# Computational Fluid Dynamics (CFD) as a Tool for Investigating Self-Organized Ascending Bubble-Driven Flow Patterns in Champagne Glasses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}), because gas-phase CO

_{2}forms together with ethanol during a second in-bottle fermentation process promoted by adding yeasts and a certain amount of sugar in bottles hermetically sealed with a crown cap or a cork stopper [1]. This second in-bottle fermentation process forces an amount equivalent to around 11–12 g L

^{−1}of CO

_{2}to progressively dissolve into the wine, according to so-called Henry’s law [1]. Immediately after uncorking a bottle of champagne, the thermodynamic equilibrium between gas-phase and dissolved CO

_{2}is broken, and the liquid phase becomes supersaturated with CO

_{2}. Dissolved CO

_{2}must therefore progressively desorb from the liquid phase. As firstly observed by Liger-Belair et al. [2], massive losses of dissolved CO

_{2}are indeed experienced by champagne during the pouring step. Immediately after pouring champagne into a glass, the dissolved CO

_{2}concentration falls to a level in the order of 6–9 g L

^{−1}, depending on several parameters, such as the champagne temperature, the bottle type, or the glass shape, for example [2,3,4,5]. This range of dissolved CO

_{2}concentration of 6–9 g L

^{−1}is nevertheless well beyond the required minimum level of dissolved CO

_{2}, close to 2.5 g L

^{−1}, to enable heterogeneous bubble nucleation under standard tasting conditions [6].

_{2}from champagne and sparkling wine glasses: (i) into the form of heterogeneously nucleated bubbles (the so-called effervescence process) and (ii) through the molecular diffusion of dissolved CO

_{2}through the air/wine interface (indeed invisible by the naked eye), as described in minute detail by Liger-Belair [6]. In recent years, glassmakers proposed to champagne and beer-drinkers a new generation of laser-etched glasses specially designed to cause the standardized conditions of effervescence [7]. In laser-etched glasses, bubble nucleation is most often triggered at the bottom of the glass with a ring-shaped structure made with adjoining laser beam impacts. Laser-etched glasses filled with champagne, beer, or any other sparkling beverage, are thus easily recognizable, with a continuously renewed bubbly flow ascending along their vertical axis of symmetry. A bubbling scenario from such laser-etched champagne glasses was indeed recently proposed [7].

_{2}to promote heterogeneous bubble nucleation [6]). The homogeneous stirring of champagne under the action of rising bubbles confers an advantage compared with a situation where the liquid phase is at rest by renewing the air/champagne subsurface layers with champagne from the bulk, thus enhancing the release of dissolved CO

_{2}, as well as the evaporation of volatile organic compounds (VOCs) in the headspace above the glass [9,10].

_{2}from the champagne surface [18]. Computational fluid dynamics (CFD) methods have also been applied to model ascending bubble-driven flow patterns in champagne glasses [19,20,21]. In stout beer glasses (such as the Guinness beer glass), a combined CFD and experimental approach was performed to demonstrate that the bubbly flow also strongly depends on the shape of the glass [22]. If it narrows downwards, as the traditional stout glass does, the flow is directed downwards near the wall and upwards in the interior (and sinking bubbles are observed), whereas, if the container widens downwards, the flow is opposite to that described above and only rising bubbles can be seen. More recently, a Japanese team has revisited this visually appealing phenomenon observed in glasses filled with this famous Irish stout beer [23]. By combining direct observations with laser-induced fluorescence and computer models, the researchers created a detailed profile of the liquid phase velocity. They found that the bubbles-going-down effect, observed with bubbles much smaller than those found in most standard beers or champagnes, is analogous to roll waves commonly observed in water sliding downhill on a rainy day [23].

## 2. Modeling the Glass and Physicochemical Parameters of Champagne

#### 2.1. Modeling the Champagne Glass

_{2}bubbles into the form of a bubbly flow (equivalent to the bubbles heterogeneously nucleated from the laser etchings presented in Figure 1) through injection points found at the bottom of the glass.

#### 2.2. Meshing of the Computational Domain

^{®}Workbench Meshing software. Because 2D modeling is carried out with axisymmetric calculations, only half of the fluid domain is meshed, as seen in Figure 3. The 2D mesh is composed of about 7700 quadrilateral and hexahedral elements, with a structured grid in the central part of the computational domain and an unstructured grid in the vicinity of the glass wall. To ensure that the results of the CFD calculation were consistent with the experimental results, a grid dependency test was performed. The convergence test consisted of improving the results by using successively smaller cell sizes. More details about the grid dependency test can be found in a previous study [19].

#### 2.3. Physicochemical Parameters of Champagne and Gas-Phase CO_{2}

^{−1}, its density ρ was found to be very close to that of water (i.e., 10

^{−3}kg m

^{−3}) and its dynamic viscosity was found to be about 50% higher than that of pure water (mainly because of ethanol). The initial concentration of dissolved CO

_{2}(denoted [CO

_{2}]

_{ini}) found in the liquid phase immediately after pouring 100 mL of champagne into the specific glass described above was also determined in a previous work [4]. Moreover, the diffusion coefficient of molecular species in a liquid phase, such as dissolved CO

_{2}molecules in champagne, for example, is therefore strongly temperature dependent. The diffusion coefficients of dissolved CO

_{2}molecules in a standard commercial Champagne wine (denoted D) was accurately determined as a function of temperature by classical molecular dynamics simulations and by

^{13}C nuclear magnetic resonance (NMR) spectroscopy, respectively [25,26]. The various physicochemical parameters of champagne (i.e., the liquid phase) and gas-phase CO

_{2}(i.e., the bubbles), as determined at 20 °C, are reported in Table 1. These parameters were retrieved in our numerical procedures.

## 3. Numerical Methods and Set-Up

_{2}bubbles and air), the use of a multiphase model was required. Firstly, 2D CFD simulations were carried out through an Eulerian–Lagrangian approach (with one Eulerian phase and one Lagrangian phase). In a second step, we used a 3D multiphase model combined with the volume-of-fluid (VOF) method (with two Eulerian phases and one Lagrangian phase) in an open channel flow configuration [27,28]. The VOF method was used to model the interaction between the liquid phase (champagne) and the gaseous phases (both ascending CO

_{2}bubbles and open atmosphere). This interface tracking method is usually used to follow the position of the interface between the fluids [29,30,31,32]. In complement, we applied the open channel flow condition [33,34] to simulate the presence of a free surface (i.e., the air/champagne interface). Because the CFD code cannot model the upward dynamics of CO

_{2}bubbles, the discrete phase model (DPM) was used as a very efficient way to follow bubbles in a Lagrangian reference frame [35]. In a Lagrangian formulation, the trajectory of each ascending bubble is calculated individually at each time-step of the simulation.

#### 3.1. Liquid-Phase Governing Equations

_{2}ascending bubbles), a volume averaging method was used to develop a set of partial differential equations that describes the conservation of mass and momentum. The conservation equations can be formulated as a volume-average equation, as follows.

**g**being the gravitational field and ${\mathit{F}}_{bf}$ being the forces that result from the interaction between each bubble and the liquid phase.

#### 3.2. Discrete Phase Modeling

_{2}bubble in the present case) by integrating the balance of forces, which is written in a Lagrangian reference frame [28]. This force balance can be written (for the vertical direction in Cartesian coordinates) according to the following relationship:

_{z}being the gravity acceleration (operating in the z direction in Cartesian coordinates), ${F}_{D}\left({u}_{l}-u\right)$ being the drag force per unit particle mass and with the additional force ${F}_{z}$ corresponding to the “virtual mass” force required to accelerate the fluid surrounding the bubble.

_{2}found in the liquid phase (i.e., the supersaturation ratio of CO

_{2}). Moreover, CO

_{2}bubbles formed through heterogeneous nucleation on the bottom of the laser-etched glass grow in size as they rise toward the champagne surface because dissolved CO

_{2}diffuses from the liquid phase toward gas-phase CO

_{2}through the bubble interface [37]. The growth rate of ascending bubbles therefore strongly depends on several parameters of the liquid phase, including the level of dissolved CO

_{2}, the diffusion coefficient of CO

_{2}and the liquid-phase viscosity, among many others, as already described by several authors working with champagne and beer [37,38,39]. More details can be found about effervescence and key parameters governing bubble nucleation and rise in champagne glasses in the review by Liger-Belair [37].

_{2}, such as champagne, sparkling wines and beers, user-defined functions (UDFs) were implemented in the CFD code. UDFs were proposed to best model the frequency of bubble formation at the bottom of the glass, the bubble growth during ascent, the drag force exerted by the liquid phase on bubbles and the subsequent continuous acceleration experienced by ascending bubbles. More details about each and every UDF implemented in the CFD code can be found in the articles by Beaumont et al. [19,20,21]. It is worth noting that bubbles are injected into the fluid domain (at the bottom of the glass) with a trajectory that faithfully reproduces the dynamics of a single ascending and growing bubble [37]. Actually, bubbles rising at small to intermediate Reynolds numbers rise in line, in a straight-line path and remain nearly spherical, as described in previous works [38,39,40]. A photograph showing a train of successive bubbles rising and growing in line in a champagne glass is displayed in Figure 4.

_{2}. However, in fact, under standard champagne tasting conditions, the myriad of ascending bubbles strongly interact with each other. Wake interactions between successive bubbles, bouncing or coalescence events [41,42,43] likely to occur in the central bubbly flow were not taken into account in our CFD simulations. The growth and rise of interacting bubbles require much more complex models, which go well beyond the scope of this paper. Our simplified procedure is therefore considered as a first step toward a more complex numerical study including the interactions between multiple bubbles, as for example, in the recent article by Lai et al. [44] and the references therein.

#### 3.3. Boundary Conditions

_{2}bubbles) were injected at the bottom of the glass and tracked with the fluid flow time step. The initial CO

_{2}bubble diameter was equal to 5 × 10

^{−5}m, in accordance with previous observations conducted through high-speed video imaging [37].

^{®}19.2 software, which is based on the finite volume approach. Ascending bubbles were subjected to low to moderate Reynolds numbers [40] and the flow was assumed to be laminar and governed by the finite volume equations. The convergence criteria were based on residuals whose values were monitored throughout the iterative calculation process. For all simulations carried out in this study, convergence of the results was achieved with residuals lower than 10

^{−5}. It is worth noting that the VOF method (needed in the 3D multiphase simulations) used a variable volume fraction to capture the air–champagne interface. To accurately represent the presence of the open atmosphere above the champagne surface, a buffer zone of air was defined as a 2 mm thick layer above the champagne surface. This buffer zone was the second Eulerian phase, the first Eulerian phase being the liquid phase (i.e., the champagne itself).

## 4. Results and Discussion

#### 4.1. The Two-Dimensional (2D) Model

_{2}champagne progressively decreases over time. Actually, the 2D CFD model numerically reproduces the global shape of the flow patterns observed through PIV and its evolution with time, with a main vortex ring that occupies nearly all of the computational domain and a second one of a much smaller size close to the glass wall. The second, smaller vortex, situated about 2 cm below the champagne surface, counter-rotates with respect to the main vortex, and progressively decreases in size before finally vanishing over time. Nevertheless, it is worth noting that small topological differences can be found between the numerical and experimental data for the flow patterns found immediately below the air/champagne interface, especially for the vortex core location.

^{−1}(in the zone marked with the red color in Figure 5b). Driven upward by the central bubbly flow, the liquid phase radially migrates close to the air/champagne interface, before plunging back downward into the liquid bulk close to the edge of the glass, thus initiating a swirling flow clearly visible in the streamline patterns displayed in Figure 5a. Sixty seconds after pouring champagne into the glass, the radial (subsurface) flow velocity ranges between 2.5 and 3.5 cm s

^{−1}(in the light blue zone in Figure 5b). Moreover, both the experimental and CFD velocity contour maps displayed in Figure 5b show that the velocities of the liquid phase progressively decrease with time, which is particularly visible in the central area of the glass.

#### 4.2. The Three-Dimensional (3D) Model

_{2}found in champagne) [13,14,15,16,17,18]. Moreover, and most interestingly, in a standard laser-etched coupe filled with champagne, self-organized and counter-rotating two-dimensional convective cells were also unveiled at the air/champagne interface [45]. Various regimes were evidenced, from a highly unstable eight-cell regime, to a very stable four-cell regime [45], as shown in Figure 6. There are indeed eight cells counter-rotating close to each other in Figure 6a (with seven big cells and a much smaller one marked with a white arrow). From the topological point of view, the four-cell regime displayed in Figure 6c looks strikingly like the so-called steady streaming flow, resulting from the action of an oscillatory cylinder in the main body of a fluid or in thin boundary layers [46,47,48].

## 5. Conclusions and Prospects

_{2}, as well as the evaporation of VOCs in the headspace above glasses.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Photographic evidence for bubble production from the ring-shaped structure made at the bottom of a laser-etched flute filled with 100 mL of champagne [7] (

**a**). Detail of the ring-shaped structure made with adjoining laser beam points of impact (

**b**,

**c**). In frame (

**a**), dimensions are indicated in mm.

**Figure 2.**Computational domain, in gray, as determined for the 2D computational fluid dynamics (CFD) approach (

**a**) and a 3D sketch of the glass, as determined for the 3D CFD approach (

**b**). In frame (

**a**), dimensions are indicated in mm for the glass filled with 100 mL of champagne.

**Figure 3.**The 2D meshing of the fluid domain (

**a**) and 3D meshing of the fluid domain (

**b**), respectively. The 2D meshing as seen in the plane of symmetry of the glass (

**c**) and detail of the mesh close to the glass wall (

**d**).

**Figure 4.**Train of successive spherical bubbles released with clockwork regularity from a specific nucleation site found at the bottom of a glass filled with champagne. Bubbles are seen growing in size and rising in line. User-defined functions (UDFs) implemented in the CFD code were based on the equations governing the dynamics of such spherical bubbles rising and growing in size in a solution supersaturated with dissolved CO

_{2}, as described in [21].

**Figure 5.**Comparison between the CFD and particle image velocimetry (PIV) streamline patterns as time proceeds (respectively 60 s, 300 s and 600 s after champagne was poured into the glass) (

**a**); comparison between the CFD and PIV velocity fields as time proceeds (respectively 60 s, 300 s and 600 s after champagne was poured into the glass) (

**b**).

**Figure 6.**Laser tomography combined with long exposure photography showing fluid in motion at the free surface of a coupe filled with 100 mL of champagne. In frame (

**a**), a highly unstable eight-cell regime is evidenced (soon after pouring the champagne), usually followed by a poorly stable six-cell regime (

**b**) and then by a highly stable and long lasting four-cell regime, several minutes after pouring champagne into the coupe, where four counter-rotating cells self-organize at the air/champagne interface (

**c**) [45].

**Figure 7.**The central bubbly flow ascending in a laser-etched glass filled with 100 mL of champagne (

**a**) has the ability to set up a network of various convective cells, invisible to the naked eye, but revealed through the 3D CFD model, in the plane of symmetry of the glass and at the air/champagne interface (

**b**). (Photograph by Alain Cornu/Collection CIVC Authors have the permission to reuse this photography).

**Figure 8.**Close-up view of the 2D eight-cell regime found at the air/champagne interface, as determined through the 3D CFD model (

**a**) and a subsequent layout showing how the fluid self-organizes at the air/liquid interface (

**b**).

**Figure 9.**Volume velocity field resulting from the 3D simulations, with the key parameters required by the UDF, such as being able to numerically reproduce the highly unstable eight-cell regime found at the air/champagne interface.

Champagne | Gas-Phase CO_{2} | |
---|---|---|

Density ρ (kg m ^{−3})
| 9.98 × 10^{2} | 1.79 |

Dynamic viscosity η (kg m ^{−1} s^{−1})
| 1.56 × 10^{−3} | 1.37 × 10^{−5} |

Surface tension γ (mN m ^{−1})
| 46.8 | / |

Dissolved CO_{2} concentration [CO_{2}]_{ini} (g L^{−1})
| 7.4 | / |

CO_{2} diffusion coefficientD (m ^{2} s^{−1})
| ≈1.4 × 10^{−9} | / |

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**MDPI and ACS Style**

Beaumont, F.; Liger-Belair, G.; Polidori, G. Computational Fluid Dynamics (CFD) as a Tool for Investigating Self-Organized Ascending Bubble-Driven Flow Patterns in Champagne Glasses. *Foods* **2020**, *9*, 972.
https://doi.org/10.3390/foods9080972

**AMA Style**

Beaumont F, Liger-Belair G, Polidori G. Computational Fluid Dynamics (CFD) as a Tool for Investigating Self-Organized Ascending Bubble-Driven Flow Patterns in Champagne Glasses. *Foods*. 2020; 9(8):972.
https://doi.org/10.3390/foods9080972

**Chicago/Turabian Style**

Beaumont, Fabien, Gérard Liger-Belair, and Guillaume Polidori. 2020. "Computational Fluid Dynamics (CFD) as a Tool for Investigating Self-Organized Ascending Bubble-Driven Flow Patterns in Champagne Glasses" *Foods* 9, no. 8: 972.
https://doi.org/10.3390/foods9080972