The η-Pairing Superconductivity in Spin-Density Wave Background

In this letter,we propose a modified attractive Hubbard model at half filling that can exhibit sup er con ductivity through η-pairing mecha.nism in spin-density wave representation and derive a concise relationship between the energy disparity and the single-particle energy spectrum.This relationship,in the two limits of U,clearly shows that the system does not display sup er con ductivity for very low doping concen・tration.but denionst.ra.tes s up ere on ductivity when the doping concentration exceeds a certain critical value.This conclusion is in quaJitative agreement with the familiar experiments on higli-T_(c) supcrconductivity.

Efficient Implicit Non-linear LU-SGS Approach for Compressible Flow Computation Using High-Order Spectral Difference Method

An implicit non-linear lower-upper symmetric Gauss-Seidel(LU-SGS)solution algorithm has been developed for a high-order spectral difference Navier-Stokes solver on unstructured hexahedral grids.The non-linear LU-SGS solver is preconditioned by a block element matrix,and the system of equations is then solved with the LU decomposition approach.The large sparse Jacobian matrix is computed numerically,resulting in extremely simple operations for arbitrarily complex residual operators.Several inviscid and viscous test cases were performed to evaluate the performance.The implicit solver has shown speedup of 1 to 2 orders of magnitude over the multi-stage Runge-Kutta time integration scheme.

High-Order Multidomain Spectral Difference Method for the Navier-Stokes Equations on Unstructured Hexahedral Grids

A high order multidomain spectral difference method has been developed for the three dimensional Navier-Stokes equations on unstructured hexahedral grids.The method is easy to implement since it involves one-dimensional operations only,and does not involve surface or volume integrals.Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner in a unit cube.The concepts of the Riemann solver and high-order local representations are applied to achieve conservation and high order accuracy.In this paper,accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions.High order of accuracy and spectral convergence are obtained for the propagation of an isotropic vortex and Couette flow.The capability of the method for both inviscid and viscous flow problems with curved boundaries is also demonstrated.

Benchmark for scale-resolving simulation with curved walls: the Taylor Couette flow

Flow between rotating concentric cylinders,or the Taylor Couette flow,has been studied extensively because of its rich physics,ranging from axisymmetric steady laminar flow,to fully developed turbulent flow.In the present study,we advocate the use of this problem as a benchmark case for scale-resolving simulation,such as large eddy simulation(LES)and direct numerical simulation(DNS).The problem is attractive because of its simple geometry,simple boundary conditions,and complex physics involving wall-shear induced and centrifugal instability.Unlike the wellknown fully developed channel flow,this problem has a curved wall boundary,and it is unnecessary to add a source term to the governing equations to sustain the fully developed turbulent flow.A p-refinement study for Re=4000 is performed first to establish DNS data,including the time history of enstrophy,which can be used as an accuracy and resolution indicator to evaluate numerical methods,and is orders of magnitude faster than using the mean flow quantities and Reynolds stresses to evaluate solution quality.Finally,an hp-refinement study is performed to establish the relative accuracy and efficiency of high-order schemes of various accuracy.

Large Eddy Simulation of Flow over a Cylinder Using High-Order Spectral Difference Method

Large eddy simulation of the flow over a circular cylinder at Reynolds number ReD=2580 has been studied with a high-order unstructured spectral difference method.Grid and polynomial refinement studies were carried out to assess numerical errors.The mean and fluctuating velocity fields in the wake of a circular cylinder were compared with PIV experimental measurements.The numerical results are in excellent agreement with the experimental data for both the mean velocity and Reynolds stresses using the high-order SD scheme.Other wake characteristics such as the recirculation bubble length,vortex formation length and maximum intensity of the velocity fluctuations have also been predicted accurately.The numerical simulations demonstrated the potential of the high-order SD method in accurate large eddy simulation of physically complex problems.

Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR(correction procedure or collocation penalty via reconstruction).The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure.In addition to being simple and economical,it unifies several existing methods including discontinuous Galerkin,staggered grid,spectral volume,and spectral difference.To discretize the diffusion terms,we use the BR2(Bassi and Rebay),interior penalty,compact DG(CDG),and I-continuous approaches.The first three of these approaches,originally derived using the integral formulation,were recast here in the CPR framework,whereas the I-continuous scheme,originally derived for a quadrilateral mesh,was extended to a triangular mesh.Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out.Finally,results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

A Parameter–Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids

A parameter-free limiting technique is developed for high-order unstructured-grid methods to capture discontinuities when solving hyperbolic conservation laws.The technique is based on a“troubled-cell”approach,in which cells requiring limiting are first marked,and then a limiter is applied to these marked cells.A parameter-free accuracy-preserving TVD marker based on the cell-averaged solutions and solution derivatives in a local stencil is compared to several other markers in the literature in identifying“troubled cells”.This marker is shown to be reliable and efficient to consistently mark the discontinuities.Then a compact highorder hierarchical moment limiter is developed for arbitrary unstructured grids.The limiter preserves a degree p polynomial on an arbitrary mesh.As a result,the solution accuracy near smooth local extrema is preserved.Numerical results for the high-order spectral difference methods are provided to illustrate the accuracy,effectiveness,and robustness of the present limiting technique.

On the mesh resolution of industrial LES based on the DNS of flow over the T106C turbine

A high-order Navier-Stokes solver based on the flux reconstruction(FR)or the correction procedure via reconstruction(CPR)formulation is employed to perform a direct numerical simulation(DNS)and large eddy simulations(LES)of a well-known benchmark problem–transitional flow over the low-pressure T106C turbine cascade.Hp-refinement studies are carried out to assess the resolution requirement.A 4th order(p3)simulation on the fine mesh is performed with a DNS resolution to establish a"converged"solution,including the mean pressure and skin-friction distributions,and the power spectral density in the wake.Then LES on the coarse and fine meshes with lower order schemes are conducted to assess the mesh and order dependence of the solution.In particular,we study the error in the transition location,the mean skin-friction distribution,and the mean lift and drag coefficients.These h-and p-refinement studies provide a much-needed guideline in h-and p-resolutions to achieve a certain level of accuracy for industrial LES applications.

An Optimized Correction Procedure via Reconstruction Formulation for Broadband Wave Computation

Recently,a new differential discontinuous formulation for conservation laws named the Correction Procedure via Reconstruction(CPR)is developed,which is inspired by several other discontinuous methods such as the discontinuous Galerkin(DG),the spectral volume(SV)/spectral difference(SD)methods.All of them can be unified under the CPR formulation,which is relatively simple to implement due to its finite-difference-like framework.In this paper,a different discontinuous solution space including both polynomial and Fourier basis functions on each element is employed to compute broad-band waves.Free-parameters introduced in the Fourier bases are optimized to minimize both dispersion and dissipation errors through a wave propagation analysis.The optimization procedure is verified with a mesh resolution analysis.Numerical results are presented to demonstrate the performance of the optimized CPR formulation.

Performance of Low-Dissipation Euler Fluxes and Preconditioned LU-SGS at Low Speeds

In low speed flow computations,compressible finite-volume solvers are known to a)fail to converge in acceptable time and b)reach unphysical solutions.These problems are known to be cured by A)preconditioning on the time-derivative term,and B)control of numerical dissipation,respectively.There have been several methods of A)and B)proposed separately.However,it is unclear which combination is the most accurate,robust,and efficient for low speed flows.We carried out a comparative study of several well-known or recently-developed low-dissipation Euler fluxes coupled with a preconditioned LU-SGS(Lower-Upper Symmetric Gauss-Seidel)implicit time integration scheme to compute steady flows.Through a series of numerical experiments,accurate,efficient,and robust methods are suggested for low speed flow computations.

An Implicit LU-SGS Scheme for the Spectral Volume Method on Unstructured Tetrahedral Grids

An efficient implicit lower-upper symmetric Gauss-Seidel(LU-SGS)solution approach has been applied to a high order spectral volume(SV)method for unstructured tetrahedral grids.The LU-SGS solver is preconditioned by the block element matrix,and the system of equations is then solved with a LU decomposition.The compact feature of SV reconstruction facilitates the efficient solution algorithm even for high order discretizations.The developed implicit solver has shown more than an order of magnitude of speed-up relative to the Runge-Kutta explicit scheme for typical inviscid and viscous problems.A convergence to a high order solution for high Reynolds number transonic flow over a 3D wing with a one equation turbulence model is also indicated.

A Mathematical Analysis of Scale Similarity

Scale similarity is found in many natural phenomena in the universe,from fluid dynamics to astrophysics.In large eddy simulations of turbulent flows,some sub-grid scale(SGS)models are based on scale similarity.The earliest scale similarity SGS model was developed by Bardina et al.,which produced SGS stresses with good correlation to the true stresses.In the present study,we perform a mathematical analysis of scale similarity.The analysis has revealed that the ratio of the resolved stress to the SGS stress is g2,where g is the ratio of the second filter width to the first filter width,under the assumption of small filter width.The implications of this analysis are discussed in the context of large eddy simulation.