Scaling Laws for Mixing in Inter-droplet Space

P_f = 4cπr²_sdr

df = 4cπr³_s[ln (1 +Bf)]³

[ln (1−f)]⁴ . (2.36) However, Eq. (2.35) vastly overpredicts the scalar dissipation from the droplet surface to lower mixture fraction regions (LES cell mean) and unduly affects any mixture fraction based model [130, 157]. In addition, with the presence of relative movement between the droplet and its surrounding gas phase, such model cannot be directly applied. In this study, the modelling strategy suggested by Klimenko and Bilger [67] are employed. Two different zones are distinguished in inter-droplet space based on different mixing characteristics: a near droplet zone, i.e. the quasi-laminar wake, of an inertial droplet dominated by molecular diffusion and a Kolmogorov scale zone determined by small scale turbulence. The determining parameters are different and different scaling laws can be employed for the mixing modelling for each zone.

2.4 Scaling Laws for Mixing in Inter-droplet Space

When droplets move relative to the surrounding gas phase in turbulent flows, wake-like structures develop. The zone in the vicinity directly behind the droplet is dominated by a quasi-laminar wake named here ”the near droplet zone” (NDZ).

Beyond this zone, the Kolmogorov scale fluctuations penetrate the wake. This zone is accordingly called ”the Kolmogorov scale zone” (KSZ). The approximate spatial distributions of these two zones in a regular droplet array are illustrated in Fig.

2.1. Klimenko and Bilger [67] proposed that the transition between the near droplet zone and the Kolmogorov scale zone can be based on the relationship between the Kolmogorov time scale τ_η and the characteristic time scale τ, τ =x₁/U_d, where x₁ is the longitudinal downstream distance from the droplet upstream and U_d is the mean relative velocity between droplets and surrounding gas, as illustrated in Fig.

2.1. Here, a normalized longitudinal downstream distance is defined as ˆx =x₁/x_u where x_u = U_dτ_η is named as the characteristic undulation length of the smallest eddy.

The derivation of scaling laws for the mixture fraction distribution and its condi-tional scalar dissipation and PDF in the near droplet zone were presented in earlier

η U _d

r x

₁

r x

1

r

c

r

_c

Inter-droplet space 1 Inter-droplet space 2

NDZ KSZ NDZ KSZ

Figure 2.1: Schematic diagram of the near droplet zone (NDZ) and the Kolmogorov scale zone (KSZ) in droplets arrays. The circles with arrows represent eddies of the Kolmogorov length scale, η.

studies in [67, 157, 159]. Following the guidelines by Klimenko and Bilger [67], the derivation of scaling laws in the Kolmogorov scale zone is complemented in the next chapter (Chapter 3). Here, the final expressions of the scaling laws are listed as a brief introduction. The variation of mixture fraction in the near droplet zone in axial and radial directions (alongx₁ andr, respectively, as visualized in Fig. 2.1) is given by

f_{N DZ} =f₂ +J_m(f_d−f₂) 4πρDx₁φ_n exp

− U_d 4Dx₁φ_nr²

. (2.37)

Here, f₂ denotes the minimum mixture fraction in the near droplet zone which is related to the average mixture fraction in inter-droplet space and also characterizes the interactions between different droplets. The computation of f₂ will be given in Chapter 3. f_d is the mixture fraction inside the droplet and it equals 1 for a single component droplet. Jm represents the evaporation rate of a single droplet. ρ and Dare the average density and Favre averaged diffusivity in inter-droplet space, respectively. The parameter r is the radial distance from the droplet center on a plane perpendicular to the flow direction (see Fig. 2.1). The modelling parameter φ_n was suggested as 1 in the derivation by Klimenko and Bilger [67] and it will be calibrated by the fully resolved DNS results in this study. The expectation of the maximum mixture fraction on each plane perpendicular to the relative mean flow

2.4. SCALING LAWS FOR MIXING IN INTER-DROPLET SPACE 20 direction in the near droplet zone can be obtained from Eq. (2.37) for r→0, viz.

f_{max,N DZ} =f₂+Jm(fd−f2)

4πρDx₁φ_n . (2.38)

Similarly, the scaling law for mixture fraction in the Kolmogorov scale zone can be derived from non-dimensional analysis and is written as

f_KSZ =f₁+ (φ_c−φ₁)J_m(f_d−f₂) ρU_dε

x₁ U_d

−3

exp

−(φ_c−φ₁)πU_d³ εx₁³ r²

. (2.39) Here, f₁ is the minimum mixture fraction in the Kolmogorov scale zone. Similar to f₂, it is also related to the average mixture fraction in inter-droplet space and characterizes the interactions between droplets. The definition of f₁ is also given in Chapter3. εdenotes the Favre averaged turbulent energy dissipation in inter-droplet space. φ_c and φ₁ are again modelling parameters and their values are calibrated in Sec. 5.3.5 by the fully resolved DNS study. The expectation of the maximum value of mixture fraction on each plane perpendicular to the relative mean flow direction in the Kolmogorov scale zone can be obtained from Eq. (2.39) for r→0 as

fmax,KSZ =f1+ (φc−φ1)J_m(f_d−f₂) ρU_dε

x₁ U_d

−3

. (2.40)

Turbulent fluctuations perpendicular to the wake tend to smooth the inhomo-geneities in radial direction, which can be observed from Fig. 2.1. Radial ho-mogeneity is assumed in [67] which is also followed by the derivation in Chapter 3.

However, in case that the turbulent mixing is not strong enough to enforce a com-pletely homogeneously mixed state in the radial direction within the Kolmogorov scale zone, mixture fraction gradients persist and require the modelling of radial de-pendence. Hence, an additional radial dependence of mixture fraction is introduced by an exponential function as given in Eq. (2.39). Equations. (2.37 -2.40) are used to evaluate the variation of mixture fraction in axial and radial directions in inter-droplet space and the transition between the near inter-droplet zone and the Kolmogorov scale zone can be determined by the fully resolved DNS of droplet arrays. The scal-ing laws for mixture fraction conditional dissipation and its PDF (Eqs. (2.41-2.45)) can be assessed in the corresponding zones. The scaling law for scalar dissipation

in the near droplet zone is given by N_{f,N DZ} = 2Ud(f −f2)²

(R_c−∆x)φ_n

ln

Jm(fd−f2) 4πρDφ_n(f−f₂)

ln Rc

∆x +ln²∆x

2 − ln²Rc

2

. (2.41) Here, Rc is the axial length of the near droplet zone. The parameter ∆x originates from an integration across the near droplet zone (cf. Appendix A), Eq. (A.7)) and denotes the lower bound. Equation (A.6) has a singularity at x₁ = 0 and becomes valid at around d/2 away from the droplet surface. In LES computations,

∆x could be specified as d/2 according to the suggestions by Klimenko and Bilger [67]. Equation (2.41) is different from the expression proposed by Klimenko and Bilger [67] where the Stefan flow is not considered. The derivation by Zoby [159] is also improved upon and the complete derivation is presented in Appendix A. The scaling law for scalar dissipation in the Kolmogorov scale zone is derived in Chapter 3 and reads

N_f,KSZ = 1

A₁(f−f₁)^7/3

J_m(f_d−f₂) ρU_dε

−1/3

, (2.42)

whereA₁is a modelling constant that will be calibrated by the fully resolved DNS in Chapter5. As conventional for scaling law analysis, a quasi-steady-state is assumed.

A simple relationship between the mixture fraction PDF and scalar dissipation was provided in [67] which is given by N_fP_fρ = cJ_m(f_d−f₂) and its derivation is complemented in Chapter 3. The scaling law for the PDF in the near droplet zone and the Kolmogorov scale zone can be obtained as

P_{f,N DZ} = cJm(fd−f2)

ρN_{f,N DZ} (2.43)

and

P_f,KSZ = cJ_m(f_d−f₂)

ρN_f,KSZ , (2.44)

respectively. The droplet number density c can be calculated from c = r_c⁻³ (r_c is the inter-droplet distance). Klimenko and Bilger [67] also claimed that it is not easy to adjust all the scaling laws to different conditions, and in practice the model for the PDF in the corresponding region may be substituted by an expression for a Gaussian core with the power tail (f −f₁)^a, viz.

P_f,G = 1 q

2πff⁰² exp





−

f −f˜ 2

2ff⁰²





(f −f₁)^a, (2.45)

2.4. SCALING LAWS FOR MIXING IN INTER-DROPLET SPACE 22 where ˜f denotes the Favre averaged mixture fraction of the corresponding region and ff⁰² represents its variance. The constant a is a modelling parameter that will also be calibrated by the fully resolved DNS.

Derivation of the Scaling Laws for the Kolmogorov Scale Zone

The derivation of the scaling laws for the Kolmogorov scale zone (KSZ) comple-ments the descriptions in Sec. 2.4. The near droplet zone (NDZ) of inertial droplets is dominated by the quasi-laminar wake structure behind the droplet. Further away from the droplet, small scale turbulent motion disturbs the wake structure. Mixing in this zone is dominated by the small turbulent eddies. In the Kolmogorov scale zone, the determining parameter ε (turbulent energy dissipation) needs to be com-plemented by the evaporation of the droplets that act as sources of mixture fraction and the relative velocity between the droplet and its surrounding gas phase that characterizes the wake development [67]. Note that the parameters that have been introduced in Sec. 2.4 will not be defined again.

3.1 Mixture Fraction Conditional Scalar Dissipa-tion

For simplicity and as implied in the derivation in [67], the Schmidt number, Sc, is set to unity and the Kolmogorov length scale, η, can then be written as

η =η_B = D³

ε 1/4

, (3.1)

where η_B is the Batchelor scale. The Kolmogorov time scale can be expressed as τ_η =

D ε

1/2

. (3.2)

23

3.1. MIXTURE FRACTION CONDITIONAL SCALAR DISSIPATION 24 The diffusion of the evaporation rate, J_m(f_d−f₂), can be normalized by ρU_dη² leading to

f_k = J_m(f_d−f₂)

ρU_dη² = Sε^1/2

D^3/2. (3.3)

Here, S characterizes the average cross-sectional area of the wake and is given by S ≡ J_m(f_d−f₂)

ρU_d . (3.4)

The parameter f₂ is the minimum value of mixture fraction in the near droplet zone that is defined as

f₂ =f_g−φ₂f₂^o, (3.5)

with

f₂^o = W

(DcU_d)^1/2 (3.6)

and

W = cJ_m(f_d−f₂)

ρ . (3.7)

Here, W is a collective evaporation term of a droplet cloud and characterizes the interactions of droplet evaporation where Jm is the average evaporation rate. f₂^o is a normalized term and the normalization is based on the characteristic parameters in the near droplet zone, viz. D, candU_d. The mixture fraction in the Kolmogorov scale zone is normalized by f_k to yield

φ_k= f−f₁

f_k , (3.8)

with

f₁ =f_g −φ₁f₁^o (3.9)

and

f₁^o = W

(εcU_d)^1/4. (3.10)

Here, f₁ is the minimum mixture fraction in the Kolmogorov scale zone which -similar to f₂ above - is composed of f_g and the normalized collective evaporation term, f₁^o. The normalization is now based on the characteristic parameters in the Kolmogorov scale zone, viz. ε, c and Ud. The mixture fraction is approximately homogeneous in the space that is of the order of inter-droplet distance and is given by

f =f_g−φ_If₁^o, (3.11)

which can be matched with the mixture fraction at the outer boundary of the Kol-mogorov scale zone that can be obtained by the rearrangement of Eq. (3.8). The matching is expressed as

f1+φkfk =fg −φIf₁^o. (3.12) Using Eqs. (3.3), (3.9) and (3.10) and the characteristic time scale, τ = x1/Ud ∼ r_c/U_d, in the space withx₁ ∼r_c, the term φ_k is yielded from Eq. (3.12) as

φ_k ∼ r_c^−3/2ε^3/4τ^−3/4

τη−3 . (3.13)

At the outer boundary of the Kolmogorov scale zone, the characteristic time scale τ can also be obtained by a similarity to the integral time scale (τ_t∼ l_t²/ε1/3

) as τ ∼

r_c² ε

1/3

. (3.14)

Equation (3.13) can be rewritten with Eq. (3.14), viz. [67]

φ_k ∼ τ

τ_η −3

. (3.15)

One constant φ₁, which is related to the Kolmogorov scale zone, is not enough to close φ_k and another constant φ_c is introduced to balance Eq. (3.15) that is given by

φ_k = (φ_c−φ₁) τ

τ_η −3

. (3.16)

Substituting Eqs. (3.3), (3.4) and (3.8) into Eq. (3.16), the mixture fraction in the Kolmogorov scale zone can be derived as

f_KSZ =f₁+ (φ_c−φ₁)J_m(f_d−f₂) ρU_dε

x₁ U_d

−3

. (3.17)

In case that the turbulent mixing is not strong enough to enforce a completely homogeneously mixed state in the radial direction of the Kolmogorov scale zone, an additional exponential function is required for modelling the radial dependence of mixture fraction that is added to Eq. (3.17), viz.

fKSZ =f1+ (φc−φ1)J_m(f_d−f₂) ρU_dε

x₁ U_d

−3

exp

−(φ_c−φ₁)πU_d³ εx₁³ r²

. (3.18)

3.2. THE PDF OF MIXTURE FRACTION 26

Im Dokument Modelling of mixing in moderately dense sprays (Seite 52-60)