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Application of Gas-Kinetic BGK Scheme for Solving 2-D Compressible Inviscid Flow around Linear TurbineCascadeSaleh N. Abdusslam a , Ong J. Chit b , Megat M. Hamdan a , Ashraf A. Omar b & Waqar Asrar ba Department of Mechanical and Manufacturing Engineering, Faculty of Engineering ,University Putra Malaysia , Selangor, D.E., Malaysiab Department of Mechanical Engineering, Faculty of Engineering , International IslamicUniversity Malaysia , Kuala Lumpur, MalaysiaPublished online: 23 Feb 2007.

To cite this article: Saleh N. Abdusslam , Ong J. Chit , Megat M. Hamdan , Ashraf A. Omar & Waqar Asrar (2006) Applicationof Gas-Kinetic BGK Scheme for Solving 2-D Compressible Inviscid Flow around Linear Turbine Cascade, International Journalfor Computational Methods in Engineering Science and Mechanics, 7:6, 403-410, DOI: 10.1080/15502280600826357

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International Journal for Computational Methods in Engineering Science and Mechanics, 7:403–410, 2006Copyright c© Taylor & Francis Group, LLCISSN: 1550–2287 print / 1550–2295 onlineDOI: 10.1080/15502280600826357

Application of Gas-Kinetic BGK Scheme for Solving 2-DCompressible Inviscid Flow around Linear Turbine Cascade

Saleh N. AbdusslamDepartment of Mechanical and Manufacturing Engineering, Faculty of Engineering,University Putra Malaysia, Selangor D.E., Malaysia

Ong J. ChitDepartment of Mechanical Engineering, Faculty of Engineering, International Islamic UniversityMalaysia, Kuala Lumpur, Malaysia

Megat M. HamdanDepartment of Mechanical and Manufacturing Engineering, Faculty of Engineering,University Putra Malaysia, Selangor D.E., Malaysia

Ashraf A. Omar and Waqar AsrarDepartment of Mechanical Engineering, Faculty of Engineering, International Islamic UniversityMalaysia, Kuala Lumpur, Malaysia

Fluid flows within turbomachinery tend to be extremely com-plex. Understanding such flows is crucial in the effort to improvecurrent turbomachinery designs. Hence, computational approachescan be used to great advantage in this regard. In this paper, gas-kinetic BGK (Bhatnagar-Gross-Krook) scheme is developed forsimulating compressible inviscid flow around a linear turbine cas-cade. BGK scheme is an approximate Riemann solver that usesthe collisional Boltzmann equation as the governing equation forflow evolutions. For efficient computations, particle distributionfunctions in the general solution of the BGK model are simpli-fied and used for the flow simulations. Second-order accuracy isachieved via the reconstruction of flow variables using the MUSCL(Monotone Upstream-Centered Schemes for Conservation Laws)interpolation technique together with a multistage Runge-Kuttamethod. A multi-zone H-type mesh for the linear turbine cascadesis generated using a structured algebraic grid generation method.Computed results are compared with available experimental dataand found to be in agreement with each other. In order to furthersubstantiate the performance of the BGK scheme, another test case,namely a wedge cascade, is used. The numerical solutions obtainedvia this test are validated against analytical solutions, which showedto be in good agreement.

Received 20 August 2004; in final form 12 May 2006.Address correspondence to A.A. Omar, Dept. of Mechanical Engi-

neering, International Islamic University Malaysia, P.O. Box 10, 50728Kuala Lumpur, Malaysia. E-mail: aao@iiu.edu.my

Keywords Finite Difference Method, Gas-Kinetic Scheme, Turbo-machinery Flow, High-Order Scheme, Multistage Runge-Kutta Method

1. INTRODUCTIONThe history of turbomachinery design has seen important

evolution in recent years. The need for efficiency and weightreduction has driven designers to investigate the details of thecomplex flow field where each component is expected to operate.Computational Fluid Dynamics (CFD) has progressed rapidlyand increased its reliability as an effective design tool. Thus,it is reasonable to see CFD taking an indispensable role in thefuture design and optimization of turbomachinery. In the pastfew years, several two- and three-dimensional codes for solvinginviscid flows have reached a high level of maturity and arecommonly used in turbomachinery applications [1, 2].

The numerical simulation of turbomachinery is a challeng-ing area that involves unsteady flow phenomena in complex ge-ometries. The problem has received the attention of researchersever since the emergence of the discipline, and several tech-niques have been applied towards its solution. Jorgenson [3, 4]presents a list of relevant work in unsteady turbomachinery flowprediction.

The development of gas-kinetic schemes has attracted muchattention in recent years. These schemes are based on theapproximate collisional Boltzmann equation [5, 6]. A particular

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404 S. N. ABDUSSLAM ET AL.

strength of the kinetic schemes lies precisely where FDS schemesoften fail, such as carbuncle phenomenon, entropy condition andpositivity [7, 8, 9]. There are mainly two kinds of gas-kineticschemes, and the differences lie within the governing equationsused in the gas evolution stage. One of the well-known kineticschemes is called the KFVS (Kinetic Flux Vector Splitting),which is based on the collisionless Boltzmann equation and theother is based on the collisional BGK model [10, 11]. Like anyother FVS method, the KFVS scheme is very diffusive and lessaccurate in comparison with the Roe-type FDS method. Thediffusivity of the FVS schemes is mainly due to the particle orwave-free transport mechanism, which sets the CFL time stepequal to particle collision time [12]. In order to reduce diffu-sivity, particle collisions have to be modeled and implementedinto the gas evolution stage. One of the distinct approaches totake particle collision into consideration in gas evolution can befound in Xu [5]. In this method, the collision effect is consid-ered by the BGK model as an approximation of the collisionintegral in the Boltzmann equation. It is found that this gas-kinetic BGK scheme possesses accuracy that is superior to theflux vector splitting schemes and avoids the anomalies of FDS-type schemes [5, 7, 13, 14].

In this paper, the authors will present the outcomes of thegas-kinetic BGK scheme for computing compressible inviscidflow around a linear turbine cascade. To the best of the authors’knowledge, this is the first time that the gas-kinetic BGK schemehas been used for analyzing the flow around turbine cascade.Besides the linear cascade simulation, a compressible inviscidsupersonic channel flow will also be included in this study.

2. NUMERICAL METHODS

2.1. Governing EquationsThe two-dimensional Euler equations in Cartesian coordi-

nates are written as

∂W

∂t+ ∂ F

∂x+ ∂G

∂y= 0 (1)

where

W =

ρ

ρU

ρV

ρε

, F =

ρU

ρU 2 + p

ρU V

(ρε + p) U

,

G =

ρV

ρU V

ρV 2 + p

(ρε + p) V

(2)

In Eq. (1), ρ, ρU , ρV and ρε are the macroscopic mass, x-momentum, y-momentum and total energy density respectively,while p is the pressure.

Equation (1) can be transformed from Cartesian coordinates(x, y) into curvilinear coordinates (ξ, η) as

∂W

∂t+ ∂ F

∂ξ+ ∂G

∂η= 0 (3)

where

W = J−1W, E = J−1(ξx F + ξyG),G = J−1(ηx F + ηyG)

(4)

In Eq. (4), J = (ξxηy − ξyηx ) is the Jacobian of transformation.The manner in which the metrics ξx , ξy, ηx , ηy and the Jacobianof transformation J are evaluated is detailed in Hoffmann andChiang [15].

2.2. Gas-Kinetic BGK SchemeA standard gas-kinetic BGK scheme is based on the colli-

sional Boltzmann equation [5] and is written in two dimensionsas

∂ f

∂t+ u

∂ f

∂x+ v

∂ f

∂y= Q ( f, f ) (5)

where f is a real particle distribution function, u and v are theparticle velocities, and the right-hand side stands for the collisionterm. The collision term is an integral function, which accountsfor the binary collisions. The BGK model of the Boltzmannequation is realized when a relaxation model [5] is suggested asan approximation for the complicated collision term in Eq. (5),which can be written as

∂ f

∂t+ u

∂ f

∂x+ v

∂ f

∂y= (g − f )

τ(6)

where g is an equilibrium particle distribution function that fapproaches through particle collisions within a collision timescale τ . Both f and g are functions of space (x , y), time (t),particle velocities (u, v), and ς . ς is a K dimensional vec-tor that accounts for the internal degrees of freedom such asmolecular rotation and vibrations. The dimensional vector, K ,

is related to the specific heat ratios and the space dimensionby the relation K = (4 − 2γ )/(γ − 1), where for a diatomicgas γ = 1.4.

The relations between the densities of mass ρ, momentum(ρU, ρV ), and total energy ε with the distribution function fare derived from the following moment relation [5]:

ρ

ρU

ρV

ε

=

∫f d (7)

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APPLICATION OF GAS-KINETIC BGK SCHEME 405

where d = dudvdς is the volume element in the phase spacewhile is the vector of moments

=

1

u

v

1

2(u2 + v2 + ς2)

(8)

With the moment relation defined in Eq. (7), a similar approachcould be adopted in obtaining the numerical fluxes across cellinterfaces and they are given as

Fx =∫

u f d

G y =∫

v f d (9)

where Fx and G y are the physical flux in the x- and y-direction,respectively. In addition, within the physical constraint of theconservation of mass, momentum and energy during particlecollisions required that the following compatibility conditionhas to be satisfied,

∫(g − f )

τ d = 0 (10)

at any point in space and time. This assumption will lead to thetwo-dimensional Euler equations.

A general solution f of Eq. (6) at the cell interface (xi+1/2,y j ) in two dimensions is obtained as

f (x, y, t, u, v, ς) = 1

τ

∫ t

0g(x ′, y′, t, u, v, ς ) e−(t−t ′)/τ dt ′

+ e−t/τ fo(x − ut, y − vt) (11)

where x ′ = xi+1/2 − u(t − t ′) and y′ = y j − v(t − t ′) areparticle trajectories and fo is the initial non-equilibrium distri-bution function at t = 0. To construct the numerical flux fromthe solution of f in Eq. (11), two unknowns g and fo must bedetermined. Hence, by means of discretizing the two distribu-tion functions g and fo as proposed by Xu [5], the followingforms are obtained

fo ={

gl , x < 0

gr , x > 0(12)

g = go (13)

where gl , gr , and go are local Maxwellian distribution functionat the left, right and middle of a cell interface, respectively. Usingthe relation between the microscopic and macroscopic descrip-tions as described in Eq. (7), the state variables that constitute

the Maxwellian distribution function gl , and gr of Eq. (12) canbe obtained as [5]

ρ∗

ρ∗U ∗

ρ∗V ∗

ε∗

=

∫g∗ d (14)

where the superscript ∗ denotes the left or right state variablesat the cell interface. The Maxwellian g∗ has the following form[5, 12, 16, 17]:

g∗ = ρ∗(

λ∗

π

)(K+2)/2e−λ∗[(u−U ∗)2+(v−V ∗)2+ς2] (15)

Then, the parameters of the Maxwellian go can be constructedby taking the limit of Eq. (11) as t → 0 and substituting it intoEq. (10) at the cell interface (xi+1/2, y j ) = (0, 0) to obtain

∫go d = ∫

fo (−ut) d

ρo

ρoUo

ρoVo

εo

= ∫

u>0 gl d + ∫u<0 gr d

(16)

The underlying physical assumption in the above equation is thatthe left and right moving particles collapse at a cell interface toform an equilibrium state go. Hence, the quantities with subscripto are termed equilibrium state variables at the cell interfaceand they could be determined by taking the moments of theMaxwellian distribution function in Eq. (16). All the momentcalculations involved in the integration of the Maxwellian inphase space from negative infinity to zero or zero to positiveinfinity are expressed in term of exponential and error functionsfound Refs. 5, 16, and 20.

With all the parameters for the Maxwellian distribution func-tions determined up to this point, the solution f of the BGKmodel at (xi+1/2,y j ) = (0, 0) in Eq. (11), after substitutingEq. (12) and (13) together with the assumption of constant equi-librium state go in space and time. The following form is obtained

f (0, 0, t, u, v, ς ) = (1 − e−t /τ

)go + e−t /τ fo (−ut, −vt) (17)

With the definition of e−t/τ = ϕ, the above distribution functionf is written as

f (0, 0, t, u, v, ς ) = (1 − ϕ) go + ϕ fo (−ut, −vt) (18)

For first-order BGK scheme, the term ϕ is assumed as a constant,where ϕ ∈ [0, 1]. When the BGK scheme is extended to high-order, the value of ϕ should depend on the real flow situation.This option is necessary to prevent numerical oscillations andphysically correct in that it accounts for the non-equilibrium

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406 S. N. ABDUSSLAM ET AL.

behavior of the gas flow in the discontinuity region. A possiblechoice for ϕ in a high-order scheme is to design a pressure-basedstencil, such as the switch function in the JST scheme [5, 12,17]. The following form is chosen for the ϕ parameter when thescheme is extended to high-order [5]:

ϕ = 1 − exp

(−β

|pl − pr |pl + pr

)(19)

where β can be some constant.Finally, the gas-kinetic BGK numerical flux across the cell

interface in the x-direction can be computed as

Fx =∫

u f (0, 0, t, u, v, ς) d

(20)Fx = (1 − ϕ)Fe

x + ϕ F fx

where Fex is the equilibrium flux function and F f

x is the non-equilibrium or free stream flux function.

The numerical flux for the BGK scheme at the cell interfacein the x-direction are obtained from Eq. (20) as,

Fi+1/2, j = (1 − ϕ)Fei+1/2, j + ϕ F f

i+1/2, j (21)

While the numerical flux at the cell interface in the y-directionis obtained in a similar manner and the resulting relation is pre-sented as

Gi, j+1/2 = (1 − ϕ)Gei, j+1/2 + ϕ G f

i, j+1/2 (22)

2.3. Reconstruction StageFor high-order spatial accuracy, a method known as the MUSCL

approach [21] is adopted. To avoid spurious oscillations in thesolution, a limiter is used to extrapolate the primitive variablesat the cell interfaces. In this paper, van Leer’s limiter is em-ployed. Hence, the left and right states of the primitive variablesρ, U, V, p at a cell interface could be obtained through the non-linear reconstruction of the respective variables and are given as

Ql = Qi, j + 1

2φ

(�Qi+1/2, j

�Qi−1/2, j

)�Qi−1/2, j

(23)

Qr = Qi+1, j − 1

2φ

(�Qi+3/2, j

�Qi+1/2, j

)�Qi+1/2, j

where Q is a primitive variable and the subscript l, and r corre-spond to the left and right side of a considered cell interface. Inaddition, �Qi+1/2, j = Qi+1, j − Qi, j . van Leer’s limiter used inthe reconstruction of flow variables in Eq. (23) is given as

φ(�) = (� + |�|)(1 + �)

(24)

By using the extrapolation relations in Eq. (23), a second-orderspatial accurate scheme is produced.

2.4. Runge-Kutta Multistage MethodIn the present study, the Runge-Kutta multistage scheme,

which is one of the most popular explicit time integration meth-ods, is used to compute the linear cascade problem. A three-stage Runge-Kutta scheme [22, 23] is selected because it has asecond-order temporal accuracy. The general form ofRunge-Kutta scheme adopted in this study is written as

W (1) = W n − κ1�t R(W n)

W (2) = W n − κ2�t R(W (1))

W n+1 = W n − �t R(W (2)) (25)

with integration constants of κ1 = κ2 = 1/2.

3. BOUNDARY CONDITIONS

3.1. Inlet and Exit Boundary ConditionsFor subsonic inlet boundary, as required by the theory of char-

acteristics, three variables are fixed, namely the total pressure,total temperature and flow angle, while the static pressure is ex-trapolated from the interior. At exit, if the exit flow is subsonic,only the static pressure is fixed, while total pressure, total tem-perature and flow angle are extrapolated from the interior. If theexit flow is supersonic, all four variables are extrapolated fromthe interior.

3.2. Periodic Boundary ConditionsThe periodicity condition on the bounding streamlines, up-

stream and downstream of the blade row, is easily satisfied bytreating the calculating points on each of the bounding streamlineas if they were interior ones by assuming that for correspondingpoints on each of the streamline all properties are equal.

3.3. Solid Boundary ConditionsAt the solid boundary, an inviscid wall boundary condition

is implemented where the velocity component that is normal tothe solid boundary is set equal to zero.

4. RESULTS AND DISCUSSIONS

4.1. Linear Turbine CascadeIn this paper, the developed numerical scheme is applied to

a linear turbine cascade flow problem and the outcomes of thesimulation will be presented and discussed in this section. Theblade profile of the cascade belongs to a gas turbine rotor bladeof the Von Karman Institute (VKI LS-59). This blade was ex-tensively tested experimentally [19, 24] and numerically [2, 24,25]. In this paper, a multi-zone structured H-type grid is adopted.The grids around the selected model are efficiently generatedusing a structure algebraic grid generation method. The meshemployed in the computations is shown in Fig. 1, and it consistsof 170 × 90 grids. In order to test the capabilities of the de-veloped scheme, two test conditions are selected. The first testcase corresponds to a subsonic-transonic flow while the other

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APPLICATION OF GAS-KINETIC BGK SCHEME 407

FIG. 1. H-type grid for the turbine cascade used in the calculations.

corresponds to subsonic-subsonic flow. The subsonic inlet is setat Mach number 0.282 whereas the outlet Mach number is setat 0.961 and 0.782, which correspond to transonic and subsonicflows, respectively.

Comparison of measured and computed values of Mach num-ber distributions on the surface of the cascade’s blade forsubsonic-transonic and subsonic-subsonic flow cases are shownin Fig. 2 and Fig. 3, respectively. From these figures, the com-puted Mach number distributions follow the outline of the mea-sured Mach number distributions in quite a similar trend, exceptfor a sharp jump at location 0.95 in the figures. These could

FIG. 2. Comparison of measured and computed Mach number distributionsfor the turbine cascade in subsonic-transonic flow condition.

FIG. 3. Comparison of measured and computed Mach number distributionsfor the turbine cascade in subsonic-subsonic flow condition.

be caused by the fact that the simulations were implementedwith compressible inviscid flow conditions, whereas the mea-sured Mach number distributions were results from turbulent,compressible viscous flow experiment. Also, these findings aresimilar to the results found in Ref. 18 as shown in Fig. 2. Exceptthat the BGK scheme shows less agreement with experimen-tal data in suction side and better agreement with experimentaldata in the pressure side than Ref. 18. In addition, the contoursof constant Mach numbers for the subsonic-transonic test caseare shown in Fig. 4, together with the Schlieren picture obtainedin experiment [19], which is plotted in Fig. 5. Comparison of

FIG. 4. Mach number contours for the subsonic-transonic turbine cascadeflow.

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408 S. N. ABDUSSLAM ET AL.

FIG. 5. Mach number contours for the subsonic-transonic turbine cascadeflow [Ref. 19].

these two contour plots showed that the Mach number contoursfrom the numerical computation agree very well with the ex-perimental result where the location of the shock is predictedaccurately. The corresponding Mach number contours plot forthe subsonic-subsonic flow test case of the turbine cascade isdepicted in Fig. 6.

The aspect of grid independence for the current case is alsostudied by considering two additional sets of grid size, namely5600 and 17190 grid points. The former accounts for about 63%

FIG. 6. Mach number contours for the subsonic-subsonic turbine cascadeflow.

FIG. 7. Computed Mach number distributions for the turbine cascade insubsonic-transonic flow condition with 17190 grid points.

of grid size reduction while the latter is about 12% of grid sizeincrement from the current grid size of 15300 points. A clearidea of grid independence can be obtained by comparing thecomputed surface Mach number distributions for three of thesegrid sizes under subsonic-transonic flow condition. The resultfor the finest mesh, i.e. 17190 grid points, is shown in Fig. 7.It clearly shows that the computed result is identical to the oneshown in Fig. 2. Lastly, the result for the coarse mesh, i.e. 5600grid points, is shown in Fig. 8. A marked difference can be seenfrom this computed result where the scheme is still unable toresolve the flow effectively. By comparing the results from Fig. 8to Fig. 2, it would be clear that the flow solution at this stage isstill dependent on the grid size. Thus, it would be appropriateto state that the grid has reached its independence with respectto the numerical solutions of the flow at mesh size of 15300.Therefore, the mesh with 170 × 90 grid points is used in the

FIG. 8. Computed Mach number distributions for the turbine cascade insubsonic-transonic flow condition with 5600 grid points.

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APPLICATION OF GAS-KINETIC BGK SCHEME 409

current study in order to avoid unnecessary extra computingtime in using a finer mesh.

4.2. Supersonic Wedge CascadeTo complete this study, another test case of internal flow is

selected to further evaluate the computational characteristics ofthe BGK scheme. This particular flow problem is taken fromRefs. 27 and 28 where a cascade of wedges is used to illustratethe shock capturing ability of the method for oblique shocks. Therationalization behind this selection is simply because analyticalresults can be calculated from characteristic theory and obliqueshock relations [28] to validate the numerical results computedfrom the BGK scheme.

The physical domain for this wedge cascade is shown inFig. 9. The compression/expansion angle for the wedge is 5.73◦.The number of grid points for the domain is 150 × 50. The fol-lowing flow conditions are used to initialize the flow:

M = 1.6 α = 60◦ P2/P01 = 0.3536

where M and α are the inlet Mach number and angle of attack,respectively.

Since the flow at the inlet is supersonic, all the flow variablesare fixed to free stream. At exit, as required by the theory ofcharacteristics, for subsonic flow static pressure is fixed whiletotal pressure, total temperature, and flow angle are extrapolatedfrom the interior. However, if the exit flow is supersonic, all fourvariables are extrapolated from the interior. The lower and upperboundaries consist of both periodic and solid boundaries. Aninviscid wall condition is applied on the wedge surface whileperiodic condition is applied to the upstream of the leading edgeand downstream of the trailing edge.

FIG. 9. Physical domain for the wedge cascade.

FIG. 10. Mach number contours by BGK scheme for the supersonic wedgecascade flow.

The leading edge of the wedge leads to an oblique shock waveinside the channel, which can be seen from the Mach numbercontours plot of Fig. 10. This shock is then reflected on the suc-tion side and cancelled at the upstream corner, giving uniformflow between the two parallel surfaces. At the downstream cor-ner, an expansion shock wave is seen to be present on the con-tours plot. Figures 11 and 12 show the surface Mach numberdistributions computed by the BGK scheme and second-ordercentral difference scheme with TVD (Total Variation Diminish-ing), together with the analytical solutions. From both of thesefigures, it can be seen that the numerical results produced by bothschemes are in agreement with the analytical result. Better flow

FIG. 11. Comparison surface Mach number distributions for BGK schemeand analytical results.

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410 S. N. ABDUSSLAM ET AL.

FIG. 12. Comparison of surface Mach number distributions for central dif-ference scheme with TVD limiter and analytical results.

resolution can be seen for the BGK scheme in comparison withthe central scheme, especially at the region of the first corner andparallel surfaces of the wedge cascade. The description of theflow at the expansion corner is rather unsatisfactory. Nonethe-less, the numerical results illustrated here are by far much betterthan the one presented in Ref. 27. Another interesting aspectthat can be observed from Figs. 11 and 12 is the shock resolu-tion capability of the BGK scheme, where it is better than thecentral difference scheme. It can be seen that BGK scheme isable to resolve the shock location much better than the centraldifference scheme.

5. CONCLUSIONIn this paper, a numerical method based on the two-dimensi-

onal gas-kinetic BGK scheme was developed and subsequentlyapplied to the simulation of the linear turbine cascade for the firsttime. The outcomes of the simulations, which consist of two testconditions for the linear cascade, namely, subsonic-transonicand subsonic-subsonic, have been derived and compared withresults from experiments and literatures. The solutions obtainedare reasonably acceptable and in agreement with the experimen-tal data considering that the simulations conducted were basedon the compressible inviscid flow condition while the experi-mental results are derived from a turbulent, compressible vis-cous flow condition. In addition, the outcomes of the supersonicwedge cascade problem further substantiated the computationalcharacteristics of the BGK scheme to be accurate and havinghigh shock resolution capabilities.

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