Shape Optimization of Labyrinth Seals to Improve Sealing Performance
Shape Optimization of Labyrinth Seals to Improve Sealing Performance
Zhao, Yizhen;Wang, Chunhua
2021-04-01 00:00:00
aerospace Article Shape Optimization of Labyrinth Seals to Improve Sealing Performance 1 1 , 2 , Yizhen Zhao and Chunhua Wang * AECC Shenyang Engine Research Institute, No. 1 Wanlian Road, Shenyang 110015, China; zyz199212@163.com College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China * Correspondence: chunhuawang@nuaa.edu.cn Abstract: To reduce gas leakage, shape optimization of a straight labyrinth seal was carried out. The six design parameters included seal clearance, fin width, fin height, fin pitch, fin backward, and forward expansion angle. The CFD (Computational Fluid Dynamics) model was solved to generate the training and testing samples for the surrogate model, which was established by the least square support vector machine. A kind of chaotic optimization algorithm was used to determine the optimal design parameters of the labyrinth seal. As seal clearance, fin width, fin height, fin pitch, fin backward and forward expansion angles are 0.2 mm, 0.1 mm, 7 mm, 9 mm, 0 , and 15 , the discharge coefficient can reach its minimum value in the design space. The chaotic optimization algorithm coupled with least square support vector machine is a promising scheme for labyrinth seal optimization. Keywords: straight labyrinth seal; discharge coefficient; chaotic optimization algorithm; least square support vector machine Citation: Zhao, Y.; Wang, C. Shape Optimization of Labyrinth Seals to Improve Sealing Performance. 1. Introduction Aerospace 2021, 8, 92. https:// With the increase of operation pressure, leakage in gas turbines has attracted more doi.org/10.3390/aerospace8040092 and more attention. To prevent fluid leakage from high- to low-pressure regions, labyrinth seals, one kind of non-contacting mechanical seal, have been widely used for many decades. Academic Editor: Lawrence Ukeiley Labyrinth seals work by throttling flow through small successive openings in series, each one of which converts pressure into velocity, which ideally is dissipated in the intervening Received: 29 January 2021 chambers. The main advantages of labyrinth seals are high flow resistance, structural Accepted: 19 March 2021 reliability, and simplicity. Due to these advantages, labyrinth seals remain competitive Published: 1 April 2021 compared with new designs such as brush and finger seals [1–3]. To analyze the performance of a labyrinth seal, many numerical and experimental Publisher’s Note: MDPI stays neutral studies have been performed. The influence of fin-shaped parameters on leakage were with regard to jurisdictional claims in studied by Du et al. [4] in straight labyrinth seals. Their experimental results showed that published maps and institutional affil- increasing the seal clearance, fin width, and fin height all lead to an increase of discharge iations. coefficient, while adding the fin pitch causes the decrease of the discharge coefficient. Anker and Mayer [5] reported that, at realistic clearance, due to negative incidence downstream stator, the flow separation of leakage flow takes place. Pychynski et al. [6] established the prediction model for discharge coefficient with a data mining method. In the model, the Copyright: © 2021 by the authors. discharge coefficient relies on the number of seal fins, fin height, fin pitch, pressure ratio, Licensee MDPI, Basel, Switzerland. seal clearance, step shift, honeycomb cell diameter, fin width, groove shift, step height, This article is an open access article and groove width. Asok et al. [7] applied an artificial neural network for the prediction distributed under the terms and and optimization of square cavity labyrinth seals. Kim and Cha [8] indicated that, as the conditions of the Creative Commons clearance increases, the performance of stepped seals becomes much better compared Attribution (CC BY) license (https:// with straight seals. Labyrinth seals are complex systems under the influence of geometric creativecommons.org/licenses/by/ and flow parameters. To find global optimal solutions for a given set of requirements, 4.0/). Aerospace 2021, 8, 92. https://doi.org/10.3390/aerospace8040092 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, x FOR PEER REVIEW 2 of 11 geometric and flow parameters. To find global optimal solutions for a given set of require- ments, researchers need to handle a high-dimensional design space [9,10]. Therefore, an effective optimization tool is necessary to be developed for labyrinth seals. Aerospace 2021, 8, 92 2 of 11 In the present optimization method, the surrogate model was established by a super- vised learning model called a least square support vector machine (LS-SVM). As one kind of SVM, LS-SVM [11,12] is an effective machine learning tool for regression analysis and researchers need to handle a high-dimensional design space [9,10]. Therefore, an effective pattern recognition. Compared with other surrogate models, LS-SVM shows better regres- optimization tool is necessary to be developed for labyrinth seals. sion accuracy and lower computation cost. LS-SVM is especially suitable for machine In the present optimization method, the surrogate model was established by a su- pervised learning model called a least square support vector machine (LS-SVM). As one learning with a small-size training sample. Moreover, a kind of chaotic optimization algo- kind of SVM, LS-SVM [11,12] is an effective machine learning tool for regression analysis rithm was applied for global and detail searches. Compared with the random ergodic and pattern recognition. Compared with other surrogate models, LS-SVM shows better searches, the chaotic optimization can perform global searches at higher efficiency due to regression accuracy and lower computation cost. LS-SVM is especially suitable for ma- non-repetition of the chaotic system [13,14] chine learning with a small-size training sample. Moreover, a kind of chaotic optimization algorithm was applied for global and detail searches. Compared with the random ergodic In this paper, the optimization model of straight labyrinth seals was introduced firstly; searches, the chaotic optimization can perform global searches at higher efficiency due to and then a CFD (Computational Fluid Dynamics) solution was performed to generate the non-repetition of the chaotic system [13,14] training and testing samples for the surrogate model; finally, the structural optimization In this paper, the optimization model of straight labyrinth seals was introduced firstly; was carried out with the chaotic optimization algorithm and LS-SVM surrogate model, and then a CFD (Computational Fluid Dynamics) solution was performed to generate the training and testing samples for the surrogate model; finally, the structural optimization and a detailed analysis of optimization results were provided. was carried out with the chaotic optimization algorithm and LS-SVM surrogate model, and a detailed analysis of optimization results were provided. 2. Objective Function and Design Variables 2. Objective Function and Design Variables In current research, the design variables include seal clearance (c), fin pitch (B), fin In current research, the design variables include seal clearance (c), fin pitch (B), fin height (H), fin width (w), fin forward expansion angle (β), and backward expansion angle height (H), fin width (w), fin forward expansion angle (b), and backward expansion angle (α). These parameters were defined in Figure 1, and their lower and upper limits are listed (). These parameters were defined in Figure 1, and their lower and upper limits are listed in Table 1. To meet the high sealing requirement in the outlet region of the high-pressure in Table 1. To meet the high sealing requirement in the outlet region of the high-pressure compressor, the seal in the present model has five fins. The inlet total temperature and compressor, the seal in the present model has five fins. The inlet total temperature and pressu pressure re arar e 300 e 30 K0 and K an 0.12 d MPa 0.12 rM espectively Pa respec . The tively. outlet The o staticu pr tlet static pr essure is 0.1 MPa. essure is 0.1 MPa. Figure 1. Structure of straight-labyrinth seal. Figure 1. Structure of straight-labyrinth seal. Table 1. Design variables and design space. Table 1. Design variables and design space. Design Variable Symbol Unit Lower Bound Upper Bound FinDe clearance sign Variable c Symbol mm Uni 0.2t Lower Boun 0.6 d Upper Bound Fin width w mm 0.1 0.7 Fin clearance c mm 0.2 0.6 Fin height H mm 3 7 Fin pitch Fin width B mmw mm 3 0.1 9 0.7 Fin backward expansion angle a 0 15 Fin height H mm 3 7 Fin forward expansion angle b 0 15 Fin pitch B mm 3 9 Fin backward expansion angle α ° 0 15 The discharge coefficient is used to evaluate the seal performance quantitatively. C is Fin forward expansion angle β ° 0 15 defined as [2,15]: C = (1) The discharge coefficient is used to evaluate the seal performance quantitatively. Cd is defined as [2,15]: C (1) Aerospace 2021, 8, x FOR PEER REVIEW 3 of 11 Aerospace 2021, 8, x FOR PEER REVIEW 3 of 11 Aerospace 2021, 8, 92 3 of 11 where ma and midenote the actual and ideal mass flow rate through the channel, respec- where ma and midenote the actual and ideal mass flow rate through the channel, respec- tively. mi can be expressed by: tively. mi can be expressed by: where m and m denote the actual and ideal mass flow rate through the channel, respec- a i tively. m can be expressed by: 2 21 k 2 21 k p A p p 2k 0 nn p A 2k pkk p 0 nn m s ( ) [1kk ( ) ] (2) im ( ) [1 ( ) ] (2) 2 k 1 kRT k1 p p p A 2k p p 0 k1 n00 p k np k kR 0T m = p ( ) [1 ( ) ] (2) k 1 p p kRT 0 0 where k is the isentropic coefficient (k = 1.4). pn and p0 denote the static and total pressure where k is the isentropic coefficient (k = 1.4). pn and p0 denote the static and total pressure where k is the isentropic coefficient (k = 1.4). p and p denote the static and total pressure at the outlet, respectively. R is the gas constant. A de0 notes the cross-section area of the at the outlet, respectively. R is the gas constant. A denotes the cross-section area of the at the outlet, respectively. R is the gas constant. A denotes the cross-section area of the channel, and T0 is the inlet total temperature. channel, and T0 is the inlet total temperature. channel, and T is the inlet total temperature. 3. Optimization Method 3. Optimization Method 3. Optimization Method 3.1. CFD Method 3.1. CFD Method 3.1. CFD Method To study the influences of geometric variables on the discharge coefficient, numerical To study the influences of geometric variables on the discharge coefficient, numerical To study the influences of geometric variables on the discharge coefficient, numerical experiments were carried out by Ansys-Fluent 14.6. Static pressure at the channel outlet, experiments were carried out by Ansys-Fluent 14.6. Static pressure at the channel outlet, experiments were carried out by Ansys-Fluent 14.6. Static pressure at the channel outlet, total pressure at the channel inlet, and adiabatic and non-slip condition at the walls are total pressure at the channel inlet, and adiabatic and non-slip condition at the walls are total pressure at the channel inlet, and adiabatic and non-slip condition at the walls are specified. The standard k-ε turbulence model with the enhanced wall function, which has specified. The standard k-ε turbulence model with the enhanced wall function, which has specified. The standard k-# turbulence model with the enhanced wall function, which has been proven to be suitable for labyrinth-seal flow by Morrison and Al-Ghasem [16], is been proven to be suitable for labyrinth-seal flow by Morrison and Al-Ghasem [16], is been proven to be suitable for labyrinth-seal flow by Morrison and Al-Ghasem [16], is used used for modeling turbulence. for used modeling for modturbulence. eling turbulence. As shown in Figure 2, structured grids are generated by the use of ICEM software. In As shown in Figure 2, structured grids are generated by the use of ICEM software. In As shown in Figure 2, structured grids are generated by the use of ICEM software. In the clearance region, there are at least 13-layer grids for all the cases. Moreover, 5-layer the the clearance clearance rregi egion, on, ther there e ar are e at at least least 13-layer 13-layer grids grids for for all all the the cases. cases. Mor Moreover eover, , 5-layer 5-layer meshes with the growing ratio of 1.2 were placed in the boundary layer region. The meshes meshes with with the the gr growin owing g ratio ratio of of 1.2 1.2 wer were e placed placedin in the the boundary boundary layer layerr egion. region. The The maximum y+ and x+ (flow direction) was less than 10.0 and 40, respectively. To determine maximum y+ and x+ (flow direction) was less than 10.0 and 40, respectively. To determine maximum y+ and x+ (flow direction) was less than 10.0 and 40, respectively. To determine the optimal grid number, a grid independent test was performed. Take one case with (c, the optimal grid number, a grid independent test was performed. Take one case with the optimal grid number, a grid independent test was performed. Take one case with (c, w, H, B, α, β) = (0.2 mm, 0.1 mm, 7 mm, 7 mm, 8°, 5°) as example. Figure 3 shows the grid (c, w, H, B, a, b) = (0.2 mm, 0.1 mm, 7 mm, 7 mm, 8 , 5 ) as example. Figure 3 shows the w, H, B, α, β) = (0.2 mm, 0.1 mm, 7 mm, 7 mm, 8°, 5°) as example. Figure 3 shows the grid independent test results. As the number of computational girds exceeded 117,741, the grid independent test results. As the number of computational girds exceeded 117,741, the independent test results. As the number of computational girds exceeded 117,741, the discharge coefficient kept stable. Therefore, the grid number of 117,741 can be accepted. discharge coefficient kept stable. Therefore, the grid number of 117,741 can be accepted. discharge coefficient kept stable. Therefore, the grid number of 117,741 can be accepted. Figure 2. Details of the grid used in the simulation. Figure 2. Details of the grid used in the simulation. Figure 2. Details of the grid used in the simulation. 0.52 0.52 CFD results 0.50 CFD results 0.50 Experimental result[4] 0.48 Experimental result[4] 0.48 0.46 0.46 0.44 0.44 0.42 0.42 0.40 0.40 0.38 0.38 0.36 0.36 0.34 0.34 0.32 4 4 4 4 5 5 5 0.32 2.0x10 4.0x 4 10 6.0x 4 10 8.0x 4 10 1.0x 4 10 1.2x 5 10 1.4x 5 10 5 2.0x10 4.0x10 6.0x10 8.0x10 1.0x10 1.2x10 1.4x10 Grid number Grid number Figure 3. Grid independence test result. Figure 3. Grid independence test result. Figure 3. Grid independence test result. Aerospace 2021, 8, 92 4 of 11 Using the Latin hypercube sampling method, 72 groups of data samples were gener- ated for the training samples for the surrogate model, and the other 20 groups generated randomly were used as the testing samples. 3.2. Surrogate Model Based on LS-SVM It was assumed that there are training points G= {(I , O ), i = 1, . . . ,N} needed to be i i fitted, where N denotes the total number of training points, which is the input vector, and I and O denote the input vector and output value, respectively. According to LS-SVM i i theory, this problem can be expressed by [17,18]: f (I ) = O a ker(I , I ) + b (3) j å i i j i i=1 where f (I ) is LS-SVM output value, and ker() denotes the kernel function. The following equations can be solved to determine a and b: 0 A b 0 = (4) T 1 A W + h E a O where h is the penalty factor, E denotes the unit matrix, a = [a , . . . , a ] , A = [1, . . . , 1], 1 N l O = [O , . . . , O ] , and W(i,j) = k(I , I ). Kernel function can be expressed by radial 1 Ns i j basis function: kI I k i j ker(I , I ) = exp( ) (5) i j 2d where d is the kernel parameter. 3.3. Chaotic Optimization Algorithm Currently, global optimization schemes including particle swarm algorithm, simulated annealing, evolutionary programming, and genetic algorithm have been widely applied for optimization practice. To escape from the local optimal values, the classical evolutionary (genetic) algorithm accepts some bad solutions under a certain probability. However, in a chaotic optimization scheme, the local optimal value can be avoided directly due to the regularity of chaotic motion [13,14,19]. In the present study, labyrinth seal geometries were optimized by a kind of chaotic algorithm. The labyrinth seal optimization model can be expressed by: c 2 [c , c ] min max w 2 [w , w ] min max H 2 [H , H ] min max minF(c, w, H, B, a, b)s.t. (6) > B 2 [B , B ] min max a 2 [a , a ] max > min b 2 [b , b ] min max where F() denotes the objective function. The chaotic time series is generated by logistic model, which is expressed by: t = lt (1 t ) l2(0, 4.0) (7) m +1 m m Adaptive mutative scale chaos optimization can be divided into 9 steps [19]: Step 1: Give N and N a large positive integer. Set m = 1 and n = 1, where, n and m 1 2 denote global and detailed searching times. Aerospace 2021, 8, 92 5 of 11 (1) (1) (1) (1) Step 2: Give t = (t , t , . . . , t ) a 6-dimensional random vector, and generate 1 2 6 (j+1) (j+1) chaotic series t , j = 1, . . . ,N according to the logistic model. Change t from the range (0,1) to (t , t ), i = 1,2, . . . ,6, by: i ,min i ,max 0(j) (j) t = t + (t t )t (8) i,min i,max i,min i i (n) (n) (n) n Step 3: Assign (t 0 , t 0 , . . . , t 0 ) to (c, w, H, B, a, b), and calculate F by LS-SVM. 1 2 6 n * If n equals 1, then assign F to F . * n n ’(n) * * Step 4: If F < F , then assign F and t to F and t . i i Step 5: Assign n + 1 to n. If n is smaller than N , return to Step 3. If n is larger than N , 1 1 ’ ’ then generate (t , t ) by: ,min ,max i i t = t j(t t ) i,max i,min i,min i (9) t = t + j(t t ) i,max i,min i,max i where j is in the interval of 0 and 0.5. If t is smaller than t , then assign t to i ,min i ,min i ,min ’ ’ ’ t . If t is higher than t , then assign t to t . ,max ,max ,max ,max i ,min i i i i Step 6: A new chaotic variable t can be generated by: 0 1 t t t t i i,min i i,min t = (1 b ) + b (10) i i 0 0 t t t t i,max i,min i,max i,min where adaptive adjustment coefficient, b can be determined by the following equation: m 1 b = 1 (11) where l equals 2 in the present stduy. * (m) (m) (m) (m) Step 7: Assign t to t , and give (t , t , . . . , t ) to (c, w, H, B, a, b). Calculate 1 2 6 i i n n * F . If m equals to 1, then assign F to F . * n n (m) * * Step 8: If F is smaller than F , assign F and t to F and t . i i Step 9: Assign m + 1 to m. If m is smaller than N , then go to 6th step. If m equals to N , the optimization processes end. 4. Analysis of Results 4.1. CFD Model Validation The open-published experimental results from Ref. [4] were used for model validation. The 25 groups of experimental data are shown in Table 2. Define the calculation error as: C C d,cal d,exp error = (12) d,exp where C is the CFD calculation results and C is the experimental results. The mean ,exp d ,cal d error is 7.9%, the maximum calculation error is 11.7%, and the minimum error is 1.8%. Overall, CFD calculated results agree with the experimental results well. Aerospace 2021, 8, x FOR PEER REVIEW 6 of 11 Aerospace 2021, 8, 92 6 of 11 7 0.3 0.5 5 7 15 0 0.365 0.426 8 0.5 0.5 6 9 0 5 0.409 0.440 Table 2. Experimental data vs. CFD data. 9 0.4 0.5 7 3 5 8 0.573 0.641 Number c (mm) w (mm) H (mm) B (mm) a( ) B( ) C C ,exp d ,cal d 10 0.6 0.5 3 4 8 12 0.608 0.551 1 0.2 0.7 3 3 0 0 0.438 0.450 211 0.3 0.20.7 0.3 4 4 5 5 9 5 0.491 5 12 0.500 0.340 0.357 3 0.5 0.7 5 6 8 8 0.530 0.581 12 0.3 0.3 6 3 8 15 0.472 0.427 4 0.4 0.7 6 7 12 12 0.403 0.389 13 0.5 0.3 7 4 12 0 0.609 0.595 5 0.6 0.7 7 9 15 15 0.500 0.449 6 0.2 0.5 4 6 12 15 0.322 0.361 14 0.4 0.3 3 6 15 5 0.514 0.476 7 0.3 0.5 5 7 15 0 0.365 0.426 15 0.6 0.3 4 7 0 8 0.511 0.494 8 0.5 0.5 6 9 0 5 0.409 0.440 9 0.4 0.5 7 3 5 8 0.573 0.641 16 0.2 0.2 6 4 15 8 0.460 0.516 10 0.6 0.5 3 4 8 12 0.608 0.551 17 0.3 0.2 7 6 0 12 0.409 0.386 11 0.2 0.3 5 9 5 12 0.340 0.357 12 0.3 0.3 6 3 8 15 0.472 0.427 18 0.5 0.2 3 7 5 15 0.342 0.36 13 0.5 0.3 7 4 12 0 0.609 0.595 19 0.4 0.2 4 9 8 0 0.412 0.369 14 0.4 0.3 3 6 15 5 0.514 0.476 15 0.6 0.3 4 7 0 8 0.511 0.494 20 0.6 0.2 5 3 12 5 0.652 0.587 16 0.2 0.2 6 4 15 8 0.460 0.516 21 0.2 0.1 7 7 8 5 0.363 0.331 17 0.3 0.2 7 6 0 12 0.409 0.386 18 0.5 0.2 3 7 5 15 0.342 0.36 22 0.3 0.1 3 9 12 8 0.292 0.274 19 0.4 0.2 4 9 8 0 0.412 0.369 23 0.5 0.1 4 3 15 12 0.557 0.509 20 0.6 0.2 5 3 12 5 0.652 0.587 21 0.2 0.1 7 7 8 5 0.363 0.331 24 0.4 0.1 5 4 0 15 0.373 0.344 22 0.3 0.1 3 9 12 8 0.292 0.274 25 0.6 0.1 6 6 5 0 0.539 0.497 23 0.5 0.1 4 3 15 12 0.557 0.509 24 0.4 0.1 5 4 0 15 0.373 0.344 25 0.6 0.1 6 6 5 0 0.539 0.497 4.2. Surrogate Model Validation The surrogate model is established by the support vector machine toolbox in Matlab 4.2. Surrogate Model Validation software. In the LS-SVM model, the kernel parameter and the penalty factor affect the The surrogate model is established by the support vector machine toolbox in Matlab sur softwar rogate e. In acc the ura LS-SVM cy. In model, current the rese kernel arch, parameter these tw and o the parpenalty ameters factor were affect determ the ined by the surrogate accuracy. In current research, these two parameters were determined by the trial-and-error method. The optimal value of the penalty factor and kernel parameter were trial-and-error method. The optimal value of the penalty factor and kernel parameter were chosen as 18 and 2.8, respectively, based on 20 groups of testing samples. The calculation chosen as 18 and 2.8, respectively, based on 20 groups of testing samples. The calculation error of LS-SVM was 4.31% (shown in Figure 4). error of LS-SVM was 4.31% (shown in Figure 4). 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 Reference value Figure 4. Validation of least square support vector machine (LS-SVM) model. Figure 4. Validation of least square support vector machine (LS-SVM) model. 4.3. Analysis of Calculation Results from LS-SVM 4.3. Analysis of Calculation Results from LS-SVM The effects of fin width and seal clearance on the discharge coefficient are shown in Figure 5. Adding seal clearance led to the decrease of the jet velocity, which lowered the The effects of fin width and seal clearance on the discharge coefficient are shown in flow resistance and turbulence induced viscous loss [3]. Therefore, with the increase of Figure 5. Adding seal clearance led to the decrease of the jet velocity, which lowered the flow resistance and turbulence induced viscous loss [3]. Therefore, with the increase of seal clearance, the seal performance deteriorated, and the discharge coefficient increased. Adding fin width resulted in the increase of jet penetration length, which mitigated the vortex loss in the downstream cavity [20,21]. Therefore, the discharge coefficient increased with the rise of fin width. Compared with the fin width, the effect of the seal clearance on discharge coefficient is predominant. Calculated value Aerospace 2021, 8, 92 7 of 11 seal clearance, the seal performance deteriorated, and the discharge coefficient increased. Adding fin width resulted in the increase of jet penetration length, which mitigated the vortex loss in the downstream cavity [20,21]. Therefore, the discharge coefficient increased with the rise of fin width. Compared with the fin width, the effect of the seal clearance on discharge coefficient is predominant. Aerospace 2021, 8, x FOR PEER REVIEW 7 of 11 (H = 5 mm, B = 6 mm, a = 7.5 , b = 7.5 ) Aerospace 2021, 8, x FOR PEER REVIEW 7 of 11 0.60 0.56 0.60 c=0.25mm c=0.4mm 0.52 0.56 c=0.55mm c=0.25mm c=0.4mm 0.52 0.48 c=0.55mm 0.48 0.44 0.44 0.40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.40 w(mm) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 w(mm) Figure 5. Influences of seal clearance and fin width on discharge coefficient. Figure 5. Influences of seal clearance and fin width on discharge coefficient. Figure 5. Influences of seal clearance and fin width on discharge coefficient. Figure 6 shows the influences of fin height and pitch on the discharge coefficient. (H = 5 mm, B = 6 mm, α = 7.5°, β = 7.5°) (H = 5 mm, B = 6 mm, α = 7.5°, β = 7.5°) The rise of fin pitch resulted in the increase of cavity area; accordingly, vortex loss in the cavity was promoted. Therefore, as fin pitch increased, the discharge coefficient decreased. Figure 6 shows the influences of fin height and pitch on the discharge coefficient. The Figure 6 shows the influences of fin height and pitch on the discharge coefficient. The Adding the fin height promoted the vortex formation and the flow stagnation [21], and rise of fin pitch resulted in the increase of cavity area; accordingly, vortex loss in the cavity thus, the discharge coefficient decreased with the rise of fin height. Compared with the fin rise of fin pitch resulted in the increase of cavity area; accordingly, vortex loss in the cavity was promoted. Therefore, as fin pitch increased, the discharge coefficient decreased. height, the effect of fin pitch on discharge coefficient was predominant. was promoted. Therefore, as fin pitch increased, the discharge coefficient decreased. Adding the fin height promoted the vortex formation and the flow stagnation [21], and Adding the fin height promoted the vortex formation and the flow stagnation [21], and thus, the discharge coefficient decreased with the rise of fin height. Compared with the (w = 0.4 mm, c = 0.4 mm, a = 7.5 , b = 7.5 ) fin height, the effect of fin pitch on discharge coefficient was predominant. thus, the discharge coefficient decreased with the rise of fin height. Compared with the fin height, the effect of fin pitch on discharge coefficient was predominant. 0.54 0.52 0.54 0.50 B=4.0mm 0.52 B=6.5mm 0.48 B=9.0mm 0.50 B=4.0mm 0.46 B=6.5mm 0.44 0.48 B=9.0mm 0.42 0.46 0.40 0.44 3.0 3.5 4.0 4.5 5.0 5.5 6.0 0.42 H(mm) Figure 6. Influences of fin height and pitch on discharge coefficient. Figure 0.406. Influences of fin height and pitch on discharge coefficient. 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Figure 7 shows the influences of fin forward and backward expansion angle on the (w = 0.4 mm, c = 0.4 mm, α = 7.5°, β = 7.5°) H(mm) discharge coefficient. As backward expansion angle increased, the cavity area decreased. Therefor Figure e, the 7 sho dischar ws th ge e coef influe ficient nces incr of fieased n forward as the and backwar backward d expansion expansiang on angle le incro eased n the Figure 6. Influences of fin height and pitch on discharge coefficient. fr dis om ch0 arge to coe 12 fficient. . However As ,ba as ckward the backwar expand siexpansion on angle increas angleeexceeded d, the cavity 12 ar , the ea d dischar ecrease ge d. coef Therefore ficient, th decr e d eased ischar with ge coe the fficient rise of incr backwar easedd as th expansion e backw angle ard exp slightly ansion . Reducing angle increase the fin d (w = 0.4 mm, c = 0.4 mm, α = 7.5°, β = 7.5°) forward expansion angle caused the decrease of jet expansion angle, which mitigated the from 0° to 12°. However, as the backward expansion angle exceeded 12°, the discharge coefficient decreased with the rise of backward expansion angle slightly. Reducing the fin Figure 7 shows the influences of fin forward and backward expansion angle on the forward expansion angle caused the decrease of jet expansion angle, which mitigated the dis vortex char loss ge an coe d fficient. flow stagn As ation baloss ckward in the exp subse an qsi uent on down angle stream increas cavity ed, [th 4]. eTh cavity ereforear , ea decreased. reducing the forward expansion angle caused the increase of discharge coefficient. Therefore, the discharge coefficient increased as the backward expansion angle increased from 0° to 12°. However, as the backward expansion angle exceeded 12°, the discharge coefficient decreased with the rise of backward expansion angle slightly. Reducing the fin forward expansion angle caused the decrease of jet expansion angle, which mitigated the vortex loss and flow stagnation loss in the subsequent downstream cavity [4]. Therefore, reducing the forward expansion angle caused the increase of discharge coefficient. d d C Aerospace 2021, 8, 92 8 of 11 vortex loss and flow stagnation loss in the subsequent downstream cavity [4]. Therefore, reducing the forward expansion angle caused the increase of discharge coefficient. Aerospace 2021, 8, x FOR PEER REVIEW 8 of 11 (w = 0.4 mm, c = 0.4 mm, H = 5 mm, B = 6 mm) 0.49 0.48 0.47 0.46 3 0.45 7.5 0.44 12 15 0.43 -2 0 2 4 6 8 10 12 14 16 () Figure 7. Influences of fin expansion angle on discharge coefficient. Figure 7. Influences of fin expansion angle on discharge coefficient. 4.4. Optimization Processes (w = 0.4 mm, c = 0.4 mm, H = 5 mm, B = 6 mm) The labyrinth seal optimization model can be expressed by: 0.2 mm c 0.6 mm 0.1 mm w 0.7 mm 4.4. Optimization Processes 3 mm H 7 mm minC (c, w, H, B, a, b)s.t. (13) The labyrinth seal optimization model can be expressed by: 3 mm B 9 mm > 0 a 15 min C (c, w, H , B, , ) 0 b 15 Set N = 4000 and N = 2000, where N and N denote the maximum global and 1 2 1 2 0.2 mm c 0.6 mm detailed search step, respectively. Two different optimization processes with different initial design values were performed. In the first optimization process, the initial values 0.1mm w 0.7 mm of c, w, H, B, a, and b were 0.4 mm, 0.7 mm, 6 mm, 7 mm, 12 , and 12 . As shown in (13) 3 mm H 7mm Figure 8a, by optimization, C decreased from 0.403 to 0.267, and the optimal design s.t. parameters were 0.202 mm, 0.103 mm, 6.950 mm, 8.991 mm, 0.007 , and 14.994 . In the 3 mm B 9mm second optimization process, the initial values of c, w, H, B, a, and b were 0.5 mm, 0.3 mm, 7 mm, 4 mm, 12 , and 0 . As shown in Figure 8 b, by optimization, C decreased from 0.509 0 15 to 0.266, and the optimal design parameters were 0.212 mm, 0.105 mm, 6.989 mm, 8.930 mm, 0.009 , and 14.989 ). This illustrates that the effects of initial values on optimization results 0 15 were notobvious. Gas-velocity distributions in the flow channel are shown in Figure 9. In the cavity, Set N1 = 4000 and N2 = 2000, where N1 and N2 denote the maximum global and detailed vortexes with different scales were formed. After optimization, the vortex effect was search step, respectively. Two different optimization processes with different initial promoted effectively, and this promoted the vortex loss and flow stagnation loss. Moreover, design by the optimization, values were the per seal forclearance med. In decr the eased, first op and timiz theati gas on leakage process was , th mitigated e initial values of c, w, effectively. The distributions of static and total pressure along the flow direction are shown H, B, α, and β were 0.4 mm, 0.7 mm, 6 mm, 7 mm, 12°, and 12°. As shown in Figure 8a, by in Figure 10. The profiles of static pressure show that, as the fluid passed through different optimization, Cd decreased from 0.403 to 0.267, and the optimal design parameters were seal elements, energy conversion took place. At each throttling location, it showed a gain 0.202 mm, 0.103 mm, 6.950 mm, 8.991 mm, 0.007°, and 14.994°. In the second optimization in kinetic energy and a reduction in static pressure. Conversely, due to the dissipation of process, the initial values of c, w, H, B, α, and β were 0.5 mm, 0.3 mm, 7 mm, 4 mm, 12°, kinetic energy, the recovery of static pressure afterwards could also be observed [1]. By the optimization, the dissipation of gas kinetic energy was enhanced and the throttling effect and 0°. As shown in Figure 8b, by optimization, Cd decreased from 0.509 to 0.266, and the was promoted. The seal performance after optimization was improved effectively. optimal design parameters were 0.212 mm, 0.105 mm, 6.989 mm, 8.930 mm, 0.009°, and 14.989°). This illustrates that the effects of initial values on optimization results were notobvious. Gas-velocity distributions in the flow channel are shown in Figure 9. In the cavity, vortexes with different scales were formed. After optimization, the vortex effect was promoted effectively, and this promoted the vortex loss and flow stagnation loss. Moreover, by the optimization, the seal clearance decreased, and the gas leakage was mitigated effectively. The distributions of static and total pressure along the flow direction are shown in Figure 10. The profiles of static pressure show that, as the fluid passed through different seal elements, energy conversion took place. At each throttling location, it showed a gain in kinetic energy and a reduction in static pressure. Conversely, due to the dissipation of kinetic energy, the recovery of static pressure afterwards could also be observed [1]. By the optimization, the dissipation of gas kinetic energy was enhanced and the throttling effect was promoted. The seal performance after optimization was improved effectively. Aer Aerosp ospac ace e 20 2021 21, , 8 8, , x FO x FOR P R PEE EER R RE REVIEW VIEW 9 9 of of 11 11 Aerospace 2021, 8, 92 9 of 11 0.65 0.65 0.40 0.40 0.60 0.60 0.55 0.55 Global Global Detailed Detailed 0.50 0.50 0.35 0.35 search search search search 0.45 0.45 Global Global search search 0.40 0.40 0.30 0.30 Deailed Deailed 0.35 0.35 search search 0.30 0.30 0.25 0.25 0.25 0.25 0 0 1 1 9 9 13 13 19 19 102 102 966 966 1428 14284043 40436000 6000 0 0 4 4 12 12 19 19 93 93 576 576 3476 3476 4448 4448 6000 6000 Iteration step Iteration step Iteratio Iteration step n step ((b b) ) cc 0 0 = = 0 0..5 5 m mm m,, w w 0 0 = = 0 0..3 3 m mm m,, H H 0 0 = = 7 7 m mm m,, B B 0 0 = = 4 4 m mm m,, α α 0 0 = = 1 12° 2° ,, β β 0 0 = = ((a a) ) cc 0 0 = = 0 0..4 4 m mm m,, w w 0 0 = = 0 0..7 7 m mm m,, H H 0 0 = = 6 6 m mm m,, B B 0 0 = = 7 7 m mm m,, α α 0 0 = = 1 12 2° ° , , β β 0 0 = = 1 12 2° ° (Reference g (Reference geomet eometry ry I) I) 0 0° ° (Reference (Reference geom geometry I etry II) I) Figure Figure 8. 8. Var Variati iation of on of C C d d with with it iteration steps eration steps.. Figure 8. Variation of C with iteration steps. ((a a)) c c = = 0 0..2 2 m mm m,, w w = = 0 0..1 1 m mm m,, H H = = 7 7 m mm m,, B B = = 9 9 m mm m,, α α = = ((b b)) c c = = 0 0..4 4 m mm m,, w w = = 0 0..7 7 m mm m,, H H = = 6 6 m mm m,, B B = = 7 7 m mm m,, α α = = 0 0° ° , , β β = = 1 15° 5° (Opt (Optiim mal al geom geometry) etry) 1 12 2° ° , , β β = = 1 12 2° ° (Re (Reference ference geom geomet etry ry I) I) ((cc)) c c = = 0 0..5 5 m mm m,, w w = = 0 0..3 3 m mm m,, H H = = 7 7 m mm m,, B B = = 4 4 m mm m,, α α = = 1 12 2° ° , , β β = = 0 0° °(Reference (Reference geom geometry I etry II) I) Figure Figure Figure 9. 9. 9. Dis Distribution Distri tribution o bution o of f f gas gas gas velocity v vel eloc ocit ity y in in in the the the second ssec econd seal ond seal seal cavity cavi cavity (U ty (U (Unit: nit: m/s) nit: m/s) m/s). .. Fin center Fin center Fin center Fin center 20,000 20,000 Static pressur Static pressur ee Static Static pressure pressure 20,000 20,000 Total pressure Total pressure Total pressure Total pressure 15,000 15,000 15,000 15,000 10,000 10,000 10,000 10,000 5,000 5,000 5,000 5,000 00 11 22 33 44 55 00 11 22 33 44 55 xx /B /B xx // B B Pressure(Pa) Pressure(Pa) Pressure (Pa) Pressure (Pa) Cd Aerospace 2021, 8, x FOR PEER REVIEW 9 of 11 0.65 0.40 0.60 0.55 Global Detailed 0.50 0.35 search search 0.45 Global search 0.40 0.30 Deailed 0.35 search 0.30 0.25 0.25 0 1 9 13 19 102 966 1428 4043 6000 0 4 12 19 93 576 3476 4448 6000 Iteration step Iteration step (b) c0 = 0.5 mm, w0 = 0.3 mm, H0 = 7 mm, B0 = 4 mm, α0 = 12°, β0 = (a) c0 = 0.4 mm, w0 = 0.7 mm, H0 = 6 mm, B0 = 7 mm, α0 = 12°, β0 = 12°(Reference geometry I) 0°(Reference geometry II) Figure 8. Variation of Cd with iteration steps. (a) c = 0.2 mm, w = 0.1 mm, H = 7 mm, B = 9 mm, α = (b) c = 0.4 mm, w = 0.7 mm, H = 6 mm, B = 7 mm, α = 0°, β = 15°(Optimal geometry) 12°, β = 12° (Reference geometry I) Aerospace 2021, 8, 92 10 of 11 (c) c = 0.5 mm, w = 0.3 mm, H = 7 mm, B = 4 mm, α = 12°, β = 0° (Reference geometry II) Figure 9. Distribution of gas velocity in the second seal cavity (Unit: m/s). Fin center Fin center 20,000 Static pressure Static pressure 20,000 Total pressure Total pressure 15,000 15,000 10,000 10,000 5,000 5,000 01234 5 Aerospace 2021, 8, x FOR PEER REVIEW 10 of 11 x/B x/B (a) c = 0.2 mm, w = 0.1 mm, H = 7 mm, B = 9 mm, α = (b) c = 0.4 mm, w = 0.7 mm, H = 6 mm, B = 7 mm, α = 0°, β = 15°(Optimal geometry) 12°, β = 12° (Reference geometry I) Fin cente r Static pressure 20,000 Total pressure 15,000 10,000 5,000 0 123 45 x/B (c)c = 0.5 mm, w = 0.3 mm, H = 7 mm, B = 4 mm, α = 12°, β = 0° (Reference geometry II) Figure 10. Distribution of pressure profile on the flow direction. Figure 10. Distribution of pressure profile on the flow direction. 5. Conclus 5. iConclusions ons A kind of optimization method was developed for straight labyrinth seals. The A kind of optimization method was developed for straight labyrinth seals. The discharge coefficient was considered for objective function, which is to be minimized. For discharge coefficient was considered for objective function, which is to be minimized. For the six design variables, namely, seal clearance, fin width, fin height, fin pitch, and fin the six design variables, namely, seal clearance, fin width, fin height, fin pitch, and fin backward and forward expansion angle, 72 groups of training samples and 20 groups of backward and forward expansion angle, 72 groups of training samples and 20 groups of testing samples were generated by solving a CFD model. testing samples were generated by solving a CFD model. The surrogate model was established by LS-SVM. The prediction of LS-SVM for dis- The surrogate model was established by LS-SVM. The prediction of LS-SVM for charge coefficient agreed well with the experimental data. By analyzing LS-SVM output discharge coefficient agreed well with the experimental data. By analyzing LS-SVM results, some meaningful conclusions are summarized: adding fin width and seal clear- output results, some meaningful conclusions are summarized: adding fin width and seal ance both lead to the increase of discharge coefficient; the rises of fin forward expansion clearance both lead to the increase of discharge coefficient; the rises of fin forward angle, fin pitch, and height all result in the decrease of discharge coefficient; as the back- expansion angle, fin pitch, and height all result in the decrease of discharge coefficient; as ward expansion angle of fin increases, the discharge coefficient increases firstly, and then the backward expansion angle of fin increases, the discharge coefficient increases firstly, decreases slightly. and then decreases slightly. A chaotic optimization algorithm was used for global searches. By optimization, the A chaotic optimization algorithm was used for global searches. By optimization, the sealing performance was improved effectively. The optimal shape of labyrinth seals has sealing performance was improved effectively. The optimal shape of labyrinth seals has a a seal clearance of 0.2 mm, fin width of 0.1 mm, fin height of 7 mm, fin pitch of 9 mm, seal clearance of 0.2 mm, fin width of 0.1 mm, fin height of 7 mm, fin pitch of 9 mm, fin fin backward expansion angle of 0 , and fin forward expansion angle of 15 . It could be backward expansion angle of 0°, and fin forward expansion angle of 15°. It could be found found that a chaotic optimization algorithm coupled with LS-SVM suggorate model is an that a chaotic optimization algorithm coupled with LS-SVM suggorate model is an effective tool for labyrinth-seal optimization. effective tool for labyrinth-seal optimization. In the present study, only the discharge coefficient was considered an optimization In the present study, only the discharge coefficient was considered an optimization goal. In fact, windage temperature rising performance is also a very important index. goal. In fact, windage temperature rising performance is also a very important index. THerefore, in the future, muti-goal optimization should be carried out, especially with the THerefore, in the future, muti-goal optimization should be carried out, especially with the condition of small fin number. condition of small fin number. Author Contributions: Conceptualization, C.W. and Y.Z.; methodology, C.W. and Y.Z.; writing— original draft preparation, Y.Z.; writing—review and editing, C.W.; funding acquisition, C.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Fundamental Research Funds for the Central Univer- sities (grant No: NS2020013) and National Science and Technology Major Project of China (grant No: 2017-III-0011-0037). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data available on request due to restrictions eg privacy or ethical. Conflicts of Interest: The authors declare no conflict of interest. Pressure(Pa) Pressure (Pa) Pressure (Pa) d Aerospace 2021, 8, 92 11 of 11 Author Contributions: Conceptualization, C.W. and Y.Z.; methodology, C.W. and Y.Z.; writing— original draft preparation, Y.Z.; writing—review and editing, C.W.; funding acquisition, C.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Fundamental Research Funds for the Central Universi- ties (grant No: NS2020013) and National Science and Technology Major Project of China (grant No: 2017-III-0011-0037). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data available on request due to restrictions e.g., privacy or ethical. Conflicts of Interest: The authors declare no conflict of interest. References 1. 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