Analysis of the Total Leakage Characteristics of Finger Seal Considering Fractal Wear and Fractal Porous Media Seepage Effects

Abstract

As an advanced flexible dynamic sealing technology, the leakage characteristics of a finger seal (FS) is one of the key research areas in this technology field. Based on the fractal theory, this paper establishes a mathematical model of the FS main leakage rate considering the fractal wear effect by taking into account the influence of the wear height on the basis of the eccentric annular gap flow equation. Based on the Hagen-Poiseuille law and the fractal geometry theory of porous media, a mathematical model of the FS side leakage rate considering the fractal porous media seepage effect is developed. Then, a mathematical model of the FS total leakage rate is established. The results show that the mathematical model of the FS total leakage rate is verified with the test results, the maximum error rate is less than 5%, and the mathematical model of the FS total leakage rate is feasible. With the gradual increase in working conditions and eccentricity, the FS main leakage rate gradually increases. In addition, the effects of the fractal dimension, fractal roughness parameters and porosity after loading on the FS main leakage rate are negligible. As the fractal dimension of tortuosity after loading gradually decreases, the fractal dimension of porosity after loading gradually increases, and the FS side leakage rate gradually increases. As the porosity after loading gradually increases, the FS side leakage rate gradually increases. Under different working conditions, different fractal characteristic parameters and different porosities after loading, the weight of the FS main leakage rate is much greater than that of the FS side leakage rate by more than 95%.

1. Introduction

As a green technology in the field of high-end equipment manufacturing, sealing technology has a decisive impact on the stable operation, energy consumption reduction, fatigue life and green cycle of high-end precision rotating machinery systems [1,2,3]. A finger seal (FS) is a kind of contact flexible dynamic sealing technology, which has excellent sealing performance and is widely used in the exhaust cavity of aircraft engine and the inlet and outlet of compressor [4]. The dynamic leakage rate of an FS is about 1/2 of that of standard carbon seal and 4/5 of that of standard brush seal. Its manufacturing cost is significantly lower than that of carbon seal and brush seals [5,6,7,8]. Since the FS can meet the strict requirements of high cycle parameters of aircraft engines, many researchers have carried out a lot of research on the sealing performance and mechanism of the FS.

As one of the critical indicators to measure the sealing performance, the flow leakage characteristic has been the research hotspot in the FS field. Wang et al. [9] found that the pressure fluid has a great influence on the sealing performance of an FS. Du et al. [10] found that when the pressure difference is more than 0.3 MPa, the leakage coefficient tends to be flat, and the pressure difference has no significant effect on the leakage coefficient in the interference fit. Bai et al. [11] first proposed a two-dimensional porous media computational fluid dynamics (CFD) model of an FS. Hu et al. [12] and Wang et al. [13] used a two-dimensional porous media CFD model to study the leakage characteristics of the FS. However, the two-dimensional porous media CFD model cannot accurately reflect the influence of the hysteresis gap on the leakage characteristics of an FS, so the prediction of the leakage performance of an FS may not be accurate. Chen et al. [14] first proposed a finite element analysis model that can properly reflect the hysteresis gap between the FS and the rotor. Su [15], Zhang et al. [16], Yin et al. [17], and Zhao et al. [18] used the finite element analysis model of the FS to study the influence of the hysteresis gap on the leakage performance of the FS. However, most of the researchers did not consider the effect of wear characteristics on the leakage performance of the FS.

In fact, the wear characteristics will change the assembly state of the FS/rotor, causing the seal structure to fail and the leakage rate to shift. Zhang et al. [19] found that the leakage rate is closely related to the amount of wear. Persson [20] found that the plastic deformation of metallic materials has a significant effect on the leakage rate of metallic seals. Du et al. [21] found that wear has a significant effect on the leakage characteristics in the early stage of the experiment; the effect of wear on the leakage characteristics is gradually reduced in the middle and late stages of the experiment. Therefore, it is crucial to construct the mathematical model of FS leakage characteristics considering the wear characteristics. Wang et al. [22] established a finite element analysis model of dynamic leakage of an FS considering the progressive wear effect based on the Archard model, and they found that the leakage rate considering the progressive wear effect was closer to the experimental results. Wang et al. [23] established a numerical analysis model for the transient leakage characteristics of an FS considering the wear effect based on the Archard model, and they numerically studied the leakage performance of the FS by using the two-dimensional porous media CFD model. The research showed that the leakage in the finger foot area was mainly in the early stage of wear, and the leakage in the finger beam area was mainly in the late stage of wear. However, the Archard model cannot accurately reflect the changing characteristics of the friction surface morphology in the wear process, and there is a certain error in the prediction of the wear rate, which may lead to the deviation of the prediction of the leakage rate.

Fractal theory can exclusively and quantitatively describe the contact behaviour of the contact surface, and it can effectively characterise the morphological changes of the contact surface and the leakage channel changes of the sealing interface during the operation of the sealing structure [24,25,26]. Maleki et al. [27] found that surface topography has a significant effect on the leakage rate of sealed parts. A study by Lv et al. [28] found that fractal feature parameters can effectively characterise the morphological features of the sealing surface and have a significant effect on leakage performance. Zhao et al. [29] established a fractal model for the mechanical seal leakage rate based on fractal theory, taking into account the characteristics of contact surface morphology. Li et al. [30] established a fractal model for mechanical seal end face leakage based on fractal theory and analysed the effect of changes in seal end face morphology on leakage performance from a microscopic point of view. Sun [31] used fractal feature parameters to describe in detail the morphological characteristics of the dynamic and static ring end faces of mechanical seals, established a fractal model for mechanical seal leakage, and investigated the relationship between leakage characteristics, fractal feature parameters, and working conditions. Feng et al. [32] established a fractal model for metal gasket leakage based on fractal theory, which accurately revealed the relationship between leakage characteristics and sealing surface morphology. Ni et al. [33] found that the fractal dimension has a significant effect on the contact surface throughout the working phase, and the fractal characteristic parameters have an effect on the contact surface only in the initial phase. The above research shows that fractal theory can effectively describe the morphological changes of contact surfaces and accurately reveal the relationship between the morphological changes of friction surfaces and leakage characteristics.

In recent years, most researchers have paid great attention to the fluid leakage caused by hysteresis and wear of FSs and rotors. In fact, the leakage channels of FSs are mainly divided into main leakage channels and side leakage channels. The main leakage channel is the hysteresis gap formed between the finger beam and the rotor after deformation, as well as the radial wear height caused by the wear of the FS. The radial gap is formed by the combination of the above two reasons; the side leakage channel is a microchannel formed by the interconnection of micropores on the rough contact surface of the adjacent finger seal annulus (FSA). Bai et al. [34,35] through experimental research and numerical analysis, found that the side leakage channel has a significant effect on the total leakage rate of the FS. Zhao et al. [36] considered adjacent FSAs as porous media structures and established a mathematical model for the side leakage channel rate of FSs based on Hagen-Poisson’s law and Darcy’s law. Through numerical research, they found that the side leakage channel dominates when the contact surface roughness is steep. The above research indicates that the side leakage channel can improve the total leakage rate of the FS, and it has a significant effect on the total leakage rate of the FS, which is of great significance for the composition of the total leakage channel of the FS. In a complete finger sealing system, several layers of FSAs are overlapped, staggered and stacked, tightly joined by riveting, and the adjacent FSAs are subjected to high axial pressure for a long time during the working process. As a result, the gap between adjacent FSAs is particularly narrow. Lv et al. [37] found through numerical research that rough surface characteristics have a significant effect on the air-tightness of sealing structures. Bao et al. [38] established a calculation method for the leakage rate of mechanical seals using the porous medium theory, and their research found that the surface roughness characteristics have a significant effect on the leakage rate. Huang et al. [39] established a prediction model for the leakage rate of metal gaskets based on the theory of fractal porous media, and they found that the geometric morphology characteristics of a rough surfaces are crucial factors affecting the leakage characteristics. The above research indicates that the morphological characteristics of rough surfaces have a significant impact on the side leakage channels and can be analysed from both microscopic and fractal perspectives.

Based on the fractal theory, this paper uses fractal characteristic parameters to characterise the wear height and establishes a mathematical model of the main leakage rate of the FS considering the fractal wear effect. Then, based on the fractal porous media transport theory, a mathematical model of the side leakage rate of the FS considering the fractal surface morphological characteristics is established. In addition, the mathematical model of the total leakage rate of the FS is established. The mathematical model of the total leakage rate not only reveals the influence of the fractal wear effect on the leakage rate from the macroscopic point of view, but also reveals the influence of the microscopic morphological features of the seal surface on the leakage rate from the microscopic point of view. The parameters in each of the mathematical models constructed above have specific physical meanings.

2. Basic Situation of Finger Seal

A standard type FS system, consisting of front and back baffles, a front and back fixed distance annulus, and a multiple contact FSA, is shown in Figure 1a. The contact FSA consists of a series of curved beams machined in the circumferential direction according to a certain rule, as shown in Figure 1b. We applied the following FSA principles: adjacent FSAs are staggered stacked, and have close contact; we always ensure that the gap between the former finger beam can be covered by the latter finger beam, effectively preventing the fluid through the gap between the finger beam axial leakage; the plates are always located on the high- and low-pressure air flow sides; the fixed distance annulus prevents the front and back plate and FSA from having direct contact to reduce the hysteresis effect; the FSA and rotor contact form an interference fit to achieve a critical sealing interface, effectively preventing fluid flow from high to low pressure. The FSA compensates for the shortcomings of the grate seal with high leakage, and the finger beam is similar to the brush wire bundle in the brush seal in the structure, which can adapt to the vortex effect formed by the rotor due to uneven material and the thermal effect. In addition, the rigidity and strength of the finger beam is greater than that of the brush wire bundle, overcoming the defect of the brush wire breakage.

There are many random and irregular asperities and gaps on the contact surfaces of the adjacent FSA. In this paper, the asperities are considered as skeletons and the gaps are considered as pores, so that the contact surfaces of the adjacent FSA can be reduced to microscopic pore structures. In this paper, the microscopic leakage channels on the contact surfaces of the adjacent FSA are assumed to be parallel capillaries of the same diameter and bent. The total flow space in the thin tube is equal to the volume of all the leakage channels in the sealing contact surface. The internal surface area of the thin tube is equal to the area of all the leakage channels in the rough contact surface. The asperity of the contact surface is circular in cross section in any direction perpendicular to the depth of roughness.

In this paper, the arc FS structure is used for the analysis. The main structural parameters are shown in Figure 2 and

Table 1. Structural parameters of arc finger seal (FS).

Parameters	Units	Values
Radius of the outer circle/Dw	mm	77
Radius of the root circle/Df	mm	68.75
Radius of the inner circle/Dr	mm	60.5
Radius of the base circle/Dcc	mm	11
Arc radius/Rc	mm	63
Rotor radius/r	mm	59.5
Gap width/Ia	mm	2
Clearance angle/α	°	0.2
Downstream protection height/gd	mm	1.2
Height of the heel/xg	mm	1.2
Number of finger beams/Nfb		42

. Here, D_w is the outer circle radius; D_f is the root circle radius; D_r is the inner circle radius; D_cc is the base circle radius; R_c is the arc radius; r is the rotor radius; I_a is the gap width; α is the clearance angle; g_d is the downstream protection height; x_g is the heel height. In this paper, the material used for the FSA is GH605, with a density of 9.13 × 10⁻⁹ t/mm³; a Young’s modulus of 2.31 × 10⁵ MPa; and a Poisson’s ratio of 0.286. The rotor is GH4169 with a density of 8.24 × 10⁻⁹ t/mm³; a Young’s modulus of 2.04 × 10⁵ MPa; and a Poisson’s ratio of 0.3. The back plate is 1Cr13 with a density of 7.75 × 10⁻⁹ t/mm3; a Young’s modulus of 2.17 × 10⁵ MPa and a Poisson’s ratio of 0.3.

3. Analysis Method of Total Leakage Characteristics of Finger Seal

3.1. Mathematical Model for the Main Leakage Rate of Finger Seal

The FS main leakage is mainly divided into leakage caused by the radial hysteresis gap and leakage caused by the radial wear height, as shown in Figure 3. The FS and the rotor are usually assembled by interference assembly, and the rotor generates a vortex effect due to uneven material. Thus, it is difficult to maintain strict concentricity between the rotor and the FS, which often has a certain amount of eccentricity, and the FS and the rotor may form an eccentric annulus gap. Therefore, the q_m mathematical model for predicting eccentric annulus gap leakage is [40].

$$q_m=\frac{2\pi r{h}^3}{12\eta t_{sr}}\Delta p\left(1+1.5{\epsilon }^2\right)+\frac{u_0}{2}\pi rh$$

(1)

where r is the rotor radius, mm; h is the leakage gap, mm; Δp is the axial pressure difference, MPa; ε is the eccentricity; u₀ is the flow rate of the seal medium; η is the dynamic viscosity of the seal gas; and t_sr is the total thickness of the FSA, mm.

According to the research results of Zhang et al. [41], it is known that the circumferential flow velocity u₀ at the rotor surface due to friction is

$$u_0=\frac{\pi nr}{60}$$

(2)

where n is the rotor speed, r/min.

$$t_{sr}=t_m{t}_b$$

(3)

where t_m is the number of layers of the FSA; t_b is the thickness of a single FSA, mm.

$$h=h_g+h_w$$

(4)

where h_g is the radial hysteresis clearance, mm, obtained by finite element calculation; and h_w is the mean radial wear height, mm.

The average radial wear height of the FS is mainly represented by the radial length variation of the finger foot, so the mathematical model for the wear height is [42]

$$h_w=\frac{\Delta V}{A_a}$$

(5)

where ΔV is the wear volume of the FS, mm³; and A_a is the nominal effective contact area between the FS and the rotor, mm².

Fractal theory can effectively describe the complex behaviour of frictional wear and contribute to the quantitative analysis of tribological behaviour [25]. Therefore, in this paper, a fractal wear model is used to describe the wear volume of the FS. In the FS system, micromotion friction exists between adjacent FSAs; sliding friction exists between the rotor and the finger boots of each FSA; and the FSA operates in a very harsh environment with high-pressure intensity for a long time. As a result, the contact surfaces of the FSAs eventually tend to become completely elastically deformed when repeatedly subjected to the above loads [43]. In this paper, the research results of Lei et al. [44] are improved, and the improved mathematical model for the volume wear of the FS is shown in Equation (6).

$$\Delta V={\left(1+\psi {\mu }^2\right)}^{0.5}{K}_e{A}_{re}\Delta S$$

(6)

where ψ is the shear stress influence coefficient on the actual contact area [45]; μ is the friction coefficient; K_e is the elastic contact wear coefficient; A_re is the elastic contact area; and ΔS is the sliding distance.

$$\Delta S=v_0\Delta t$$

(7)

where v₀ is the relative linear sliding linear velocity between the FS and the rotor; and Δt is the relative sliding time.

$$v_0=\frac{\pi nr}{30}$$

(8)

where n is the rotor speed; and r is the rotor radius.

$$A_{re}=\frac{\lambda D}{2-D}\left(a_l-a_l^{0.5D}{a}_c^{1-0.5D}\right)$$

(9)

where λ is the contact coefficient between the FS and the rotor [44]; D is the fractal dimension; a_l is the maximum contact area of the asperity; and a_c is the critical contact area for plastic deformation of the asperity [46].

$$\lambda ={\left[\frac{720N{\left(t_{sr}F{r}^{*}\right)}^{0.5}{\left(E{\pi }^3\right)}^{-0.5}}{\left(r+D_r\right)\left(360-N_{fb}\alpha \right)N_n{t}_{sr}}\right]}^{{\left(r^{*}\right)}^{-1}}$$

(10)

where N_fb is the total number of finger beams in a single FSA; F is the total pressure on the contact surface; r^* is the comprehensive radius of curvature; E is the composite Young’s modulus; D_r fis the inner radius of the FSA; α is the clearance angle between the finger feet; and N_n is the number of finger beams in contact with a single FSA and rotor.

$${\left(r^{*}\right)}^{-1}=\frac{r_i}{r{d}_r}$$

(11)

where r_i is the interference amount.

$$E={\left[\frac{\left(1-v_1^2\right)}{E_1}+\frac{\left(1-v_2^2\right)}{E_2}\right]}^{-1}$$

(12)

where E₁ and E₂ are the Young’s modulus of the two materials, respectively; v₁ and v₂ are Poisson’s ratio of the two materials, respectively.

$$a_c=G^2{\left(\frac{33{\pi }^{0.5}\phi k_{\mu }}{40}\right)}^{\frac{2}{1-D}}$$

(13)

where G is the fractal roughness parameter that reflects the height of the asperity contour; φ is the characteristic parameter of the material, φ = ơ_y/E, where ơ_y is the yield strength of the softer material, and E is the composite Young’s modulus; and k_μ is the correction factor for friction force.

$$k_{\mu }=\left\{\begin{array}{ll}1-0.228\mu \hfill & 0\le \mu \le 0.3\hfill \\ 0.932e^{-1.58(\mu -0.3)}\hfill & 0.3<\mu \le 0.9\hfill \end{array}\right.$$

(14)

where μ is the friction coefficient.

$$A_a=\frac{\left(360-\alpha N_{fb}\right)\pi r}{180N_{fb}}{N}_n{t}_{\mathrm{sr}}$$

(15)

Therefore, if Equations (6) and (15) are substituted into Equation (5), the wear height h_w is

$$h_w=\frac{180N_{fb}{K}_e{A}_{re}\Delta \mathrm{S}{(1+\gamma {\mu }^2)}^{\frac{1}{2}}}{\left(360-N_{fb}\alpha \right)\pi r{N}_n{t}_{sr}}$$

(16)

Therefore, by substituting Equation (16) into Equation (4), it can be seen that the leakage gap h is

$$h=h_g+\frac{180N_{ fb}{K}_e{A}_{re}\Delta S{(1+\gamma {\mu }^2)}^{\frac{1}{2}}}{\left(360-N_{ fb}\alpha \right)\pi r{N}_n{t}_{sr}}$$

(17)

In summary, the mathematical model Q_m for the FS main leakage rate, taking into account the fractal wear effect, is as follows

$$Q_m=\frac{\pi r{h}^3\Delta p}{6\eta t_m{t}_b}\left(1+1.5{\epsilon }^2\right)+\frac{nh{r}^2{\pi }^2}{120}$$

(18)

3.2. Mathematical Model for the Side Leakage Rate of Finger Seal

The volumetric flow of fluid through the microscopic channels of the adjacent FSA rough surface is referred to as the FS side leakage. The G-W model [47] first considered the contour height distribution of the contact surface as a random variable and assumed that the mean radius of curvature is closely related to the experimental instrument resolution. Sayles et al. [48] found that the contour height distribution of the contact surface has a non-stationary random characteristic and the standard deviation of its height distribution is the sampling length. Therefore, the statistical parameters of the surface topography are closely related to the instrument resolution and sampling length, and the characterisation of the contact surface is not unique and cannot fully reflect all the information of the surface roughness. Mandelbrot et al. [49] showed that the rough surface has self-similarity and self-simulation, the local and overall characteristics of the profile are self-similar, and this feature can be characterised by the fractal characteristic parameters, independent of the experimental instrument resolution and scale. This fractal characteristic parameter can provide full roughness information on the fractal surface for all the scale ranges.

The materials used for the FS are mostly cobalt-based superalloys. According to reference [42], cobalt-based superalloys have self-similarity and fractal characteristics, so the FS contact surface is a contact surface with fractal characteristics. The W-M function is an ideal fractal curve which is non-differentiable and has self-radiating fractal characteristics. It has uniqueness and certainty in characterising the height of the contact surface profile and its mathematical model is [50]

$$\mathrm{Z}\left(x_0\right)=G^{D-1}{\displaystyle \sum _{n=n_{min}}^{\infty }\frac{\cos \left(2\pi {\gamma }^n{x}_0\right)}{{\gamma }^{\left(2-D\right)n}}}$$

(19)

Here, Z(x) is the contour curve of the rough surface; x₀ is the coordinate of the contour; G is the fractal roughness parameter which reflects the magnitude of the rough contour curve; D is the fractal dimension which can describe the irregularity of the rough contour curve; γ is a constant, and for the rough surface obeying the normal distribution, γ = 1.5; γⁿ is the spatial frequency of the contour curve; and n_min is the lowest cut-off frequency of the contour curve corresponding to the order number.

In this paper, based on Equation (19), a 2D rough surface scheme with profile height varying with different fractal dimensions (D = 1.4, 1.5, 1.6) and the same fractal roughness parameter (G = 1 × 10⁻¹²) is generated as shown in Figure 4, and a 2D rough surface scheme with profile height varying with different fractal roughness parameters (G = 1 × 10⁻¹¹, 1 × 10⁻¹⁰, 1 × 10⁻⁹) and the same fractal dimension (D = 1.3) is generated as shown in Figure 5.

From Figure 4, it can be seen that as the fractal dimension gradually increases, the contour height of the rough surface gradually decreases, because the larger the fractal dimension, the smaller the height of the asperity on the rough surface, and therefore, the lower the contour height of the rough surface. This conclusion can be further extended to indicate that the larger the fractal dimension, the lower the profile height of the rough surface, the smoother the contact surface, the greater the number of asperities in contact in the contact surface, and the larger the contact area, which is consistent with the conclusion in the reference [30,51]. From Figure 5, it can be seen that the height of the profile of the rough surface gradually increases with the increase in the fractal roughness parameter, because the larger the fractal roughness parameter is, the higher the height of the asperity on the rough surface is, and therefore the height of the profile of the rough surface is larger. The conclusion can be further extended that the greater the fractal roughness parameter, the greater the contour height of the rough surface, the rougher the contact surface, the smaller the number of asperities in the contact surface that are in contact with each other, and the smaller the contact area. Therefore, the fractal dimension D and the characteristic roughness parameter G can affect the contour size of the rough surface and are some of the important parameters to characterise the contour features of the fractal surface.

If each asperity in the contact surface satisfies Equation (19), the profile curve before deformation can be described as [50]

$$\begin{array}{c}z\left(x_0\right)=G^{D-1}{l}^{2-D}\cos \left(\frac{\pi x_0}{l}\right)\hspace{1em}-\frac{l}{2}<x_0<\frac{l}{2}\end{array}$$

(20)

where, l is the diameter of the bottom surface of the asperity profile.

For an asperity satisfying Equation (20), the height of the contact gap h_b before loading can be expressed as [52]

$$h_b=G^{D-1}{l}^{2-D}$$

(21)

The Roth model [53] provides the relationship between the contact gap height h_a and the axial pressure difference Δp after loading as follows

$$h_a=h_a{e}^{\left(-\frac{\Delta p}{R_c}\right)}$$

(22)

where, R_c is the sealing performance coefficient [37], which depends on the yield strength of the material and the characteristic parameters of the rough surface.

$$\begin{array}{cc}{R}_c=64.14\Delta p^{0.3406}& \Delta p\le 90\mathrm{MPa}\end{array}$$

(23)

Therefore, the deformation of the asperity after being loaded ω by

$$\omega =h_a-h_b$$

(24)

According to the research results of Yu et al. [54], the initial porosity of porous media before loading ϕ₀ is

$${\varphi }_0={\left(\frac{{\lambda }_{min}}{{\lambda }_{max}}\right)}^{D_E-D_{fb}}$$

(25)

where, λ_min is the minimum pore diameter [54]; According to reference [55], (λ_min/λ_max) < 10⁻²; D_E is the Euclidean dimension, which takes the value of 2 in this paper; and D_fb is the pore fractal dimension before loading.

According to the research results of Ji et al. [52], the porosity of porous media after loading ϕ by

$$\varphi =\frac{h_b{\varphi }_0-\omega }{h_a}$$

(26)

Therefore, the pore fractal dimension D_f0 after loading is

$$D_{f0}=D_E-\frac{ln\varphi }{\ln \left({\lambda }_{min}/{\lambda }_{max}\right)}$$

(27)

The FS side leakage channel is shown in Figure 6. From Figure 6, it can be seen that the FS side gap leakage channel is mainly composed of interconnected microscopic leakage channels on the contact surfaces of adjacent FSAs, the width of the leakage channel is related to the thickness of the FSA, and its length is the outer radius of the FS minus the inner radius.

In this paper, the microscopic pore structure on the contact surface of adjacent FSAs is considered as a fractal porous medium, and the capillary bundle model is used to describe the fractal porous structure. Extensive references [55,56,57,58] show that the pore area and pore size distribution of porous media satisfy the fractal scalar law, and the relationship between the number of pores N and the pore diameter λ₀ is [54].

$$N\left(L\ge {\lambda }_0\right)={\left(\frac{{\lambda }_{max}}{{\lambda }_0}\right)}^{D_{fb}}$$

(28)

where L is the measured length; and λ_max is the maximum pore diameter.

Therefore, from Equation (28), the total number of pores N_t is

$$N_t\left(L\ge {\lambda }_{min}\right)={\left(\frac{{\lambda }_{max}}{{\lambda }_{min}}\right)}^{D_{fb}}$$

(29)

Differentiating Equation (29) gives the number of pores −dN in the interval λ to λ + dλ.

$$-dN=D_{f_b}{\lambda }_{max}^{D_f}{\lambda }^{-\left(D_f+1\right)}d\lambda$$

(30)

where −dN > 0.

Considering the fractal geometry of the microscopic pore structure of the contact surface, the leakage channel can be viewed as a bundle of curved capillaries with different cross-sectional areas. According to the Hagen-Poiseuille equation, the fluid flow rate q(λ) in a single curved capillary is [59]

$$q\left(\lambda \right)=\frac{\pi \Delta p{\lambda }^4}{128L_t\left(\lambda \right)\eta }$$

(31)

where L_t(λ) is the actual length of the bent capillary before loading [59]; and η is the viscosity of the fluid.

$$L_t\left(\lambda \right)={\lambda }^{1-D_{Tb}}{L}_0^{D_{Tb}}$$

(32)

where L₀ is the length of the straight line along the macroscopic pressure gradient direction, mm; D_Tb is the fractal dimension of tortuosity before loading; and the greater the D_Tb value, the more tortuous the capillary [55].

$$L_0=D_w-D_r$$

(33)

where D_w is the outer circle radius of the FS; and D_r is the inner circle radius of the FS.

$$D_{Tb}=1+\frac{\ln {\tau }_{ave0}}{\ln \left(L_0/2{\lambda }_{ave0}\right)}$$

(34)

where τ_ave0 is the average tortuosity before loading [60]; and λ_ave0 is the average pore radius before loading [55,61].

$${\tau }_{ave0}=\frac{1}{2}\left[1+\frac{1}{2}\sqrt{1-{\varphi }_0}+\sqrt{1-{\varphi }_0}\frac{\sqrt{{\left(\frac{1}{\sqrt{1-{\varphi }_0}}-1\right)}^2+\frac{1}{4}}}{1-\sqrt{1-{\varphi }_0}}\right]$$

(35)

where ϕ₀ is the initial porosity of the porous medium before loading.

$${\lambda }_{ave0}=\frac{D_{fb}{\lambda }_{min}}{{\mathrm{D}}_{fb}-1}$$

(36)

where D_fb is the fractal dimension of the pore before loading.

Therefore, the fractal dimension of the tortuosity D_T after loading is

$$D_T=1+\frac{\ln {\tau }_{ave}}{\ln \left(L_0/2{\lambda }_{ave}\right)}$$

(37)

where τ_ave is the average tortuosity after loading; and λ_ave is the average pore radius after loading.

$${\tau }_{ave}=\frac{1}{2}\left[1+\frac{1}{2}\sqrt{1-\varphi }+\sqrt{1-\varphi }\frac{\sqrt{{\left(\frac{1}{\sqrt{1-\varphi }}-1\right)}^2+\frac{1}{4}}}{1-\sqrt{1-\varphi }}\right]$$

(38)

where ϕ is the initial porosity of the porous medium after loading.

$${\lambda }_{ave}=\frac{D_{f0}{\lambda }_{min}}{D_{f0}-1}$$

(39)

where D_f0 is the fractal dimension of the pore after loading.

Integrating Equation (31), the FS side leakage rate Q_b, taking into account the fractal porous media seepage effect before loading, is given by

$$Q_b=-{\displaystyle {\int }_{{\lambda }_{min}}^{{\lambda }_{max}}q\left(\lambda \right)}dN\left(\lambda \right)=\frac{\pi \Delta p{D}_{fb}{\lambda }_{max}^{3+D_{Tb}}}{128\eta L_0^{D_{Tb}}\left(3+D_{Tb}-D_{fb}\right)}$$

(40)

According to the actual working conditions of the FS system, it is known that the contact surface of the adjacent FSA is subjected to high-strength pressure for a long time, so it is necessary to improve Equation (40) by replacing D_fb with D_f0 and D_T with D_Tb; then, the mathematical model of the FS side leakage rate Q_s, taking into account the fractal porous media seepage effect after loading, is

$$Q_s=-{\displaystyle {\int }_{{\lambda }_{min}}^{{\lambda }_{max}}q\left(\lambda \right)}dN\left(\lambda \right)=\frac{\pi \Delta p{D}_{f0}{\lambda }_{max}^{3+D_T}}{128\eta L_0^{D_T}\left(3+D_T-D_{f0}\right)}$$

(41)

3.3. Mathematical Model for Total Leakage Rate of Finger Seal

According to the above analysis, the FS total leakage rate Q

$$\begin{array}{c}Q=Q_m+Q_s=\left[\frac{\pi r{h}^3\Delta p}{6\eta t_m{t}_b}\left(1+1.5{\epsilon }^2\right)+\frac{nh{r}^2{\pi }^2}{120}\right]+\\ \frac{\pi \Delta p{D}_{f0}{\lambda }_{max}^{3+D_T}}{128\eta L_0^{D_T}\left(3+D_T-D_{f0}\right)}\end{array}$$

(42)

4. Numerical Calculation Method for Dynamic Performance of Finger Seal

4.1. Finite Element Calculation Model for Dynamic Performance of Finger Seal

Most of the performance analyses of the FS belong to static performance analyses, and fewer FS dynamic performance analyses are considered, so most of the theoretical results have large errors with the actual test results [62,63,64]. In this paper, the numerical calculation method of the dynamic performance of the FS is used to obtain the value of the hysteresis gap in Equation (18) by the finite element method. The FS structure has the characteristics of cyclic symmetry and staggered superposition, so the finite element calculation model of the FS dynamic performance can be simplified into four parts. In the first part, there is one complete finger beam in the circumferential direction of the FSA and two half-finger beams, which is referred to as the high-pressure FS. In the second part, there are two complete finger beams in the circumferential direction of the FS, which is referred to as the low-pressure FS. In the third part includes part of the back plate. The fourth part includes part of the rotor. The finite element calculation model of the dynamic performance of the FS is shown in Figure 7.

4.2. Boundary Condition

According to the actual situation of the FS structure in operation, fixed constraint boundary conditions must be applied to the circumferential surfaces of the high-pressure FSA and low-pressure FSA, the circumferential surface and back of the back plate, and the axial direction of the rotor when performing finite element calculations. The fixed constraint boundary condition is usually understood as a displacement of 0, which means that the three components u, v and w along the x, y and z axes are 0.

$$\left\{q\right\}=\left\{\begin{array}{c}u\left(x,y,z\right)\\ v\left(x,y,z\right)\\ w\left(x,y,z\right)\end{array}\right\}={\left\{u,v,w\right\}}_{\left(x,y,z\right)}^{\mathrm{T}}=0$$

(43)

where {q} is the displacement.

Then it is also necessary to apply the cyclic symmetry constraint to the high-pressure FSA, the low-pressure FSA cross-section and the rotor cross-section, and the mathematical model of the cyclic symmetry constraint boundary condition is

$$\left\{\begin{array}{c}{u}_A^{\prime }\\ u_B^{\prime }\end{array}\right\}=\left[\begin{array}{cc}\cos k{\alpha }_n& -\sin k{\alpha }_n\\ \sin k{\alpha }_n& \cos k{\alpha }_n\end{array}\right]\left\{\begin{array}{c}{u}_A\\ u_B\end{array}\right\}$$

(44)

where u_a and u_b are the basic sector displacement and the replica sector displacement of the low-angle edge, respectively; u’_a and u’_b are the basic sector displacement and the replica sector displacement of the high-angle edge, respectively; k denotes the harmonic index and α_n is the sector angle.

$${\alpha }_n=\frac{360}{N_s}$$

(45)

where N_s is the number of sectors.

4.3. Setting of Load Step and Extraction of Calculation Results

The calculation process of the FS needs to be divided into three load steps: the first load step, which applies an axial pressure to the high-pressure FSA; the second load step, which applies a periodic displacement function in the radial direction of the rotor [65], which is used to simulate the vortex effect of the rotor due to unbalance effects; and the third load step, which resets the rotor.

The mathematical model of periodic displacement function y(t) is [65]

$$y\left(t\right)=\Delta r\sin \left(\frac{n\pi t}{30}\right)+\Delta r_x\sin \left(\frac{n\pi t}{30}\right)\pm r_i$$

(46)

where Δr is the radial displacement excitation of the rotor; Δr_x is the inclination of the rotor; “+” is the is the interference amount; and “−” is the amount of clearance.

At the end of the third load step, the gap between the deformed FSA and the rotor, the hysteresis gap, is extracted from the result file and substituted into Equation (18) to obtain the main leakage rate of the FS.

5. Results and Discussion

5.1. Verification of the Accuracy of Theoretical Models

It should be noted that the FS is a shaft seal component, as shown in Figure 1. Zhao et al. [36] first divided the FS total leakage (Q) into the FS main leakage (Q_m) and the FS side leakage (Q_s). In this paper, the FS main leakage (Q_m) is divided into the leakage caused by the hysteresis gap and the leakage caused by the radial wear height, and in this paper, the FS side leakage (Q_s) is considered as the flow rate of the adjacent FSA microscopic leakage channel after loading. Due to the limited testing equipment, only the FS total leakage (Q) can be measured at present [65,66,67]. Therefore, the test value of the main leakage volume of the FSA cannot be measured by the test alone and cannot be compared individually; similarly, the test value of the side leakage volume of the FSA cannot be measured by the test alone and cannot be compared individually. Furthermore, the mathematical model of the FS main leakage rate (Q_m) and the FS total leakage rate (Q) established in this paper is only applicable to the fully elastic deformation stage of the asperity.

The test results in reference [67] were used to compare the analysis with the mathematical models Q (Equation (42)) for the FS total leakage, Q_m (Equation (18)) for the FS main leakage, Q_s (Equation (41)) for the FS side leakage and the mathematical model of the FS total leakage developed in reference [68] and established in this paper, as shown in Figure 8. The finite element calculation model used in the theoretical model validation was consistent with the structural parameters of the FS specimen for the FS bench test in the reference [67].

The calculation results show that the leakage gradually decreases with the increase in rotor speed, and the numerical calculation results keep the same trend and order of magnitude of change compared with the experimental results of the reference [67]. The maximum error between the numerical calculation results of the reference [68] and the experimental results of the reference [67] is 4.56%. The maximum error between the numerical calculation results of the FS total leakage rate mathematical model (Equation (42)) established in this paper and the experimental results is 2.195%. The maximum error between the numerical calculation results of the FS main leakage rate mathematical model (Equation (18)) and the experimental results is 2.196%. Meanwhile, for the FS side leakage rate mathematical model, the error rate between the numerical calculation results and the experimental results is very large. The numerical calculation results of the mathematical model of the FS total leakage rate established in this paper are closer to the experimental results than the numerical calculation results of the reference [68], because the eccentricity of the rotor, the wear height of the FS/rotor, and the FS side leakage are considered in this paper. The above three factors have an influence on the calculation of the FS total leakage rate, so considering the influence of these three factors in the calculation of the leakage rate will be more consistent with the actual operating condition of the FS. The comparison of the above numerical calculation results proves the reasonableness and correctness of the numerical calculation method of the FS considering the change of fractal surface shape.

5.2. Analysis of Factors Influencing the Main Leakage Rate of Finger Seals

5.2.1. Working Conditions

The influence law of the FS main leakage rate was analysed under different axial pressure differences, different rotor speeds and different radial displacement excitations, as shown in Figure 9. When n = 13,000 r/min, D = 1.3, G = 1 × 10⁻⁹, from Figure 9a, it can be seen that the FS main leakage rate gradually increases with the continuous increase in the axial pressure difference; when the axial pressure difference is constant, the FS main leakage rate also gradually increases with the increase of the radial displacement excitation. When Δp = 0.3 MPa, D = 1.3, and G = 1 × 10⁻⁹, it can be seen from Figure 9b that the FS main leakage rate gradually Increases with the gradual increase in the radial displacement excitation; when the radial displacement excitation is constant, the FS main leakage rate gradually decreases with the increase in the rotor speed. When Δr = 0.03 mm, D = 1.3, G = 1 × 10⁻⁹, it can be seen from Figure 9c that the FS main leakage rate gradually decreases with the increase in the rotor speed; when the rotor speed is constant, the FS main leakage rate gradually increases with the increase in the axial pressure difference. The variation laws in Figure 9a–c are consistent with the trend of the leakage rate under the actual working conditions of the FS.

5.2.2. Fractal Dimension

Under different fractal dimension D, different axial pressure differences Δp and different rotor speeds n, the influence law of the main leakage rate of FS is analysed, as shown in Figure 10. When Δr = 0.03 mm, n = 13,000 r/min, and G = 1 × 10⁻⁹, it can be seen from Figure 10a that the FS main leakage rate gradually increases with the increasing axial pressure difference; when the axial pressure difference is constant, the FS main leakage rate gradually increases with the increasing fractal dimension, because the larger the fractal dimension, the larger the contact area is, and according to Equation (6), the wear rate of the FS is larger, and then according to Equation (5), the wear height of the FS is higher, so the FS main leakage rate is greater than the leakage rate of the FS. When Δr = 0.03 mm, Δp = 0.3 MPa, and G = 1 × 10⁻⁹, it can be seen from Figure 10b that the FS main leakage rate decreases as the rotor speed continues to increase; at constant rotor speed, the FS main leakage rate increases as the fractal dimension increases, and the reason for this phenomenon is the same as in Figure 10a.

5.2.3. Fractal Roughness Parameter

The influence law of the FS main leakage rate was analysed for different fractal roughness parameters at different axial differential pressures and different rotor speeds, as shown in Figure 11. When Δr = 0.03 mm, n = 13,000 r/min, and D = 1.3, it can be seen from Figure 11a that the FS main leakage rate gradually increases with the increasing axial pressure difference; when the axial pressure difference is constant, the FS main leakage rate gradually decreases with the increasing fractal roughness parameter, because the larger the fractal roughness parameter, the smaller the contact area, and according to Equation (6), the smaller the wear rate of the FS, and then according to Equation (5), the smaller the wear height of the FS, so the FS main leakage rate is smaller. When Δr = 0.03 mm, Δp = 0.3 MPa, and D = 1.3, from Figure 11b, it can be seen that the FS main leakage rate gradually decreases as the rotor speed continues to increase; at constant rotor speed, the FS main leakage rate gradually becomes smaller as the fractal roughness parameter increases, and the reason for this phenomenon is consistent with Figure 11a.

5.2.4. Eccentricity

Under different values of eccentricity ε, axial pressure difference Δp and rotor speed n, the influence law of the FS main leakage rate is analysed, as shown in Figure 12. When Δr = 0.03 mm, n = 13,000 r/min, D = 1.3, and G = 1 × 10⁻⁹, the effect of axial pressure difference Δp and eccentricity ε on the main leakage rate is shown in Figure 12a. From Figure 12a, it can be observed that the main leakage rate gradually increases as the axial pressure difference continues to increase. When the axial pressure difference is constant, the main leakage rate gradually increases with the increase in eccentricity, because the larger the eccentricity, the larger the radial displacement excitation of the rotor, and the larger the leakage channel between the finger foot and the rotor, so the main leakage rate gradually increases. When Δr = 0.03 mm, Δp = 0.3 MPa, D = 1.3, and G = 1 × 10⁻⁹, the effect of rotor speed n and eccentricity ε on the main leakage rate is shown in Figure 12b. It can be seen from Figure 12b that as the rotor speed continues to increase, the main leakage rate gradually decreases; when the rotor speed is constant, the main leakage rate gradually increases with the increase in eccentricity. The reason for this phenomenon is consistent with Figure 12a.

5.3. Analysis of Factors Influencing the Side Leakage Rate of Finger Seals

5.3.1. Axial Pressure Difference and Fractal Characteristic Parameters

Under different values of axial pressure difference Δp, tortuosity fractal dimension D_T and pore fractal dimension D_f0, the influence law of the FS side leakage rate is analysed, as shown in Figure 13. When D_f0 = 1.6, D = 1.3, and G = 1 × 10⁻⁹, the effects of axial pressure difference Δp and tortuosity fractal dimension D_T on the side leakage rate are shown in Figure 13a. From Figure 13a, it can be observed that as the axial pressure difference continues to increase, the side leakage rate gradually increases; when the axial pressure difference is constant, the side leakage rate gradually decreases as the tortuosity fractal dimension continues to increase, because the larger the tortuosity fractal dimension, the higher the actual leakage channel bends, the longer the actual leakage channel, the more obvious the change trend of the side leakage rate, and the smaller the flow of the side leakage rate. When D_T = 1.3, D = 1.3, G = 1 × 10⁻⁹, the axial pressure difference Δp and pore fractal dimension Df0 on the side leakage rate are shown in Figure 13b. From Figure 13b, it can be seen that the side leakage rate gradually increases as the axial pressure difference gradually increases; when the axial pressure difference is constant, the side leakage rate gradually increases with the increase in the pore fractal dimension, because the larger the pore fractal dimension, the larger the porosity, so the larger the side leakage channel, the larger the side leakage rate.

5.3.2. Porosity

When D_T = 1.3, D = 1.3, and G = 1 × 10⁻⁹, the influence law of the FS side leakage rate was analysed under different axial pressure differences Δp and porosity ϕ, as shown in Figure 14. From Figure 14, it can be observed that as the axial pressure difference continues to increase, the side leakage rate gradually increases; when the axial pressure difference remains constant, as the porosity continues to increase, the side leakage rate gradually increases and decreases. This is because the larger the porosity, the larger the micro-leakage channel between the contact surfaces, resulting in a larger flow rate and a higher side leakage rate.

5.4. Analysis of Influencing Factors on Leakage Performance of Finger Seal

5.4.1. Working Conditions

Under different values of axial pressure difference, radial displacement excitation and rotor speed, the influence laws of the total leakage, main leakage and side leakage performance of the FS are analysed, as shown in Figure 15. When D = 1.3, G = 1 × 10⁻⁹, ϕ = 0.5, Δr = 0.03 mm, and n = 13,000 r/min, the effects of total leakage, main leakage, and side leakage under different axial pressure differences Δp are shown in Figure 15a. It can be seen from Figure 15a that with the continuous increase in axial pressure difference, the increase in the total leakage rate and main leakage rate is large, and the increase in the side leakage rate is small. The above phenomenon is consistent with the actual working conditions of the FS. When D = 1.3, G = 1 × 10⁻⁹, ϕ = 0.5, Δp = 0.3 MPa, and n = 13,000 r/min, the effects of total leakage, main leakage, and side leakage under different radial displacement excitation Δr are shown in Figure 15b. It can be seen from Figure 15b that with the continuous increase in the radial displacement excitation, the total leakage rate and the main leakage rate increase significantly, and the side leakage rate remains unchanged because the side leakage rate is not affected by the radial displacement excitation. When D = 1.3, G = 1 × 10⁻⁹, ϕ = 0.5, Δp = 0.3 MPa, and Δr = 0.03 mm, the effects of total leakage, main leakage, and side leakage under different values of rotor speed n are shown in Figure 15c. It can be seen from Figure 15c that with the continuous increase in the rotor speed, the total leakage rate and the main leakage rate gradually decrease, and the side leakage rate remains unchanged, because the side leakage rate is not affected by the rotor speed.

Corresponding to Figure 15,

Table 2. The proportion of the FS main leakage rate and the FS side leakage rate to the FS total leakage rate under different axial pressure differences.

Axial Pressure Differences (MPa)	Main Leakage Rate Ratio (%)	Side Leakage Rate Ratio (%)
0.1	97.90	2.10
0.15	98.63	1.37
0.2	98.71	1.29
0.25	98.77	1.23
0.3	99.09	0.91
0.35	99.25	0.75
0.4	99.39	0.61
0.45	99.51	0.49
0.5	99.59	0.41

shows the ratio of the main leakage rate and the side leakage rate of the FS under different values of axial pressure difference Δp, and it can be seen that the axial pressure difference Δp has a significant effect on the ratio of the main leakage rate.

Table 3. The proportion of the FS main leakage rate and the FS side leakage rate to the FS total leakage rate under different radial displacement excitation values.

Radial Displacement Excitation (mm)	Main Leakage Rate Ratio (%)	Side Leakage Rate Ratio (%)
0.01	96.21	3.79
0.02	96.96	3.04
0.03	96.97	3.03
0.04	97.54	2.46
0.05	98.03	1.97
0.06	98.31	1.69
0.07	98.57	1.43
0.08	98.78	1.22
0.09	98.78	1.22

shows the ratio of the main leakage rate and side leakage rate of the FS under different radial displacement excitation Δr. It can be seen that the radial displacement excitation Δp has a significant effect on the ratio of the main leakage rate ratio.

Table 4. The proportion of the FS main leakage rate and the FS side leakage rate to the FS total leakage rate under different rotor speed values.

Rotor Speed (r/min)	Main Leakage Rate Ratio (%)	Side Leakage Rate Ratio (%)
10,000	98.72	1.28
10,500	98.71	1.29
11,000	98.60	1.40
11,500	98.49	1.51
12,000	98.35	1.65
12,500	98.18	1.82
13,000	97.97	2.03
13,500	97.70	2.30
14,000	97.35	2.65
14,500	96.89	3.11
15,000	96.30	3.70

shows the ratio of the main leakage rate and the side leakage rate of the FS at different rotor speeds n. It can be seen that the rotor speed n has a significant effect on the ratio of the main leakage rate.

5.4.2. Fractal Feature Parameters

Under different fractal dimensions and fractal roughness parameters, the influence law of the total leakage, main leakage and side leakage performance of the FS is analysed, as shown in Figure 16. When G = 1 × 10⁻⁹, ϕ = 0.5, Δp = 0.3 MPa, n = 13,000 r/min, and Δr = 0.03 mm, the effects of total leakage, main leakage, and side leakage under different values of fractal dimension D are shown in Figure 16a. It can be seen from Figure 16a that the total leakage rate and the main leakage rate are increasing with the increasing of fractal dimension, and the reasons for this conclusion are consistent with those in Figure 10a. The side leakage rate has practically no shift, because the influence of the fractal dimension on the side leakage rate is extremely small and can be ignored. When D = 1.3, ϕ = 0.5, Δp = 0.3 MPa, n = 13,000 r/min, and Δr = 0.03 mm, the effects of total leakage, main leakage, and side leakage under different values of fractal roughness parameter G are shown in Figure 16b. From Figure 16b, it can be observed that as the feature scale coefficient continues to increase, the total leakage rate and the main leakage rate continue to decrease, and the reasons for this conclusion are consistent with those in Figure 11a. The side leakage rate remains practically unchanged because the influence of the fractal roughness parameter on the side leakage rate is extremely small and can be ignored.

Corresponding to Figure 16,

Table 5. The proportion of the FS main leakage rate and the FS side leakage rate to the FS total leakage rate under different fractal dimensions.

Fractal Dimension	Main Leakage Rate Ratio (%)	Side Leakage Rate Ratio (%)
1.3	97.97	2.03
1.4	97.98	2.02
1.5	97.99	2.01
1.6	98.01	1.99
1.7	98.05	1.95
1.8	98.11	1.89
1.9	98.29	1.71

shows the ratio of the main leakage rate to the side leakage rate of the FS under different values of fractal dimension D. It can be seen that the fractal dimension has a significant effect on the ratio of the main leakage rate.

Table 6. The proportion of the FS main leakage rate and the FS side leakage rate to the FS total leakage rate under different fractal roughness parameters.

Fractal Roughness Parameter	Main Leakage Rate Ratio (%)	Side Leakage Rate Ratio (%)
1 × 10−7	97.88	2.12
1 × 10−8	97.97	2.03
1 × 10−9	97.97	2.03
1 × 10−10	97.97	2.03
1 × 10−11	97.97	2.03
1 × 10−12	97.97	2.03

shows the ratio of the main leakage rate to the side leakage rate of the FS under different values of fractal roughness parameter G, and it can be found that the fractal roughness parameter has a significant effect on the ratio of the main leakage rate.

5.4.3. Porosity

When D = 1.3, G = 1 × 10⁻⁹, Δp = 0.3 MPa, n = 13,000 r/min, and Δr = 0.03 mm, the influence laws of the total leakage, main leakage, and side leakage performance of the FS under different values of porosity ϕ were analysed as shown in Figure 17. From Figure 17, it can be seen that as the porosity increases, the total leakage rate and the side leakage rate continue to increase. This is because the larger the porosity, the larger the leakage channel, and therefore the larger the leakage rate. Meanwhile, the main leakage rate remains virtually unchanged as it is not affected by porosity.

Corresponding to Figure 17,

Table 7. The proportion of the FS main leakage rate and the FS side leakage rate to the FS total leakage rate under different porosity values.

Porosity	Main Leakage Rate Ratio (%)	Side Leakage Rate Ratio (%)
0.1	97.25	2.75
0.2	97.09	2.91
0.3	96.99	3.01
0.4	96.92	3.08
0.5	96.86	3.14
0.6	96.81	3.19
0.7	96.77	3.23
0.8	96.74	3.26
0.9	96.70	3.30

shows the ratio of the main leakage rate to the side leakage rate of the FS under different porosities, and it can be seen that porosity has a significant effect on the ratio of the side leakage rate.

6. Conclusions

In this study, based on the fractal theory and the fractal geometry theory of porous media, a numerical analysis method is proposed for the finger seal (FS) total leakage characteristics considering the fractal wear and fractal porous media seepage effects, and a mathematical model for the FS main leakage rate considering the fractal wear effect and a mathematical model for the FS side leakage rate considering the fractal porous media seepage effect are proposed as well. The influence of working conditions, fractal characteristic parameters and other parameters on the total, main and side leakage rates of the FS is numerically investigated. The main conclusions are as follows:

• The maximum error rate between the numerical calculation and the experimental value of the mathematical model of the FS total leakage rate is 2.2%, which is less than 5%, and the mathematical model of the FS total leakage rate is feasible. However, the model is only applicable to the study of the leakage rate during the fully elastic deformation phase of the asperity, and the use of this model assumes that the contact surface has fractal characteristics. The FS main leakage rate mathematical model is used in the same way as the FS total leakage rate mathematical model.
• A mathematical model for the FS side leakage rate considering the seepage effect of fractal porous media takes into account the physical characteristics of isotropic porous media before and after loading, and the model can better reflect the FS side leakage characteristics, but the model does not take into account the physical characteristics of each anisotropic porous media and the non-Darcy seepage characteristics of porous media.
• In the analysis of the FS total leakage performance under various parameters, the FS main leakage rate is above 95% and the FS side leakage rate is below 5%. Thus, the FS’s main leakage rate is much larger than the FS’s side leakage rate, so the density of the material used for the FS can be improved to reduce the porosity, or non-metallic materials (such as C/C composite materials, etc.) can be used to reduce the FS wear rate and thus improve the sealing performance.
• In this study, the fractal characteristic parameters are used to characterise the contact surface morphology, which in turn reflects the leakage characteristics of the FS. It is recommended that in subsequent studies, the arithmetic mean roughness Ra or the effective root mean square surface roughness σ be used to characterise the contact surface topography and then Ra or σ be used to characterise the leakage performance.