Adaptation of general purpose CFD code for fusion MHD applications

Analysis of many fusion applications such as liquid metal blankets requires application of Computational Fluid Dynamics (CFD) methods for electrically conductive liquids in the geometrically complex regions and in the presence of a strong magnetic field. Current state of the art general purpose CFD code allows modeling of the flow in complex geometric regions, with simultaneous conjugated heat transfer analysis in liquid and surrounding solid parts. Together with Magneto Hydro Dynamics (MHD) capability general purpose CFD code will be a valuable tool for design and optimization of fusion devices. This presentation describes an introduction of MHD capability into a general purpose CFD code CFX part of the ANSYS Workbench. The code was adapted for MHD problems using magnetic induction approach. CFX allows introduction of user defined variables using transport, or Poisson equations. For MHD adaptation of the code three additional transport equations were introduced for the components of the magnetic field, with additional Poisson equation for electric potential. Lorentz force is included in the momentum transport equation as a source term. Fusion applications usually involve very strong magnetic field with the values of the Hartmann number of up to tens of thousands. In this situation system of MHD equations became very rigid with very large source terms, and very strong gradients of the variables. To increase system robustness, special measures were introduced during iterative convergence process, such as under-relaxation using source coefficient for momentum and magnetic field equations. MHD implementation in general purpose CFD code was tested against benchmarks specifically selected for liquid metal blanket applications. Results of numerical simulations using present implementation closely match analytical solution for the Hartmann number of up to 15000 for two-dimensional laminar flow in the duct of square cross-section, with conducting and non-conducting walls. Results for three dimensional test cases are also included in the presentation.


Abstract-Analysis of many fusion applications such as liquid metal blankets requires application of Computational Fluid
This presentation describes an introduction of MHD capability into a general purpose CFD code CFX part of the ANSYS Workbench. The code was adapted for MHD problems using magnetic induction approach. CFX allows introduction of user defined variables using transport, or Poisson equations. For MHD adaptation of the code three additional transport equations were introduced for the components of the magnetic field, with additional Poisson equation for electric potential. Lorentz force is included in the momentum transport equation as a source term. Fusion applications usually involve very strong magnetic field with the values of the Hartmann number of up to tens of thousands. In this situation system of MHD equations became very rigid with very large source terms, and very strong gradients of the variables. To increase system robustness, special measures were introduced during iterative convergence process, such as under-relaxation using source coefficient for momentum and magnetic field equations.
MHD implementation in general purpose CFD code was tested against benchmarks specifically selected for liquid metal blanket applications. Results of numerical simulations using present implementation closely match analytical solution for the Hartmann number of up to 15000 for two-dimensional laminar flow in the duct of square cross-section, with conducting and non-conducting walls. Results for three dimensional test cases are also included in the presentation.

I. INTRODUCTION
ANSYS CFX is a general purpose CFD code, allowing solving hydrodynamics and heat transfer problems. It is used at PPPL for thermal analysis of the complex systems involving fluid flow and heat transfer in liquids and solids [1]. The code was adapted for MHD problems using magnetic induction approach. The modified code was benchmark tested against fusion relevant cases presented in [2].

II. MODIFICATION OF THE CODE
To include magnetic flux density induction three additional transport equations were added: Lorentz force is added to momentum transport equations as an additional source: Momentum transport equations also use modified pressure: Poisson equation is added to insure solenoidal nature of magnetic field using additional scalar: This scalar is used to project magnetic field obtained by equation (1)  This paper presents validation of the code against some of the test cases presented in [2]:

A. Square Non-Conducting Duct flow
Fully developed flow of conducting fluid subject to uniform transverse magnetic field, walls are non-conducting, case A1. Analytical solution is available for this case [3]. Problem set-up is presented on Fig. 1. Incompressible fully developed flow is considered: The following wall boundary conditions are imposed: where: Very thin boundary layers are created on the channel walls, requiring grid compression towards the wall for proper resolution. Table 1 shows comparison of numerical results with analytical solution. Results show that for higher Ha numbers compression ratio need to be increased to improve accuracy.

B. Square Duct flow with Conducting Walls
Fully developed flow of conducting fluid subject to uniform transverse magnetic field, conducting walls are perpendicular to magnetic field, case A2 in [2]. Walls parallel to magnetic field are non-conducting. Analytical solution is available for this case [4]. Fig. 5 presents problem set-up. Incompressible fully developed flow is considered using conditions (8,9). Boundary conditions are the same as in case A1 (10 -13) except for conducting walls where conditions proposed by Shercliff in [5] are used for part of the magnetic field parallel to the wall: where: -relative conductance of the wall -wall thickness -wall electric conductivity Table 2 shows comparison of numerical results with analytical solution. Results show that for higher Ha numbers compression ratio need to be increased to improve accuracy.  High gradients in near wall layers require very dense mesh. Mesh with compression ratio of up to 20000 was used.
Since source terms have adverse effect on solution stability, under laxation was needed proportional to the source term magnitude and convergence was very slow especially for higher Ha numbers.