Alexander Fuchs
email: alexander.fuchs@ltf.mw.tum.de

Oskar J Haidn

S Y Mpo

O Tatin

Effects of uncertainty and quasi-chaotic geometry on the leakage of brush seals

Keywords:

is paper presents a brief review of the experimental and theoretical state of the art regarding the leakage flow prediction of brush seals. e authors model a computational fluid dynamics based approach for the leakage flow of brush seals. e brush seal is treated by its real geometrical structure, namely numerous bristles in an array in transverse flow. For the inner composition of the package, two differing textures are chosen to study their influence on the leakage behavior. Furthermore, for each texture multiple inter-bristle distances, pressure ratios and turbulence models are evaluated. In addition, the influence of not regular arrangement configurations, which form a quasi-chaotic inner structure, are studied. e gained results are compared to other authors' experimental and numerical data.

INTRODUCTION

To accomplish the goal of improving the overall efficiency of thermal turbomachinery, the development and improvement of the secondary flow system of this machinery is a crucial task. Using advanced sealing technologies for this demanding task, has been shown to be very beneficial [ ]. Brush seals are one of these advanced seals.

To lay the foundation for a deeper understanding of brush seals, a bo om-up strategy gaining an empirically validated numerical fluid-structure interaction model is required. e first cornerstone of this strategy -generating a sub-system flow model of the bristle package -is to be accomplished by creating a parameterized computational fluid dynamic model and analyzing the flow through the bristle package without its movement. Validating this model, results from different sources are used and briefly discussed. In the cross sectional view (figure (a)) the main components of this sealing type can be seen, containing sealing element (consisting of the bristle pack, central wire and c-tube) and housing. To reduce and protect the sealing element from flow distortions of the upstream flow, the outer part of the housing -the cover ring -is used. Downstream of the bristle pack the support ring is installed to support the sealing element from bending due to the pressure differential ∆p across the sealing. Both rings combined act as housing of the brush seal and clamp the sealing element inside them. e main part of this sealing is the sealing element between the both rings. It consists of a vast number of individual bristles and a clamping device (figure (b)).

System description

e main difference between the shown and other manufacture designs is the fixation of the bristles. While these are commonly welded together, the MTU Aero Engines design uses a cut open tube, called c-tube due to its shape, and a wire where the bristles are laying around of. e c-tube is clamped over the upper part of the sealing element, thus locks the bristles from moving. Due to the support rings' geometrical shape, the radial outer part of the support ring does not touch the bristle pack.

erefore, in the arising cavity, the pressure relief chamber, the pressure p adapts to the upstream pressure p us .

is effect is intended to generate a counter acting force on the bristle pack which adapts automatically to the pressure differential ∆p and therefore reduces the bristles deformation.

Because of the great number of individual wires, which deform reversibly under the a ack of external forces, this sealing type is challenging to build, apply, and even more to model due to the interaction of the flow and and the bristle package.

Experimental work from other authors

Several studies regarding the leakage flow behaviour of brush seals have been carried out experimentally. Bayley

Theoretical work from other authors

Multiple approaches have been applied to determine the leakage mass flow across a brush seal, differing in the treatment of the bristle pack and the way the computational work is performed. Braun et al. [ , ] and Hendricks et al. [ , ] used analytical bulk flow models which treat the bristle pack as a porous medium. ese models are related to experimental results to compute the leakage mass flow. Bulk flow models have the main disadvantage to be not capable of predicting the pressure distribution inside the bristle pack but are fairly accurate in close range with respect to the experimental basis. With increasing computational capacity the numerical investigation of brush seals has become a promising approach. Chew et al. [ , ] have implemented a porous media approach based on an anisotropic Darcian medium in a two dimensional axis-symmetric CFD model. To solve the porous media region properly, source terms in the momentum equations are introduced. is methodology was used and refined by many authors. Pröstler [ , ]

Objectives of this paper

In this paper a CFD based model is presented which is based on modeling an array of bristles in transverse flow. e influence of the inter-bristle distance δ, pressure ratio Π, multiple turbulence models and inner bristle composition are evaluated and tested against other authors' results. In addition, the effects of quasi-chaotic arrangement within the bristle pack are investigated.

. METHODOLOGY . Geometrical structure and modeling e geometrical structure is gained by a MATLAB ® tool which has been developed for this purpose. is tool allows to configure the number of axial (N a ) and quasi-circumferential (N c ) bristle rows, the diameter d of the bristles and their distance δ among each other. Furthermore, the shape of the circumscribing area of a bristle can be defined. Due to geometrical limitations (exclusively triangles, squares and hexagons are space-filling in D euclidean space) only squares and hexagons are implemented. is leads to two differing inner bristle pack compositions (see figure ). One of the most notable distinctions is that the hexagonal shaped structure, besides its representation of the most compact packing possible, allows back tapering of the bristles, while the square shaped inner structure does not. Also the hexagonal shaped circumscription is the usually assumed inner composition of a brush seal [ ], the square shaped surrounding in contrast leads to significantly simpler numerical meshes. erefore, a comparison between both is carried out.

In addition, a probability density function P i regarding the axial and circumferential position of the center point of each individual bristle can be applied within their circumscribing area. In this study a normal distribution is used (equation ) as probability density function P i . As expected value µ the center point position of the regular arrangement is set.

P i = 1 √ 2πσ 2 e -(x i -µ) 2 2σ 2 ( )
Depending on the variance σ 2 the geometrical structure can be altered from a regular arrangement with σ 2 = 0 (figure (a) and (c)) to a quasi-chaotic arrangement with σ 2 > 0 (figure (b) and (d)). e resulting displacement is limited so, that all bristles stay within their circumscribing area.

erefore, a limiting curve can be defined which must not be passed by the center points. is curve is a smaller concentric circle related to the in-circle of the considered surrounding structure. Due to numerical limitations, the maximal displacement is set to be slightly smaller than the mentioned limiting curve. Figure represents an exemplary distribution of the center points of individual bristles which leads to a quasi-chaotic arrangement. Furthermore, the first and last row in circumferential direction c are set to feature the same deviation vector to ensure that the periodic boundary conditions set later on perform well. By enlarging the variance σ 2 , the quasi-chaotic behaviour of the inner composition of the bristle pack increases. If variances σ 2 are chosen which tend to generate most of the center points beyond the limiting curve, the inner composition of a bristle pack reaches a frozen state regarding the average displacement.

Furthermore, the tool allows to vary the inlet and outlet path length easily be varying input parameters. In table geometry andmodel of the tool is a JavaScript ™ with ANSYS ® DesignModler ™ commands.

is JS is processed within DM to gain a D computer aided design model.

. Meshing and grid convergence study

Using ANSYS ® Workbench ™ the D CAD model is interconnected with ANSYS ® ICEM CFD ™ where, based on user generated templates, a structured multi-block mesh is generated automatically for both surrounding types (see figure

). All grids considered in this study represent quasi two dimensional domains, with only one cell in radial direction of the brush seal. Furthermore, all generated meshes satisfy Figure Due to the Q D domain setup, boundary conditions are applied on all boundary faces. e inlet face is considered as a pressure inlet where the inlet total pressure p tot is set to multiple pressure levels within this study. For the outlet face a pressure outlet boundary condition with a constant static pressure of p st at = 1.0bar is set. For both faces in circumferential domain direction transitional matching periodic boundary conditions, in radial dimension symmetry boundary conditions are used. erefore, the domain se ing represents a section of a linear brush seal element underneath the support ring. Furthermore, as fluid dry air is used and modeled as an ideal gas.

. RESULTS

In this study multiple parameters are varied and their influence on the leakage mass flow rate m is investigated. Valida-tion of the results is a demanding task, since only limited data is available for this purpose. Experimental data regarding the flow characteristics typically only provide the upstream and downstream pressure as well as the resulting leakage mass flow rate m. Usually no statements are made relating to the geometrical inner composition, movement and resulting displacement, or the pressure and velocity field within the bristle pack. eoretical data from other authors' are based on and calibrated with these experimental results. erefore, for a validation purpose only the resulting leakage mass flow rate m for a given pressure ratio Π can be used.

An initial study investigated the influence of the considered number of bristle rows N c in circumferential direction c. No impact could be observed by variation of 1 < N c < 4. For all further simulations N c = 4 was chosen to lay the foundation for the investigation of quasi-chaotic displacement, where a minimum number of rows in circumferential direction c is essential to provide a sufficiently large domain.

. Influence of inter-bristle distance δ and shape of circumscribing area e inter-bristle distance δ is defined as distance between two bristles in circumferential direction c. Depending on the considered surrounding type, the distances between bristles in other directions than circumferential in the a-c-plane is controlled by the geometrical composition. Due to the fact that the main pressure gradient acts in axial direction, the inter-bristle distance δ defines, for both considered circumscribing shapes, the narrowest gap transverse to the flow direction. erefore, the inter-bristle distance δ influences the resulting leakage mass flow rate m intensely (see figure

). e deviation of the leakage mass flow rate m related to the resulting leakage mass flow rate m of a inter-bristle distance of δ = 8 × 10 -6 reach from % up to %. Further-more, figure shows the influence of the bristle surrounding shape. Square shaped surroundings result in a higher leakage mass flow rate m for low pressure ratios Π. With increasing pressure ratios Π the simulated leakage mass flow rates for both surrounding types converge to one another, until they intersect. A er this intersection, the simulations with hexagonal shaped circumscribing area show higher leakage mass flow rates m. In addition, the point of intersection of both leakage mass flow rate trends shi s to lower pressure ratios Π with increasing inter-bristle distance δ. the literature are only available in a range of 1 < Π < 4, only this detail is shown here and considered for validation. Generally, the presented results correlate qualitatively well with data from the literature.

e leakage mass flow rate m shows the supposed linear dependency relating to the pressure ratio Π for both investigated bristle surrounding types and the same order of magnitude. For a quantitative comparison of the results it has to be noted, that only the experimental data from MTU Aero Engines features the same bristle diameter of d = 1.5 × 10 -4 m like in the performed numerical simulations of this study. e results of Bayley and Long [ ] Generally, all three models show very similar results. e k-ǫ RNG model performs be er for square shaped bristle surroundings than for hexagonal ones, even be er than the k-ω SST model for the square shaped circumscription. For both bristle surrounding types the Transition-SST model correlates best with the experimental data. In case of the hexagonal surrounding the k-ω SST model leads to almost identical results than the Transition-SST model. Table

shows the relative errors e rel of all considered turbulence models related to the experimental data from MTU Aero Engines. Based on this results for all further simulations the Transition-SST model is chosen.

. Influence of quasi-chaotic displacement

For the investigation on the influence of quasi-chaotic displacement of each individual bristle on the leakage mass flow rate m the method described in section . is used. erefore, in both surrounding types a probability density function P i (see equation ) is applied and the variance σ 2 is varied. In figure the results of this analysis are shown. e square shaped circumscribing area as well as the hexagonal shaped one show the same trend towards a decline in the leakage mass flow rate m with increasing variance σ 2 relating to the results for regular arrangement with σ 2 = 0. Multiple simulations with the same variance σ 2 but differing in the resulting geometrical displacement, due to the quasi-chaotic methodology, were performed. is analysis indicates that for equally chosen variance values the resulting leakage mass flow rate m stays almost identical. erefore, it can be concluded that only the order of deviation due to the variance σ 2 plays a role. e resulting deviated geometrical composition only acts on a minor scale. Nevertheless the decrease of the leakage mass flow rate m is verifiable, the quasi-chaotic displacement only has a minor impact on the overall results. As can be seen in table , the deviation of the leakage mass flow rate m lies within the error margin of the simulation (see table ).

Although the influence on the leakage mass flow rate m of a quasi-chaotic displacement of the inner bristle composition Rivering describes a fluid flow within a brush seal where, caused due to the non regular arrangement of the bristle pack, parts of the brush seal are partially or completely blocked for fluid flow due to contacting bristles. erefore, in other sectors of the bristle pack enlarged flow channels emerge where a significantly larger mass flow rate passes by; the resulting flow pa ern looks like a river winding through the bristle pack. Figure shows an exemplary contour plot where this phenomena clearly can be observed. In addition the effects of je ing, partial accelerated fluid flow in regions of arising narrow gaps, also can be seen in figure . Lateral flow pa erns primarily appear in simulations with square shaped bristle surroundings, wherein hexagonal ones this flow phenomena is prevented by the back tapering of the bristles.

.

CONCLUSION

In this study a CFD based leakage mass flow model for brush seals has been developed. Validating this model has shown that it is in accordance with experimental and numerical data from multiple sources. e influence of different geometrical and modeling parameters has been investigated, identifying the inter-bristle distance δ as the main leakage mass flow rate m affecting factor. Its representation of the narrowest gap transverse to the main flow path causes that 10% variation of this parameter results in a deviation of the leakage mass flow rate m from 15% up to 30%.

Studying two differing arrangement setups, square and hexagonal shaped bristle surroundings, shows only a minor effect on the leakage mass flow rate m. Nevertheless, the arrangement influences the observed flow pa erns intensively. Furthermore, square shaped bristle surroundings perform be er at low, hexagonal shaped circumscriptions be er at high pressure ratios Π. For further work regarding the modeling of brush seals a hexagonal setup has been chosen, as that this setup represents the experimentally confirmed and more realistic composition.

To clarify the influence of turbulence modeling, the k-ǫ RNG model, the k-ω SST model with low-Reynolds correction, and the Transition-SST model have been tested against each other and experimental results. is analysis has proven the Transition-SST model to agree best with experimental results.

To consider realistic quasi-chaotic inner composition of a brush seal, a probability density function P i within the bristle surrounding shapes was implemented. Compared to a regular arrangement only minor deviations in the leakage mass flow rate m could be observed using this method. Due to the highly increased modeling effort required for this setup, its consideration in designing and evaluation of brush seals is appropriate when a in-depth analysis is aimed. Furthermore, with this approach experimentally observed flow phenomena like rivering, je ing, and lateral flow can be reproduced.

Figure

  Figure shows a brush seal in MTU Aero Engines design [ ]. In the cross sectional view (figure (a)) the main components of this sealing type can be seen, containing sealing element (consisting of the bristle pack, central wire and c-tube) and housing. To reduce and protect the sealing element from flow distortions of the upstream flow, the outer part of the housing -the cover ring -is used. Downstream of the bristle pack the support ring is installed to support the sealing element from bending due to the pressure differential ∆p across the sealing. Both rings combined act as housing of the brush seal and clamp the sealing element inside them. e main part of this sealing is the sealing element between the both rings. It consists of a vast number of individual bristles and a clamping device (figure (b)).e main difference between the shown and other manufacture designs is the fixation of the bristles. While these are commonly welded together, the MTU Aero Engines design uses a cut open tube, called c-tube due to its shape, and a wire where the bristles are laying around of. e c-tube is clamped over the upper part of the sealing element, thus locks the bristles from moving. Due to the support rings' geometrical shape, the radial outer part of the support ring does not touch the bristle pack.erefore, in the arising cavity, the pressure relief chamber, the pressure p adapts to the upstream pressure p us .is effect is intended to generate a counter acting force on the bristle pack which adapts automatically to the pressure differential ∆p and therefore reduces the bristles deformation.

  and Long [ ] have measured the leakage on a static (no sha rotation) (a) Cross section of a brush seal. (b) Axial view of bristle pack without housing.

Figure .

 . Figure . Brush seal in MTU Aero Engines design [ ].

  (a) Square shaped bristle surrounding. (b) Hexagonal shaped bristle surrounding.

Figure .

 . Figure .Exemplary regular bristle pack arrangements in a a-c-cross-section.

Figure .

 . Figure . Exemplary inner bristle pack compositions without and including displacement.

  Figure .Exemplary center point distribution and limiting curve of a quasi-chaotic arrangement.

  (a) Square shaped circumscription. (b) Hexagonal shaped circumscription.

Figure .

 . Figure . Influence of inter-bristle distance δ and shape of circumscribing area. For all results shown in this figure the Transition-SST model [ ] for turbulence modeling and N c = 4 are used.

Figure

  Figure shows the numerical results of this study correlated against the results of Bayley and Long [ ], Chew et al. [ ], Flouros et al. [ , ], and experimental data from MTU Aero Engines. Due to the fact that the results from

Figure .

 . Figure . Correlation of the numerical results of this study against experimental and theoretical findings from other authors. For all results shown in this figure the Transition-SST model [ ] for turbulence modeling and N c = 4 are used.

  are based on measurements of a brush seal with a bristle diameter of d = 7.622 × 10 -5 m. Chew et al. [ ] and Flouros et al. [ , ] calibrated their porous media approach based simulations on the experimental findings of Bayley and Long [ ]. Considering this fact, the numerical results for both investigated surrounding types with a inter-bristle distance of δ = 8 × 10 -6 m show the best agreement with the reference data from MTU Aero Engines. erefore, all further mentioned results use an inter-bristle distance of δ = 8 × 10 -6 m as their basis. . Influence of turbulence modeling Initial simulations showed a significant turbulent intensity I within the bristle pack. erefore, the influence of turbulence modeling on the flow field and especially on the resulting leakage mass flow rate m of the simulated bristle pack has been investigated in this study. ree widely used turbulence models are tested against each other. e first two turbulence models are two equation models; the k-ǫ RNG model [ ] and the k-ω SST model with low-Reynolds correction [ ]. e third turbulence model is the four equation Transition-SST model [ ]. Figure shows the results for both bristle surrounding types and all considered turbulence models.

  (a) Square shaped bristle surrounding. (b) Hexagonal shaped bristle surrounding.

Figure .

 . Figure . Comparison of different turbulent models. For all results shown in this figure a inter-bristle distance of δ = 8 × 10 -6 m and N c = 4 are used.

  (a) Square shaped bristle surrounding. (b) Hexagonal shaped bristle surrounding.

Figure .

 . Figure . Influence of quasi-chaotic displacement. For all results shown in this figure the Transition-SST model [ ] for turbulence modeling, δ = 8 × 10 -6 m and N c = 4 are used.

Figure .

 . Figure . Exemplary contour plots of fluid flow domain for an quasi-chaotic arrangement with hexagonal bristle surrounding and a variance of σ 2 = 0.2 × 10 -12 , Transition-SST model [ ] for turbulence modeling, an inter-bristle distance of δ = 8 × 10 -6 m, a pressure ratio of Π = 8.90, and N c = 4.

[

  ] Y. Dogu, M. F. Aksit, M. Demiroglu, and O. S. Dinc. Evaluation of flow behavior for clearance brush seals. Journal of Engineering for Gas Turbines and Power, ( ): , .

  [ ] M. J. Braun and V. V. Kudriavtsev. A numerical simulation of a brush seal section and some experimental results. Journal of Turbomachinery, ( ): , . [ ] V. V. Kudriavtsev and M. J. Braun. Model developments for the brush seal numerical simulation. Journal of Propulsion and Power, ( ): -, January . [ ] R. C. Hendricks, V. V. Kudriavtsev, M. J. Braun, and M. M. Athavale. Flows in pinned arrays simulating brush seals. NASA Technical Memorandum, . [ ] M. C. Sharatchandra and D. L. Rhode. Computed effects of rotorinduced swirl on brush seal performance-Part I: Leakage analysis. Journal of Tribology, ( ): , . [ ] M. C. Sharatchandra and D. L. Rhode. Computed effects of rotorinduced swirl on brush seal performance-Part II: Bristle force analysis. Journal of Tribology, ( ): , . [ ] D. Lelli, J. W. Chew, and P. Cooper. Combined d fluid dynamics and mechanical modelling of brush seals. In Volume : Turbo Expo , Parts A and B. ASME, . [ ] D. Lelli, J. W. Chew, and P. Cooper. Combined three-dimensional fluid dynamics and mechanical modeling of brush seals. Journal of Turbomachinery, ( ): , . [ ] M. J. Pekris, G. Franceschini, and D. R. H. Gillespie. Effect of geometric changes in an idealised contacting brush seal bristle pack on typical key performance measures. In Volume : Heat Transfer, Parts A and B. ASME, . [ ] M. J. Pekris, G. Franceschini, and D. R. H. Gillespie. An investigation of flow, mechanical and thermal performance of conventional and pressure-balanced brush seals. In Volume : Heat Transfer, Parts A and B. ASME, June . [ ] M. J. Pekris, G. Franceschini, and D. R. H. Gillespie. An investigation of flow, mechanical, and thermal performance of conventional and pressure-balanced brush seals. Journal of Engineering for Gas Turbines and Power, ( ): , January . [ ] D. Sun, N.-N. Liu, C.-W. Fei, G.-Y. Hu, Y.-T. Ai, and Y.-S. Choy. eoretical and numerical investigation on the leakage characteristics of brush seals based on fluid-structure interaction. Aerospace Science and Technology, : -, November .

[

  ] J. W. Chew and C. Guardino. Simulation of flow and heat transfer in the tip region of a brush seal. International Journal of Heat and Fluid Flow, ( ): -, August . [ ] S. Huang, S. Suo, Y. Li, and Y. Wang. eoretical and experimental investigation on tip forces and temperature distributions of the brush seal coupled aerodynamic force. Journal of Engineering for Gas Turbines and Power, . B. Celik, U. Ghia, P. J. Roache, C. J. Freitas, H. Coleman, and P. E. Raad. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. Journal of Fluids Engineering, ( ): , . [ ] F. R. Menter, R. B. Langtry, S. R. Likki, Y. B. Suzen, P. G. Huang, and S. Völker. A correlation-based transition model using local variables-Part I: Model formulation. Journal of Turbomachinery, ( ): , . [ ] V. Yakhot and S. A. Orszag. Renormalization group analysis of turbulence. I. Basic theory. Journal of Scientific Computing, ( ): -, . [ ] F. R. Menter. Zonal two equation k-w turbulence models for aerodynamic flows. In rd Fluid Dynamics, Plasmadynamics, and Lasers Conference. American Institute of Aeronautics and Astronautics, July .

  To extend the valid range, Chupp et al. [ ] generalized the results over a wider range of conditions implementing the effective thickness B ef f of a brush seal introduced in a pre-vious work of Dowler et al. [ ].To overcome the drawback of missing the pressure distribution within the bristle pack, porous media models with a differing theoretical basis were developed. Bayley and Long [ ] used an analytical porous media approach where they solved Laplace's equations. Chew et al. [ ] states, this approach is equivalent of treating the bristle pack as an anisotropic Darcian porous medium.

  has considered a three dimensional CFD model, Dogu et al. [ -] has researched the influence of geometry variations, Chen et al. [ , ] and Flouros et al. [ , ] have used the resulting pressure distribution to predict bristle pack deformation, Qiu et al. [ , ] and Li et al. [ -] studied the heat generation due to frictional forces on the rotor, and Helm et al. [ ] has simulated a test bench including labyrinth and brush seals.

	Beside porous media models, approaches have been ap-
	plied which treat the bristle pack as an array of cylinders in
	transverse flow. Braun et al. [ , ] discretize the cylinders in
	a rather rough manner, being refined later on by Kudriavtsev
	and Braun [ ]. Hendricks et al. [ ] and Sharatchandra and
	Rhode [ , ] have computed only a partial region between
	the bristles and have applied periodic boundary conditions
	to get results of the entire bristle pack. Lelli et al. [ , ],
	Pekris et al. [ -], and Sun et al. [ ] have coupled their
	CFD results to a computational structural mechanics tool
	to distinguish the deformation of the bristle pack. Further-
	more, Chew and Guardino [ ] and Huang et al. [ ] have
	investigated the frictional heat due to friction with the rotors'
	surface.

Table .

 . Geometry and model parameters

	Parameter	Symbol	Value	Unit
	Bristle diameter Inter bristle distance Number of axial rows Number of circumferential rows Variance Upstream total pressure	d δ N a N c σ 2 p us	1.5 × 10 -4 7 -9 × 10 -5 8 1 -4 1 -2 × 10 -12 1.5 -9	m m ---bar
	Downstream static pressure	p ds	1	bar

  . Details of exemplary numerical meshes for both considered arrangements. the numerical requirement of y + ≤ 1 to resolve the viscous sublayer on the bristle surfaces. A grid convergence study is carried out applying the grid convergence index [ ]. As the variable of interest φ the leakage mass flow rate m of the bristle pack is defined. For both bristle surrounding types a Results of the grid convergence study with the grid convergence index [ ]. For all results presented in this table the Transition-SST model [ ] for turbulence modeling, a pressure ratio Π = 8.90, and N c = 4 are used.

	Square shaped bristle surrounding
		Coarse	Middle		Fine
	71680 7.03 4.55 Hexagonal shaped bristle surrounding 99840 6.02 h [10 -6 m] Cells [-] 1.17 1.11 128000 5.41 r [-] φ = m [10 -7 kg 4.58 4.58 s ] 30.5 p [-] φ ext [10 -7 kg 4.78 4.57 s ] 0.74 0.03 e a [%] 4.23 0.25 e ext [%] 5.30 0.29 GCI [%]
		Coarse	Middle		Fine
	Cells [-] h [10 -6 m] r [-] φ = m [10 -7 kg s ] p [-] φ ext [10 -7 kg s ] e a [%] e ext [%] GCI [%]	41800 6.78 4.78 13.93 1.15 4.12 1.81 14.67	60800 5.91 4.69 14.1	1.12 4.84 0.35 2.73 3.36	85724 5.26 4.71
	. Solving the governing equations and do-
	main se ings

geometrical setup without displacement and a pressure ratio of Π = 8.90 is used. As table clearly states for both meshes the finest refinement leads to an acceptable GCI. Based on this results for both surrounding types the finest mesh with 128000 cells and 85724 cells respectively are used for further investigations.

Table .

For the purpose of solving the governing flow field equations the commercial CFD solver ANSYS ® Fluent ® is used. Turbulence is modeled by Reynolds averaged Navier-Stokes equations, using two (k-ǫ RNG [ ] and k-ω SST with low-Reynolds correction [ ]) and four equation (Transition-SST [ ]) turbulence models.

Table .

 . Relative error e rel of leakage mass flow rate m of turbulence models relating to experimental data from MTU Aero Engines.

	Square shaped bristle surrounding
	Turbulence model	e rel,min [%] e rel,max [%] e rel [%]
	k-ǫ RNG	0.04	7.55	3.18
	k-ω SST low-Re	0.98	7.66	3.73
	Transition-SST	0.96	7.66	2.21
	Hexagonal shaped bristle surrounding
	Turbulence model	e rel,min [%] e rel,max [%] e rel [%]
	k-ǫ RNG	0.11	23.67	5.73
	k-ω SST low-Re	0.31	23.03	4.85
	Transition-SST	0.53	23.03	4.79

Table .

 . Relative error e rel of leakage mass flow rate m of variance σ 2 > 0 relating to regular arrangment with σ 2 = 0. Square shaped bristle surrounding σ 2 [×10 -12 ] e rel,min [%] e rel,max [%] e rel [%] [×10 -12 ] e rel,min [%] e rel,max [%] e rel [%]

	1	0.05	0.52	0.27
		Hexagonal shaped bristle surrounding
	σ 2 1	0.12	2.95	0.89
	2	1.92	8.37	3.76
	is almost negligible, with this method phenomena observed
	in experiments can be reproduced numerically. As Braun et
	al. [ -] have stated based on their experimental work, the
	flow within brush seals shows special flow pa erns. Most
	notable are the rivering, je ing and lateral flow pa erns.

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ACKNOWLEDGMENTS

is study was performed in cooperation with MTU Aero Engines. e support for this work by MTU Aero Engines and the permission to publish this paper and the containing data are highly acknowledged.


Laura Branchetti
email: laura.branchetti@unipr.it

Olivia Levrini

Eleonora Barelli

Michael Lodi

Giovanni Ravaioli

Laura Rigotti

Sara Satanassi

Giulia Tasquier

STEM analysis of a module on Artificial Intelligence for high school students designed within the I SEE Erasmus+ Project

Keywords: Artificial intelligence, big ideas, STEM education

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The I SEE ERASMUS+ Project

The EU ERASMUS+ project I SEE (inclusive STEM Education to enhance the capacity to aspire and imagine future careers) stems from the awareness that one of the main challenges for mankind today, above all for the young, is to recover a good relationship with time, merely with the future. We live in a multi-speed society in which the capacity to predict sudden changes are decreasing and this causes a compulsive search for opportunities. The educational system is not able to keep up with social acceleration. This project bet on STEM education as a breeding ground for preparing young people for uncertainty and making them developing future scaffolding skills, i.e. skills that enhance their capacity to aspire, envisage themselves as agents of change, and push their imagination towards future careers in STEM [START_REF] Branchetti | The I SEE project: An approach to futurize STEM education[END_REF]. STEM education can give two main contributions in this direction, one epistemological and the other technical. From the epistemological point of view, the issue of uncertainty related to future predictions is intrinsic to science: the perspective of complexity values uncertainty and probability as sources of a non-deterministic and non-linear paradigm which opens horizons of possibilities, instead of a unique inescapable path and ground the discourses about possibilities on a rational basis. On the technical side, STEM disciplines investigate potentialities, limits and applications of innovations in the present, hypothesizing their possible impact on the future; this enlarges students' and teachers' imagination towards future challenges and careers.

The structure of I SEE modules and the case of Artificial Intelligence

Among the results of the first two years of research within the project, there are concrete teaching/learning modules for high-school, already tested in real contexts with 18-19 years old students. Although the issues addressed are very different (e.g. climate change, artificial intelligence, quantum computing), all the modules have a common structure in which the students are encouraged to: i) encounter the focal issue; ii) engage with the interaction between scientific ideas and future; iii) synthesise the ideas and put them into practice [START_REF] Branchetti | The I SEE project: An approach to futurize STEM education[END_REF]. The design and implementation of the Italian module about Artificial Intelligence (AI) were carried out by a team of researchers in Mathematics, Physics and Informatics education, together with high school teachers and experts in engineering, applied physics, epistemology of science and complex systems. The students encounter the focal issue thanks to a group activity, about AI applications in many fields, and two seminars, about: i. the relationship between AI and complexity science; ii. the history of AI. In the second phase, they are showed different approaches to AI (imperative, logical/declarative, machine learning) applied to the Tic-Tac-Toe game; then, they are introduced to applications of AI in STEM research and to the future-relevant concepts of complexity (system, feedback, projection, scenario, non-linearity, agent/model-based simulations). In the last phase, students are involved in group activities about the imagination of a future ideal city, more or less influenced by AI, and are asked to design actions to do in the present with an eye to desirable futures, thinking about future careers, also in the STEM fields.

STEM analysis of the I SEE module on AI: Integration and "Big Ideas"

Since the project aims to propose guidelines and examples of inclusive and future-oriented STEM education, we wondered what aspects of our research could be relevant for contributing to the emerging debate about how STEM education should be conceived and concretely realised. As framework we chose the STEM "big ideas", defined as key ideas that link various discipline understandings into coherent wholes, mediating the construction of in-depth knowledge [START_REF] Chalmers | Implementing "big ideas" to advance the teaching and learning of science, technology, engineering, and mathematics (STEM)[END_REF]. We analysed our module separating its S-T-E-M components, discussing their integration and searching for different kinds of big ideas underlying it (encompassing, cross-disciplines, and within-disciplines big ideas).

The STEM disciplines are all included and structurally integrated within the module, since the topic allows to show the dialectic relationship between the technology/engineering and pure scientific/mathematical issues. Mathematics is relevant, not only in terms of concept and methods technically necessary, but also of thinking tools and skills. As content encompassing big ideas, we recognised AI and Future. As conceptual encompassing big ideas, we identified paradigms like linearity/complexity or practices like problem posing and solving. The cross-discipline big ideas we used are the concepts like algorithm, which are transversal to the STEM domains. There also within-disciplines big ideas that have an application in the other disciplines, like cost function, which is a mathematical optimization concept relevant to other STEM disciplines.

Results and conclusions

The main results of our analysis are three: a) the lens of the "big ideas" is effective to flesh out the connections among the S-T-E-M disciplines within our module on AI; b) the analytic separation of the contributions of the different disciplines and the search for their integration allows to emphasise the specificity of each of them; c) our module can be considered an example of a meaningful integration between disciplines, with STEM "big ideas" connecting them.