Smooth Grid Research Articles

Generation of orthogonal grids with control of spacing

A methodology for generating orthogonal curvilinear grids is applied to two-dimensional domains. An important feature of the methodology is its ability to control effectively the grid spacing, especially near the boundaries. This paper summarizes the governing equations used for grid generation. Then the numerical procedure is described, with special emphasis on the scheme used to enhance stability and accuracy of the solution. The significance of the distortion function and the way it is used to control grid spacing are illustrated in geometries commonly found in engine combustion chambers. The influence of various parameters, including number of grid points, relaxation factor, and range of values of the distorion function, on the performance of the method are also investigated. It is concluded that the methodology can successfully produce smooth orthogonal grids with control of spacing in symmetric and nonsymmetric domains.

Simulation of transonic flow in radial compressors

A collection of computer codes for the generation of grids and the solution of the Euler equations have been developed for the purpose of simulating the complex three-dimensional transonic and rotational flow through high pressure ratio radial compressors. The grid generation procedure is based on transfinite interpolation and generates smooth grids of H-, C-, and O-type with a minimum of operations. The Euler solution procedure is based on a centered finite volume scheme with explicit Runge-Kutta time integration and absorbing inflow/outflow boundary conditions. An example solution for a high-speed radial compressor with a total pressure ratio of 1:12 demonstrates that the method is robust and preserves both mass and rothalpy through strong shocks. It also demonstrates that shock-induced separations with reverse flow can be captured by the numerical procedure.

A combination conformal-transfinite mapping method for grids about fin-afterbody combinations

An algebraic procedure is presented for the generation of a smooth computational grid about an afterbody-fin configuration. The method makes use of a sequence of conformal transformations to unwrap the geometry and remove the corner singularities at the fin trailing edge and tail of the afterbody. A 3-D grid is generated by stacking a sequence of 2-D grids of the C type on predetermined, smooth tubular surfaces. Clustering is accomplished by a sequence of one-dimensional stretching functions in physical space. Examples are presented to show the character of the resulting grid.

Blending method for grid generation

A systematic procedure is presented for synthesizing a complex computational grid out of a number of simpler “elementary” grids. This method is useful when a grid is required for a region which, though complex, consists of a number of simpler sub-regions. Frequently, in such cases, validated grid generation methods already exist for the sub-regions, such as the individual lifting surfaces of an airplane. The procedure presented allows a smooth complex grid to be generated which becomes exactly equal to each elementary grid as the surface corresponding to that elementary grid is approached. In this way, the existing generation methods do not have to be changed and can be used as “black boxes,” whether they are algebraic, partial differential equation based, or just given numerically. A number of examples are described in detail.

A geometric approach to two-dimensional finite strain compatibility: interpretation and review

The purpose of this paper is to collect, clarify, augment and modify the authors' previous work on the subject of finite strain compatibility. The derivations of the fundamental equations are reviewed so that the geometric meaning of each step can be explained. Besides providing a basis for the geological interpretations of the equations, these derivations also lead to a useful new form of the strain compatibility equations. We begin by showing that compatibility is a geometric property of continuous and smooth coordinate grids, and we derive and explain the coordinate grid compatibility equations. We then use the fact that every finite deformation may be described by two coordinate grids to derive finite strain compatibility equations in principal coordinates and Cartesian coordinates. The resulting strain compatibility equations are not easily solved for general strain fields in any coordinate system. Nonetheless, we show that many common geological strain patterns have simple geometries for which the compatibility equations can be interpreted. For example, if a deformation has constant strain in one direction, as most shear zones do, then compatibility provides an iterative method for determining the strain throughout the deformed region if the strain is initially known at any one point. Some of the other strain geometries to which we apply compatibility in this paper include simple shear, inhomogeneous pure shear, parallel and similar folding.

Wind tunnel experiments on the effects of turbulence on the aerodynamic behaviour of bridge road decks

Experimental results are presented for several different road deck systems including a truss system, a plate girder, and two box sections. The measurements have been made with a variety of test techniques including a full model in smooth and turbulent shear flow, taut strip models in smooth and turbulent flow and sectional models in smooth flow and grid generated turbulence. Comparisons are made between behaviour in the different flows and between two dimensional and three dimensional response.

Automatic mesh-point clustering near a boundary in grid generation with elliptic partial differential equations

Elliptic partial differential equations are used to generate a smooth grid that permits a one-to-one mapping in such a way that mesh lines of the same family do not cross. Problems that arise due to lack of clustering at crucial points or intersections of mesh lines at highly acute angles, are examined and various forcing or source terms are used (to correct the problems) that are either compatible with the maximum principle or are so locally controlled that mesh lines do not intersect. Attention is given to various schematics of unclustered grids and grid detail about (highly cambered) airfoils.