Implicit Finite-difference Algorithm Research Articles

Transonic flutter analysis of aerodynamic surfaces in the presence of structural nonlinearities

The influence of structural nonlinearities on the transonic aeroelastic behavior of a two-degree-of-freedom system consisting of an airfoil which is plunging and pitching is considered. The preload and freeplay nonlinearity in the spring model are treated using the asymptotic expansion method which takes into account the influence of the higher harmonics. The aerodynamic load vector is obtained by solving the unsteady transonic small-disturbance (TSD) equation using an alternating-direction implicit (ADI) finite-difference algorithm. The frequency domain solution of the aeroelastic equation is obtained using the U-g method, and the aeroelastic response is carried out by applying Newmark-β and Wilson-ϑ methods. A comparison is made between the results obtained from these two solutions. The effects of preload and freeplay magnitudes on the stability of the system and the corresponding reduced frequency and mass density ratio at neutrally stable response are studied.

Simulation of leading-edge vortex flows

An implicit upwind-relaxation finite-difference algorithm solving the incompressible Navier-Stokes equations is employed to simulate low-speed, three-dimensional, laminar, leadingedge vortex flows over three round-edged low-aspect-ratio wings. The effects of grid density, angle of attack, Reynolds number, and wing planform on the flowfield structures and integral values are studied. Computed results are presented and compared with experimental data.

Laser balloon angioplasty: An optical and thermal analysis

Our theoretical assessment of laser balloon angioplasty (LBA) begins with a solution of the popular P1 (diffusion) approximation to the radiative transport equation to map the optical field in an axisymmetric, finite annulus surrounding an infinitely long hollow cylindrical vessel. The inner surface is irradiated by diffuse light at 1.06 μm and delivered by a spiral segment of fiberoptic positioned centrally inside an inflated balloon. The present model considers only a single layer of optically homogenous arterial wall because the hollow is presumbly filled with a non-absorbing and non-scattering fluid. The cylindrical enclosure collects reflected light and redirects it back onto the inner surface. Our model accounts for this ‘integrating’ behavior, unique to enclosures. Absorption of the laser light then raises tissue temperature. The resultant transient and spatial temperature distribution is calculated with an implicit Finite Difference algorithm based on the heat conduction equation. Multiple concentric layers, each with different thermal properties from the other, are considered. However, changes in optical and thermal properties with temperature, and thermal phase transitions have been omitted in the present model. Finally, the Arrhenius rate process integral is applied for damage calculations.

Solution Procedure for Unsteady Two-Dimensional Boundary Layers

An accurate and reliable solution procedure is presented for solving the two-dimensional, compressible, unsteady boundary layer equations. The procedure solves the governing equations in a coupled manner using a fully implicit finite-difference numerical algorithm. Several unsteady compressible and incompressible laminar flows are considered. Example results for two unsteady incompressible turbulent flows are also included. An algebraic mixing length closure model is used for the turbulent flow calculations. The computed results compare favorably with experimental data and available analytical/numerical solutions.

Formation and Destruction of Vortices in a Motored Four-Stroke Piston-Cylinder Configuration

THE effects of the compression ratio, engine speed, boreto-stroke ratio, and valve seat angle on the turbulent flowfield within an axisymmetric piston-cylinder configuration have been studied by means of an implicit finite difference method which solves the conservation equations of mass, momentum, and energy, and two additional equations for the turbulent kinetic energy and its dissipation rate. The numerical results indicate that the turbulent intensity at topdead-center of the compression stroke is independent of the rpm and decreases with decreasing compression ratio. The turbulent intensity also increases when decreasing the bore-tostroke ratio. It has been found that the valve seat angle has the most important effect on the flowfield. A valve seat angle of 45 deg produces two vortical structures which break down and merge by the end of the compression stroke, and persist into the expansion stroke. For smaller valve seat angles there is no vortex breakdown and the vortical structures created during the intake stroke disappear by the end of the compression stroke. The importance of vortical structures on fuel mixing and turbulence levels within an internal combustion engine is also discussed. Contents The purpose of this paper is to report some numerical results concerning the effects of the compression ratio, boreto-stroke ratio, rpm, and valve seat angle on the formation and destruction of vortices, and turbulent levels within an axisymmetric piston-cylinder configuration equipped with a centrally located valve. The calculations reported here were performed with an implicit finite difference algorithm that solves the conservation equations of mass, axial and radial momentum components, and energy. Turbulence has been modeled by means of a two-equation model for the turbulent kinetic energy and its dissipation rate. The motion of the valve has been accounted for by means of the two-domain technique developed in Ref. 1. Calculations were performed at several engine speeds, i.e., rpm, and showed that as long as the valve seat angle is less than 45 deg (this angle is measured with respect to the engine cylinder centerline) the flowfield in the intake stroke is characterized by the formation of two vortical structures. One of these structures is located at the corner between the cylinder

Two-Dimensional Inlet Simulation Using a Diagonal Implicit Algorithm

A modification of an implicit approximate-factorization finite-difference algorithm applied to the two-dimensional Euler and Navier-Stokes equations in general curvilinear coordinates is presented for supersonic freestream flow about and through inlets. The modification transforms the coupled system of equations Into an uncoupled diagonal form which requires less computation work. For steady-state applications the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. Solutions are given for inviscid and laminar flow about a two-dimensional wedge inlet configuration. Comparisons are made between computed results and exact theory.

A diagonal form of an implicit approximate-factorization algorithm

A modification of an implicit approximate-factorization finite-difference algorithm applied to partial differential equations is presented. This algorithm is applied to the two- and three-dimensional Euler equations in general curvilinear coordinates. The modification transforms the coupled system of equations into an uncoupled diagonal form that requires less computational work. For steady-state applications, the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. The diagonal algorithm reduces the storage requirement of the implicit solution process and therefore has an important effect on the application of implicit finite-difference schemes to vector processors. Results are presented for realistic two-dimensional transonic flow fields about airfoils. Computation costs are reduced 24–34%.

Numerical Analysis of Turbulent Flow Along an Abruptly Rotated Cylinder

A parabolic system of equations governing turbulent, incompressible fluid motion along a stationary cylinder fitted with a rotating aft-section is solved with an implicit finite-difference algorithm. The mean flow is steady, axisymmetric, and characterized by abrupt skewing of the boundary layer. The equations of mean motion are closed with an eddy viscosity model based on Cebeci’s axisymmetric formulation. The analysis is compared with the experimental results of Bissonnette and Mellor, who have obtained data for ratios of cylinder wall speed to free-stream speed equal to 0.936 and 1.800. Predicted components of wall shear stress agree with the experimental values within the uncertainty associated with the experiment, and early boundary-layer development is predicted reasonably well. However, the analysis fails to predict the experimentally observed acceleration of the mean, axial flow.

Implicit Finite-Difference Simulation of Flow about Arbitrary Two-Dimensional Geometries

Finite-difference procedures are used to solve either the Euler equations or the thin-layer Navier-Stokes equations subject to arbitrary boundary conditions. An automatic grid generation program is employed, and because an implicit finite-difference algorithm for the flow equations is used, time steps are not severely limited when grid points are finely distributed. Computational efficiency and compatibility to vectorized computer processors is maintained by use of approximate factorization techniques. Computed results for both inviscid and viscous flow about airfoils are described and compared to viscous known solutions.

An implicit finite-difference algorithm for hyperbolic systems in conservation-law form

An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.