Influence of coherent vortex structures in subgrid scale motions on particle statistics in homogeneous isotropic turbulence

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Influence of coherent vortex structures in subgrid scale motions on particle statistics in homogeneous isotropic turbulence Yan Xiong a, Jing Li a,∗, Fei Fei b, Zhaohui Liu a,∗, Wei Luo c a

State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China c Wuhan second ship design and research institute, Wuhan 430205, China b

a r t i c l e

i n f o

Article history: Received 26 January 2018 Revised 31 October 2018 Accepted 31 October 2018 Available online xxx Keywords: Particle-laden flow Large eddy simulation Subgrid-scale structures Coherent vortex extraction Wavelet filter

a b s t r a c t Subgrid scale (SGS) structures in large eddy simulations (LES) of turbulent particle-laden flows significantly influence particle dynamics, especially those of small inertial particles. In this study, homogenous isotropic turbulence with Taylor’s Reynolds number of 102.3 is generated by a direct numerical simulation (DNS) and a wavelet-based coherent vortex extraction method is implemented to extract the coherent SGS structures and then investigate their effects on particle dynamics, including single-particle and particle-pair statistics. Compared to the classical spectral-filtered DNS (FDNS), which cuts off only the high wavenumber components regardless of the turbulence structures in the SGS motions, the waveletfiltered DNS (WFDNS) can retain more coherent vortex structures in the SGS flow field with the help of the high compression rate characteristics of wavelet transformation. Comparing the results of WFDNS and FDNS at the identical effective grid number, it can be found that the single-particle statistics are mainly controlled by the macro energy-containing structures, and the SGS coherent vortex structures play important roles in the particle-pair dynamics, including the radial distribution function, radial relative velocity, and collision kernel. Therefore, in view of the characteristic of wavelet filtering that preserves the SGS coherent structure, the wavelet-based structural filter should be particularly suitable for LES modeling of particle-laden flow. © 2018 Elsevier Ltd. All rights reserved.

List of symbols CVE coherent vortex extraction DNS direct numerical simulation DPM discrete particle model FDNS filtered direct numerical simulation HIT homogeneous isotropic turbulence LES large eddy simulation MRA multilevel resolution analysis PDF probability density function RDF (g) radial distribution function (dimensionless) RRV radial relative velocity (particle-pair) SGS sub-grid scale WFDNS wavelet filtered direct numerical simulation L length of the simulation domain Ncell number of cells in the simulation field Np number of particles β particle collision kernel dp diameter of particle

∗

Corresponding authors. E-mail addresses: [email protected] (J. Li), [email protected] (Z. Liu).

j k R r Rij,p Reλ Rep r.m.s Sp 3 S Stk Sij,p u’ wr w− r

ε εT εν τp τk TE

the level of MRA analysis wavenumber in spectral space separation distance of particle-pair at collision radius separation distance of particle-pair rotation tensor rate of the flow at particle location Taylor microscale Reynolds number particle Reynolds number root mean square third-order structure function of particle radial relative velocity skewness of particle radial relative velocity particle Stokes number based on the Kolmogorov scale strain tensor rate of the flow at particle location r.m.s value of the turbulent velocity radial relative velocity (dimensionless) radial relative velocity in inward direction compression ratio of effective node number in WFDNS threshold value in wavelet filtering turbulent viscosity dissipation particle relaxation time Kolmogorov time scale of turbulence Eulerian integral time scale of turbulence

https://doi.org/10.1016/j.ijmultiphaseflow.2018.10.021 0301-9322/© 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Xiong, J. Li and F. Fei et al., Influence of coherent vortex structures in subgrid scale motions on particle statistics in homogeneous isotropic turbulence, International Journal of Multiphase Flow, https://doi.org/10.1016/j.ijmultiphaseflow.2018. 10.021

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volume of the domain volume of the annulus with radius r Kolmogorov length scale of turbulence Kolmogorov velocity of turbulence fluid viscosity

1. Introduction Particle-laden flows are important in a wide variety of environmental and industrial applications in which particle motions play a key role in atmospheric pollutant dispersion (Shaw, 2003), rain droplet formation (Grabowski and Wang, 2013), turbulent mixing in spray combustion (Jenny et al., 2012), and protoplanetary formation (Pan and Padoan, 2014). The multiscale characteristic of turbulence strongly affects the characteristics of particle motions (Grabowski and Wang, 2013; Shaw, 2003). For the single-particle diffusion process, the turbulent transport affects the interaction between the inertial particles and carrier flow, which are characterized by the underlying fluid turbulent kinetic energy and the Lagrangian time scale seen by particles (Fede et al., 2006). Particlepair dispersion not only depends on turbulent transport effects but also on inter-particle collisions (Voßkuhle et al., 2014b; Wang et al., 2008; Wang et al., 20 0 0a; Zhou et al., 1998) and covers a broad range of phenomena such as particle preferential concentration (Andrew and Lance, 2014; Wood et al., 2005; Zaichik and Alipchenkov, 2005), particle-pair relative dispersion (Leonid and Vladimir, 2009; Zaichik and Alipchenkov, 2003), collision (Pan and Padoan, 2014), and agglomeration (Reeks, 2014). Although direct numerical simulation (DNS) has been proven to be a reliable tool to analyze the behaviors of particle pairs, it inevitably suffers from unaffordable computation limitations on high Reynolds numbers that can be achieved. The limitation of DNS motivates the development of large eddy simulation (LES) in particle-laden flow. Within the framework of LES particle-laden flows, some fundamental and industrial issues on the interactions between small scale turbulence and particle dispersion have been investigated (Bianco et al., 2012; Chibbaro et al., 2014; Pitton et al., 2012; Zhou et al., 2017). However, the absence of subgrid scale (SGS) motions leads to evident discrepancies in the particle-pair statistics, such as the radial distribution function (RDF) and the radial relative velocity (RRV), especially for particles with Stk ∼ O(1) in both monodisperse systems (Fede and Simonin, 2006; Jin et al., 2010a; Pozorski and Apte, 2009; Ray and Collins, 2011; Xiong et al., 2018) and bidisperse conditions (Chen and Jin, 2017). Therefore, modelling the SGS motions and predicting the particle variables correctly in LES remains a major challenge. Currently, there are two main SGS particle model frameworks, that is, stochastic models (Minier et al., 2014) and structural models (Marchioli, 2017). In the first framework, most models are derivatives of Langevin-type formulations. Previous studies implemented this model to reconstruct the particle seen SGS velocity, which exhibited excellent agreement with the particle kinetic energy and the integral time seen by the particles (Berrouk et al., 2007; Fede et al., 2006; Shotorban and Mashayek, 2006). Extensions of this model, such as replacing the SGS velocity with a corresponding particle seen value (Jin et al., 2010b) and considering the influence of the space-time correlation (Jin and He, 2013), have also been reported. Recently, the developments of the Lagrangian-filtered mass density function (LFMDF) approach (Innocenti et al., 2016) have shown improved predictions of the particle concentration. However, they have few effects on predicting the RDF because the random force term in the model would introduce additional numerical diffusions that might destroy the microstructures in the SGS motions (Pozorski and Apte, 2009). Furthermore, the Lagrangian subgrid model (LSGS) proposed by

Mazzitelli et al. (2014) works well for the particle pair and tetrad dispersion due to its ability to describe the turbulent temporal and spatial correlations. But these models are not currently aimed at recovering the coherent structures in the SGS motions. Moreover, structural models that aim to mimic the important structural features in the SGS motions should address this issue. The approximate deconvolution model (ADM) could partially recover the subgrid kinetic energy by enhancing the scales near the cutoff wavenumber in LES. Shotorban et al. (2007) and Marchioli et al. (2008) showed the advantages of the model in predicting particle-pair dynamics such as the RDF, relative particle dispersion, and collisions compared to stochastic models. Nevertheless, this model does not work for a coarse LES or a high Reynolds configuration because it cannot recover the contributions below the subgrid scales associated with the filter size. Recently, a similar concept to ADM that implements a differential filter (Park et al., 2017) exhibited improvement in predicting particle preferential concentrations. Extensions of ADM were reported by Marchioli et al. (2008) and Gobert and Manhart (2011), which was expected to reconstruct the velocity field at scales smaller than the filter size as much as possible. However, the current structural models are based on the numerical extensions of the kinetic turbulent energy spectra and lack of explicit physical descriptions of the coherent vortex structures in subgrid scale motions (hereafter referred to as SGS coherent vortex structures), which would lead to severe limitations for high Reynolds conditions. Appropriate SGS structural models for particles, especially for predicting particle-pair dynamics such as the RDF and the particle collision rate, should recover the effects of SGS coherent vortex structures as much as possible. The wavelet-based vortex extraction technique provides a good opportunity to retain SGS structures. Wavelet decompositions were introduced in the early 1990s to analyze turbulent flows (Farge, 1992; Meneveau, 1991), and the ability of this technique to extract local flow structures was also exploited. It shows significant potential to characterize those structures at a comparably low cost due to the high compression rate for wavelet filtering. The extension of this method is also called coherent vortex simulation (CVS) and is used to compute and model the coherent part of the turbulent flows (De Stefano and Vasilyev, 2012; Farge and Schneider, 2001; Jacobitz et al., 2016; Sakurai et al., 2017; Yoshimatsu et al., 2009). The preservation of coherent vortex structures at low computational costs indicates the potential of this method to address issues of particle-pair statistics in LES. For instance, De Stefano and Vasilyev (2010) have introduced a so-called SCALES (stochastic coherent adaptive large eddy simulation) method, which utilizes the wavelet filter in the adaptive LES framework through the algorithm of adaptive wavelet collocation method (AWCM) proposed by Vasilyev et al. (1995). Recently, Bassenne et al. (2016) proposed a new method that aims to extract the particle cluster with the help of wavelet filters and exhibited potential ability to capture the preferential concentration effect with a significant reduction in the computation costs. In view of this, an orthogonal wavelet decomposition algorithm is adopted in the present study to extract the coherent vortex structures in DNS flow fields with different specified threshold values. The influence of SGS structures on particle dynamics are obtained by comparing the results of wavelet-filtered DNS (WFDNS) and classical spectral-filtered DNS (FDNS). These findings provide the foundation for the further development of the ideal SGS structural model of two-phase turbulence LES. The rest of this paper is organized as follows: The coherent vortex extraction (CVE) method with wavelet filtering and the numerical setup are described in Section 2. The particle statistics are shown in Section 3. An analysis of the influence of the SGS structures on particle dynamics is reported with the comparison

Please cite this article as: Y. Xiong, J. Li and F. Fei et al., Influence of coherent vortex structures in subgrid scale motions on particle statistics in homogeneous isotropic turbulence, International Journal of Multiphase Flow, https://doi.org/10.1016/j.ijmultiphaseflow.2018. 10.021

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Table 1 Characteristics of the homogeneous isotropic turbulence in the DNS.

Reynolds number RMS fluid velocity Turbulent kinetic energy Dissipation rate Kolmogorov length scale Kolmogorov time scale Eulerian integral time scale Molecule viscosity kmax η

Symbols

DNS

Jin et al. (2010a)

Reλ u, k ɛ

102.3 18.76 527.9 3210 0.0138 0.0039 0.048 0.0488 1.7

102.05 19.34 561.1 3771.4 0.0135 0.0037 0.050 0.0488 1.73

η τk TE

ν

3

WFDNS (ε = 0.01) WFDNS (ε = 1/83 ) FDNS ( = 8)

N

k

ε

η

τk

0.01×2563 323 323

523.8 512.8 498.4

2470 1777 1311

0.0147 0.0160 0.0173

0.00445 0.00524 0.00610

wavelet filtering framework is expected to provide more information based on both the physical and scale space. The velocity field u(x) can be represented in terms of the wavelet basis function as

of DNS and FDNS in Section 4. Finally, conclusions are provided in Section 5.

u (x ) =

 l∈L

+∞ 2 −1   n

cl0 φl0 (x ) +

0

j=0 μ=1 k∈Kμ, j

μ, j

dk

ψkμ, j (x )

(4)

μ, j

2. Numerical methodology 2.1. Fluid phase (DNS and FDNS) The dimensionless governing equations for the fluid field shown below were solved on a cubical uniform mesh with 2563 grid points, and the side length of the cube L was 2π :

∂ ui = 0 and ∂ xi

(1)

∂ ui ∂ ui ∂p 1 ∂ ui + uj =− + + fi . ∂t ∂xj ∂ xi Re ∂ x j ∂ x j



(2)

The fluid field was solved using a pseudo-spectral method to solve the N–S equation. The Fourier coefficients were advanced in time using a second-order explicit Runge–Kutta method and the time step was chosen to ensure that the CFL number was approximately 0.5 for numerical stability and accuracy. The aliasing errors were almost removed by the 3–2 rule (Rogallo, 1981), where the velocity field was driven and maintained by a random √ artificial force, which is non-zero only at low wavenumbers |k| < 8 in the Fourier space. The forcing scheme was analogous to that developed and tested by Eswaran and Pope (1988). In the present paper, a homogeneous isotropic turbulence (HIT) was evolved through approximately 10 eddy turnover times to reach a stationary state with Reλ = 102.3. The main characteristics of the HIT are listed in Table 1. It’s clear that the main fluid statistics in the present paper are close to those of Jin et al. (2010a). The small discrepancies should be attributed to the differences in additional numerical viscosity of the time integration scheme and also the random force added on the right-hand side of the N–S equations. A filtered DNS field can be obtained by applying a cutoff filter in spectral space with various filter scales, which truncated the Fourier coefficients larger than the cutoff wavenumber (kc ) in the turbulent energy spectra and is expressed as follows:



u(k, t ), if|k| ≤ kc , 0, otherwise.

εT (x ) = u¯ > i

l∈L0

2

u˜ (k, t ) =

where φl0 and ψk are three-dimensional (3D) scale functions and wavelet coefficients of different families μ and levels of scale j, respectively. The wavelet decomposition procedure mainly consists of two steps. First, we use an orthogonal 3D multiresolution analysis (MRA) of the field u(x) with the help of wavelet transformation. Second, wavelet filtering is performed through a wavelet coefficient threshold within the wavelet space. The wavelet filter is defined as:

(3)

In this paper, the kc of the spectral filters is set to 16 (corresponding to the filter width ࢞ = 8 in the physical space), and the filter scales selected are located in the range of inertial subzone, which is always resolved in LES. Detailed parameters can be found in Table 2. 2.2. Coherent vortex extraction with wavelet filtering Wavelets are basis functions that are localized in physical and wavenumber space (scale space). For comparison, the classical Fourier transform used in FDNS provides only frequency information below a chosen cutoff wavenumber kc not in the physical space. Therefore, this space/scale localization characteristic in the

+∞ 2 −1  n

cl0 φl0 (x ) +



j=0 μ=1 k∈Kμ, j μ, j

μ, j

dk

ψkμ, j (x )

(5)

|dk |>εT ui W T F

where ε T is the nondimensional threshold parameter and  · WTF represents the wavelet threshold filtering (WTF) norm. For all of the simulations presented in the following, it has been specified as the L2 norm, that is, ui W T F = ui 2 . As demonstrated by Farge and Schneider (2001) and Goldstein et al. (2005), the implementation of a wavelet filter with the specified threshold value ε T to the homogeneous isotropic turbulence velocity or vorticity field can lead to the decomposition of the corresponding field into coherent uc and incoherent part ui (nearly Gaussian white noise):

u = uc + ui .

(6)

 = ∇ × v. This algorithm is implemented in a 3D vorticity field w  are developed into an orthonorFirst, the three components of w mal wavelet series that ranges from the largest scale lmax = 20 to the smallest scale lmax = 2−J +1 . The vorticity field is then decomposed into coherent and incoherent parts with the threshold εT = ( 43 Z ln N )1/2 , where Z is the vorticity magnitude and N is the sum of effective grid number 2563 . We can reconstruct the velocity field from the coherent part through the 3D inverse wavelet  ). For the transformation and the Biot–Savart law v = ∇ × (∇ −2 w specific threshold value ε T , it could also be called the CVE, which is consistent with the work of Jacobitz et al. (2008). Fig. 1 shows the ratios of the turbulent energy and enstrophy retained in the WFDNS field with the corresponding results of the DNS as a function of the compression rate ε , for which ε = neffective /noriginal is defined as the ratio of the number of effective nodes in the WFDNS to the original DNS one. It exhibits a high compression rate of turbulent statistics, for which nearly 87% of the enstrophy and 99% of the turbulent kinetic energy are retained with only 1% of the DNS nodes, that is, ε = 0.01. Apparently, the enstrophy is more difficult to capture than the turbulent kinetic energy since it contains more structural information, as shown in Fig. 1. Moreover, we decrease the compression rate ε to 1/83 (called WFDNS ε = 1/83 hereafter) to be consistent with the FDNS resolution. The turbulent energy and dissipation spectrum for the DNS and the WFDNS (ε = 0.01 and 1/83 ) are presented in Fig. 2. The results of the FDNS (࢞ = 8) that has identical effective nodes to the WFDNS (ε = 1/83 ) are also included for comparison. Obviously, the

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constant value dp of 0.5η. In this case, only the drag forces working on the particle center of the mass are considered and the governing equation of the particle motion can be simplified described as below. These were solved using the second-order Runge–Kutta method for the momentum equation of the particle velocity and a second-order Adams–Bashforth integral scheme for the particle position equation in time:

dxpi = u pi and dt

(7)

(ugi, p − upi ) dupi = CD , dt τp

(8)

where xpi and upi are the instantaneous particle position and velocity, respectively, and ugi,p is the instantaneous fluid velocity at the particle location. In the DNS and FDNS, the particle seen fluid velocity is obtained through a three-dimensional partial Hermite interpolation (PHI) scheme developed by Balachandar and Maxey (1989). However, for the case of the WFDNS, we use the fourth-order Lagrangian interpolation method instead. The particle residual time τ p is expressed as follows: Fig. 1. Ratios of the kinetic energy and enstrophy retained in the wavelet filtering to the corresponding DNS results along with the compression rate.

curves of the WFDNS (ε = 0.01) agree well with the DNS results, with only a certain bias in the high wavenumber region. For the smaller threshold WFDNS, that is, the WFDNS (ε = 1/83 ), the curve dropped much faster than that of the WFDNS (ε = 0.01). By contrast, the WFDNS (ε = 1/83 ) still preserved a comparable amount of coherent vortex structures in the high wavenumber region compared to the FDNS (࢞ = 8), where an obvious cutoff exists in the inertial subzone, as shown in Fig. 2. Thus, the wavelet filter exhibits a distinct advantage over the classical spectral filter for the protection of the structural information. 2.3. Discrete particle motion In this study, the dispersed phase was assumed to be diluted and the one-way momentum coupling was adequate. The density of the particles was assumed to be much larger than that of the fluid, that is, ρ p /ρ f >> 1, and the diameter of the particles dp was assumed not to exceed the Kolmogorov spatial microscale η with a

τp =

ρ p d2 . ρ f 18ν

(9)

The particle drag force coefficient (CD ) and particle Reynolds number (Rep ) are expressed by

CD = 1 + 0.15Re p 0.687 and

Re p =

  u pi − ugi,p  v

d p.

(10)

(11)

Meanwhile, the particle Stokes number (Stk ) was defined as:

Stk = τ p /τk ,

(12)

where τ k is the Kolmogorov time scale. When the velocity field of the fluid reaches a stationary state, 1.2 × 106 particles are released with random distribution in the computational domain. The initial velocity of the particles was set to be equal to the fluid velocity in the same location. The statistics were calculated after a stationary condition was achieved, which was sustained for approximately six eddy turnover times.

Fig. 2. The ensemble average spectra in homogeneous isotropic turbulence as a function of the wavenumber k for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8): (a) the turbulent kinetic energy and (b) the turbulent dissipation rate.

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2.4. Particle-pair statistics Particle collision detection was conducted using the efficient cell index method by Sundaram and Collins (1997) and Allen and Tildesley (1987) with the concept of linked lists, and the collision detection procedure per step needs N p (27N p /Ncell − 1 )/2 operations. Then, the collision kernel is calculated using

βTDNS =

2NTDNS , n2

(13)

where n is the number density in this field and NTDNS represents the number of collisions detected in a time step. In addition, the estimated results can be provided using the following formula (Sundaram and Collins, 1997):

βT EST = 2π R2 |wr |g(r ).

(14)

In the preceding equation, g(r) is the RDF and |wr | is the ensemble value of the modulus of the particle-pair RRV, which can be expressed using

N pcol l iding × V f ield g( r ) = and VS × Nt × N pnor

|wr | = where



l iding i=N col p

i=1

Vfield

is

(15)

wri · R col l ding /NP , R the

volume

(16) of

the

domain,

Vs =

4π [(R + δ /2) − (R − δ /2 ) ] is the volume of the annulus for detection, δ is width of the detection buffer zone, and N nor is the p number of particle pairs in the detection zone when the particles are uniformly distributed. 3

3

3. Effect of SGS structures on particle dynamics In this section, inertial particles were released into the flows described and evolved to obtain steady statistics. The particle Stk number ranges from 0.1 to 10, and the particle-pair separation ratio r/η also varies from 0.25 to 4.5. In the following paragraphs, Stk = 0.1, 1.0, and 3.0 are especially selected for simplicity to represent the small, intermediate, and comparably large particles, respectively. Furthermore, r/η = 0.25, 1.0, and 4.0 are chosen to discover the underlying effects on the particle-pair dynamics in the range roughly covering the dissipation subzone. 3.1. One-point particle statistics First, one-point particle statistics, including the particle seen fluid turbulent kinetic energy, that is, kpg , and the particle seen fluid Lagrangian integral time scale, that is, TLp@f , are examined in detail within the framework of the DNS, WFDNS, and FDNS, as shown in Fig. 3. In fact, there is a close relationship between the values of kpg and TLp@f and the single-particle dispersion statistics. As reported by Fede and Simonin (2006), particle dispersion coefficient can be expressed as the product of the particle kinetic energy kp and particle Lagrangian integral time scale TLP , that is, D p = 23 k p TLp . At the same time, the particle kinetic energy can also be further expressed as k p = f¯ k pg (Tchen, 1947), where f¯ is the fp

fp

filtered fluid-particle correlation function which depends on the particle seen fluid Lagrangian integral time scale TLp@f . Fig. 3a presents the distributions of kpg (Stk ) ≡ vpg (t) · vpg (t)/2, where vpg (t) is the instantaneous fluid velocity interpolated at the particle position. It is clear that kpg of the WFDNS (ε = 0.01) almost overlaps with the DNS result, and the peak value occurs at Stk ≈ 1.0 due to the preferential concentration effects. As the compression ratio ε decreases, that is, the WFDNS (ε = 1/83 ), the curve gradually deviates from the DNS, but the maximum error is still limited

5

to 5% in this range. Moreover, Fig. 3 also indicates that the discrepancy between the WFDNS (ε = 1/83 ) and FDNS (࢞ = 8) is quite small, especially for Stk ≤ 3.0. The distributions of TLp@f are plotted in Fig. 3b. The value of TLp@f remains in a plateau for Stk ≤ 1.0 and then grows sharply with increasing Stk . Theoretically, two limit values of TLp@f exist, that is, TL (the turbulent Lagrangian integral time scale) for fluid tracer particles and TE (the turbulent Eulerian integral time scale) for large particles with Stk > > 1. Similar to the results of kpg , the influence of filtering is limited, and the relative errors between the cases of WFDNS (ε = 1/83 ) and FDNS (࢞ = 8) are below 3%. Consequently, with the identical effective grid number, the wavelet filtering does not significantly improve the one-point particle statistics compared to the classical spectral filtering, as shown in Fig. 3. This because kpg and the TLp@f are both dominated by the resolved energy-contained vortex structures (Fede and Simonin, 2006). Thus, the filtered coherent structures in the SGS motions have small effects on the one-point particle statistics. 3.2. Particle-pair collision statistics In the following section, the influence of the SGS coherent vortex structures on the particle-pair dynamics are discussed, including the RDF, RRV, and collision kernel. For a given effective threshold value ε = 0.01 with the wavelet filter, the differences between the results of the WFDNS and DNS should be attributed to the contribution of the incoherent vortex structures discarded by the wavelet filtering method. In addition, due to the optimized preservation of the SGS structures in the WFDNS framework, the effects of the coherent vortex structures in the SGS motions on the particles can be revealed by comparing with the results of the WFDNS (ε = 1/83 ) and the FDNS (࢞ = 8). Fig. 4 presents the contoured slice of the vorticity field at plane of y = π and the preferential concentration effect of the inertia particles with Stk = 1.0 in the cases of the DNS, WFDNS (ε = 1/83 ), and FDNS (࢞ = 8). As shown in Fig. 4a, the vortex structures retained in the WFDNS (ε = 1/83 ) field with the black lines nearly match the contoured vorticity field of the DNS. Meanwhile, the preferential concentration effect of the particles within the WFDNS (ε = 1/83 ) flow field is highly similar to that in the DNS flow field, as seen in Fig. 4b. In addition, the corresponding vorticity field and particle distribution in the FDNS (࢞ = 8) are also presented in Figs. 4c and d for comparison. It is clear that the preservation of the vortex structures in the WFDNS is much better than in the FDNS (see Figs. 4a and c). The particle distribution in the WFDNS is also much closer to that in the DNS flow field than the result in the FDNS (see Figs. 4b and d). These results indicate that the filtered field based on the wavelet filters can better capture the structural information to recover the local accumulation effect of the particles. The RDF is an effective variable used to investigate the particle preferential concentration effects. Fig. 5 presents the variation of the RDF as a function of Stk and the particle-pair separation distances. The curves decrease with the increasing separation r/η since the inertial particles in the HIT tend to accumulate in the regions of high strain rates but low vorticity as a result of the “centrifuge” effect, which grows stronger at smaller scales. Using precise capturing of the coherent structures, the results of the WFDNS (ε = 0.01) are best consistent with those of the DNS for all of the separation distances considered in this study. Furthermore, the curves of the WFDNS (ε = 1/83 ) exhibit improvements over the cases of FDNS (࢞ = 8) at all of the Stokes numbers and separations considered. Fig. 6a compares the RDF at the collision radius Rcol = dp for different Stk with different filters, which is a special case presented in Fig. 5. The absence of SGS motion leads to a large characteristic time scale of the flow that in turn drives the correspond-

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Fig. 3. One-point particle statistics as a function of Stk for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8): (a) the particle seen fluid turbulent kinetic energy kpg and (b) the particle seen fluid Lagrangian integral time scale TLp@f .

ing vortex to interact with large inertia particles. In contrast, no similar trend has been found in the WFDNS (ε = 1/83 ). This infers that a better preservation of SGS coherent vortex structures in the WFDNS could maintain the characteristic space/time scales and prevent these values from being extremely large. Fig. 6b shows the relative errors of the RDF compared to the results in the DNS. Evidently, the WFDNS (ε = 0.01) almost captures the particle preferential concentration effect, with some discrepancies only at Stk = 0.3. As a whole, the errors decrease to zero, oscillating as the Stk number increases. For the results in the WFDNS (ε = 1/83 ), the relative errors reach their maximum value at Stk = 0.3 as much as 30% but still much smaller than those of the FDNS (࢞ = 8). For Stk < 1.2, the curves of the WFDNS (ε = 1/83 ) and FDNS (࢞ = 8) share a similar trend; however, the error curve of the WFDNS (ε = 1/83 ) also exhibits considerable improvement. The method introduced in this study can retain more coherent vortex structures than the classical FDNS, especially for SGS motions. This is a key point for precise descriptions of particle clustering. In the FDNS and LES, the filtered turbulent vortex structures become more correlated and stronger, leading to a relatively large RRV error as reported by Jin et al. (2010a). Fig. 7a shows the variations of the RRV with Stk , and the curves in the DNS, WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8) share the same trend. All of the curves remain flat for Stk ≤ 1.0 and increase sharply for larger Stk . The turbulent transport effect, which is dominated by the macro vortex structures in the field, strongly affects the RRV for large Stk particles. The relative errors of the RRV in the WFDNS and FDNS compared to the results of the DNS are displayed in Fig. 7b. In the WFDNS (ε = 0.01), the relative errors are below 10% in the range of Stk = O(0.1 ∼ 10 ) because of its ability to maintain the coherent vortex structures at most of the scales resolved. When the threshold ε decreases, that is, the WFDNS (ε = 1/83 ), the relative errors increase as expected, especially for small inertial particles with Stk ≤ 1.0, which could be as large as 20%. For Stk > 1.0, the errors decreased with oscillation since the macro vortex structures are almost resolved for both the WFDNS (ε = 1/83 ) and FDNS. For all Stk , the results of the WFDNS (ε = 1/83 ) exhibit obvious improvements over those of the FDNS. The collision kernel represents the joint effects of particle preferential concentration and the turbulent transport, which could

be described as the RDF and RRV, respectively. Fig. 8 presents the distribution of the collision kernel, that is, β col , as a function of Stk . The collision kernel increased rapidly for Stk ≤ 1.0 until it approached a plateau for Stk > 1.0, as shown in Fig. 8a. Fig. 8b presents the relative errors of the collision kernel compared to corresponding DNS results as a function of Stk . The absence of smallscale motions in the FDNS would lead to large errors for Stk ≤ 3.0. For the WFDNS (ε = 0.01), the relative errors are limited and below 10% in this range. Notably, the errors increase with decreasing ε , that is, the WFDNS (ε = 1/83 ), and the peak value reaches 40%. However, they remain much smaller than those of the FDNS (࢞ = 8). This result implies that the preservation of the coherent vortex structures in the SGS motions is also of vital importance for the prediction of β col . Moreover, both the FDNS (࢞ = 8) and WFDNS (ε = 1/83 ) can provide reliable predictions of the collision kernel for Stk > 3.0, which is consistent with the findings of Jin et al. (2010a). This is due to the complementary effects of the over-prediction of g(r) and the under-prediction of |wr | after filtering. As previously shown, particle-pair statistics benefit significantly from wavelet filtering, especially for the low Stk range. The implementation of wavelet filtering proves it is able to better preserve the coherent vortex structures in the SGS motions than the FDNS, which contributes to the predictions of the processes that the local vortex structures dominate. 4. Effect of SGS structures on particle seen flow statistics with different filters In this section, the underlying mechanism regarding the effects of SGS vortex structures on particle-pair dynamics are further discussed by examining the particle seen flow statistics that reflect the detailed information of the local flow structures. 4.1. Vortex structures of the filtered SGS flow field In order to more clearly understand the influencing mechanism of SGS flow field on particle-pair dynamics, Fig. 9 first presents the structures of the filtered SGS flow field at the plane of y = π in both the WFDNS (ε = 1/83 ) and FDNS (࢞ = 8), which have the identical effective grid number. It is clear that the SGS flow field

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Fig. 4. Instantaneous snapshots of the vorticity field and particle distributions with Stk = 1.0: (a), (b) WFDNS (ε = 1/83 ), (c), and (d) FDNS ( = 8). The color contours are the results of the DNS, and black lines are the results of the WFDNS or FDNS. The solid and dashed lines represent the positive and negative values, respectively. The red points represent the particles in the DNS, and the black points represent the particles in the WFDNS or FDNS.

obtained from the spectral cutoff filter (see Fig. 9b) contained more microstructures than that from the wavelet filtering method (see Fig. 9a). These microstructures, that is, SGS coherent vortex structures, play a considerable role in the particle preferential concentration effect (Ray and Collins, 2013). In traditional LES, the SGS coherent vortex structures (as shown in Fig. 9) are always modelled using the eddy viscosity models, for which their structural characteristics would not be captured. However, the effect of these vortex structures of the filtered SGS flow field on particle motion cannot be ignored, especially for the particle-pair dynamics, and the wavelet filter also exhibits advantages in preserving such structural information. These results were mentioned in Section 3 and will be further discussed. 4.2. Effect of the absence of SGS coherent vortex structures on particle seen fluid derivatives In this paper, following the ideas of Chun et al. (2005), the gradient of the particle seen velocity, i.e. A(xp (t), t) ≡ ∇ u(xp (t), t), is

decomposed into a symmetric strain rate tensor, i.e. S(x p (t ), t ) = [A(x p (t ), t ) + AT (x p (t ), t )]/2 and an antisymmetric rotation rate tensor, i.e. R(x p (t ), t ) = [A(x p (t ), t ) − AT (x p (t ), t )]/2. And then, the normalized second invariant variable of the rate of strain and rotation tensor at the particle location are defined as:

 2p S

 2p R

= S(x p (t ), t ) : S(x p (t ), t )

(17)

= R(x p (t ), t ) : R(x p (t ), t ).

(18)

As pointed out by Maxey (1987) that the preferential concentration of particle field was directly related to the difference between the second invariant of the rate of strain and rate of rotation tensors sampled by the particles, that is, τη2 S2  p − τη2 R2  p , at low Stk . Fig. 10 presents the distributions of this variable, and its relative errors compared to the DNS results. As shown in Fig. 10a, all the curves own a mono-peak at intermediate Stk , and the DNS result increases with increasing Stk for the range of Stk ≤ 0.5, which is identical with the theoretical prediction (Andrew and Lance, 2014).

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herent vortex structures, especially for particle clustering at small Stk . 4.3. Effect of the absence of SGS coherent vortex structures on high-order moments of RRV The probability distribution function (PDF) of the particle RRV is a major component used to investigate high-order statistical characteristics (de Jong et al., 2010). Fig. 11 presents the distributions of the PDF of the particle RRV normalized by its r.m.s value, that is, 1/2

Fig. 5. Distribution of the RDF as a function of Stk for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8) with particle-pair separation distances r/η = 0.25, 1.0, and 4.0, respectively.

It is noted that these curves of τη2 S2  p − τη2 R2  p share the same tendency with the RDF presented in Figs. 5 and 6. Conversely, the peaks of the curves of τη2 S2  p − τη2 R2  p and the RDF are not coincident. This shift of the peak should be attributed to the clustering mechanism for large inertial particles, that is, non-local caustic (Bragg et al., 2015). In Fig. 10b, the relative errors of WFDNS (ε = 0.01) are the smallest as expected. And, the curve of WFDNS (ε = 1/83 ) also exhibits considerable improvement compared to the result of FDNS (࢞ = 8), for the whole range of the considered Stk . Furthermore, the critical Stk is located at 4.0, and the SGS coherent vortex structures have few effects on τη2 S2  p − τη2 R2  p as Stk > 4.0. From the significant influence of the coherent part of the SGS motions on particle seen variables, it might give us a clue that the interactions between the SGS motions and inertia particles would benefit a lot from the methods aimed at adaptively reconstructing the local co-

wr / w2r  with r/η = 1.0 and 4.0 for the DNS, WFDNS (ε = 1/83 ), and FDNS (࢞ = 8). Notably, the PDFs share a similar shape. For a small particle-pair separation ratio, that is, r/η = 1.0, the tails of the PDFs become more pronounced and deviate from the Gaussian shape, especially for the negative part. This indicates that the intermittency is mainly affected by small-scale events in the turbulence. Moreover, with increasing Stk , the deviation between the PDF distribution of the RRV and the Gaussian distribution also gradually increases, especially for the positive part of the PDFs, as shown in Figs. 11b and c. The phenomenon called “caustic” (Falkovich et al., 2002; Gustavsson and Mehlig, 2014; Wilkinson et al., 2006), the “crossing trajectory” or the “sling effect” (Voßkuhle et al., 2014a) should be responsible for this extra intermittency since the multi-value of the particle velocities at a given point could lead to large relative velocities at small separations as Stk increases (Gustavsson et al., 2012). Notably, the effects of filtering seem negligible for most cases with wavelet filtering in Fig. 11, except for the FDNS cases at Stk = 0.1 and 1.0, which means that the macro coherent vortex structures dominate the distribution of the RRV and the small inertial particles suffer more from the absence of the SGS coherent vortex structures. The skewness of the RRV is used to analyze the effect of the absence of SGS coherent vortex structures on the non-Gaussian characteristics of the PDFs of the RRV, which is defined as the normalized third-order moments of the particle RRV:

 3 / 2

S = S3p / S2p



= [w p (t )]

3

  r

/ [w p (t )]

2

 3/2 r

.

(19)

Fig. 12 presents the distribution of the skewness of the RRV as a function of r/η for the DNS, WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8). Three patterns corresponding to three different Stk are shown in Fig. 12. As reported by Wang et al. (20 0 0b) and

Fig. 6. (a) Distribution of the RDF at the contact radius Rcol , that is, g(r), as a function of Stk for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8). (b) Relative errors of the RDF compared to the corresponding results of the DNS after filtering.

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Fig. 7. (a) Distribution of ensemble mean absolute value of the particle RRV, that is, wr as a function of Stk for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8). (b) Relative errors of |wr | compared to the corresponding results of the DNS after filtering.

Fig. 8. (a) Distribution of the collision kernel, that is, β col , as a function of Stk for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8). (b) Relative errors of the collision kernel compared to the corresponding results of the DNS after filtering.

Ray and Collins (2011), the RRV are negatively skewed because of two contributions. First, the velocity derivatives of the underlying turbulence are always negatively skewed as a result of the turbulent energy cascade, which is the case for particles with Stk ≈ 0.1 since they mimic the behavior of fluid particles, as shown in Fig. 12a. Second, the absolute value of skewness increases for intermediate inertial particles, that is, particles with Stk ≈ 1.0 (see Fig. 12b), due to the path history effect proposed by Bragg et al. (2015). For particles with much larger Stk , that is, Stk = 3.0 (see Fig. 12c), the effects of the two mechanisms vanish since the particle motions are gradually decoupled from the fluid velocity field as with free molecules. In addition, the influence of the wavelet filtering on the skewness vary with Stk , but they all reverse the erroneous trend for the cases of Stk = 0.1, 1.0, and 3.0 compared to the FDNS. The distribution of the skewness of the RRV as a function of Stk and its relative errors compared to the DNS result are plotted in Fig. 13. Because the non-Gaussian characteristics of the PDFs of the RRV (as shown in Fig. 11) can be attributed to the particle preferential concentration (Xiong et al., 2018), the

peak value of the skewness of the RRV appears near Stk = 1.0, as shown in Figs. 13a and c. In addition, Figs. 13b and d also indicate that the precise preservation of the SGS coherent vortex structures in the WFDNS (ε = 0.01) ensures that the distribution of the skewness of the RRV can be accurately captured. For the same reason, the results of the WFDNS (ε = 1/83 ) have also improved significantly compared to the FDNS (࢞ = 8), especially for Stk < 3.0. Although the relative error of the skewness in the framework of the WFDNS (ε = 0.01 and 1/83 ) remains large for Stk > 3.0, especially for the case of r/η = 1.0 (see Figs. 13b and d), this result should be attributed to fact that the absolute values of the skewness nearly approach zero for Stk > 3.0 (see Figs. 13a and c). Intermittency is sensitive to the SGS coherent vortex structures and plays a vital role in the multiscale interactions with the inertial particles, especially in high-order statistics. The improved predictions of the skewness verify its ability to reconstruct such structures with wavelet filtering. The wavelet filtering is aimed at preserving the SGS coherent vortex structures, not just complimenting the SGS kinetic energy or dissipation.

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Fig. 9. Instantaneous vorticity snapshot of the filtered field, that is, the SGS flow field: (a) WFDNS (ε = 1/83 ) and (b) FDNS (࢞ = 8).

Fig. 10. (a) Difference between the second invariant of the rate of strain and the rate of the rotation tensors sampled by the particles, that is, τη2 S2  p − τη2 R2  p , as a function of Stk for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8). (b) Relative errors of τη2 ( S2  p − R2  p ) compared to the corresponding results of the DNS after filtering.

1/2

Fig. 11. Distribution of the PDF of the normalized particle RRV, that is, wr / w2r  , with r/η = 1.0 and 4.0 for the DNS (2563 ), WFDNS (ε = 1/83 ), and FDNS (࢞ = 8): (a) Stk = 0.1, (b) Stk = 1.0, and (c) Stk = 3.0. The Gaussian-style PDF is displayed as the dashed lines for comparison.

Please cite this article as: Y. Xiong, J. Li and F. Fei et al., Influence of coherent vortex structures in subgrid scale motions on particle statistics in homogeneous isotropic turbulence, International Journal of Multiphase Flow, https://doi.org/10.1016/j.ijmultiphaseflow.2018. 10.021

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Fig. 12. Distribution of the skewness of the RRV as a function of r/η for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8): (a) Stk = 0.1, (b) Stk = 1.0, and (c) Stk = 3.0.

Fig. 13. Distribution of the skewness of the RRV as a function of Stk for the DNS (2563 ), WFDNS (ε = 0.01 and 1/83 ), and FDNS (࢞ = 8): (a) and (c). Its relative errors compared to the DNS results: (b) and (d). (a) and (b) r/η = 1.0 and (c) and (d) r/η = 4.0.

5. Conclusion In this study, a wavelet-based coherent vortex extraction method was implemented to extract the SGS vortex structures and then investigate the effects of such structures on the particle statistics in the LES particle-laden flow. The WFDNS can enable good preservation of SGS structures due to the high compressibility and local fidelity of the wavelet filter. Therefore, it leads to an accu-

rate prediction of the particle-pair collision statistics. Better results could be attributed to the precise reflection in the particle seen fluid structural variables. The present work is a posterior analysis based on the wavelet filter. On this basis, this wavelet filter will be applied to the normal LES of particle laden flow, via the methods like CVS or AWCM as introduced in the Introduction, and used to improve existing SGS models. Actually, such works is still ongoing. Additional details are listed as follows:

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1. The effects of the SGS motions on the particle-pair collision statistics are much greater than on the one-point particle statistics. Different from the single-particle dynamics, which are mainly driven by the macrostructures in the flow, the particlepair statistics are dominated by small-scale structures. Compared to the classical FDNS with a spectral cutoff filter, particlepair statistics such as the RDF and the collision kernel can benefit significantly from the effective preservation of the SGS structures with the help of the wavelet-filtering process. 2. Particle seen structural variables, that is, the difference between the second invariant of the rate of strain and the rate of the rotation tensors and the skewness of RRV are sensitive to the SGS structures and strongly influence the particle preferential concentration effect and particle-pair relative dispersion, respectively. Clearly, the relative errors of such variables are reduced in the WFDNS framework, especially for small inertia particles. 3. The SGS structures play considerable roles in the particle-pair dynamics, and the preservation of such structures could recover the essential physical mechanisms regarding the collision process for small inertial particles. Moreover, a particle SGS model aimed at recovering the coherent vortex structures in SGS motions remains necessary.

Acknowledgments This work was supported by the National Natural Science Foundation of China (51876076, 51390494 and 51506063), the Foundation of the State Key Laboratory of Coal Combustion (FSKLCCB1702), and the Fundamental Research Funds for the Central Universities (2014QN183). The computing hours were supported by the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under Grant No. U1501501.

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Please cite this article as: Y. Xiong, J. Li and F. Fei et al., Influence of coherent vortex structures in subgrid scale motions on particle statistics in homogeneous isotropic turbulence, International Journal of Multiphase Flow, https://doi.org/10.1016/j.ijmultiphaseflow.2018. 10.021