Grid In Physical Space Research Articles

Mode multigrid - A novel convergence acceleration method

This paper proposes a mode multigrid (MMG) method, and applies it to accelerate the convergence of the steady state flow on unstructured grids. The dynamic mode decomposition (DMD) method is used to analyze the convergence process of steady flow field according to the solution vectors from the previous time steps. Unlike the traditional multigrid method, we project the flowfield solutions from the physical space into the modal space, and truncate all the high-frequency modes but only the first-order mode are retained based on the DMD analysis. The real solutions in the physical space can be obtained simply by the inverse transformation from the modal space. The developed MMG method ingeniously avoids the complicated process of coarsening computational mesh, and does not need to make any change for the grid in physical space. Therefore, it is very convenient to be applied to any numerical schemes with just a little change for the flow solver, which is also suitable for unstructured grids and easy for parallel computing. Several typical test cases have been used to verify the effectiveness of the proposed method, which demonstrates that the MMG can dramatically reduce the number of iterative steps for the different mesh types, different accuracy of spatial discretization and different time-marching schemes. The method is 3 to 6 times faster than the baseline method while ensuring the computational accuracy.

Numerical Computation of Doubly-Periodic Stokes Flow Bounded by a Plane with Applications to Nodal Cilia

AbstractA numerical method is presented for the computation of externally forced Stokes flows bounded by the planez=0 and satisfying periodic boundary conditions in thexandydirections. The motivation for this work is the simulation of flows generated by cilia, which are hair-like structures attached to the surface of cells that generate flows through coordinated beating. Large collections of cilia on a surface can be modeled using a doubly-periodic domain. The approach presented here is to derive a regularized version of the fundamental solution of the incompressible Stokes equations in Fourier space for the periodic directions and physical space for thezdirection. This analytical expression forû(k,m;z) can then be used to compute the fluid velocityu(x,y,z) via a two-dimensional inverse fast Fourier transform for any fixed value ofz. Repeating the computation for multiple values ofzleads to the fluid velocity on a uniform grid in physical space. The zero-flow condition at the planez=0 is enforced through the use of images. The performance of the method is illustrated by numerical examples of particle transport by nodal cilia, which verify optimal particle transport for parameters consistent with previous studies. The results also show that for two cilia in the periodic box, out-of-phase beating produces considerablemore particle transport than in-phase beating.

An adaptive multiresolution method for combustion problems: application to flame ball–vortex interaction

In this article, we present numerical simulations of flame ball–vortex interactions using adaptive multiresolution methods. The numerical scheme for the convection–diffusion–reaction equations modelling this problem is based on a finite volume discretization coupled with a discrete multiresolution analysis. The grid in physical space is adapted dynamically to track the evolution of the solution in scale and space. Time integration is done by an explicit Runge–Kutta scheme. In space we use second-order centered schemes. The implementation is based on a graded tree data structure, which improves both CPU and memory performances, as no fine gridding is required in regions where the solution is smooth. To illustrate the features and the efficiency of the method, we compute several flame ball–vortex interactions and study the role played by the fluid flow on the evolution of the flame ball. We observe the roll-up of the flame ball around the vortex into a snail-like structure. We also put into evidence the flammability limit of the flame ball in function of both vortex and radiation intensities.

A time self-adaptive multilevel algorithm for large-eddy simulation

An extension of the multilevel method applied to LES proposed in Terracol et al. [J. Comput. Phys. 167 (2001) 439] is introduced here to reduce the CPU times in unsteady simulation of turbulent flows. Flow variables are decomposed into several wavenumber bands, each band being associated to a computational grid in physical space. The general framework associated to such a decomposition is presented, and a new adapted closure is proposed for the subgrid terms which appear at each filtering level, while the closure at the finest level is performed with a classical LES model. CPU time saving is obtained by the use of V-cycles, as in the multigrid terminology. The main part of the simulation is thus performed on the coarse levels, while the smallest resolved scales are kept frozen (quasi-static approximation [Comput. Methods Appl. Mech. Engrg. 159 (1998) 123]). This allows to reduce significantly the CPU times in comparison with classical LES, while the accuracy of the simulation is preserved by the use of a fine discretization level. To ensure the validity of the quasi-static approximation, a dynamic evaluation of the time during which it remains valid is performed at each level through an a priori error estimation of the small-scales time variation. This leads to a totally self-adaptive method in which both the number of levels and the integration times on each grid level are evaluated dynamically. The method is assessed on a fully unsteady time-developing compressible mixing layer at a low-Reynolds number for which a DNS has also been performed, and in the inviscid case. Finally, a plane channel flow configuration has been considered. In all cases, the results obtained are in good agreement with classical LES performed on a fine grid, with CPU time reduction factors of up to five.

A Multilevel Algorithm for Large-Eddy Simulation of Turbulent Compressible Flows

A multilevel method for large-eddy simulation of turbulent compressible flows is proposed. The method relies on the splitting of the turbulent flowfield into several frequency bands in space and time, each band being associated to a specific computational grid in physical space. This allows to take into account in a deterministic way the information contained on finer grids. A subgrid model adapted to such a decomposition—based on a generalization of the Germano's identity to multilevel decomposition—is also introduced. The approach is validated by several multilevel simulations in a subsonic plane channel flow configuration for a low and a high value of the Reynolds number, while reductions of the CPU times up to 80% are obtained.

Computation of a Solenoid as a Ratio of Areas and a Geometric Interpretation of the Arakawa Jacobian

Abstract A formula for the computation of a solenoidal term, for example, k · (∇ψ × ∇ϒ) or ∂(ψ,ϒ)/∂(x, y) in Jacobian form, from three or more noncollinear stations or grid points is presented. The formula is based on a geometric interpretation of the solenoid as the ratio of an elementary area (expressed as a line integral) in ψ–ϒ dependent-variable or phase space to the corresponding area in the x–y plane. The Arakawa finite-difference Jacobian on a rectangular grid in physical space is shown to be a linear combination of such ratios. Thus the Arakawa method is a line-integral method. The new interpretation readily provides the forms of the Jacobian at boundary points needed to maintain the integral constraints in a closed domain. Elementary properties of Jacobians ensure that the Arakawa Jacobian can be used on a regular mesh in any general orthogonal curvilinear coordinate system, thus permitting the use of “stretched” grids that have a lesser density of grid points away from boundaries and also away ...

Numerical simulation of a mixing layer in an adaptive wavelet basis

This note presents an adaptive wavelet method to compute two-dimensional turbulent flows. The Navier–Stokes equations in vorticity–velocity form are discretized using a Petrov–Galerkin scheme. The vorticity field is developed into an orthogonal wavelet series where only the most significant coefficients are retained. The testfunctions are adapted to the linear part of the equation so that the resulting stiffness matrix turns out to be the identity. The nonlinear term is evaluated on a locally refined grid in physical space. This numerical scheme is applied to simulate a temporally developing mixing layer. A comparison with a classical pseudo-spectral method is used for validation of the new method. The results show that the formation of Kelvin–Helmholtz vortices is well captured and all scales of the flow are well represented.

Une méthode multiniveau pour la simulation des grandes échelles des écoulements turbulents compressibles

A multilevel method for Large-Eddy Simulation of turbulent unsteady compressible flows is proposed. It relies on the splitting of the turbulent flowfield into several frequency bands in space and time, each band being associated to a computational grid in physical space, allowing to take into account in a deterministic way the information contained on finer grids. A subgrid-scale model adapted to such a decomposition — based on a generalization of the Germano's identity to multilevel decomposition — is also introduced. The approach is validated by a simulation in a subsonic plane channel flow configuration.

An automatic grid generation scheme for pseudospectral self-consistent field calculations on polyatomic molecules

In a series of previous papers,'-3 a new algorithm for solution of self-consistent electronic structure equations has been proposed and implemented. The algorithm involves a synthesis of conventional quantum chemical techniques with the pseudospectral meth~d,~ an efficient numerical method originally applied to large-scale hydrodynamic simulations5 and recently employed in various contexts in chemical physics problems.6 The essential feature of the approach is use of both a basis set and a numerical grid representation of the solution, allowing accurate evaluation of derivatives and integrals but also permitting multiplications to be carried out efficiently on the physical space grid. The net results for the Hartree-Fock equations are elimination of twoelectron integrals and an scaling of both integral evaluation and Fock matrix assembly, thus potentially yielding 2-3 orders of magnitude reduction in computation time for large molecules. In ref 3, quantitatively accurate results (as compared to equivalent calculations done via the Roothaan procedure') for the

Calculation of annular nozzle trisonic flowfields by the method of characteristics

A numerical algorithm based on the unsteady two-dimensional method of characteristics is presented for calculating axisymmetric inviscid trisonic (i.e., subsonic-transonic-supersonic) flowfields in annular nozzles. The algorithm is an explicit second-order accurate predictor-corrector procedure. An inverse marching method based on a local network of wavelines and the pathline is employed wherein the solution grid in physical space is prespecified. An orthogonal uniform transformed space, obtained by an algebraic grid generation method, is used for flow property interpolation in the initial-data surface. Steady flow solutions are obtained as the asymptotic solution in time. Results for four annular nozzles are presented. Comparison of the predicted and measured wall pressure distributions and nozzle efficiencies shows good agreement.

Calculation of three-dimensional flowfields by the unsteady method of characteristics

A numerical algorithm based on the unsteady three-dimensional method of characteristi cs is presented for calculating unsteady three-dimensional subsonic/transonic inviscid flowfields. The flowfield unit processes are based on a local network consisting of six bicharacteristics and the pathline. The algorithm is an explicit secondorder accurate predictor-corrector procedure. An inverse-march ing method is employed, wherein the solution grid in physical space is prespecified. Steady flow solutions are obtained as the asymptotic solution in time. Results are presented for spherical supersonic source flow, for axisymmetric supersonic flow in a conical nozzle, for three-dimensional supersonic flow in a superelliptical nozzle, and for three-dimensional subsonic/transonic flow in a superelliptical nozzle. Solutions obtained by the present method agree well with solutions obtained by steady flow method of characteristi cs algorithms and experimental data.