# Numerical Analyses of Static Characteristics of Liquid Annular Seals Based on 2D LBM-LES Model

Authors: Zhenjie Zhang; Lulu Zhai; Jia Guo; Zuchao Zhu; Guoyou Chen
Journal: Shock and Vibration
 (2021)

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## Abstract

Static characteristics and leakage flow rates of liquid annular seals have great influences on the hydraulic efficiency of turbomachinery. In this paper, a two-dimensional (2D) mathematical model for predicting the leakage flow rates and static characteristics of liquid seal is established, based on the lattice Boltzmann method (LBM) combined with the D2G9 velocity model for incompressible fluid and large eddy simulation (LES) turbulence model, in which the transformation equation of reference pressure is developed with the Bernoulli equation. Moreover, the proposed model is validated by comparing with the experimental results, calculation results based on the finite volume method (FVM), and the results based on the empirical method of three seals under different operating conditions. The comparisons show that the maximum deviation in leakage prediction of the calculating model based on 2D LBM is 4%, and this calculating model will effectively improve the leakage prediction accuracy of the seals compared with the FVM and theoretical method.


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## Body

## 1. Introduction

In turbomachinery, there exist several sets of liquid annular seals, including neck-ring seals, interstage seals, and balance-piston seals. These seals prevent leakage flow while increasing the volumetric efficiency of the machinery. At present, there are mainly three methods to investigate the static and dynamic characteristics of these seals, including empirical formulas, bulk-flow method, and computational fluid dynamics (CFD) method. However, these three methods are all based on the hypothesis of continuous media, which means that the methods are all developed from a point of macroscopic view. With the development of the lattice Boltzmann method (LBM), many researchers started to use this method to solve different fluid flow problems of compressible and incompressible fluids. Compared with the widely used finite volume method (FVM), LBM has the advantages of simple algorithm, accurate calculation, and strong adaptability of boundary conditions, which also has provided an effective method for calculating fluid flow from a point of mesoscopic view. And in recent years, many researchers have analyzed the static and dynamic characteristics of sliding bearings, which are structurally similar to liquid annular seals based on LBM. Kucinschi and Afjeh [1] analyzed the internal flow characteristics and the fluid-film lubrication details within model sliding bearings based on 2D LBM. Kim et al. [2] have used LBM to study the flow conditions in a nano-sized air bearing, and Ramirez et al. [3] used this method to analyze the switching flow characteristics between different channels in a micron-sized air bearing. The simulation results showed that LBM has high accuracy in simulating the fluid flow of the sliding bearing with mini clearance.Although the structures of sliding bearings are similar to those of liquid annular seals, there are major differences between them. The internal flow within sliding bearings is laminar flow with small Reynolds number, while the internal flow of seals in the centrifugal pumps is highly turbulent with large Reynolds number. Recently, Posa and Lippolis [4], Li et al. [5], and Si et al. [6] analyzed the internal flow characteristics of centrifugal pumps considering liquid annular seals channels based on different turbulent models. Andres et al. [7], Saber and Abdou [8, 9], and Nagai et al. [10] proposed the theoretical calculation method for static and dynamic characteristics of different kinds of labyrinth seals and investigated the effects of operating conditions on these characteristics. As the maximum Reynolds number that fit the single-relaxation-time LBM (SRT-LBM) is 104 and the maximum Reynolds number that fit the multiple relaxation-time LBM (MRT-LBM) [11] is 4 × 104, analyses on the fluid flow with large Reynolds number based on LBM are often performed with the turbulent model, such as LBM-k-ε model established by Succi et al. [12], mixed length algebraic turbulent model established by Teixeira [13], and LBM-LES turbulent model established by Hou et al. [14]. However, the LBM-k-ε turbulent model shows poor performances on describing flow separation, reattachment, and recovery effects [15], and mixed length algebraic turbulent model requires a large number of empirical coefficients, which make the two models not so widely used.In contrast, the LBM-LES turbulence model combines the advantages of LBM and LES, in which the turbulent viscosity coefficient is calculated based on the Smargorinsky eddy viscosity model of LES. Premnath et al. [16] investigated the turbulent kinetic energy for the separation and attachment phenomenon of complex turbulent flow within a back-step model based on the LBM-LES turbulence model. Schneider et al. [17] used the LBM-LES turbulence model to solve the inlet flow characteristics of a centrifugal pump firstly. Chang et al. [18] analyzed the heat transfer between square cavity flow and back-step flow with the LBM-LES turbulent model. Comparisons between the numerical and experimental results showed that the method could describe the turbulent heat transfer flow well. Hamane et al. [19] simulated the flow within a 2D cylindrical channel with a large Reynolds number based on the LBM-LES turbulent model. Eitel et al. [20] introduced hierarchically refined lattice technology to the LBM-LES turbulent model and contrasted simulations of the cylinder flow model with different Reynolds number. The refinement of lattice reduced the number of lattice and significantly improved the computational efficiency. Nadim et al. [21] analyzed the flow of airfoil using LBM which combined the Smagorinsky SGS model, while parallel calculations were also used in the paper.With the development of LBM, static characteristics and leakage analyses of liquid annular seals from a mesoscopic point of view become possible. In this paper, a two-dimensional (2D) mathematical model for predicting the leakage flow rates and static characteristics of liquid seal is established, based on the lattice Boltzmann method (LBM) and large eddy simulation (LES) turbulence model. Because the internal fluid of the seals is incompressible, the D2G9 velocity model for incompressible fluid is applied. Nonequilibrium extrapolation scheme is used for the boundary conditions of the wall. Moreover, on comparisons of the experimental results with the results of LBM, FVM and empirical formulas are conducted and analyzed to verify the effectiveness and accuracy of LBM in calculating the results of static characteristics and leakages of seals.


## 2. Simulation Method

As the internal flow of the seal is mainly affected by the pressure-induced force, the lattice Boltzmann equation (LBE) can be expressed as follows:(1)∂f∂t+c•∇f=Mf.In equation (1), M is the rate of change from initial state to final state of the particle distribution function f, which is called collision operator. The basic thought of the Boltzmann equation is not to determine the motion state of each molecule but to find the probability of each molecule in a certain state and to obtain the macroscopic parameters of the system through statistical methods [22]. The distribution function f acts to describe the number of molecules in a certain position in a certain range of speed at a particular time. The macroscopic values of velocity and pressure can be obtained by f, which is related to discrete velocity model, equilibrium distribution function, and its evolution equation [23].


### 2.1. Governing Equations

The governing equation of the LBM calculating model can be written as equation (2) under the SRT-LBM model. It has second-order accuracy at low Mach number. In equation (2), γ is collision frequency, τ is relaxation time, and the relationship between γ and τ is γ = 1/τ. τ is mainly related to the lattice kinematic viscosity ν0 as shown in equation (3).(2)fαx+cαΔt,y+cαΔt,t+Δt−fαx,y,t=−γfαx,y,t−fαeqx,y,t,(3)ν0=2τ−16Δx2Δt.For the LBM calculating model, the distribution function evolution process is the most important component and the evolution process consists of two steps: the collision step that is described as equation (4) and the streaming step, which is described as equation (5).(4)fαx,y,t+Δt=fαx,y,t1−γ+γfαeqx,y,t,(5)fαx+Δx,y+Δy,t+Δt=fαx,y,t+Δt.


#### 2.1.1. Discrete Velocity Model

The discrete velocity model is one of the most important components of LBM calculating models. For compressible fluid flow calculation, Kataoka and Tsutahara [24] developed a new lattice Boltzmann velocity model for the compressible fluid simulation. Yang et al. [25] studied the compressible fluid flow which is around NACA0012 airfoil by the D1Q4 velocity model. Yang analyzed the 3D viscous compressible flows [26] by using the lattice Boltzmann flux solver. Meanwhile, for incompressible fluid flow calculation, Qian et al. [27] established the D2Q9 velocity model for weakly compressible. In this model, fluid flow is driven by density gradient, and the relationship between pressure P and ρ is shown as equation (6). On the basis of the D2Q9 velocity model, Guo et al. [28] developed the D2G9 velocity model for incompressible fluid flow. The fluid flow is driven by pressure gradient in D2G9. The equilibrium distribution function satisfies Σfαeq=const. The macroscopic equations of the D2G9 velocity model can be derived by Chapman–Enskog expansion [29], and the equations can be written as equations (7) and (8) [28].(6)P=cs2ρ,(7)∇·u=0,(8)∂u∂t+∇·uu=−∇p+ν∇2u.The form of equations (7) and (8) is equivalent to the incompressible Navier–Stokes equations.In this paper, the D2G9 velocity model is used to solve the fluid flow within the liquid annular seals. The velocity directions of the model are shown in Figure1. The speed configurations of the D2G9 model are shown as equation (9).where ea is the discrete velocity, c is the lattice speed and it is defined as c = Δx/Δt, Δx is the lattice step, and Δt is the time increment.Figure 1
2D, nine-velocity lattice direction.(9)eα=0,0,α=0,ccosα−1π2,sinα−1π2,α=1,2,3,4,2cos2α−1π4,sin2α−1π4,α=5,6,7,8,


#### 2.1.2. Equilibrium Distribution Function

The equilibrium distribution function of the D2G9 model is shown as equation (10). The equilibrium distribution itself is calculated from the macroscopic values which themselves are low order moments of f [28].(10)fαeq=ρ0−4d0Pc2+ρ0sαuα=0,d1Pc2+ρ0sαuα=1,2,3,4,d2Pc2+ρ0sαuα=5,6,7,8,sαu=ωαeα•ucs2+eα•u22cs4−u22cs2.In equation (10), the parameters d0, d1, and d2 satisfy the relationship as equation (11) [28]. The macroscopic velocity and pressure equations can be given as equations (12) and (13), respectively.(11)d1+d2=d0,d1+2d2=12,(12)u=1ρ0∑αeαfαeq,(13)P′=ρ0c24d0∑α≠0fα+s0u.


### 2.2. Turbulence Model

According to the lattice Reynolds number equation shown as shown in equation (14), the lattice kinematic viscosity ν0 will decrease with the increase in Reynolds number. Since the flow within seals is highly turbulent with a Reynolds number of more than 104, the relaxation factor τ is close to 0.5, which will make the computation difficult to converge. To solve this problem, the LBM calculating model combined with the 2D LES Smagorinsky model is established specially for the static characteristics and leakage of liquid annular seals. As shown in equation (15), the total lattice viscosity νt consists of two parts: the lattice kinematic viscosity ν0 and the Smargorinsky eddy viscosity νt,LES. The value of ν0 can be described in equation (14), and the value of νt,LES is described in equation (16), in which Cs is the Smagorinsky constant, Δ is the mean lattice spacing defined as Δ = (ΔxΔy)1/2, and s¯ij is the strain rate tensor that could be obtained by equation (17). In this paper, the value of Smagorinsky constant Cs is fixed at 0.1 [18]. Besides, near the walls, νt,LES is damped, which means νt,LES will decrease and approach to zero as the wall is encountered.(14)ν0=Ulattice•HRe,(15)νt=ν0+νt,LES,(16)νt,LES=CsΔ2s¯ij,(17)s¯ij=∑α=08eαieαjfα−fαeq.


### 2.3. Boundary Conditions

Boundary conditions have great influences on accuracy, stability, and computational efficiency of LBM simulation. In this paper, velocity inlet, open outlet, and nonequilibrium extrapolation scheme are used as the initial boundary conditions.


#### 2.3.1. Velocity Boundary Condition

The modified velocity boundaries based on bounce back part of the nonequilibrium extrapolation scheme are shown as equation (18). Extrapolation is used to calculate the outlet functions defined in equation (19) [30].(18)f10,j=f30,j+23p0,ju0,j,f50,j=f70,j−12f20,j−f40,j+16p0,ju0,j,f80,j=f60,j+12f20,j−f40,j+16p0,ju0,j,(19)f3,6,7NX,j=2•f3,6,7NX−1,j−f3,6,7NX−2,j.


#### 2.3.2. Wall Boundary Condition

Nonequilibrium extrapolation scheme [30] is chosen as the wall boundary condition. Pressure is used as the dynamic variable for eliminating compressibility errors in the simulation. As shown in Figure 2, Node A, Node O, and Node C are located at the boundaries, and Node E, Node B, and Node D are located in the flow field. The distribution function of the boundary Node O can be calculated by the following equation:(20)fαO,t=fαeqO,t+1−γfαB,t−fαeqB,t.Figure 2
Nonequilibrium extrapolation equations.


### 2.4. Code Program

Calculation code for leakage flow rate of liquid annular seals is accomplished by using C++ on the basis of 2D LBM combined with the governing equations, turbulence model, and boundary conditions described above. The overall program block diagram is shown in Figure3.Figure 3
LBM Program block diagram.


## 2.1. Governing Equations

The governing equation of the LBM calculating model can be written as equation (2) under the SRT-LBM model. It has second-order accuracy at low Mach number. In equation (2), γ is collision frequency, τ is relaxation time, and the relationship between γ and τ is γ = 1/τ. τ is mainly related to the lattice kinematic viscosity ν0 as shown in equation (3).(2)fαx+cαΔt,y+cαΔt,t+Δt−fαx,y,t=−γfαx,y,t−fαeqx,y,t,(3)ν0=2τ−16Δx2Δt.For the LBM calculating model, the distribution function evolution process is the most important component and the evolution process consists of two steps: the collision step that is described as equation (4) and the streaming step, which is described as equation (5).(4)fαx,y,t+Δt=fαx,y,t1−γ+γfαeqx,y,t,(5)fαx+Δx,y+Δy,t+Δt=fαx,y,t+Δt.


### 2.1.1. Discrete Velocity Model

The discrete velocity model is one of the most important components of LBM calculating models. For compressible fluid flow calculation, Kataoka and Tsutahara [24] developed a new lattice Boltzmann velocity model for the compressible fluid simulation. Yang et al. [25] studied the compressible fluid flow which is around NACA0012 airfoil by the D1Q4 velocity model. Yang analyzed the 3D viscous compressible flows [26] by using the lattice Boltzmann flux solver. Meanwhile, for incompressible fluid flow calculation, Qian et al. [27] established the D2Q9 velocity model for weakly compressible. In this model, fluid flow is driven by density gradient, and the relationship between pressure P and ρ is shown as equation (6). On the basis of the D2Q9 velocity model, Guo et al. [28] developed the D2G9 velocity model for incompressible fluid flow. The fluid flow is driven by pressure gradient in D2G9. The equilibrium distribution function satisfies Σfαeq=const. The macroscopic equations of the D2G9 velocity model can be derived by Chapman–Enskog expansion [29], and the equations can be written as equations (7) and (8) [28].(6)P=cs2ρ,(7)∇·u=0,(8)∂u∂t+∇·uu=−∇p+ν∇2u.The form of equations (7) and (8) is equivalent to the incompressible Navier–Stokes equations.In this paper, the D2G9 velocity model is used to solve the fluid flow within the liquid annular seals. The velocity directions of the model are shown in Figure1. The speed configurations of the D2G9 model are shown as equation (9).where ea is the discrete velocity, c is the lattice speed and it is defined as c = Δx/Δt, Δx is the lattice step, and Δt is the time increment.Figure 1
2D, nine-velocity lattice direction.(9)eα=0,0,α=0,ccosα−1π2,sinα−1π2,α=1,2,3,4,2cos2α−1π4,sin2α−1π4,α=5,6,7,8,


### 2.1.2. Equilibrium Distribution Function

The equilibrium distribution function of the D2G9 model is shown as equation (10). The equilibrium distribution itself is calculated from the macroscopic values which themselves are low order moments of f [28].(10)fαeq=ρ0−4d0Pc2+ρ0sαuα=0,d1Pc2+ρ0sαuα=1,2,3,4,d2Pc2+ρ0sαuα=5,6,7,8,sαu=ωαeα•ucs2+eα•u22cs4−u22cs2.In equation (10), the parameters d0, d1, and d2 satisfy the relationship as equation (11) [28]. The macroscopic velocity and pressure equations can be given as equations (12) and (13), respectively.(11)d1+d2=d0,d1+2d2=12,(12)u=1ρ0∑αeαfαeq,(13)P′=ρ0c24d0∑α≠0fα+s0u.


## 2.1.1. Discrete Velocity Model

The discrete velocity model is one of the most important components of LBM calculating models. For compressible fluid flow calculation, Kataoka and Tsutahara [24] developed a new lattice Boltzmann velocity model for the compressible fluid simulation. Yang et al. [25] studied the compressible fluid flow which is around NACA0012 airfoil by the D1Q4 velocity model. Yang analyzed the 3D viscous compressible flows [26] by using the lattice Boltzmann flux solver. Meanwhile, for incompressible fluid flow calculation, Qian et al. [27] established the D2Q9 velocity model for weakly compressible. In this model, fluid flow is driven by density gradient, and the relationship between pressure P and ρ is shown as equation (6). On the basis of the D2Q9 velocity model, Guo et al. [28] developed the D2G9 velocity model for incompressible fluid flow. The fluid flow is driven by pressure gradient in D2G9. The equilibrium distribution function satisfies Σfαeq=const. The macroscopic equations of the D2G9 velocity model can be derived by Chapman–Enskog expansion [29], and the equations can be written as equations (7) and (8) [28].(6)P=cs2ρ,(7)∇·u=0,(8)∂u∂t+∇·uu=−∇p+ν∇2u.The form of equations (7) and (8) is equivalent to the incompressible Navier–Stokes equations.In this paper, the D2G9 velocity model is used to solve the fluid flow within the liquid annular seals. The velocity directions of the model are shown in Figure1. The speed configurations of the D2G9 model are shown as equation (9).where ea is the discrete velocity, c is the lattice speed and it is defined as c = Δx/Δt, Δx is the lattice step, and Δt is the time increment.Figure 1
2D, nine-velocity lattice direction.(9)eα=0,0,α=0,ccosα−1π2,sinα−1π2,α=1,2,3,4,2cos2α−1π4,sin2α−1π4,α=5,6,7,8,


## 2.1.2. Equilibrium Distribution Function

The equilibrium distribution function of the D2G9 model is shown as equation (10). The equilibrium distribution itself is calculated from the macroscopic values which themselves are low order moments of f [28].(10)fαeq=ρ0−4d0Pc2+ρ0sαuα=0,d1Pc2+ρ0sαuα=1,2,3,4,d2Pc2+ρ0sαuα=5,6,7,8,sαu=ωαeα•ucs2+eα•u22cs4−u22cs2.In equation (10), the parameters d0, d1, and d2 satisfy the relationship as equation (11) [28]. The macroscopic velocity and pressure equations can be given as equations (12) and (13), respectively.(11)d1+d2=d0,d1+2d2=12,(12)u=1ρ0∑αeαfαeq,(13)P′=ρ0c24d0∑α≠0fα+s0u.


## 2.2. Turbulence Model

According to the lattice Reynolds number equation shown as shown in equation (14), the lattice kinematic viscosity ν0 will decrease with the increase in Reynolds number. Since the flow within seals is highly turbulent with a Reynolds number of more than 104, the relaxation factor τ is close to 0.5, which will make the computation difficult to converge. To solve this problem, the LBM calculating model combined with the 2D LES Smagorinsky model is established specially for the static characteristics and leakage of liquid annular seals. As shown in equation (15), the total lattice viscosity νt consists of two parts: the lattice kinematic viscosity ν0 and the Smargorinsky eddy viscosity νt,LES. The value of ν0 can be described in equation (14), and the value of νt,LES is described in equation (16), in which Cs is the Smagorinsky constant, Δ is the mean lattice spacing defined as Δ = (ΔxΔy)1/2, and s¯ij is the strain rate tensor that could be obtained by equation (17). In this paper, the value of Smagorinsky constant Cs is fixed at 0.1 [18]. Besides, near the walls, νt,LES is damped, which means νt,LES will decrease and approach to zero as the wall is encountered.(14)ν0=Ulattice•HRe,(15)νt=ν0+νt,LES,(16)νt,LES=CsΔ2s¯ij,(17)s¯ij=∑α=08eαieαjfα−fαeq.


## 2.3. Boundary Conditions

Boundary conditions have great influences on accuracy, stability, and computational efficiency of LBM simulation. In this paper, velocity inlet, open outlet, and nonequilibrium extrapolation scheme are used as the initial boundary conditions.


### 2.3.1. Velocity Boundary Condition

The modified velocity boundaries based on bounce back part of the nonequilibrium extrapolation scheme are shown as equation (18). Extrapolation is used to calculate the outlet functions defined in equation (19) [30].(18)f10,j=f30,j+23p0,ju0,j,f50,j=f70,j−12f20,j−f40,j+16p0,ju0,j,f80,j=f60,j+12f20,j−f40,j+16p0,ju0,j,(19)f3,6,7NX,j=2•f3,6,7NX−1,j−f3,6,7NX−2,j.


### 2.3.2. Wall Boundary Condition

Nonequilibrium extrapolation scheme [30] is chosen as the wall boundary condition. Pressure is used as the dynamic variable for eliminating compressibility errors in the simulation. As shown in Figure 2, Node A, Node O, and Node C are located at the boundaries, and Node E, Node B, and Node D are located in the flow field. The distribution function of the boundary Node O can be calculated by the following equation:(20)fαO,t=fαeqO,t+1−γfαB,t−fαeqB,t.Figure 2
Nonequilibrium extrapolation equations.


## 2.3.1. Velocity Boundary Condition

The modified velocity boundaries based on bounce back part of the nonequilibrium extrapolation scheme are shown as equation (18). Extrapolation is used to calculate the outlet functions defined in equation (19) [30].(18)f10,j=f30,j+23p0,ju0,j,f50,j=f70,j−12f20,j−f40,j+16p0,ju0,j,f80,j=f60,j+12f20,j−f40,j+16p0,ju0,j,(19)f3,6,7NX,j=2•f3,6,7NX−1,j−f3,6,7NX−2,j.


## 2.3.2. Wall Boundary Condition

Nonequilibrium extrapolation scheme [30] is chosen as the wall boundary condition. Pressure is used as the dynamic variable for eliminating compressibility errors in the simulation. As shown in Figure 2, Node A, Node O, and Node C are located at the boundaries, and Node E, Node B, and Node D are located in the flow field. The distribution function of the boundary Node O can be calculated by the following equation:(20)fαO,t=fαeqO,t+1−γfαB,t−fαeqB,t.Figure 2
Nonequilibrium extrapolation equations.


## 2.4. Code Program

Calculation code for leakage flow rate of liquid annular seals is accomplished by using C++ on the basis of 2D LBM combined with the governing equations, turbulence model, and boundary conditions described above. The overall program block diagram is shown in Figure3.Figure 3
LBM Program block diagram.


## 3. Simulation Parameters

In order to verify the method proposed in this paper, the predicted leakage flow rates of three different liquid annular seals under different operating conditions based on LBM are compared with FVM results, experimental results, and theoretical results of empirical formulas. Figure4 illustrates the basic geometry and coordinate system used in this paper. The geometric parameters of the three seals are listed in Table 1, and the operating conditions are illustrated in Table 2.Figure 4
Basic geometry of liquid annular seal.(22)P=Pc−Pr=Pc−P′ρrur2/2.Table 1
Geometric parameters of three model seals.
Model123ReferenceLindsey and Childs [31]Darden et al. [32]Marquette et al. [33]Seal clearanceC (mm)0.0760.530.11Seal lengthL (mm)13.1345.734.93Seal diameterD (mm)76.291.476.29Open areaS (mm2)18.2152.9926.39Table 2
Operating conditions of the model seals.
Model1Pressure difference (MPa)1.382.413.45Inlet velocity (m/s)50.4469.9182.96Rotor speed (rpm)102001020010200Kinematic viscosity (m2/s)5.215e − 75.215e − 75.215e − 7Reynolds numbers6038971312182Model2Pressure difference (MPa)9.2911.512.6Inlet velocity (m/s)130.082144.6151.36Rotor speed (rpm)5500158005950Kinematic viscosity (m2/s)1.0048e − 61.0048e − 61.0048e − 6Reynolds numbers690137627079835Model3Pressure difference (MPa)4.145.526.89Inlet velocity (m/s)87.37100.881112.7Rotor speed (rpm)102001020010200Kinematic viscosity (m2/s)5.215e − 75.215e − 75.215e − 7Reynolds numbers184262127723771


### 3.1. Parameters of LBM

The geometric parameters of model seals should be transformed to reference parameters with lattice unit [34], and the reference parameters must be the same value in the same model. The reference length Hr, reference velocity ur, and reference density ρr are defined as follows:(21)Hr=H′H,ρr=ρ′ρ,ur=cscs.In equation (21), where cs’ and cs are the sound speed in physical unit and lattice unit, respectively, in this study, the value of ur is 2678.44 m/s. Andρ′ and ρ are the density in physical unit and lattice unit, respectively; the value of the ρr is 992 kg/m3. The reference length Hr is defined by grid resolution; for example, if the lattice number of the model gets 9047 and the physical length H′ is 13.13 mm, the reference length Hr should be 1.451 × 10−6m. Besides, the transformation equation of lattice pressure P and reference pressure Pr is shown as follows.In equation (22), Pc is the stagnation point pressure in the lattice unit, and it must take the same value for the stimulating of same model seal. The value of Pc generally takes less than 1 but more than 0.09 for accelerating convergence.Moreover, as the lattice Reynolds numberRelattice shown in equation (14), it is equal to the macroscopic Reynolds number Re, and it is possible to choose proper values for Ulattice and ν0 under the condition that Relattice equals to Re [35]. Recently, ν0 remains consistent in different stimulation cases for the same model. Therefore, the value of ν0 in this work remains the same for the same model seal when stimulations for different operation conditions are conducted. The simulation parameters of the model seals are listed in Table 3.Table 3
LBM-LES simulation parameters of the model seals.
Model123Number of grids in radial direction527029Number of grids in axial direction904761789374Length to height ratio17286318Pc0.60.70.4Ulattice0.00869, 0.0125, 0.01570.065, 0.0722, 0.07560.042, 0.0485, 0.0542v06.639e − 56.639e − 56.639e − 5A grid independence study for each of the three models using seven different sizes of lattices has been carried out, as shown in Figure5. The simulation time and convergence accuracy are taken into account, the lattice number of Childs’ model is chosen as 470,444, Darden’s model is chosen as 432,460, and Marquette’s model is chosen as 271,846. Leakage error ε shown in Figure 5 is calculated based on equation (23), where Qn is the numerical result and Qe is the experimental result:(23)ε=Qn−QeQe×100%.Figure 5
Grid sensitivity.


### 3.2. Parameters of FVM

The calculation results based on FVM which is realized by FLUENT software are compared with the results based on the proposed method in this study. Transient 2D model, 2D coupled solver, and implicit difference scheme are applied. The LES Smagorinsky model is introduced to describe the fluid stress term in the control equations and is discretized with the first-order upwind style. The convergence condition is that the residuals of all the equations are less than 1 × 10−5.


### 3.3. Parameters of Empirical Formulas

As the empirical formulas shown in equation (24) are the most commonly used method for calculating the leakage flow rates of the seals in engineering, in this work, the theoretical results of empirical formulas also act as an important part of the comparisons. In the empirical formulas, C is the seal clearance, µ is the flow coefficient, L is the seal length, S is the open area of seal clearance, and Q is the leakage of seal. As shown in the above equations, µ is related to the input rounded coefficient, and the friction coefficient number is determined by the Reynolds number and the surface roughness. The coefficient λ generally takes the value between 0.04 and 0.06, and the coefficient η takes the value between 0.5 and 0.9.(24)S=2πRC,μ=11+0.5η+λL/2C,Q=μS2gPin−Pout.


## 3.1. Parameters of LBM

The geometric parameters of model seals should be transformed to reference parameters with lattice unit [34], and the reference parameters must be the same value in the same model. The reference length Hr, reference velocity ur, and reference density ρr are defined as follows:(21)Hr=H′H,ρr=ρ′ρ,ur=cscs.In equation (21), where cs’ and cs are the sound speed in physical unit and lattice unit, respectively, in this study, the value of ur is 2678.44 m/s. Andρ′ and ρ are the density in physical unit and lattice unit, respectively; the value of the ρr is 992 kg/m3. The reference length Hr is defined by grid resolution; for example, if the lattice number of the model gets 9047 and the physical length H′ is 13.13 mm, the reference length Hr should be 1.451 × 10−6m. Besides, the transformation equation of lattice pressure P and reference pressure Pr is shown as follows.In equation (22), Pc is the stagnation point pressure in the lattice unit, and it must take the same value for the stimulating of same model seal. The value of Pc generally takes less than 1 but more than 0.09 for accelerating convergence.Moreover, as the lattice Reynolds numberRelattice shown in equation (14), it is equal to the macroscopic Reynolds number Re, and it is possible to choose proper values for Ulattice and ν0 under the condition that Relattice equals to Re [35]. Recently, ν0 remains consistent in different stimulation cases for the same model. Therefore, the value of ν0 in this work remains the same for the same model seal when stimulations for different operation conditions are conducted. The simulation parameters of the model seals are listed in Table 3.Table 3
LBM-LES simulation parameters of the model seals.
Model123Number of grids in radial direction527029Number of grids in axial direction904761789374Length to height ratio17286318Pc0.60.70.4Ulattice0.00869, 0.0125, 0.01570.065, 0.0722, 0.07560.042, 0.0485, 0.0542v06.639e − 56.639e − 56.639e − 5A grid independence study for each of the three models using seven different sizes of lattices has been carried out, as shown in Figure5. The simulation time and convergence accuracy are taken into account, the lattice number of Childs’ model is chosen as 470,444, Darden’s model is chosen as 432,460, and Marquette’s model is chosen as 271,846. Leakage error ε shown in Figure 5 is calculated based on equation (23), where Qn is the numerical result and Qe is the experimental result:(23)ε=Qn−QeQe×100%.Figure 5
Grid sensitivity.


## 3.2. Parameters of FVM

The calculation results based on FVM which is realized by FLUENT software are compared with the results based on the proposed method in this study. Transient 2D model, 2D coupled solver, and implicit difference scheme are applied. The LES Smagorinsky model is introduced to describe the fluid stress term in the control equations and is discretized with the first-order upwind style. The convergence condition is that the residuals of all the equations are less than 1 × 10−5.


## 3.3. Parameters of Empirical Formulas

As the empirical formulas shown in equation (24) are the most commonly used method for calculating the leakage flow rates of the seals in engineering, in this work, the theoretical results of empirical formulas also act as an important part of the comparisons. In the empirical formulas, C is the seal clearance, µ is the flow coefficient, L is the seal length, S is the open area of seal clearance, and Q is the leakage of seal. As shown in the above equations, µ is related to the input rounded coefficient, and the friction coefficient number is determined by the Reynolds number and the surface roughness. The coefficient λ generally takes the value between 0.04 and 0.06, and the coefficient η takes the value between 0.5 and 0.9.(24)S=2πRC,μ=11+0.5η+λL/2C,Q=μS2gPin−Pout.


## 4. Results and Discussion

The comparisons of predicted leakage flow rates of three different liquid annular seals under different operating conditions based on LBM, FVM results, empirical formulas, and experiences are conducted and discussed in this section.


### 4.1. Results of Model 1

As shown in Table4, the simulation results of Model 1 based on LBM, FVM, and empirical formulas are compared with the experimental results conducted by Childs [28]. Since the clearance of the annular seal is small, enlarged velocity contour diagrams at the seal exit based on LBM under different operating conditions are shown in Figure 6. From Figure 6, it can be found that the fluid velocity increases as the pressure difference increases. In the seal clearance, the fastest fluid velocity is at the center of the clearance. The comparisons show that the empirical formulas are the most time efficient method but with the lowest prediction accuracy of leakage flow rates. The maximum error of the empirical formulas is nearly 15%. LBM and FVM have a good accuracy with a maximum error of less than 3.86% and 6.42%, respectively. For this model, the accuracy of LBM is higher than that of FVM. In contrast with the complex modeling and meshing process of FVM, for different model seals using LBM, only a few initial parameters such as geometric parameters, inlet velocity, and pressure difference need to be changed, which has made LBM more time efficient and more suitable for engineering application.Table 4
Childs leakage annular seals at different simulate methods.
Operating conditionsPressure difference (MPa)1.382.413.45Leakage flow rates of LBM (L/s)0.510.7060.883Leakage flow rates of FVM (L/s)0.4920.6940.857Leakage flow rates of empirical formulas (L/s)0.4680.6190.741Leakage flow rates of test results (L/s)0.4920.70.864Leakage flow error of LBM (%)0.3620.7992.16Figure 6
Different pressure difference velocity contour using LBM of Model 1. Velocity contour under (a) 1.38 MPa, (b) 2.41 MPa, and (c) 3.45 MPa using LBM.
(a)(b)(c)


### 4.2. Results of Model 2

Experimental results are conducted by Darden et al [32], and the calculation results which are calculated by the three methods for leakage flow rates of Model 2 are shown in Figure 7 and Table 5. It is observed that the leakage rates based on empirical formulas were under-predicted with an error of more than 10% compared to the experimental results. The prediction error of the leakage rate based on FVM is 2.85% under the pressure difference of 9.29 MPa but with error of more than 5% under the other two operating conditions. Among the three methods, LBM has obvious advantages in leakage prediction accuracy, especially under high pressure differences. The prediction error of LBM-LES decreases from 3.48% to 2.32% as the pressure difference increases, while the error of FVM shows an opposite variation.Figure 7
Comparisons of leakage flow rates of Model 2 with different methods.Table 5
Darden leakage annular seals at different simulate methods.
Operating conditionsPressure difference (MPa)9.2911.512.6Leakage flow rates of LBM (L/s)14.9615.6716.41Leakage flow rates of FVM (L/s)15.0617.1918.15Leakage flow rates of empirical formulas (L/s)12.9714.4315.11Leakage flow rates of test results (L/s)15.516.316.8Leakage flow error of LBM (%)3.483.872.32Leakage flow error of FVM (%)2.855.468.01Leakage flow error of empirical (%)16.311.510.09


### 4.3. Results of Model 3

Experimental results are conducted by Marquette et al [33], and the calculation results based on three methods for predicting leakage flow rates of Model 3 are shown in Figure 8 and Table 6. It is observed that the calculation errors of the empirical formula are all around 8%. Calculation errors of FVM decrease with the increasing pressure differences. The maximum error is 12.48%, and the minimum error is 2.55%. By contrast, prediction results based on LBM show good agreement with the experimental results. All of the results have a margin of error of less than 4%. Even more, the prediction error is less than 2% under the pressure difference of 5.52 MPa and 6.89 MPa.Figure 8
Comparisons of leakage flow rates of Model 3 with different methods.Table 6
Marquette leakage annular seals at different simulate methods.
Operating conditionsPressure difference (MPa)4.145.526.89Leakage flow rates of LBM (L/s)0.871.0021.12Leakage flow rates of FVM (L/s)0.750.951.124Leakage flow rates of empirical formulas (L/s)0.921.061.18Leakage flow rates of test results (L/s)0.831.001.09Leakage flow error of LBM (%)3.610.521.95Leakage flow error of FVM (%)12.485.042.55Leakage flow error of empirical (%)106.147.9


## 4.1. Results of Model 1

As shown in Table4, the simulation results of Model 1 based on LBM, FVM, and empirical formulas are compared with the experimental results conducted by Childs [28]. Since the clearance of the annular seal is small, enlarged velocity contour diagrams at the seal exit based on LBM under different operating conditions are shown in Figure 6. From Figure 6, it can be found that the fluid velocity increases as the pressure difference increases. In the seal clearance, the fastest fluid velocity is at the center of the clearance. The comparisons show that the empirical formulas are the most time efficient method but with the lowest prediction accuracy of leakage flow rates. The maximum error of the empirical formulas is nearly 15%. LBM and FVM have a good accuracy with a maximum error of less than 3.86% and 6.42%, respectively. For this model, the accuracy of LBM is higher than that of FVM. In contrast with the complex modeling and meshing process of FVM, for different model seals using LBM, only a few initial parameters such as geometric parameters, inlet velocity, and pressure difference need to be changed, which has made LBM more time efficient and more suitable for engineering application.Table 4
Childs leakage annular seals at different simulate methods.
Operating conditionsPressure difference (MPa)1.382.413.45Leakage flow rates of LBM (L/s)0.510.7060.883Leakage flow rates of FVM (L/s)0.4920.6940.857Leakage flow rates of empirical formulas (L/s)0.4680.6190.741Leakage flow rates of test results (L/s)0.4920.70.864Leakage flow error of LBM (%)0.3620.7992.16Figure 6
Different pressure difference velocity contour using LBM of Model 1. Velocity contour under (a) 1.38 MPa, (b) 2.41 MPa, and (c) 3.45 MPa using LBM.
(a)(b)(c)


## 4.2. Results of Model 2

Experimental results are conducted by Darden et al [32], and the calculation results which are calculated by the three methods for leakage flow rates of Model 2 are shown in Figure 7 and Table 5. It is observed that the leakage rates based on empirical formulas were under-predicted with an error of more than 10% compared to the experimental results. The prediction error of the leakage rate based on FVM is 2.85% under the pressure difference of 9.29 MPa but with error of more than 5% under the other two operating conditions. Among the three methods, LBM has obvious advantages in leakage prediction accuracy, especially under high pressure differences. The prediction error of LBM-LES decreases from 3.48% to 2.32% as the pressure difference increases, while the error of FVM shows an opposite variation.Figure 7
Comparisons of leakage flow rates of Model 2 with different methods.Table 5
Darden leakage annular seals at different simulate methods.
Operating conditionsPressure difference (MPa)9.2911.512.6Leakage flow rates of LBM (L/s)14.9615.6716.41Leakage flow rates of FVM (L/s)15.0617.1918.15Leakage flow rates of empirical formulas (L/s)12.9714.4315.11Leakage flow rates of test results (L/s)15.516.316.8Leakage flow error of LBM (%)3.483.872.32Leakage flow error of FVM (%)2.855.468.01Leakage flow error of empirical (%)16.311.510.09


## 4.3. Results of Model 3

Experimental results are conducted by Marquette et al [33], and the calculation results based on three methods for predicting leakage flow rates of Model 3 are shown in Figure 8 and Table 6. It is observed that the calculation errors of the empirical formula are all around 8%. Calculation errors of FVM decrease with the increasing pressure differences. The maximum error is 12.48%, and the minimum error is 2.55%. By contrast, prediction results based on LBM show good agreement with the experimental results. All of the results have a margin of error of less than 4%. Even more, the prediction error is less than 2% under the pressure difference of 5.52 MPa and 6.89 MPa.Figure 8
Comparisons of leakage flow rates of Model 3 with different methods.Table 6
Marquette leakage annular seals at different simulate methods.
Operating conditionsPressure difference (MPa)4.145.526.89Leakage flow rates of LBM (L/s)0.871.0021.12Leakage flow rates of FVM (L/s)0.750.951.124Leakage flow rates of empirical formulas (L/s)0.921.061.18Leakage flow rates of test results (L/s)0.831.001.09Leakage flow error of LBM (%)3.610.521.95Leakage flow error of FVM (%)12.485.042.55Leakage flow error of empirical (%)106.147.9


## 5. Conclusions

In this paper, a 2D LBM calculating model which consists of 2D LES turbulence model, D2G9 velocity model, and nonequilibrium extrapolation scheme is established for predicting of leakage flow rates and static characteristics of liquid annular seals. The leakage rates of three different plain annular seals under three different operating conditions are calculated, and the results of them are compared with the experimental results to prove the validity and accuracy of the proposed model. Comparisons show that all of the prediction errors of the 2D LBM calculating model are less than 5%, some of them even less than 2%, which indicates that the calculating model satisfies the requirements of engineering application better compared with the other commonly used methods. The accuracy of the calculating model for predicting the leakage flow rates of seals is mainly related to the clearance and pressure differences of the model seals. The accuracy decreases with the increasing clearance of the model seals which increases with the increase in pressure difference.The transformation equation of reference pressurePr is developed based on the Bernoulli equation for incompressible fluids within annular seals. Reference velocity, reference density, and stagnation point pressure Pc in lattice unit are introduced in this transformation equation, and the optimal value range of Pc for leakage rates predictions of annular seals is proposed.



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*Source: 6631557-2021-03-03.xml*