# The Effect of a Backward-Facing Step on Flow and Heat Transfer in a Polydispersed Upward Bubbly Duct Flow

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}= 1–3 mm and their volumetric gas flow rate ratios of β = 0%–10%. The effect of the gas volumetric flow rate ratios on the flow structure and heat transfer in the two-phase flow is numerically studied in a bubbly polydispersed flow in a horizontal duct with a single-side BFS [13]. The model based on the Eulerian two-fluid approach.

## 2. Measurement Setup

_{1}= 8 mm, and after the sudden expansion were L2 = 400 mm and h

_{2}= 20 mm. The step height H = 12 mm, and the expansion ratio was ER = (H + h

_{1})/h

_{1}= h

_{2}/h

_{1}= 2.5. A honeycomb was mounted on the duct inlet in front of the plate to establish a uniform fluid flow. The test section is fixed on a frame made of machine-tool aluminum profiles, which provides convenient fastening of the stand elements and the optical system and easy readjustment.

_{m}

_{1}= 0.55 m/s and Reynolds number based on step height Re

_{H}= U

_{m}

_{1}H/ν = 6600. Gas bubbles were introduced into the liquid flow through 9 capillaries with an inner diameter of 0.7 mm located in the lower part of the working section.

## 3. Mathematical Model and Method of Numerical Realization

#### 3.1. D RANS + SMC “In-House” Numerical Code

_{l}and Φ

_{b}are the volume fractions of the liquid phase and bubbles, respectively; K is the number of groups of bubbles; P and P

_{in}≈ P

_{b}are the pressures in the liquid phase and on the surface of bubble [1,29], respectively;

**σ**

^{BI}is the influence of the kth group of bubbles on the tensor of averaged Reynolds stresses in the liquid phase [1,27]; τ, $\langle {u}^{\prime}{u}^{\prime}\rangle $ and $\langle u\theta \rangle $ are the tensors of viscous stress and Reynolds stress and turbulent heat flow in the carrier phase, respectively; ${M}_{l}={\displaystyle {{\displaystyle \sum}}_{k=1}^{K}}{M}_{lk}=-{M}_{b}$ interfacial term [23,27]; g is the gravity acceleration; ${g}_{ut,k}$ is the coefficient of involvement of gas bubbles of the kth fraction in the thermal fluctuation movement of the carrier fluid [27]; ${\tau}_{\Theta k}$ is the thermal relaxation time. The right-hand side of the momentum equation includes the pressure gradient in carrier phase, gravity, the viscous stress, Reynolds stresses in bubbly flow, interfacial pressure gradient and momentum exchange between carrier fluid and gas phases that arise from the actions from interfacial forces.

_{K}and S

_{ε}in the right part of the turbulence model take into account the effect of the dispersed phase on transport processes [31].

#### 3.2. D RANS + SST and LES (Ansys CFD Package)

_{+}< 1. The profiles of the velocity, turbulent kinetic energy and the rate of their dissipation are set at the entrance to the duct with a backward-facing step. They are simulated in the upstream section with the length 5H before the section of flow separation. The wall functions are not used, i.e., the predictions are carried out directly to solid surfaces. Boundary conditions at the duct entrance profiles—velocity, turbulent kinetic energy and their dissipation rate—are set after the preliminary simulations of the flow in the upstream section. The temperature of the incoming water flow is constant T

_{1}= 298 K, and the change in the thermophysical properties is neglected in the predictions.

_{+}< 1. The temperature of the incoming water flow was constant T

_{1}= 298 K, and the change in the thermophysical properties was neglected in the calculations. The boundary conditions on the wall behind the step is T

_{W}= const = 313 K. The end wall of the step is not heated.

_{max}is defined as: $e\underset{i=1,N}{max}{\left|{\mathit{Nu}}_{i}^{n}-{\mathit{Nu}}_{i}^{n-1}\right|}^{-6}{}_{max}$, where N is the total number of CVs in corresponding direction, the subscript i is the specific CV and the superscript n is the iteration level. The computational grid is nonuniform both in the streamwise and transverse directions. A more refined grid is applied in the recirculation region and in the zones of flow detachment and reattachment and in the inlet region of the duct. The coordinate transformation is suitable for such a two-dimensional problem:

_{j}= K × Δψ

_{j}

_{-1},

_{j}and Δψ

_{j}

_{-1}are the current and previous steps of the grid in the axial or radial directions and K = 1.05 (longitudinal direction) and K = 1.03 (transverse direction). The first cell is located at a distance y

_{+}= yU

_{*}/ν = 0.3–0.5 from the wall surfaces, where U

_{*}is the friction velocity obtained for the single-phase flow in the inlet of the duct (other parameters being identical). At least 10 CVs were generated to ensure resolution of the mean velocity field and turbulence quantities in the viscosity-affected near-wall region (y

_{+}< 10). The time step was Δt = 0.1 ms. The Courant number, which is a necessary condition for the stability of numerical solution of differential equations in partial derivatives, does not exceed 1 for all simulations.

## 4. Comparison with Experimental and Numerical Results

#### 4.1. Single-Phase Flow in the BFS

_{1}= 298 K. The mean upstream fluid velocity U

_{m}

_{1}= 11 m/s and Reynolds number was Re

_{H}= 2.8 × 10

^{4}. The Stanton number is calculated using the expression: St = h/(ρC

_{p}U

_{m}

_{1}). Here h is the heat transfer coefficient, ρ is the density of the carrier fluid flow and C

_{p}is the specific heat at constant pressure.

_{f,}

_{min}in the separation region for single-phase fluid flow vs. the Reynolds number Re

_{H}based on the step height are given in Figure 4b. The measurements of [37] were performed in the duct with single-side backward-facing step with dimensions: length was 1600 mm, height 70 mm after sudden expansion and width was 220 mm. The step height was H = 20 mm and expansion ratio was ER = 1.4. The carrier fluid was water with temperature T

_{1}= 298 K. The mean upstream fluid velocity U

_{m}

_{1}= 0.024–0.24 m/s and Reynolds number varied Re

_{H}= (0.12—1.22) × 10

^{4}. The wall shear stress was measured using the electrodiffusion method. Line 1 is the semi-empirical correlation [37] for calculating the wall friction ${C}_{f}{}_{,\mathrm{min}}=-0.38R{e}_{H}{}^{-0.57}$, and points (2) represent the author’s predictions. The strong dependency of minimal values of wall friction in the reverse zone on the Reynolds numbers was obtained as well as in experiments [37] and our numerical predictions. Good agreement was obtained between the results [36,37] and the data of authors’ numerical predictions.

#### 4.2. Two-Phase Bubbly Flow in Round Vertical Pipe

#### 4.3. Two-Phase Bubbly Flow in the BFS

_{m}

_{1}= 1.78 m/s, which corresponds to the Reynolds number Re

_{H}= 1.1 × 10

^{5}. The pipe diameter, before the abrupt, was 2R

_{1}= 50 mm, while after expansion, 2R

_{2}= 100 mm, which corresponded to the step height of H = 25 mm, and ER = (R

_{2}/R

_{1})

^{2}= 4 and the inlet bubble Sauter diameter was d

_{1}= 2 mm. The authors results agree well with experiments and LES data. We have conducted the comparison with present measurements for the mean fluid axial velocity profiles and heat transfer enhancement [23]. The maximal difference between our and other authors measured results and our predictions is up to 20%.

#### 4.4. Two-Phase Bubbly Flow behind a Backward-Facing Step. Our Measurements and Numerical Simulation and Their Discussion

_{H}= 6600 in the single-phase fluid flow (lines 1) and at gas volumetric flow rate ratio β = 3% (lines 2) along the duct length are shown in Figure 6. The carrier fluid velocity was normalized to the maximum value of liquid velocity. The point of flow reattachment in the single-phase fluid flow is located at a distance of (7–8)H, which is in good agreement with the literature data [41,42]. The following characteristic features of the bubbly flow can be distinguished. Immediately behind the step, the flow recirculation zone is observed. The gas bubbles badly penetrate this region and the effect of gas bubbles on the flow is rather weak in this area. The position of the flow reattachment point shifts towards the step for the two-phase flow and, in general, the establishment of a profile in the channel after the expansion occurs faster in a two-phase flow than in a single-phase flow, which was previously reported, for example, in [23]. Our numerical results capture the main features of the bubbly flow behind the BFS described above. The main difference between the experimental and numerical results for the two-phase bubbly is obtained in the near-wall zone in the flow relaxation region.

## 5. Numerical Results and Its Discussion

#### 5.1. Single-Phase Fluid Flow in a Backward-Facing Step Flow

#### 5.2. Upward Bubbly Flow in a Backward-Facing Step Flow

_{m}

_{1}H/ν = 6600, and the mean-mass velocity at the inlet U

_{m}

_{1}= 0.55 m/s. The height of the duct before separation h

_{1}= 8 mm, height of the duct after separation h

_{2}= 20 mm, and the step height H = 12 mm, the expansion ratio ER = (H + h

_{1})/h

_{1}= h

_{2}/h

_{1}= 2.5, the wall temperature T

_{W}= const = 313 K and the initial temperatures of carrier liquid and gas bubbles T

_{1}= T

_{b}

_{1}= 293 K. In the inlet section, the mean phase velocities have the same value. The initial gas volumetric flow rate ratios are varied β = 0–10% and the mean initial Sauter diameter of air bubbles d

_{1}= 3 mm. All simulations are performed for four monodispersed δ-functions (modes) of air bubbles at the inlet cross-section: 1—d/d

_{1}= 0.33, 2—0.66, 3—1, and 4—1.33. The volume fraction of each fraction is Φ

_{1}(1 mm) = 0.031Φ, Φ

_{2}(2 mm) = 0.12Φ, Φ

_{3}(3 mm) = 0.6Φ, and Φ

_{4}(4 mm) = 0.15Φ, where Φ is the total volume fraction of gas bubbles. Authors did not carry out the measurements of the distributions of the diameters of gas bubbles at the inlet cross-section. Therefore, the choice of just such values of gas bubbles diameters and their volume fractions is based on our preliminary numerical predictions. There is no steam generation on wall surface. The same assumptions were used in our previous recent numerical simulations [12,13,29], but they may be important in other thermal boundary conditions on the wall [43].

_{1}≤ 0.33 (line 1) are found over the entire section of the channel, both in the flow recirculation zone and in the flow core. The smallest air bubbles can come closer to the duct wall than the larger bubbles due to the effect of transverse forces (lift, turbulent migration, turbulent diffusion and wall force). The largest bubbles d/d

_{1}≥ 1 (line 5) are located mainly in the near-wall part of the channel, which confirms the data of our numerical predictions presented in Figure 11a,b. Bubbles of the other two monodispersed fractions (d/d

_{1}= 0.33–0.66, 0.66–1, lines 2 and 3) also practically do not penetrate the flow separation region.

_{1}= 3 mm). The additional production of carrier fluid phase turbulence is explained by vortex formation upon streamlining of the gas bubbles by the carrier fluid flow. The profiles of TKE at a small value of β agree qualitatively with those for the case of one-phase flow in a duct with single-side BFS. The same tendencies were obtained in our previous works [12,23] for a pipe with sudden expansion.

_{W}is the gradient of the fluid temperature on the wall, T

_{W}is the wall temperature and T

_{m}is the mean temperature of the carrier fluid (water) in this section. Lines 1 are Nusslet number in the fully developed single-phase duct flow and dashed curves 2 are the predictions of heat transfer in the single-phase duct flow with sudden expansion for the fluid (water) flow with other conditions being identical.

_{1}= 3 mm in comparison with the single-phase fluid flow) with an increase in the volumetric flow rate ratio β is observed (see Figure 14a). This is explained by an increase in the velocity and temperature gradients and turbulization of the carrier fluid (water) phase in the near-wall region of the duct. The position of the peak of heat transfer shifts upstream and at β = 5% is x

_{Nu_max}/H = 4.5, and the length of the recirculation zone approximately coincides with it, x

_{R}/H = 4.8 at β = 5%. The same values for the single-phase fluid flow are x

_{Nu_max}/H = 6.8 and x

_{R}/H = 7. The position of the peak of heat transfer rate x

_{Nu_max}is close to the position of the flow reattachment point x

_{R}in two-phase bubbly flow. The same tendency was observed in [41,42] for single-phase fluid flows in a BFS. The effect of bubble diameter in the inlet on the Nusselt number distributions along the duct length is given in Figure 14b. The largest increase in heat transfer (approximately 35% at fixed value β = 5%) is characteristic for the largest gas bubbles with the initial diameter d

_{1}= 3 mm.

_{f},

_{min}/C

_{f},

_{min,0}, maximal heat transfer enhancement ratios Nu

_{max}/Nu

_{max,0}, recirculation length x

_{R}, maximal values of turbulence modification ratios k

_{max}and position of heat transfer maximum x

_{max}are shown in Figure 15. All these variables are normalized on the value in the single-phase fluid flow and subscript “0” is the in the single-phase flow parameter with other conditions are identical to the two-phase bubbly flow.

_{f},

_{min}/C

_{f},

_{min,0}≈ 0.75. The reduction of minimal value of wall friction ratio is small (up to 10%) for the β ≤ 5%. Almost linear convective heat transfer enhancement is observed for the range studied of gas volumetric flow rate ratios and for β = 10% the heat transfer augmentation ratio is Nu

_{max}/Nu

_{max,0}≈ 1.75. The addition of gas bubbles leads to the significant shortening of the recirculation length ratio of bubbly flow. The shortening of the recirculating zone is almost twice in comparison with the case of the single-phase fluid flow. The main reason is the flow turbulization by flowing around the gas bubbles. The maximal value of turbulent kinetic energy modification is up to 50% at β = 10%. It causes the intensification of mixing process. The same trends were obtained for the single-phase separated flows [41,42]. It is well known [41,42] that the position of maximal value of heat transfer is close to the flow reattachment point for the single-phase for both gas (air) and fluid (liquid) flows behind a BFS and a sudden pipe expansion. The addition of relatively large air bubbles shifted the position of maximal value of heat transfer far from the reattachment point for the bubbly flow in the pipe with sudden expansion [12]. In the case of bubbly flow in the backwards-facing step we observe the same trends. The position of the heat transfer maximum is located after the reattachment point.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${C}_{f}=2{\tau}_{W}/{U}_{1}^{2}$ | wall friction coefficient |

C_{P} | heat capacity |

d | mean bubble Sauter diameter |

h_{1} | height of the duct before the sudden expansion |

h_{2} | height of the duct after the sudden expansion |

H | step height |

J | and J_{b} superficial velocity of carrier fluid (water) and gas bubbles respectively |

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

L | duct length |

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

Re_{H} = U_{m}_{1}H/ν | the Reynolds number |

St = h/(ρC_{p}U _{m}_{1}) | Stanton number |

T | temperature |

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

U_{*} | friction velocity |

x | streamwise coordinate |

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

x_{Nu_max} | position of the peak of heat transfer rate |

y | distance normal from the wall |

Subscripts | |

0 | single-phase fluid (water) flow |

1 | initial condition |

W | wall |

b | bubble |

l | liquid |

m | mean-mass |

Greek | |

Φ | volume fraction |

α | void fraction |

β | gas volumetric flow rate ratio |

ε | dissipation of the turbulent kinetic energy |

λ | thermal conductivity |

ρ | density |

ν | kinematic viscosity |

τ_{W} | wall shear stress |

Acronym | |

BFS | backward-facing step |

CV | control volume |

DNS | direct numerical simulation |

ER | expansion ratio |

LES | large eddy simulation |

PIV | particle image velocimetry |

PLIF | planar laser induced fluorescence |

RANS | Reynolds-averaged Navier-Stokes |

ROI | region of interests |

SMC | second moment closure |

SST | shear stress tensor |

TKE | turbulent kinetic energy |

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**Figure 1.**The scheme of the experimental setup (

**a**): 1 is the water tank, 2 is the pump, 3 is the flowmeter, 4 is the test section, 5 is the PIV/PLIF system. (

**b**) The scheme of a two-phase bubbly upward flow behind a backward-facing step.

**Figure 2.**Single frames of the flow at Re = 6600 and β = 0.03: (

**a**) “small” bubbles; (

**b**) “medium” bubbles; (

**c**) “large” bubbles; (

**d**) “small” bubbles without Rhodamine in the test fluid.

**Figure 3.**Grid convergence test of 2D RANS + SMC “in-house” code for the bubbly flow at T

_{W}= 313 K, T

_{1}= T

_{b}

_{1}= 293 K, β = 5%, d

_{1}= 3 mm (

**a**), single-phase fluid flow 3D RANS+SST (

**b**) and LES+WALE (

**c**) in a duct with sudden expansion. Re

_{H}= 6600, ER = 2.5.

**Figure 4.**Heat transfer coefficient distributions along the streamwise coordinate (

**a**) and minimal wall friction values in the reverse zone vs. Reynolds number (

**b**). (

**a**): Symbols are the experiments of [36], line is our RANS + SMC “in-house” code simulations. (

**b**): 1 is the semi-empirical correlation [37], 2 is our RANS+SMC “in-house” code simulations.

**Figure 5.**Radial profiles of the carrier fluid and gas bubbles (

**a**) and mean Sauter gas bubbles diameters (

**b**). β = 4.6%, Re = U

_{m}

_{1}2R/ν = 4.9 × 10

^{4}, x/(2R) = 53.5, 2R = 50.8 mm, U

_{m}

_{1}= 0.986 m/s, d

_{1}= 2.4 mm. Symbols are the measurements of [38], curves are the authors predictions. (

**a**): 1, 2 are the carrier fluid axial velocities in the single-phase and two-phase bubbly flows respectively; 3 is the gas bubbles axial velocity. (

**b**): 1—J

_{b}= 0.0473 m/s, β = 4.6%, d

_{1}= 2.4 mm; 2—0.113 m/s, 10.3%, 2.5 mm; 3—0.242 m/s, 19.7%, 2.8 mm.

**Figure 6.**The transverse profiles of the carrier phase mean streamwise velocity along the duct length. Symbols and curves are the authors’ measurements and predictions respectively. Re

_{H}= 6600, ER = 2.5, β = 3%. 1—single-phase flow (β = 0), 2—β = 3%. (

**a**)—x/H = 2, (

**b**)—4, (

**c**)—6, (

**d**)—8, (

**e**)—10.

**Figure 8.**The 3D RANS + SST predicted contours of the longitudinal velocity component and streamlines (

**a**) and turbulent kinetic energy (

**b**) at the z = 10, 50, 100, 190 mm (z/H = 0.8, 4.2, 8.3 and 15.8).

**Figure 9.**The transverse profiles of single-phase mean longitudinal velocity behind a backward-facing step. Symbols and curves are the authors’ measurements and predictions, respectively. Re

_{H}= 6600, ER = 2.5.

**Figure 10.**The transverse profiles of total local void fraction (

**a**) and bubble diameter (

**b**) along the duct length in the upward bubbly flow behind the backward-facing step. Re

_{H}= 6600, ER = 2.5, β = 3%. (

**a**): 1—x/H = 0, 2—2, 3—4, 4—6, 5—10; (

**b**): 1—x/H = 2, 2—4, 3—6, 4—10.

**Figure 11.**The transverse profiles of local void fractions (

**a**) and bubble diameter (

**b**) for various bubble size modes. Re

_{H}= 6600, ER = 2.5, β = 3%, x/H = 4. (

**a**): 1—d/d

_{1}= 0–0.33, 2—0.33–0.66, 3—0.66–1, 4—>1, 5—sum of all modes; (

**b**): 1—d = 0–0.33, 2—0.33–0.66, 3—0.66–1, 4—>1.

**Figure 12.**Profiles of mean axial fluid velocity in bubbly flow in the backward-facing step. 1—β = 0 (single-phase fluid flow, other conditions being identical), 2—2%, 3—5%.

**Figure 13.**The effect of the gas volumetric flow rate ratios (

**a**) and gas bubbles diameter at the inlet (

**b**) on the carrier fluid phase turbulence.

**Figure 14.**Heat transfer in the bubbly flow behind the backward-facing step along the streamwise coordinate. (

**a**): d

_{1}= 3 mm, 1—Nusslet number value in the fully developed single-phase duct flow, 2—β = 0 (single-phase fluid flow, other conditions being identical), 3—2%, 4—5%; (

**b**): β = 5%, 1—Nusslet number value in the fully developed single-phase duct flow, 2—d

_{1}= 0 (single-phase fluid flow, other conditions being identical), 3—0.5 mm, 4—2 mm, 5—3 mm.

**Figure 15.**The magnitudes of minimal wall friction coefficient (1), maximal heat transfer enhancement ratios (2), recirculation length (3), maximal values of turbulence modification ratios (4) and position of heat transfer maximum (5) depending on gas volumetric flow rate ratios β. Re

_{H}= 6600, ER = 2.5.

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## Share and Cite

**MDPI and ACS Style**

Bogatko, T.V.; Chinak, A.V.; Evdokimenko, I.A.; Kulikov, D.V.; Lobanov, P.D.; Pakhomov, M.A. The Effect of a Backward-Facing Step on Flow and Heat Transfer in a Polydispersed Upward Bubbly Duct Flow. *Water* **2021**, *13*, 2318.
https://doi.org/10.3390/w13172318

**AMA Style**

Bogatko TV, Chinak AV, Evdokimenko IA, Kulikov DV, Lobanov PD, Pakhomov MA. The Effect of a Backward-Facing Step on Flow and Heat Transfer in a Polydispersed Upward Bubbly Duct Flow. *Water*. 2021; 13(17):2318.
https://doi.org/10.3390/w13172318

**Chicago/Turabian Style**

Bogatko, Tatiana V., Aleksandr V. Chinak, Ilia A. Evdokimenko, Dmitriy V. Kulikov, Pavel D. Lobanov, and Maksim A. Pakhomov. 2021. "The Effect of a Backward-Facing Step on Flow and Heat Transfer in a Polydispersed Upward Bubbly Duct Flow" *Water* 13, no. 17: 2318.
https://doi.org/10.3390/w13172318