Filtered gas-solid momentum transfer models

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FILTERED GAS-SOLID MOMENTUM TRANSFER MODELS

Juray De Wilde *

*Université catholique de Louvain, Materials and Process Engineering Department (IMAP), Réamur, Place Sainte Barbe 2, B-1348 Louvain-la-Neuve, Belgium,

e-mail: dewilde@imap.ucl.ac.be

Key words: Gas-solid flow, Filtered models, Momentum transfer, Generalized added mass Abstract. To account for meso-scale phenomena in coarse grid simulations, correlation

terms appearing in the filtered gas-solid flow equations have to be modeled. Filtered models for the gas-solid momentum transfer are focused on and possible approaches are derived and evaluated.

1 INTRODUCTION

Real fluidized beds contain structures of different length and time scales or different spatial and temporal frequencies ω. The smallest structures that can be calculated using a continuum approach are meso-scale structures, like clusters [1]. Due to the finite spatial and temporal grid dimensions used for solving the continuum gas-solid flow models, a filter frequency ωf is

introduced. Phenomena with a frequency ω lower than the filter frequency ωf (ω < ωf) are

explicitly calculated. Phenomena with a frequency ω higher than the filter frequency ωf (ω > ωf) are filtered out and not explicitly calculated. Such unresolved sub-grid-scale phenomena

may, however, affect the lower frequency behaviour explicitly calculated and should, therefore, be accounted for by using so-called filtered models and including sub-grid models [1,2]. The development of sub-grid models is particularly important and complex for gas-solid flows by the intrinsic lack of scale separation in such flows [1,3].

One crucial effect of meso-scale structuring is in the way gas and solids interact, in particular with respect to momentum transfer [4]. Therefore, the description of momentum transfer in filtered gas-solid flow models is focused on.

2 NON-FILTERED MODEL

Because the number of solid particles in practical gas-solid flows is very large, the Eulerian-Eulerian approach is usually taken [1,2,5,6]. The basic model equations express the conservation of mass and momentum for each phase [5]. Using the Eulerian-Eulerian approach, expressions for the solid phase physico-chemical properties have to be derived. If the concentration of solid particles is sufficiently large, direct interaction of particles through collisions occurs easily and rapidly and the kinetic theory of granular flow (KTGF) can be used [7]. The constitutive equations used in the present work and an extensive discussion on the transport equations is found in Agrawal et al. [1].

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contribution to the gas-solid momentum transfer is the distribution over the phases of the gas phase stresses, i.e. the gas phase pressure gradient and/or the gas phase shear stress, according to their volume fraction. Due to the large density difference between gas and particles, other inter-phase forces, as for example the added mass and history forces, are usually neglected.

3 FILTERED MODEL

When performing coarse grid calculations, the gas-solid flow model equations have to be filtered on a scale, i.e. with a filter frequency ωf, typical for the computational grid that is

used. Following Zhang and VanderHeyden [2], Reynolds-averaging is applied to the solids volume fraction and the gas phase pressure, whereas Favre-averaging is applied to the other variables. The filtered model is shown in Table 1.

Filtered mass conservation solid phase:

(

)

+ ⋅

(

v

)

=0 r t s ∂ φ ρs ∂ ρ φ ∂ ∂ ₍₁₎

Filtered mass conservation gas phase:

(

1−

)

+ ⋅

(

1− u

)

=0 r t g _∂ φ ρg ∂ ρ φ ∂ ∂ ₍₂₎

Filtered momentum conservation solid phase:

(

)

(

)

(

)

(

u v

)

r P g s r P r v v r v t A s s s s s − + ∂ ∂ − + ⋅ − − = ⋅ + β φ ρ φ φ ∂ ∂ ∂ ∂ ρ φ ∂ ∂ ρ φ ∂ ∂ ~ (3)

Filtered momentum conservation gas phase:

(

)

(

)

(

u v

)

r P g s r r P u u r u t A g g g g − − ∂ ∂ + − + ⋅ − − = − ⋅ + − β φ ρ φ ∂ ∂ ∂ ∂ ρ φ ∂ ∂ ρ φ ∂ ∂ 1 ~ 1 1 ₍₄₎

Table 1. Filtered conservation equations.

By using Favre-averaging, no correlation terms appear in the mass conservation equations. This is not the case in the momentum equations. Currently, no reliable closure models are available for the convection related correlation terms. These terms are, however, of minor importance for the present work. Commonly, they are incorporated in the viscous shear terms. In what follows, the closure models for the filtered gas-solid interaction terms are focused on.

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(

u−v

)

=β_e

(

u − v

)

β (5)

The filter frequency is not explicitly accounted for in the effective interphase momentum transfer coefficient formulations proposed. Andrews et al. [10] investigated both the time-averaged βe approach (Eq. (5)) and a stochastic correction. Most formulations predict a

reduction of the interphase momentum transfer coefficient by a factor 1.5 to 4, depending on the filtered solids volume fraction, in agreement with the reduction calculated from dynamic mesoscale simulations on a periodic 2x8 cm2_{domain [1]. With respect to the latter} simulations, it should, however, be remarked that the ratio of the domain size used for averaging or filtering and the mesh size is only 32, i.e. only a limited amount of effects is filtered out. Hence, it is possible that for lower filter frequencies ωf, e.g. in case of

steady-state simulations, as more effects are filtered out, the value of the effective interphase momentum transfer coefficient is further reduced. In the present work, the effective interphase momentum transfer coefficient formulation of Heynderickx et al. [9] and the interphase momentum transfer coefficient of Wen and Yu [11] reduced with a constant factor, are used to investigate the effect of a reduced interphase momentum transfer coefficient approach.

The Reynolds-average of the solids volume fraction of the gas phase pressure gradient can be easily decomposed, introducing the correlation between the solids volume fraction and the gas phase pressure gradient φ'∂ 'P ∂r :

r P r P r P ∂ ∂ + ∂ ∂ = ∂ ∂ ' ' φ φ φ (6)

Zhang and VanderHeyden [2] were the first to study the φ'∂ 'P ∂r term and found it to be

surprisingly important. A generalized added mass closure model is proposed:

        ∂ ∂ ⋅ − ∂ ∂ − ∂ ∂ ⋅ + ∂ ∂ = ∂ ∂ r u u t u r v v t v C r P m a ρ φ' ' (7)

which scales with the volume fraction based mixture density ρm and the generalized added

mass coefficient Ca for which a formulation as a function of the filter frequency ωf is yet to be

derived. For what follows, the 1D inviscid form of the filtered momentum equations obtained with a generalized added mass approach is given here:

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(

)

(

)

(

)

g Dt u D C Dt v D C P Dt u D g m a m a g ρ φ ρ ρ φ ρ φ − + − + ∇ − − = − 1 1 1 ₍₉₎

According to Zhang and VanderHeyden [2] and De Wilde [12], Ca can be much larger than

one, resulting in a generalized added mass that is surprisingly large compared to the well-known added mass.

4 MIXTURE SPEED OF SOUND TEST FOR THE FILTERED MOMENTUM TRANSFER MODELS

Filtered models should allow to calculate the low frequency behavior without explicitly calculating the high frequency behavior. Gas-solid flows exhibit an interesting behavior with respect to frequency dependence. The propagation speed of gas phase pressure waves in gas-solid mixtures, the so-called mixture speed of sound, strongly depends on the gas-solids volume fraction and, more importantly for the present investigation, on the sound frequency [13,14,15]. In practical gas-solid flows, a broad range of pressure wave (or sound) frequencies and solid volume fractions occur. The remarkable difference between the single gas phase speed of sound (± 330 m/s) and the minimum mixture speed of sound (± 10 m/s) on the one hand, and the frequency and solid volume fraction dependence of the mixture speed of sound on the other hand, are at the origin of physical and calculational complex behavior of gas-solid flows [6].

The complex mixture speed of sound behavior is a result of gas-solid momentum transfer. One of the early successes of the non-filtered Eulerian-Eulerian gas-solid flow model was its capability of describing the complex mixture speed of sound behavior over the entire solids volume fraction and frequency ranges [14,15]. The drag was found to play a crucial role in the calculated mixture speed of sound behavior. The frequency dependence of the mixture speed of sound then suggests an interesting necessary test for filtered momentum transfer models. Such models should still be able to describe the mixture speed of sound for frequencies lower than the filter frequency, but not necessarily for frequencies higher than the filter frequency. It is the purpose of this paragraph to investigate the mixture speed of sound behavior of some filtered momentum transfer models presented in literature and to demonstrate the information that can be obtained from a mixture speed of sound test.

4.1 Non-filtered model

Figure 1 shows the mixture speed of sound cm as a function of the solids volume fraction φ

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solids volume fraction fre qu en cy [H z] m ixtu re s p ee d of s ou nd [m/

s] mixture speed_{of sound [m/s]}

Figure 1: Mixture speed of sound as a function of the sound frequency and the solid volume fraction calculated with the non-filtered model. Conditions: ρs = 2650 kg m-3, ρg = 0.934 kg m-3, dp = 310 µm, <P> = 104800 Pa.

solids volume fraction

fre qu en cy [H z] m ix ture s pee d of sou nd [ m

/s] mixture speedof sound [m/s]

fre qu en cy [H z] m ix ture s pee d of sou nd [ m

/s] mixture speedof sound [m/s]

(a) βe (b) βe = β/10

fre qu en cy [H z] m ix tur e s p ee d of sound [ m /s ] mixture speed of sound [m/s]

fre qu en cy [H z] m ix tur e s p ee d of s ou nd [m /s ] mixture speed of sound [m/s] (c) βe = β/100 (d) βe = β/1000

Figure 2: Mixture speed of sound as a function of the sound frequency and the solid volume fraction calculated with the filtered model (Eqs. (1)-(5)) with: (a) effective interphase momentum transfer coefficient βe

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frequencies below 0.1 Hz. At such low frequencies, the mixture speed of sound gradually decreases from the single gas phase speed of sound for solid volume fractions lower than 10-5 to a minimum mixture speed of sound for solid volume fractions higher than about 0.1.

4.2 Effective interphase momentum transfer coefficient approach for the filtered drag

Figure 2 shows the effect of a decreasing effective interphase momentum transfer coefficient approach for the filtered drag force (Eq. (5)) on the mixture speed of sound behavior calculated from the filtered gas-solid flow model (Eqs. (1)-(4)). Such an approach is seen to affect the mixture speed of sound over the entire frequency range, independent of the filter frequency. In particular, the mixture speed of sound at frequencies ω lower than the filter frequency ωf is also altered. Furthermore, as the effective interphase momentum transfer

coefficient βe decreases, the high frequency mixture speed of sound behavior is gradually

imposed over lower frequencies. As such, it is impossible to obtain a grid independent solution with filtered models based on an effective interphase momentum transfer coefficient approach for the filtered drag force.

4.3 Generalized added mass approach for the correlation between the solids volume fraction and the gas phase pressure gradient

Figure 3 shows the effect of a generalized added mass approach (Eq. (7)) for the correlation between the solids volume fraction and the gas phase pressure gradient φ' P∇ ' (Eq.

(6)) on the mixture speed of sound behavior calculated from a filtered model. As the filter frequency ωf decreases and φ' P∇ ' grows in importance, the generalized added mass coefficient Ca increases, and the mixture speed of sound behavior is progressively affected

from the high frequencies on. In fact, the filter frequency mixture speed of sound behavior cm(ωf) is imposed to frequencies ω higher than the filter frequency ωf. The mixture speed of

sound behavior for frequencies ω lower than the filter frequency ωf is, however, not affected

and remains being predicted correctly by the filtered model. The filter frequency mixture speed of sound behavior cm(ωf) being imposed to frequencies ω higher than the filter

frequency ωf allows to obtain grid independent solutions. It should be emphasized that the

impact of the generalized added mass closure term (Eq. (7)) on the mixture speed of sound behavior calculated from the filtered model is in agreement with the behavior generally expected from filtered models. Figure 4 presents, for two solid phase densities ρs of

respectively 250 and 2650 kg m-3_{, the maximum allowable generalized added mass coefficient}

Ca as a function of the filter frequency ωf, determined from the analysis of the calculated

mixture speed of sound behavior (Figure 3). As the solids density ρs decreases, the maximum

allowable generalized added mass coefficient Ca for a given filter frequency ωf increases

(Figure 4). Figures 3 and 4 furthermore teach that generalized added mass coefficient values larger than 1, as stated by Zhang and VanderHeyden [2], are indeed possible and allowable for filter frequencies ωf lower than 10 Hz. Such low filter frequencies are unlikely to be

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solids volume fraction fre qu en cy [H z] m ix tur e s p ee d of sound [ m /s ] mixture speed of sound [m/s]

fre qu en cy [H z] m ix tur e s p ee d of sound [ m /s ] mixture speed of sound [m/s] (a) Ca= 0.002 (b) Ca = 0.02

fre qu en cy [H z] m ix tur e s p ee d of sound [ m /s ] mixture speed of sound [m/s]

fre qu en cy [H z] m ix tur e s p ee d of sound [ m /s ] mixture speed of sound [m/s] (c) Ca= 0.2 (d) Ca = 2

Figure 3: Mixture speed of sound as a function of the sound frequency and the solid volume fraction calculated with the filtered model (Eqs. (1)-(4), (6)-(7)) with a generalized added mass [2,12] with: (a) Ca =

0.002, (b) Ca = 0.02, (c) Ca = 0.2, (d) Ca = 2. Conditions: see Fig. 1. (a) → (d): Ca ↑ as ωf ↓.

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

filter frequency [Hz] max im u m al lo w a b le C a filter frequencyωf[Hz] ρ_s = 26 50 k_{g m} ^-3 ρs = 25 0 kg _m ^-3

Figure 4: Maximum allowable generalized added mass coefficient Ca (Eq. (7)) as a function of the filter

frequency ωf. Determined from a mixture speed of sound analysis with the filtered model (Eqs. (1)-(4), (6)) with

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Finally, it should be remarked that the effect of filtering the drag force and the effect of filtering the solids volume fraction of the gas phase pressure gradient (Eqs. (5)-(7)) seem, somehow, related. As the filter frequency ωf decreases and the effective interphase

momentum transfer coefficient βe (Eq. (5)) decreases, the correlation between the solids

volume fraction and the gas phase pressure gradient φ' P∇ ' increases. A possible explanation

is that, as the filter frequency ωf decreases, the microscopic description of the gas-solid

interaction force provided by the drag force is progressively replaced by a more macroscopic description provided by the generalized added mass. The latter is better understood by reformulating the generalized added mass as a generalized distribution of the filtered gas phase pressure gradient over the phases.

5 ALTERNATIVE FORMULATION FOR THE GENERALIZED ADDED MASS

An alternative formulation for the generalized added mass closure model can be obtained from the Reynolds-Favre-averaged momentum equations. The Reynolds-averaged pressure gradient can be written as:

g z P m ρ = ∂ ∂ ₍₁₀₎

Substracting the hydrostatic contribution, φ∂~P ∂z can be defined as: g z P g z P z P m m φ φρ ρ φ φ − ∂ ∂ =       ₋ ∂ ∂ = ∂ ∂~ ₍₁₁₎

It can be shown that:

z P z P ∂ ∂ = ∂ ∂ ~' ' ~ φ φ (12)

Hence, using Eq. (12), Eq. (11) is rewritten as:

g z P z P m φρ φ φ + ∂ ∂ = ∂ ∂ ~' ' (13)

Further decomposition of φρmg yields:

(

)

g

g _m _s _g

m φ ρ φφ ρ ρ

φρ = + ' ' − (14)

Introducing Eqs. (10) and (14) in Eq. (13) results in the following expression for φ∂P∂z :

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A more practical reformulation of Eq. (15) is: z P z P m z P m g s _∂ ∂ + ∂ ∂                 − + = ∂ ∂ ' ' _φ_' ~' ρ ρ φ φ φ φ φ φ (16)

, introducing the Reynolds-averaged solids mass fraction <ms>.

Because of the large density difference between the phases, Eq. (16) can be simplified to:

z P z P m z P s _∂ ∂ + ∂ ∂         + = ∂ ∂ ' ' _' ~' φ φ φ φ φ φ (17)

By rearranging Eqs. (8) and (9), De Wilde [12] showed that the effect of the meso-scale generalized added mass correction is a redistribution of the filtered gas phase pressure gradient over the phases, i.e. a change of the prefactor in front of the filtered continuous phase pressure gradient term. Assuming ρ_s>>ρ_g , Eq. (8) can be rewritten as:

(

)

(

)

g P C m C Dt v D s a m g s s a s ρ φ ρ ρ ρ φ φ φ φ ρ φ + ∇               +         − − + − = 2 1 1 ₍₁₈₎

An analogeous reformulated equation for the gas phase Eq. (9) can be obtained.

From Eq. (17), it is clear that a direct contribution to the generalized added mass appears in Eq. (18) via s m φ φ φ' ' _{. In case C}

a is large, the distribution of the gas phase pressure gradient

over the phases is seen to become almost mass fraction based and independent of Ca.

According to Newton’s second law of motion, a mass fraction based distribution of the filtered gas phase pressure gradient over the phases implies equal phase accelerations and, to avoid a more pronounced solid phase than gas phase acceleration, requires the drag to vanish. Hence, as the filter frequency ωf decreases, the microscopic drag description of the gas-solid

momentum transfer is progressively replaced by a more macroscopic description that basically consists of distributing the filtered gas phase pressure gradient, the ultimate macroscopic driving force of both the phases, over the phases.

In what follows, the importance of the correlation terms

z P m_s ∂ ∂ φ φ φ' ' _and_φ_'_{∂ '}_P~ _∂_z _is

investigated, as well as a possible indirect contribution of φ'∂ 'P~ ∂z to the generalized added

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6 QUANTIFYING THE GENERALIZED ADDED MASS: DYNAMIC MESO-SCALE SIMULATIONS (DMS)

To investigate the importance of the φ' P∇ ' term, 2D periodic domain simulations,

so-called dynamic meso-scale simulations (DMS), of gas-solid flow in a segment of a vertical channel are performed using a fine grid [1]. The MFIX code is used for the calculations.

The basic 2D DMS simulation in this work is for a Reynolds averaged solids volume fraction of 0.05 and uses a specific box size of 2 cm by 8 cm and a grid resolution of 32 by 128 grid nodes, respectively in the lateral and the axial direction. From the spatial and temporal grid resolution used, the smallest phenomena explicitly calculated have a frequency

ω of 5⋅105_{Hz, whereas the correlation terms, resulting from averaging over the calculation} domain, are calculated for a filter frequency ωf of 2⋅103 Hz. To calculate the correlation terms

for low filter frequencies ωf, large domain sizes for the DMS simulations have to be used.

Due to the computational load of DMS, this is beyond the scope of the present work. As previously discussed, the maximum of φ' P∇ ' and of the generalized added mass coefficient

Ca (Eqs. (7) and (18)) can be calculated as a function of the filter frequency ωf via the mixture

speed of sound test, allowing to investigate to a certain extent low filter frequencies ωf.

Starting from the initial uniformly fluidized state, it takes about two seconds before perturbations in the flow field grow into instabilities and the uniformly fluidized state breaks by the formation of meso-scale structures, i.e. clusters. At this stage, persisting fluctuations in time of all the flow quantities occur, as well of the local as of the averaged values. Figure 5 shows a snapshot of the calculated solids volume fraction and vertical gas phase pressure gradient fields. Figure 6 shows −φ'∂P'∂z and −φ'∂P'∂x versus time. A time averaged value

Q of the domain averaged quantity Q can be defined. As seen from Figure 6,

2.76E-01 2.57E-01 2.39E-01 2.21E-01 2.02E-01 1.84E-01 1.66E-01 1.47E-01 1.29E-01 1.10E-01 9.21E-02 7.38E-02 5.54E-02 3.70E-02 1.87E-02 x (m) y( m ) 0.0 0.02 0.0 0.04 _4.63E+03 4.32E+03 4.01E+03 3.71E+03 3.40E+03 3.09E+03 2.78E+03 2.48E+03 2.17E+03 1.86E+03 1.55E+03 1.25E+03 9.38E+02 6.31E+02 3.23E+02 y y( m ) x (m) (Pa/m) 0.0 0.02 0.0 0.04

(a) solids volume fraction (b) –dP/dy

Figure 5: 2D Dynamic Meso-scale Simulation (non-filtered model) with periodic boundary conditions. Snapshot profiles: (a) solids volume fraction, (b) gas phase pressure gradient in y-direction. Domain size: 2 cm x

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Figure 6: (a) −φ'∂P'∂z versus time; Figure 7: (a) φ'φ' versus time;

(b) −φ'∂P'∂x versus time. (b) −φ'∂P~'∂z (Eq. (11)) versus time.

Conditions: see Fig. 5. Conditions: see Fig. 5.

x

P ∂

∂

−φ' ' is small, whereas −φ'∂P'∂z is large. In fact, −φ'∂P'∂z is larger than − φ ∂P∂z

(Eq. (6)). The large value of −φ'∂P'∂z is further examined. Figure 7 shows φ'φ' and

z

P ∂

∂

−φ' ~' (Eqs. (15) and (17)) versus time. −φ'∂P~'∂z is small, in particular compared to

        ∂ ∂ − z P ms φ φ

φ' ' _{, but is roughly one order of magnitude larger than} ₋_φ_'_∂_P_'_∂_x _{. It is}

important to note that the contribution of the φ'φ' related term to −φ'∂P'∂z (Eq. (17)) is quite

pronounced and even accounts for about 90% of it.

To verify the grid independency of the results and the dependency on the domain size (i.e. the filter frequency dependency), DMS with an increased grid resolution, i.e. 64 by 256 grid nodes for a 2 cm by 8 cm domain, and with an increased domain size with the basic grid resolution, i.e. 64 by 256 grid nodes for a 4 cm by 16 cm domain, are carried out. Table 2 summarizes the results for an average solid volume fraction φ of 0.05. Improving the grid resolution increases both −φ'∂P'∂z and φ'φ' . This is expected, as the meso-scale structures,

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frequency for which the correlation terms are calculated. This lack of scale separation is intrinsic to gas-solid flows [1,3]. Although the variation of the domain size is only limited, the results confirm the expected gradual increase of the correlations −φ'∂P'∂z and φ'φ' and of

the generalized added mass coefficient Ca with decreasing filter frequency ωf.

To investigate the influence of the solids volume fraction on −φ'∂P'∂z and φ'φ' ,

simulations with Reynolds averaged solid volume fractions φ of respectively 0.05, 0.10, 0.20, 0.30, and 0.50 are carried out on a 2 cm by 8 cm domain with a grid resolution of 32 by 128 nodes. Table 3 summarizes the results.

Below solid volume fractions of about 0.3, −φ'∂P'∂z increases with increasing solids

volume fraction, but the increase becomes less pronounced with increasing solids volume fraction. At very high solids volume fractions, −φ'∂P'∂z is seen to decrease. For all solid

volume fractions investigated, the φ'φ' related term in Eq. (17) accounts for about 90% of

z

P ∂

∂

−φ' ' .

In contrast with the φ'φ' related term (Eq. (17)), the relation between −φ'∂P~'∂z and a

generalized added mass is not direct and its statistical significance was tested based on a linear model: ( ) t v u b v u a z P z z z z _∂ − ∂ + − = ∂ ∂ '~ ' φ (19) 05 . 0 = φ domain size: number of nodes: [ ] [/×/] × cm cm 2 x 8 32 x 128 64 x 256 2 x 8 64 x 256 4 x 16 filter frequency ωf [ ]s−1 2⋅103 2⋅103 6.8⋅102 z P ∂ ∂ −φ [_Pa_m−1] _37.38 _37.38 _37.38 z P ∂ ∂ −φ [_Pa_m−1] 129 ± 29 ± 54159 ± 47187 z P ∂ ∂ −φ' ' [_Pa_m−1] 91 ± 29 ± 54122 ± 47150 ' 'φ φ [ ]/ _{± 0.001}0.0072 _{± 0.001}0.0091 _{± 0.001}0.0096 z P ms ∂ ∂ − φ φ φ'' _[_Pa_m−1_] 106 ± 16 ± 14135 ± 18142 z P ∂ ∂ −φ' ~' [_Pa_m−1] -9 ± 24 ± 50-9 ± 4013 Qu an tity = T ime av er ag ed q ua ntity ± va ri an ce x P x P ∂ ∂ − = ∂ ∂ −φ' ' φ [_Pa_m−1] -0.75 ± 16 -1.44 ± 21 -1.79 ± 25 Ca (Eq. (45)) [ ]/ 2.8⋅10-3 3.8⋅10-3 4.0⋅10-3 domain size: 2 cm x 8 cm number of nodes: 32 x 128 φ [ ]/ 0.05 0.10 0.20 0.30 0.50 z P ∂ ∂ −φ [_Pa_m−1] _37.38 _149.52 _598.09 _1345.70 _3738.05 z P ∂ ∂ −φ [_Pa_m−1] 129 ± 29 ± 69434 1280 ± 87 ± 1162043 3851 ± 30 z P ∂ ∂ −φ' ' [_Pa_m−1] 91 ± 29 ± 69285 ± 87690 ± 116717 ± 30170 ' 'φ φ [ ]/ _{± 0.001}0.0072 _{± 0.002}0.019 _{± 0.003}0.041 _{± 0.006}0.045 _{± 0.002}0.011 z P ms _∂ ∂ −φ_φ'φ' [_Pa_m−1] 106 ± 16 ± 28283 ± 51608 ± 94676 ± 28171 z P ∂ ∂ −φ' ~' [_Pa_m−1] -9 ± 24 ± 5916 ± 65109 ± 4774 ± 812 Qu an tity = Ti m e av er ag ed q uantity ± va ri an ce x P x P ∂ ∂ − = ∂ ∂ −φ' ' φ [_Pa_m−1] -0.75 ± 16 ± 342.7 ± 42-1.8 -2.7 ± 36 -0.41 ± 12 Ca (Eq. (45)) [ ]/ 2.8⋅10-3 2.1⋅10-3 1.2⋅10-3 5.5⋅10-4 4.0⋅10-5

Tables 2 and 3. Calculated correlation terms and parameters.

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-8 -6 -4 -2 0 2 4 3.6 3.8 4 4.2 4.4 E P S _dP ^/ dy ta vg( E P S _dP ^/ dy) Time (s) DMS result Model prediction 3.6 3.8 4 4.2 4.4 -6 -2 0 2 4 -4 -8 Time [s] z P ∂ ∂ −φ' ~'

Figure 8: Comparison of −φ'∂P~'∂z (Eq. (11)) versus time from DMS and from

model equation Eq. (19). Conditions: see Fig. 5.

The parameters a and b are estimated via linear regression using about 82000 data points. In case of a Reynolds averaged solid volume fraction φ of 0.05, values of 0.0887 and 0.0007 were obtained for respectively a and b. For higher solid volume fractions, comparable and even smaller values for a and b were found. Figure 8 shows for φ = 0.05 the fit between the

z

P ∂

∂

−φ' ~' versus time values calculated from the DMS simulations and calculated via model equation Eq. (19). In general, the agreement is satisfactory and statistically significant.

Compared to the direct contribution to Ca resulting from the φ'φ' related term (Eq. (17))

(Tables 2 and 3), the indirect contribution to Ca from −φ'∂P~'∂z is, however, about one order

of magnitude smaller.

Figures 3 and 4 teach that for filter frequencies ωf of 500 – 2⋅103 Hz, corresponding to the

filter frequencies for which the correlation terms and Ca are calculated from the DMS

simulations presented, Ca values of the order of 10-3 are allowed. Good agreement is found

with the order of magnitude of Ca obtained from the DMS simulations (Tables 2 and 3).

Figures 3 and 4 furthermore teach that Ca values larger than 1, as stated by Zhang and

VanderHeyden [2], can be obtained and are allowable for filter frequencies ωf lower than

10-20 Hz.

7 CONCLUSIONS

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phases. The distribution becomes mass fraction based for a large generalized added mass coefficient. The mixture speed of sound test shows that such a large value is allowed if the filter frequency is sufficiently low, i.e. if the mesh is spatially or temporally sufficiently coarse, as e.g. in steady-state simulations. A further quantification of the generalized added mass effect is obtained from dynamic meso-scale simulations.

REFERENCES

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