Numerical simulations of MHD flow in the wake of a magnetic obstacle in laminar and turbulent flow regimes. Sybren ten Cate

Transcription

1 Numerical simulations of MHD flow in the wake of a magnetic obstacle in laminar and turbulent flow regimes Sybren ten Cate October 20, 2009

2 Abstract Magnetohydrodynamics (MHD), the interaction between an electrically conducting fluid and a magnetic field, offers an interesting cross-point study between electrodynamics and hydrodynamics. Objective of this study is to calculate an MHD flow, affected by a nonhomogeneous localized magnetic field (magnetic obstacle), through a dielectric duct in the laminar (Re = 100 and Re = 400) and a dielectric channel in the turbulent (Re τ = 180) regime, and for different values of the magnetic interaction parameter (N = 4 and N = 11.25). Flow in the turbulent regime is calculated using the large eddy simulation (LES) with dynamical Smagorinsky sub-grid model, and all calculations are performed on multiple processors using the message parsing interface (MPI). Channel turbulence is created on the minimal flow unit, which is also used as a generator for performing turbulent MHD. The results obtained in the laminar regime are compared to direct numerical simulation (DNS) by Votyakov et al. (Phys.Rev.Let vol. 98, 2007), and results obtained for channel turbulence are compared to DNS data by Moser et al. (Phys.Fluids vol. 11, 1998). A fully unstructured computational fluid dynamics (CFD) code was extended for solving the MHD equations in the low magnetic Reynolds number regime (Re m 1). In this report, we compare the laminar MHD flow patterns as calculated by independent codes, we show that our LES simulations closely approximates turbulence DNS data, and demonstrate the effect of a magnetic obstacle on a fully turbulent channel flow. The significance of the agreement in the laminar case is that 1) it validates the prediction of a zero, two, and six vortex pattern as created behind a magnetic obstacle, 2) the effect of the sub-grid model is negligible in this regime, and 3) it validates our parallel MHD solver. The turbulent channel flow as performed 1) validates our parallel solver in the fully turbulent regime, and 2) provides further support for the validity of the LES with dynamical Smagorinsky approach for turbulent flows.

3 Contents 1 Introduction 3 2 Theory Turbulence Magnetohydrodynamics Numerical Method Generic scalar-transport equation Discretization Linear system of equations Parallel computation Solver comparison Turbulence Laminar duct flow in the wake of a magnetic obstacle Computational domain Grid Method Results Channel turbulence Computational domain Grid Method Results Turbulent channel flow in the wake of a magnetic obstacle Computational domain Grid Method Results Conclusions 45 8 Acknowledgement 47 A The Navier-Stokes equation 48 1

4 B Solutions from literature 49 B.1 Laminar duct flow B.2 Magnetic obstacle C Code 54 C.1 Solver C.2 Grid C.3 Visualization

5 Chapter 1 Introduction The electrodynamic problem of a conductor moving in a magnetic field was seminal in physics, in the sense that it was the problem that led Einstein to develop the theory of relativity [4]. However, the problem regained interest in another form: that in which the moving conductor is an electrically conductive fluid. This field is called magnetohydrodynamics (MHD 1 ), and is related to historical problems such as the origin of the earth magnetic field, which is concluded to be caused by a moving liquid metal core, and astronomic phenomena such as northern light and sun flares. The theory describing MHD in full comprises both fluid dynamics and electromagnetism. The field is initiated by Hartmann in his 1937 publication introducing the subject as Hg dynamics. The term magnetohydrodynamics describing this was first used by H. Alfén in Different terms have also been used such as magnetofluiddynamics and hydromagnetics. H.O.G. Alfvén, shared the Nobel prize for physics in 1970 with L.E.F. Néel. Alfvén for fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics, and Néel for fundamental work and discoveries concerning antiferromagnetism and ferrimagnetism which have led to important applications in solid state physics. MHD applies to fluids which are electrically conducting, or ferrous, such as plasmas, liquid metals, or salt water. The characteristic regimes of fluid flow are known as laminar and turbulent, where the fluid flow behaves in a stable and fully unstable way, respectively. This means that in the laminar regime, fluid flows in an ordered way, and in the turbulent regime, fluid motion is chaotic and disordered. The velocity of a fluid flow next to any solid surface takes on the value relative zero, the term for this is the no-slip boundary condition. It is shown that the roughness of surfaces on an atomic scale can explain this condition (Richardson, 1973 [15]). Numerical calculations of the fluid/solid interface result in a general boundary condition (Thompson, 1997 [20]). The limits of this condition are experimentally tested (Zhu, 2002 [26]). The term magnetic obstacle is introduced by Y. Kolesnikov in the 1970 s to indicated the obstacle-like properties that a magnetic field can induce in a flowing conductor. The analogy between a magnetic obstacle and a physical obstacle is discussed in literature [23], and includes the effects of a braking of the fluid velocity, and the formation of recirculations in the wake behind the obstacle. 1 A 1961 educational movie on MHD can be found at 3

6 5 a: N=4, Re=100 5 c: N=11.25, Re=400 y 0 y x x 5 b: N=11.25, Re=100 5 d: exp, N=11.25, Re=2000 y 0 y x x Figure 1.1: Top view of laminar MHD flow patterns, black indicates the magnetic obstacle (Votyakov et al., [24]). Votyakov et al. [24] have performed three-dimensional calculations on the flow pattern of an electrically conducting fluid as it passes a magnet pair. The three-dimensionality requires a larger computational expense as opposed to a quasi-two-dimensional approach 2, however, the latter can not catch the three-dimensional flow patterns, and, important for the present research, it does not seem compatible with turbulent flows. The simulations are performed in the laminar flow regime, as can be seen by the ordered flow patterns in Fig The numerical results presented in the article of Votyakov et al.[24] are selected for use as calibration. The geometry used in this work is illustrated in figure A quasi-two-dimensional case has been studied by Cuevas et al.[3], in which the effect of boundary layers is introduced through a friction term. Due to the localization of the applied magnetic field, this term models either the Hartmann braking within the zone of high magnetic field strength or a Rayleigh friction in zones where the magnetic field is negligible. The arguments they give to justify their method are that studies have confirmed the well-known tendency of uniform cylindric magnetic obstacle flow to become quasi-twodimensional (Mück et al. (2000) and Frank, Barleon, and Müller (2001) Theoretical study by Bühler (1996) of a quasi two-dimensional flat channel flow). 4

7 3 side 2 front conductor (f ) top 4 10 dielectric (s) hard magnet (s) soft ferromagnet (s) 3 Figure 1.2: A sketch of how our geometry may look experimentally. The direction of magnetization of the permanent magnets is indicated by arrows. Note that length scales are dimensionless in this sketch. The scope of this report is the laminar and turbulent flow of a liquid conductor around a magnetic obstacle. One-way coupling between magnetic field and fluid is assumed, which is that of negligible magnetic induction compared to the external magnetic field, i.e. the externally applied magnetic field remains constant. Two assumptions are made concerning fluids in this report: 1) the fluid is incompressible, and 2) the fluid is Newtonian. These assumptions, however, do not drastically limit the range of fluids which can be accurately modeled. For example, most natural fluids are practically incompressible. The assumption of treating Newtonian fluids, means that there exists a linear relation between the stress and strain, and again, this includes most natural fluids. Further, the no-slip boundary condition is assumed for solid/fluid interfaces. All material properties such as mass density ρ, dynamic viscosity µ, and electrical conductivity σ are considered constant in time and space. Goal The goal is to confirm the results of Votyakov et al. [12] and to extend them into the turbulent regime. Turbulent MHD is no new subject because both experimental and numerical work have already been done. One example of such experiment is the magnetohydrodynamic turbulence experiment (MATUR) in Grenoble. A review of low-magnetic-reynolds-number turbulent MHD flow in a uniform magnetic field (Knaepen and Moreau, [7]). However, the extension of the realistic geometry as used by Votyakov et al. into the turbulent regime is a new contribution. 5

8 The goal is thus two-fold: to confirm and to extend on [24]. Confirming requires the calibration simulations of the laminar MHD case after Votyakov et al. Extending requires the benchmark case to be performed for turbulent channel flow without magnetic obstacle, for which that of Moser et al. [10] is used. They have performed direct numerical calculation on the minimal flow unit, which is the smallest turbulence-supporting computational geometry. From the perspective of doing turbulent simulations with an applied magnetic field, the method of choice for solving the flow equation is the large eddy simulation, with the dynamical Smagorinsky model. The applicability of LES with the dynamical Smagorinsky model to perform simulations of magnetohydrodynamics at low magnetic Reynolds number, Re m, has been verified [6, 21]. Laminar MHD research questions Research questions for laminar MHD are: 1. Can different numerical methods agree? It is assumed that fundamentally different methods produce identical results given enough accuracy both in discretization and solver tolerance. 2. Does the flow pattern really contain zero, two and six vortices? Experimental verification is not performed for the exact flow parameters as calculated by Votyakov et al. Turbulent MHD research questions The main research question to be answered is: Can a turbulent flow of conducting fluid be locally controlled by a magnetic obstacle? 6

9 Chapter 2 Theory The Navier-Stokes equation (named after C.L.M.H. Navier, and G. Stokes) for an incompressible Newtonian fluid is written as 1 ρ( t v i + v j j v i ) = i p + µ 2 j v i + f E i. (2.1) The gravitational force is not included in this thesis since a system is considered such that the gravitational force as working on the fluid is canceled by the normal force as exerted by the container. The Navier-Stokes equation (Eq. 2.1) can be nondimensionalized, in which it is shown that any fluid flow can be characterized by a single parameter. It is observed that the SI base units for weight, length, and time can be composed of the characteristic quantities of the system, which are the mass density ρ, a characteristic velocity v 0, a characteristic length scale L 0, and the dynamic viscosity µ. Separating into a dimensional and dimensionless part is achieved by defining the dimensional quantities as p ρv0 2 p, v i v 0 ṽ i, i L 0 ĩ, t (L 0 /v 0 ) t, and the operators as i L 1 0 i and t (v 0 /L 0 ) t, where a tilde indicates nondimensionality. Substituting these definitions and dividing through by ρv0 2/L 0 (a characteristic force density of unit N m 3 ) results in the nondimensionalized Navier-Stokes equation for incompressible Newtonian fluids t ṽ i + ṽ j j ṽ i = i p + 1 Re j 2 ṽ i + f i E, (2.2) where the characteristic dimensionless number is the Reynolds number (named after O. Reynolds who introduced it in 1883 [14]) written as 2.1 Turbulence Re ρv 0L 0 µ. (2.3) Because in this section mention is made of cell volume, the concept of a numerical grid is introduced here. A grid, also known as mesh, consists of a certain division of the geometry on which equations are to be solved numerically. The division of the geometry creates cells, each having a cell center, a finite number of cell faces, and a cell volume. 1 Notation: SI units, index notation, summation convention, Euler notation for derivatives. 7

10 2.1.1 Large-eddy simulation In the turbulent regime, vortical, or eddy motion appears in an incompressible fluid. In calculating such flows numerically, there is a discretization required of such resolution that all of these eddies are properly captured in time and space. Fully resolving every scale of turbulent motion can thus require a large computational expense. The idea of the large eddy simulation (LES) is that a filter is applied to exclude the smallest scales of motion which have a universal behaviour, and to model the effect of these on the resolved velocity field. It is opposed to direct numerical simulation (DNS) which solves the Navier-Stokes equation without additional modeling. LES is the method of solving Eq. 2.1 in the case that a filtering operation has been applied to it. The physical interpretation of this operation is that only flow structures larger than a certain size are solved, and that the influence of any smaller scale flow structures on the resolved ones are modeled. A filtering operation is applied after which the Navier-Stokes equation has retained its structure apart from one difference which is the ρ j v i v j term. It has a benefit of arranging the filtered equation in such a way that the structure of the Navier-Stokes equation is regained with an additional term, which is done by defining the residual stress tensor as τij R v iv j v i v j, and subtracting ρ j τij R from both sides of the filtered equation, which then becomes ρ( t v i + v j j v i ) = i p + µ 2 j v i + f E i ρ j τ R ij, (2.4) which has the structure of the Navier-Stokes equation with an added term which represents the influence of the unresolved eddies on the resolved velocity field. It is observed that as 0, v v and therefore the added term disappears, i.e. LES becomes DNS for sufficiently small grid sizes [5]. This is an occurrence of the closure problem since an additional unknown is introduced. The filtered set of equations contains more unknowns than equations and is therefore called unclosed. It is impossible to solve the set of equations unless one unknown is removed. Closure is achieved by specifying the residual-stress tensor by means of a model, which removes one unknown. This modeled residual-stress tensor is known as the subgrid-scale model (SGS) in reference to the physical interpretation of LES Smagorinsky SGS model The model proposed by Smagorinsky in 1963 is the first and still most widely used SGS model. The reason for its popularity lies mainly with its simplicity [5]. The anisotropic (traceless) residual-stress tensor is the term to be modeled (Pope, [13]), and is τ r ij τ R ij 1 3 τ R ij δ ij. (2.5) The Smagorinsky SGS model is based on the Boussinesq linear eddy-viscosity model, which relates the anisotropic residual stress to the filtered rate of strain as τ r ij = 2ν rs ij, where the eddy viscosity ν r remains to be quantified. This eddy viscosity is then modeled in analogy to the mixing-length hypothesis to be proportional to the characteristic filtered rate of strain S (2S ij S ij ) 1/2 as ν r = l 2 S S, where l S is the Smagorinsky length scale, which is taken proportional to the filter width, with prefactor C S the Smagorinsky coefficient to be 8

11 determined. The definition most widely used for the filter width is = ( V ) 1/3, where V is the volume of a computational grid cell [5], and it is the definition used in this work. In effect, the Smagorinsky model is thus τ r ij τ R ij 1 3 τ R ij δ ij = 2C 2 S 2 S S ij, (2.6) where the only yet undetermined quantity is c s. The value of the Smagorinsky coefficient can be taken as a constant, however, the problems associated with a fixed value of c s are that 1) The value is non-universal, and has to be specified depending on the type of flow considered, 2) In near-wall regions, the value has to be attenuated by means of a damping function, and 3) the transfer of energy from the residual motions to the filtered velocity field (backscatter, or, ν r < 0) is not taken into account Dynamical Smagorinsky model As a solution to the aforementioned problems the dynamical Smagorinsky procedure is proposed by Germano et al. in 1991, where the idea is to use two filters: a grid-scale filter of size, and a coarser test filter of size ˆ >. Residual stresses as based on the filtering operation, and those as based on the test filtering applied to the filtering operation (double filtering) are respectively τ R ij v i v j v i v j (2.7) T ij v i v j ˆv iˆv j, (2.8) where the terms containing v i v j may be removed, by following a procedure due to Germano, as follows. Applying the test filter to the already filtered model τij R and subtracting the result from T ij is defined as L ij (resolved stress) L ij T ij τ ˆ ij R = v i v j ˆv iˆv j (Germano identity), which has the benefit of solely depending on v whereas τij R and T ij do not. The resolved stress L ij represents the resolved stresses of the scales between the grid filter and the test filter. The Smagorinsky model (Eq. 2.6, as written corresponding to filter width ) is written, corresponding to filter width ˆ, as T d ij T ij 1 3 T kkδ ij = 2c s ˆ 2 Ŝ Ŝij, (2.9) where the squared Smagorinsky coefficient CS 2 is replaced by c s to allow for negative real numbers. Applying Germano s procedure to the traceless residual stresses τij r and T ij d returns the traceless resolved stress L d ij, where an approximation to its corresponding Smagorinsky model is simply found subtracting the corresponding Smagorinsky models. Under the definition M ij 2 2 Ŝ S ij 2 ˆ 2 Ŝ Ŝij, the Smagorinsky model for the traceless resolved stress tensor L d ij can thus be written as L d ij L ij 1 3 L kkδ ij T d ij ˆ τ r ij = c sm ij, (2.10) from which the only unknown is c s, solely depending on the known value of v through the functions M ij and L d ij. The value of c s can be determined from this equation, however, 9

12 since tensors are involved, the value of c s can principally take on several different values. It is shown (Lilly, 1992 [8]) that the mean-square error is minimized by specifying the Smagorinsky coefficient as c s = M ijl ij M kl M kl. (2.11) With the dynamical Smagorinsky procedure 1) the value of c s follows from the model, 2) a correct near-wall behaviour is contained in the model, and 3) the coefficient c s is able to become negative, which accounts for the effects of backscatter. The model is well-received, and is reformulated in more consistent forms by the scientific community [12]. For a description of the dynamic model and its implementation into our parallel solver, the reader is referred to the thesis of Hadžiabdić [5] Turbulence statistics The time-averaged quantities of turbulence, also called turbulence statistics, are used to compare calculation to measurement. The quantities used for this purpose are the mean stream-wise velocity, and the mean velocity correlations, called Reynolds stresses. A viscous velocity is defined τw u τ ρ, (2.12) where τ w is the wall shear stress, given as τ w µ (d y u) y=0. (2.13) The profile of the mean stream-wise velocity follows a universal behaviour, known as the log-law. It is displayed in a semi-log plot as in Fig.2.1, where it is shown that for three different values of friction Reynolds number, Re τ, the values follow the same trend. The log law is given as u + = ln(y + )/κ + B (Pope, [13]), where κ is the Von Kàrmán constant (κ 0.41) and B is an integration constant (B 5.2). The near-wall region is dominated by u + = y +. Quantities non-dimensionalized in wall-units are denoted by an upper index +. The procedure is for respectively a length y, a mean velocity u, and a mean velocity correlation uu as follows[13] y + u τ ρy µ, u+ u, uu + uu. (2.14) u τ The friction Reynolds number, Re τ, is defined as the normal Reynolds number, with the velocity u substituted by the friction velocity u τ u 2 τ Re τ ρu τ L 0 µ. (2.15) 10

13 u u + = lny = y + u y + 10 uu ww y + y y + = 590 = 395 = 180 Re Re τ Re τ τ 1.2 vv + 0 uw y Figure 2.1: The time-averaged velocity magnitude u + and Reynolds stresses as function of wall distance. As collected from direct numerical calculation by Moser et al.[10]. On the axes are used variables non-dimensionalized in wall-units. 2.2 Magnetohydrodynamics The equations governing magnetohydrodynamic flow are the Navier-Stokes equation coupled to Maxwell s equations. For the governing equations considered in this report, the assumptions apply as described in the introduction. The Lorentz force as induced by only a magnetic field on a moving electron [9] is f i = Qε ijk v j B k ε ijk J j B k, noting that J i Qv i, and given that Q is the charge density of conduction electrons in the conductor. Since the electrons continuously collide with the atoms in the conductor, this Lorentz force, in effect, works on the conductor. Substituting the Lorentz force as the external force f E i into Eq. 2.1, this produces the MHD equation ρ( t v i + v j j v i ) = i p + µ 2 j v i + ε ijk J j B k, (2.16) where the electric current density is given by the generalized Ohm s law for a conductor moving locally at velocity v relative to a magnetic field. Under the assumptions [11] 1) v c, 2) electron velocity v e v, thus neglecting the Hall effect, 3) electrons move without inertia, thus electric displacement currents are neglected, and 4) no thermo-electric voltage sources are present, this is J i = σ( i ϕ + ε ijk v j B k ), (2.17) where, applying conservation of charge i J i = 0 and E i = i ϕ, the electric potential ϕ is determined by suitable boundary conditions together with 2 j ϕ = i ε ijk v j B k. (2.18) This equation can be nondimensionalized in analogy with that of the Navier-Stokes equation. The procedure is identical, with the difference the SI base unit for electric current is additionally required. It can be composed of characteristic system quantities identical to those present in the definition for the Reynolds number, along with the characteristic magnet field 11

14 strength B 0, and the electric conductivity σ. The additional definitions required for nondimensionalization of the MHD equation are made for the electric current density J i σv 0 B 0 Ji, and for the magnetic field B i B 0 Bi. Substituting the definitions and dividing through by the characteristic force density as before, this results in t ṽ i + ṽ j j ṽ i = i p + 1 Re 2 j ṽ i + Nε ijk Jj Bk, (2.19) where Re is as defined in equation 2.3, and N is the interaction parameter, describing the amount of interaction between the electrically conducting fluid and the magnetic field, and is defined as N σlb2 0 ρu 0. (2.20) To make explicit the fact that the magnetic field remains constant as mentioned in the introduction, the induction equation [11] is considered t B i + v i j B j = 1 µ 0 σ 2 j B i + B j j v i, (2.21) where nondimensionalization will again reveal a characteristic number. The routine procedure, with the exception of dividing through by B 0 v 0 /L 0 (unit T s 1 ) instead, produces where the magnetic Reynolds number is defined as t Bi + ṽ i j Bj = 1 Re m 2 j Bi + B j j ṽ i, (2.22) Re m µ 0 σl 0 v 0. (2.23) It can be seen that in terms of a transport equation, the limit in which Re m tends to zero corresponds to the case in which the transport by convection is negligible in respect to the transport by diffusion. What diffusion of a magnetic field means physically, is that the magnetic field tends to its lowest energy configuration. For the case relevant to the research presented in this thesis, the focus lies with Re m 1. 12

15 Chapter 3 Numerical Method The aim is to solve the MHD equations for low magnetic Reynolds number. These differential equations are transformed, using suitable approximations, into algebraic equations which can then be arranged into the form of a linear system of equations. A discussion of the numerical method is given, as a framework from which to explain the solving of the electrical potential equation. Parts of the existing code which were merely used are excluded from discussion, and will only be mentioned with a reference to either Ničeno s thesis[12] or other sources. 3.1 Generic scalar-transport equation The electrical potential equation will be solved within the framework in which the code was built up, which is based upon a generic scalar-transport equation. For a complemental discussion, the reader is directed to the thesis of Ničeno[12]. The finite volume method is used, which is based on the integral formulation. Divergence terms are written as a surface integration using Gauss theorem. The reason for the preference of the surface integration over volume integration, is that numerically this offers a greater accuracy as a result of involving neighbouring cells into the calculation, as opposed to using only the information from one cell. The transport equation can be derived logically as follows. We start with the unknown being the rate of change. The only two methods of change are production and transport, because if a quantity does not exist it can only be created and if a quantity exists it can be destroyed or transported. If there are carriers present, transport can be achieved by either convection or transfer among the carriers. rate of change = convection + transfer + production. (3.1) This transport equation may have less terms depending on the quantity being transported. For example, since mass can not be produced, can not be transferred from one particle to another, but can only be convected along with particles, the balance becomes: rate of change = convection. The continuity equation ρu j ds j = 0, (3.2) S 13

16 the Navier-Stokes equation including the Lorentz force due to a one-way magnetic field interaction ρ t u i dv + ρu j u i ds j = i p dv + j u i ds j + ε ijk J j B k dv, (3.3) V S V and the differential equation for determining the electrical potential i ϕ ds j = ε ijk v j B k ds i. (3.4) S S These equations are interpreted as sharing the common structure of a transport equation. The generic scalar-transport equation is introduced in the following form t ρb ϕ dv + ρu j ϕ ds j = Γ ϕ j ϕ ds j + q ϕs ds j + dv, (3.5) V } {{ } Inertia S } {{ } Convection S S } {{ } Diffusion S V V q ϕv } {{ } Sources where ϕ is the scalar field, B ϕ is the inertial constant, and Γ ϕ is the diffusion constant, and q ϕs and q ϕv are the surface and volume source constants, respectively. The abstraction of using one generic equation can benefit both the theoretical treatment, as well as the programming. Field ϕ B ϕ Γ ϕ q ϕs q ϕv Mass density Momentum density u i 1 µ µ i u j ε ijk J j B k Electric potential ϕ 0 1 ε ijk u j B k 0 Table 3.1: Constants for the generic scalar-transport equation. The reason for treating the viscous stress tensor as two different contributions lies in computational economy, and is known as the segregated approach[12]. The dimensions of the electric potential equation are verified as follows (gradient adds unit m 1 )[18] diffusion term V m 1 m 2 = m 3 kg s 3 A 1 (SI base units) source term m s 1 T m 2 = m 3 kg s 3 A 1 (SI base units). Since there is no unsteady term in Eqt.3.4, Incomplete Cholesky preconditioning is used, and the number of inner iterations for the electric potential solver is set to a value comparable to that used in calculating the pressure field [19]. 3.2 Discretization The T-FlowS solver discretizes the governing equations with second-order accuracy in time and space[12], where, in order to allow unstructured and hybrid grids, a collocated data structure is used. Equation 3.5 is evaluated on polyhedral cells by means of the finite-volume 14

17 method. Because the creation of the MHD extension requires the transport equation to be solved for the case of the electric potential, an overview of the discretization approach used in T-FlowS follows, with specific attention for the treatment of the diffusion term. Because the values are only kept in the cell center, calculating the integrals exactly requires the values at different locations in the cell to be calculated. For keeping these calculations to a minimum, the mid-point rule is applied for approximating these integrals. For volume integrals this means that the value in the cell center is multiplied with the cell volume ϕ(r) dv ϕ(r 0 ) V. (3.6) V Assuming that all information concerning the cells is known, this approximation makes volume integrals straight-forward to evaluate. For surface integrals the mid-point rule means that the value at the surface center is multiplied with the corresponding surface area ϕ(r) ds ϕ(r i ) S i. (3.7) i S Again assuming the information related to the cells is known, to evaluate this still requires the calculation of the values at the face centers. This is done by taking the average of the first-order Taylor expansions of the two neighbouring cell centers at the face separating them ϕ(r i ) [ ] ϕ(r A ) + ϕ(r A ) (r i r A ) [ ] + ϕ(r B ) + ϕ(r B ) (r i r B ). (3.8) 2 In evaluating equation 3.8, the only unknown value remaining is the gradient of the field in the cell centers. For evaluating these gradients, T-FlowS uses a least-squares method[12]. Diffusion term The only non-trivial term to be discretised for solving the electrical potential equation is the diffusion term from equation 3.5, which, using equation 3.7, is written as Γ ϕ j ϕ(r) ds j Γ ϕ i ϕ(r j ) S i. (3.9) i S The unknown to be approximated is i ϕ(r j ): the gradient of ϕ at the face center. i ϕ i ϕ + ϕ(r B) ϕ(r A ) x i 3.3 Linear system of equations The linear system of equations for velocity is + x j j ϕ x i. (3.10) A ij u j = b i, (3.11) 15

18 where A ij is called the system matrix, u j is the vector to be solved for, and b i is called the solution vector. All implicit terms are collected in the system matrix while all explicit terms contribute to the solution vector. Solving the linear system of equations is done via an iterative process in which the (bi)conjugate gradient method is used. As an indication of the accuracy of the obtained solution of this system, the residuals can be used. These represent the amount of change of the current to the previous estimate solution. Usually, when the residuals are two orders of magnitude smaller than the predicted values they correspond to, this indicates that the results can be trusted. However, for the electrical potential, a much stricter criterium has to be used as is illustrated in figure tolerance 1E-1 1E-2 1E-3 1E-4 1E-5 1E-10 line x/x 0 Error Error comparison e-04 1e-06 1e-08 1e-10 1e-12 1e-14 Phi 1e-16 1e-18 1e-12 1e-10 1e-08 1e-06 1e Tolerance Figure 3.1: As a test to determine the necessary convergence criterium for the electrical potential solver, a case was performed in which the diffusive term without a source term was evaluated. The boundary conditions were set such that a steady-state diffusion should be produced with an expected straight line behaviour. As can be seen, the accuracy of the solution needs to be relatively high for the diffusion term in order to trust the predicted result. 3.4 Parallel computation Since a certain resolution is required for the discretization step in order to obtain accurate solutions, the computational time increases rapidly. One way to increase calculation power is to split the problem over multiple processors. All simulations are performed in parallel using four processors. For calculating on multiple processors, the Message Parsing Interface (MPI) is used. Communication using MPI is limited to nearest neighbours only to keep overhead memory for buffer regions low, and to keep the procedure more straight-forward[12]. The domain is decomposed into four sub-domains from the fact that communication between subdomains forms the bottleneck for computational speed increase, and four seems to be the optimum for speed increase. The procedure of parallel computation is as follows. The domain is divided into a number of sub-domains, see figure 3.2 for one example of such division. 16

19 process: 1 process: 2 process: 3 process: 4 Figure 3.2: A computational domain divided into four sub-domains. Note that the method of division ensures an equal number of cells per sub-domain, this is because for computational speed, the size of the cells is not important but the amount is. The reason why process four occupies a larger part of the domain, for example, is because larger cells are contained in that region. During execution, each sub-domain is treated by a single processor, which can only access the cells within its sub-domain. To treat gradients near the boundaries of the sub-domain, it is however necessary to have information of the corresponding neighbouring sub-domains. This is due to the method of calculating cell-centered gradients. Note that the processors work independently by default and that communication between them requires commands to be given. Data files are written per sub-domain individually, and connected during postprocessing to form a data set spanning the complete domain. Boundaries are left untouched, except for the creation of buffer boundaries. These buffer areas are filled during the exchange of values between sub-domains. The importance of these boundaries is illustrated in figure 3.3. Figure 3.3: The effect of overwriting the buffer boundaries during parallel computation. The predicted velocity field is shown in the upper half, the lower half shows the corresponding domain with indicated sub-domains. The effects occur at the interface between the two sub-domains, indicating that it is a problem due to parallelization. Certain tasks are to be performed approximately simultaneously by each processor. Such as, for example, the writing of data files on manual command. This method of issuing a write command while the code is running is done by the creation of a file. The code tries to read this file at the end of every time step, and depending on its success in doing so, starts a writing sequence. This sequence involves an MPI call to collect all processors involved before 17

20 proceeding to write. What can happen is that, because processor speeds may vary, some processors arrive at the point of the code earlier than others. The consequence of this is that, if the file is manually created at that moment at which some processors are to arrive at the point of the code, and other processors already passed it, the faster processors will wait in another part of the code where also a wait statement is given. In this way, both groups of processors are waiting for each other, and the code freezes. 3.5 Solver comparison Because the laminar MHD case requires a comparison of data between separate codes, a solver comparison is in order here. The non-dimensionalized Navier-Stokes equation is directly solved for by DNS method in the code employed by Votyakov, in our case, the equations to solved are chosen to be the filtered dimensional Navier-Stokes equations solved for by LES method using dynamic Smagorinsky method. In table 3.2 a comparison between NaSt3DGP 1 and T-FlowS 2 is made. Feature Votyakov et al. Present study Solver Name a NaSt3DGP T-FlowS Language C++ Fortran 90/95 Parallelization MPI MPI Data structure Staggered Collocated Discretization technique Finite difference Finite volume Governing equations b Dimensionless Dimensional Solver settings Solver type DNS LES, dynamical Smagorinsky Discretization scheme VONOS CDS Time discretization explicit Adams-Bashforth Fully Implicit, parabolic interpolation Matrix preconditioning Incomplete Cholesky Scheme for p-v coupling SIMPLE Table 3.2: Numerical method comparison. a, The names correspond to the base solvers which were extended for solving MHD equations. b, The governing equations include in both cases Eq.2.17,2.18 and??. As a dimensionless governing equation is meant Eq.2.19, and as a dimensional one Eq Turbulence For performing a turbulent simulation, mass density and geometry half-height are set to unity. A way to obtain the Reynolds number of our choice, the viscosity is changed. This leads to not necessarily realistic material properties, however, it is expected that this is no immediate problem since flow behaviour is believed to be characterized by the Reynolds number alone. 1 Data based on NaSt3DGP s online manual[1]. 2 Data based on Ničeno s PHD thesis[12]. 18

21 This was left unchanged because it is not clear at exactly which points corrections would need to be made in order to generalize this, and the motivation of working this way was not known. After noting that this assumption occurred several times, it was decided to specify our problem in these terms. 19

22 Chapter 4 Laminar duct flow in the wake of a magnetic obstacle Goal is to construct and test a solver for one-way coupling of magnetic field on electrically conducting fluid for Re m 1. It is expected that the results of Votyakov and me will be comparable, even though our numerical methods differ, because identical equations are solved. Any differences can be contributed to the following possibilities 1) The distance of the inlet plane is 7.5 cm away from the center of the magnetic obstacle, and this distance should ideally be infinite. The effect of having a longer upstream region is expected to be a lower velocity inside the magnetic obstacle. 2) The lack of accuracy, such as grid independence or convergence criteria. 3) Boundary conditions at outlet. 4.1 Computational domain Since numerical validation will follow from comparison with [24], the geometry and conditions have to be equal in order to produce equal results. The case used is thus the rectangular duct with centered magnet pair, described in more detail in [22], with the difference that the center of the magnet pair is shifted upstream to 7.5 cm from inlet in order to capture the flow pattern in the wake region. 20

23 a b side 3 cm 2 cm 3 cm 7.5 cm top 4 cm 10 cm 3 cm 50 cm Figure 4.1: a, Sketch illustrating the magnet-pair induced flow pattern in the plane of our interest. b, Two projections of the geometry used in the numerical calculations; hatching indicates the magnet pair. The duct consists of solid walls on all four sides, one inlet, and one outlet. The walls are electrically insulating, and the fluid is electrically conducting. The thickness of the walls does not influence their electrical properties (Müller and Bühler, [11]). The center of the magnetic obstacle is placed 7.5 cm from the inlet of the duct, and the magnetic field is modeled as that of two magnets enclosed in a soft iron yoke. 4.2 Grid The amount of nodes is chosen to be doubled in each direction with respect to the amount of Votyakov et al. due to the lower-order numerical scheme used. The number of nodes is compared to Votyakov s 64 3, where the power-three indicates that this is a rectangular grid with an equal amount of nodes in each direction. The need for more(less) nodes corresponds to using lower(higher)-order numerical schemes. A rectangular non-uniform grid was used after Votyakov. 21

24 Figure 4.2: A full depiction of the numerical grid used. The grid used is chosen to correctly resolve the Hartmann layers, following the function describing the node distribution [22] [ ] L x arctanh r tanh(r) x = R [ ] 2L y arctan s tan(sπ/2) y = (4.1) [ Sπ ] 2L z arctan t tan(t π/2) z =. T π The values used were not available from published writing [24, 25, 22], so they were determined directly from the grid points as received from the first author[19]. The procedure to obtain these values was as follows. The node values were mapped onto a range [-1,1] and evenly distributed over a domain [-1,1]. The node-distribution functions were then fit to the data points using gnuplot. The resulting fits are shown in figure 4.3, where the dummy variables on the axes are chosen as alpha and beta [ ] arctanh α tanh(r) x : β = y : β = z : β = [ R ] 2 arctan α tan(sπ/2) [ Sπ ] 2 arctan α tan(t π/2), α, β [ 1, 1]. T π 22

25 x: R=1.25 y,z: S=T=0.85 y,z: S=T=0.75 grid 1,2 x grid 1 y,z grid 2 y,z Figure 4.3: Fit to obtain the function parameters as used by Votyakov et al. Two rectangular grids were received referred to as grid 1 and grid 2, which were mentioned to have produced identical results[19]. Note that in this mapping the y and z directions are equal. Due to this, also the values S and T are equal. Both grids have identical stream-wise node distributions. 23

26 xz,xy yz Figure 4.4: Two mappings of the reduced grid (32 3 cells shown instead of ) onto the unit square for clarity. Note that the top view xz and the side view xy share one mapping. The grid used consists of cells, and has nodal distribution function (Eq.4.1) parameters R = 1.25, S = T = 0.85, L x = 42.5, and L y and L z equal to the half width and half height of the duct, respectively. The value of L x different from the half length of the duct is due to the shifting of the magnet pair from a centered position to the position 7.5 cm downstream of the inlet. To have the grid refinement remain around the magnet pair, the domain of the stream-wise distribution function (x(r) in Eq. 4.1) is linearly mapped from r [ 1, 1] onto r [ r min, 1]. Setting L x to (50 7.5) cm = 42.5 cm, the value of r min is calculated by substituting the relevant values into the inverse function r(x): r min = tanh[1.25( 7.5/42.5)]/ tanh(1.25) Method The inflow boundary condition is supplied with Eq. B.1, for which the mean velocity (the bulk velocity) is set to the correct value by using the definition of the Reynolds number Eq. 2.3 and the analytical mean value of the two-dimensional inflow profile Eq. B.2. The outflow boundary is set to a zero-gradient condition. The initial velocity field is also set to the analytical profile in order to increase the initial convergence rate. Physically, this set of initial conditions represents a laminar duct flow onto which a magnetic field is applied instantly. The boundary conditions for the electric potential are such that n ϕ wall = 0 (insulating walls), for the inlet and outlet boundary the electric potential is set zero ϕ in,out = 0. 24

27 Feature Material properties Value Case A Case B Case C Material Ga 0.68 In 0.20 Sn 0.12 (Galinstan) Mass density, ρ (kg m 3 ) Dynamic viscosity, µ (Pa s) Electrical conductivity, σ (S m 1 ) Solver settings Iterations per time step 10 Under-relaxation factor a (v, p, ϕ) (0.3, 0.1, 1) (0.5, 0.2, 1) (0.2, 0.1, 0.9) Tolerance (v,p,ϕ,simple) (10 13,10 11,10 15,10 10 ) Time step b, t (s) Conditions Reynolds number, Re (-) Magnetic interaction parameter, N (-) Derived values Inverse magnetic field strength c, B 0 (T) Bulk velocity, u 0 (m s 1 ) Table 4.1: Solver settings for the laminar test cases A, B and C. Material properties are taken as in article, and are verified to be taken at standard temperature and pressure (STP). Some specification of the data presented in table 4.1 is given. a, The under-relaxation factors (URF s) for case B are higher than those for case A. It is expected that case A also converges well for the URF s of case B. For case C convergence could only be reached with the URF s as specified. b, In calculating the different cases, time stepping has been disabled since it is expected that the final solution will be time-independent. This is done by setting the time step to 10 10, which effectively removes the time-derivative from the equation. Doing this has increased convergence speed for case A and B, but has caused divergence for case C. Case C uses a five per cent of the bulk flow-through time, using the stream-wise domain length is 50 cm.c, It should be noted that 1/B 0 is the prefactor to the magnetic field, which means that a smaller value for B 0 corresponds to a larger magnetic field. 25

28 4.4 Results Votyakov et al. (2007 ) P r e s e nt Ca s e A Re = 100 N =4 Ca s e B Re = 100 N = Ca s e C Re = 400 N = Z (cm) X (cm) Figure 4.5: A qualitative comparison of the flow patterns as produced in the central horizontal plane. 26

29 Present case A Present case B Present case C Votyakov et al. (2007) case A Votyakov et al. (2007) case B Votyakov et al. (2007) case C 0.6 v x /v x-x 0 (cm) Figure 4.6: Comparison of stream-wise velocity magnitude along the duct centerline. For case B, the simulation was performed as a direct numerical simulation. The result of this was identical to the result as shown above, proving that the contribution of the sub-grid model is negligible in this case. 27

30 Figure 4.7: The velocity magnitude together with stream lines for cases A, B and C (top to bottom) in the central horizontal plane. A grid refinement study was performed until case A showed good agreement with the data presented by Votyakov et al. Any disagreement between the two numerical studies as compared in Fig. 4.6 remain to be investigated by means of experiment. 28

31 Figure 4.8: Electric potential and pressure fields for cases A, B and C (top to bottom). In the same plane as in Fig The pressure is taken relative to a gauge pressure. A discussion of the observed behaviour follows. The obstacle-like effect of the magnetic field on the laminar flow is clearly observed in case A. This observation already justifies the existence of the term magnetic obstacle, since it proves to be a useful way to consider the effects of such a localized magnetic field. The recovery region (behind the magnetic obstacle) of velocity as seen in all cases considered here indicates that the magnetic field has a very localized influence. In case B, a small plateau region is observed behind the magnetic obstacle. The back flow inside the magnetic obstacle is a feature which is seen to be conserved up to Re = 400, and it only appears at a certain value of the Reynolds number. This indicates that the back flow is in part due to inertial effects. It can be understood that a back flow is necessary since the magnetic obstacle creates two jets of fluid on either side of the magnetic obstacle. The jets have an increased velocity which can be understood in terms of mass conservation and the obstacle-like effect of the magnetic field. When these jets are deflected by the walls, the wake region is subjected to such a fluid flow for which back flow is a necessary effect. The second back flow in case C appears due to the increasing of the magnetic field strength. The difference between case B and C is furthermore seen to be completely due to the post-magnetic-obstacle region (wake region) since the first back flow is almost identical in both cases. This suggests that the magnetic field is strong enough to influence not only the flow inside the magnetic obstacle, but also the flow inside the wake region. The antisymmetry of the electric potential field may be attributed to the antisymmetry of the x and z components of the magnetic field, since the y component of the magnetic field, the geometry and velocity inflow are completely symmetrical. It is seen that a positive and a 29

32 negative potential are located on the walls surrounding the magnetic obstacle. This voltage difference may be a measurable effect, which can determine the flow strength (principle of a magnetic flow meter). The pressure field is seen to increase in front of the magnetic obstacle. Interesting is that the pressure drop over the magnetic obstacle increases most due to increase of Reynolds number (case B to case C), and not considerably due to an increase in magnetic field strength (case A to case B). 30

33 Chapter 5 Channel turbulence The existing solver for turbulence is used to reproduce a benchmark turbulence case. 5.1 Computational domain π span normal stream 2 π/2 Figure 5.1: The minimal flow unit. In stream- and span-wise directions the geometry is periodic. The minimal flow unit is used here. It is the smallest numerical domain which can support turbulence, and is widely documented, both experimentally and numerically. The geometry consists of a rectangular numerical domain, having two opposite no-slip boundaries, and for the remaining boundary conditions periodicity in their respective directions. 31

34 5.2 Grid Y X Z Figure 5.2: A full depiction of the numerical grid used. A proper grid distribution is vital for the solution of wall-bounded turbulence. correctly resolving the near-wall behaviour, self-sustained turbulence can not arise. Without 32

35 xy xz,yz Figure 5.3: Two mappings of the full grid (64 3 cells) onto the unit square for clarity. Note the top view xz and the front view yz share one mapping. The grid nodes are uniformly distributed in the stream-wise and span-wise directions. In the wall-normal direction, the nodes are clustered so as to correctly resolve the near-wall regions. The clustering is specified by setting the distance from the wall to the node closest to the wall, the number of nodes to be distributed, and using a successive ratio for cell sizes. The size of the first cell at the wall is calculated as follows. Using Re τ = 180 and z + = 0.5 [17], making use of the definition of the wall unit Eq. 2.14, the distance in the wall-normal direction of the cell center closest to the wall is z m. 5.3 Method The settings used in generating a turbulent channel flow are described in table

36 Feature Imposed quantities Value Mass density, ρ (kg m 3 ) 1 Channel half height, L (m) 1 Friction velocity, u τ (m s 1 ) Friction Reynolds number, Re τ (-) 180 Turbulence solver settings Iterations per time step 10 Under-relaxation factor (v, p) 0.5, 0.5 Tolerance (v, p, SIMPLE) 10 5, 10 5, 10 4 Perturbation Random multiplication a Derived material properties Dynamic viscosity b, µ (Pa s) Derived solver settings Pressure gradient c, x p (kg m 2 s 2 ) Bulk velocity d, u 0 (m s 1 ) Time step e, t (s) Table 5.1: Settings to generate turbulent channel flow for Re τ = 180. The values of the Reynolds number Re τ is considered to fully characterize any non-mhd flow, and therefore the realism of the set of material properties is not evaluated. Some details concerning the derivation of certain quantities as indicated in table 5.1 follows. a, Perturbation of the velocity field is applied every 100 time steps until time step 1000, and comprises a random multiplication of each velocity component per cell with a maximum magnitude of 5% of the largest component currently in the cell. The perturbation is applied to only the first time step of each set of a 100, after which the remaining 99 time steps are used to relax the fluid. b, Using the imposed values and the definition of the friction Reynolds number 2.15 this value is derived to be exactly 1/ Pa s of which a decimal representation with four significant digits is used. c, The pressure gradient is calculated from the exact relationship x p = ρu 2 τ /L (S.B. Pope, [13]) making use of the imposed quantities as specified. d, Using the definition of the Reynolds number (Eqt.2.3) together with the approximate relation Re (Re τ /0.09) (1/0.88) (S.B. Pope, [13]) and imposed values as mentioned above, the bulk velocity is derived. e, Five per cent of the flow-through time (defined as the time it takes the bulk to travel the stream-wise length of the geometry) is taken as a time step. 34

37 2 1.5 v / v v x center v y center v z center Figure 5.4: The instantaneous velocity components as calculated in the central point of the geometry. Several regions can be distinguished according to Fig. 5.4: 1) perturbation of the velocity field (flow-through times 0 50), 2) build up (flow-through times ), in which no perturbation is performed and the velocity field is allowed to develop. 3) onset of turbulence (flow-through time 150), and 4) development into a statistically stable turbulent flow (flowthrough times ). 5) The collecting of mean velocity field quantities is performed from the moment that stable turbulence is reached, and until the moment that mean fields seem to be converged to their final values (flow-through times ). 5.4 Results Because turbulence is time-dependent, a comparison of instantaneous features is not made, instead, the time-averaged values are compared. The mean velocity and Reynolds stresses are compared with the data shown in figure??. 35

38 DNS 20 u Figure 5.5: The mean velocity along the wall-normal direction. The numbers indicate the amount of averaging steps taken, where each step corresponds to a number of one thousand time steps. Multiple averaging results are displayed together to show the convergence of the statistics. z + 36

39 8 0.8 uu + vv + 0 y y ww + uw y y + Figure 5.6: The Reynolds stresses calculated displayed against the DNS data (symbols). Z x= y= Y Z X v Figure 5.7: Instantaneous field of mean velocity in Re τ = 180 turbulence. The deviation from DNS results in Fig. 5.5 is seen to only appear in the region away from 37

40 the wall. Considering that the grid is the coarsest in this region, this deviation is attributed simply to grid coarseness. 38

41 Chapter 6 Turbulent channel flow in the wake of a magnetic obstacle The idea in applying magnetic forcing to an electrically conducting fluid at high Reynolds number, is to extend the published numerical results[24] into the turbulent regime. 6.1 Computational domain Requirements to perform such a calculation are to have a fully developed turbulent flow on which the magnetic forcing can act. Here, a two-domain geometry can be used, having a periodic first domain, which projects it values onto the inlet of the second domain. The first domain will be used for producing fully developed turbulence, and the magnetic forcing will only affect the second domain. 4x 50 m top 2.5 m 4 m 10 m 3 m 50 m side 2 m 2 m 3 m Figure 6.1: A schematic illustration of the two domains. Two projections of the geometry used for generating the fully-developed turbulent flow which serves as an inflow condition to the second domain. The first domain has periodic boundaries, and, because the second domain also has periodicity in the span-wise direction, can be projected multiple times onto the second domain s inlet. 39

42 a π b 2 span normal 2 stream π/2 2.5 π Figure 6.2: The modification of the minimal flow unit. a, the minimal flow unit (cf. chapter 5). b, the modified domain used here as a generator of turbulent channel flow. As a generator of a fully turbulent flow, a modification of the minimal flow unit is used. The domain is slightly extended in the span-wise direction in order to be projected seamlessly onto the second domain four times. 6.2 Grid Figure 6.3: A full depiction of the numerical grid used. 40

43 z x process: 1 process: 2 process: 3 process: 4 Figure 6.4: The top view of the generated grid after division amongst four processors. The reason for different size of sub-domains is discussed in figure 3.2. Laminar MHD Turbulence Present case Domain one Domain two stream-wise (50/128) (3.1416/64) (3.1416/16) (50/64) span-wise (10/128) (1.5708/64) (2.5/16) (10/64) wall-normal (2/128) (2/64) (2/128) (2/128) Table 6.1: Comparison of grid coarseness. The format used is: numerical value (length scale/number of grid cells). In the case of the duct geometry, the z direction is referred to as span-wise direction, and the y direction is referred to as wall-normal direction here to have easy comparison. From table 6.1 it is seen that the two-domain grid as used here is coarser than both the grid for laminar MHD, and the grid for channel turbulence. 41

44 6.3 Method Feature Imposed quantities Value Mass density, ρ (kg m 3 ) 1 Electrical conductivity, σ (S m 1 ) 1 Channel half height, L (m) 1 Friction velocity, u τ (m s 1 ) Friction Reynolds number, Re (-) 180 Magnetic interaction parameter, N (-) Turbulence solver settings Iterations per time step 10 (20 with magnetic obstacle) Under-relaxation factor (v, p, ϕ) 0.5, 0.5, 0.9 Tolerance (v, p, ϕ, SIMPLE) 10 5, 10 5, 10 5, 10 4 Perturbation Random multiplication Derived material properties Dynamic viscosity, µ (Pa s) Derived solver settings Pressure gradient, x p (kg m 2 s 2 ) Bulk velocity, u 0 (m s 1 ) Magnetic field strength a, B 0 (T) Time step b, t (s) Table 6.2: Settings to generate two-domain turbulent MHD flow for Re τ = 180. The values of the Reynolds number Re and magnetic interaction parameter N are considered to fully characterize any MHD flow, and therefore the realism of the set of material properties is not evaluated. Several quantities as indicated in table 6.2 are now explained. For quantities not derived here, please refer to table 5.1. a, Based on Eqt.2.20, and the imposed quantities as mentioned above, the magnetic field strength is calculated. b, The stream-wise length of the first domain ( m) is used. 42

45 6.4 Results Figure 6.5: The turbulent field without any magnetic field present. Top figure shows the secondary motion in a vertical slice in x = 29 (center of second domain). The bottom figure shows the top view on the central horizontal plane (y = 0). It is observed that the projection method from domain one onto domain two works properly. Although flow develops independently in the second domain, the periodicity due to the first domain remains visible. The upper figure of Fig. 6.5 clearly shows the periodicity of velocity magnitude, and even the secondary motions close to the wall as well as in the center, where mixing is expected to occur on a faster time scale are seen to retain something less than, but still close to periodicity. However, these effects are not considered to be vitally important, as long as the velocity field satisfies the governing equations. Figure 6.6: The turbulent field with magnetic obstacle located in the second domain. Top figure shows the secondary motion in a vertical slice in x = 29 (center of second domain). The bottom figure shows the top view on the central horizontal plane (y = 0). 43

46 Figure 6.7: The pressure and electric potential fields in the central horizontal plane. The influence of the magnetic obstacle (Fig. 6.6) immediately disrupts the seeming periodicity as expected. A discussion of the results follows. It is observed that inside the magnetic obstacle all turbulence is suppressed. Two jets are created as in the laminar cases, and within the head of the jets (close to the magnetic obstacle), secondary motion is completely suppressed. The turbulence reappears inside the wake region. The pressure distribution (Fig. 6.7) around the magnetic obstacle is seen to be identical in shape to the one as found for laminar cases. The pressure drop over the magnetic obstacle, however, is larger, which is in line with the observation as made before that the magnitude of the pressure drop depends most strongly on the Reynolds number, and in to lesser degree on the magnetic interaction parameter. For this case, the full domain is shown for both pressure and electric potential field due to the fact that in turbulence the whole domain remains interesting. Comparing the electric potential (Fig. 6.7) as found for the turbulent case here, to that as found for the laminar cases, it is observed that the overall structure looks the same. The difference is that in this turbulent case, there are no side walls. The reason for the smooth distribution of the electrical potential field is the suppression of turbulence in the region of strong magnetic field. The reason for negligible electrical potential field in the wake region is that the magnetic field strength is negligible in that region. Conditions which apply under which the results obtained from this grid will are considered acceptable are that 1) turbulence is self-sustained, which means that the grid is fine enough to support a coarse version of a turbulent flow, and 2) the solution converges and satisfies the governing equations. 44