Direct numerical simulation of turbulence modulation

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J. Fluid Mech. (2017), vol. 832, pp. 438–482. doi:10.1017/jfm.2017.672

c Cambridge University Press 2017

438

Direct numerical simulation of turbulence modulation by particles in compressible isotropic turbulence Qi Dai1 , Kun Luo1, †, Tai Jin1 and Jianren Fan1 1 State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, PR China

(Received 23 January 2017; revised 4 September 2017; accepted 18 September 2017)

In this paper, a systematic investigation of turbulence modulation by particles and its underlying physical mechanisms in decaying compressible isotropic turbulence is performed by using direct numerical simulations with the Eulerian–Lagrangian point-source approach. Particles interact with turbulence through two-way coupling and the initial turbulent Mach number is 1.2. Five simulations with different particle diameters (or initial Stokes numbers, St0 ) are conducted while fixing both their volume fraction and particle densities. The underlying physical mechanisms responsible for turbulence modulation are analysed through investigating the particle motion in the different cases and the transport equations of turbulent kinetic energy, vorticity and dilatation, especially the two-way coupling terms. Our results show that microparticles (St0 6 0.5) augment turbulent kinetic energy and the rotational motion of fluid, critical particles (St0 ≈ 1.0) enhance the rotational motion of fluid, and large particles (St0 > 5.0) attenuate turbulent kinetic energy and the rotational motion of fluid. The compressibility of the turbulence field is suppressed for all the cases, and the suppression is more significant if the Stokes number of particles is close to 1. The modifications of turbulent kinetic energy, the rotational motion and the compressibility are all related with the particle inertia and distributions, and the suppression of the compressibility is attributed to the preferential concentration and the inertia of particles. Key words: compressible turbulence, isotropic turbulence, multiphase and particle-laden flows

1. Introduction

Turbulent flows laden with particles or droplets are common phenomena in various natural and engineering applications, such as atmospheric dispersal of pollutants, pulverized coal in furnaces and liquid sprays in engines. These areas involve turbulence modulation in the carrier phase, as well as dispersion and polydispersity of the dispersed phase, which are very important and complicated issues in multiphase flows. When the turbulence is highly compressible, the rotational motion and the compressibility of fluid are both significant. Besides, eddy shocklets can also exist † Email address for correspondence: [email protected]


Turbulence modulation by particles in compressible isotropic turbulence

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in the carrier phase, which makes the problems of two-way coupled turbulent flows more complex. Researches on the mechanisms of the complex interactions between particles and compressible turbulence are very meaningful, which can promote the understandings of multiphase flows and ultimately improve the industrial devices involving these flows. Isotropic turbulence is the simplest and representative turbulence, which are extensively used to study turbulence modulation by particles. In the past decades, turbulence modulation by dispersed particles are mostly conducted in incompressible isotropic turbulence based on direct numerical simulations (DNSs) with the Eulerian– Lagrangian point-source approach, where the diameter of particles is smaller than the Kolmogorov length scale. Ferrante & Elghobashi (2003) studied the physical mechanisms of the modulation to decaying isotropic turbulence by dispersed particles, and found that microparticles (the initial Stokes number of particles St0 1) would augment turbulence and the rotational motion of fluid, while large particles (St0 > 1) would attenuate turbulence and the rotational motion of fluid. In addition, they also found that the kinetic energy spectrum was attenuated at low wavenumbers and augmented at high wavenumbers. Boivin, Simonin & Squires (1998) investigated turbulence modulation in stationary isotropic turbulence, where the attenuation of the turbulent kinetic energy and dissipation rate was observed, and the kinetic energy spectrum was attenuated at low wavenumbers and augmented at high wavenumbers. Abdelsamie & Lee (2012, 2013) clarified the distinction of turbulence modulation between decaying turbulence and stationary turbulence through investigating the turbulence modulation mechanisms and heavy particles statistics. They found turbulence modulation in decaying turbulence is qualitatively and quantitatively discrepant from that in stationary turbulence for microparticles, while they were qualitatively similar for large particles. Furthermore, they concluded that stationary turbulence was not appropriate for the research of turbulence modulation by microparticles owing to the artificial forcing therein. Different from incompressible turbulence, temperature, pressure and other thermodynamic quantities fluctuate sharply in some cases such as hypersonic aircrafts’ engines and interstellar clouds, where the compressibility of the flow field cannot be neglected. Compressible turbulence, especially for those with high turbulent Mach number (Mt ), is more complicated compared with the incompressible turbulence, because of the nonlinear interactions between the velocity field and the pressure field, as well as the existence of shocklets. Passot & Pouquet (1987) simulated a two-dimensional compressible homogeneous turbulence with high Mach number, and found shocklets existed in the field when Mt > 0.3. Samtaney, Pullin & Kosovi´c (2001) used a tenth-order compact finite difference scheme to study decaying compressible isotropic turbulence with direct numerical simulations, and analysed the influence of turbulent Mach number and Taylor Reynolds number on random shocklets in the flow field. Wang et al. (2011, 2012a,b, 2013) simulated a highly compressible forced isotropic turbulence with turbulent Mach number around 1.0, and investigated the scaling and statistical properties, the effect of shocklets on the velocity gradients and the effect of compressibility on the small-scale structures. Although research on incompressible two-phase turbulence and compressible one-phase turbulence have made great progress over the past decades, there are few studies on compressible two-phase turbulence, especially those with high turbulent Mach number. Xia et al. (2016) studied modifications of forced compressible homogenous turbulence by heavy point particles with different particle densities, where the turbulent Mach number is around 1.0, and analysed the influence of


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particle densities on the statistics of the turbulence. They found that turbulence was suppressed by particles and the suppression was more obvious when the density of particles increased, which is similar to that in forced incompressible turbulence. However, the underlying turbulence modulation mechanism was not clarified. Zhang et al. (2016) investigated the preferential concentration of heavy particles in stationary compressible isotropic turbulence with one-way coupling, and found particles tended to collect in high-density and low-vorticity regions. Critical particles with St = 1.0 were also found to accumulate in high-vorticity regions behind the shocklets. As the turbulent Mach number decreased, the degree of particle clustering in high-vorticity regions was weakened. So far, turbulence modulation by small inertia particles in compressible turbulence has not been completely understood, especially when the turbulent Mach number is high. In particular, the interaction between particles and decaying compressible isotropic turbulence has not been investigated, and the modifications of the compressibility in the turbulence filed remain unknown. Therefore, it is of great importance to perform the investigation of turbulence modulation by particles in decaying compressible isotropic turbulence. In the present work, we apply direct numerical simulations with the Eulerian– Lagrangian point-source method to investigate turbulence modulation by inertia particles in compressible isotropic turbulence. The diameter of particles is smaller than the Kolmogorov length scale and particles interact with turbulence through two-way coupling. As stationary turbulence is not appropriate for the study of turbulence modulation, especially for particles with St 6 1 (Lucci, Ferrante & Elghobashi 2010; Abdelsamie & Lee 2012), we employ decaying turbulence to avoid the influence of the artificial forcing. The initial turbulent Mach number is up to 1.2 and the Taylor Reynolds number is 51.7. Five different simulations with different particle diameters (or initial Stokes numbers) are conducted. The main objectives of the present work is to investigate the modifications of turbulent features, compressibility and energy spectra of the flow field by particles, as well as analyse the compressible turbulence modulation mechanism. 2. Numerical approaches

2.1. Governing equations of the fluid flow The carrier phase is supposed to be compressible Newtonian fluid, so the governing equations of the carrier phase employ the Eulerian forms of three-dimensional compressible Navier–Stokes equations, which include the conservation equations of continuity, momentum and energy, as well as the state equation. The governing equations are non-dimensionalized based on the following characteristic parameters: L0 for length, u0 for velocity, √ L0 /u0 for time, T0 for temperature, ρ0 for density, c0 for the speed of sound (c0 = γ RT0 for an ideal gas, where γ = Cp /Cv is the ratio of specific heat at constant pressure Cp to that at constant volume Cv and R = Cp − Cv is the specific gas constant), p0 = ρ0 c20 /γ for pressure, ρ0 u20 for energy per unit volume, µ0 for dynamic viscosity and κ0 for conductivity. Therefore, the dimensionless governing equations can be formulated as ∂ρ ∂(ρuj ) + = 0, ∂t ∂xj ∂(ρui ) ∂[ρui uj + pm δij ] 1 ∂σij + = + ϕui , ∂t ∂xj Re ∂xj

(2.1) (2.2)


Turbulence modulation by particles in compressible isotropic turbulence ∂T 1 ∂(σij ui ) ∂E ∂[(E + pm )uj ] 1 ∂ + = k + + ϕe , ∂t ∂xj α ∂xj ∂xj Re ∂xj p = ρT.

441 (2.3) (2.4)

Here, ui , ρ, p and T are dimensionless velocity, density, pressure and temperature respectively. The ideal gas model is used for the thermodynamic equation of state. The modified pressure pm , viscous stress σij and total energy per unit volume E are defined as p ∂ui ∂uj 2 pm = , σij = µ + − µθ δij , (2.5a,b) 2 γM ∂xj ∂xi 3 p 1 + ρuj uj , E= (2.6) (γ − 1)γ M 2 2 where θ = ∂uk /∂xk is the dimensionless velocity divergence or dilatation. The other dimensionless characteristic parameters in the governing equations are as followings: Prandtl number Pr = µ0 Cp /κ0 , Reynolds number Re = ρ0 u0 L0 /µ0 , Mach number M = u0 /c0 and α = Pr Re(γ − 1)M 2 . The dimensionless dynamic viscosity µ and dimensionless thermal conductivity κ are associated with temperature, and computed by the Sutherland law (Sutherland 1893) as µ=κ =

1.40417T 1.5 . T + 0.40417

(2.7)

The effects of particles on the fluid are considered through the source terms ϕui and ϕe which describe the momentum and energy contributions from particles. 2.2. Governing equations of particle motion It is assumed that particles are solid spheres with identical diameter dp , temperature tp and density ρp in a specific case. Gravitational settling, inter-particle collisions and heat transfer between turbulence and particles are negligible. The trajectories of particles are tracked in the Lagrangian manner and follow the Basset–Boussinesq– Oseeen equations in compressible flow (Parmar, Haselbacher & Balachandar 2012). Since the density of particles is much larger than that of the fluid at the particle positions (ρp /ρf ≈ 1000), only the Stokes drag force is significant. The buoyancy force, the pressure-gradient force, the Basset force, the virtual-mass force and other unsteady drag forces can be neglected since they are all much smaller than the Stokes drag force (Crowe, Sommerfeld & Tsuji 1998). Therefore, the governing equations for each particle motion can be given by dxp,i = vp,i , dt dvp,i Fp,i f1 = = (up,i − vp,i ). dt mp τp

(2.8) (2.9)

Here, xp,i is the particle displacement, vp,i is the particle velocity, up,i is the fluid velocity at the particle position, Fp,i is the Stokes drag force acting on the particle, τp is the particle relaxation time which is defined as τp = Reρp dp2 /(18µ) and mp is the mass of an individual particle which is computed by mp = πdp3 ρp /6. f1 is the empirical correction to the Stokes drag force which is expressed as f1 = 1 + 0.15Re0.687 p,slip ,

(2.10)



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where Rep,slip = Reρ|up,i − vp,i |dp /µ is the relative particle Reynolds number based on the slip velocity between the particle and the fluid. Since the relative particle Mach number Mp = |up,i − vp,i |/cp is small, where cp is the speed of gas sound at the particle position, the compressibility correction is neglected in our numerical simulations (Zhang et al. 2016). 2.3. Coupling terms between the fluid and the particles The momentum and energy couplings between the particles and the carrier fluid are shown as the momentum source term ϕui and the energy source term ϕe in the governing equations of the fluid flow. Based on the point source in cell (PSIC) assumption, the source terms are calculated from the Lagrangian particle variables by volume averaging the contributions from all the particles existing in the computational cell centred on each grid point, which are given by ϕui = −

1 X Fp,i , 1V n

(2.11)

p

1 X Fp,i vp,i , ϕe = − 1V n

(2.12)

p

where np is the number of particles residing within each computational cell and 1V is the volume of the computational cell. 2.4. Fundamental parameters definition In this section, some fundamental parameters of the compressible isotropic turbulence are defined as follows (Wang et al. 2012a; Xia et al. 2016). The root p mean square (r.m.s.) of the fluctuation velocity magnitude is defined as u = hu21 + u22 + u23 i, where hi indicates the Eulerian ensemble average over the number of grid points. The turbulent kinetic energy is Ek = hρui ui i/2, the internal energy is Ei = hpi/[(γ − 1)γ M 2 ], and the total energy is E = Ek + Ei . The vorticity magnitude is computed by q ω = (ω12 + ω22 + ω32 ), (2.13) where ω1 = ∂u3 /∂x2 − ∂u2 /∂x3 is vorticity in the x1 direction, ω2 = ∂u1 /∂x3 − ∂u3 /∂x1 is vorticity in the x2 direction and ω3 = ∂u2 /∂x1 − ∂u1 /∂x2 is vorticity in the x3 direction. Two fundamental parameters are adopted to characterize the flow filed, which are the Taylor Reynolds number and the turbulent Mach number, expressed as uλhρi Reλ = Re √ , 3hµi u Mt = M √ , T

(2.14) (2.15)

where the Taylor microscale is s λ=

u2 . h(∂u1 /∂x1 )2 + (∂u2 /∂x2 )2 + (∂u3 /∂x3 )2 i

(2.16)


Turbulence modulation by particles in compressible isotropic turbulence The energy spectrum of the velocity field is given by X 2 ˆ E(k) = |u(k)| /2,

443

(2.17)

k6|k| 5.0 (Case E, F). We study the influence of the particle diameter (or initial Stokes number St0 ) on turbulence through changing the particle number Np and the particle diameter dp . Since the motions of all kinds of particles can reach the equilibrium with the surrounding fluid after about two large eddy turnover times, which will be shown in § 4.4.1,



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F IGURE 2. (Colour online) Time development of the averaged Stokes number of particles.

the statistics of turbulence modulation are mainly investigated after t = 3τ0 to avoid the sudden perturbation effect due to the coupling (Abdelsamie & Lee 2012) and the influence of particle initial conditions. 4.1. Turbulent features of the flow field In this section, we will investigate the influence of particle diameters on the turbulent features of the flow field, including total energy, turbulent kinetic energy, vorticity magnitude, characteristic length scales and coherent structures. Figure 3 shows the temporal evolution of the total energy normalized by its initial value, E(t)/E(0), for all the six cases (A–F). The total energy of the particle-free flow (Case A) is constant, while the addition of particles provides a positive contribution to the turbulence field, causing the enhancement of the total energy. As St0 increases, the normalized total energy decreases at the time when t 6 1.5τ0 , but increases at the time when t > 3.5τ0 . Because the fluid velocity is decaying and the density of particles is far larger than that of the fluid, the particle velocity is also decaying but the decay rate is smaller than that of fluid, resulting in the continuous positive work of particles on the fluid and the energy transfer from particles to the turbulence field. As the Stokes number of particles decreases with temporal evolution, the vector alignment between particles and fluid increases (Abdelsamie & Lee 2013). Thus, the difference between particle and fluid velocity decreases and the increase rate of the total energy is reduced. Figure 4 shows the temporal evolution of the turbulent kinetic energy (TKE) normalized by its initial value Ek (t)/Ek (0) and the value of the corresponding particle-free case Ek (t)/EkA (0) for all the six cases (A–F). Figure 5 displays the time development of the decay rate of TKE dEk (t)/dt for the same cases. Since TKE of compressible turbulence decays faster than that of incompressible turbulence, the normalized TKE of the particle-laden flows are close to that of the particle-free flow in figure 4(a) although the turbulence modulation by particles is not weak. When the two-way coupling is turned on at t/τ0 = 1, the force introduced by the particles



447

1.04 Case A Case B Case C Case D Case E Case F

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F IGURE 3. (Colour online) Time development of the total energy normalized by its initial value.

suddenly accelerates the fluid (Abdelsamie & Lee 2012) and the dissipation rate is not modified at that moment, which will be shown in figure 22, resulting in the reduction of the magnitude of dEk (t)/dt at t/τ0 = 1, especially for microparticles (Case B). Microparticles with St0 = 0.1 (Case B) initially reduce the decay rate of TKE, which results in TKE larger than that of the particle-free flow (Case A) at all the times. Microparticles with St0 = 0.5 (Case C) initially (1.1τ0 < t 6 1.5τ0 ) slightly enhance the decay rate of TKE and then reduce it, leading to TKE first (τ0 < t 6 2.2τ0 ) smaller and then larger than that of Case A. However, critical and large particles with St0 > 1.0 (Cases D, E, F) initially enhance the decay rate of TKE, resulting in TKE smaller than that of Case A at all the times. As St0 increases, TKE decreases at the time when t > 3τ0 . The modifications are similar to that in decaying incompressible turbulence, where microparticles augment TKE and large particles attenuate TKE (Ferrante & Elghobashi 2003; Abdelsamie & Lee 2012). Nevertheless, they are different from that in forced incompressible and compressible turbulence, where TKE is attenuated by particles for all the cases with 0.1 6 St0 6 10 (Boivin et al. 1998; Xia et al. 2016). In addition, the normalized TKE and the decay rate of TKE have fluctuations in figures 4(b) and 5 compared with the incompressible flows (Abdelsamie & Lee 2012), and the amplitude of fluctuation decreases with the increase of particle diameters. Finally at the time t = 20, compared with TKE in Case A, TKE in Case B is larger by 9 % and TKE in Case E is smaller by 14.5 %, which are larger than 3 % and 12 % in the incompressible cases under the same conditions (He 2004). Figure 6 shows the temporal evolution of the r.m.s. of the vorticity magnitude ωrms for all the six cases (A–F). Microparticles and critical particles (Cases B, C, D) enhance the r.m.s. of the vorticity, while large particles (Cases E, F) attenuate the r.m.s. of the vorticity. As St0 increases, ωrms decreases. Compared with the incompressible turbulence, critical particles (Case D) have more influence on the vorticity in compressible turbulence with a difference 7.7 % at time t = 5τ0 , which is larger than 1.4 % at the same time in incompressible turbulence (Ferrante & Elghobashi 2003).



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F IGURE 4. (Colour online) Time development of the turbulent kinetic energy: (a) normalized by its initial value; (b) normalized by the value of the corresponding particle-free case.

Figure 7 displays the temporal evolution of Kolmogorov length scale η, Taylor microscale λ and integral length scale Lf for all the six cases (A–F). Microparticles (Cases B, C) reduce the three characteristic length scales. Critical particles (Case D) hardly modify the three characteristic length scales and make them nearly identical to those of the particle-free flow (Case A) at all the times. Large particles (Cases E, F) enhance the three characteristic length scales. Besides, all the three characteristic length scales increase as St0 increases. Figure 8 shows the distribution of vortical structures in the turbulence field at the time t = 4τ0 in the particle-free flow (Case A) and the particle-laden flow (only Cases B, D, F for clarity of the presentation). The vortical structures are identified using the Q criterion with Q = 0.15 (Hunt, Wray & Moin 1988) and coloured by the local vorticity magnitude ω. The Q criterion is the isosurfaces of the second invariant of


Turbulence modulation by particles in compressible isotropic turbulence 0


–0.002

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F IGURE 5. (Colour online) Time development of the turbulent kinetic energy decay rate dEk(t)/dt.

1.6 Case A Case B Case C Case D Case E Case F

1.4 1.2 1.0 0.8 0.6 0.4 0.2 1

2

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F IGURE 6. (Colour online) Time development of the r.m.s. of the vorticity magnitude.

the instantaneous velocity gradient tensor and given by 1 Q= 2

ω2 θ − sij sji + k 2 2

1 = 2

∂ui ∂xi

2

∂ui ∂uj − ∂xj ∂xi

! ,

(4.1)

where sij = (∂ui /∂xj + ∂uj /∂xi )/2 is the strain rate. The particle-free field (Case A) is filled with vermicular vortex tubes. Microparticles with St0 = 0.1 (Case B) increase the number of vortex tubes, because their trajectories are almost aligned



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F IGURE 7. (Colour online) Time development of characteristic length scales: (a) the Kolmogorov length scale; (b) the Taylor microscale; (c) the integral length scale.

with the surrounding fluid points’ trajectories and the vorticity of fluid is enhanced. Critical particles with St0 = 1.0 weakly influence the vortical structures, because particles mainly accumulate in the peripheries of the vortex cores and the change of the vorticity is relatively small. Large particles with St0 = 10.0 (Case F) break the


Turbulence modulation by particles in compressible isotropic turbulence (a)

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F IGURE 8. (Colour online) Turbulence vortical structures at time t = 4τ0 for different cases: (a) Case A, (b) Case B, (c) Case D and (d) Case F.

vortex tubes and destroy the coherence of the turbulence field, because they can cross the vortex cores and reduce the vorticity. 4.2. Compressibility of the turbulence field In this section, we are going to analyse the effect of particles on the compressibility of the turbulence field. In compressible turbulence, the dilatation is not zero and shocklets can exist, which differ from incompressible flows. The dilatation is associated with the fluctuations of thermodynamic quantities (Donzis & Jagannathan 2013), and thus usually employed to represent the compressibility of the flow field. Figure 9 shows the temporal evolution of the turbulent Mach number Mt for all the six cases (A–F). The decay of Mt shows that the flow field evolves from the highly compressible turbulence to the weakly compressible turbulence. Microparticles with St0 6 0.5 (Cases B, C) enhance the turbulent Mach number, while critical and large particles with St0 > 1.0 (Cases D, E, F) attenuate the turbulent Mach number. These modifications are similar to that of turbulent kinetic energy in figure 4. Figure 10 shows the temporal evolution of the r.m.s. of the dilatation θrms for all the six cases (A–F). The addition of particles reduces the r.m.s. of the dilatation compared



452 1.2


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F IGURE 9. (Colour online) Time development of the turbulent Mach number. 0.8 Case A Case B Case C Case D Case E Case F

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F IGURE 10. (Colour online) Time development of the r.m.s. of the dilatation.

with the particle-free flow (Case A), which means the compressibility of the turbulence field is attenuated. θrms first decreases and then increases with the increment of St0 , and θrms is the smallest in Case D (critical particles with St0 = 1.0) compared with the other cases. According to Samtaney et al. (2001), the shocklets of compressible isotropic turbulence are a set of disjoint surfaces, where the compression rate is high and the local density exhibits inflection along the normal direction. Each surface may be simply or multiply connected, which is defined as S = {Sl |∇ 2 ρ = 0, θ < −3θrms }.

(4.2)


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F IGURE 11. (Colour online) Probability density functions (PDF) of (a) velocity divergence; (b) normalized velocity divergence at time t = 4τ0 .

Therefore, the shocklets are identified by the isosurfaces of θ = −3θrms (Wang et al. 2012a). For the clarity of the presentation, the results only from Cases A, B, D and F are shown. Figure 11 provides the probability density functions (PDFs) of the velocity divergence θ and the normalized velocity divergence θ /θrms at time t = 4τ0 . The addition of particles attenuates the fluctuations of dilatation and the attenuation becomes more significant if St0 gets close to 1. In addition, there is a strong tendency for the PDFs of the velocity divergence to be skewed to the negative value: the right tail of the PDFs falls fast, while the left tail is very long (Wang et al. 2012a). Microparticles with St0 = 0.1 (Case B) augment the skewness which makes the left tail longer; critical particles with St0 = 1.0 (Case D) hardly change the skewness although the velocity divergence is reduced; large particles with St0 = 10.0 (Case F) attenuate the skewness and shorten the left tail.


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F IGURE 12. (Colour online) Shocklet structures at time t = 4τ0 for different cases: (a) Case A, (b) Case B, (c) Case D and (d) Case F.

Figure 12 shows the shocklet structures coloured by the local Mach number at the same time t = 4τ0 . Blob-like shocklet structures are sparsely distributed in the particle-free turbulence field (Case A), which is similar to the findings of Wang, Gotoh & Watanabe (2017a). Microparticles (Case B) hardly change the structures of isosurfaces. Critical particles (Case D) enlarge the area of isosurfaces and reduce the number of isosurfaces, illustrating that the compressibility of the turbulence field is attenuated. Large particles (Case F) make the isosurfaces more scattered, which means the destruction of the coherence. 4.3. Energy spectra of the turbulence field Turbulent kinetic energy is the integral of the three-dimensional density-weighted P 2 0.5 u) at all scales ˆ ˆ = ρ[ kinetic energy spectra Eρ (k) = k6|k| 5.0 (Cases E and F). As the particles’ diameter increases, the small eddy energy increases but the energy of intermediate eddies and large eddies decreases, which is different from some researches in incompressible turbulence (Ferrante & Elghobashi 2003; Abdelsamie & Lee 2012), where the energy of small eddies, intermediate eddies and large eddies all decreases with the increase of the particles’ diameter. Microparticles with St0 = 0.1 (Case B) increase Eρ (k) at wavenumbers k > 3, and reduce Eρ (k) only at wavenumbers k = 1 and 2; critical particles with St0 = 1.0 (Case D) increase Eρ (k) at wavenumbers k > 6, and reduce Eρ (k) at wavenumbers k 6 5; large particles with St0 = 10.0 (Case F) increase Eρ (k) at wavenumbers k > 22, and reduce Eρ (k) at wavenumbers k 6 21. At the increase of St0 , the crossing point of Eρ (k) between the two-way coupled simulation and the one-way coupled simulation (the domain where the energy spectrum is attenuated) shifts to the right (toward high wavenumbers), which nevertheless appears at the wavenumber lower than that of the forced compressible turbulence (Xia et al. 2016). In Cases B and C (St0 6 0.5), the reduction at low wavenumbers is less than the augment at high wavenumbers, resulting in the enhancement of TKE. However, in Cases D, E and F (St0 > 1.0), the reduction at low wavenumbers surpasses the augment at high wavenumbers, leading to the attenuation of TKE, as shown in figure 4. In order to further understand the modification of Eρ (k), the evolution equation for the density-weighted kinetic energy spectrum (Xia et al. 2016) is investigated, ∂Eρ (k) = T(k) + P(k) − D(k) + F(k), ∂t

(4.3)

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32 48 64

k

F IGURE 14. (Colour online) (a) Spectral nonlinear energy transfer rate due to the convection T(k); (b) spectral energy transfer rate due to the pressure gradient P(k); (c) the opposite of viscous dissipation rate spectrum D(k); (d) two-way interaction energy rate spectrum F(k) at time t = 4τ0 .

where T(k) is the spectral nonlinear energy transfer rate due to the convection, P(k) is the spectral energy transfer rate due to the pressure gradient and D(k) is the spectral viscous dissipation rate. F(k) is the spectral two-way interaction energy rate, which can be expressed as X d ϕui (4.4) F(k) = Re wˆ ∗i (k) 0.5 (k) , ρ k6|k| 2.35τ0 ), which is relatively small and slightly enhances the decay rate of TKE at early times. The modifications of Ψp (t) are similar to that in incompressible turbulence (Ferrante & Elghobashi 2003), which is perhaps due to the same drag force and governing equations for particle motion (2.8)–(2.10). Figure 24 shows the PDFs of the two-way interaction energy rate of a single particle, which is given by ϕp = −mp

f1 (up,i up,i − up,i vp,i ). τp

(4.11)

With the increment of the particle diameter, the two-way interaction energy rate of a single particle also increases. For microparticles (Case B), the PDFs are skewed to the



467

1.014 1.012 1.010 1.008 1.006 1.004 1.002 1.000 0.998 1

2

3

4

5

6

F IGURE 25. (Colour online) Time development of the fluid velocity autocorrelation hhup,i up,i ii, the correlation hhup,i vp,i ii between fluid velocity and particle velocity and the particle velocity autocorrelation hhvp,i vp,i ii, all normalized by hhup,i up,i ii, for Case B. 0.06 0.05 0.04 0.03 0.02 0.01 0 1

2

3

4

5

6

F IGURE 26. (Colour online) Time development of the fluid velocity autocorrelation hhup,i up,i ii, the correlation hhup,i vp,i ii between fluid velocity and particle velocity and the particle velocity autocorrelation hhvp,i vp,i ii for Case F.

positive value, resulting in the whole two-way interaction energy rate being positive. For large particles (Case F), the left tail of the PDFs is longer than the right tail, and Ψp (t) is hence negative. For critical particles (Case D), the PDFs shift in the negative direction at t = 1.5τ0 but become skewed to the positive side at t = 4τ0 , so Ψp (t) is negative at early times (t < 2.35τ0 ) and positive later (t > 2.35τ0 ). As given in (4.9), the sign of Ψp (t) is determined by the relative magnitude of the fluid velocity autocorrelation hhup,i up,i ii and the correlation hhup,i vp,i ii between fluid



468 (a)

0.06

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0.011

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0.008 4.0

4.5

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F IGURE 27. (Colour online) Time development of the fluid velocity autocorrelation hhup,i up,i ii, the correlation hhup,i vp,i ii between fluid velocity and particle velocity and the particle velocity autocorrelation hhvp,i vp,i ii for Case D during time: (a) τ0 6 t 6 2.5τ0 ; (b) 4τ0 6 t 6 5τ0 .

velocity and particle velocity. Figures 25–27 show the temporal evolution of the fluid velocity autocorrelation hhup,i up,i ii, the correlation hhup,i vp,i ii between fluid velocity and particle velocity and the particle velocity autocorrelation hhvp,i vp,i ii, which are the Lagrangian one-point one-time correlations (Libby 1996). hhup,i up,i ii is the Lagrangian ensemble average of the square of the fluid velocity at the particle position, hhup,i vp,i ii is the Lagrangian ensemble average of the scalar product of the fluid velocity and the particle velocity at the particle position and hhvp,i vp,i ii is the Lagrangian ensemble average of the square of the particle velocity. Due to the higher inertia of microparticles compared with the carrier fluid, the decay rate of particle velocities is smaller than that of the surrounding fluid velocities. Therefore, the particle velocity autocorrelation hhvp,i vp,i ii is larger than the surrounding fluid velocity autocorrelation hhup,i up,i ii as shown in figure 25. Because microparticles



469

mostly follow the fluid points’ trajectories, the correlation hhup,i vp,i ii between fluid velocity and particle velocity keeps larger than hhup,i up,i ii, resulting in the two-way interaction energy rate Ψp (t) being positive. Large particles, as microparticles, also keep their momentum longer than the surrounding fluid due to the high inertia, and the particle velocity autocorrelation hhvp,i vp,i ii is larger than the surrounding fluid velocity autocorrelation hhup,i up,i ii, as shown in figure 26. However, the correlation hhup,i vp,i ii between fluid velocity and particle velocity is smaller than the surrounding fluid velocity autocorrelation hhup,i up,i ii, as large particles move randomly and can cross the trajectories of the fluid points. Consequently, large particles are obstacles to fluid motion and the two-way interaction energy rate Ψp (t) becomes negative. At early times (t < 2.35τ0 ), critical particles move out of the initial vortical structures of surrounding fluid and cross the fluid points’ trajectories, as large particles do. Therefore, hhup,i vp,i ii is smaller than hhup,i up,i ii, as shown in figure 27(a), and the two-way interaction energy rate Ψp (t) is negative at the early times. At later times (t > 2.35τ0 ) after critical particles have escaped from the vortex cores, they do not have enough inertia to enter new vortex cores, but rather accumulate in the peripheries of the vortex cores. Because of the high inertia, particles keep their kinetic energy longer than the surrounding fluid and the particle velocity autocorrelation hhvp,i vp,i ii is larger than the surrounding fluid velocity autocorrelation hhup,i up,i ii. Once the particles accumulate in the peripheries of the vortex cores, they tend to follow the surrounding fluid points’ trajectories like microparticles. Thus, the correlation hhup,i vp,i ii between fluid velocity and particle velocity is larger than the surrounding fluid velocity autocorrelation hhup,i up,i ii in figure 27(b), and the two-way interaction energy rate Ψp (t) is positive at the later times. 4.4.3. Mechanisms for the modification of the rotational motion of fluid The rotational motion of fluid is associated with the vorticity, so we analyse the temporal evolution of the vorticity in different cases to reveal the underlying physical mechanisms for the modification of the rotational motion of fluid. In this subsection, particles are treated as a kind of hypothetical fluids, and the velocity derivatives and the vorticity of Lagrangian particles can be considered. The vorticity of particles is positive if particles rotate in the clockwise direction; the vorticity of particles is negative if particles rotate in the anticlockwise direction. The transport equation of vorticity in compressible turbulence can be obtained by taking the curl of the momentum equation (2.2) Dω ∇P × ∇ρ 1 1 2 = −ω∇ · V − + ν∇ ω + ∇ × ϕu , (4.12) Dt γ M2ρ 2 Re ρ where the bulk viscosity is negligible and the shear viscosity is supposed to be constant. Taking the component in x1 direction of the vorticity vector, we can obtain the transport equation for ω1 dω1 ∂u1 1 ∂p ∂ρ ∂p ∂ρ = ωj − − dt ∂xj γ M 2 ρ 2 ∂x3 ∂x2 ∂x2 ∂x3 1 ∂ 1 ∂σ3j ∂ 1 ∂σ2j ∂ ϕu3 ∂ ϕu2 − + − . (4.13) + Re ∂x2 ρ ∂xj ∂x3 ρ ∂xj ∂x2 ρ ∂x3 ρ



470

Particles

Clockwise vortex

Drag force on fluid

F IGURE 28. (Colour online) Schematic of the drag interaction between microparticles and a simplified two-dimensional clockwise vortex (ω1 > 0).

Since microparticles (Case B) respond fast to the fluid motion, most particles remain in the initial vortex cores of the surrounding fluid shown in figure 17(b), and rotate as the fluid points’ trajectories. Figure 28 shows a schematic of the interaction between microparticles and the surrounding fluid when particles remain in a simplified twodimensional clockwise vortex in the x2 x3 plane. Due to the high inertia of particles, microparticles keep their momentum longer than the surrounding fluid and the vorticity of particles is larger than that of the fluid. Therefore, microparticles exert a force with positive effect on the surrounding fluid, and augment the rotational motion of fluid compared with the particle-free flow (Case A). The enhancement of the fluid vorticity also causes the reduction of the characteristic length scales in figure 7. Critical particles (Case D) move out of the vortical structures with high vorticity soon after injection and mostly accumulate in the peripheries of the vortex cores with low vorticity, so particles seldom remain in the high-vorticity regions. The two-way coupling force ϕu ≈ 0 inside these vortex cores and the source term in the transport equation (4.12) ∇ × ϕu is very small. Therefore, the r.m.s. of vorticity is closest to that of Case A compared with the other cases in figure 6. On the other hand, due to the high compressibility of the turbulence field, some critical particles also tend to collect in the downstream regions of the shocklets with high vorticity. Consequently, these particles, as microparticles, exert a positive work on the vorticity of the surrounding fluid, and the rotational motion of fluid is slightly augmented. Since the vortical structures are hardly modified, the characteristic length scales are nearly identical to those of Case A in figure 7. Large particles (Case F) move randomly with nearly uniform velocity and cross the vortex cores of the surrounding fluid, which is different from the microparticles, and the influence on the vorticity is also different. Figure 29 shows a schematic of the interaction between large particles and the surrounding fluid when particles moving in the x3 direction encounter a simplified two-dimensional clockwise vortex in the x2 x3 plane. Large particles exert a reverse drag force mp f1 /τp (vp,3 − up,3 ) on the fluid in the x3 direction, which is proportional to the relative velocity (vp,3 − up,3 ). As the gradient of the drag force is negative in the x2 direction, the interaction produces a counterclockwise torque on the vortex, which results in the reduction of the vorticity. Therefore, the rotational motion of fluid is also attenuated.



Particles

Clockwise vortex

471

Drag force on fluid

F IGURE 29. (Colour online) Schematic of the drag interaction between large particles and a simplified two-dimensional clockwise vortex (ω1 > 0).

For further analysis, the source term in the transport equation for ω1 (4.13) is investigated, which is the curl of the drag force exerted on the fluid ∂ ϕu3 ∂ ϕu2 − ∂x2 ρ ∂x3 ρ X ∂vp,3 ∂vp,2 ∂up,3 ∂up,2 1 3πνdp f1 − − − . (4.14) ≈ 1V · Re n ∂xp,2 ∂xp,3 ∂xp,2 ∂xp,3 p

Since large particles move randomly in the field, their motion can be treated as a kind of hypothetical irrotational flow, which means the vorticity of particles is approximate to 0 and ∂vp,3 /∂xp,2 − ∂vp,2 /∂xp,3 ≈ 0. In addition, ω1 is positive in the clockwise vortex and ∂up,3 /∂xp,2 − ∂up,2 /∂xp,3 > 0. Therefore, we can infer that the source term ∂(ϕu3 /ρ)/∂x2 − ∂(ϕu2 /ρ)/∂x3 in (4.14) is negative in the clockwise vortices (or positive in the anticlockwise vortices), resulting in the enhancement of the decay rate of |ω1 |. Accordingly, large particles attenuate the vorticity and reduce the lifetime of vortices, which also causes the enhancement of the characteristic length scales in figure 7. 4.4.4. Mechanisms for the modification of the compressibility of the turbulence field The compressibility of the turbulence field is related to the dilatation, so the temporal evolution of the dilatation in different cases is investigated to reveal the underlying physical mechanisms for the modification of the compressibility of the turbulence field. In this subsection, particles are also treated as a kind of hypothetical fluids, and the velocity divergence of Lagrangian particles can be considered, which represents the particle compressibility (Maxey 1987; Goto & Vassilicos 2008). The difference between the particle dilatation and the fluid dilatation at the particle positions ∇ · v p − ∇ · up is positive if particles spread out; ∇ · v p − ∇ · up is negative if particles accumulate. Therefore, particles have the tendency to collect in the regions where ∇ · v p − ∇ · up is negative (see appendix B for the verification). The transport equation of the dilatation in compressible turbulence can be obtained by taking the divergence of the momentum equation (2.2) dθ ∂uj ∂ui 1 ∂ 1 ∂p 4ν ∂ 2 θ ∂ 1 =− − + + ϕui , (4.15) dt ∂xi ∂xj γ M 2 ∂xi ρ ∂xi 3Re ∂xi2 ∂xi ρ



472

Particles

Control volume

Drag force on fluid

F IGURE 30. (Colour online) Schematic of the drag interaction between particles and a simplified one-dimensional expansion control volume (θ > 0).

where the influence of density fluctuations on the viscous term is neglected to simplify the discussions (Wang et al. 2012b). In the one-dimension circumstances, the transport equation of the velocity divergence, taking x1 direction for example, is reduced to ∂ ∂u1 ∂u1 2 1 ∂ 1 ∂p 4ν ∂ 3 u1 ∂ ϕu1 ∂ 2 u1 + u1 2 = − − + + . 3 2 ∂t ∂x1 ∂x1 ∂x1 γ M ∂x1 ρ ∂x1 3Re ∂x1 ∂x1 ρ (4.16) Considering a one-dimensional control volume, fluid enters the volume from the left and exits from the right. Taking the situation of the positive velocity divergence as an example, the fluid expands in the control volume and the velocity up,1,out at the outlet is larger than the velocity up,1,in at the inlet. Figure 30 shows a schematic of the interaction between particles and the surrounding fluid when a group of particles encounter a simplified one-dimensional expansion control volume in the x1 direction. The effect of this interaction is to exert a reverse drag force mp f1 /τp (vp,1 − up,1 ) on the fluid in the x1 direction. In addition, because ∇ · v p − ∇ · up is negative in the regions where particles accumulate, the dilatation of particles is smaller than that of the surrounding fluid and vp,1,out − vp,1,in < up,1,out − up,1,in . Therefore, the drag force at the inlet is larger than the drag force at the outlet. This means particles, when interacting with an expansion control volume, exert a compression force on the fluid, which in turn prevents the expansion of fluid and hence reduces the fluid dilatation. For further analysis, the source term in the transport equation for θ (4.15) is investigated, which is the divergence of the drag force exerted on the fluid X 1 1 ∇· ϕu ≈ 3πνdp f1 (∇ · v p − ∇ · up ). (4.17) ρ 1V · Re n p

When τp > τη , large particles (Case F), taken as an example, move randomly in the turbulence field and can cross the compression and expansion regions, whose preferential concentration is weak but inertia is large. Therefore, the motion of large particles can be treated as a kind of hypothetical flow without sinks or sources and ∇ · v p ≈ 0. In the expansion regions, the velocity divergence of fluid is positive, ∇ · up > 0, so the source term ∇ · (ϕu /ρ) < 0, results in the enhancement of the decay rate of θ. In the compression regions, the velocity divergence of fluid is negative, ∇ · up < 0, so the source term ∇ · (ϕu /ρ) > 0, leading to the enhancement of the decay rate of |θ |. In conclusion, the inertia of large particles causes the suppression of the compressibility of the turbulence field, as shown in figure 10.



473

When τp 6 τη , particles can respond fast to the fluid motion and the acceleration of particle closely matches that of fluid, dup,i /dt ≈ dvp,i /dt (Maxey 1987; Picano et al. 2011). Therefore, the governing equation for particles (2.9) yields v p − up ≈ −

τp dup . f1 dt

(4.18)

Taking divergence on both sides of this equation (Zhang et al. 2016), one can obtain " # τp dθp ∂uj ∂ui , (4.19) ∇ · v p − ∇ · up ≈ − + f1 dt ∂xi ∂xj p so

τp ∇ · v p ≈ ∇ · up − f1

"

# ωk2 dθp + sij sij − , dt 2 p

(4.20)

where θp = ∇ · up is the local fluid dilatation at the position of particles and ‘()p ’ denotes the value of fluid parameters seen by particles. Since the transport equation of the fluid dilatation at the particle positions is ∂uj ∂ui 4ν ∂ 2 θ 1 ∂ 1 ∂p ∂ 1 dθp =− + − + ϕui , (4.21) dt ∂xi ∂xj p 3Re ∂xi2 γ M 2 ∂xi ρ ∂xi p ∂xi ρ p the dilatation of particles ∇ · v p also equals ( ) 1 ∂ 1 ∂p ∂ 1 τp 4ν ∂ 2 θ − + ϕui ∇ · v p ≈ ∇ · up − . (4.22) f1 3Re ∂xi2 γ M 2 ∂xi ρ ∂xi p ∂xi ρ p Therefore, the source term in the transport equation for θ can be obtained by (4.20) and (4.22) " # X mp 1 dθp ωk2 ∂ , (4.23) ϕui ≈ − + sij sij − ∂xi ρ ρ · 1V dt 2 p n p

or ∂ ∂xi

ϕui ρ

≈−

X np

mp ρ · 1V

(

4ν ∂ 2 θ 1 ∂ − 2 3Re ∂xi γ M 2 ∂xi

1 ∂p ρ ∂xi

∂ + ∂xi p

ϕui ρ

) . p

(4.24) In addition, particles tend to cluster in the regions where " # τp dθp ωk2 − + sij sij − < 0, f1 dt 2 p

(4.25)

or τp − f1

(

4ν ∂ 2 θ 1 ∂ − 2 3Re ∂xi γ M 2 ∂xi

1 ∂p ρ ∂xi

∂ + ∂xi p

1 ϕui ρ

) p

< 0.

(4.26)



474 (a)

40 Case B (fluid) Case B (particle) Case C (fluid) Case C (particle) Case D (fluid) Case D (particle)

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2 F IGURE 31. (Colour online) Conditional average of (a) h(sij sij − ωk2 /2)/θrms |θ/θrms i at fluid 2 points and particle positions; (b) h(∂/∂p/∂xi /ρ)/∂xi /γ M 2 /θrms |θ/θrms i at fluid points and particle positions at time t = 4τ0 for Cases B, C and D.

The local low vorticity where sij sij − ωk2 /2 > 0 and maximum pressure (or fluid density) where ∂(∂p/∂xi /ρ)/∂xi < 0 result in the negative value of ∇ · v p − ∇ · up and the concentration of particles, as shown in figure 31. The average 2 (sij sij − ωk2 /2)/θrms conditioned on the fluid velocity divergence at the particle positions is positive and larger than that at the fluid points. Besides, the value of 2 h∂(∂p/∂xi /ρ)/∂xi /γ M 2 /θrms |θ/θrms i at particle positions is negative and smaller than that at fluid points. Thus, particles tend to collect in the regions with high density and low vorticity, as displayed in figures 16 and 17. Such a tendency becomes more apparent when the particle relaxation time τp increases and gets close to the Kolmogorov time scale τη .


Turbulence modulation by particles in compressible isotropic turbulence Multiplying (4.15) by 2θ, one can obtain the transport equation of θ 2 dθ 2 ∂uj ∂ui 1 ∂ 1 ∂p 4ν ∂ 2 θ ∂ 1 = −2θ − 2θ + 2θ + 2θ ϕui , dt ∂xi ∂xj γ M 2 ∂xi ρ ∂xi 3Re ∂xi2 ∂xi ρ

475

(4.27)

and the source term is ( ) X mp ϕui 4ν ∂ 2 θ 1 ∂ 1 ∂p ∂ ϕui ∂ ≈ −2θ − + . 2θ 2 2 ∂xi ρ ρ · 1V 3Re ∂xi γ M ∂xi ρ ∂xi p ∂xi ρ p n p

(4.28) The Eulerian ensemble average of 2θ · ∂(ϕui /ρ)/∂xi is ( ∂ ϕui 1 X 2θp mp 4ν ∂ 2 θ 1 ∂ 1 ∂p 2θ ≈ − − ∂xi ρ Ng N ρ · 1V 3Re ∂xi2 γ M 2 ∂xi ρ ∂xi p p ) ∂ ϕui + , (4.29) ∂xi ρ p so ** ( ϕui 1 ∂ 1 ∂p ∂ 4ν ∂ 2 θ ≈ −φm − 2θp 2θ ∂xi ρ 3Re ∂xi2 γ M 2 ∂xi ρ ∂xi p )++ ϕui ∂ + . ∂xi ρ p Because

∂ ϕui 1 X 1 ∂ 2θ ≈− 2θp Fp,i , ∂xi ρ Vf N ∂xp,i ρf

(4.30)

(4.31)

p

and ** 2θp

∂ ∂xi

ϕui ρ

++ p

" X # X 1 ∂ Fp,i 2 =− 2θp · βijk , 1V · Np N ∂xp,i ρf p

(4.32)

p

where βijk is the distribution coefficient of drag force on the surrounding grid point 2 and the sum of βijk over the eight grid points of the particle surrounding satisfies P 2 0 < p βijk 6 1, ** 2θp

∂ ∂xi

ϕui ρ

++ p

Ng ≈β Np

∂ 2θ ∂xi

ϕui ρ

,

(4.33)

where 0 < β 6 1. Therefore, h2θ · ∂(ϕui /ρ)/∂xi i meets the relationship ** ++ Ng ∂ ϕui 4ν ∂ 2 θ 1 ∂ 1 ∂p 1 + β · φm 2θ ≈ −φm 2θp − . Np ∂xi ρ 3Re ∂xi2 γ M 2 ∂xi ρ ∂xi p (4.34)



476 0.03

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F IGURE 32. (Colour online) Time development of the Lagrangian ensemble average of 2θp [4v/3Re · ∂ 2 θ/∂xi2 − ∂(∂p/∂xi /ρ)/∂xi /γ M 2 ]p at the particle positions.

Figure 32 shows the time evolution of the Lagrangian ensemble average of 2θp [4ν/3Re · ∂ 2 θ/∂xi2 − ∂(∂p/∂xi /ρ)/∂xi /γ M 2 ]p at particle positions. The initial time after particles are injected is not considered because particles do not follow the fluid points’ trajectories and dup,i /dt 6= dvp,i /dt at the initial injection time, when the inertia of particles plays a major role in suppressing the compressibility of the turbulence field. As a whole, 2θp [4ν/3Re · ∂ 2 θ /∂xi2 − ∂(∂p/∂xi /ρ)/∂xi /γ M 2 ]p is positive in the regions where particles accumulate. Consequently, the source term h2θ · ∂(ϕui /ρ)/∂xi i is negative, leading to the enhancement of the decay rate of θ 2 and the reduction of θrms compared with Case A, as shown in figure 10. Generally speaking, the preferential concentration of particles also play a role in the suppression of the compressibility. Microparticles (Case B) can respond quickly to the fluid motion and follow the fluid points’ trajectories, so the velocity divergence of particles is close to that of the surrounding fluid, ∇ · v p ≈ ∇ · up . Consequently, the source term ∇ · (ϕu /ρ) ≈ 0, resulting in the r.m.s. of the fluid velocity divergence nearly equal to that of the particle-free flow (Case A) in figure 10. As the diameter of particles dp increases, the particle relaxation time τp increases and the preferential accumulation becomes stronger. Thus, the magnitude of hh2θp [4ν/3Re · ∂ 2 θ /∂xi2 − ∂(∂p/∂xi /ρ)/∂xi /γ M 2 ]p ii becomes larger and the ensemble-averaged source term h2θ · ∂(ϕui /ρ)/∂xi i decreases, resulting in the reduction of the r.m.s. of the dilatation. Critical particles (Case D) with τp = τη have the strongest preferential concentration, and the ensemble-averaged source term h2θ · ∂(ϕui /ρ)/∂xi i is hence smallest compared with the other cases. Accordingly, the reduction of the r.m.s. of the dilatation is most obvious in figure 10. 5. Summary and concluding remarks

We have presented a systematic investigation of turbulence modulation by particles, whose diameters are smaller than the Kolmogorov length scale, and the underlying physical mechanisms in decaying compressible isotropic turbulence with the initial turbulent Mach number up to 1.2. Particles interact with turbulence through two-way



477

coupling, and direct numerical simulations with the Eulerian–Lagrangian point source approach are adopted. Five simulations with different particle diameters (or initial Stokes numbers, St0 ), while fixing both their volume fraction and particle densities, are conducted to investigate the influence of particle diameters on turbulence modulation. The statistics of turbulence modulation are mainly investigated after t = 3τ0 to avoid the influence of particle initial conditions. The addition of particles results in the increase of the total energy. When St0 6 0.5, microparticles enhance turbulent kinetic energy and the r.m.s. of the vorticity, but reduce the characteristic length scales. In addition, the number of vortex tubes is also increased. When St0 = 1.0, critical particles enhance the decay rate of TKE at early times after injection, resulting in slight attenuation of TKE, and the r.m.s. of the vorticity is also augmented. However, the characteristic length scales and the coherent structures are hardly modified. When St0 > 5.0, large particles reduce TKE and the r.m.s. of the vorticity, but enhance the characteristic length scales. In addition, the vortex tubes are broken and the coherence of the turbulence field is destroyed. The compressibility of the turbulence field is suppressed for all the cases, and the suppression is more significant when the Stokes number of particles is close to 1. The addition of particles also reduces the fluctuations of dilatation compared with the particle-free flow. Microparticles hardly change the shocklet structures; critical particles decrease the number of the shocklet structures; large particles make the shocklet structures more scattered. The density-weighted kinetic energy spectra exhibits attenuation at low wavenumbers and augmentation at high wavenumbers for the particle-laden flow compared with the particle-free flow. For the medium wavenumbers, the energy spectra is augmented by microparticles and critical particles, and attenuated by large particles. As the Stokes number of particles increase, the small eddy energy increases but the energy of intermediate eddies and large eddies decreases, and the location of the cross-over of the kinetic energy spectra between the two-way coupled simulation and the one-way coupled simulation shifts toward high wavenumbers. Furthermore, we analyse the distributions of particles and the transport equations of TKE, vorticity and dilatation, especially the two-way coupling terms, to reveal the underlying mechanisms responsible for turbulence modulation. The fluctuations of normalized TKE and the decay rate of TKE result from the pressure-dilatation transfer rate. The modification of the viscous dissipation rate is similar to that of TKE, which is associated with the change of the rotational motion and the compressibility. The modulations to turbulence, including TKE, the rotational motion and the compressibility, are all related to the particle inertia and distributions. Particles have the tendency to collect in the regions with high density and low vorticity, and such a tendency is most significant when St0 = 1.0. On the other hand, critical particles would like to accumulate in the downstream regions of shocklets with both high fluid density and high vorticity especially in the initial period when the turbulent Mach number is high. Therefore, the rational motion of fluid is slightly enhanced by critical particles compared with the particle-free flow. The suppression of the compressibility of the turbulence field is attributed to the preferential concentration and the inertia of particles, and particles are obstacles to the expansion or compression of fluid. When the degree of the preferential accumulation of particles strengthens, the suppression of the compressibility becomes more significant. Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant numbers 51576176 and 51390491) and the Special Program for Applied Research on



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F IGURE 33. (Colour online) Time development of the turbulent kinetic energy normalized by the value of the corresponding particle-free (one-way) case in compressible turbulence with (a) Reλ,0 = 120 and (b) Reλ,0 = 51.7. (a) 0.005

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F IGURE 34. (Colour online) Time development of the viscous dissipation rate ε(t) in compressible turbulence with (a) Reλ,0 = 120 and (b) Reλ,0 = 51.7.

Super Computation of the NSFC-Guangdong Joint Fund (the second phase). We are grateful to Dr Zhenhua Xia and Dr Zhu He for many useful discussions. Appendix A

To justify the current used Reynolds number, we conduct another four simulations with the initial Taylor Reynolds number up to 120 through increasing the characteristic length. A resolution of 2563 uniform grids is employed and the resolution parameter kmax η = 2.95, where kmax = 256/2. The initial turbulent Mach number is also 1.2 and the initial Stokes numbers (St0 ) of particles are 0.1, 1.0 and 10.0, while the initial energy spectrum, particle volume fraction and particle densities remain unchanged. The modifications of turbulence kinetic energy and its dissipation rate, vorticity and compressibility in compressible turbulence with Reλ,0 = 120 is qualitatively similar to those in compressible turbulence with Reλ,0 = 51.7, as shown in figures 33–36. Besides, compared with the high Reynolds number turbulence, the inertial subranges are small, but the dissipation ranges are obvious in low Reynolds number turbulence (Wang et al. 2017b). As the diameters of particles are smaller than the Kolmogorov


Turbulence modulation by particles in compressible isotropic turbulence (a) 2.0

479

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F IGURE 35. (Colour online) Time development of the r.m.s. of the vorticity in compressible turbulence with (a) Reλ,0 = 120 and (b) Reλ,0 = 51.7. (b) 0.8

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5

6

F IGURE 36. (Colour online) Time development of the r.m.s. of the dilatation in compressible turbulence with (a) Reλ,0 = 120 and (b) Reλ,0 = 51.7.

length scale, particles directly act on small eddies and the interaction between particles and turbulence is mainly occurred in the dissipation range of turbulent motion. Despite the low Reynolds number, it allows one to investigate the interaction between turbulence and inertial particles in compressible isotropic turbulence with relatively low computational cost (Zhang et al. 2016). Therefore, we believe the current used Reynolds number is reasonable for our research of turbulence modulation by point-source particles in compressible isotropic turbulence, considering the current computational capability. Appendix B

In this appendix, we present the verification of the relationship between the particle dilatation ∇ · v p and the concentration of particles. When τp 6 τη , according to the definition of material derivative (4.20) can also be expressed as τp ∇ · v p ≈ ∇ · up − f1

"

# ∂θp ∂θp ωk2 + uj + sij sij − . ∂t ∂xj 2 p

(B 1)



480 4 2 0 –2

Case B (fluid) Case B (particle) Case C (fluid) Case C (particle) Case D (fluid) Case D (particle)

–4 –6

–8

–6

–4

–2

0

2

4

F IGURE 37. (Colour online) Conditional average of the fluid dilatation at the particle positions hh∇ · up /θrms |θ/θrms ii (lines) and the particle dilatation hh∇ · v p /θrms |θ/θrms ii (lines + symbols) at time t = 4τ0 for Cases B, C and D. t/τ0 Case B Case C Case D

1

2

3

4

−0.15336 −0.16564 −0.1558

−0.06039 −0.05949 −0.06366

−0.02498 −0.02185 −0.02209

−0.01136 −0.00906 −0.00885

5 −0.00862 −0.00664 −0.00572

TABLE 3. The decay rate of the r.m.s. of the fluid dilatation dθrms /dt for Cases B, C and D.

From the temporal evolution of the r.m.s. of the fluid dilatation θrms in figure 10, one can obtain the decay rate of θrms , which is shown in table 3. As turbulence is homogenous and the magnitude of the fluid dilatation decay rate is small compared with the fluid dilatation, ∂θp /∂t can be approximated by dθrms /dt in the expansion regions and −dθrms /dt in the compression regions. Consequently, the approximation of ∇ · v p can be obtained through (B 1) and shown in figure 37. In the compression regions where particles accumulate, ∇ · up < 0 and the particle dilatation is negative. However, in the expansion regions where particles collect, ∇ · up > 0 and the particle dilatation is positive, which is against the model proposed by Maxey (1987) in incompressible turbulence where the negative value of the particle dilatation ∇ · v p is closely related to the concentration of particles. When the inertia of particles τp is zero, ∇ · v p is equal to ∇ · up and particles amount to the tracers, whose preferential concentration is very weak. Tracers just tend to accumulate in high-density zones owing to the compressibility of turbulence, rather than in the regions where ∇ · up is negative (Zhang et al. 2016). Furthermore, particles tend to cluster in the regions with low vorticity and high fluid density where sij sij − ωk2 /2 > 0 and ∂(∂p/∂xi /ρ)/∂xi < 0, as shown in figure 31. In figure 37, the particle dilatation ∇ · v p is smaller than the fluid dilatation at the particle positions ∇ · up in both compression and expansion regions. Therefore, particles tend to cluster



481

in the regions where τp − f1 or

τp − f1

(

"

# ωk2 dθp + sij sij − < 0, dt 2 p

4ν ∂ 2 θ 1 ∂ − 2 3Re ∂xi γ M 2 ∂xi

1 ∂p ρ ∂xi

∂ + ∂xi p

(B 2)

1 ϕui ρ

)

< 0.

(B 3)

p

The concentration of particles is associated with the negative value of the difference between the particle dilatation and the fluid dilatation at the particle positions ∇ · v p − ∇ · up in compressible turbulence. ∇ · v p − ∇ · up is positive if particles spread out; ∇ · v p − ∇ · up is negative if particles accumulate. REFERENCES

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Direct numerical simulation of turbulence modulation

Direct numerical simulation of turbulence modulation

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