# RANS Modeling of Turbulent Flow and Heat Transfer in a Droplet-Laden Mist Flow through a Ribbed Duct

^{*}

## Abstract

**:**

## 1. Introduction

_{L}

_{1}= 15%. Their initial diameter was d

_{1}= 50–60 µm, the flow Reynolds number based on the mean mass flow velocity at the inlet and the hydraulic diameter was Re = U

_{m}D

_{h}/ν = (0.8–2.4) × 10

^{4}, the heat flux density was q

_{W}= const = 2.6 kW/m

^{2}, and p/h = 10 and 20 for the ribs installed at an angle to the free-stream flow of φ = 90°. Heat transfer measurements in the case of a gas-droplet flow between two ribs in a system of continuous V-shaped ribs and broken V-shaped ribs were carried out in [18]. The study was carried out at M

_{L}

_{1}= 10%, d

_{1}= 50–60 μm, Re = (0.8–2.4) × 10

^{4}, q

_{W}= const = 1–10 kW/m

^{2}, p/h = 10 and 20, and φ = 45°.

_{L}

_{1}= 2%, d

_{1}= 5 μm, Re = (0.5–4) × 10

^{4}, q

_{W}= const = 10 kW/m

^{2}, and p/h = 10 in a gas-droplet flow. The range of variation in the initial parameters in [20] was as follows: M

_{L}

_{1}= 1%, d

_{1}= 10 μm, Re = (1–6) × 10

^{4}, q

_{W}= const = 4.8 kW/m

^{2}, and p/h = 10 in the flow of steam water droplets.

## 2. Mathematical Models

#### 2.1. Governing Equations for the Two-Phase Turbulent Mist Phase

_{i}(U

_{x}≡ U, U

_{y}≡ V) and ${u}^{\prime}{}_{i}$ (${u}^{\prime}{}_{x}\equiv {u}^{\prime},{u}^{\prime}{}_{y}\equiv {v}^{\prime}$) are components of mean gas velocities and their pulsations; x

_{i}are projections on the coordinate axis; $2k=\langle {u}_{i}{u}_{i}\rangle ={u}^{\prime 2}+{v}^{\prime 2}+{w}^{\prime 2}\approx {u}^{\prime 2}+{v}^{\prime 2}+0.5\left({u}^{\prime 2}+{v}^{\prime 2}\right)\approx 1.5\left({u}^{\prime 2}+{v}^{\prime 2}\right)$ is the kinetic energy of gas-phase turbulence; $\tau ={\rho}_{L}{d}^{2}/(18\rho \nu W)$; $W=1+{\mathrm{Re}}_{L}^{2/3}/6$ is the particle relaxation time, taking into account the deviation from the Stokes power law; and ${\mathrm{Re}}_{L}=\left|\mathbf{U}-{\mathbf{U}}_{L}\right|d/\nu $ is the Reynolds number of the dispersed phase.

_{T}and Sc

_{T}equal to 0.9, is used in this work.

#### 2.2. Evaporation Model

_{L}is the coefficient of heat conductivity of the droplet; α and α

_{P}are the heat transfer coefficient for the evaporating droplet and non-evaporating particle, respectively; T

_{L}is the temperature of the droplet; J is the mass flux of steam from the surface of the evaporating droplet; L is the latent heat of evaporation; ρ is the density of the gas–steam mixture; D is the diffusion coefficient; and ${K}_{V}^{*}$ is the steam mass fraction at the “steam-gas mixture–droplet” interface, corresponding to the saturation parameters at droplet temperature T

_{L}. Subscript “L” corresponds to the parameter on the droplet surface. The Jacob number, $\mathrm{Ja}={C}_{P}\left(T-{T}_{L}\right)/L$, is the ratio of sensible heat to latent heat during droplet evaporation. It characterizes the rate of the evaporation process and is the reciprocal of the Kutateladze number, Ku. For our conditions, the Jakob number is Ja ≤ 0.01.

_{L}d

_{1}/λ

_{L}<< 1 and the Fourier number is Fo = τ

_{eq}/τ

_{evap}<< 1. Here, τ

_{eq}is the period when an internal temperature gradient inside a droplet exists, and τ

_{evap}is the droplet’s lifetime. In this case, a droplet evaporates at the saturation temperature, and the temperature distribution inside a droplet is uniform.

#### 2.3. The Elliptic Blending Reynolds Stress Model (RSM) for the Gas Phase

_{ij}is the stress-production term, T

_{T}and L

_{T}are the turbulent time and geometrical macroscales, and ϕ

_{ij}is the velocity–pressure–gradient correlation, well-known as the pressure term. The blending model (8) presented in [28] is used to predict ϕ

_{ij}in Equations (6) and (7), where β is the blending coefficient, which goes from zero at the wall to unity far from the wall; ${\varphi}_{{}_{ij}}^{H}$ is the “homogeneous” part (valid away from the wall) of the model; and ${\varphi}_{{}_{ij}}^{W}$ is the “inhomogeneous” part (valid in the wall region).

#### 2.4. Governing Equations for the Dispersed Phase

_{Lij}and D

^{Θ}

_{Lij}are the turbulent diffusivity tensor and the particle turbulent heat transport tensor [29,30], ${\mathrm{\tau}}_{\Theta}={C}_{PL}{\rho}_{L}{d}^{2}/\left(12\mathrm{\lambda}Y\right)$ is the thermal relaxation time, and $Y=\left(1+0.3{\mathrm{Re}}_{L}^{1/2}{\mathrm{Pr}}^{1/3}\right)$.

_{1}< 10

^{−4}), and the droplets are finely dispersed (d

_{1}< 100 μm); therefore, the effects of interparticle collisions and break-up are neglected [25,32,33]. Droplet bag break-up is observed at We = ρ(

**U**

_{S}−

**U**

_{L})

^{2}d/σ ≥ We

_{cr}= 7 [33]. Here, ${U}_{S}=U+\langle {u}_{S}^{\prime}\rangle $ and

**U**

_{L}are the gas velocity seen by the droplet [34] and the mean droplet velocity, respectively, U is the mean gas velocity (derived directly from the RANS predictions), $\langle {u}_{S}^{\prime}\rangle $ is the drift velocity between the fluid and the particles [34], and ρ and ρ

_{L}are the densities of the gas and dispersed phases. For all droplet sizes investigated in the present paper, the Weber number is very small (We << 1). Droplet fragmentation at its contact with a duct wall also is not considered. The effect of break-up and coalescence in the two-phase mist flow can be neglected due to a low droplet volume fraction at the inlet (Φ

_{1}= M

_{L}

_{1}ρ/ρ

_{L}< 2 × 10

^{−4}). Here, M

_{L}

_{1}is the initial droplet mass fraction, and ρ

_{L}is the density of the dispersed phase.

## 3. Numerical Solution and Model Validation

#### 3.1. Numerical Solution

_{+}= u

_{*}y/ν ≈ 0.5 (the friction velocity u

_{*}was determined for a single-phase air flow with other identical parameters). Additionally, simulations were carried out on grids containing “coarse” 128 × 60 and “fine” 512 × 200 CVs. The difference in the results of the calculations of the wall friction coefficient (a) and the Nusselt number (b) for two-phase flow did not exceed 0.1% (see Figure 3). The Nusselt number at T

_{W}= const was determined by the formula:

_{W}and T

_{m}are the wall and the mass-averaged temperatures of the gas in the corresponding cross section.

_{0}= 10p, where p is the rib pitch (the spacing between upstream and downstream ribs). The 1st rib was installed at the end of this domain. The output parameters from section L

_{0}were the input values for section L

_{1}= 10p, located between the 1st and 2nd ribs (see Figure 1). All simulations were performed for the two-dimensional case of a gas-droplet flow for the 2nd and 3rd obstacles. Drops were fed into a single-phase turbulent air flow along the entire cross section of the duct in the inlet cross section behind the 2nd rib. The initial temperatures of the gas and dispersed phases at the inlet to the computational domain were T

_{1}= T

_{L}

_{1}= 293 K. The boundary condition T

_{W}= const = 373 K was set on the ribbed wall; the opposite smooth (without obstacles) wall of the flat duct was adiabatic. The entire ribbed duct surface and all the ribs were heated to eliminate the influence of the possible formation of liquid spots during the deposition of droplets on the wall from a two-phase mist flow. The impermeability and no-slip conditions for the gas phase were imposed on the duct walls. For the dispersed phase on the duct wall, the boundary condition of the “absorbing wall” [30] was used when a droplet did not return to the flow after contact with the wall surface. All droplets deposited from two-phase flow onto the wall momentarily evaporated. Thus, the pipe surface was always dry, and there was no liquid film or spots of deposited droplets formed on the wall [25,31,35]. This assumption for the heated surface is valid (see, for example, papers [25,35]). Furthermore, this condition is valid if the temperature difference between the wall and the droplet is greater than ${T}_{W}-{T}_{L}\ge 40K$ [38]. In the outlet cross section, the conditions for the equality to zero of the derivatives of all variables in the streamwise direction were set.

#### 3.2. Model Validation

_{m}

_{1}(a), and the velocity of its fluctuations, u’/U

_{m}

_{1}(b), along the duct length. The averaged and fluctuating components of the streamwise velocity were normalized by the value of the average mass velocity of a single-phase flow at the duct inlet U

_{m}

_{1}. Comparisons with the data of [39] were made for the 17th and 18th obstacles. The height of the duct with a square cross section was H = 60 mm. The profiles of the mean longitudinal velocity component agreed well with the experimental data (the difference did not exceed 5–7%). The agreement between the measurements and numerical predictions for longitudinal velocity pulsations was also quite good (the difference did not exceed 10%) except for the near-wall region.

_{0}is the Nusselt number in a smooth duct for a single-phase flow. The Nusselt number at a constant value of heat flux density (q

_{W}= const) is determined by the formula:

## 4. The RANS Results and Discussion

_{m}

_{1}= 5–20 m/s, and the Reynolds number for the gas phase, constructed from the mass-average gas velocity at the inlet and the duct height, was Re

_{H}= HU

_{m}

_{1}/ν ≈ (0.6–5) × 10

^{4}. The initial average droplet diameter was d

_{1}= 5–50 µm, and their mass concentration was M

_{L}

_{1}= 0–10%. The initial temperature of the gaseous and dispersed phases was T

_{1}= T

_{L}

_{1}= 293 K.

#### 4.1. Flow Structure

_{m}

_{1}(a); turbulent kinetic energy (TKE), $k/{U}_{m1}^{2}$ (b); and gas-phase temperature, $\mathsf{\Theta}=({T}_{W}-T)/({T}_{W}-{T}_{1,m})$, in a single-phase flow (M

_{L}

_{1}= 0), a gas-droplet flow at M

_{L}

_{1}= 0.05, and liquid drops ${\mathsf{\Theta}}_{L}=({T}_{L,max}-{T}_{L})/({T}_{L,max}-{T}_{L1})$ (c) as well as vorticity, ${\mathrm{\Omega}}_{z}={\omega}_{z}h/{U}_{m}{}_{1}$ (d). The solid lines are the single-phase flow at M

_{L}

_{1}= 0, the dotted line is the gas phase at M

_{L}

_{1}= 0.05, and the dash-dot line is the dispersed phase behind the two-dimensional obstacle. Here, T

_{L}

_{,max}и T

_{L}

_{1,m}are the droplet temperatures, which were highest in the corresponding cross section and at the inlet.

_{R}

_{1}≈ 4.1h, and the length of the second recirculation region in front of the step ahead was x

_{R}

_{2}≈ 1.1h. The lengths of the recirculation zones were determined from the zero value of the flow velocity.

_{L}/M

_{L}

_{1}, for various droplet mass fractions (a) and their initial diameters (b). Obviously, due to the evaporation of droplets, their mass fraction decreased continuously, both streamwise and in traverse directions, when approaching the wall of the heated duct between the ribs. This was typical of the numerical data given in Figure 8a,b. The distributions of the mass fraction of droplets with changes in their initial amounts had qualitatively similar forms (see Figure 8a).

_{L}/M

_{L}

_{1}→ 1. This is explained by the almost complete absence of droplet evaporation. Fine particles at Stk < 1 penetrated into the region of flow separation and were observed over the entire cross section of the duct. Large inertial droplets (d

_{1}= 100 μm, Stk > 1) almost did not penetrate into the flow recirculation zone, and they were present in the mixing layer and the flow core. In the near-wall zone, large drops were observed only behind the reattachment point. The largest and inertial droplets (d

_{1}= 100 μm) accumulated in the near-wall region towards the downstream obstacle. Finely dispersed low-inertia droplets could leave the region between the two ribs due to their low inertia, while large drops could not leave this region. This led to an increase in the droplet mass fraction in this flow region and towards the downstream obstacle.

_{m}

_{1}(a), and the temperature, $\mathsf{\Theta}=({T}_{W}-T)/({T}_{W}-{T}_{1,m})$ (b), in two-phase mist flow are shown in Figure 9. Large-scale and small-scale flow recirculation zones behind the upwind rib (BFS) and before the downstream rib (FFS) can be found in Figure 9a. The small corner vortex directly behind the upstream rib was observed. The length of the main recirculation zone of the flow was x

_{R}

_{1}≈ 4.1h, and the length of the second recirculation region in front of the step ahead was x

_{R}

_{2}≈ 1.1h. The lengths of the recirculation zones were determined from the zero value of the mean streamwise flow velocity (U = 0). In this region, the gas temperature increased, and it led to the suppression of heat transfer (see Figure 9b). These conclusions agree with the data of Figure 6 and Figure 7a,c.

#### 4.2. Heat Transfer

_{1}= 5 µm) evaporated most intensely, and the largest ones evaporated least intensely (d

_{1}= 100 µm) (see Figure 10b). The sizes of the zone of two-phase flow and the zone of HTE also decreased. This was an obvious fact for the evaporation of droplets in the two-phase mist flows, which was associated with a significant interface reduction; it was first shown by the authors of this work for a gas-droplet flow in a system of two-dimensional obstacles. Heat transfer was attenuated and trended toward the corresponding value for the single-phase flow in the region of flow separation for the most inertial droplets. These drops did not penetrate into the flow separation region behind the upstream rib (BFS). An increase in heat transfer was obtained in the region behind the point of flow reattachment. A decrease in heat transfer was shown in the section of flow separation towards the downstream rib (FFS). The most inertial droplets also did not leave the region between the two ribs and accumulated in front of the downstream obstacle.

_{L}

_{1}, on the thermal hydraulic performance parameter is shown in Figure 11. The wall friction coefficient, C

_{f}, was calculated using the formula ${C}_{f}/2={\tau}_{W}/\left(\rho {U}_{m1}^{2}\right)$. Here, Nu

_{0}and C

_{f}

_{0}are the maximal Nusselt number and wall friction coefficient in the two-phase mist flow of a fully developed smooth duct, other conditions being equal. Nu/Nu

_{0}/(C

_{f}/C

_{f}

_{0}) is the thermal hydraulic performance parameter. This is the ratio of the maximal Nusselt numbers divided by the maximal wall friction coefficient ratio. The ribbed surface provided a much better thermohydraulic performance than a smooth duct in the case of a droplet-laden turbulent mist flow, with other conditions being identical. This effect was quite pronounced at small Reynolds number values of Re < 10

^{4}. It should be noted that the wall friction coefficient ratio, C

_{f}/C

_{f}

_{0}, was taken to the first power.

## 5. Comparison with Results of Other Authors

_{P}

_{1}= U

_{m}

_{1}/25, the Reynolds number was plotted from the obstacle height, Re = HU

_{m}

_{1}/ν = 740, ρ

_{P}/ρ = 769.2, and ρ

_{P}was the particle material density. The height of the boundary layer for a single-phase flow in the inlet section of the computational domain was δ = 7h, and the carrier phase was atmospheric air at T = 293 K (see Figure 12). Here, h = 7 mm was the obstacle height, H = 1 mm, U

_{m}

_{1}= 1.59 m/s was the free flow velocity, and V

_{P}

_{1}= 0.06 m/s. The two-dimensional obstacle was square in cross section and was mounted on the bottom wall. The flow of solid particles was blown vertically through a flat slot along the normal surface at distance h from the trailing edge of the obstacle. The number of solid particles during the LES calculation was 2 × 10

^{5}. The calculations were performed for three Stokes numbers, St

^{+}= τu

_{*}

^{2}/ν = 0.25, 1, 5, and 25, where τ = ρ

_{P}d

^{2}/(18µ) was the particle relaxation time and u

_{*}= 0.5 m/s was the friction velocity for a single-phase flow without particles, other things being equal. This corresponded to the solid particle diameters d = 8, 15, 34, and 76 µm. The calculations were carried out in a two-dimensional formulation for an isothermal two-phase flow around a single obstacle.

_{b}is the mean concentration of particles over the hole (slot) width at the inlet to the computational domain. As the Stokes number, St

_{+}, increased, heavier particles stopped penetrating into the recirculation region, resulting in lower concentrations along the obstacle wall. The local maximum concentration at x/h ≈ 1 for all studied Stokes numbers (particle diameters) is explained by the injection of the particle flow. A characteristic feature of the low-inertia particles was a significant increase in the concentration of particles near the obstacle wall, according to the LES data (C/C

_{b}≈ 10–20). Most likely, such an accumulation of particles in the corner near the wall of the obstacle can be explained by the effect of the accumulation of particles in [42]. For our Eulerian simulations, an increase in concentration was also obtained, but the values were much smaller (by a factor of approximately 8–10). For inertial particles at St

_{+}= 5, the region turned out to be almost completely free of solid particles. This was typical for both the data of the LES calculations [10] and our numerical calculations. Behind the obstacle, a decrease in the particle concentration in the near-wall region was observed, and here our numerical predictions agreed satisfactorily with the LES data (the difference did not exceed 20% at St

_{+}= 1 and 5 and did not exceed 100% at St

_{+}= 0.25).

_{+}, is varied along the length of the channel behind a two-dimensional obstacle. Particles at St

_{+}= 0.25 accumulated in the near-wall region near the bottom wall. Further downstream, heavier particles gradually left the recirculation region, and at St

_{+}> 1 the decrease in their distribution profile was similar to a Gaussian distribution. For the largest particles at St

_{+}= 25, according to the results of our numerical predictions, an underestimation of the position of the concentration maximum was observed, and in general the particles rose lower than according to the LES results [10].

_{+}, was varied along the length of the duct behind a two-dimensional obstacle. Particles at St

_{+}= 0.25 accumulated in the near-wall region near the bottom wall. Further downstream, heavier particles gradually left the recirculation region, and at St

_{+}> 1 the decrease in their distribution profile was similar to a Gaussian distribution. An underestimation of the position of the concentration maximum was observed, according to the results of our numerical predictions for the largest particles at St

_{+}= 25. The maximal penetration coordinate in the transverse directions in our RANS predictions was smaller than that in the LES results [10].

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${C}_{f}/2={\tau}_{W}/\left(\rho {U}_{m1}^{2}\right)$ | wall friction coefficient |

D | droplet diameter |

H | rib height |

$2k=\langle {u}_{i}^{\prime}{u}_{i}^{\prime}\rangle $ | turbulent kinetic energy |

M_{L} | mass fraction |

$\mathrm{Nu}=-{\left(\partial T/\partial y\right)}_{W}H/\left({T}_{W}-{T}_{m}\right)$ | Nusselt number |

p | rib pitch |

q_{W} | heat flux density |

Re_{D} = U_{m}D_{h}/ν | Reynolds number, based on hydraulic diameter |

Re = U_{m}_{1}H/ν | Reynolds number, based on the duct height |

Stk = τ/τ_{f} | the mean Stokes number |

T | temperature |

U_{L} | the mean droplet velocity |

U_{m}_{1} | mean mass flow velocity |

U_{S} | the fluid (gas) velocity seen by the droplet |

u_{*} | wall friction velocity |

x | streamwise coordinate |

x_{R} | position of the flow reattachment point |

y | distance normal from the wall |

Subscripts | |

0 | two-phase mist flow in a smooth duct |

1 | initial condition |

W | wall |

L | liquid |

M | mean mass |

Greek | |

Φ | volume fraction |

Ρ | density |

μ | the dynamic viscosity |

ν | kinematic viscosity |

τ | the droplet relaxation time |

τ_{W} | wall shear stress |

Acronym | |

BFS | backward-facing step |

CV | control volume |

FFS | forward-facing step |

THE | heat transfer enhancement |

RANS | Reynolds-averaged Navier–Stokes |

SMC | second-moment closure |

TKE | turbulent kinetic energy |

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**Figure 1.**Scheme of flow in a two-phase turbulent flow in a ribbed flat duct (not to scale). Abbreviations L

_{0}and L

_{1}are the computational domains for the preliminary simulations; C.D. is the computational domain; and 1, 2, and 3 are the 1st, 2nd, and 3rd ribs.

**Figure 3.**Grid independence test for the wall friction (

**a**) and heat transfer (

**b**) distributions along the duct length. M

_{L}

_{1}= 5%, d

_{1}= 15 µm, Re = HU

_{m}

_{1}/ν = 1.6 × 10

^{4}.

**Figure 4.**Profiles of the mean longitudinal velocity (

**a**) and its fluctuations (

**b**) in the flow around a two-dimensional square obstacle. (

**a**): h/H = 0.067, Re = U

_{m}

_{1}H/ν = 5 × 10

^{4}, p/h = 9, H = 60 mm, h = e = 4 mm, p = 36 mm, U

_{m}

_{1}= 12.5 m/s. The symbols are the measurements of [39]; the lines are the authors’ simulations.

**Figure 5.**Distribution of heat transfer enhancement ratio in the flow around a two-dimensional square obstacle. Re

_{D}= 0.8 × 10

^{4}, h = e = 4 mm, p = 40 mm, H = 30 mm, H/h = 7.5, T

_{1}= 300 K, q

_{W}= 1 kW/m

^{2}, Tu

_{1}= 5%. The symbols are the experiments of [40]; the lines are predictions: v2f, k–ω SST, and k–ε are predictions of [41], and RSM is the authors’ simulations. Reprinted with permission from (Liu, J. et al.).

**Figure 6.**The streamlines for a gas-droplet flow between two ribs. Re = U

_{m}

_{1}H/ν = 1.6 × 10

^{4}, h/H = 0.1, p/h = 10, H = 40 mm, h = 4 mm, p = 40 mm, U

_{m}

_{1}= 6 m/s, T

_{1}= 293 K, T

_{W}= 373 K, d

_{1}= 15 μm, M

_{L}

_{1}= 0.05.

**Figure 7.**Transverse profiles of averaged streamwise velocities (

**a**), turbulent kinetic energy (

**b**), and gas-phase temperature in a single-phase flow (M

_{L}

_{1}= 0), a gas-droplet flow at M

_{L}

_{1}= 0.05, and liquid drops (

**c**) as well as vorticity (

**d**). Re = 1.6 × 10

^{4}, h/H = 0.1, p/h = 10, d

_{1}= 15 μm, M

_{L}

_{1}= 0.05.

**Figure 8.**Transverse profiles of the water droplet mass fraction vs. the initial mass fraction (

**a**) and the droplet diameter (

**b**). Re = 1.6 × 10

^{4}. (

**a**) d

_{1}= 15 μm; (

**b**) M

_{L}

_{1}= 0.05.

**Figure 9.**The contour plots of the mean streamwise velocity (

**a**) and temperature (

**b**). Re = 1.6 × 10

^{4}, d

_{1}= 15 μm, M

_{L}

_{1}= 0.05.

**Figure 10.**The effect on the heat transfer rate of the droplet mass fraction at the inlet (

**a**) and their diameter (

**b**). Re = 1.6 × 10

^{4}. (

**a**) d

_{1}= 15 μm; (

**b**) M

_{L}

_{1}= 0.05.

**Figure 11.**The effect on the thermal hydraulic performance parameter of the Reynolds number of the gas flow and the mass fraction of the dispersed phase at the inlet. d

_{1}= 15 µm.

**Figure 12.**Scheme of two-phase flow behind a 2D square obstacle. The 1 is the single-phase air flow with the mean mass velocity U

_{m}

_{1}, and the 2 is the dispersed phase stream U

_{P}

_{1}.

**Figure 13.**Concentration profiles of the dispersed phase at y = 0.02h vs. the Stokes number St

_{+}along the length of the channel behind a 2D obstacle. The points are LES calculations [10]; the lines are the authors’ predictions. h = H/7, V

_{P}

_{1}= U

_{m1}/25, Re = HU

_{m1}/ν = 740, ρ

_{P}/ρ = 769.2. Reprinted with permission from (Grigoriadis, D.G.E. et al.)

**Figure 14.**The transverse profiles of particle concentrations for various Stokes numbers, St

_{+}, after a 2D obstacle. The points are LES calculations [10]; the lines are the authors’ predictions.

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**MDPI and ACS Style**

Pakhomov, M.A.; Terekhov, V.I. RANS Modeling of Turbulent Flow and Heat Transfer in a Droplet-Laden Mist Flow through a Ribbed Duct. *Water* **2022**, *14*, 3829.
https://doi.org/10.3390/w14233829

**AMA Style**

Pakhomov MA, Terekhov VI. RANS Modeling of Turbulent Flow and Heat Transfer in a Droplet-Laden Mist Flow through a Ribbed Duct. *Water*. 2022; 14(23):3829.
https://doi.org/10.3390/w14233829

**Chicago/Turabian Style**

Pakhomov, Maksim A., and Viktor I. Terekhov. 2022. "RANS Modeling of Turbulent Flow and Heat Transfer in a Droplet-Laden Mist Flow through a Ribbed Duct" *Water* 14, no. 23: 3829.
https://doi.org/10.3390/w14233829