Chunlei Liang

Spectral difference method for compressible flow on unstructured grids with mixed elements

This paper presents the development of a 2D solver for inviscid and viscous compressible flows using the spectral difference (SD) method for unstructured grids with mixed elements. A mixed quadrilateral and triangular grid is first refined using one-level h-refinement to generate a quadrilateral grid while keeping the curvature of boundary edges. The SD method designed for quadrilateral meshes can subsequently be applied for the refined unstructured grid. Results obtained with the SD method for both inviscid and viscous compressible flows compare well with analytical solutions and other published results.

Large Eddy Simulation of Compressible Turbulent Channel Flow with Spectral Difierence method

This paper presents the development of a three-dimensional high-order solver with unstructured spectral difierence method. The solver employs the formulations of Sun et al. 22 It is implemented on unstructured hexahedral grid elements. It is flrstly validated using test problems of 2D and 3D subsonic inviscid ∞ows past a circular cylinder. The spectral difierence method constructs element-wise continuous flelds. Five difierent types of Riemann solvers are employed to deal with the discontinuity across elements. We demonstrate the spatial accuracy up to fourth-order using the viscous compressible Couette ∞ow with analytic solution. The 3D SD method is flnally applied to a compressible turbulent channel ∞ow at Re? = 400. The predicted mean and r.m.s velocity proflles are in good agreement with DNS results of Moser et al. 15

Simulation of Transitional Flow over Airfoils using the Spectral Difference Method

This work addresses the simulation of transitional flow over airfoils under low Reynolds number conditions (Rec � 60000). The flow solutions are obtained by means of an Implicit Large Eddy Simulation (ILES) using a newly developed unstructured, parallel solver that employs the high-order spectral difference (SD) method for spatial discretization. The calculations are performed on the SD7003 airfoil section at an angle of attack of 4 ◦ at Reynolds number of 10000 and 60000. The SD7003 airfoil was selected due to the availability of experimental and computational results. The use of the SD method without an added subgrid-scale model appears to be capable of accurately predicting the laminar separation, transition and reattachment locations. To the authors’s knowledge, the present study is the first att empt to analyze transitional flow using the SD method.

Computation Of Flows with Shocks Using Spectral Dierence Scheme with Articial Viscosity

The current work focuses on applying an arti cial viscosity approach to the Spectral Di erence (SD) method to enable high-order computation of compressible uid ows with discontinuities. The study modi es the arti cial viscosity approach proposed in the earlier work. Studies show that a dilatation sensor for arti cial viscosity, combined with a dilatation-based switch and lter for smoothing, works well for curvilinear and unstructured grids. The arti cial viscosity model is found to stabilize numerical calculations and reduce oscillations near discontinuities. Promising results are demonstrated for 2-D test problems. Adaptive mesh re nement is used in conjunction with arti cial viscosity to obtain improved shock pro les. A mortar element method is used to handle the non-conforming interfaces generated from mesh-re nement of quadrilateral elements.

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 dif- ference method. Grid and polynomial refinement studies were carried out to as- sess numerical errors. The mean and fluctuating velocity fields in the wake of a circular cylinder were compared with PIV experimental measurements. The nu- merical 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. AMS subject classifications: 76F65

A p-Multigrid spectral difference method for viscous compressible flow using 2D quadrilateral meshes

The work focuses on the development of a 2D quadrilateral element based Spectral Difference solver for viscous flow calculations, and the application of the p-multigrid method and implicit time-stepping to accelerate convergence. This paper extends the previous work by Liang et al (2009) on the p-multigrid method for 2D inviscid compressible flow, to viscous flows. The high-order spectral difference solver for unstructured quadrilateral meshes is based on the formulation of Sun et al for unstructured hexahedral elements. The p-multigrid method operates on a sequence of solution approximations of different polynomial orders ranging from one upto four. An efficient preconditioned Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit scheme is also implemented. The spectral difference method is applied to a variety of inviscid and viscous compressible flow problems. The speed-up using the p-Multigrid and Implicit time-stepping techniques is also demonstrated.

A Spectral Difference Method for Viscous Compressible Flows With Shocks

The current work focuses on applying an artificial viscosity approach to the Spectral Difference (SD) method to enable high-order computation of compressible fluid flows with discontinuities. The study uses an artificial viscosity approach similar to the high-wavenumber biased artificial viscosity approach introduced by Cook and Cabot, 5,6 and modified by Kawai and Lele. 9 The model employs a bulk viscosity for treating shocks, a shear viscosity for treating turbulence, and an artificial conductivity to handle contact discontinuities. The high-wavenumber biased viscosity is found to stabilize numerical calculations and reduce oscillations near discontinuities. Promising results are demonstrated for 1-D and 2-D test problems.

High-Order Spectral Difference Method for the Navier-Stokes Equation on Unstructured Moving Deformable Grid

In this paper the high-order accurate spectral difference method for the Navier-Stokes equations is applied to moving boundary problems. Boundary movements are achieved, firstly, through rigid displacement of the entire flow domain. In order to account for the dynamic rigid mesh motion, the Navier-Stokes equations are modified through an unsteady coordinate transformation. Airfoils in pitching and plunging motions are studied. In both cases, computation results are compared with existing experimental data, and favorable results have been obtained. Secondly, spectral difference method is extended to include capability for handling dynamic deforming grids. The physical boundary movement is achieved through a time dependent unsteady transformation that allows part of the flow domain to be rigidly displacing, part of it fixed, and the rest deforming smoothly in between. The time dependent transformation preserves spectral difference method’s high order accuracy by solving the governing equations in a steady reference domain where the same shape functions are used, and introducing the unsteady perturbation in the physical space only through the changes in the transformation metrics and Jacobian. The blended deforming mesh allows the far field boundary or some desirable portions of the flow domain to be unaltered. These together make the overall solver accurate, flexible, and simple to implement. The order of accuracy of the spectral difference method in highly distorted mesh has been demonstrated through simulation of euler vortex problem. Simulations for flow over a plunging cylinder with rigid displacing and dynamic deforming meshes have yielded nearly identical results.

High-Order Spectral Difference Simulation of Laminar Compressible Flow Over Two Counter-Rotating Cylinders

Flow past a single rotating cylinder has been studied, both numerically and experimentally, by many authors. In contrast is the flow past rotating cylinders in a side-by-side arrangement, which has only very limited numerical and experimental data available, despite the pioneering theoretical study by Jeffery 1 as early as 1922. In this paper, we discuss a numerical investigation of the steady laminar viscous flow past two infinite rotating cylinders in a side-by-side configuration. The solution of the compressible two-dimensional Navier-Stokes equations is determined numerically using the high-order spectral difference scheme over an unstructured qudralateral grid. Third order accurate results in both time and space were obtained and compared with existing data. In addition to obtaining a highorder accurate result for flow past two rotating cylinders, we extend the current numerical effort to investigate the effects of Reynolds number, compressibility, and high rotation speed, which have not been comprehensively studied in the past.

An Artificial Compressibility Method for the Spectral Difference Solution of Unsteady Incompressible Navier-Stokes Equations on Multiple Grids

This paper presents the development of a 2D high-order solver with unstructured spectral difference method for unsteady incompressible Navier-Stokes equations. Timemarching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach. 10,25 A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented 14 in conjunction with the dual time stepping method for convergence acceleration. The SD method added with the ACM (SD-ACM) is able to accurately simulate 2D incompressible steady and unsteady viscous flows.