Gustaaf Jacobs

High-order nodal discontinuous Galerkin particle-in-cell method on unstructured grids

We present a high-order particle-in-cell (PIC) algorithm for the simulation of kinetic plasmas dynamics. The core of the algorithm utilizes an unstructured grid discontinuous Galerkin Maxwell field solver combining high-order accuracy with geometric flexibility. We introduce algorithms in the Lagrangian framework that preserve the favorable properties of the field solver in the PIC solver. Fast full-order interpolation and effective search algorithms are used for tracking individual particles on the general grid and smooth particle shape functions are introduced to ensure low noise in the charge and current density. A pre-computed levelset distance function is employed to represent the geometry and facilitates complex particle-boundary interaction. To enforce charge conservation we consider two different techniques, one based on projection and one on hyperbolic cleaning. Both are found to work well, although the latter is found be too expensive when used with explicit time integration. Examples of simple plasma phenomena, e.g., plasma waves, instabilities, and Landau damping are shown to agree well with theoretical predictions and/or results found by other computational methods. We also discuss generic well known problems such as numerical Cherenkov radiation and grid heating before presenting a few two-dimensional tests, showing the potential of the current method to handle fully relativistic plasma dynamics in complex geometries.

A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks

A high-order particle-source-in-cell (PSIC) algorithm is presented for the computation of the interaction between shocks, small scale structures, and liquid and/or solid particles in high-speed engineering applications. The improved high-order finite difference weighted essentially non-oscillatory (WENO-Z) method for solution of the hyperbolic conservation laws that govern the shocked carrier gas flow, lies at the heart of the algorithm. Finite sized particles are modeled as points and are traced in the Lagrangian frame. The physical coupling of particles in the Lagrangian frame and the gas in the Eulerian frame through momentum and energy exchange, is numerically treated through high-order interpolation and weighing. The centered high-order interpolation of the fluid properties to the particle location is shown to lead to numerical instability in shocked flow. An essentially non-oscillatory interpolation (ENO) scheme is devised for the coupling that improves stability. The ENO based algorithm is shown to be numerically stable and to accurately capture shocks, small flow features and particle dispersion. Both the carrier gas and the particles are updated in time without splitting with a third-order Runge-Kutta TVD method. One and two-dimensional computations of a shock moving into a particle cloud demonstrates the characteristics of the WENO-Z based PSIC method (PSIC/WENO-Z). The PSIC/WENO-Z computations are not only in excellent agreement with the numerical simulations with a third-order Rusanov based PSIC and physical experiments in [V. Boiko, V.P. Kiselev, S.P. Kiselev, A. Papyrin, S. Poplavsky, V. Fomin, Shock wave interaction with a cloud of particles, Shock Waves, 7 (1997) 275-285], but also show a significant improvement in the resolution of small scale structures. In two-dimensional simulations of the Mach 3 shock moving into forty thousand bronze particles arranged in the shape of a rectangle, the long time accuracy of the high-order method is demonstrated. The fifth-order PSIC/WENO-Z method with the fifth-order ENO interpolation scheme improves the small scale structure resolution over the third-order PSIC/WENO-Z method with a second-order central interpolation scheme. Preliminary analysis of the particle interaction with the flow structures shows that sharp particle material arms form on the side of the rectangular shape. The arms initially shield the particles from the accelerated flow behind the shock. A reflected compression wave, however, reshocks the particle arm from the shielded area and mixes the particles.

Implicit-Explicit Time Integration of a High-Order Particle-in-Cell Method with Hyperbolic Divergence Cleaning.

A high-order implicit-explicit additive Rung-Kutta time integrator is implemented in a particle-in-cell method based on a high-order discontinuous Galerkin Maxwell solver for simulation of plasmas. The method satisfies Gauss law using a hyperbolic divergence cleaner that transports divergence out of the computational domain at several times the speed of light. The stiffness in the field equations induced by high transport speeds is alleviated by an implicit time integration, while an explicit time integration ensures a computationally efficient particle update. Simulations on a plasma wave and a Weibel instability show that the implicit-explicit solver is computationally efficient, allowing for computations with high divergence transport speeds that ensure an accurate representation of the governing plasma equations. The high-order method only requires two time steps per plasma wave period. Numerical instability appears when the time step exceeds the plasma frequency time scale. A divergence transport speed of approximately ten times the speed of light is shown to be optimal, since it combines an accurate representation of Gauss law with a small influence of numerical noise on the solution.

Experimental and numerical investigation of the kinematic theory of unsteady separation

c 2008 Cambridge University Press J. Fluid Mech. (2008), vol. 611, pp. 1–11. doi:10.1017/S0022112008002395 Printed in the United Kingdom Experimental and numerical investigation of the kinematic theory of unsteady separation M. W E L D O N 1 , T. P E A C O C K 1 , G. B. J A C O B S 2 , M. H E L U 1 A N D G. H A L L E R 1 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Rm 1-310, Cambridge, MA 02139, USA tomp@mit.edu Department of Aerospace Engineering & Engineering Mechanics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182, USA (Received 13 December 2007 and in revised form 14 May 2008) We present the results of a combined experimental and numerical study of flow separation in the unsteady two-dimensional rotor-oscillator flow. Experimentally detected material spikes are directly compared to separation profiles predicted from numerical shear-stress and pressure data, using a recent kinematic theory of unsteady separation. For steady, periodic, quasi-periodic and random forcing, fixed separation is observed, and experimental observations and theoretical predictions are in close agreement. The transition from fixed to moving separation is also reported. 1. Introduction Prandtl (1904) initiated the idea that in steady two-dimensional flow, laminar boundary-layer separation occurs at a point of zero skin-friction and negative skin-friction gradient on a no-slip boundary. He stated that at this point ‘a fluid-sheet projects itself into the free flow and effects a complete alteration of the motion.’ If the x-axis is tangent and the y-axis normal to the boundary (cf. figure 1), then Prandtl’s steady theory predicts boundary-layer separation at the (x, y) point p = (γ , 0) satisfying: τ x (γ ) < 0, where τ (x) = μu y (x, 0) is the skin-friction, (u (x, y) , v (x, y)) the two-dimensional velocity field, and μ the dynamic viscosity. More generally, the criterion (1.1) applies to flow separation of any scale, i.e. in any situation where a fluid-sheet projects itself into an incompressible and steady mean flow. Examples other than boundary-layer separation include: separating streamlines in Stokes flows, such as Moffatt corner-eddies (Moffatt 1964) and the rotor-oscillator flow (Hackborn, Ulucakli & Yuster 1997); and small-scale separation structures within a boundary layer, such as a separation bubble (Horton 1968). In all these cases, fluid particles separating away from the boundary form a material spike with some spatial scale. The separating streamline (which is the backbone of the material spike) makes an angle α with the wall that can be determined from the skin-friction gradient, τ x , and wall-pressure gradient, p x , (Lighthill 1963) using α = tan −1 −3τ x (γ ) p x (γ , 0)

Validation study of a Multidomain Spectral code for simulation of turbulent flows

A Chebyshev multidomain staggered-grid method (CMSM) presented by Kopriva for numerical simulation of three-dimensional compressible turbulent flows, is validated. Computations with the CMSM of an isotropic turbulence and a fully developed turbulent channel flow are compared with previously published results. Turbulence initiation of the isotropic turbulence on the Chebyshev grid is discussed. Low-resolution simulations ensure transition of an initial laminar channel flow to a turbulent channel flow. The turbulent kinetic energy spectrum has sufficient dropoff (indicating a resolved solution) at a minimum of 3 to 6 points per wave number for a polynomial approximation of 13 to 10, respectively

Evaluation of convergence behavior of metamodeling techniques for bridging scales in multi-scale multimaterial simulation

The effectiveness of several metamodeling techniques, viz. the Polynomial Stochastic Collocation method, Adaptive Stochastic Collocation method, a Radial Basis Function Neural Network, a Kriging Method and a Dynamic Kriging Method is evaluated. This is done with the express purpose of using metamodels to bridge scales between micro- and macro-scale models in a multi-scale multimaterial simulation. The rate of convergence of the error when used to reconstruct hypersurfaces of known functions is studied. For sufficiently large number of training points, Stochastic Collocation methods generally converge faster than the other metamodeling techniques, while the DKG method converges faster when the number of input points is less than 100 in a two-dimensional parameter space. Because the input points correspond to computationally expensive micro/meso-scale computations, the DKG is favored for bridging scales in a multi-scale solver.

An exact theory of three-dimensional fixed separation in unsteady flows

We develop a nonlinear theory for separation and attachment on no-slip boundaries of three-dimensional unsteady flows that have a steady mean component. In such flows, separation and attachment surfaces turn out to originate from fixed lines on the boundary, even though the surfaces themselves deform in time. The exact separation geometry is not captured by instantaneous Eulerian fields associated with the velocity field, but can be determined from a weighted average of the wall-shear and wall-density fields. To illustrate our results, we locate separation surfaces and attachment surfaces in an unsteady model flow and in direct numerical simulations of a time-periodic lid-driven cavity.

A Conservative Isothermal Wall Boundary Condition for the Compressible Navier---Stokes Equations

We present a conservative isothermal wall boundary condition treatment for the compressible Navier-Stokes equations. The treatment is based on a manipulation of the Osher solver to predict the pressure and density at the wall, while specifying a zero boundary flux and a fixed temperature. With other solvers, a non-zero mass flux occurs through a wall boundary, which is significant at low resolutions in closed geometries. A simulation of a lid driven cavity flow with a multidomain spectral method illustrates the effect of the new boundary condition treatment.

Extraction of Separation and Attachment Surfaces from Three-Dimensional Steady Shear Flows

We apply a recent analytic theory of three-dimensional steady separation to direct numerical simulations of backward-facing-step and lid-driven-cavity flows. We determine the exact location and slope of separation and attachment surfaces that have only been approximated heuristically in earlier studies. We also visualize the corresponding global separation and attachment surfaces, which reveal highly complex separated flow geometry.

Large-eddy simulation using a discontinuous Galerkin spectral element method

In this paper we discuss the development of a robust, high-order discontinuous Galerkin (DG) spectral element method for large-eddy simulation (LES) of compressible ∞ows. The method secures geometrical ∞exibility through a fully unstructured grid (triangles in 2D and tetrahedral elements in 3D), allows for arbitrary order of accuracy and has excellent stability properties. An element based flltering technique is used in conjunction with the dynamic procedure to model the efiect of sub-grid scales. We aim to use the LES methodology for large-scale simulation in geometrically complex dump combustors. As a flrst step towards these simulations, we perform validation simulations of compressible, turbulent ∞ow in a plane channel with isothermal walls.