Investigation of Compressible Electromagnetic Flute Mode Instability in Finite Beta Plasma in Support of Z-Pinch and Laboratory Astrophysics Experiments

Flute mode turbulence plays an important role in numerous applications, such as tokamak, Z-pinch, space and astrophysical plasmas. In a low beta plasma flute oscillations are electrostatic and in the nonlinear stage they produce large scale density structures co-mingling with short scale oscillations. Large scale structures are responsible for the enhanced transport across the magnetic field and appearance of short scales leads to ion heating, associated with the ion viscosity. In the present paper nonlinear equations which describe the nonlinear evolution of the flute modes treated as compressible electromagnetic oscillations in a finite beta inhomogeneous plasma with nonuniform magnetic field are derived and solved numerically. For this purpose the 2D numerical code FLUTE was developed. Numerical results show that even in a finite beta plasma flute mode instability can develop along with formation of large scale structures co-existing with short scale perturbations in the nonlinear stage.     

Implementation of 2D Domain Decomposition in the UCAN Gyrokinetic Particle-in-Cell Code and Resulting Performance of UCAN2

The massively parallel, nonlinear, three-dimensional (3D), toroidal, electrostatic, gyrokinetic, particle-in-cell (PIC), Cartesian geometry UCAN code, with particle ions and adiabatic electrons, has been successfully exercised to identify non-diffusive transport characteristics in present day tokamak discharges. The limitation in applying UCAN to larger scale discharges is the 1D domain decomposition in the toroidal (or z-) direction for massively parallel implementation using MPI which has restricted the calculations to a few hundred ion Larmor radii or gyroradii per plasma minor radius. To exceed these sizes, we have implemented 2D domain decomposition in UCAN with the addition of the y-direction to the processor mix. This has been facilitated by use of relevant components in the P2LIB library of field and particle management routines developed for UCLA's UPIC Framework of conventional PIC codes. The gyro-averaging specific to gyrokinetic codes is simplified by the use of replicated arrays for efficient charge accumulation and force deposition. The 2D domain-decomposed UCAN2 code reproduces the original 1D domain nonlinear results within round-off. Benchmarks of UCAN2 on the Cray XC30 Edison at NERSC demonstrate ideal scaling when problem size is increased along with processor number up to the largest power of 2 available, namely 131,072 processors. These particle weak scaling benchmarks also indicate that the 1 nanosecond per particle per time step and 1 TFlops barriers are easily broken by UCAN2 with 1 billion particles or more and 2000 or more processors.     

Improved Unlike-Particle Collision Operator for delta-f Drift-Kinetic Particle Simulations

Plasmas in modern tokamak experiments contain a significant fraction of impurity ion species in addition to main deuterium background. A new unlike-particle collision operator for δf particle simulation has been developed to study the nonlocal effects of impurities due to finite ion orbits on neoclassical transport in toroidal plasmas. A new algorithm for simulation of cross-collisions between different ion species includes test-particle and conserving field-particle operators. An improved field-particle operator is designed to exactly enforce conservation of number, momentum and energy.     

Comparative Study of Three High Order Schemes for LES of Temporally Evolving Mixing Layers

Three high order shock-capturing schemes are compared for large eddy simulations (LES) of temporally evolving mixing layers for different convective Mach numbers ranging from the quasi-incompressible regime to highly compressible supersonic regime. The considered high order schemes are fifth-order WENO (WENO5), seventh-order WENO (WENO7) and the associated eighth-order central spatial base scheme with the dissipative portion of WENO7 as a nonlinear post-processing filter step (WENO7fi). This high order nonlinear filter method of Yee & Sjogreen is designed for accurate and efficient simulations of shock-free compressible turbulence, turbulence with shocklets and turbulence with strong shocks with minimum tuning of scheme parameters. The LES results by WENO7fi using the same scheme parameter agree well with experimental results compiled by Barone et al., and published direct numerical simulations (DNS) work of Rogers & Moser and Pantano & Sarkar, whereas results by WENO5 and WENO7 compare poorly with experimental data and DNS computations.     

Synchronization of Weighted Essentially Non-Oscillatory Methods

Weighted essentially non-oscillatory (WENO) methods have been developed to simultaneously provide robust shock-capturing in compressible fluid flow and avoid excessive damping of fine-scale flow features such as turbulence. Under certain conditions in compressible turbulence, however, numerical dissipation remains unacceptably high even after optimization of the linear component that dominates in smooth regions. Of the nonlinear error that remains, we demonstrate that a large fraction is generated by a "synchronization deficiency" that interferes with the expression of theoretically predicted numerical performance characteristics when the WENO adaptation mechanism is engaged. This deficiency is illustrated numerically in simulations of a linearly advected sinusoidal wave and the Shu-Osher problem [J. Comput. Phys., 83 (1989), pp. 32-78]. It is shown that attempting to correct this deficiency through forcible synchronization results in violation of conservation. We conclude that, for the given choice of candidate stencils, the synchronization deficiency cannot be adequately resolved under the current WENO smoothness measurement technique.     

Parareal-Richardson Algorithm for Solving Nonlinear ODEs and PDEs

The parareal algorithm, proposed firstly by Lions et al. [J. L. Lions, Y. Maday, and G. Turinici, A "parareal" in time discretization of PDE's, C.R. Acad. Sci. Paris Sér. I Math., 332 (2001), pp. 661-668], is an effective algorithm to solve the time-dependent problems parallel in time. This algorithm has received much interest from many researchers in the past years. We present in this paper a new variant of the parareal algorithm, which is derived by combining the original parareal algorithm and the Richardson extrapolation, for the numerical solution of the nonlinear ODEs and PDEs. Several nonlinear problems are tested to show the advantage of the new algorithm. The accuracy of the obtained numerical solution is compared with that of its original version (i.e., the parareal algorithm based on the same numerical method).     

Numerical Simulation of Airfoil Vibrations Induced by Turbulent Flow

The subject of the paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil with large amplitudes. The airfoil with three degrees of freedom performs rotation around an elastic axis, oscillations in the vertical direction and rotation of a flap. The numerical simulation consists of the finite element solution of the Reynolds averaged Navier-Stokes equations combined with Spalart-Allmaras or k−ω turbulence models, coupled with a system of nonlinear ordinary differential equations describing the airfoil motion with consideration of large amplitudes. The time-dependent computational domain and approximation on a moving grid are treated by the Arbitrary Lagrangian-Eulerian formulation of the flow equations. Due to large values of the involved Reynolds numbers an application of a suitable stabilization of the finite element discretization is employed. The developed method is used for the computation of flow-induced oscillations of the airfoil near the flutter instability, when the displacements of the airfoil are large, up to ±40 degrees in rotation. The paper contains the comparison of the numerical results obtained by both turbulence models.     

Progress of Pattern Dynamics in Plasma Waves

This paper is concerned with the pattern dynamics of the generalized nonlinear Schrödinger equations (NSEs) related with various nonlinear physical problems in plasmas. Our theoretical and numerical results show that the higher-order nonlinear effects, acting as a Hamiltonian perturbation, break down the NSE integrability and lead to chaotic behaviors. Correspondingly, coherent structures are destroyed and replaced by complex patterns. Homoclinic orbit crossings in the phase space and stochastic partition of energy in Fourier modes show typical characteristics of the stochastic motion. Our investigations show that nonlinear phenomena, such as wave turbulence and laser filamentation, are associated with the homoclinic chaos. In particular, we found that the unstable manifolds W^(u)possessing the hyperbolic fixed point correspond to an initial phase θ =45^◦and 225^◦, and the stable manifolds W^(s) correspond to θ=135^◦ and 315^◦.     

A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model

In this paper we propose and analyze a (temporally) third order accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $−A∆t^2∆^2_N (u^{n+1}−u^n)$, is added in the numerical scheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the $ℓ^∞(0,T;ℓ^2)∩ℓ^2(0,T;H^2_h)$ norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for $ε = 0.02$ (up to $T = 3×10^5$) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.     

An Improved Second-Order Finite-Volume Algorithm for Detached-Eddy Simulation Based on Hybrid Grids

A hybrid grid based second-order finite volume algorithm has been developed for Detached-Eddy Simulation (DES) of turbulent flows. To alleviate the effect caused by the numerical dissipation of the commonly used second order upwind schemes in implementing DES with unstructured computational fluid dynamics (CFD) algorithms, an improved second-order hybrid scheme is established through modifying the dissipation term of the standard Roe's flux-difference splitting scheme and the numerical dissipation of the scheme can be self-adapted according to the DES flow field information. By Fourier analysis, the dissipative and dispersive features of the new scheme are discussed. To validate the numerical method, DES formulations based on the two most popular background turbulence models, namely, the one equation Spalart-Allmaras (SA) turbulence model and the two equation k−ω Shear Stress Transport model (SST), have been calibrated and tested with three typical numerical examples (decay of isotropic turbulence, NACA0021 airfoil at 60^◦incidence and 65^◦swept delta wing). Computational results indicate that the issue of numerical dissipation in implementing DES can be alleviated with the hybrid scheme, the resolution for turbulence structures is significantly improved and the corresponding solutions match the experimental data better. The results demonstrate the potentiality of the present DES solver for complex geometries.     

AWECS: A Linear Gyrokinetic δf Particle-in-Cell Simulation Code for the Study of Alfvénic Instabilities in High-β Tokamak Plasmas

A1-Dlinear gyrokinetic code called AWECS is developed to study the kinetic excitation of Alfvénic instabilities in a high-β tokamak plasma, with β being the ratio of thermal to magnetic pressure. It is designed to describe physics associated with a broad range of frequencies and wavelengths. For example, AWECS is capable of simulating kinetic ballooning modes, Alfvénic ion-temperature-gradient-driven modes, as well as Alfvén instabilities due to energetic particles. In addition, AWECS may be used to study drift-Alfvén instabilities in the low-β regime. Here, the layout of the code and the numerical methods used are described. AWECS is benchmarked against other codes and a convergence study is carried out.     

Elbow joint laxity after experimental radial head excision and lateral collateral ligament rupture: efficacy of prosthetic replacement and ligament repair

The objectives of this experimental study were to investigate the effect of radial head excision and lateral collateral ligament (LCL) division on elbow joint laxity and to determine the efficacy of radial head prosthetic replacement and LCL repair. Valgus, varus, internal rotation, and external rotation of the ulna were measured during passive flexion-extension and application of a 0.75-Nm torque in 6 intact cadaveric elbows and after (1) either excision of the radial head or division of the LCL, (2) removal of both constraints, (3) isolated radial head prosthetic replacement, (4) isolated LCL repair, and (5) radial head replacement combined with LCL repair. Isolated radial head excision increased varus (mean, 4.8 degrees) and external rotatory laxity (mean, 7.1 degrees), as did isolated LCL division (mean, 14.1 degrees for varus; mean, 14.7 degrees for external rotation). After removal of both constraints, varus and external rotatory laxities were increased by 19.0 degrees and 20.1 degrees, respectively, compared with the intact specimens. Isolated radial head replacement reduced mean varus laxity to 14.6 degrees and mean external rotatory laxity to 14.8 degrees. Isolated LCL repair normalized varus laxity but resulted in a 2.9 degrees increase in external rotatory laxity. The combined procedures restored laxity completely. The radial head is a constraint to varus and external rotation in the elbow joint, functioning by maintaining tension in the LCL. Still, removal of both constraints induces severe laxity, and in this case, prosthetic replacement may substitute for the constraining capacity of the native radial head. The combination of LCL repair and radial head replacement restores laxity completely, but an isolated LCL repair performs almost as well, probably by compensating for the ligamentous tension lost from radial head excision.     

Computational investigation of hemodynamics in hardshell venous reservoirs: A comparative study

Extracorporeal circulation using heart-lung-machines is associated with a profound activation of corpuscular and plasmatic components of circulating blood, which can also lead to deleterious events such as systemic inflammatory response and hemolysis. Individual components used to install the extracorporeal circulation have an impact on the level of activation, most predominantly membrane oxygenators and hardshell venous reservoirs as used in extracorporeal systems. The blood flows in two different hardshell reservoirs are computationally investigated. A special emphasis is placed on the prediction of an onset of transition and turbulence generation. Reynolds-averaged numerical simulations (RANS) based on a transitional turbulence model, as well as large eddy simulations (LES) are applied to achieve an accurate prediction. In the LES analysis, the non-Newtonian behavior of the blood is considered via the Carreau model. Blood damage potential is quantified applying the Modified Index of Hemolysis (MIH) based on the predicted flow fields. The results indicate that the flows in both reservoirs remain predominantly laminar. For one of the reservoirs, considerable turbulence generation is observed near the exit site, caused by the specific design for the connection with the drainage tube. This difference causes the MIH of this reservoir to be nearly twice as large as compared to the alternative design. However, a substantial improvement of these performance criteria can be expected by a local geometry modification.     

A model of osteon closure in cortical bone

Summary A model of osteon closure is presented that incorporates some physiologic features of cortical bone remodeling, such as matrix synthesizing activity of osteoblasts, their burial as osteocytes, and elimination of cells. A simplified version of the model assumes a constant osteoblast activity throughout radial closure. An extended version allows for varible osteoblasts activity and is based on Lee's law of radial closure kinetics. The model calculations (extended model) show that both in humans and beagle dogs osteoblast activity steadily decreases during radial closure of the osteon. The potential of the model is also illustrated by calculating the dynamic change of the geometrical shape of the closing cone and the time dependence of the osteoid seam width, number of osteoblasts and bone formation rate in the closing cone.     

Effects of turbulence on gas transport during high frequency oscillation

To clarify the enhancing effect of turbulence on gas transfer in high frequency oscillation (HFO), the role of turbulence during oscillatory flow was studied. Oscillatory flow was generated in a straight circular tube as a trachea model, over the range of Reynolds' number (Re: 2000-16000) and Stokes parameter (lambda : 3-7). This simulates clinical settings. Measurements of the axial velocity were made by means of hot wire anemometer, and turbulence intensity (I) as well as ensemble mean velocity was calculated. At the same time, 1 ml of 100% CO2 was injected into the tube, and CO2 concentration was measured by a CO2 analyzer to yield the effective diffusivity (Deff). As Re increases and lambda decreases to a certain level, "conditional turbulence" appears in the whole cross section to produce an efficient lateral mixing. Deff increases with increasing I as well as Re, and increasing lambda results in an increase in the difference of measured and theoretical Deff (Watson's prediction for laminar oscillatory flow). It is concluded that turbulence during oscillatory flow plays an important role in lateral mixing to facilitate gas transport, and decision of ventilator settings is essential so as to give rise to "conditional turbulence" for the purpose of Deff augmentation during HFO.     

An Approximate Second-Order Closure Model for Large-Eddy Simulation of Compressible Isotropic Turbulence

In this paper, the detailed dynamic characteristics of the subgrid scale (SGS) stress tensor and heat flux are investigated through Taylor series expansion in numerical simulations of compressible isotropic turbulence. A new approximate second-order closure (ASOC) model is introduced based on the transport equations of the first-order Taylor series approximation of SGS stress tensor and heat flux. The proposed model is implemented in large eddy simulation (LES) of compressible isotropic turbulence. Detailed comparisons with direct numerical simulation (DNS) dataset using both a priori and a posteriori approaches are carried out. A priori tests show that, SGS stress tensor and heat flux have high correlations with the first-order Taylor series approximation. Their root mean square (rms) values are close to those of the first-order Taylor series approximation. In a posteriori tests, the proposed ASOC model yields good agreement with DNS dataset. Compared with the results of the dynamic Smagorinsky model (DSM) and dynamic mixed model (DMM), the ASOC model predicts better energy spectra at high wavenumbers. The probability density function (PDF) and the structure functions of velocity and thermodynamic variables are further studied, demonstrating that the statistical properties of the simulated flows are improved by the ASOC model. The numerical results illustrate the ability of the model to improve the statistical properties of the simulated flows in the context of LES. Finally, a simplified ASOC model can be derived by neglecting the effect of density gradient for low turbulent Mach number turbulence.     

Subspace Trajectory Piecewise-Linear Model Order Reduction for Nonlinear Circuits

Despite the efficiency of trajectory piecewise-linear (TPWL) model order reduction (MOR) for nonlinear circuits, it needs large amount of expansion points for large-scale nonlinear circuits. This will inevitably increase the model size as well as the simulation time of the resulting reduced macromodels. In this paper, subspace TPWL-MOR approach is developed for the model order reduction of nonlinear circuits. By breaking the high-dimensional state space into several subspaces with much lower dimensions, the subspace TPWL-MOR has very promising advantages of reducing the number of expansion points as well as increasing the effective region of the reduced-order model in the state space. As a result, the model size and the accuracy of the TWPL model can be greatly improved. The numerical results have shown dramatic reduction in the model size as well as the improvement in accuracy by using the subspace TPWL-MOR compared with the conventional TPWL-MOR approach.     

On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix. The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws, while retaining the attractive features of the original solver: the method is entropy-satisfying, differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field, in particular to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system is used. To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics with ideal gas and real gas equation of state, classical and relativistic MHD equations as well as the equations of nonlinear elasticity. To the knowledge of the authors, apart from the Euler equations with ideal gas, an Osher-type scheme has never been devised before for any of these complicated PDE systems. Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of P_NP_M schemes on unstructured meshes recently proposed in [9].     

Nonlinear analysis of arterial oscillated flow in experimental stenosis and microsurgical anastomosis.

BACKGROUND: Thrombotic vascular occlusion is one of the main complications that can occur during microsurgical anastomosis and is frequent when the blood becomes turbulent. The aim of this ex vivo study was to test the use of nonlinear mathematical tools to detect turbulence flow upstream and downstream of an arterial stenosis and of a microsurgical anastomosis technique in arteries with diameters in the range of microsurgical practice. MATERIALS AND METHODS: Rat carotid arteries (0.8 to 1.2 mm diameter) were transferred to a flow chamber and perfused with Krebs solution. An oscillated vascular flow was initiated with a peristaltic pump and a transit time flowmeter was used to measure flow with two probes. An arterial stenosis was created by a ligature and progressively increased ranging from 0 to 95%. For each flow signal three nonlinear analytical procedures were applied: time-delayed procedures, correlation dimension, and computing of the largest Lyapunov exponent. RESULTS: Our results indicated that the level of turbulence flow is correlated with the area reduction stenosis. In the range of 60-95% area reduction stenosis, we noted an experimental increase of turbulence flow. We also founded that a classical end-to-end anastomosis technique induced an increase of the turbulence flow in comparison with a control artery. CONCLUSIONS: Thus nonlinear analysis can be useful in characterizing the complexity of an oscillated flow in small arteries submitted to stenosis or microsurgical anastomosis and may have clinical uses in detecting high level turbulent flow after microsurgery.