STABILIZED FEM SOLUTIONS OF MHD EQUATIONS AROUND A SOLID AND INSIDE A CONDUCTING MEDIUM

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Available online: March 09, 2018 Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Volume 68, Number 1, Pages 197 208 (2019) DOI: 10.1501/Commua1_0000000901 ISSN 1303 5991 http://communications.

1 Available online: March 09, 2018 Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Volume 68, Number 1, Pages (2019) DOI: /Commua1_ ISSN STABILIZED FEM SOLUTIONS OF MHD EQUATIONS AROUND A SOLID AND INSIDE A CONDUCTING MEDIUM S.H. AYDIN A. In this stud, the numerical solution of the magnetohdrodnamic (MHD) flow is considered in a circular pipe around a conducting solid and in an insulating or conducting medium. An external magnetic field is applied through axis of the pipe with an angle α with through the x-axis. The mathematical model of the considered phsical problem can be defined in terms of coupled MHD equations in the pipe domain and the Laplace equations on the solid and external mediums. The coupled equations are transformed into decoupled inhomogeneous convection-diffusion tpe equations in order to appl stabilization in the finite element method solution procedure. Obtained stabled solutions for the high values of the problem parameters displa the well-known characteristics of the MHD pipe flow. I The electricall conducting, viscous and incompressible fluid is driven down a straight pipe of suffi cient length and of circular cross-section b a constant pressure gradient under an external magnetic field applied perpendicular to the axis of the pipe [1]. MHD pipe flow finds some engineering and biomedical applications as MHD generators, pumps and instruments for measuring blood pressure. In this stud, a suffi cientl long annular pipe around a clindrical conducting solid and inside a medium is considered which ma be electricall insulated or conducting. There is an electricall conducting, viscous and incompressible fluid in the annular region. The fluid moves b a constant pressure gradient under an external magnetic field applied perpendicular to the axis of the pipe. The considered problem models the MHD turbines and nuclear fusion apparatus. Therefore there are man studies, most of them are numerical in the literature. Hartmann firstl investigated the MHD flow of viscous, incompressible flow between two plates [2]. Then, man Received b the editors: Februar 09, 2017; Accepted: November 27, Mathematics Subject Classification. Primar 65N30, 76W05; Secondar 74S30. Ke words and phrases. MHD, stabilized FEM. c 2018 A nkara U niversit Communications Facult of Sciences Universit of Ankara-Series A1 M athematics and Statistics. Communications de la Faculté des Sciences de l Université d Ankara. Séries A1. M athematics and Statistics. 197

2 198 S.H. AYDIN number of authors have interested in the solutions of the MHD pipe flow equations both analticall and numericall. The MHD pipe flow problem without a solid in the pipe but in a conducting medium has been solved b using BEM [3] for square and circular pipes, b using DRBEM [4] and FEM [5] for a circular pipe with the assumption that the induced magnetic field of outside conducting medium tends to zero at infinit. The standard Galerkin FEM is one of the most usable numerical solution technique for the linear/nonlinear or coupled boundar value problems. However, it is well known that numerical solution of the MHD problems for the high values of the problem parameters (Renolds number, Hartmann number, etc.) bring some well-known numerical instabilities. In order to eliminate these numerical diffi culties, either a finer mesh should be used or a stabilized finite element methods should be considered in the numerical solution procedure. Using the finer mesh increases the size of the resulting sstem of equations so the computational time and computational cost which is not preferred. Therefore, as an alternative, some stabilization techniques should be used in the solution procedure. Streamline Upwind Petrov-Galerkin (SUPG) method [6] is the most popular and widel used. In 1979, Hughes and Brooks [6] proposed the SUPG method in order to solve the incompressible Navier-Stokes equations as a first application. The proposed method has better stabilit and accurac futures compared to previousl used standard method. The method has been alread applied for man different problems such as convection-diffusion tped equations, Navier-Stokes equations, MHD equations, natural convection equations. Also some variants of the method are developed such as GLS (Galerkin Least Square), DWG(Douglas-Wang), RFB(Residual Free Bubble), TLFEM(Two Level Finite Element Method), SSM(Stabilizing Subgrid Method). The SUPG method reduces the oscillations in the standard Galerkin FEM solutions b adding the mesh-dependent perturbation terms to the formulation [8]. In this stud we have used SUPG tped stabilization for the considered problem. 1. M M Mathematical problem is derived from Navier-Stokes equations of continuum mechanics and Maxwell s equations of electromagnetic field through Ohm s law. Assume that the cross-section of the pipe contains solid Ω s, annular fluid domain Ω f and conducting outside region is Ω ex (see Fig. 1). Also, we assumed that magnetic field is applied through with an angle α with x-direction [1]. Therefore, governing coupled differential equations in non-dimensional form are written as [3] 2 B s = 0 in Ω s (1.1)

3 STABILIZED FEM SOLUTIONS OF MHD EQUATIONS 199 F 1. Domain of the problem and sample mesh used in the numerical scheme 2 V f + Re Rh [cos α Bf x 2 B f + Rm f [cos α V f x with boundar conditions ] Bf + sin α ] f V + sin α = 1 = 0 in Ω f (1.2) 2 B ex = 0 in Ω ex (1.3) V f (x, ) = 0 and B f (x, ) = B s (x, ) 1 B f Rm f n = 1 B s Rm s n V f (x, ) = 0 (x, ) Γ 1 (1.4) B f (x, ) = B ex (x, ) 1 B f Rm f n = 1 B ex Rm ex n (x, ) Γ 2 (1.5) where V f (x, ), B s (x, ), B f (x, ) and B ex (x, ) are the velocit of the fluid, induced magnetic field of the solid, induced magnetic field of the fluid and induced magnetic field of the external medium, respectivel. The superscripts s, f and ex correspond to solid, fluid and external medium, respectivel. The parameters Re, Rh, Rm s, Rm f and Rm ex are the Renolds number, magnetic pressure, magnetic Renolds number of the solid, magnetic Renolds number of the fluid and magnetic Renolds number of the external medium, respectivel. The unit normal

4 200 S.H. AYDIN vectors n and n are the inward normal vectors, and n and n are the outward normal vectors. In order to appl FEM, an artificial boundar (Γ 3 ) should be defined for the unbounded external region. Using the behavior of the real potential solution of B ex, we can assume that B ex 0 as x Therefore one can define the following boundar condition for the external region. B ex (x, ) = 0 (x, ) Γ 3. (1.6) Alternativel, one can also use the free exit condition for B ex at infinit. Therefore, it is also possible to use B ex (x, ) n = 0 (x, ) Γ 3 (1.7) as a boundar condition. However, if this tpe of boundar condition is used, induced magnetic field solutions will be obtained up to a constant. Therefore, these values should be normalized b using the identit. BdΩ = 0. (1.8) Ω As a more simple case, if there is no solid inside the fluid, then equation (1.3) and the boundar conditions (1.4) are not considered. 2. FEM F It is seen that both equations and also boundar conditions are in a coupled form. Therefore all of the equations should be solved simultaneousl. However, as indicated in the introduction using the standard Galerkin finite element method for these coupled equations, brings some numerical instabilities. Therefore we should consider the SUPG tped stabilization technique. In order to appl SUPG stabilization to the coupled MHD equations (1.2), the should be in decoupled convection-diffusion tped equations. Therefore, the are decoupled first denoting V = V f and B 1 = ReRhBf M, where M = ReRhRm f is the Hartmann number of the fluid. Then the coupled equations are rewritten as 2 B 1 V + M x x + M B 1 = 1 2 V B 1 + M x x + M V = 0 (2.1) where M x = M cos α and M = M sin α. In order to decouple the equations, the new variables U 1 (x, ) and U 2 (x, ) are defined as U 1 = V + B 1 U 2 = V B 1 (2.2)

5 STABILIZED FEM SOLUTIONS OF MHD EQUATIONS 201 then equations become 2 U 1 U 1 + M x x + M U 1 = 1 2 U 2 U 2 M x x M U 2 = 1. (2.3) Now, it is possible to use the SUPG tpe stabilized FEM technique to these decoupled equations. Before obtaining the SUPG tped formulation, let s write standard Galerkin FEM tpe weak formulation of the equations (1.1), (2.3) and (1.3) b emploing the linear function space L = (H 1 0 (Ω)) 2 as: Find {B s, U 1, U 2, B ex } {L L L L} such that B(B s ; U 1 ; U 2 ; B ex, s; v 1 ; v 2 ; e) l(u 1 ; U 2, v 1 ; v 2 ) = (1, v 1 ) + (1, v 2 ) (2.4) {s, v 1, v 2, e} {L L L L} where and B(B s ; U 1 ; U 2 ; B ex, s; v 1 ; v 2 ; e) = ( B s, s) U 1 + ( U 1, v 1 ) (M x x + M U 1, v 1) U 2 + ( U 2, v 2 ) + (M x x + M U 2, v 2) + ( B ex, e) l(u 1 ; U 2, v 1 ; v 2 ) = ( U 1 n, v 1) + ( U 2 n, v 2) where B(u, v) and l(u, v) are the usual bi-linear and linear forms for the domain and boundar integrals as follows; B(u, v) = Ω uvdω and l(u, v) = Ω uvds. Then, the variational formulation is written b the choice of finite dimensional subspaces L h L, defined b triangulation of the domain. Therefore, specifing a finite element method [7]: Find {Bh s, U 1 h, U 2h, Bh ex} {L h L h L h L h } such that B(B s h; U 1h ; U 2h ; B ex h, s h ; v 1h ; v 2h ; e h ) l(u 1h ; U 2h, v 1h ; v 2h ) = (1, v 1h ) + (1, v 2h ) (2.5) {s h, v 1h, v 2h, e h } {L L L L}.

6 202 S.H. AYDIN Now, it is read to write the SUPG tped variational formulation of these equations using linear elements as [6]; B(Bh; s U 1h ; U 2h ; Bh ex, s h ; v 1h ; v 2h ; e h ) + l(u 1h ; U 2h, v 1h ; v 2h ) U +τ K {(M 1h x x + M U 1h v + 1, M 1h x x + M v 1h ) + (M x U 2h x + M U 2h v 1, M 2h x x + M } v 2h ) = (1, v 1h ) + (1, v 2h ) (2.6) with the stabilization parameter τ K = h K 2M if P e k 1 h 2 K 12 if P e k < 1 (2.7) M where h K is the diameter of the element K, P e K = h K is the Peclet number. 6 For the value of h K there are two different approaches in the literature which are the area of the element or the longest edge. In this stud the value is taken as the length of the longest edge. The Peclet number is mesh element size dependent positive number and controls the convective and diffusive effects as a ratio. If the flow is convective dominated or small diffusive the value of the Peclet number is larger than 1. Let s turn back to original unknowns with inverse transformations V = U 1 + U 2 2 B 1 = U 1 U 2 2 B f = M ReRh B 1 (2.8) and use the relations in the coupled boundar conditions (1.4) and (1.5) as B f n = Rm f B s Rm s n on Γ 1 and B f n = Rm f B ex Rm ex n on Γ 2,

7 STABILIZED FEM SOLUTIONS OF MHD EQUATIONS 203 then one gets the final variational form of the equations as ( B s h, s h ) + ( V h, v 1h ) ReRh(cos α Bf h x + sin α Bf h, v 1 h ) V h +τ K (M x x + M V h, M v 1h x x + M v 1h ) + ( Bf h, v 2 h ) Rm f (cos α V h x + sin α V h, v B f h 2 h ) + τ K (M x x + M B f h, M v 2h x x + M v 2h ) Rm f ( Bs h Rm s n, v 2 h ) Rm f ( Bex h Rm ex n, v 2 h ) + ( Bh ex, e h ) = (1, v 1h ) Rm f τ K (1, cos α v 2 h x + sin α v 2 h ). (2.9) We will obtain the sstem of linear equations in matrix-vector form as K K + S ReRhC 0 Rm f Rm s Q Rm f C K + S Rm f Rm ex Q K B s V f B f B ex = 0 R 1 R 2 0 (2.10) where K, C, S and Q are the matrices and R 1 and R 2 are the vectors with the entries ( Ni N j K ij = Ω f x x + N ) i N j dω, ( C ij = cos α N ) i Ω f x + sin α N i N j dω, ( ) ( ) N i S ij = τ K M x Ω f x + M N i N j M x x + M N j dω, N i Q ij = Γ f n N jdγ, R 1i = N i dω. Ω f ( ) N i R 2i = M x x + M N i dω. Ω f τ K From the solution of the sstem 2.10, the induced magnetic field of the solid, the velocit of the fluid, the induced magnetic field of the fluid and the induced magnetic field of the external medium are obtained at the discretization points.

204 S.H. AYDIN 3. N R D In this section, some test results of the considered stabilized FEM formulation are given in terms of contour plots of the velocit and induced magnetic field.

8 204 S.H. AYDIN 3. N R D In this section, some test results of the considered stabilized FEM formulation are given in terms of contour plots of the velocit and induced magnetic field. For the small value of the Hartmann number (M = 100) linear triangular elements obtained from 8985 discretization points are used. However, for the large value of the Hartmann number (M = 1000) the number of the linear triangular elements is (from discretization points). The obtained sstem of linear equations are solved b using the linear sstem solver from Netlib librar called dgsev Case 1: No solid inside the pipe. As the first test problem, it is assumed that there is no solid inside the pipe. The radius of the circular pipe is taken as 1 and the radius of the artificial boundar is 3. Both of the pipe domain and the external region is discretized using linear triangular elements. F 2. The effect of the stabilization velocit (left) and induced magnetic fields (right) for Rm f = 100, Rm ex = 1, Re = 10, Rh = 10 In Figure 2, the effect of the stabilization in both velocit and induced magnetic fields for the values Rm f = 100, Rm ex = 1, Re = 10 and Rh = 10, so Hartmann number becomes M = 100. In Figure 2(a), standard Galerkin FEM solutions,

9 STABILIZED FEM SOLUTIONS OF MHD EQUATIONS 205 in Figure 2(b), SUPG tped stabilized solutions are displaed. The stabilization effect is clearl seen in fluid velocit and induced current values inside the pipe domain. Since the Laplace equation is solved in the external region, both Galerkin and SUPG solutions are regular in the external region. The moderate Hartmann value (M = 1000) results are displaed in Figure 3. The accurac of the proposed numerical scheme is seen clearl even for the ver large values of the problem parameters. Also, the well known characteristic of MHD flow which is boundar laer formation is seen both in the fluid velocit and in the induced magnetic field as Rm f getting larger. The velocit values are getting smaller inside the pipe center for the large values of M. F 3. Velocit of the fluid and induced magnetic fields for Rm f = 1000, Rm ex = 1, Re = 100, Rh = 10 In Figure 4, the problem is solved with the same parameters as in Figure 3 but with the angle α = π/4. It is seen that, the same behaviors are obtained with rotating angle α. The normal derived boundar condition for B ex on Γ 3 is also tested and displaed in Figure 5. As expected, induced magnetic field contours are perpendicular to the artificial external boundar, and the have tendenc to become zero at infinit Case 2: A conducting solid inside the pipe. In the second case, let s assume that there is a conducting solid inside the pipe with unit circular cross section and outer radius of the circular pipe is 3. Therefore the fluid moves inside the annular pipe domain. Similar to Case 1, we have tested the proposed numerical scheme for the values Rm f = 1000, Rm s = 10, Rm ex = 1, Re = 10, Rh = 10 so M = 100 using boundar condition 1.6 in Figure 6 and boundar condition 1.7 in Figure 7 with α = π/2. The continuation of the induced magnetic contours and the effect of ratios Rm f Rm s and Rm f Rm ex is clearl seen from the figure. Existence of different circulations in the velocit is also observed.

10 206 S.H. AYDIN F 4. Velocit of the fluid and induced magnetic fields for Rm f = 1000, Rm ex = 1, Re = 100, Rh = 10, α = π/4 F 5. Velocit of the fluid and induced magnetic fields for the normal derivative boundar condition and for Rm f = 500, Rm ex = 1, Re = 50, Rh = 10, M = C The magnetohdrodnamic flow inside a circular cross-section duct around a conducting solid in an insulating or conducting medium is solved using stabilized finite element method. Even the considered problem has been alread solved with some other numerical solution procedures, the proposed formulation enables to obtain the stable solution for the high values of the problem parameters. Obtained solutions are in good agreement with the previousl obtained results and displa the well-known characteristics of the MHD flow [3, 4].

11 STABILIZED FEM SOLUTIONS OF MHD EQUATIONS 207 F 6. Velocit of the fluid and induced magnetic fields for Rm f = 1000, Rm s = 10, Rm ex = 1, Re = 10, Rh = 10, α = π/2 F 7. Velocit of the fluid and induced magnetic fields for the normal derivative boundar condition and for Rm f = 500, Rm ex = 1, Re = 50, Rh = 10, α = π/2 A The author wish to thank Prof.Dr. M. Tezer-Sezgin for her valuable support, comments, suggestions and corrections. R [1] Dragoş, L., Magnetofluid Dnamics, Abacus Pres, [2] Hartmann, J., Theor of the laminar flow of an electricall conductive liquid in a homogeneous magnetic field, K. Dan. Vidensk. Selsk. Mat. Fs. Medd., 15(6) (1937), [3] Tezer-Sezgin, M. and Han Adın, S., BEM Solution of MHD Flow in a Pipe Coupled with Magnetic Induction of Exterior Region, Computing, 95(1) (2013),

12 208 S.H. AYDIN [4] Han Adın, S. and Tezer-Sezgin, M., DRBEM Solution of MHD Pipe Flow in a Conducting Medium, J. Comput. Appl. Math., 259(B) (2014), [5] Tezer-Sezgin, M. and Han Adın, S., FEM Solution of MHD Flow Equations Coupled on a Pipe Wall in a Conducting Medium, PAMIR [6] Brooks, A.N. and Hughes, T.J.R., Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), [7] Redd, J.N., An Introduction to the Finite Element Method, McGraw-Hill, New York, [8] Adın, S. H. The Finite Element Method Over a Simple Stabilizing Grid Applied to Fluid Flow Problems (Ph.D. Thesis) Middle East Technical Universit, Institute of Applied Mathematics, Current address: S.H. Adın: Department of Mathematics, Karadeniz Technical Universit, Trabzon, Turke address: shadin@ktu.edu.tr ORCID Address:

Numerical Solution for Coupled MHD Flow Equations in a Square Duct in the Presence of Strong Inclined Magnetic Field

Numerical Solution for Coupled MHD Flow Equations in a Square Duct in the Presence of Strong Inclined Magnetic Field International Journal of Advanced Research in Physical Science (IJARPS) Volume 2, Issue 9, September 2015, PP 20-29 ISSN 2349-7874 (Print) & ISSN 2349-7882 (Online) www.arcjournals.org Numerical Solution