# Simulation of Air Ingestion in a Mechanical Seal With Inward Pumping Spiral Grooves

Abdel-Salem Medjahed, Antoinette Blouin, Balint Pap, Noël Brunetière

► To cite this version:

Abdel-Salem Medjahed, Antoinette Blouin, Balint Pap, Noël Brunetière. Simulation of Air Ingestion in a Mechanical Seal With Inward Pumping Spiral Grooves. Journal of Tribology, 2023, 145 (11), 10.1115/1.4062899. hal-04266389

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Submitted on 31 Oct 2023

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# Simulation of air ingestion in a mechanical seal with inward pumping spiral grooves

**Abdel-Salem Medjahed**

Safran Transmission Systems
Colombes
France

Email:abdel.salem.medjahed@univ-poitiers.fr

**Antoinette Blouin**

Institut Pprime

CNRS, University of Poitiers
Dept GMSC

Poitiers
France

Email:antoinette.blouin@univ-poitiers.fr

**Balint Pap**

Safran Transmission Systems
Colombes
France

Email:balint.pap@safrangroup.com

**Noël Brunetière***

Institut Pprime

CNRS, University of Poitiers
Dept GMSC

Poitiers
France

Email:noel.brunetiere@univ-poitiers.fr

## ABSTRACT

Inward pumping spiral groove seals are used in many applications, such as transmission gear boxes or electric vehicles, as they can provide zero leakage and very low operating friction. The spiral grooves are connected to the low pressure side (typically atmosphere) and pump the fluid inward that is to say toward the high pressure side (inlet). If the rotational speed is high enough, the low pressure fluid (air) completely occupies the sealing area and is used as a low friction lubricant. However, for lower operating speeds or higher fluid pressures, the seal can be lubricated by both fluids simultaneously. This occurs particularly during start-up and shut down. In this work a numerical simulation of the two phases in the sealing gap of an upstream pumping spiral groove seal during start-up is performed. The establishment of

<footnote>*Corresponding Author.</footnote>
the film is analyzed and the effect of the rotational speed and operating pressure on the fluid composition (air and liquid) in the sealing interface is studied. For the analyzed seal, it is shown that the amount of air pumped in the sealing area is controlled by the duty parameter.

# 1 Introduction

A part of the power from the aircraft engines is transferred to accessories with the help of a gearbox. As it is important to ensure a good reliability of the system while avoiding oil leakage, mechanical seals were chosen as the sealing solution due to their ability to operate under harsh conditions [1]. Mechanical face seals are composed of several elements, including a stator linked to the housing, a rotor linked to the shaft, a spring, and secondary seals. The sealing is ensured by a flat on flat contact of the stator and rotor surfaces, which can be lubricated by the sealed fluid. To reduce friction and wear, shallow grooves, such as spirals, are machined on the sealing surface since the 1970s in mechanical gas face seals [2, 3, 4]. The grooves allow enhancing the load carrying capacity due to hydrodynamic effects by pumping the gas located at the high pressure side. If these seals are used with liquids, prohibitive flow rates are obtained when compared to flat smooth surfaces due to the large thickness of the film and pumping promoted by the spiral grooves [5].

In 1992, Salant and Holmiller [6] conducted a parametric numerical study to understand the influence of the geometry of the grooves on the behavior of a liquid lubricated seal. The same authors showed that it is possible to enhance the stiffness of the film while creating a negative leakage (reverse or upstream flow) from the low pressure to the high pressure side [7]. For that, the grooves are connected to the low pressure side of the seal. If the grooves are well designed, mechanical seals operating with zero leakage and low friction can be obtained [8, 9, 10]. In 2014, Zhao et al. [11] carried out a transient numerical study of a liquid lubricated spiral groove seal. Their model considers centrifugal forces as well as cavitation and was compared to experiments. The grooves are machined from the inner radius to the high pressure lubricating oil. The grooves allow reaching full film regime due to hydrodynamic pressure spikes whereas cavitation areas develop on the opposite side of the grooves. The importance of cavitation in the operation of grooved seals was confirmed by Ma et al. in 2019 [12]. They studied different groove geometries on a mechanical face seal immersed in water and observed the cavitation area through a transparent disk. They show that zero leakage is promoted by the suction effect of cavitation. However, in real situations, the seals are not fully immersed in a single liquid and are generally connected to the atmosphere on one side. Indeed, it was experimentally shown that reverse pumping grooves leads to air ingestion in the sealing area [13, 14]. As the mechanical seal is no longer fully lubricated by a liquid, some areas are lubricated by the ingested air. This gas lubrication regime can significantly reduce the seal friction during operation [13]. According to Tokunaga et al. [13], there is a transition from liquid lubrication to gas lubrication when the operating speed increases.

The modeling of the flow in the sealing area of a mechanical seal with upstream pumping grooves requires a two-phase flow approach. The two-phase flow problem in mechanical seals has been addressed for the case of liquid boiling in the sealing area in ref. [1]. In many studies, the fluid is described as a homogeneous mixture whose relative amount of liquid and vapor varies with the local enthalpy [15], [16], [17], [18]. The case of different fluids at the seal boundaries has been rarely studied. Lebeck, in 2008 [10], numerically studied this problem using a mass flow equation for each phase. In 2016, Brunetière [19] published the premises of the model which is used in the present paper. It couples the Reynolds equation for the homogeneous fluid flow with a transport equation to get the mixture constitution in steady-state.

In the present work, a mechanical seal with reverse pumping grooves is studied during start-up. The conditions model [19] is extended to consider transient problems. The mechanical seal has two rings: one rotating and one stationary. The non-rotating rings has a single degree of freedom considered, which is a translation along the axis of rotation of the shaft and its faces are aligned allowing the study of a single groove by employing periodic boundary conditions. The finite element method solves the Reynolds and transport equations. The asperity contact and the axial force balance are considered as well. The establishment of the film is analyzed and the effect of the rotational speed and operating pressure on the composition of the fluid (air and liquid) in the sealing interface are studied.
<figure><img src="image_2.png" /><figcaption>Fig. 1: Configuration of the seal: a) Cross section - b) Spiral grooves</figcaption></figure>

## 2 Material and methods

### 2.1 Studied Seal

The mechanical seal is composed of several elements presented in figure 1). The bottom element represents the rotor rigidly attached to the shaft. A floating stator, attached to the housing, has one degree of freedom in the z direction to ensure contact with the rotor. The sealing interface is limited by the inner and outer radii, respectively $r_i$ and $r_o$. The spiral grooves machined on the rotating ring are defined by their outer radius $r_g$, angle $\alpha$, depth $h_g$ and relative width $w = \frac{w_g}{w_g+w_d}$. The pressure at the outer radius, $p_o$, is greater than the inner radius, $p_i$. The gas mass fraction at the boundaries is defined by $\lambda_i$ at the inner radius and $\lambda_o$ at the outer radius.

In this work, the seal faces are aligned. Thus, only one period of the groove is studied. Some periodic boundary conditions are applied in the circumferential direction.

### 2.2 Sealing interface model

The fluid in the sealing area is described using the following assumptions [20]:

1. The fluid is a homogeneous mixture of an incompressible liquid of density $\rho_l$ and a perfect gas of density $\rho_g$.

2. The surface tension at the liquid-gas interface is neglected: the gas pressure is equal to the liquid pressure.

3. The liquid and gas phases move at the same speed.

The first three assumptions are typical of the homogeneous models used in two-phase flow [21]. The density of the gas is assumed to vary according to the ideal gas law:

$$ \rho_g = \frac{p}{r_g T} \qquad (1) $$

where $p$ and $T$ are, respectively, the absolute pressure and temperature of the mixture, $r_g$ is the specific perfect gas constant of the considered gas, which is the product of the molar mass of the gas times the universal gas constant. The density $\rho$ of the mixture is given by [21]:

$$ \rho = \frac{1}{\frac{\lambda}{\rho_g} + \frac{1-\lambda}{\rho_l}} \qquad (2) $$
A similar expression, which was shown to be accurate for fluid films [22], is used for the viscosity $\mu$ of the mixture:

$$ \mu = \frac{1}{\frac{\lambda}{\mu_g} + \frac{1-\lambda}{\mu_l}} \qquad (3) $$

In these equations, $\lambda$ is the gas mass fraction. It is possible to have a pure gas when $\lambda = 1$ as well as a pure liquid when $\lambda = 0$. Using the standard lubrication hypothesis and assuming the fluid is Newtonian with a constant viscosity and density along the z axis, the Reynolds equation can be obtained:

$$ \frac{\partial}{\partial x}(\rho h^3 \frac{\partial p}{\partial x}) + \frac{\partial}{\partial y}(\rho h^3 \frac{\partial p}{\partial y}) = -\frac{y\omega}{2} \frac{\partial \rho h}{\partial x} + \frac{x\omega}{2} \frac{\partial \rho h}{\partial y} + \frac{\partial}{\partial x}(\frac{\rho h^3 \omega^2 y}{36\mu}) + \frac{\partial}{\partial y}(\frac{\rho h^3 \omega^2 x}{36\mu}) + \frac{\partial \rho h}{\partial t} \qquad (4) $$

where $\omega$ is the rotational speed of the rotor and $h$ the thickness of the film. Its solution gives the distribution of the fluid pressure $p$.

The gas mass fraction $\lambda$ is a scalar quantity that is advected by the fluid in the sealing area. Assuming the two phases are not miscible, there is no diffusion term. Adding the transient term to the steady state equation proposed by Brunetière [19], the following transport equation is obtained:

$$ \rho h \frac{\partial \lambda}{\partial t} + q_x \frac{\partial \lambda}{\partial x} + q_y \frac{\partial \lambda}{\partial y} = 0 \qquad (5) $$

where $q_x$ and $q_y$ are the local mass flow rate per unit length:

$$ q_x = -\frac{y\omega}{2}\rho h + \frac{\rho h^3}{12\mu} \left( \frac{\rho x \omega^2}{3} - \frac{\partial p}{\partial x} \right) \qquad (6) $$

$$ q_y = \frac{x\omega}{2}\rho h + \frac{\rho h^3}{12\mu} \left( \frac{\rho y \omega^2}{3} - \frac{\partial p}{\partial y} \right) \qquad (7) $$

The transport equation allows us to know the distribution of the gas mass fraction in the sealing area. Note that the gas mass fraction of a fluid particle at a given location depends only on its origin (inner or outer radius).

## 2.3 Surface roughness

When the thickness of the film is less than three times the standard deviation $S_q$ of the surface roughness [23], asperity contact of the surfaces can occur. The asperity contact model used in this work is based on the Greenwood and Williamson model [24] and assumes an exponential distribution of the peak heights. The contact pressure is of the form:

$$ p_c(h) = \frac{E'}{8Sal} S_q \sqrt{\frac{\pi}{8}} e^{-\frac{2h}{S_q}} \qquad (8) $$

where $E'$ is the equivalent Young's modulus and $Sal$ the surface roughness correlation length. The surface roughness parameters combine the effect of the two surfaces. Note that the exponential distribution is justified by the fact the contact pressure is generally weak in mechanical seals (the relative contact area being less than 0.1%, [23]), leading to contact only on the top summits.

In the present work, the effect of the surfaces roughness on the fluid flow is neglected. As the surfaces remain parallel, the usual methods based on flow factors [25, 26], homogenization [27] or artificial viscosity [28] would
lead to constant factors modifying the pressure terms and the shear term. These approaches are in addition not relevant for two-phase flows. This is why it was decided to not consider surfaces roughness in the fluid flow. Another solution would be the use of a deterministic approach or a multi-scale technique [29]. As the lateral scale of the roughness is small compared to the size of the grooves, a very large mesh would be necessary and prohibitive for our problem.

## 2.4 Axial force balance

To determine the axial position of the stator, the axial force balance is needed:

$$ -F_{\text{hydraulic}} - F_{\text{spring}} + F_{\text{contact}} + F_{\text{film}} = m \frac{\partial^2 h}{\partial t^2} \qquad (9) $$

where m is the mass of the stator. Two main forces are involved in the balance:

The closing force is composed of the mechanical action of the surrounding fluid pressure $F_{\text{hydraulic}}$ and the spring force $F_{\text{spring}}$ applied on the back of the stator.

The opening force is composed of the fluid film pressure $F_{\text{film}}$ and the contact force $F_{\text{contact}}$ applied on the seal face of the stator.

The distribution of the fluid pressure is integrated on the sealing area S to obtain $F_{\text{film}}$

$$ F_{\text{film}} = \int_{S} p dS \qquad (10) $$

In the same way, we have $F_{\text{contact}}$, the force due to contact of asperities (eq. 8)

$$ F_{\text{contact}} = \int_{S} p_c dS \qquad (11) $$

The hydraulic force is

$$ F_{\text{hydraulic}} = p_o BS + p_i (1 - B)S \qquad (12) $$

where B is the balance ratio. The acceleration is obtained by a backward temporal finite difference scheme:

$$ \frac{\partial^2 h}{\partial t^2} = \frac{h(t) - 2h(t - \Delta t) + h(t - 2\Delta t)}{\Delta t^2} \qquad (13) $$

where $\Delta t$ is the step time. Once the forces are computed, the Newton-Raphson method is used to find the new position of the stator.

## 2.5 Boundary conditions and numerical method

The boundary conditions used in the Reynolds and transport equations solutions are presented in figure 2.

For the Reynolds equation, the pressure is imposed on the inner and outer radii.

For the transport equation, the boundary conditions are affected by the sign of the local mass flow rate $q_r$. It is the projection of $q_x$ (eq. 6) and $q_y$ (eq. 7) along the external normal to the boundary. If the mass flow rate at a given location on the inner or/and outer radii is negative, the fluid enters the contact and the gas mass fraction is imposed. In the case of a positive mass flow rate, the fluid exits the contact and the value of $\lambda$ is unknown and calculated from the transport equation.
<figure><img src="image_3.png" /><figcaption>Fig. 2: Boundary conditions of a portion of the sealing zone : a) Gas mass fraction boundary conditions b) Pressure boundary conditions</figcaption></figure>

For both equations, the boundary conditions in the circumferential direction are periodic.

The Reynolds equation and the transport equation must be solved numerically. Here, the finite element method is used. For both equations the meshing is constituted of linear four-nodes elements. The solution of the Reynolds equation is given in detail in [20]. The transport equation is a pure convective equation that needs an upwind scheme. In the previous work of Brunetiere, the SUPG (Streamline Upwind Petrov Galerkin) method was used [19]. However, some numerical oscillations were still visible perpendicularly to the pressure flow. These oscillations are due to streamline diffusion introduced by the SUPG method and can be eliminated by a crosswind dissipation [30]. By using this approach, the solution is free of oscillations.

The code workflow is presented in fig. 3. After the initialization of the variables, the Reynolds equation is solved using the Newton-Raphson method. The fluid and contact pressure are computed and used to calculate the opening force. The thickness of the film is then adjusted to ensure force balance. Then the transport equation is solved and the fluid properties are updated. The computation goes to the next time step, and the entire process repeats.
<figure><img src="image_4.png" /><figcaption>Fig. 3: Code Workflow</figcaption></figure>

## 2.6 Spatial and temporal discretization

As the nodes located on the inner and outer radii can change from inlet to outlet conditions during one time step period $\Delta t$, it is necessary to ensure that during this time step, enough fluid enters the element to have a consistent solution. In the radial direction and for linear spatial variation, the local mass flow balance can be written in this way:

$$ \Delta t > \frac{\rho h \Delta r}{2 \dot{m}_r} \qquad (14) $$

where $\Delta r$ is the radial spatial step and $\dot{m}_r$ the radial mass flow rate. The time step was chosen to verify this condition. The spatial discretisation was chosen to ensure a mesh independent solution and a reasonable computation time. It was found that 81 nodes in the circumferential direction and 121 nodes in the radial direction are a good compromise.
For each rotational speed, the time step used is reported in table 1. As expected, the time step used is smaller when the speed is higher due to the increase in the flow rate $\dot{m}_r$ with angular velocity. The only exception is the lowest rotational speed for which, the sealing area is almost full of liquid. The time step used is of the same order of magnitude as the time needed to move from one groove to the other.

<table><caption>Table 1: Time step as a function of the speed and number of time steps per full revolution and per period (one period per groove).</caption><thead><tr><th>Rotational speed</th><th>Time step</th><th>Time steps per revolution</th><th>Time steps per period</th></tr></thead><tbody><tr><td>10 rpm</td><td>0.005 s</td><td>1200.0</td><td>40.0</td></tr><tr><td>75 rpm</td><td>0.05 s</td><td>16.0</td><td>0.533</td></tr><tr><td>200 rpm</td><td>0.025 s</td><td>12.0</td><td>0.4</td></tr><tr><td>400 rpm</td><td>0.008 s</td><td>18.8</td><td>0.626</td></tr><tr><td>600 rpm</td><td>0.003 s</td><td>33.3</td><td>1.11</td></tr><tr><td>800 rpm</td><td>0.003 s</td><td>25.0</td><td>0.833</td></tr><tr><td>1000 rpm</td><td>0.003 s</td><td>20.0</td><td>0.667</td></tr></tbody></table>

# 3 Results

## 3.1 Comparison with experimental results

To validate the model, the results are compared with the experimental findings of Sato and Ochiai [14]. They studied an outward pumping spiral groove seal working with oil. In their case the distance between the seal faces was set at 50 µm. The stator and seal chamber were made of acrylic resin for ease of visual observations and the rotor was made of stainless steel. The properties of the seal rings are given in table 2 and the operating conditions are given in table 3.

<table><caption>Table 2: Structural properties for the seal ring used by Sato and Ochiai [14]</caption><thead><tr><th>Properties</th><th>Value</th></tr></thead><tbody><tr><td>Outside radius r<sub>o</sub> [m]</td><td>0.032</td></tr><tr><td>Inside radius r<sub>i</sub> [m]</td><td>0.0256</td></tr><tr><td>Groove radius r<sub>g</sub> [m]</td><td>0.030</td></tr><tr><td>Groove width w [%]</td><td>50</td></tr><tr><td>Depth of the spiral groove h<sub>g</sub> [µm]</td><td>50</td></tr><tr><td>Spiral angle β [degree]</td><td>70</td></tr><tr><td>Number of the spiral grooves N</td><td>15</td></tr></tbody></table>
<table><caption>Table 3: Experimental conditions used by Sato and Ochiai [14]</caption><thead><tr><td>Properties</td><td>Value</td></tr></thead><tbody><tr><td>Rotational speed ω [RPM]</td><td>2000</td></tr><tr><td>Pressure at inner radius p<sub>i</sub> [Pa]</td><td>100 000</td></tr><tr><td>Pressure at outer radius p<sub>o</sub> [Pa]</td><td>120 000</td></tr><tr><td>Film thickness h [μm]</td><td>50</td></tr><tr><td>Outer fluid</td><td>Oil</td></tr><tr><td>Viscosity of oil at 40 °C μ<sub>l</sub> [mPa.s]</td><td>27.5</td></tr><tr><td>Density of oil ρ<sub>l</sub> [kg.m<sup>-3</sup>]</td><td>860</td></tr><tr><td>Inner fluid</td><td>Ambient air</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>Fig. 4: Comparison of the sealing interface: measurements from [14] and present model with h = 50 μm</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5: Comparison of the sealing interface: measurements from [14] and present model for different values of the film thickness</figcaption></figure>

As can be seen in figure 4, the authors have captured two different areas in the sealing interface during steady state operation of the seal at 2000 rpm. At the inner radius, there is air and oil located at the outer radius. The spiral grooves suck the air and push the pressurized oil toward the outer radius. The results obtained with the present model are also reported in figure 4.

For a better comparison of the results, density profiles are extracted along the black solid lines in figure 4. The calculated density profile for $h = 50 \ \mu m$ is presented in figure 5 as well as the density profile in the experiments. In the simulations, the density varies more smoothly than in the experiments because of the mixing due to the sliding speed and because the surface tension between air and oil is neglected. The interface is considered to be located at the average density value (thin black dashed line). The location of the interface in the simulation appears to be at a slightly smaller radius.

A possible explanation for the difference between the simulation and the test is that the surface tension of the oil is neglected in the model. A second possibility is that the distance between the seal rings varied during the tests. Indeed, during their tests, Sata and Ochiai [14] measured a temperature rise of more than 10 °C in the stator. These temperature gradients can lead to deformations of the surfaces and a variation of the sealing gap. To explore this scenario, the initial film thickness of 50 µm was varied by ± 10%. The results with $h = 45 \ \mu m$ and 55 µm are also presented in figure 5. When $h$ is reduced, the pumping of the grooves is more efficient leading to less oil in the sealing area and a better correlation with the experiments. The opposite is observed when $h$ is increased. Based on this comparison, it can be considered that the model is faithful enough to simulate two-phase flows in outward pumping seals.

## 3.2 Operating conditions and configuration of the seal

The design parameters of the studied seal are given in table 4. The operating conditions are detailed in table 5. In this work, start-up simulations are performed. At $t = 0$ s, the rotational speed increases suddenly from 0 to the specified value. At $t = 0$ s, it is assumed that the interface is filled by the fluid from the low pressure side (air, $\lambda = \lambda_i$ everywhere). The film thickness is initially set at $h = 0.1 \ \mu m$.
<table><caption>Table 4: Geometrical properties for the seal ring</caption><thead><tr><th>Properties</th><th>Value</th></tr></thead><tbody><tr><td>Outer radius r<sub>o</sub> [m]</td><td>0.02875</td></tr><tr><td>Inner radius r<sub>i</sub> [m]</td><td>0.02575</td></tr><tr><td>Groove radius r<sub>g</sub> [m]</td><td>0.02755</td></tr><tr><td>Groove width w [%]</td><td>50</td></tr><tr><td>Depth of the spiral groove h<sub>g</sub> [µm]</td><td>2</td></tr><tr><td>RMS roughness height S<sub>q</sub> [µm]</td><td>0.1</td></tr><tr><td>Correlation length Sal [µm]</td><td>8.5</td></tr><tr><td>Dry friction coefficient between asperities</td><td>0.2</td></tr><tr><td>Spiral angle β [degree]</td><td>82</td></tr><tr><td>Number of grooves N</td><td>30</td></tr><tr><td>Balance ration B</td><td>1.23</td></tr><tr><td>Spring force F [N]</td><td>240</td></tr><tr><td>Stator mass m [g]</td><td>92.17</td></tr></tbody></table>

<table><caption>Table 5: Operating conditions for the seal ring</caption><thead><tr><th>Properties</th><th>Value</th></tr></thead><tbody><tr><td>Outer pressure p<sub>o</sub> [MPa]</td><td>0.15–0.4</td></tr><tr><td>Inner pressure p<sub>i</sub> [MPa]</td><td>0.1</td></tr><tr><td>Rotational speed ω [Rpm]</td><td>10-1000</td></tr></tbody></table>

<table><caption>Table 6: Fluid's parameter</caption><thead><tr><th>Properties</th><th>Viscosity [Pa.s]</th><th>Density [Kg/m<sup>3</sup>]</th></tr></thead><tbody><tr><td>Air</td><td>18.10-2</td><td>1.292</td></tr><tr><td>Eau</td><td>1.10-3</td><td>1000</td></tr><tr><td>Oil</td><td>2.10-2</td><td>860</td></tr></tbody></table>

### 3.3 Results at $\omega = 800$ rpm and $p_o = 0.4$MPa

The results obtained at $\omega = 800$ rpm with three types of fluids are firstly presented. In the first case, the seal operates with air only ($\lambda = \lambda_i = \lambda_o = 1.0$). The second case is a mixture of water only ($\lambda = \lambda_i = \lambda_o = 0.01$), and the last case is with a two-phase flow where the fluids at the boundaries are different ($\lambda_i = 1.0$ and $\lambda_o = 0.01$). The evolution of the thickness of the film and of the area weighted average fluid viscosity $\bar{\mu}$ with the number of revolutions during start-up are presented respectively in fig 6 a) and b). For the two first cases, the average viscosity obviously does not vary.

In the case of air (dotted grey line), the load carrying capacity is not high enough to maintain the thickness of the film at its initial value, and this thickness decreases with time. In the mixture of water (dashed purple line) having a higher viscosity, an increase in this thickness is observed with time due to the hydrodynamic effect provided by the spiral grooves. In the case of two phases (solid red line), a non monotonic evolution is observed. As the fluid is initially air, the evolution is similar to that of the air lubricated case during the first milliseconds.
<figure><img src="image_7.png" /><figcaption>Fig. 6: Variation with time at $\omega = 800$ rpm and $p_o = 0.4$ MPa - a) film thickness - b) average viscosity</figcaption></figure>

After 0.01 s, the average dynamic viscosity starts to increase. Some water mixture is pushed from the outer radius to the sealing area by the pressure difference. The viscosity of the fluid in the sealing area increases and, after a few oscillations, reaches a stable value. This higher viscosity allows a slight increase in the thickness of the film compared to the case with air only.

The mass flows at the inner and outer radii are plotted as functions of the number of revolutions for the case of a two-phase flow in figure 7. When the flow rate is positive, the fluid is going out of the sealing area and the opposite when it is negative. At the beginning, the fluid is squeezed out from both sides due to the decrease in the film's thickness. After 0.2 revolution (i.e. 0.01 s) the fluid starts to enter from the outer (high pressure) side. This is consistent with the increase in viscosity with time. The amount of water mixture entering the sealing area starts to decrease while at the same time the air flow from the inner radius becomes negative, meaning the air is pumped into the seals due to the grooves. After 100 revolutions, the inner and outer flow stabilize to opposite very low values. The direction of the observed flow is what is desired since it indicates that there is no leakage at the inner radius.
<figure><img src="image_8.png" /><figcaption>Fig. 7: Mass flow rate as a function of number of revolutions at ω = 800 rpm, $p_o$ = 0.4 MPa, $\lambda_i$ = 1 and $\lambda_o$ = 0.01</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8: Fluid density distribution in the sealing area at different times at ω = 800 rpm, $p_o$ = 0.4 MPa, $\lambda_i$ = 1 and $\lambda_o$ = 0.01</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 9: Fluid density profiles along the seal radius at different times at $\omega = 800$ rpm, $p_o = 0.4$ MPa, $\lambda_i = 1$ and $\lambda_o = 0.01$</figcaption></figure>

To have a better view on the way the outer fluid enters the sealing area, the distribution of the density is presented in figure 8 at different times. The sealing area is initially full of air. Then some mixture of air and water enters at the outer radius and propagates toward the inner radius. Due to the pumping effect of the air provided by the grooves, the inward movement of the interface is stopped. A solid line in yellow was added at the location where the density value is the average of air and mixture density values.

For a better view of the interface location, radial density profiles are extracted in the sealing interface, out of the grooves and presented in figure 9. The interface tends to move toward the inner when times is running. a slight step back of the interface at 20 revolutions is observed. It corresponds to the minimum in average viscosity (figure 6 b). At about 270 revolutions, the interface position is stabilized.

## 3.4 Influence of the rotational speed

The same simulations were performed at different rotational speeds from 10 to 1000 rpm. The evolution of the thickness of the film with the number of revolutions is presented in figure 10 a) and the evolution of the average viscosity as a functions of the number of revolutions in 10 b). Generally speaking, the thickness of the film is higher when the rotational speed and the hydrodynamic effect are higher. For all the considered speed values, the thickness of the film first decreases because the hydrodynamic generated pressure is not high enough to maintain a full film. Except for 1000 rpm, the thickness of the film reaches a minimum and then increases to a peak and stabilizes after a few oscillations. The magnitude of the oscillation increases with speed.

The increase in film is correlated with an increase in viscosity because of the mixture of water entering the seal from high pressure side. The occurrence of minimum and maximum in thickness of the film is shifted to higher number of revolutions when the speed is increased. The higher number of revolutions however corresponds to lower time. This is probably due to the fact that the air pumping due to the grooves is less efficient at low speed, more time is thus needed to reach equilibrium. A second effect of the less efficient pumping is that the amount of water mixture increases when the speed is lower (see figure 10 ). On the opposite, when the speed is high enough, that is to say 1000 rpm, the pumping is so efficient that the fluid from the high pressure side is not able to enter the contact. The seal is then only lubricated with air.
<figure><img src="image_11.png" /><figcaption>Fig. 10: Variation with the number of revolutions for different rotational speeds at $p_o = 0.4$ MPa, $\lambda_i = 1$ and $\lambda_o = 0.01$ - a): Film thickness b): Average viscosity</figcaption></figure>

The fluid density distributions in the sealing area at different rotation speeds are presented in figure 11 for the steady state condition. As discussed before, the amount of water mixture in the contact decreases when the operating speed is increased. The location of the interface between air and mixture is pushed to higher radius and eventually out of the sealing area when the speed is raised. For a better view of the interface location, radial density profiles are extracted in the sealing interface, out of the grooves and presented in figure 12. The significant impact of speed on the fluid density distribution in the sealing interface is clearly shown.

<figure><img src="image_12.png" /><figcaption>Fig. 11: Fluid density distribution in the sealing area at steady state for $p_o = 0.4$ MPa, $\lambda_i = 1$, $\lambda_o = 0.01$ and different rotational speeds: 10-1000 rpm.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 12: Fluid density profiles along the radius at steady state for $p_o = 0.4$ MPa, $\lambda_i = 1$, $\lambda_o = 0.01$ and different rotational speeds: 10-1000 rpm.</figcaption></figure>

## 3.5 Influence of the fluid pressure

The same simulations were performed for different pressures of the sealed fluid but at a rotational speed of 600 rpm. For these case, the time step was set at $\Delta t = 0.003$ s. The evolution of the thickness of the film and average viscosity with the number of revolutions are presented in figure 13.

<figure><img src="image_14.png" /><figcaption>Fig. 13: Variation with the number of revolutions for different pressures at $\omega = 600$ rpm, $\lambda_i = 1$ and $\lambda_o = 0.01$ - a) Film thickness - b) Average viscosity</figcaption></figure>

As expected, the thickness of the film decreases if the pressure difference is increased. Moreover, the average dynamic viscosity in the sealing area increases with the pressure. As the pressure difference increases, the pumping effect of the groove becomes less efficient in comparison. The balance between the hydrostatic pressure of the mixture and the hydrodynamic pressure of the air is reached at a smaller radius. In contrast, when the pressure
difference is reduced down to 1.5 bar, the mixture is not able to enter the sealing area and the seal works only with air.

<figure><img src="image_15.png" /><figcaption>Fig. 14: Fluid density distribution in the sealing area at steady state for $\omega = 600$ rpm, $\lambda_i = 1$ and $\lambda_o = 0.01$ at different pressures</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15: Fluid density profiles along the seal radius area at steady state for $\omega = 600$ rpm, $\lambda_i = 1$ and $\lambda_o = 0.01$ at different pressures</figcaption></figure>
The fluid density distributions in the sealing area at different outer pressures are presented in figure 14. As can be seen, the amount of water mixture in the contact increases when the pressure is increased. The location of the interface between the air and the mixture is pushed to a larger radius and eventually out of the sealing area when the pressure is decreased. For a better view of the interface location, radial density profiles are extracted in the sealing interface, out of the grooves and presented in figure 15. The significant impact of speed on the fluid density distribution in the sealing interface is clearly shown.

## 4 Discussion

As the results were obtained for different speeds and pressures, it is convenient to present the result as a function of the duty parameter G:

$$G = \frac{\mu\omega}{2\pi B(p_o - p_i)} \qquad (15)$$

As the fluid viscosity significantly changes during operation, more particularly for two-phase operation, a dimensional duty parameter $G^* = \frac{G}{\mu}$ without viscosity was used. The friction coefficient is obtained from the calculated friction torque $C_f$, the closing force $F$ and the mean radius of the sealing interface $R_{mean}$:

$$f = \frac{C_f}{F R_{mean}} \qquad (16)$$

<figure><img src="image_17.png" /><figcaption>Fig. 16: Friction coefficient as a function of the duty parameter</figcaption></figure>

The apparent friction coefficient $f$ of the seal is plotted as a function of the dimensional duty parameter in figure 16. Results obtained with air lubrication, mixture lubrication and two-phase flow are presented. Since the mixture has a much higher viscosity, there can be a rapid transition from a mixed regime (where $f$ is a decreasing function of the duty parameter) to a hydrodynamic regime. For the latter regime, a low level of the friction is reached. When the seal is lubricated with air, it is not possible to reach a full film regime for the conditions used
in the present work. The real behavior of the seal is described by the two-phase curve. At very low values of the duty parameter $G* < 4.5 \times 10^{-7} Pa^{-1}.s^{-1}$, the seal behaves as a mixture lubricated seal. When the duty parameter is increased, some air is pumped in the sealing area, and there is a progressive transition from the mixture to the air operations. Full air operation is reached when $G* = 4.53 \times 10^{-5} Pa^{-1}.s^{-1}$.

## 5 Conclusion

A two-phase flow model able to deal with air ingestion in outward pumping spiral groove seals has been developed. A transport equation is solved to determine the distribution of the gas mass fraction between the faces of the seal. Some simulations of start-up with different final speeds and several pressure differences were performed assuming isothermal operation and no deformation of the seal face. It was shown that the distribution of the fluid in the sealing interface evolves with time. When the speed is increased or the pressure difference is reduced, the amount of air ingested in the sealing area increases. An air lubrication regime is reached when $G* \ge 4.53 \times 10^{-5} Pa^{-1}.s^{-1}$. For lower values of this parameter, the seal operates in two-phase flow, showing the interest of the present approach. When $G* < 4.53 \times 10^{-5} Pa^{-1}.s^{-1}$, the seal if fully lubricated with the high pressure fluid. For the next steps, it will be interesting to investigate this mechanical seal with thermal effect or faces misalignment.

## Acknowledgments

This work pertains to the French government program France 2030 (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01 and EUR INTREE, reference ANR-18-EURE-0010).
## Nomenclature

$\Delta t, t_f$ = Time step of the temporal scheme, final time (s)

$\dot{m}_r$ = Radial mass flow (kg.s⁻¹)

$\lambda$ = gas mass fraction of the mixture

$\mu, \mu_t, \mu_g$ = mixture, liquid, and gas viscosity (Pa.s)

$\omega$ = rotational speed (rad.s⁻¹)

$\rho, \rho_t, \rho_g$ = mixture, liquid, and gas density (kg.m⁻³)

$B$ = Balance ratio

$F$ = Closing force

$f$ = Friction coefficient

$h$ = film thickness (m)

$p$ = Absolute fluid pressure (Pa)

$p_i, p_o$ = Inner and outer fluid pressure (Pa)

$q_x, q_y$ = Mass flow rate per unit length in x and y directions (kg.s⁻¹.m⁻¹)

$q_{ri}, q_{ro}$ = mass flow rate per unit length at inner and outer radii (kg.s⁻¹.m⁻¹)

$r_g$ = ideal gas constant (J.kg⁻¹.m⁻¹)

$r_i, r_o, r_g$ = inner, outer, and groove radii (m)

$S$ = Total surface area of the contact interface (m²)

$Sal$ = Roughness correlation length $Sal$ (m)

$Sq$ = Rms roughness height (m)

$T$ = Absolute temperature (K)

$x, y, z$ = Space coordinate (m)

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