The simulations are performed for the droplet-laden flow at atmospheric pressure. The duct height before sudden expansion is

h_{1} = 20 mm; after expansion,

h_{2} = 40 mm, the step height

H = 20 mm and the expansion ratio ER = (

h_{2}/

h_{1}) = 2 (see

Figure 1). The mean-mass gas velocity at the inlet is

U_{m}_{1} = 10 m/s, and the Reynolds number Re

_{H}=

HU_{m}_{1}/

ν ≈ 1.33 × 10

^{4}. We add the droplets to the hydrodynamically fully developed single-phase air flow in the inlet cross-section (the section of sudden expansion) and keep the initial droplet velocity constant across the duct height:

U_{L}_{1} = 0.8

U_{m}_{1}. The initial size of droplets in our studies is

d_{1} = 1–100 μm, and the mass concentration of droplets is

M_{L}_{1} = 0–0.1. The vapor mass fraction at the inlet is

M_{V}_{1} = 0.005. The temperature of the air and droplets at the inlet is

T_{1} =

T_{L}_{1} = 293 K, and the wall temperature is

T_{W} = const = 373 K. The mean Stokes number is Stk =

τ/

τ_{f} = 0.03–2.9,

τ_{f} = 5

H/

U_{1} = 0.01 s, and the Stokes number is

${\mathrm{Stk}}_{K}=\tau /{\tau}_{K}$ = 0.2–19.

#### 4.1. Flow Structure, Wall Friction and Turbulence Quantities

The transverse profiles of the mean longitudinal velocities of gas and dispersed phases at different distances from the flow separation point are shown in

Figure 2a. The flow reattachment point is located at

x_{R}/H ≈ 5.8 for a single-phase air flow and

x_{R}/H ≈ 5.83 at

M_{L}_{1} = 0.05, where

x_{R} is the length of the recirculating region. The first three cross-sections are located in the recirculation zone, the fourth cross-section corresponds to the flow reattachment area and the last two correspond to the droplet-laden flow relaxation area. A sharp change in the flow structure is observed downstream of the separation cross-section. The profiles of the streamwise velocities of phases in a two-phase flow correspond to those for a single-phase flow. The modification of the mean flow velocity with such a small addition of the dispersed phase is not observed. This qualitatively agrees with other conclusions for both gas-droplet [

17,

22] and gas-dispersed [

6,

7,

9,

10,

11,

12] turbulent separated flows. At a large distance from the point of flow reattachment, the two-phase flow takes the form of a fully developed flow in the duct. In the first cross-sections, the gas velocity is higher than the corresponding value for the dispersed phase; this is explained by the initial conditions for the addition of the droplets to the gas phase and their acceleration in the downward direction. Further, the droplet velocity is almost identical to the gas velocity.

The wall friction coefficient (

${C}_{f}=2{\tau}_{W}/{U}_{1}^{2}$) distributions along the streamwise coordinate

$\left(x-{x}_{R}\right)/{x}_{R}=x/{x}_{R}-1$ are presented in

Figure 2b, where

τ_{W} is the shear wall friction and

x_{R} is the recirculation length. The line

1 represents the simulations for the single-phase flow without droplets and with other identical conditions. The addition of a dispersed phase to the turbulent separated single-phase flow has no significant effect on the value of

C_{f} in the flow separation and relaxation (after the reattachment point) regions (see

Figure 2b). We can see a slight increase in the wall friction coefficient in a droplet-laden flow.

The distribution of the minimum value of the wall friction coefficient in the separation region for single-phase (

2) and gas-droplet flows at

M_{L}_{1} = 0.05 (

3) vs. the Reynolds number Re

_{H} is given in

Figure 3. Line

1 is the semiempirical correlation [

37] for calculating the wall friction

Points (2 and 3) represent the author’s predictions. The effect of droplets on the wall friction in the studied range of the mass fraction is small, up to 5%.

Figure 4a presents the transverse distributions of the droplet mass fraction at

x/

H = 2. The predictions are performed for three different mean Stokes numbers, Stk = 0.03, 0.28 and 2.9. The first two values of mean Stokes numbers (

1 and

2) correspond to the flow regime with droplet entrainment into the mean turbulent motion, and they are observed in the whole cross-section of the duct (flow core, shear layer and recirculation region). The third Stokes number (3) represents the flow regime, when the droplets are not involved into the mean motion, and they almost do not penetrate into the recirculation zone [

6]. These conclusions are confirmed by our numerical simulations. The thin layer close to the duct wall for the small particles is free from droplets due to their evaporation. The height of this zone depends on the droplet diameter, and this height is largest for the smallest particles studied (

1). The evaporation of droplets in the flow core is insignificant, and the predicted profiles of mass concentrations for all droplet diameters have almost constant values. The slight increase in the profiles of the droplet mass fraction toward the “upper” duct wall is computed. This explains the process of droplet accumulation in the near-wall region and their deposition on the vertical wall surface. The same tendency was numerically obtained in [

12] for the gas-dispersed flow behind a BFS.

The droplet mass concentration decreases drastically toward the duct wall. The transverse profiles of vapor mass fractions show the opposite tendency for the initial droplet diameter studied (see

Figure 4b). Here,

${M}_{V}/{M}_{V1}=\left({M}_{V1}+\Delta {M}_{V}^{evap}\right)/{M}_{V1}=1+\Delta {M}_{V}^{evap}/{M}_{V1}$, where

$\Delta {M}_{V}^{evap}$ is the additional mass of steam from the droplet surface created by vaporization. The values of the steam mass concentration are varied from unity (without evaporation) up to 11 at

M_{L}_{1} = 0.05 (the maximal possible steam magnitude and

${M}_{L1}\equiv \Delta {M}_{V}^{evap}$). The maximum values of the steam mass fraction are found close to the wall of the duct, and the smallest values occur in the duct flow core.

The distributions of the dimensionless diameter of the droplets with a change in their initial size are shown in

Figure 5 at a distance

x/

H = 2 downward from the cross-section of the sudden expansion of the flow. A region free of both the smallest (

1) and the largest (

3) particles can be seen close to the wall. In the first case, the reason for their absence close to the wall of the duct is the process of their evaporation. For the second case, this is explained by their absence in the entire flow recirculation region due to the large mean Stokes number; such particles are observed only in the shear layer and in the central and “upper” parts of the duct. Droplets of intermediate diameter (

2) are observed throughout almost the whole cross-section of the duct. Obviously, in this zone, the droplets’ diameters have the smallest values due to the process of their evaporation, whereas in the flow core, their size almost does not differ from the initial value. This confirms the conclusions in

Figure 4a.

A modification of the gas-phase TKE vs. the mean Stokes numbers at distance

x/

H = 2 is presented in

Figure 6. Here,

k_{0,max} is the maximum level of turbulence of the gas phase in a single-phase air flow. The level of gas turbulence in a two-dimensional flow is estimated using the following expression:

The maximum value of the turbulent kinetic energy is predicted in the shear mixing layer in both single- and two-phase flows behind the backward-facing step. The level of turbulence in a two-phase flow attenuates due to addition of tiny droplets in comparison with the single-phase flow. An increase in the droplet mass concentration leads to a significant decrease in the level of the carrier-phase turbulence. This correlates with the previously mentioned experimental [

7] and numerical [

9,

22,

23] works for separated flows with solid particles and liquid droplets.

The distributions of the turbulence modification ratio (TMR)

k/

k_{0} depending on the average Stokes number are shown in

Figure 7, where

k_{0} is the turbulence level in the single-phase flow. The carrier-phase turbulence is calculated using relation (11). It is shown that the smallest suppression of gas turbulence is obtained near the wall at the distance

y/

H = 0.1 (

1), where the diameters of droplets are the smallest due to their evaporation (see

Figure 4a and

Figure 5). The largest value is predicted at the distance

y/

H = 1 (

3). There is almost no evaporation in this region, and the diameters of droplets are maximal. They are roughly equal to their initial size. The different mechanisms of the influence of the average Stokes number on turbulence modification for the cross-sections operate in the recirculation zone (

1 and

2) and after flow reattachment (

3). There is a sharp bend in the TMR distribution at Stk ≈ 1 for lines (

1 and

2). This is caused by the fact that particles at Stk > 1 scarcely penetrate into the separation region, and they are found only in the shear layer and the duct core. Without a dispersed phase, an increase in turbulence was obtained in these two cross-sections for the level

k/

k_{0} → 1 in a single-phase flow. In the cross-section

y/

H = 1 (

3), a decrease in the turbulence level of the gas phase is obtained with growth in the mean Stokes number (initial droplet diameter). For the investigated range of the initial droplet diameters, this cross-section is characterized only by turbulence level suppression, while for more inertial particles, an additional generation of the TKE level can also be obtained.

The transverse distributions of the temperatures of the gas,

$\mathsf{\Theta}=\left({T}_{W}-T\right)/\left({T}_{W}-{T}_{m}\right)$ (lines

1 and

2), and droplets,

${\Theta}_{L}=\left({T}_{L,\mathrm{max}}-{T}_{L}\right)/\left({T}_{L,\mathrm{max}}-{T}_{L,m}\right)$ (line

3), are presented in

Figure 8. Here,

T is the temperature and

T_{L}_{, max} is the maximal temperature of the droplets in the corresponding cross-section. The subscripts “

W”, “

m” and “

L” correspond to the wall, mean and droplets terms, respectively, and

T_{m} and

${T}_{L,m}=\frac{2}{{U}_{1}{h}_{2}}{\displaystyle \underset{0}{\overset{{h}_{2}}{\int}}{T}_{L}Udy}$ are the mean-mass gas and droplet temperatures in the corresponding cross-section, respectively. The normalized temperature Θ

_{L} is based on the maximal value of the droplet temperature

T_{L}_{,max}. It is obvious that the minimal value of the droplet temperature is predicted in the turbulent flow core, and the maximal value of the droplet temperature is determined close to the wall. The gas temperature in the mist flow (

2) is lower than that in the single-phase flow (

1) due to the vaporization of droplets. The droplet temperature in the flow core is slightly lower than at the inlet due to droplet cooling.

#### 4.2. Heat Transfer

The distributions of local Nusselt numbers along the streamwise coordinate are shown in

Figure 9. The Nusselt number at

T_{W} = const is obtained by the following formula:

where

${T}_{m}=\frac{2}{{U}_{1}{h}_{2}}{\displaystyle \underset{0}{\overset{{h}_{2}}{\int}}TUdy}$ is the mean-mass gas temperature. The considerable heat transfer augmentation (more than twofold) in the two-phase mist flow as compared to the single-phase separated flow is obtained. Heat transfer augmentation is predicted in the flow recirculation and relaxation zones as well for Stk ≤ 1. The heat transfer coefficient is maximal in the flow reattachment zone. The coordinate of the maximal value of heat transfer in a gas-droplet flow corresponds approximately to the flow reattachment point. It should be noted that a similar conclusion is also typical of a single-phase flow [

3,

4]. The heat transfer reduction is similar to that occurring in a single-phase flow due to the growth in thickness of the dynamic and thermal boundary layers at

$\left(x-{x}_{R}\right)/{x}_{R}$ > 1, where

x_{R} is the length of flow recirculation. The value Nu

_{fd} =

h_{2}U_{m}_{2}/ν ≈ 38 (

5) is the magnitude of the heat transfer in a fully developed single-phase air flow.

An increase in the inlet size of droplets has a complicated effect on the heat transfer. The finest droplets (

d_{1} = 10 μm, Stk = 0.03,

2) evaporate faster and closer to the flow detachment section (see

Figure 9b). The latter conclusion confirms the results in

Figure 7 showing that droplets are entrained into the separated motion of the gas flow, and they scatter across the recirculating region. The largest droplets at

d_{1} = 100 μm and Stk = 2.9 (

4) are badly entrained into the mean motion of the gas phase. The heat transfer enhancement is revealed mainly after the reattachment point due to droplet evaporation. In the recirculation zone, the values of heat transfer are similar to those in the single-phase separated flow. It should be noted that the maximal value of the Nusselt number in the two-phase flow (

d_{1} = 100 μm, Stk = 2.9) is smaller than that determined for finer drops (

d_{1} = 30 μm, Stk = 0.28).

The influence of the average Stokes number on the maximum local value (

Figure 10a) and average (

Figure 10b) heat transfer is shown in

Figure 10. There are two distinctive regions in the Nu

_{max} distribution within the whole range of changes in the mass fraction of droplets studied in this work. In the region of small droplet diameters at the inlet (Stk ≈ 0.2,

d_{1} ≈ 20 μm), a slight increase in heat transfer is observed, and for large droplets, a sharp reduction is observed. An intensive evaporation of small droplets leads to a decrease in the rate of their deposition. The large droplets almost are not present in the recirculation region. The increase in the droplet mass fraction at the inlet leads to a significant heat transfer enhancement in the two-phase mist flow as compared to a single-phase flow. The largest augmentation of heat transfer in the gas-droplet flow is observed for small droplets.

The influence of the inlet droplet diameter on the average Nusselt number is presented in

Figure 10b. The average Nusselt number is calculated as

$\overline{\mathrm{Nu}}=\left({\displaystyle \underset{0}{\overset{X}{\int}}\mathrm{Nu}dx}\right)/X$, where

X = 25

H is the length of the averaging zone, and Nu is the local Nusselt number. The magnitude of the average heat transfer for the single-phase flow is

${\overline{\mathrm{Nu}}}_{0}$ ≈ 47. The increase in the droplet mass fraction causes the substantial augmentation of the average Nusselt number (it more than twofold for

M_{L}_{1} = 0.1 as compared to the single-phase air flow).