# Do the Volume-of-Fluid and the Two-Phase Euler Compete for Modeling a Spillway Aerator?

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}toolbox. As expected, the TPVoF results depend highly on the mesh, not showing convergence in the maximum chute bottom pressure and the lower-nappe aeration, tending to null aeration as resolution increases. The CTPE combined with the k–$\omega $ SST Sato turbulence model exhibits the most accurate results and mesh convergence in the lower-nappe aeration. Surprisingly, intermediate mesh resolutions are sufficient to surpass the TPVoF performance with reasonable calculation efforts. Moreover, compressibility, flow bulking, and several entrained air effects in the flow are comprehended. Despite not reproducing all aspects of the flow with acceptable accuracy, the complete two-phase Euler demonstrated an efficient cost-benefit performance and high value in spillway aerated flows. Nonetheless, further developments are expected to enhance the efficiency and stability of this model.

## 1. Introduction

## 2. Methodology

#### 2.1. Laboratory Data

_{s}) varies from $0.025$ m to $0.045$ m. The upstream emergence angle (θ

_{0}) inclination is variable from 0° to $14.1$° and the channel bottom angle (θ

_{b}) ranges from $5.7$° to $14.1$°. The channel bed and side-walls are made of smooth polyethylene with a roughness height of $1\times {10}^{-5}$ m.

_{0}) of 9 m s${}^{-1}$, water depth (h

_{0}) of $0.15$ m, equal emergence and bottom angle of $14.1$° and offset height of $0.045$ m are selected to mitigate the scale effects. Froude number (Fr) is $7.4$, $Re=1.2\times {10}^{6}$ and ${We}^{0.5}=405$.

#### 2.2. Mathematical Models

^{®}toolbox version 2012 [46]. The volume-of-fluid solver is named interFoam. The complete two-phase Euler solver is named twoPhaseEulerFoam.

#### 2.2.1. Two-Phase Volume-of-Fluid

_{t}is the eddy viscosity coefficient (i.e., turbulent dynamic viscosity) and S is the strain rate tensor. ${\overrightarrow{F}}_{\sigma}$ is the surface tension force term for the momentum Equation (3). σ is the surface tension, $\overrightarrow{\kappa}$ is the curvature of the interface and α is the water volume fraction.

^{®}, the fluids volume fraction in each cell is defined by a scalar function (α) ranging from 0 to 1 that allows the interface tracking: $\alpha =1$ is a water cell and $\alpha =0$ is an air cell. Other α values identify interface cells. The phase advection Equation (5) comprehends an artificial compression meant to preserve a sharp interface (third term).

_{α}coefficient that usually ranges from 0 to 1.

#### 2.2.2. Complete Two-Phase Euler

_{D,d}is the dispersed phase turbulent dispersion term and T

_{D,c}is the continuous phase turbulent dispersion term.

_{d}is the drag coeficient and C

_{υm}is the virtual mass coeficient. R

^{eff}is the stress rate tensor (10), υ

_{eff}= υ + υ

_{t}is the effective viscosity and k is the turbulent kinetic energy.

^{eff}is the effective thermal diffusivity, k

_{h}is the convective heat transfer coefficient, ${k}_{h}^{eff}$ is the effective volumetric convective heat transfer coefficient and C

_{pv}is the specific heat capacity. The IF superscript refers to an interfacial property.

_{D}is the non-dimensional drag coefficient and σ

_{α}is the turbulent Prandtl number for interfacial area density. The turbulent dispersion force is a function of the blending interfacial model chosen and its settings. The solver calculates the turbulent dispersion at each cell for one of the previous three blending model scenarios. Hence, the turbulence dispersion action depends totally on the blending model’s chosen parameters, especially on the maximum volume-fraction value to a phase be considered as dispersed. For example, the user may define this value as 0.6, i.e., if the volume-fraction values of a phase range from 0 to 0.6, that phase is considered dispersed. Higher values, consider the phase continuous or with no obvious dispersed phase, depending on the model applied.

#### 2.2.3. Turbulence Models

_{t}), is calculated the kinematic eddy viscosity (15) [54,55]. The default coefficients are employed.

_{k1}= υ + υ

_{t}/σ

_{k}and D

_{ϵ}= υ + υ

_{t}/σ

_{ϵ}are the effective viscosity, υ is the kinematic viscosity, G

_{k}is the k production rate, S

_{k}and S

_{ϵ}are source terms. σ

_{k}= 1.0, σ

_{ϵ}= 1.3, C

_{1}=1.44, C

_{3,RDT}= 0 and C

_{2}= 1.92 and C

_{μ}= 0.09 are the default coefficients.

_{k2}= υ + a

_{k}υ

_{t}and D

_{ω}= υ + a

_{ω}υ

_{t}are the effective viscosity, P

_{k}and P

_{ω}are production terms. S

_{k}and S

_{ω}are source terms. β* = 0.09, a

_{1}= 0.31, b

_{1}= 1.0, c

_{1}= 10.0, F

_{23}are coefficients. a

_{k}, a

_{ω}, a

_{ω}

_{2}, β, and γ blend inner and outer coefficient values using the F

_{1}coefficient.

_{t,w}and υ

_{t,a}are water and air tubulent kinematic viscosity. G

_{m}is the k

_{m}production rate, ${\overrightarrow{V}}_{m}$ is the mixture velocity vector, G

_{b}is the k

_{m}production rate by air-bubble, ρ

_{w}is the water density, ρ

_{a}is the air density and ρ

_{m}is the mixture density (22). C

_{vm}is the virtual mass coeficient. C

_{3}$=1.92$. The remaining constants share same values with the k–$\u03f5$ model. Furthermore, the k–$\u03f5$ mixture model is improved following Weller et al. [64]. Thus, a phase fraction limiter is applied to the bubble-generated turbulence (G

_{b}), avoiding spurious turbulence generation where bubbles are not present. In the current work, G

_{b}is only activated if ${\alpha}_{d}>0.3$, a standard value.

^{+}is the dimensionless distance from the wall, c

_{b}$=0.6$ and d

_{b}is the characteristic bubble size. Hereinafter, this model is mentioned as k–$\omega $ SST Sato.

#### 2.3. Numerical Setup

_{m}) relative to the nozzle height (${h}_{0}$) is considered (Table 1). The cell edge length (d

_{m}) ranges from $2.5$ mm (${R}_{m}=60$) to 15 mm (${R}_{m}=10$). Mesh resolution is limited to 60 cells due to the extremely small time-step needed to verify the Courant–Friedrichs–Lewy condition ($\Delta t<{d}_{m}/{V}_{0}=2.5\times {10}^{-3}/9=2.8\times {10}^{-4}$ s) that leads to impractical calculation times, even in high-performance computing clusters (HPC). To calculate in parallel at the HPC, the mesh was decomposed into 16 to 256 sub-domains, using the Scotch method.

_{0}) is 9 m s${}^{-1}$; the pressure is automatically calculated to assure the flux. The characteristic turbulent mixing length is $0.0105$ m ($0.07{h}_{0}$). At the top, outlet and air vents, a total pressure condition is defined. At the outlet, the velocity condition is zero gradient for outflow, and the inflow is blocked. Both top and air vents have a binary velocity condition: zero gradient for outflow and exclusive pressure-driven normal air inflow. Turbulence for the previous three boundaries is zero gradient for outflow and for inflow $k=3.1\times {10}^{-5}$ m${}^{2}$ s${}^{-2}$, $\u03f5=1.6\times {10}^{-6}$ m${}^{2}$ s${}^{-3}$ and $\omega =0.58$ s${}^{-1}$, considering a turbulent mixing length equal to the channel width (0.25 m) and an intermediate turbulent intensity of 0.05. Walls have no-slip tangent velocity and a turbulent wall function for the roughness of $1\times {10}^{-5}$ m, as described in the physical modeling. Due to the absence of specification of the inflows’ turbulence intensity (TI) in the physical modeling, two values were tested: $0.005$ and $0.2$. In the complete two-phase Euler, the temperature of the fluids is intended to be 300 K ($26.85$ °C). Hence, the initial temperature of the domain and the inflow temperature are set to 300 K, and the walls have a zero-gradient condition.

_{α}) of $0.5$. For the CTPE model, among the available closures, the following models were used: Schiller and Naumaan [66] for drag, Ranz and Marshall [67] for heat transfer, Tomiyama et al. [68] for lift and the turbulent dispersion method based on Burns et al. [52] and Otromke [53]. This study applies a hyperbolic blending function that considers the fluid continuous if the phase volume-fraction value is larger than 0.6.

^{b}

_{max}), open-boundaries air and water flow-rate, and several other domain properties: water and air volume, total kinetic energy, total turbulent kinetic energy and minimum, and maximum pressure. The presented profile plots and bottom data are collected in the central plane along the chute, i.e., the mid-plane of the chute.

## 3. Results

#### 3.1. Air-Water Mixture

#### 3.2. Lower-Nappe Aeration

_{al}) and water flow-rate at the inlet (Q

_{w}), see Equation (24). Q

_{al}is measured by the air flow-rate entering the cavity zone through both lateral air-vents (Figure 2).

#### 3.3. Pressure Increment at the Spillway Bottom

^{b}) are presented in Figure 7. This analysis is focused on two properties: the value of the maximum pressure increment (Δp

^{b}

_{max}) and its location (L

_{m}).

^{b}

_{max}, see Figure 8), both the TPVoF and CTPE models yield an increase of Δp

^{b}

_{max}as the mesh resolution increases, which is even more noticed in the k–$\omega $ SST type models. Roughly, for the model combinations, the Δp

^{b}

_{max}is only near to the reference value (Δp

^{b}

_{max}

^{ref}$=6.7\times {10}^{3}$ Pa in [11]) for the higher mesh resolutions (${R}_{m}\ge 40$). Although the CTPE with k–$\u03f5$ mixture shows simillar Δp

^{b}

_{max}for the lower mesh resolutions (${R}_{m}=\{10,\phantom{\rule{3.33333pt}{0ex}}15\}$), at higher resolutions Δp

^{b}

_{max}suffers an abrupt drop. For the different model combinations and the tested mesh resolutions, it is impossible to foresee a convergence of the Δp

^{b}

_{max}.

_{0}/1000 to respect the k–$\u03f5$ y+ condition and even more to respect the k–$\omega $ SST y+ condition. Therefore, any significant resolution increase is incompatible with practical engineering, even refining only near the walls. Nevertheless, the authors consider it is important to assess the turbulence models that are undoubtedly the most used with these solvers, in mesh resolutions representative of the commonly applied in research and practical engineering of hydraulic structures.

_{m}is the distance along the chute bottom from the offset step to a point where the pressure increment is maximum (Δp

^{b}= Δp

^{b}

_{max}), see Figure 9.

_{m}is similar for all model combinations, except for the CTPE with k–$\u03f5$ mixture, increasing in higher mesh resolutions. L

_{m}starts to converge in intermediate mesh resolutions ${R}_{m}\ge 30$ to a value approximately 15 to 35% larger than the reference ($L{m}^{ref}=0.62$ m in [11]). The CTPE with k–$\u03f5$ mixture shows abnormally small values for ${R}_{m}\ge 20$, due to the exacerbated jet diffusion. The TPVoF with k–$\u03f5$ and the CTPE with k–$\omega $ SST Sato present the peak of pressure nearer the reference value.

#### 3.4. Cavity Zone Length

^{b}= 0.1 × Δp

^{b}

_{max}. L also characterizes the jet trajectory.

_{m}(Figure 9).

#### 3.5. Turbulence

_{t}in [11]. Nevertheless, the four turbulence models are compared.

_{t}) presented in Figure 13 and Figure 14 expose the lower-nappe aeration is more significant where the υ

_{t}is higher, which is displayed clearly by the k–$\u03f5$ and k–$\u03f5$ mixture models.

_{t}as defined in Equation (12). Oppositely, the TPVoF model equations do not comprehend the turbulent dispersion. Hence, any air-water mixing is due to the numerical diffusion in the interface, which is reduced with higher mesh resolution.

#### 3.6. Computational Cost

_{m}) of 10, 15, 20, 30, 40 and 60, there is an average increment of 4.5 times between two sequential resolutions. Hence, from R

_{m}= 10 to R

_{m}= 60, calculation time increases by a factor of $2\times {10}^{3}$. The CTPE with k–$\omega $ SST Sato shows very high levels of turbulence near the interface, which is probably due to the overestimation by two-fluid per-phase turbulence models of the turbulent kinetic energy near large scale interfaces with significant shear [71].

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational fluid dynamics |

CTPE | Complete two-phase Euler |

DNS | Direct numerical simulations |

DES | Detached eddy simulation |

HPC | High-performance computing |

LS | Level-set method |

MX | Mixture model |

RANS | Reynolds-averaged Navier–Stokes equations |

SST | Shear stress transport |

SGB | Sub-grid air-bubble density equation model |

TPVoF | Two-phase Volume-of-fluid |

VoF | Volume-of-fluid method |

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**Figure 1.**Laboratory setup ([45]-authorized reproduction). (

**a**) Physical model (unknown flow conditions). (

**b**) Flow characteristics.

**Figure 3.**3D flow—surface ($\alpha =0.5$, solid blue), interface region ($0.01<\alpha <0.99$, transparent blue), air vents streamlines (red), water inlet streamlines (dark-blue) (${R}_{m}=40$, except ${R}_{m}=30$ for CTPEk–$\u03f5$mixture). (

**a**) TPVoF, k–$\u03f5$. (

**b**) CTPE, k–$\u03f5$mixture. (

**c**) TPVoF, k–$\omega $SST. (

**d**) CTPE, k–$\omega $ SST Sato.

**Figure 5.**Lower-nappe aeration: ratio β between the flow-rate of air-entrained at the cavity zone and water flow-rate, for distinct mesh resolutions.

**Figure 6.**Water volume fraction vertical profile at $x=\{0.2,0.7,1.0\}$ m; ${R}_{m}=30$. (

**a**) $x=0.2$ m. (

**b**) $x=0.7$ m. (

**c**) $x=1.0$ m.

**Figure 7.**Pressure increment along the spillway bottom. (

**a**) TPVoF, k–$\u03f5$. (

**b**) CTPE, k–$\u03f5$mixture. (

**c**) TPVoF, k–$\omega $SST. (

**d**) CTPE, k–$\omega $ SST Sato.

**Figure 11.**k at vertical central plane of the TPVoF model with R

_{m}= 20; interface ($\alpha $ = 0.5, black line). (

**a**) k–$\u03f5$. (

**b**) k–$\omega $ SST.

**Figure 12.**${k}_{w}$ at vertical central plane of the CTPE model with R

_{m}= 20; interface ($\alpha $ = 0.5, black line). (

**a**) k–$\u03f5$mixture. (

**b**) k–$\omega $ SST Sato.

**Figure 13.**υ

_{t}at vertical central plane of the TPVoF model with R

_{m}= 20; interface ($\alpha $ = 0.5, black line). (

**a**) k–$\u03f5$. (

**b**) k–$\omega $ SST.

**Figure 14.**υ

_{t,w}at vertical central plane of the CTPE model with R

_{m}= 20; interface ($\alpha $ = 0.5, black line). (

**a**) k–$\u03f5$mixture. (

**b**) k–$\omega $ SST Sato.

R_{m} | Edge Length [mm] | Total Cells |
---|---|---|

10 | ${h}_{0}/10\approx 15$ | 85,510 |

15 | ${h}_{0}/15\approx 10$ | 273,750 |

20 | ${h}_{0}/20\approx 7.5$ | 684,080 |

30 | ${h}_{0}/30\approx 5$ | 2,190,000 |

40 | ${h}_{0}/40\approx 3.75$ | 5,472,640 |

60 | ${h}_{0}/60\approx 2.5$ | 17,520,000 |

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## Share and Cite

**MDPI and ACS Style**

Mendes, L.S.; Lara, J.L.; Viseu, M.T.
Do the Volume-of-Fluid and the Two-Phase Euler Compete for Modeling a Spillway Aerator? *Water* **2021**, *13*, 3092.
https://doi.org/10.3390/w13213092

**AMA Style**

Mendes LS, Lara JL, Viseu MT.
Do the Volume-of-Fluid and the Two-Phase Euler Compete for Modeling a Spillway Aerator? *Water*. 2021; 13(21):3092.
https://doi.org/10.3390/w13213092

**Chicago/Turabian Style**

Mendes, Lourenço Sassetti, Javier L. Lara, and Maria Teresa Viseu.
2021. "Do the Volume-of-Fluid and the Two-Phase Euler Compete for Modeling a Spillway Aerator?" *Water* 13, no. 21: 3092.
https://doi.org/10.3390/w13213092