# Numerical analysis of the texture effect on the hydrodynamic performance of a mechanical seal

Mohand Adjemout, Noël Brunetière, Jean Bouyer

► To cite this version:

Mohand Adjemout, Noël Brunetière, Jean Bouyer. Numerical analysis of the texture effect on the hydrodynamic performance of a mechanical seal. Surface Topography: Metrology and Properties, 2016, 4 (1), pp.014002. 10.1088/2051-672X/4/1/014002. hal-03626579

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Submitted on 16 Apr 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Numerical analysis of the texture effect on the
hydrodynamic performance of a mechanical seal

M. ADJEMOUT, N. BRUNETIERE, J. BOUYER

Institut Pprime, CNRS - Université de Poitiers - ENSMA, UPR 3346, Département Génie mécanique
et systèmes Complexes, SP2MI, 11 Boulevard Marie et Pierre Curie, BP 30179, F86962 Futuroscope
Chasseneuil Cedex, France

**Abstract**

The purpose of this paper is to analyze the effect of the main geometrical characteristics of texture on
the hydrodynamic lubrication of a mechanical seal. A parametric study was carried out in order to
improve the performance of a mechanical seal. The numerical model used in this study solves the
Reynolds equation coupled with a mass conservative model which takes into account the cavitation
phenomenon. It is shown that among the six dimples shape tested herein, namely: cylinder, square,
triangle, truncated cone, truncated pyramid and spherical cap, the triangular dimples placed
symmetrically with respect to their bases are more effective to enhance the hydrodynamic
performance of the mechanical seal. The effect of the area and depth ratios are studied and optimized
as well. The optimized solution is able to minimize friction and leakage on a range of operating
condition.

**Keywords**

Face Seals, Hydrodynamic Lubrication, Surface texture, Triangular dimples

**1. Introduction**

Several experimental and theoretical studies showed that surface texturing is an effective approach to
improve tribological performance of sliding parallel surfaces. It was stated that there are three main
effects of textures: acting as pocket to entrap debris and reduce abrasive seizure [1], reserving
lubricant to supply the mating surfaces [2], enhancing the hydrodynamic pressure and build a thicker
film between the mating surfaces [3]. The two first contributions concern mixed lubrication which is
not considered in the present paper. Many experimental and theoretical studies, dedicated to the third
effect, showed a beneficial effect of textures on the hydrodynamic performances. Hamilton et al. [4]
experimentally demonstrated that micro-irregularities or deterministic micro-asperities can
significantly improve hydrodynamic lubrication between two parallel sliding surfaces. The authors
attribute the effectiveness of micro-textured surface to the cavitation phenomenon which appears at the
lubricated interface, preventing the occurrence of negative film pressures. Anno et al. [5] analyzed the
effect of micro-asperities with inclined top surface and found that this geometrical characteristic
enhances the hydrodynamic load generation. Two years later, the same authors studied the effect of both micro-cavities and micro-asperities [6]. They showed that micro-cavities can provide loads similar as those obtained with surfaces with micro-asperities; however the leakage in this case was drastically reduced. Etsion et al. [7] published an experimental study carried out with a textured mechanical seal. The authors tested spherical pores of 90 µm in diameter, with a depth varying from 2 to 20 µm and covering 25% of the whole surface. They stated that textured surfaces can enhance the hydrodynamic pressure and can increase the film thickness compared to untextured surfaces. Two years later the same authors [8] realized an analytical and experimental investigation with laser-textured surfaces applied to a mechanical seal lubricated with water. Their theoretical model showed that the pores depth over diameter ratio is very significant and that an optimum value of this parameter can be found in order to maximize the film pressure. However, when they compared the theoretical pressure values with those of experiments, they found that the theoretical model underestimates the pressure. Wang et al. [9] tested textured SiC against untextured SiC in water. Their results showed that area density between 4.9% and 14.4% and dimple diameter over depth ratio from 1 to 3 increased hydrodynamic pressure generation of about 2 times compared to the untextured surfaces. Yu et al. [10] found that pores with an area ratio of 20% and a dimple diameter over depth ratio of 3.6 decreases the friction torque with a factor of 2 to 3 compared to the untextured surfaces. Yan et al. [11] experimentally found an optimal configuration that minimized the friction with dimple diameter from 100 to 200µm, dimple depth ranging between 5 to 10 µm, and area ratio of 5%. Qiu et al. [12] realized a series of experiments with laser textured stainless steel lubricated with oil. The authors showed that even if the texture can reduce the friction coefficient, it can increase the wear of the tested specimens. Most of the works mentioned previously recognize that area ratio or density ratio (ratio of the total dimples area to the whole surface area) and aspect ratio are the important parameters which enhance the lubrication performance.

However, in recent years, the effect of the texture geometric shape and orientation on the lubrication performance has been the focus of research, many works being thus aimed on this topic. Siripuram et al. [13] demonstrated that the friction coefficient is independent of asperities shape and orientation, while the leakage and film thickness are very sensitive to the shape and orientation of the texture patterns. In the work of Costa et al. [14], chevron dimples pointing along the sliding direction were found to give a higher film thickness compared to the circular and groove dimples. Qiu et al. [15] numerically investigated the effect of circular and ellipsoidal shapes applied on parallel slider bearings. The authors pointed out that, ellipsoidal dimples with a major axis perpendicular to the sliding direction give the maximum load carrying capacity. Yu et al. [16] studied several dimple shapes including circle, ellipse and triangle, with different orientation regarding to the sliding direction. They showed that ellipse and triangle respectively with a major axis and the base perpendicular to the sliding direction give the largest load carrying capacity. Guo et al. [17] showed that oriented triangles have a significant ability to control the fluid flow in the case of lip seal. Thus,
the researchers also recognize that the shape of the texture patterns and their orientation have a very
important effect to enhance the performance of hydrodynamic lubrication.

In the present work, a theoretical model is used to investigate the behavior of a mechanical seal operating under hydrodynamic lubrication regime when one of its friction surfaces is micro-textured. The model is based on the Reynolds equation coupled with a mass conservative cavitation algorithm. In the first part of this study, various geometrical shapes of dimple are analyzed to investigate the local (with a single pattern) and the global (with two patterns) effect of texture. Both area ratio and depth ratio of texture are the studied and optimized parameters in this section. These results are then used in the second part of this study which is dedicated to the analysis of the distribution effect of triangular patterns in the sealing area of a mechanical seal. The running conditions have been chosen to match with real operating conditions of a mechanical seal. An optimization of the implementation of triangular patterns on the matting rings is performed in order to minimize friction and leakage.

**2. Theoretical model**

**2.1 Governing equations**

The geometrical configuration of the contact model consists in two rings having a relative rotational velocity as is shown in Figure 1. The problem is assumed to be stationary. This is achieved by considering that the upper ring is smooth and rotates with (ω) velocity while the lower ring is at rest and textured.

<figure><img src="image_2.png" /><figcaption>Figure 1: Geometric configuration of the problem</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Fluid flow model [18-19]</figcaption></figure>

Assuming that the fluid is Newtonian and in laminar flow regime as well as usual lubrication assumptions and taking into account the cavitation occurrence, a modified Reynolds equation is obtained:

$$F \frac{\partial}{\partial r} \left( \frac{r h^3}{\mu} \frac{\partial D}{\partial r} \right) + F \frac{\partial}{\partial \theta} \left( \frac{r h^3}{\mu r} \frac{\partial D}{\partial \theta} \right) = 6 V \left[ \frac{\partial h}{\partial \theta} + (1-F) \frac{\partial}{\partial \theta} (h D) \right] \quad (1)$$

where h is the film thickness, μ the viscosity and V denotes the speed of the untextured ring. F and D are the variables related to the cavitation algorithm. More details about the model are to be found in [18,19].

$$\left\{ \begin{array}{l} D = p - p_{cav} > 0 \\ \rho = \rho_0 \\ F = 1 \end{array} \right. \qquad \left\{ \begin{array}{l} p = p_{cav} \\ D = \frac{\rho}{\rho_0} - 1 < 0 \\ F = 0 \end{array} \right. \qquad (2)$$

Because of circumferential periodicity, only one small radial area band from the inner radius $R_i$ to the outer radius $R_0$ is studied, as shown in Figure 1. Thus, a periodic condition in the lateral boundaries is necessary to ensure the continuity of the domain. The fluid flow problem is discretized by a control volume method (Figure 2). For a control volume, the continuity equation is given by:

$$q_w - q_e + q_s - q_n = 0$$

This equation represents the mass flow balance in a micro-cell. According to [18, 19], the terms of the mass flow equation can be expressed by:

At the west and east sides:
$$q_w = -\frac{h_w^3}{12\mu} R_w \Delta\theta \frac{\partial}{\partial r} (FD)_w \approx -\frac{h_w^3}{12\mu} R_w \Delta\theta \frac{F_p D_p - F_w D_w}{\Delta R} \quad (4)$$

$$q_e = -\frac{h_e^3}{12\mu} R_e \Delta\theta \frac{\partial}{\partial r} (FD)_e \approx -\frac{h_e^3}{12\mu} R_e \Delta\theta \frac{F_e D_e - F_p D_p}{\Delta R} \quad (5)$$

At the south and north sides:

$$q_s = -\frac{h_s^3}{12\mu} \frac{\Delta R}{R_s} \frac{\partial}{\partial \theta} (FD)_s + 0.5h_s \Delta R R_s \omega [1 - D_s(1 - F_s)] \approx -\frac{h_s^3}{12\mu} \frac{\Delta R}{R_s} \frac{F_p D_p - F_N D_N}{\Delta \theta} + 0.5h_s \Delta R R_s \omega [1 - D_s(1 - F_s)] \quad (6)$$

$$q_n = -\frac{h_n^3}{12\mu} \frac{\Delta R}{R_n \Delta \theta} \frac{\partial}{\partial \theta} (FD)_n + 0.5h_n \Delta R R_n \omega [1 - D_p(1 - F_p)] \approx -\frac{h_n^3}{12\mu} \frac{\Delta R}{R_n} \frac{F_N D_N - F_p D_p}{\Delta \theta} + 0.5h_n \Delta R R_n \omega [1 - D_p(1 - F_p)] \quad (7)$$

In these equations, h is the local film thickness, ΔR and Δθ are the sampling interval of the mesh.

## 2.2 Texture Geometry

Figure 3 shows the six different geometric shapes of the texture analyzed in this study: cylinder, triangle, truncated cone, truncated pyramid, spherical cap and square. This figure represents an elementary texture cell. It should be noted that the positions of the lower and the upper surfaces are inverted in this figure in order to facilitate the presentation of the dimple geometries. The common geometrical characteristic of the dimples is that they are symmetrical according to the radial direction. This constraint on orientation has been decided to allow the mechanical seal to work in both sliding directions. The area ratio δ corresponding to each dimple's shape is defined in table1. In figure 3 and Table 1, R<sub>d</sub> is the radius of the circular dimple shape (cylinder, spherical cap and truncated cone), a represents the side length of noncircular dimple, R<sub>b</sub> and b represent respectively the radius at the bottom of the truncated cone and the width at the bottom of the truncated pyramid. L<sub>θ</sub> is the curvilinear length of the cell measured at the mean radius (R<sub>o</sub>+R<sub>i</sub>)/2 of the cell while L<sub>r</sub> is the radial length. For the truncated cone and the truncated pyramid, the ratio between the width (radius) at the top and the width (radius) at the bottom of the truncated pyramid (truncated cone) is set at 2 (a/b= R<sub>d</sub>/R<sub>b</sub>=2).
<figure><img src="image_4.png" /><figcaption>Figure 3: Geometric characteristics of the dimple shapes: (a) cylinder, (b) triangle, (c) truncated cone, (d) truncated pyramid, (e) spherical cap, (f) square</figcaption></figure>

<table><caption>Table1: Area ratio of each dimple's shape</caption><thead><tr><td>Dimple shape</td><td>Area ratio δ</td></tr></thead><tbody><tr><td>Spherical cap</td><td>π R<sub>d</sub><sup>2</sup> / L<sub>r</sub>L<sub>θ</sub></td></tr><tr><td>Truncated cone</td><td>π R<sub>d</sub><sup>2</sup> / L<sub>r</sub>L<sub>θ</sub></td></tr><tr><td>Cylinder</td><td>π R<sub>d</sub><sup>2</sup> / L<sub>r</sub>L<sub>θ</sub></td></tr><tr><td>Truncated pyramid</td><td>a<sup>2</sup> / L<sub>r</sub>L<sub>θ</sub></td></tr><tr><td>Square</td><td>a<sup>2</sup> / L<sub>r</sub>L<sub>θ</sub></td></tr><tr><td>Triangle</td><td>ab / 2 L<sub>r</sub>L<sub>θ</sub></td></tr></tbody></table>
### 3. Local and global effect of texture

During the first part of this study, the local and the global effects of the textures are investigated. Thus, two configurations have been analyzed as described in Figure 4. The first configuration, Figure 4 (a) is dedicated to study the local effect of texture. In this configuration, the studied domain contains one dimple across the width of the mechanical seal. The second configuration is aimed to study the global effect of the texture, the studied domain containing two dimples (Figure 4 (b)). Texture's parameters such as area ratio and depth ratio are discussed and optimized. The sides of an elementary virtual cell are arbitrary chosen: $L_r=200$ µm and $L_θ=200$ µm. According to Adjemout et al. [20] each virtual cell is discretized with a uniform grid of 200x200. During this part, the simulations are realized with a constant clearance of $h_f = 1$ µm between the mating faces of the mechanical seal. The mechanical seal dimensions, texture parameters and running conditions are summarized Table 2.

<figure><img src="image_5.png" /><figcaption>Figure 4: Two studied configurations of the problem: (a) local effect configuration, (b) global effect configuration</figcaption></figure>
<table><caption>Table 2: Data used to study the local and the global effect of texture</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Unit</th><th>Value</th></tr></thead><tbody><tr><td>Inner radius</td><td>R<sub>i</sub></td><td>mm</td><td>25.75</td></tr><tr><td>Outer radius</td><td>R<sub>o</sub></td><td>mm</td><td>25.95, 26.15</td></tr><tr><td>Inner pressure</td><td>p<sub>i</sub></td><td>MPa</td><td>0.01</td></tr><tr><td>Outer pressure</td><td>p<sub>o</sub></td><td>MPa</td><td>0.01</td></tr><tr><td>Cavitation pressure</td><td>p<sub>cav</sub></td><td>MPa</td><td>0</td></tr><tr><td>Viscosity</td><td>μ</td><td>Pa.s</td><td>0.001</td></tr><tr><td>Fluid density</td><td>ρ<sub>o</sub></td><td>kg/m<sup>3</sup></td><td>1000</td></tr><tr><td>Depth ratio</td><td>λ=h<sub>ρ</sub>/h<sub>f</sub></td><td>....</td><td>1, 5 - 2, 5</td></tr><tr><td>Area ratio</td><td>δ</td><td>....</td><td>0.1 - 0.3</td></tr><tr><td>Velocity</td><td>ω</td><td>rad/s</td><td>30-500</td></tr><tr><td>Clearance</td><td>h<sub>f</sub></td><td>μm</td><td>1</td></tr></tbody></table>

Seal performance such as hydrodynamic load carrying capacity, leakage and cavitation occurrence are calculated and presented in a non-dimensional form:

* Dimensionless load

$$ \bar{p} = \frac{1}{S \times p_o} \int_S p.dS - 1 \qquad (8) $$

* Dimensionless leakage

$$ \bar{Q} = \frac{12\mu(R_o - R_i)}{2\pi p_o h_f^3 R_m} Q \qquad (9) $$

* Cavitation Ratio

$$ \bar{\zeta} = \frac{N_c}{\delta \times N} \qquad (10) $$

In the following sections, the evolution of these variables is presented versus a dimensionless hydrodynamic parameter which is given by:

$$ \psi = 6 \frac{\mu\omega R_m (R_o - R_i)}{p_o h_f^2} \qquad (11) $$

where $R_m$ is the average radius, $S$ represents the total area of the domain, $Q$ the radial leakage, $N_c$ the number of cavitated nodes and $N$ the total number of nodes used in the simulations.

### 3.1 Local effect of the dimple

Due to large number of simulations, the results presented here are averaged on the whole range of the hydrodynamic parameter [96.2:1620].
The result of average dimensionless load carrying capacity is illustrated in Figure 5. Among the six dimples tested herein, the cylindrical shape shows the highest ability to increase the hydrodynamic load. At the opposite, the triangular dimple was not able to generate a positive load carrying capacity. Regarding the area ratio of the patterns, all of the texture shapes provide a maximum load with a ratio of 0.3. In terms of depth ratio, it is found that the edge profile of the dimples has a considerable influence on the generation of load carrying capacity: the patterns having a vertical edge (cylinder, square) are more efficient with a depth ratio of 5/2, while patterns presenting a tilt or curve at the edge (truncated cone, truncated pyramid and spherical cap) give higher load carrying capacity with a depth ratio of 5.

<figure><img src="image_6.png" /><figcaption>Figure 5: Average dimensionless load carrying capacity with one dimple</figcaption></figure>

Figure 6, shows the result of average dimensionless leakage. It can be observed that the highest value of average dimensionless leakage is obtained with the triangular pattern. This value is about 10³ greater than those observed with the other shapes of dimple. The maximum leakage is obtained with an area ratio of 0.3 and depth ratio of 5/2. The reason of this significant increase in leakage with the triangular pattern is due to the pumping rate which is increased with this kind of dimple. Indeed, except the triangular dimple, all of the dimple shapes tested herein have a symmetry axis in the circumferential direction. This will logically result in a symmetrical pressure distribution in the radial direction (in the case of perfect square cell), leading the pumping rate to approach zero, thus limiting the leakage. On the other hand, the non-symmetrical shape of the triangle provides an asymmetrical pressure distribution leading to a significant leakage.
<figure><img src="image_7.png" /><figcaption>Figure 6: Average dimensionless leakage with one dimple</figcaption></figure>

The average cavitation ratio is displayed in Figure 7. The results show that the cavitation occurrence is less dependent on the area ratio of the dimples than the other parameters. Regarding the depth ratio, the average cavitation reaches a maximum value around 0.9 with a depth ratio of 5/2 for the dimples having a vertical edge while it remains relatively constant (around 0.85) for the patterns presenting a tilted or a curved edge.

<figure><img src="image_8.png" /><figcaption>Figure 7: Average cavitation with one dimple</figcaption></figure>

The results issued from this analysis show that the cylindrical dimple is more appropriate to generate load carrying capacity while the triangle is not able to generate a sufficient load carrying capacity and leads to a high leakage. All the dimples provide a maximum load carrying capacity with an area ratio of 0.3 while the depth ratio seems to depend on the shape profile of the texture pattern. Thus, in order to reduce the number of the simulations in the following analysis, the area ratio and depth ratio are fixed respectively at 0.3 and 5/2
### 3.2 Global effect of the dimples

In the second section of this part, the studied domain contains two dimples of texture. Figure 4 (b) shows the configurations tested in this section. The triangular dimples are placed symmetrically with respect to their bases in the radial direction.

<figure><img src="image_9.png" /><figcaption>Figure 8: (a) Dimensionless load carrying capacity for the configurations with two dimples (b) Dimensionless load carrying capacity for configuration with one dimple</figcaption></figure>

Results of dimensionless load carrying capacity obtained with two dimples are illustrated in Figure 8(a) and compared to the cases with one dimple in the studied domain in Figure 8(b). In contrast to what was observed with one dimple (Figure 8 (b)) where the triangle was not able to generate a positive load carrying capacity, the maximum load carrying capacity is here (Figure 8 (a)) obtained with the triangular dimples. Its value is around 4 times greater than that of cylindrical dimple for the highest value of the hydrodynamic parameter. Furthermore, it can be seen from Figure 8 (a) that unlike the triangular dimples, all other shapes lead to a similar level of the load carrying as that with one dimple (Figure 8 (b)). The result of the triangular shape can be explained by the pumping effect created with the triangular patterns. Indeed, the way that the triangles are oriented helps in pumping the fluid along the two directions, thus leading to build-up hydrodynamic pressure between the two triangles. This positive effect is not observed with the other texture shapes as shown in Figure 9.
<figure><img src="image_10.png" /><figcaption>Figure 9: Example of pressure field with the spherical cap and the triangular dimples</figcaption></figure>

The variation of the dimensionless leakage with the hydrodynamic parameter is presented in Figure 10. It should be noted that a positive value of the leakage means that the fluid is pumped from the inner side to the outer side of the mechanical seal and a negative value an inverse leakage flow direction. If one compares the dimensionless leakage of the configurations containing two triangles (Figure 10 (a)) with that obtained with one triangle (Figure 10 (b)), a very significant reduction is observed when triangles are used. This is due to the reverse pumping of the outer triangle which balances the pumping effect of the inner triangle which then leads to an almost symmetrical distribution of the pressure field in the radial direction (Figure 9). It is thus possible to reach leakage flow values as low as those obtained with other dimple types

<figure><img src="image_11.png" /><figcaption>Figure 10: (a) Dimensionless leakage for the two dimples (b) Dimensionless leakage for the one dimple</figcaption></figure>
The results of cavitation ratio are displayed in Figure 11. The cavitation is very sensitive to the hydrodynamic parameter and, as expected, it increases when increasing the hydrodynamic parameter. It tends to reach asymptotically 100% of the textured area. Another remark can be done about the triangular patterns which give a slightly lower cavitation ratio compared to the other shapes. This is probably due to the high level of pressure created between the two triangles which reduce the spread of the cavitation zone.

<figure><img src="image_12.png" /><figcaption>Figure 11: Cavitation ratio for the two dimple configurations</figcaption></figure>

As shown throughout the analysis, triangular dimples placed symmetrically with respect to their bases lead to an improvement in the lubrication performance of the mechanical seal. Thus, the triangular dimples are retained for the following study and we propose to investigate further this shape of dimple.

### a) Effect of triangles orientation

In this part of the analysis, three orientations with two triangles are realized as illustrated in Figure 12 in order to evaluate the efficiency of the chosen configuration (A).

<figure><img src="image_13.png" /><figcaption>Figure 12: configuration with different orientations of the triangles</figcaption></figure>
Figure 13 illustrates the dimensionless load carrying capacity corresponding to the four orientations of the triangles: A, B, C and D. Configuration A appears to be the best in load carrying capacity except for low hydrodynamic parameter values where configuration B can give a higher load.

<figure><img src="image_14.png" /><figcaption>Figure 13: Dimensionless load carrying capacity with the different orientations of the triangles</figcaption></figure>

This can be explained by the fact that triangle A configuration leads the pumping effect to appear, providing high pressure generation between the triangles when increasing the hydrodynamic parameter. For configuration D, the pumping effect is inversed and promotes the generation of cavitation between triangles and thus negative load values. On the other hand, the pumping effect does not appear with configuration B and C for which the global pressure is that generated at the downstream of the dimples. When comparing the B and C triangles configurations, the B one exhibits a higher ability to generate load carrying capacity compared to the C triangles. This is due to the fact that with the triangle B configuration, the apexes of the dimples are in the downstream position, which creates a higher resistance to the outlet fluid flow leading then to a higher pressure at the downstream of the dimples. The findings prove that the triangle's orientations with respect to the sliding direction of the motion surface have a significant effect on the hydrodynamic pressure. It confirms that configuration A is the best because of the collective effect resulting from pumping.

### b) Effect of distance between the symmetrical triangles

In this section the operating conditions remain similar to those presented in Table 1, except the outer radius which is set at 26.75 mm. The computational domain in this case can contain five virtual cells of 200µm x 200µm. Two triangles with a local area ratio of 0.3 are implemented in the computational domain. Thereafter, the distance separating the triangles is varied as shown in Figure 14.
The corresponding dimensionless load carrying capacity for each distance is illustrated in Figure 15. It reveals that the load carrying capacity generated with triangles placed in symmetric position is increased by increasing the distance separating the triangles. This can be explained by the fact that, by increasing the distance between the two triangles, the area where the pumping effect takes place will increase, leading then to a large area with high pressure, as it can be seen on the pressures fields presented in Figure 16.

<figure><img src="image_15.png" /><figcaption>Figure 14: Configurations with varying the distance between the two triangles</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 15: Dimensionless load carrying capacity when varying the distance between the two triangles</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 16: Examples of dimensionless pressure fields when varying the distance between the two triangles</figcaption></figure>

**4. Effect of the triangular dimples distribution**

In the second part of this study, the configuration with triangles placed symmetrically with respect to their bases is extended to the full sealing interface of a mechanical seal as shown in Figure 17 (a). The inner and the outer radii of the mechanical seal are respectively 25.75 and 28.75 mm. The studied domain has an averaged width of 200 µm. The computational domain can be divided in 15 virtual square cells of about 200µm x 200µm.

<figure><img src="image_18.png" /><figcaption>Figure 17: Configuration with several distributions of the triangles on the radial bound</figcaption></figure>

The aim of this part of the study is to analyze the effect of varying the implementation of the dimples on the studied domain on the performance of a mechanical seal. To achieve this goal, several triangles distributions are investigated as presented in Figure 17 (b). It should be noted that the global
area ratio is not maintained constant and can vary by changing the number of dimples used. However, for all the configurations the size of the dimples is maintained constant corresponding to an average local area ratio of 0.3. Table 3 presents the global area ratio corresponding to each implementation (configuration).

<table><caption>Table 3: Global area ratio for each configuration of the triangles</caption><thead><tr><td>Configuration</td><td>Global area ratio</td></tr></thead><tbody><tr><td>7i-7o</td><td>0.28</td></tr><tr><td>10i-4o</td><td>0.28</td></tr><tr><td>9i-5o</td><td>0.28</td></tr><tr><td>9i-4o</td><td>0.26</td></tr><tr><td>8i-3o</td><td>0.22</td></tr><tr><td>7i-3o</td><td>0.2</td></tr><tr><td>9i-2o</td><td>0.22</td></tr></tbody></table>

In this part, the load $F_{close}$ applied to the mechanical seal is imposed and the film thickness is calculated to reach axial force balance. Seal performance such as radial leakage, friction and film thickness are calculated and presented as a function of the duty parameter :.

$$G = \frac{\mu \Delta r \omega R_m}{F_{close}} \quad (12)$$

The running conditions and the main characteristics of the mechanical seal are summarized in Table 4. Before addressing the study of the dimples implementation, a preliminary investigation was carried out to define an optimal depth of the triangular dimples.

<table><caption>Table 4: Data used to study the effect of the distribution of the triangles</caption><thead><tr><td>Parameter</td><td>Symbol</td><td>Unit</td><td>Value</td></tr></thead><tbody><tr><td>Inner radius</td><td>R<sub>i</sub></td><td>mm</td><td>25.75</td></tr><tr><td>Outer radius</td><td>R<sub>o</sub></td><td>mm</td><td>28.75</td></tr><tr><td>Inner pressure</td><td>p<sub>i</sub></td><td>MPa</td><td>0.1</td></tr><tr><td>Outer pressure</td><td>p<sub>o</sub></td><td>MPa</td><td>2.1</td></tr><tr><td>Cavitation pressure</td><td>p<sub>cav</sub></td><td>MPa</td><td>0</td></tr><tr><td>Viscosity</td><td>μ</td><td>Pa.s</td><td>0.001</td></tr><tr><td>Fluid density</td><td>ρ<sub>o</sub></td><td>kg/m<sup>3</sup></td><td>1000</td></tr><tr><td>Velocity</td><td>Ω</td><td>rad/s</td><td>30-500</td></tr><tr><td>Closing force</td><td>F<sub>close</sub></td><td>N</td><td>1318</td></tr><tr><td>Number of nodes</td><td>N<sub>i</sub></td><td>....</td><td>200x3000</td></tr></tbody></table>

It is well known that in case of hydrodynamic lubrication the film thickness varies monotonically with speed. Hence, the effectiveness of a given dimple depth depends on the velocity of the mechanical seal. During this investigation the depth of the triangular dimples was varied while the implementation of the triangles was set at a configuration 7i-7o (7 dimples at the inner rim and 7 dimples at the outer
rim). The results of friction coefficient as a function of the dimple depth are presented in Figure 18 for different rotational speed values.

<figure><img src="image_19.png" /><figcaption>Figure 18: Effect of dimple depth on the friction coefficient</figcaption></figure>

It is shown that for the lowest values of the rotational velocities, the dimples with small depth are more efficient in reducing the friction coefficient, while deeper dimples works better at high speed values. A dimple depth of 0.2 µm appears to provide a moderate friction on the whole considered speed range and was selected for the following study.

## 4.1 Results and discussion

Figure 19 shows the results of leakage when varying the implementation of the triangles on the seal width.

<figure><img src="image_20.png" /><figcaption>Figure 19: Effect of the triangular dimples distribution on leakage</figcaption></figure>
As could be expected, the fluid flow is very sensitive to the distribution of the triangles. It can be seen that the level and the direction of the fluid flow can be controlled by the number of triangles implemented beside the inner and outer rim of the mechanical seal. Thus, a benefit can be derived from this property in order to minimize the leakage. Indeed, over the six configurations tested herein, distributions 8i-3o and 9i-4o exhibit the smallest leakages, very close to zero, which is one of the ultimate goals of a mechanical seal. However, the results obtained with the configuration 8i-3o are situated in the positive values of the leakage, which means that this configuration has a tendency to promote the fluid pumping from the external environment (atmosphere) to the mechanical seal. This mechanism will deprive the contacting faces of lubricant, which is not beneficial to the safe operation of a mechanical seal. In fact, it is primordial to maintain a weak leakage to ensure a good lubrication of the matting faces, as it can be seen with the configuration 9i-4o.

The findings can be explained with the reverse pumping of the triangles at the inner rim which leads to mostly balance the pumping effect of the triangles at the outer rim and the leakage due to the pressure differential. In order to accurately explain this behavior, examples of radial pressure profiles for configurations 7i-3o, 8i-3o and 9i-2o are plotted in Figure 20. It appears that the slope of the pressure field in the central untextured area, situated between the two groups of dimples influences the direction of the fluid flow rate. Indeed, if the slope is positive (increase in pressure from the inside to the outside) (7i-3o), this means that the seal is leaking, while if the slope is negative (9i-2o) it means that fluid is pumped towards the process fluid. However, when the slope is close to zero (8i-3o) it corresponds to a case with very weak leakage.

<figure><img src="image_21.png" /><figcaption>Figure 20: Effect the dimple distributions on the pressure field</figcaption></figure>
The relationships between friction coefficient and distribution of the triangular dimples are displayed in Figure 21 as a function of the duty parameter. The results show that the friction coefficient is increased by increasing the duty parameter. However, the resulting friction values are nearly at the same level for all the configurations.

<figure><img src="image_22.png" /><figcaption>Figure 21: Effect of the triangular dimples distribution on friction coefficient</figcaption></figure>

This result was expected, since this configuration has the smallest global area ratio (0.2). Indeed, decreasing the number of the triangles in the sliding interface will affect the pumping effect which reduces the hydrodynamic pressure generation and it leads to a slightly lower film thickness value, as shown in Figure 22. The other tested configurations provide a very similar film thickness value which increases with the duty parameter.

<figure><img src="image_23.png" /><figcaption>Figure 22: Effect of triangular dimples distributions on film thickness</figcaption></figure>
<table><caption>Table 4: Main performance of the mechanical seal with different configurations at G=1.86 10⁻⁸</caption><thead><tr><td>Configuration</td><td>Leakage (kg/s)</td><td>Friction coefficient</td><td>Film thickness (m)</td></tr></thead><tbody><tr><td>7i-7o</td><td>-1.76 e-6</td><td>1.03e-2</td><td>2.57e-7</td></tr><tr><td>10i-4o</td><td>3.85 e-8</td><td>1.08e-2</td><td>2.46e-7</td></tr><tr><td>9i-5o</td><td>-4.11e-8</td><td>1.06e-2</td><td>2.52e-7</td></tr><tr><td>9i-4o</td><td>-1.24e-9</td><td>1.08e-2</td><td>2.49e-7</td></tr><tr><td>8i-3o</td><td>6.79e-9</td><td>1.12e-2</td><td>2.46e-7</td></tr><tr><td>7i-3o</td><td>-2.40e-9</td><td>1.14e-2</td><td>2.46e-7</td></tr></tbody></table>

Table 4 summarizes the performances obtained with the different dimples configurations at a target duty parameter (G) of 1.86 10⁻⁸. This value corresponds to the nominal velocity of the studied mechanical seal which is equal 300 rad/s. By excluding the configurations giving a positive leakage the choice is then reduced to the configurations 7i-7o, 9i-5o, 9i-4o and 7i-3o. Considering that the leakage and the friction coefficient are the main performance to take into account to define the best configuration, the choice is then done on configuration 9i-4o, which gives the smallest absolute leakage as well as a reasonably low friction coefficient.

## Conclusion

A numerical analysis has been carried out to investigate the effect of surface texturing on the performance of a mechanical seal in the case of isothermal hydrodynamic lubrication.

In the first part of this study, local and global effects of several texture shapes and orientations on the hydrodynamic performance of a mechanical seal have been analyzed. Concerning the area ratio and the depth ratio, the results showed that local area ratio of 0.3 gives the highest load carrying capacity for each dimple shape while the depth ratio seems to depend on the edge profile of the dimple. The dimples with vertical edge are more effective with a depth ratio of 5/2 and the dimples with a curve or a tilt at the edge are better with a depth ratio of 5. With a single dimple, the cylindrical shape exhibits high load carrying capacity while the triangle has a negative load carrying capacity and gives the highest leakage. On the other hand, when two dimples are used in the radial band, the triangles placed symmetrically with respect to their bases showed the highest load carrying capacity together with a significant reduction of the leakage due to a collective effect. This collective effect does not exist with other dimple shapes tested here.

In the second part of this study, the effect of the triangle distributions on the performance of a mechanical seal has been investigated. It was found that the number of triangles that are implemented beside of the inner and outer rim of the mechanical seal have a significant effect and permit to control
the flow rate of the lubricant. Triangle’s depth of 0.2 µm is suitable to maintain a low friction coefficient value throughout the range of the rotational velocity used in this study.

A parametric analysis was conducted in order to define an optimal configuration of the texture which reduces at the same time friction and leakage for the mechanical seal tested in this study. Ultimately, the optimum configuration was found to be triangular dimples with a distribution 9i-4o (9 triangle at the inner rim and 4 at the outer rim), having a local area ratio of 0.3 (global area ratio of 0.26) and a depth of 0.2 µm.

## Funding

The work presented in this study was supported by the French National Research Agency (grant no ANR-11-RMNP-0008). This work pertains to the French Government program “Investissements d’Avenir” (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).

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