCLUSION
In this work, we presented a microscopic analysis of turbulence topology in regular porous media with the help of DNS. Unlike the common macroscopic modeling, DNS resolves the full-spectrum of turbulent scales in the entire geometry, which could contribute to a more complete understanding of turbulence topology. Besides the fact that this regular porous 31
0.2
1
0.2
0.8
0.15
1
0.8
0.15
0.6
0.6
0.1
0.1 0.4
0.05
0.4 0.05
0.2
0 0
0 5 10 15 20 25 30 35 40 45 50
0.2
0 0
(a) Re568P88
0 5 10 15 20 25 30 35 40 45 50
(b) Re500P75
FIG. 20: Modal energy fraction and accumulated energy in turbulence
structure itself has extensive technical applications, the observed characteristic flow patterns of convergent channel, impinging surface and wakes are also identified in other homogeneous/heterogeneous porous media experiments7 and has been studied qualitatively. Compared to experiments, DNS is convenient in sampling data and making detailed observations in constrained spaces such as in porous media. Applying triply periodic boundary conditions enables to consider porous media in a representative elementary volume (REV) in an efficient way. Six DNS cases build up a systematic study with the Reynolds number Re = 500, 1000 and 1500 and porosity ϕ = 0.56 and 0.75. The reference case Re568P88 inherits a sparse porous arrangement from Jin et al. 12 . Three basic flow patterns: confined channel with shear flow, impinging jet and wake can be recognized in DNS. In the macroscopic statistics, turbulent kinetic energy k increases with increasing Reynolds number or decreasing porosity, which is mostly contributed by the shear layer. Confined channel flow exhibits a favorable pressure gradient followed by an adverse pressure gradient. The vortex structures captured by the λ2 criteria are characterized with local low pressure and shed with the adverse pressure gradient. A full decomposition of the turbulent kinetic energy transport equation indicates a high production rate after sudden contraction and a strong dissipation on both shear layer and impinging jet layer, whereas the wake region shows low level of these turbulence statistics. One-dimensional energy spectra and pre-multiplied spectra detect the large scale energetic structure in the shear layer and a breakup of scales in the impinging layer at low Re for the high porosity case. However, 32
this large scale structure in the shear layer breaks into less energetic small structures as Re increases. This suggests an absence of coherent structures in highly turbulent dense-packed porous media flows as in canonical wall-bounded flows such as channel and pipe flows. The macroscopic and microscopic anisotropy are discussed with the visualization based on a barycentric map. Turbulent flows in porous media are highly isotropic in the macroscopic sense, which is clearly not the case in the micro-scale. One-component and two-component turbulence are identified close to the wall surfaces with nearly isotropic state distant to the walls. This local anisotropy is attenuated with increasing Reynolds numbers and porosity. Additionally, a modal analysis based on POD is included to highlight the energy-dominant coherent structures. The results generally support the validity of the pore scale prevalence hypothesis (PSPH). However, it should be noted that energetic coherent structures are captured in the case with sparsely-packed porous medium.
ACKNOWLEDGMENTS The current research is supported by MWK (Ministerium f¨ ur Wissenschaft und Kunst) of Baden-W¨ urttemberg as a part of the project DISS (Data-integrated Simulation Science). We are sincerely thankful to the Deutsche Forschungsgemeinschaft (German Research Foundation) for the project SFB-1313. The authors gratefully appreciate the access to high performance computing facility ‘Hazel Hen’ in HLRS, Stuttgart and their kind support.
Appendix A: Code validation and grid sensitivity study The code validation based on a previous DNS work in an REV-scale porous media by Jin et al. 12 is presented in this section. The geometry and simulation conditions are chosen to represent case A in Table 4 of Jin et al. 12 . The Reynolds number, based on the definition of Jin et al. 12 , is Rep = uint ϕds /ν = 500 with a porosity of ϕ = 0.88. A high-resolution in small REV size 12ds × 8ds × 4ds and a low-resolution in large REV 24ds × 16ds × 4ds are demonstrated (Fig. 21). In high-resolution DNS, the employed mesh resolution 768 × 512 × 150 is comparable to the reference DNS work with LBM solver. In large-domain DNS, a reduced mesh resolution of 768 × 512 × 75 for larger computational domain is used. The dimensionless time step ∆t/(ds /uint ) = 0.001 is found to be close to the reference DNS 33
with the LBM solver and one magnitude smaller than the same FVM solver due to CFL (Courant-Friedrichs-Lewy) number restriction. Fig. 21 shows the two-point correlation ˆ ii components in streamwise direction x1 and transverse direction x2 as in Jin et al. 12 . R Both high-resolution DNS and large-Domain DNS show no significant difference with the reference LBM results. It indicates that the numerical solver used in the current work is reliable. Besides, the REV size 12ds × 8ds × 4ds is large enough to represent coherent motion in the porous media. The effect of computational domain is marginal.
REFERENCES 1
H. P. G. Darcy, Les Fontaines publiques de la ville de Dijon. Exposition et application des principes `a suivre et des formules `a employer dans les questions de distribution d`eau, etc (1856).
2
A. Dybbs and R. V. Edwards, “A new look at porous media fluid mechanicsdarcy to turbulent,” in Fundamentals of transport phenomena in porous media (Springer, 1984) pp. 199–256.
3
M. H. Pedras and M. J. De Lemos, “Macroscopic turbulence modeling for incompressible flow through undeformable porous media,” International Journal of Heat and Mass Transfer 44, 1081–1093 (2001).
4
N. Horton and D. Pokrajac, “Onset of turbulence in a regular porous medium: An experimental study,” Physics of fluids 21, 045104 (2009).
5
D. Seguin, A. Montillet, and J. Comiti, “Experimental characterisation of flow regimes in various porous media-I: Limit of laminar flow regime,” Chemical engineering science 53, 3751–3761 (1998).
6
D. Seguin, A. Montillet, J. Comiti, and F. Huet, “Experimental characterization of flow regimes in various porous media-II: Transition to turbulent regime,” Chemical engineering science 53, 3897–3909 (1998).
7
V. A. Patil and J. A. Liburdy, “Turbulent flow characteristics in a randomly packed porous bed based on particle image velocimetry measurements,” Physics of Fluids 25, 043304 (2013).
8
V. A. Patil and J. A. Liburdy, “Flow structures and their contribution to turbulent dispersion in a randomly packed porous bed based on particle image velocimetry measurements,” 34
Physics of Fluids 25, 113303 (2013). 9
V. A. Patil and J. A. Liburdy, “Scale estimation for turbulent flows in porous media,” Chemical Engineering Science 123, 231–235 (2015).
10
D. Hlushkou and U. Tallarek, “Transition from creeping via viscous-inertial to turbulent flow in fixed beds,” Journal of Chromatography A 1126, 70–85 (2006).
11
R. J. Hill and D. L. Koch, “The transition from steady to weakly turbulent flow in a close-packed ordered array of spheres,” Journal of Fluid Mechanics 465, 59–97 (2002).
12
Y. Jin, M.-F. Uth, A. V. Kuznetsov, and H. Herwig, “Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study,” Journal of Fluid Mechanics 766, 76–103 (2015).
13
M.-F. Uth, Y. Jin, A. V. Kuznetsov, and H. Herwig, “A direct numerical simulation study on the possibility of macroscopic turbulence in porous media: Effects of different solid matrix geometries, solid boundaries, and two porosity scales,” Physics of Fluids 28, 065101 (2016).
14
Y. Jin and A. V. Kuznetsov, “Turbulence modeling for flows in wall bounded porous media: An analysis based on direct numerical simulations,” Physics of Fluids 29, 045102 (2017).
15
D. L. Penha, S. Stolz, J. Kuerten, M. Nordlund, A. Kuczaj, and B. Geurts, “Fullydeveloped conjugate heat transfer in porous media with uniform heating,” International journal of heat and fluid flow 38, 94–106 (2012).
16
F. Kuwahara, T. Yamane, and A. Nakayama, “Large eddy simulation of turbulent flow in porous media,” International Communications in Heat and Mass Transfer 33, 411–418 (2006).
17
K. Suga, “Understanding and modelling turbulence over and inside porous media,” Flow, Turbulence and Combustion 96, 717–756 (2016).
18
P. Kundu, V. Kumar, and I. M. Mishra, “Numerical modeling of turbulent flow through isotropic porous media,” International Journal of Heat and Mass Transfer 75, 40–57 (2014).
19
W. Breugem and B. Boersma, “Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach,” Physics of fluids 17, 025103 (2005).
20
G. Yang, B. Weigand, A. Terzis, K. Weishaupt, and R. Helmig, “Numerical simulation of turbulent flow and heat transfer in a three-dimensional channel coupled with flow through porous structures,” Transport in Porous Media 122, 145–167 (2018). 35
21
R. I. Issa, “Solution of the implicitly discretised fluid flow equations by operator-splitting,” Journal of computational physics 62, 40–65 (1986).
22
J. L. Lumley, “The structure of inhomogeneous turbulent flows,” Atmospheric turbulence and radio wave propagation 790, 166–178 (1967).
23
K. S. Choi and J. L. Lumley, “The return to isotropy of homogeneous turbulence,” Journal of Fluid Mechanics 436, 59–84 (2001).
24
M. Emory and G. Iaccarino, “Visualizing turbulence anisotropy in the spatial domain with componentality contours,” Cent. Turbul. Res. Annu. Res. Briefs , 123–138 (2014).
25
S. Banerjee, R. Krahl, F. Durst, and C. Zenger, “Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches,” Journal of Turbulence 8, N32 (2007).
26
L. H. Hellstr¨om and A. J. Smits, “The energetic motions in turbulent pipe flow,” Physics of Fluids 26, 125102 (2014).
27
J. Baltzer, R. Adrian, and X. Wu, “Structural organization of large and very large scales
in turbulent pipe flow simulation,” Journal of Fluid Mechanics 720, 236–279 (2013). 28 ¨ K. E. Meyer, J. M. Pedersen, and O. Ozcan, “A turbulent jet in crossflow analysed with proper orthogonal decomposition,” Journal of Fluid Mechanics 583, 199–227 (2007). 29
P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, coherent structures, dynamical systems and symmetry (Cambridge university press, 1998).
30
B. R. Noack, M. Morzynski, and G. Tadmor, Reduced-order modelling for flow control, Vol. 528 (Springer Science & Business Media, 2011).
31
L. Sirovich, “Turbulence and the dynamics of coherent structures. i. coherent structures,” Quarterly of applied mathematics 45, 561–571 (1987).
32
D. A. Nield and A. Bejan, “Mechanics of fluid flow through a porous medium,” in Convection in Porous Media (Springer, 2017) pp. 1–35.
33
M. De Paoli, F. Zonta, and A. Soldati, “Dissolution in anisotropic porous media: Modelling convection regimes from onset to shutdown,” Physics of Fluids 29, 026601 (2017).
36
0.2
0.2 Hi-Rs DNS La-Do DNS Jin et al.
0.16
0.16 0.12 ˆ 11 /u2m R
ˆ 11 /ˆ R u2m
0.12 0.08 0.04
0.04
0
0 -2
-1
0 1 (x1 − x10 )/s
2
3
-2
0.2
0.2
0.16
0.16
0.12
0.12 ˆ 22 /u2m R
ˆ 22 /ˆ R u2m
-3
0.08
0.04
0
0 -2
-1
0 1 (x1 − x10 )/s
2
3
-2
0.2
0.16
0.16
0.12
0.12 ˆ 33 /u2m R
0.2
0.08
0.04
0
0 -2
-1
0 1 (x1 − x10 )/s
2
3
0 (x2 − x20 )/s
1
2
-1
0 (x2 − x20 )/s
1
2
-1
0 (x2 − x20 )/s
1
2
0.08
0.04
-3
-1
0.08
0.04
-3
ˆ 33 /ˆ R u2m
0.08
-2
ˆ ii /ˆ FIG. 21: Validation of two-point correlation R u2m in two directions x (left side) and y (right side) with Jin et al .12 in the LBM solver, both high-resolution DNS and large-domain DNS included, x1,2 correspond to the longitude and transverse direction as in Jin et al. 12 . Here the definition of uˆm , x10 , x20 and s are the same as in Jin et al. 12 .
37
