Boundary-fitted Curvilinear Coordinates Research Articles

Strain rate on curved surfaces: A differential-geometric approach

Utilizing the geometrical characteristics of a simple smooth curve embedded in an orientable surface of arbitrary shape, expressions are derived in boundary-fitted curvilinear coordinates, for the normal and tangential strain rate of a fluid which contacts the surface. The derived expressions can be applied to stream-surfaces, solid walls with hydrodynamic slippage and two-fluid interfaces. The use of the expressions is discussed with specific examples and critical examination of the literature.

Application of a high-resolution scheme in simulation of flow in curved channel using boundary-fitted curvilinear coordinates

This paper demonstrates the effectiveness of a high-resolution numerical scheme in boundary-fitted curvilinear coordinates for simulation of flow at channel bends. Two-dimensional shallow water equations, which are transformed from physical domain to computational domain, are solved by a robust high-resolution scheme particularly developed for boundary-fitted curvilinear coordinates. Finite-volume method is adopted for discretization of governing equations in computational domain. Turbulence shear stresses are included in the model by eddy-viscosity approach. Moreover, the effect of secondary flow, which is significant in channel bends, has been included in the model and the results have been compared with the results achieved by neglecting this phenomenon. Secondary flow intensity is evaluated by an analytical approach; the deviation of bed shear stress from main flow is related to secondary flow intensity and is considered in the model as well. All numerical results are compared with experimental data and they show satisfactory agreement.

Adapting Boundary-Fitted Curvilinear Coordinates for Mesoscale Model and Verification of its Validity

Adapting Boundary-Fitted Curvilinear Coordinates for Mesoscale Model and Verification of its Validity

A coupled boundary element-finite difference solution of the elliptic modified mild slope equation

The modified mild slope equation of [5] is solved using a combination of the boundary element method (BEM) and the finite difference method (FDM). The exterior domain of constant depth and infinite horizontal extent is solved by a BEM using linear or quadratic elements. The interior domain with variable depth is solved by a flexible order of accuracy FDM in boundary-fitted curvilinear coordinates. The two solutions are matched along the common boundary of two methods (the BEM boundary) to ensure continuity of value and normal flux. Convergence of the individual methods is shown and the combined solution is tested against several test cases. Results for refraction and diffraction of waves from submerged bottom mounted obstacles compare well with experimental measurements and other computed results from the literature.

A shock-capturing model based on flux-vector splitting method in boundary-fitted curvilinear coordinates

In this paper, a robust and accurate high-resolution finite-volume scheme is presented which employs flux-vector splitting (FVS) as the building block for solution of shallow water equations in boundary-fitted curvilinear coordinates. Eddy viscosity approach is used to accommodate shear stresses due to turbulence. Splitting of the convective terms is achieved via flux Jacobians whereas Liou–Steffen Splitting (LSS) technique, but in transformed coordinates, is used to split pressure terms. Limited flux gradients are also used to increase the computational accuracy of evaluation of interface fluxes and decrease the excessive numerical dissipation associated with FVS. This will completely remove spurious oscillations in high-gradient regions without introducing too much numerical dissipations. The method is tested for some classic simulations including hydraulic jump, 1D dam break and 2D dam break problems. The results show very satisfactory agreement with experimental data, analytical solutions and other numerical results.

Numerical modelling of barotropic tidal dynamics in the strait of Messina

A two-dimensional model for simulation of the intensive tidal dynamics in the Strait of Messina is presented. An initial boundary-value problem for the viscous shallow-water equations in the complex domain representing the Strait is transformed to the boundary-fitted curvilinear coordinates mapping the domain onto a rectangle with two opposite open boundaries. The transformed equations in the form of contravariant fluxes are integrated by the semi-implicit difference method. The results of computation include tidal maps of the M 2- and M 4-waves, currents induced by the M 2-tide and the sum of the main harmonics, a tidal energetic balance, main eddies, and residual circulation.

Two-dimensional numerical equilibria of field-reversed configuration in the strong mirror field

Two-dimensional numerical equilibria of field-reversed configuration (FRC) plasmas in the strong mirror field applied externally are studied by means of the Grad–Shafranov equation. Appropriate choice of the pressure function is necessary to obtain a thin and elongated equilibrium, as observed in the FRC Injection Experiment (FIX) [H. Himura et al., Phys. Plasmas 2, 191 (1995)]. To solve the Grad–Shafranov equation, the finite difference method is used, applying the boundary-fitted curvilinear coordinates and the attracted grids near the separatrix. The outstanding feature of the equilibria is the presence of a narrow and sharp spike in the toroidal current profile near the separatrix.

Analysis of Yield-Power-Law Fluid Flow in Irregular Eccentric Annuli

Fully developed laminar axial flow of yield-power-law fluids in eccentric annuli has been investigated numerically. The annuli may be fully open or partially blocked. General nonorthogonal, boundary-fitted curvilinear coordinates have been used to accurately model the irregular annular geometry due to the presence of a flow blockage. A computer code has been developed using a second-order finite-difference scheme. An exponential model for the shear stress, valid for both yielded and un-yielded regions of the flow, is used in the computation. The effects of pressure gradient, eccentricity, and blockage height on the flow rate have been studied and the results are presented. The flow rate is found to increase with increasing eccentricity for eccentric annuli without any blockage. For partially blocked eccentric annuli, the flow rate at a particular eccentricity decreases as the blockage height is increased.

Three-Dimensional Numerical Model for Flow through Natural Rivers

This paper presents a three-dimensional numerical model for simulating flow through natural river reaches. The model solves the Reynolds-averaged Navier-Stokes (RANS) equations closed with the standard \ik-e turbulence model. Large-scale roughness and multiple islands are directly resolved by employing boundary-fitted curvilinear coordinates in conjunction with a multiblock approach. Small-scale bed roughness is accounted for using a two-point wall functions approach. Calculations are carried out for flow through a 4 km stretch of the Columbia River, downstream of the Wanapum Dam, for which detailed field and laboratory measurements were collected for a range of power plant operating conditions. Measurements at one operating discharge are employed to calibrate the small-scale roughness distribution in the numerical model. Subsequently, the calibrated model is validated by comparing the computed results with laboratory and field measurements for other discharge combinations. These comparisons demonstrate the ability of the model to capture most experimental trends with remarkable accuracy.

A Nonhydrostatic Mesoscale Ocean Model. Part II: Numerical Implementation

Abstract The nonhydrostatic model with a free surface is numerically implemented in boundary-fitted curvilinear coordinates to model the mesoscale circulation in an ocean basin with natural topography. A semi-implicit numerical scheme is used, and the directional inhomogeneity in the elliptic equation for pressure is exploited to speed up the computation of its solution while using the multigrid method. The model is used to simulate the circulation in the Gulf of Mexico. We observe the formation of the Loop Current and several eddies. The flow is very strongly controlled by the topography and our numerical experiments reveal that in the bottom layers, the flow along topographic contours is in the opposite direction of the anticyclonic circulation in the top layers.

Orthogonal mapping in two dimensions

A method for the generation of orthogonal boundary-fitted curvilinear coordinates for arbitrary simply- and doubly-connected domains is developed on the basis of the theory of quasi-conformal mappings of quadrilaterals and of previous work by Ryskin and Leal. The method has useful applications in orthogonal grid generation in two-dimensional and axi-symmetric domains and in the extension of rapid elliptic solvers and spectral methods to complex geometries. A new technique for the calculation of the conformal module of quadrilaterals is also presented.

Numerical Solution of Incompressible Flow of Power-Law Fluid Using Boundary-Fitted Curvilinear Coordinates.

A numerical method using boundary-fitted curvilinear coordinates (BFC) is applied to compute non-Newtonian flow equations in primitive variable formulations for an incompressible inelastic power-law fluid. The flow equations are solved on a special staggered grid by the method of lines using the rational Runge-Kutta time integration scheme and the central finite difference method for spatial discretization, while the Poisson equation is solved by means of the successive overrelaxation (SOR) method using the modified simplified marker and cell (MSMAC) method. It is concluded that the results using BFC are comparable to those using Cartesian coordinates, and the proposed method can be applied to the numerical solution of flow problems of a power-law fluid in arbitrary computational domains.

Use of boundary-fitted curvilinear coordinates for the numerical simulation of complex viscoelastic flows

The method of boundary-fitted curvilinear coordinates is used for the removal of the problem of boundary complexity from finite difference simulations of viscoelastic flow. The flow of an upper convected Maxwell fluid is then analysed in various geometrically complex channels using numerically generated curvilinear computational meshes. Convergent numerical solutions are obtained fro an extended range of the elasticity parameter by use of a third-order upwinding scheme in the numerical solution of the constitutive model. The corner vortex is predicted to grow with elasticity in 4:1 planar contractions with smooth edges; this growth is largely independent of the mesh size and strongly dependent on the particular shape of the entry section, which also strongly affects the stress behaviour near the wall and the range of convergence of the computational algorithm. The elastic character of the fluid is further manifested by the suppression of the secondary flow in a converging—diverging channel.

A Mac Scheme in Boundary-Fitted Curvilinear Coordinates

A Mac Scheme in Boundary-Fitted Curvilinear Coordinates

Three-Dimensional Inviscid Analysis of the Scramjet Inlet Flowfield

A computer code has been developed to analyze the inviscid flow field in a supersonic combustion ramjet (scramjet) inlet. The code uses the three-dimensional Euler equations in full conservation form to describe the inlet flow. An algebraic numerical coordinate transformation is used to generate a set of boundary-fitted curvilinear coordinates. The governing equations are solved by a time-asymptotic, unsplit, two-step, finite-difference method. This method is highly efficient on the vector processing computers for which the current code is written. Detailed results are presented for two scramjet inlet configurations over a range of Mach numbers. The calculated results are compared with the available experimental and theoretical results.

Generation of orthogonal and nearly orthogonal coordinates with gridcontrol near boundaries

A NUMERICAL procedure is presented for the generation of boundary-fitted curvilinear coordinates with controllable mesh spacing and orthogonality. The technique is based on a straightforward two-step procedure involving the solution of Poisson's equation. The method is capable of generating orthogonal grids with partial control of the mesh spacing, or nearly orthogonal grids with strict control of the mesh spacing. The inhomogeneous terms in the Poisson equations are automatically adjusted during the grid generation. The technique is applicable to two-dimensional simply-connected regions typical of airfoils, cascades, diffusers, and inlets.

Numerical Solution for Airfoils near Stall in Optimized Boundary-Fitted Curvilinear Coordinates

Boundary-fitted curvilinear coordinate systems are optimized for viscous flows about arbitrary airfoils at angles of attack such that boundary-layer-dependent grid systems for high Reynolds numbers are generated efficiently. The grid systems are utilized in implicit finite-difference solutions. Solution of a one-dimensional model equation is compared with the theoretical solution. The unsteady Navier-Stokes equations are solved for the incompressible flow around a cylinder and around NACA airfoils approaching stall. The predicted flows around a NACA 6412 airfoil near stall at Reynolds numbers of 4x 10 and 2x 10 are compared with the experimental observations obtained in a smoke tunnel.