SPARSE—A subgrid particle averaged Reynolds stress equivalent model: testing with a priori closure

Abstract

A Lagrangian particle cloud model is proposed that accounts for the effects of Reynolds-averaged particle and turbulent stresses and the averaged carrier-phase velocity of the subparticle cloud scale on the averaged motion and velocity of the cloud. The SPARSE (subgrid particle averaged Reynolds stress equivalent) model is based on a combination of a truncated Taylor expansion of a drag correction function and Reynolds averaging. It reduces the required number of computational parcels to trace a cloud of particles in Eulerian–Lagrangian methods for the simulation of particle-laden flow. Closure is performed in an a priori manner using a reference simulation where all particles in the cloud are traced individually with a point-particle model. Comparison of a first-order model and SPARSE with the reference simulation in one dimension shows that both the stress and the averaging of the carrier-phase velocity on the cloud subscale affect the averaged motion of the particle. A three-dimensional isotropic turbulence computation shows that only one computational parcel is sufficient to accurately trace a cloud of tens of thousands of particles.

1. Introduction

The Eulerian–Lagrangian (EL) model, introduced by Crowe et al. [1,2], is one of the major approaches used for computing the interaction of a large number of particles with a turbulent flow. In the EL model, each particle is traced in its Lagrangian frame, i.e. the frame moving with the particle. Because particles are treated as volumeless mathematical points in the EL ‘point-particle’ approach, the tracing of many particles becomes much more computationally efficient. With a point-particle assumption, the simulation of a large number of particles in process-scale environments becomes feasible.

If the number of particles in a computation is prohibitively large, groups of physical particles are amalgamated into a single computational particle to further economize the computational cost. This type of method is also known as cloud-in-cell (CIC) [3]. In CIC methods, groups of particles are modelled as points and their motion is forced by the drag exerted on them by the fluid [4–7]. The CIC method as conventionally implemented does not account for subparticle cloud dynamics resulting from turbulent fluctuations or particle–particle interactions.

For the modelling of subgrid and/or subparticle cloud scales of turbulent carrier-phase flows, large-eddy simulation (LES) is commonly employed. In LES, filtered Navier–Stokes equations resolve large turbulent structures, while dissipation of energy from the subgrid scales (SGSs) is modelled. Commonly used SGS models include the classic Smagorinsky [8] and dynamic Smagorinsky [9] models.

Similar to the modelling of SGSs with eddy viscosity models in the carrier phase, SGS models are required for an accurate particle tracer [10–14]. To this end, models have been formulated with either a deterministic approach or stochastic modelling of subgrid scales. In the deterministic approach, the instantaneous velocities are reconstructed for use in particle equations through defiltering [15–19]. Shotorban and co-workers [15,17,19] employed approximate deconvolution (AD) [20] for defiltering. Deconvolution is a mathematical method to approximately reconstruct the instantaneous velocity through consecutively applying a filtering operation on the filtered velocities. The consecutive application of the filtering operator is a result of a series expansion for the deconvolution. Filtering, itself, is a convolution product of the instantaneous velocity and the filter kernel. Kuerten & Verman [16] employed a defiltering technique in which the filtering inversion is carried out in the Fourier space for the streamwise and spanwise directions while the inversion is approximated by a Taylor series for the cross-stream direction. Although defiltering is efficient to implement, it can only be carried out for the represented modes.

A Langevin-type stochastic differential equation can be used to compute the evolution of particles when the carrier phase is simulated by LES [15,21,22]. A similar model was previously proposed for particles by Pozorski & Minier [23,24] and Minier & Peirano [25] when the carrier phase is simulated by Reynolds-averaged Navier–Stokes (RANS) equations. Shotorban & Balachandar [22] extended the application of the Langevin equation from a simple RANS framework [23–25] to a higher resolution LES model by using the stochastic differential equation employed to solve LES equations through a filtered density function approach [26]. A modified version of this model was proposed by Berrouk et al. [27] to account for crossing-trajectory effects.

While modelling the effect of subgrid stresses on the dispersion of particles has been studied, the effect of subscale particle fluctuations under the computational particle assumptions of CIC has received much less attention. Several articles [28–30] have studied the inclusion of small-scale particle–fluid energy transfer in their turbulence models. However, these studies focus on the effect that the particles have on the fluid rather than the influence of individual modelled particles on the averaged computational particle dynamics.

In recent work, we have undertaken a multi-scaling modelling effort [31] in which macro-scale models such as the LES and CIC models are closed using results from full resolution mesoscale simulations that solve for the turbulence and flow over particles with moving boundaries. The general framework enables direct closure of averaging terms in a wide parameter space that otherwise would take a great range of challenging and meticulous experimentation to obtain empirical formulae. Within this framework, we have more freedom to develop macro-models, for example for particle cloud dynamics.

This paper presents a model that accounts for subscale interphase velocity perturbations on computational particle dispersion patterns in a CIC framework. The model is general and is a good fit for a multi-scale framework. The drag forcing term in the Lagrangian governing equation for the particle momentum employs a correction factor for high Reynolds and Mach numbers. The particle drag correction factor is expanded using a Taylor expansion, which enables a simple Reynolds averaging of the governing equations to yield a second-order perturbation term. These higher order terms are deemed ‘Reynolds stress equivalent’ terms, which capture the effects of mesoscale interphase velocity perturbations. We therefore refer to the model as the subgrid particle averaged Reynolds stress equivalent, which leads to the appropriate acronym SPARSE for this model which requires a small number of computational particles to simulate a larger number of real particles. The objective of this paper is to derive the model and demonstrate that SPARSE improves upon the tracing of the averaged location of a cloud of particles in CIC. One-dimensional verification tests as well as a three-dimensional homogeneous turbulence flow case demonstrate the efficacy of the model.

The derivation of the SPARSE model is provided in the next section. In §3, we present a priori testing of the SPARSE model on a one-dimensional analytical carrier-phase field followed by testing in a three-dimensional periodic box with decaying isotropic turbulence in §4. The conclusion and recommendations are reserved for the final section.

2. Derivation of the model

The Lagrangian equations that govern the particle motion under the point-particle assumption in general form are

$$\frac{\mathrm{d}{\mathbf{\text{x}}}_{\mathbf{\text{p}}}}{\mathrm{d}t}={\mathbf{\text{v}}}_{\mathbf{\text{p}}}$$ 2.1

and

$$\frac{\mathrm{d}{\mathbf{\text{v}}}_{\mathbf{\text{p}}}}{\mathrm{d}t}=\frac{C_{D_{\mathrm{s}}}}{{\tau }_{\mathrm{p}}}(\mathbf{\text{u}}-{\mathbf{\text{v}}}_{\mathbf{\text{p}}}),$$ 2.2

where v_p is the particle velocity vector, u is the carrier-phase velocity vector and τ_p is the particle time constant [2]. The drag correction factor, C_Ds, is necessarily an empirical function that modifies the Stokes drag for a number of nonlinear physical effects, such as Reynolds number [2], Mach number [32] or particle number density [33]. As many of the physical corrections such as Reynolds number and Mach number correction depend on the relative interphase velocity, a=u−v_p, we rewrite (2.2) as

$$\begin{array}{rl}\frac{\mathrm{d}{\mathbf{\text{x}}}_{\mathbf{\text{p}}}}{\mathrm{d}t}& ={\mathbf{\text{v}}}_{\mathbf{\text{p}}}\\ \text{and}\frac{\mathrm{d}{\mathbf{\text{v}}}_{\mathbf{\text{p}}}}{\mathrm{d}t}& =f(\mathbf{\text{a}})\cdot \mathbf{\text{a}}\end{array}\}$$ 2.3

to derive the SPARSE model. We have taken the correction function, f(a)=C_Ds/τ_p, that depends on the relative interphase velocity only. The derivation of the SPARSE model, however, is easily extended for a correction function that is dependent on more variables (see also Remarks below).

CIC models a group of particles with a single computational parcel. The properties of the group of particles are ensemble averaged as

$$\overline{\eta }=\frac{1}{N}\sum _{i=1}^N{\eta }_i.$$ 2.4

To derive a governing equation for the averaged single-parcel motion a Reynolds decomposition, $\eta =\overline{\eta }+{\eta }^{\mathrm{\prime }}$, is performed to split particle properties into an averaged, $\overline{\eta }$, and a fluctuating, η^′, component. Reynolds averaging (2.3) leads to

$$\begin{array}{rl}\frac{\mathrm{d}\overline{({\overline{\mathbf{\text{x}}}}_{\mathrm{p}}+{\mathbf{\text{x}}}_{\mathrm{p}}^{\mathrm{\prime }})}}{\mathrm{d}t}& =\overline{{\overline{\mathbf{\text{v}}}}_{\mathrm{p}}+{\mathbf{\text{v}}}_{\mathrm{p}}^{\mathrm{\prime }}}\\ \text{and}\frac{\mathrm{d}\overline{({\overline{\mathbf{\text{v}}}}_{\mathrm{p}}+{\mathbf{\text{v}}}_{\mathrm{p}}^{\mathrm{\prime }})}}{\mathrm{d}t}& =\overline{f(\overline{\mathbf{\text{a}}}+{\mathbf{\text{a}}}^{\mathrm{\prime }})}\cdot (\overline{\mathbf{\text{a}}}+{\mathbf{\text{a}}}^{\mathrm{\prime }}).\end{array}\}$$ 2.5

Using

$$\overline{\frac{\mathrm{d}\eta }{\mathrm{d}t}}=\frac{\mathrm{d}\overline{\eta }}{\mathrm{d}t},$$ 2.6

for linear derivative operators, the averaged computational particle velocity and position can be written as

$$\begin{array}{rl}\frac{\mathrm{d}{\overline{\mathbf{\text{x}}}}_{\mathrm{p}}}{\mathrm{d}t}& ={\overline{\mathbf{\text{v}}}}_{\mathrm{p}}\\ \text{and}\frac{\mathrm{d}{\overline{\mathbf{\text{v}}}}_{\mathrm{p}}}{\mathrm{d}t}& =\overline{f(\overline{\mathbf{\text{a}}}+{\mathbf{\text{a}}}^{\mathrm{\prime }})\cdot (\overline{\mathbf{\text{a}}}+{\mathbf{\text{a}}}^{\mathrm{\prime }})}.\end{array}\}$$ 2.7

Here, $\overline{\mathbf{\text{a}}}=\overline{\mathbf{\text{u}}}-{\overline{\mathbf{\text{v}}}}_{\mathrm{p}}$, where $\overline{\mathbf{\text{u}}}$ is the average of the carrier-phase velocity at all particle locations, and a^′=u^′−v′_p, where u^′ are the carrier-phase velocity fluctuations at the particle positions and v′_p are the fluctuations in the particle phase.

In the tradtional CIC approach [3], the average of the correction factor term is simply set as the correction factor for the average interphase velocity,

$$\overline{f(\overline{\mathbf{\text{a}}}+{\mathbf{\text{a}}}^{\mathrm{\prime }}})=f(\overline{\mathbf{\text{a}}}),$$ 2.8

leading to

$$\begin{array}{rl}\frac{\mathrm{d}{\overline{\mathbf{\text{x}}}}_{\mathrm{p}}}{\mathrm{d}t}& ={\overline{\mathbf{\text{v}}}}_{\mathrm{p}}\\ \text{and}\frac{\mathrm{d}{\overline{\mathbf{\text{v}}}}_{\mathrm{p}}}{\mathrm{d}t}& =f(\overline{\mathbf{\text{a}}})\cdot (\overline{\mathbf{\text{a}}}),\end{array}\}$$ 2.9

where the averaged carrier-phase velocity is taken as the carrier-phase velocity at the average particle position, $\overline{\mathbf{\text{u}}}=\mathbf{\text{u}}({\overline{\mathbf{\text{x}}}}_{\mathrm{p}})$.

Instead of the assumption in (2.8), we propose to Taylor expand $f(\overline{\mathbf{\text{a}}}+{\mathbf{\text{a}}}^{\mathrm{\prime }})$ around $\overline{\mathbf{\text{a}}}$. With the three-dimensional vector components, $\overline{\mathbf{\text{a}}}=(a_1,a_2,a_3)=({\overline{a}}_1+a_1^{\mathrm{\prime }},{\overline{a}}_2+a_2^{\mathrm{\prime }},{\overline{a}}_3+a_3^{\mathrm{\prime }})$; this leads to

$$\begin{array}{rl}f(\mathbf{\text{a}})& =f(\overline{\mathbf{\text{a}}})+\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}{a}_1}(({\overline{a}}_1+a_1^{\mathrm{\prime }})-{\overline{a}}_1)+\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}{a}_2}(({\overline{a}}_2+a_2^{\mathrm{\prime }})-{\overline{a}}_2)\dots \\ & +\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}{a}_3}(({\overline{a}}_3+a_3^{\mathrm{\prime }})-{\overline{a}}_3)+O({a^{\mathrm{\prime }}}^2)\\ & =f(\overline{\mathbf{\text{a}}})+\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}{a}_1}{a}_1^{\mathrm{\prime }}+\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}{a}_2}{a}_2^{\mathrm{\prime }}+\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}{a}_3}{a}_3^{\mathrm{\prime }}+O({a^{\mathrm{\prime }}}^2).\end{array}$$ 2.10

The Taylor series is truncated at second-order terms O(a^′2) for interphase velocity fluctuations that are small when compared with the averaged interphase velocity.

Substituting (2.10) in the particle momentum equation (2.3) yields

$$\frac{\mathrm{d}{\mathbf{\text{v}}}_p}{\mathrm{d}t}=(\hspace{0.17em}f(\overline{\mathbf{\text{a}}})+\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}\mathbf{\text{a}}}{\mathbf{\text{a}}}^{\mathrm{\prime }})(\overline{\mathbf{\text{a}}}+{\mathbf{\text{a}}}^{\mathrm{\prime }}).$$ 2.11

After expanding and averaging, we obtain

$$\frac{\mathrm{d}{\overline{\mathbf{\text{v}}}}_{\mathrm{p}}}{\mathrm{d}t}=f(\overline{\mathbf{\text{a}}})\overline{\mathbf{\text{a}}}+\frac{\mathrm{d}f(\overline{\mathbf{\text{a}}})}{\mathrm{d}\mathbf{\text{a}}}\overline{{\mathbf{\text{a}}}^{\mathrm{\prime }}{\mathbf{\text{a}}}^{\mathrm{\prime }}}.$$ 2.12

In (2.12) the terms $\overline{{\mathbf{\text{a}}}^{\mathrm{\prime }}{\mathbf{\text{a}}}^{\mathrm{\prime }}}$ are recognized as stresses that typically arise in Reynolds averaging and that require closure. In this study, these terms will be closed using an a priori approach where the results of Eulerian–Lagrangian simulations are averaged to find the stresses in the SPARSE model.

Remarks —

• (i) The average carrier-phase velocity in the cloud is not equal to the carrier-phase velocity at the averaged particle location, $\overline{\mathbf{\text{u}}({\mathbf{\text{x}}}_{\mathrm{p}})}\ne \mathbf{\text{u}}({\overline{\mathbf{\text{x}}}}_{\mathrm{p}})$. A model is required to close the averaged carrier-phase velocity of the particles. In this paper, we use an a priori closure.

• (ii) Equation (2.9) results if, in the SPARSE derivation, the first-order term in the Taylor expansion in (2.10) is truncated. The CIC is hence a first-order model, whereas SPARSE is a second-order model. Depending on the truncation of the Taylor series any order model may be derived. In the remainder of this paper, we refer to the combination of the CIC model and the carrier-phase velocity at an averaged particle location taken (erroneously) to be equal to $\mathbf{\text{u}}({\overline{\mathbf{\text{x}}}}_{\mathrm{p}})$ as a first-order model. We shall refer to the combination of a second-order model and $\overline{u(\mathbf{\text{x}})}$ as the carrier-phase velocity as the SPARSE model.

• (iii) Taylor expansion of the correction function leads to terms in the particle momentum equation that have integer powers only. Reynolds averaging on terms with integer powers is straightforward, whereas Reynolds averaging on terms with real and fractional powers (typical for the empirical functions such as C_Ds) is very challenging.

• (iv) It is assumed that the interphase velocity perturbations are small. This is not always the case. For example, when the particle response time, τ_p, is large (heavy particles) and the initial particle velocities are large and have a large variation, then the fluctuations of the particles are large compared with the computational particle velocity. In this case, the validity of CIC and/or SPARSE will only hold for short times.

• (v) If f(a) depends on variables other then the interphase velocity, for example particle diameter or number density, then a Taylor expansion has to be performed around those variables also. Averaging leads to additional second-order correlations that require closure.

• (vi) For a posteriori tests, the interphase stress terms require modelling and extensive knowledge of f(a). In [31], we present a multi-scale framework in which we use full-scale simulations to obtain detailed information for f(a).

3. One-dimensional verification tests

The SPARSE model improves upon the first-order model in two ways. Firstly, SPARSE accounts for the subscale particle fluctuations through stress terms. Secondly, SPARSE averages the carrier-phase velocity at the particle locations correctly. In this section, we verify the impact of these two model improvements with one-dimensional test cases.

In the tests, the carrier field is analytically specified to prevent errors in the carrier-phase solver from polluting the model error investigation. Ten thousand particles initialized in a cloud are individually traced according to (2.3). We refer to this solution as the ‘exact’ solution. SPARSE and the first-order model trace the averaged location and velocity of this cloud with a single computational parcel. The stress terms and carrier-phase velocity at the particle location are a priori obtained from the ‘exact’ solution.

(a). Effect of the Reynolds stress equivalent terms

To isolate the effect of the stress closure from the effect of the carrier-phase velocity closure, we first consider a constant carrier-phase velocity,

$$u(x,t)=10,$$ 3.1

in which case $\overline{u(x_{\mathrm{p}})}=u({\overline{x}}_{\mathrm{p}})$, i.e. the carrier-phase velocity modelled by the first-order model and the SPARSE model are the same in this case. The difference between the two models can hence only be caused by the stress terms in SPARSE.

For the exact reference case, 10 000 particles are initialized at the following locations and velocities:

$$\begin{array}{rl}{x}_{\mathrm{p}}& ={\sigma }_x\cdot d_{\mathrm{cloud}}\\ \text{and}{v}_{\mathrm{p}}& =5.0+{\sigma }_v\cdot \gamma ,\end{array}\}$$ 3.2

where the cloud diameter, d_cloud=1, and σ_x,σ_y are uniformly distributed random numbers between −1 and 1. The maximum amplitude of the initial velocity perturbations is γ=10. The modelled parcel’s location and velocity is initialized with the mean of the exact locations and velocities, respectively. Time integration is performed using a first-order explicit time-stepping routine with a time step of Δt=10⁻⁵, which ensures that numerical time integration errors are smaller than the model errors.

The particle correction factor, f(a), is taken to be a linear function of a,

$$f(a)=\frac{a}{10},$$ 3.3

similiar to low Reynolds number correction factors [34]. As derivatives of order higher than 1 are zero, the second-order expansion in (2.10) and hence the SPARSE model is exact. With exact a priori closure the SPARSE model is the same as the exact model, as confirmed in figure 1. Figure 2a,b shows the errors in the location and velocity in time determined with

$$\begin{array}{rl}{ϵ}_{x_{\mathrm{p}}}& =\frac{|{\overline{x}}_{\mathrm{p},\mathrm{exact}}-x_{\mathrm{p},\mathrm{model}}|}{{\overline{x}}_{\mathrm{p},\mathrm{exact}}}\\ \text{and}{ϵ}_{v_{\mathrm{p}}}& =\frac{|{\overline{v}}_{\mathrm{p},\mathrm{exact}}-v_{\mathrm{p},\mathrm{model}}|}{{\overline{v}}_{\mathrm{p},\mathrm{exact}}}.\end{array}\}$$ 3.4

Figures 1 and 2 highlight the importance of inclusion of the second-order stress term in SPARSE. The truncation of the expansion at O(a′) in the first-order model leads to errors in the averaged particle velocity of the order of tens of percentage (a maximum of 17%) at time t=1, whereas the SPARSE model is exact. The first-order model does not account for the subscale kinetic energy in the particle phase, causing the modelled particle to lag behind the average particle position of the exact cloud.

Figure 1.

Particle (a) velocities and (b) locations using the mean of the exact particle locations as well as the first-order and SPARSE models with a uniform background fluid velocity and a linear correction factor. (Online version in colour.)

Figure 2.

Modelling error of the mean particle (a) velocity and (b) location using the first-order and SPARSE models with a uniform background fluid velocity and a linear correction factor. (Online version in colour.)

For the initial conditions in (3.2), the interphase velocity difference can be negative and hence the correction factor in (3.3) can also have negative values. For a negative correction factor particles are non-physically propelled.

Using the following correction factor of

$$f(a)=\frac{|a|}{10},$$ 3.5

based on the absolute velocity, |a| is more physical because f(a) is always positive. Note that f(a) in (3.5) is nonlinear because of discontinuous derivatives at a=0.

The nonlinearity leads to errors in the modelling of the average particle velocity with the SPARSE model (figures 3b and 4). Because the SPARSE model accounts for the perturbations in the particle phase, it is still significantly more accurate than the first-order model.

Figure 3.

Particle (a) velocities and (b) locations using the mean of the exact particle locations as well as the first-order and SPARSE models with a uniform background fluid velocity and an absolute value correction factor. (Online version in colour.)

Figure 4.

Modelling error of the mean particle (a) velocity and (b) location using the first-order and SPARSE models with a uniform background fluid velocity and an absolute value correction factor. (Online version in colour.)

(b). Effect of modelling the average cloud velocity

To test the erroneous assumption that $\overline{u}(x_{\mathrm{p}})$ = $\overline{u}({\overline{x}}_{\mathrm{p}})$ in the first-order model, a constant correction factor comparable to the Stokes drag [35] is used,

$$f(a)=\frac{24}{\mathrm{St}}.$$ 3.6

The combination of the constant correction factor, for which the Taylor expansion terms are zero, and a spatially varying carrier-phase velocity,

$$u(x)=\cos (\pi x)+x,$$ 3.7

enables an investigation into the errors caused by the incorrect sampling of the carrier-phase velocity at a single point in the first-order model.

Figures 5a and 6a show that first-order modelling of the fluid velocity can lead to errors as high as 17% whereas the averaged velocity and location of the cloud is a priori closed in an exact manner with the SPARSE model. Because the perturbations in the fluid velocity are periodic, the errors in the modelled particle velocity are oscillatory and only cause a maximum 1% error in the particle location in figures 5b and 6b.

Figure 5.

Particle (a) velocities and (b) locations using the mean of the exact particle locations as well as the first-order and SPARSE models with a spatially varying background fluid velocity. (Online version in colour.)

Figure 6.

Modelling error of the mean particle (a) velocity and (b) location using the first-order and SPARSE models with a spatially varying background fluid velocity. (Online version in colour.)

4. Validation of SPARSE: decaying isotropic turbulence

To more rigorously test SPARSE a group of particles traced in a decaying isotropic turbulence according to the ‘exact’ model is compared with the first-order model and a priori closed SPARSE.

The isotropic turbulence simulation is performed in a cube with periodic boundary conditions on all sides. Following [36–38] an initial correlated flow field is determined based on specified energy spectra. Computations are performed with a compressible Navier–Stokes solver based on a fourth-order central difference method for the fluxes (as described [4]). To verify the Navier–Stokes solver, we compare computations with this code with 128 cubed number of grid points with the case referred to as ‘iga96’ in [36,38] computed with an N=96 Fourier spectral method. The average initial fluctuating Mach number,

$$M_0=\frac{\sqrt{\overline{u_i^{\prime }{u}_i^{\prime }}}}{c({\overline{T}}_0)},$$ 4.1

where u_i′ is the fluctuating fluid velocity, is set to M₀=0.05. The reference fluid Reynolds number is set to Re_f=2357. The decay of turbulent kinetic energy, $\mathrm{TKE}=\frac{1}{2}{\overline{u}}_{i^{\prime }}^2$, in figure 7 compares well with the results from Blaisdell et al. [36,38]. The oscillatory trend in TKE is well known and documented and is caused by pressure dilation [39].

Figure 7.

Comparison of the turbulent kinetic energy versus time in isotropic turbulence using a high-order EL code from Jacobs & Don [4] and a Fourier spectral method performed by Blaisdell et al. [36,38]. (Online version in colour.)

To test the particle models an exact case that initializes 30³=27 000 particles uniformly over 3×3×3 grid cells serves as a reference against which the first-order model and SPARSE are tested. When compared with iga96 the initial velocity field is scaled by a factor of 5. This ensures that the carrier phase disperses the particle cloud more significantly than the iga96 velocity field and hence the effects of the cloud modelling are more visible. The initial particle velocity is set according to a uniform random number around zero with an absolute maximum of 2.5. We use the following drag correction function:

$$C_{D_{\mathrm{s}}}=(0.38+\frac{24}{R{e}_{\mathrm{p}}}+\frac{4}{R{e}_{\mathrm{p}}(1/2)})(1+{\mathrm{e}}^{-0.43/M_{\mathrm{p}}^{4.67}}),$$ 4.2

which corrects the Stokes drag for high relative particle Reynolds number, Re_p=|v_f−v_p|d_p/ν, and Mach number, $M_{\mathrm{p}}=|v_{\mathrm{f}}-v_{\mathrm{p}}|/\sqrt{T_{\mathrm{f}}}$, according to Boiko et al. [4,6,7,32,40]. The particles’ response time is set to τ_p=0.01. Matlab was used to compute the partial derivatives of the drag coefficient equation, which are needed for the SPARSE model in (2.12).

The time lapse in figure 8 shows that the cloud modelled with SPARSE (red sphere) closely follows the average location of the exact cloud (green sphere), while the first-order model (red sphere) deviates significantly from the exact cloud. The temporal dispersion of the exact cloud is visualized by the large diameter of the exact model (green) sphere when compared with the initial diameter of the (red and blue) cloud approaches. Dispersion, as measured by for example the rate of change of the cloud radius, is not modelled in the SPARSE and first-order model and it is the subject of ongoing investigation.

Figure 8.

Helicity projected onto the cubical computational domain faces and particle cloud locations for the three-dimensional isotropic decaying turbulence case at times t=(a) 0.0, (b) 1.6, (c) 3.2, (d) 4.8, (e) 6.4 and (f) 8.0. The first-order modelled particle is visualized by a blue sphere, the SPARSE is modelled by a red (sphere) and the exact model by a green sphere. The root mean square dispersion of the the exact location is visualized by the diameter of the green sphere. The diameters of the red and blue spheres visualize the initial RMS dispersion. (a) t=0, (b) t=2, (c) t=4, (d) t=6, (e) t=8 and (f) t=10. (Online version in colour.)

A comparison of the average particle cloud distance from the origin,

$$|{\mathbf{\text{x}}}_{\mathbf{\text{p}}}|=\sqrt{x_{\mathrm{p}}^2+y_{\mathrm{p}}^2+z_{\mathrm{p}}^2},$$

and averaged velocity,

$$|{\mathbf{\text{v}}}_{\mathbf{\text{p}}}|=\sqrt{u_{\mathrm{p}}^2+v_{\mathrm{p}}^2+w_{\mathrm{p}}^2},$$ 4.3

in figure 9 confirms the improved modelling by SPARSE when compared with the first-order model. While both the exact cloud and the computational clouds are initially entrained in the same turbulent eddy, the average velocity over the cloud used in SPARSE is near zero, while the local velocity used in the first-order model is large. The latter cloud is hence displaced more. With increasing time, the exact cloud disperses and the fluid velocity is sampled over a larger area. Because the turbulence is isotropic and decaying the averaged fluid velocity goes to zero over a larger area and increasing time, respectively. The local velocity decays in time also, but deviates from the near-zero, averaged velocity, leading to the deviation of the first-order model in time.

Figure 9.

The magnitude of the average (a) velocity and (b) particle distance from the origin is shown when computed using the first-order model, SPARSE model and average over the physical particles. (Online version in colour.)

Errors in the magnitude of the distance from the origin and velocity plotted in figure 10 show that SPARSE modelling errors are non-zero, because the drag correction in (4.2) is nonlinear and hence the Taylor truncation error in (2.10) used for SPARSE is non-zero. The error however is small, within 0.5% of the exact model.

Figure 10.

The error in computing the magnitude of the average (a) velocity and (b) particle distance from the origin is shown when computed using the first-order model, SPARSE model and average over the physical particles. (Online version in colour.)

The impacts of two modelling components of the SPARSE model, including the averaging of the fluid velocity of the cloud and the stress modelling, are compared in figure 11. Plotted are the two terms on the r.h.s. of (2.12). The first term that is affected by the cloud velocity averaging is significantly larger than the second term, which is proportional to the subcloud stresses. The second term is of the order of 10⁻⁵. In general, the effect of the subcloud stresses is not necessarily smaller than the effect of velocity averaging, as was seen in the one-dimensional tests.

Figure 11.

Comparison of the two terms on the r.h.s. of (2.12), $f(\overline{\mathbf{\text{a}}})\overline{\mathbf{\text{a}}}$ and $(\mathrm{d}f(\overline{\mathbf{\text{a}}})/\mathrm{d}\mathbf{\text{a}})\overline{{\mathbf{\text{a}}}^{\mathrm{\prime }}{\mathbf{\text{a}}}^{\mathrm{\prime }}}$. The first term (blue line) depends on the SPARSE fluid velocity modelling and the second term (red line) is proportional to the interphase stresses. (Online version in colour.)

5. Conclusion and future work

SPARSE provides a model that traces a group of point particles through a single computational point parcel. The model improves upon CIC methods by, firstly, accounting for fluid and particle stress (i.e. the Reynolds stress equivalent) terms and, secondly, averaging of the fluid velocity at the Lagrangian computational particle. One-dimensional and three-dimensional a priori tests show that both improvements yield an excellent comparison of the averaged trace of a cloud of particles and the computational parcel.

This paper is only a first step in the development of a closed SPARSE model. Current investigation focuses on closure based on a multi-scale approach as reported in [31].

Data accessibility

One-dimensional Matlab codes are all figured in Matlab format (including data) and are available at http://attila.sdsu.edu/~jacobs/PUBDATA/PROCA/17/.

Authors' contributions

S.L.D. programmed codes, ran codes and processed data. G.B.J. and H.S.U. co-supervised S.L.D. and reviewed the article. O.S. aided in the testing of the model and reviewed the article. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

We gratefully acknowledge the financial support by the Air Force Office of Scientific Research under grant no. FA9550-16-1-0008, a collaborative grant that supported all authors. S.L.D. and O.S. are PhD students. G.B.J. and H.S.U. are supervising professors.