Numerical performance of smoothers in coupled multigrid methods for the parallel solution of the incompressible Navier–Stokes equationsJohn, Volker; Tobiska, Lutz
doi: 10.1002/1097-0363(20000630)33:4<453::AID-FLD15>3.0.CO;2-0pmid: N/A
In recent benchmark computations (Schäfer M, Turek S. The benchmark problem ‘Flow around a cylinder’. In Flow Simulation with High‐Performance Computers II, Hirschel EH (ed.), vol. 52 of Notes on Numerical Fluid Mechanics. Vieweg: Wiesbaden, 1996; 547–566), coupled multigrid methods have been proven as efficient solvers for the incompressible Navier–Stokes equations. A numerical study of two classes of smoothers in the framework of coupled multigrid methods is presented. The class of Vanka‐type smoothers is characterized by the solution of small local linear systems of equations in a Gauss–Seidel manner in each smoothing step, whereas the Brass–Sarazin‐type smoothers solve a large global saddle point problem. The behaviour of these smoothers with respect to computing times and parallel overheads is studied for two‐dimensional steady state and time‐dependent Navier–Stokes equations.
Numerical simulation and analysis of three‐dimensional turbulent impinging square twin‐jet flow field with no‐crossflowChuang, Shu‐Hao; Nieh, Tsu‐Jui
doi: 10.1002/1097-0363(20000630)33:4<475::AID-FLD16>3.0.CO;2-Qpmid: N/A
The three‐dimensional turbulent impinging square twin‐jet flow with no‐crossflow is analyzed by employing the computational fluid dynamics (CFD) code PHOENICS. The SIMPLEST algorithm and the Jones–Launder k–ε two‐equation turbulence model are used to simulate the strong turbulence of the three‐dimensional impinging twin‐jet flow field. The transport properties of velocity, pressure, and structure of exhausted nozzles at a space of S=5D, jet exit height of H=3D, and main nozzle jet Reynolds number of 105 000 are solved in this paper. The axial velocities of the present calculated results are found to be in good agreement with the experimental data of Barata et al. (Barata JMM, Durao DFG, Heitor MV. Impingement of single and twin turbulent jets through a crossflow. AIAA Journal 1991; 29: 595–602). The calculated results show that the flow field structure of twin‐jet impinging on a flat surface is strongly affected by the depth of geometry. Also, the calculated results show that several recirculating zones are distributed around the flow field. Their size and location are different from the two‐dimensional flow field due to the effect of flow stretching in the y‐direction. In addition, fountain upwash flow is extended to the narrow region of the outer boundary. The phenomena in the present analysis provide a fundamental numerical study of three‐dimensional impinging twin‐jet flow fields and a basis for the further analysis of three‐dimensional impinging twin‐jet flow fields with a variable angle nozzle and plate.
Control of Navier–Stokes equations by means of mode reductionPark, H. M.; Lee, M. W.
doi: 10.1002/1097-0363(20000630)33:4<535::AID-FLD20>3.0.CO;2-Hpmid: N/A
In a previous work (Park HM, Lee MW. An efficient method of solving the Navier–Stokes equation for the flow control. International Journal of Numerical Methods in Engineering 1998; 41: 1131–1151), the authors proposed an efficient method of solving the Navier–Stokes equations by reducing their number of modes. Employing the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear sub‐space that is sufficient to describe the observed phenomena, and consequently, reduce the Navier–Stokes equations defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. In the present work, we apply this technique, termed the Karhunen–Loève Galerkin procedure, to a pointwise control problem of Navier–Stokes equations. The Karhunen–Loève Galerkin procedure is found to be much more efficient than the traditional method, such as finite difference method in obtaining optimal control profiles when the minimization of the objective function has been done by using a conjugate gradient method.
A boundary‐only approach to the deformation of a shear‐thinning drop in extensional Newtonian flowKhayat, Roger E.
doi: 10.1002/1097-0363(20000630)33:4<559::AID-FLD22>3.0.CO;2-1pmid: N/A
The influence of shear thinning on drop deformation is examined through a numerical simulation. A two‐dimensional formulation within the scope of the boundary element method (BEM) is proposed for a drop driven by the ambient flow inside a channel of a general shape, with emphasis on a convergent–divergent channel. The drop is assumed to be shear thinning, obeying the Carreau–Bird model and the suspending fluid is Newtonian. The viscosity of the drop at any time is estimated on the basis of a rate‐of‐strain averaged over the region occupied by the drop. The viscosity thus changes from one time step to the next, and it is strongly influenced by drop deformation. It is found that small drops, flowing on the axis, elongate in the convergent part of the channel, then regain their spherical form in the divergent part; thus confirming experimental observations. Newtonian drops placed off‐axis are found to rotate during the flow with the period related to the initial extension, i.e. to the drop aspect ratio. This rotation is strongly prohibited by shear thinning. The formulation is validated by monitoring the local change of viscosity along the interface between the drop and the suspending fluid. It is found that the viscosity averaged over the drop compares, generally to within a few per cent, with the exact viscosity along the interface.
The anti‐dissipative, non‐monotone behavior of Petrov–Galerkin upwindingBradford, Scott F.; Katopodes, Nikolaos D.
doi: 10.1002/1097-0363(20000630)33:4<583::AID-FLD23>3.0.CO;2-Wpmid: N/A
The Petrov–Galerkin method has been developed with the primary goal of damping spurious oscillations near discontinuities in advection dominated flows. For time‐dependent problems, the typical Petrov–Galerkin method is based on the minimization of the dispersion error and the simultaneous selective addition of dissipation. This optimal design helps to dampen the oscillations prevalent near discontinuities in standard Bubnov–Galerkin solutions. However, it is demonstrated that when the Courant number is less than 1, the Petrov–Galerkin method actually amplifies undershoots at the base of discontinuities. This is shown in an heuristic manner, and is demonstrated with numerical experiments with the scalar advection and Richards' equations. A discussion of monotonicity preservation as a design criterion, as opposed to phase or amplitude error minimization, is also presented. The Petrov–Galerkin method is further linked to the high‐resolution, total variation diminishing (TVD) finite volume method in order to obtain a monotonicity preserving Petrov–Galerkin method.