eu NUMERICAL STUDY OF 2 D MHD CONVECTION AROUND PERIODICALLY PLACED CYLINDERS

Abstract: In this paper 2D stationary boundary value problem for the system of magnetohydrodynamic (MHD) equations along with the heat transfer equation is considered. The viscous electrically conducting incompressible liquid-electrolyte is to move between infinite cylinders placed periodically. Similarly 2D MHD channel flow with periodically placed obstacles on the channel walls is examined. We analyze the 2D MHD convection around the cylinders and obstacles subject to homogeneous external magnetics field. The cylinders, obstacles and walls of channel with constant temperature are heated. The goal of such investigation is to obtain the distributions of stream function, temperature, velocity and the vortex formation in the cross-section plane of the cylinders and obstacles depending on the external magnetic field and on the direction of the gravitation. For the numerical treatment finite difference method is used.


Introduction
In many physical experiments and technological applications it is important to mix and heat an electroconductive liquid: liquid-metals, electrolyte, water, air.In the developed mathematical models vortex-type structures appearing in liquid flows, as well as in problems related to energy conversion in new technological devices.
In the developed mathematical models, vortex-type structures appear in liquid flows, as well as in problems related to energy conversion in new technological devices.MHD convection flow of a viscous incompressible fluid around cylinder with combined effects of heat and mass transfer is an important problem prevalent in many engineering applications [1], [4], [5].
Heat exchanger systems are employed in numerous industries.Steam generation in boiler, air cooling within the coil of air conditioner and automotive radiators represent just some of the conventional applications of this mechanical system.Of particular importance in the design of heat exchanger is the pivotal understanding of heat transfer in flow across a bank of tubes.Tube banks, as used in many heat exchangers, can be systematically arranged in an in-line or staggered manner.
For mathematical model the flow around infinite periodically placed tube banks (cylinders with cross-section of the circle) can be considered [7].In this paper the heat transfer significant influence on the fluid flow behaviour without the magnetic field and gravity is investigated .The problem is calculated using finite-volume CFD computer code ANSYS for the in-line and staggered arrangements of tube banks.The working fluid is taken to be water.Streamlines with two recirculation vortices at different Reynolds numbers (Re = 114 − 2735) are obtained.The lowest temperature is located within the mainstream bulk cold fluid while the highest temperature is located close to the circumferential wall.
We consider the viscous electrically conducting incompressible liquid-electrolyte.In [6] the flow and temperature of electrolyte in a conducting cylinder with alternating current is calculated.
The electrolyte is between infinite cylinders placed periodically in the(x, y) plane.For the in-line arrangements of cylinders, the liquid has prescribed inlet ambient temperature T 0 much lower than the wall temperature T w .By taking the advantage of special geometrical features the computational fluid domain allows the possible exploitation of symmetric and periodic boundary conditions in speeding up the computations and in turn enhancing the computational accuracy of the simplified geometries.Flow around finite arrangement of cylinders occurs in many applications (heat exchanges, fuel rod cooling etc).The formulation of single periodicity cell with periodic BCs is a reasonable approximation.Consider a closed flow circuit around cylinders in a periodic arrangement, then the temperature stationary field is going to be periodic in space.
The cross-sections of the cylinders are square.We consider 2D stationary boundary value problems for the system of magnetohydrodynamic (MHD) equation.We analyze the 2D free convection between these cylinders in homogeneous external magnetic field, depending on the direction of the gravitation.
By taking the advantage of special geometrical features using the conditions of symmetry and periodicity, we can consider only two cylinders.Here, the cross -flow exchanger of cooler fluid re-wowing the heat from the warmer fluid within the tubes flowing perpendicular to the 2D plane.
This process of the magnetohydrodynamics (MHD) is considered with the so-called inductionless approximation.This would mean that the action of a moving liquid on the external magnetic field is insignificant [1].
The external magnetic field, Lorentz force, dimensionless stationary Navier-Stokes equations, numerical domain with two cylinders and the system of three equations for calculating the stream function, vorticity and temperature are formulated.The distribution of electromagnetic fields, forces, velocity and temperature around the cylinders has been calculated using the finite difference method, Seidel iterations and specific boundary conditions for the vorticity function.

Mathematical Model
The external homogeneous 2D magnetic field has two components of the induction: where α is the angle between the Ox-axis and direction of the induction vector, B 0 is the magnitude of the magnetic field.The magnetic field creates the F x (t, x, y), F y (t, x, y) components of the Lorentz force F.
Considering the vector of Lorentz force where E z = const, J z are the azimuthal components of the electric field vector E and the density vector of the electric current J,B x , B y are the components of the magnetic induction vector B, σ is the electric conductivity, V x , V y are the components of velocity vector V.
We analyze the free convection flow depending on two settings of the homogeneous magnetic field: the field parallel to Ox-axis (α = 0) and the transverse field (α = π 2 ) The z-component ∂Fy ∂x − ∂Fx ∂y of the vector curlF = ∇ × F affects the liquid motion, which can be described by Navier-Stokes equations in Boussinesq approximation and the heat transfer equation [2], [4]: where ∆ is the Laplacian, p, T, σ, ρ, ν, k, C p , β t are static pressure, temperature, electrical conductivity, fluid density, kinematic viscosity, heat conductivity, specific heat, acceleration due to gravity, volumetric coefficient of thermal expansion, T 0 is the initial fluid temperature, g = g(sin(β), − cos(β), 0) is the vector of gravitation, g is the gravitational acceleration, β is the angle between the Oy-axis and direction of the gravitation.The dynamic pressure is p + p h , where p h is hydrostatical pressure.The surface of the cylinders is subject to heat loss modelled by Newtonian cooling from the cylinders which is at a temperature T w > T 0 with heat transfer coefficient h[ J s m 2 K ] and there is resulting in the boundary conditions where n is the external normal to the surface of the cylinders.The equations (1) were made dimensionless by using characteristic values L 0 (side length of a square), U 0 (velocity), B 0 (magnetic field), P 0 = U 2 0 ρ (pressure), L 0 /V 0 (time), U 0 B 0 (density vector of the electric current).The dimensionless temperature is equal to T −T 0 Tw−T 0 ∈ [0, 1].The dimensionless boundary conditions for temperature on r = 1 is where Bi = h L 0 λ is the Biot number.Using the vorticity function ζ = ∂Vy ∂x − ∂Vx ∂y , one obtains where j z = e z + V x sin(α) − V y cos(α), e z is the dimensionless form of azimuthal component for electric current density and electric field, p = p + 0.5V 2 , are Reynolds, Stuart, Grashof and Hartman numbers, P e = P rRe, P r are Prandtl number and heat source parameter.The hydrodynamic stream function ψ can be determined by relations Eliminating the pressure from Eq. ( 4), one obtains where is the z-component of the vector curlF, is the Jacobian of the functions ψ, v, v = ζ; T , is the derivative in the direction Using the boundary conditions (BCs) of symmetry and periodicity, we can consider only the domain containing quarters of two cylinders, see [9].
For the MHD convection using in-line arrangement cylinders and periodic flow (PCF) we consider the domain (see [2]) Ω = Ω 1 Ω 2 (see Figures 1, 2), where Here are the walls of the cylinders with the non-slip BCs In the case of free MHD convection, ψ 0 = 0.For the additional periodic channel flow with symmetry (CF), is the wall of the half-channel Ω with non-slip BCs, V x = V y = ψ = 0.The cylinders are electrically non-conducting and From .
On the walls we use the following BCs [3]: where m is the number of iterations with ζ 0 = 0, γ > 0 is the parameter, n is the outer normal to the walls.We consider following parameters for electrolyte [8]: The characteristic length scale is L 0 = 0.063[m] (the diameter of cylinder).For flow through periodically placed cylinders arranged in-line, with Gr = 0, similarly to [2], in a space between cylinders the vortexes are created.
In order to obtain the dimensional values, we need to multiply the dimensionless values by the following scalar factors:

Numerical Algorithm for Solution of the Problem
We consider an uniform grid ((N + 1) × M ): Subscripts (i, j) refer to x, y indices with the mesh spacing h.
Equations ( 5) in the uniform grid (x i , y j ) are replaced by difference equations of second order approximation in a 5-point stencil and the numerical calculations are made using Seidel iterations with under-relaxation for vorticity and temperature (see Appendix, where Ψ i,j ≈ ψ(x i , y j ), ζ i,j ≈ ζ(x i , y j ), T i,j ≈ T (x i , y j )).The difference of pressure P = p(l, L) − p(l, 0), p(l, 0) = 0 and integral heat quantity Q T = Ω T (x, y)dxdy are calculated (the maximal value is Q T = 1.500 for maximal dimensionless temperature T ≡ 1).
• If α = ± π 2 , then vortexes and integral heat quantity decrease in the magnetic field; for α = 0 the vorticity increases but the integral heat quantity decreases in the magnetic field.
• Depending on orientation of gravity we have different behaviour of flow: The gravitation in x-direction determine the direction of velocity, temperature and orientation of vortexes in the flow (see , for β = π 2 the central flow is moving in x-direction and the vortex between cylinder rotated clockwise, but for β = − π 2 the central flow is moving in opposite direction and the vortex between cylinder rotated counter-clockwise, similar behaviour can be observed for temperature. • The integral heat quantity for CF convection is greater compared with PCF convection (see Tab. 1).
• In the parallel magnetic field vortices diminish and for α = 0, S = 1000, β = ± π 2 we obtain the vortex-free flow with two opposite flows in x-direction (see Figure 14 The pressure p = p * − 0.5V 2 is calculated using trapezoidal quadrature formula in the direction from the point (0, L) to point (l, L), assumed that p(0, L) = 0.