In this article the principles of the field operation and manipulation (FOAM) C++ class library for continuum mechanics are outlined. Our intention is to make it as easy as possible to develop reliable and efficient computational continuum-mechanics codes: this is achieved by making the top-level syntax of the code as close as possible to conventional mathematical notation for tensors and partial differential equations. Object-orientation techniques enable the creation of data types that closely mimic those of continuum mechanics, and the operator overloading possible in C++ allows normal mathematical symbols to be used for the basic operations. As an example, the implementation of various types of turbulence modeling in a FOAM computational-fluid-dynamics code is discussed, and calculations performed on a standard test case, that of flow around a square prism, are presented. To demonstrate the flexibility of the FOAM library, codes for solving structures and magnetohydrodynamics are also presented with appropriate test case results given. © 1998 American Institute of Physics.

A tensorial approach to computational continuum mechanics using object-oriented techniques

A comprehensive review of zonal flow phenomena in plasmas is presented. While the emphasis is on zonal flows in laboratory plasmas, planetary zonal flows are discussed as well. The review presents the status of theory, numerical simulation and experiments relevant to zonal flows. The emphasis is on developing an integrated understanding of the dynamics of drift wave–zonal flow turbulence by combining detailed studies of the generation of zonal flows by drift waves, the back-interaction of zonal flows on the drift waves, and the various feedback loops by which the system regulates and organizes itself. The implications of zonal flow phenomena for confinement in, and the phenomena of fusion devices are discussed. Special attention is given to the comparison of experiment with theory and to identifying directions for progress in future research.

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Zonal flows in plasma—a review

The dynamic subgrid-scale (SGS) model of Germano et al. (1991) is generalized for the large eddy simulation (LES) of compressible flows and transport of a scalar. The model was applied to the LES of decaying isotropic turbulence, and the results are in excellent agreement with experimental data and direct numerical simulations. The expression for the SGS turbulent Prandtl number was evaluated using direct numerical simulation (DNS) data in isotropic turbulence, homogeneous shear flow, and turbulent channel flow. The qualitative behavior of the model for turbulent Prandtl number and its dependence on molecular Prandtl number, direction of scalar gradient, and distance from the wall are in accordance with the total turbulent Prandtl number from the DNS data.

A dynamic subgrid‐scale model for compressible turbulence and scalar transport

Astrophysical magnetic fields and nonlinear dynamo theory

The numerical simulation of high Reynolds number flows is hampered by model accuracy if the Reynolds-averaged Navier–Stokes (RANS) equations are used, and by computational cost if direct or large-eddy simulations (LES) that resolve the near-wall layer are employed. The cost of a calculation scales like the Reynolds number to the power 3 for direct numerical simulations, or 2.4 for LES, making the resolution of the wall layer at high Reynolds number infeasible even with the most advanced computers. In LES, an attractive alternative to compute high-Re flows is the use of wall-layer models, in which only the outer layer is resolved, while the near-wall region is modeled. Three broad classes of approaches are presently used: bypassing this region altogether using wall functions, solving a separate set of equations in the near-wall region, weakly coupled to the outer flow, or simulating the near-wall region in a global, Reynolds-averaged, sense. These approaches are discussed and their ranges of applicability are highlighted. Various unresolved issues in wall-layer modeling are presented.

Wall-layer models for large-eddy simulations

A statistical theory for compressible turbulent shear flows subject to buoyancy effects is developed. Important correlation functions in compressible shear flows are calculated with the aid of a multiscale direct‐interaction approximation. They are expressed in the gradient‐diffusion form similar to the eddy‐viscosity representation for the Reynolds stress in incompressible flows. The results obtained are applicable to subgrid modeling, and a Smagorinsky‐type model in compressible flows is constructed.

Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling

A subgrid-scale (SGS) kinetic energy model for the large-eddy simulation (LES) of turbulent flows is constructed with the aid of the statistical results obtained from the two-scale direct-interaction approximation. In this model, the SGS Reynolds stress is written in terms of a generalized SGS eddy-viscosity representation which is expressed by using the SGS kinetic energy and the characteristic grid width. A model equation for the SGS kinetic energy is combined with the grid-scale Navier-Stokes and continuity equations to lead to an LES model of one-equation type. The usual Smagorinsky model is derived as its special case, and critical comparison is made between them.

A Statistically-Derived Subgrid-Scale Kinetic Energy Model for the Large-Eddy Simulation of Turbulent Flows

An improvement of the eddy‐viscosity representation for Reynolds stress is made from the statistical viewpoint. The Reynolds stress is calculated with the aid of the two‐scale direct‐interaction formalism, and its deviation from the eddy‐viscosity representation is found under general mean flows. This result theoretically elucidates the noncoincidence of the zeros of Reynolds stress and mean strain, which is frequently observed in asymmetric turbulent shear flows.

Statistical analysis of the deviation of the Reynolds stress from its eddy‐viscosity representation

Turbulent dynamo modeling in a rotating frame is performed to enable studies of reversed field pinches of fusion plasma and planetary magnetic fields. The Reynolds stresses, turbulent electromotive force, and other important correlation functions, which appear in the equations for the mean velocity and magnetic field, are modeled using the mean fields and four bulk turbulence quantities (the magnetohydrodynamic turbulent energy, its dissipation rate, the cross helicity, and the residual helicity). Four equations for those bulk quantities are combined with the mean‐field equations to lead to a self‐consistent dynamo model. Specifically, the importance of the cross‐helicity effect is pointed out. This model shows that the equilibrium state of reversed field pinches of plasma is a neutral state under the condition of vanishing cross helicity. The relevance to the study of the Earth’s and other planetary magnetic fields is discussed.

Self-consistent turbulent dynamo modeling of reversed field pinches and planetary magnetic fields

Turbulent channel and Couette flows are studied numerically by using an anisotropic A>e model. A feature of this model lies in an anisotropic expression for the Reynolds stress and the deviation of the Reynolds stress from an isotropic eddy-viscosity representation is incorporated. Only one kind of wall damping function is introduced to impose the no-slip boundary condition on solid walls. The results obtained show that turbulence quantities of channel and Couette flows are in good agreement with experimental data and numerical results from large-eddy simulation. The anisotropy of turbulent intensities, which the usual A:-e model cannot predict, is well reproduced.

Akira Yoshizawa

Papers

Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling

A Statistically-Derived Subgrid-Scale Kinetic Energy Model for the Large-Eddy Simulation of Turbulent Flows

Statistical analysis of the deviation of the Reynolds stress from its eddy‐viscosity representation

Self-consistent turbulent dynamo modeling of reversed field pinches and planetary magnetic fields

Turbulent channel and Couette flows using an anisotropic k-epsilon model