A numerical approach for plasma based flow control

Abstract

In the present study, a novel numerical method has been developed to solve incompressible magnetohydrodynamics (MHD) and electrohydrodynamics (EHD) flow problems in a parallel monolithic (fully-coupled) approach. To solve the fluid flow, incompressible Navier-Stokes equations are discretized using face/edge centered unstructured Finite Volume Method (FVM). The same formulation is used for the magnetic transport equation to model the magnetic effects. The side-centered approach, where the velocity and magnetic field components are placed at the center of each cell face while pressure and Lagrange variables are placed at the center of the control volume, provides a stable numerical algorithm without the need of modifications for pressure-velocity coupling. The discretization of both MHD and EHD equations described above results in saddle point problem in fully coupled (monolithic) form. In order to solve this problem an upper triangular right preconditioner is used and restricted additive Schwarz preconditioner with FGMRES algorithm is employed to solve the system. Domain decomposition is handled by METIS library. For these numerical algorithms PETSc software package is used. For the solution of incompressible MHD flow problems, the continuity, incompressible Navier-Stokes, magnetic induction equation are solved along with the divergence free condition of magnetic field. Due to the interaction between magnetic field and conducting fluids, Lorentz force term is added to the fluid momentum equation. For the numerical stability, a Lagrange multiplier term is used in the magnetic induction equation, which has no physical meaning nor effect on the solution. The original approach satisfies the mass conservation within each element but it is not necessarily satisfied in the momentum control volume. Two modifications are proposed as a remedy. First, the convective fluxes are computed over the two-neighbouring elements which then resulted in improved mass conservation over the momentum control volume and increased stability. The second modification applies to only two-dimensional MHD flows. The Lorentz force term in the momentum equation is replaced with $\sigma [\textbf{E} + \textbf{u} \times \textbf{B}] \times \textbf{B}$. Neglecting $\textbf{E}$ makes this term similar to mass matrix if $\textbf{B}$ is taken from the previous time step. Therefore, this modification improves the preconditioning of the monolithic approach. The developed solver is first validated for two-dimensional Hartmann flow of which the analytical solution is known. Then lid-driven cavity and backward facing step problems are investigated under external magnetic field both in 2D and 3D with insulating walls. Three-dimensional MHD flow in ducts is another case where analytic solutions exist. Both conducting and insulating wall boundary conditions are employed and validated. Finally two-dimensional flow over circular cylinder and NACA 0012 profile are investigated for vertical/horizontal external magnetic field and insulating/conducting boundaries. The eletrohydrodynamics (EHD) flow problems involve the interaction between electric field and charged particles inside the fluid. In the present study, the effect of plasma on the flow over lifting bodies is investigated and the working fluid is air, which is neutral at standard conditions. Therefore, a device called Dielectric Barrier Discharge (DBD) is used to ionize the air in a small volume near the surface. DBD consists of two electrodes separated by a dielectric layer. When a voltage is applied to the electrodes, ionization takes place. In order to simulate this phenomenon, Suzen\&Huang model is employed in which Poisson equation is solved for electric potential and charge density, separately. Once potential and charge density are known Coulumb force can be calculated and added as a body force term in the incompressible Navier-Stokes equation. The side-centered approach is used for the velocity components and pressure is placed at the element center for the momentum and continuity equations. For the solution of Poisson equation the charge density and electric potential are placed at the element center while gradients are defined at the edge centers. The solver is first applied to an EHD flow in quiescent air and compared with both experimental and numerical solutions. Then, two electrodes are placed at the bottom wall of 2D cavity with a moving lid to investigate the effect of electric field on classical cavity problem. Finally, EHD flow over NACA 0012 airfoil at angle of attacks up to $\alpha=7$ is investigated in terms of flow structure, lift and drag coefficients.