Improved solution for potential flow about arbitrary axisymmetric bodies by the use of a higher-order surface source method

In recent years the surface-source method of calculating potential flow about arbitrary bodies has been developed extensively and has proved to be a useful tool in a wide variety of low-speed design applications ranging from simple shapes to complicated inlets with centerbodies, multi-element airfoils, and wing-fuselage-pylon-nacelle combinations. Two-dimensional, axisymmetric, and three-dimensional methods have been developed. While the method is generally quite satisfactory, it is desirable to increase computational speed and accuracy for certain applications, particularly interior flows and exterior flows about complicated multiple-body combinations. Such improvements can be realized by refining the formulation. In the basic method the profile curve of a two-dimensional or axisymmetric body is approximated by a large number of straight-line elements over each of which the source density is constant. The so-called higher-order refinement consists of using curved surface elements and a source density that varies over an element. This paper describes the analysis for the axisymmetric case and presents a number of test cases to show the large increases of speed and accuracy that can be obtained with the higher-order formulation.     

Drag minimization and lift maximization in laminar flows via topology optimization employing simple objective function expressions based on body force integration

This paper deals with topology optimization of body shapes in fluid flows, where some new ideas for drag minimization and lift maximization problems are proposed. For drag minimization problems, the objective function is expressed as a body force integration in the flow domain. Also a similar expression of objective function is given for lift maximization problems. Employing those objective function expressions, optimum shapes of bodies in incompressible axisymmetric and two-dimensional flows are numerically investigated.     

A computerized analysis of supersonic nonuniform flows over sharp and spherically blunted cones at angle of attack

A system of computer programs has been developed to predict supersonic inviscid and viscous nonuniform flow fields over sharp and spherically blunted cones at angle of attack. For blunt cones the flow fields considered were axisymmetric wake flows positioned such that the flow in the subsonic nose region remained axisymmetric. For sharp cones, both axisymmetric wake flows and two-dimensional shear flows were considered. The programs used in solving inviscid flow fields incorporate a modified inverse method for solving subsonic flow regions and modified axiymmetric and three-dimensional method of characteristics procedures for solving the supersonic flow regions. Body properties predicted by the inviscid solutions were used as edge data for solution of the corresponding laminar boundary-layers over the bodies. The viscous flow solutions were obtained using axisymmetric and full three-dimensional boundary-layer programs. Typical results from inviscid calculations have shown the development of strong adverse pressure gradients over both sharp and blunt cones in wake flows. In addition a thin entropy layer was found near the surface of both bodies; however, the normal pressure gradient was found to be negligible for the nonuniform flows considered. For the sharp cone in shear flow, property variation along the body was found to be almost linear. In all cases the aerodynamic coefficients were found to be significantly affected by the free-stream nonuniformity. Typical viscous flow field results have shown that relative to uniform flow values the skin friction and heat transfer increase along the windward streamline of both blunt and sharp cones in the nonuniform flows considered. Decreasing the width of the wake in wake flow increases the heat transfer and skin friction.     

Investigation of different isoparametric axisymmetric crack tip elements applied to a complete circumferential surface crack in a pipe

Standard isoparametric finite elements can be used as special crack tip elements in fracture mechanical computations by appropriately shifting the middle nodes in the neighbourhood of the crack tip. Such elements have already been applied to several plane and three-dimensional problems so that this method can be considered as commonly well accepted. In this paper the application of isoparametric axisymmetric elements as crack tip elements to a particular axisymmetric problem is studied. For that reason a complete circumferential crack at the inner surface of a pipe under axial tension is considered. The calculated stress intensity factors are compared with results from the literature. The general purpose finite element programs ASKA and ADINA have been used. In the first case triangular and quadrilateral elements were investigated, in the latter case quadrilateral and collapsed quadrilateral elements. In spite of the rather coarse grids good results for the stress intensity factor were found with the only exception of the collapsed quadrilateral elements.     

A control-volume finite element method for dilute gas-solid particle flows

The formulation of a co-located equal-order Control-Volume-based Finite Element Method (CVFEM) for the solution of two-fluid models of 2-D, planar or axisymmetric, incompressible, dilute gas-solid particle flows is presented. The proposed CVFEM is formulated by borrowing and extending ideas put forward in earlier CVFEMs for single-phase flows. In axisymmetric problems, the calculation domain is discretized into torus-shaped elements and control volumes: in a longitudinal cross-sectional plane, or in planar problems, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the midpoints of the sides. In each element, mass-weighted skew upwind functions are used to interpolate the convected scalar dependent variables and the volume concentrations. An iterative variable adjustment algorithm is used to solve the discretized equations. The capabilities of the proposed CVFEM are illustrated by its application to two test problems and one demonstration problem, using a simple two-fluid model for dilute gas-solid particle flows. The results are quite encouraging.     

Simulation of floating bodies with the lattice Boltzmann method

This paper is devoted to the simulation of floating rigid bodies in free surface flows. For that, a lattice Boltzmann based model for liquid–gas–solid flows is presented. The approach is built upon previous work for the simulation of liquid–solid particle suspensions on the one hand, and on an interface-capturing technique for liquid–gas free surface flows on the other. The incompressible liquid flow is approximated by a lattice Boltzmann scheme, while the dynamics of the compressible gas are neglected. We show how the particle model and the interface capturing technique can be combined by a novel set of dynamic cell conversion rules. We also evaluate the behaviour of the free surface–particle interaction in simulations. One test case is the rotational stability of non-spherical rigid bodies floating on a plane water surface–a classical hydrostatic problem known from naval architecture. We show the consistency of our method in this kind of flows and obtain convergence towards the ideal solution for the heeling stability of a floating box.     

Some theorems about spectrum and finite element approach for eigenvalue problems for elastic bodies with voids

The paper concerns eigenvalue problems for elastic bodies with voids in contact with massive rigid plane punches. The linear theory of elastic materials with voids according to the Cowin–Nunziato model is used. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical and “porous” boundary conditions and the modulus of elastic solid with voids change, are established. A finite element method is proposed for numerical solution of eigenvalue problems for elastic media with voids. Some effective block algorithms for finite element eigenvalue problems with partial coupling are described. Numerical experiments are presented for determining the first eigenfrequencies of an axisymmetric elastic body with voids.     

Numerical study of interference between simple-shape bodies in hypersonic flows

Hypersonic rarefied-gas flows near two side-by-side plates and cylinders, toroidal balloon, plate and cylinder over a plane surface, and plate behind a cylinder in argon, nitrogen, oxygen, and carbon dioxide have been studied numerically using the direct simulation Monte-Carlo technique under the transition flow conditions at Knudsen numbers from 0.004 to 10. Strong influences of the geometrical factor (the ratio of a distance between bodies to a body length) and the Knudsen number on the flow structure about the bodies (shock-wave shapes, the configuration of subsonic flow zones), skin friction, pressure distribution, lift, and drag have been found.     

Lattice Boltzmann simulation of liquid–gas flows through solid bodies in a square duct

The lattice Boltzmann method for two-phase immiscible fluids with large density differences proposed by Inamuro et al. [T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys. 198 (2004) 628–644] is applied to the problem of liquid–gas flows through solid bodies in a square duct. A wetting boundary condition is introduced so that partial wetting on solid surfaces is realized to agree with Cahn theory. Using this method, we investigate the characteristics of wettability in terms of dynamic contact angles between two fluids and a solid wall. Also, we carry out simulations of liquid–gas rising flows through solid bodies in a square duct. It is found from these simulations that the present method can be useful for the problems of liquid–gas flows through complicated geometries.     

Higher-order axial singularity distributions for potential flow about bodies of revolution

The approach of using axial singularity distributions of different orders for representing bodies of revolution in axial flow to solve both the direct and inverse problems has been developed and critically evaluated. A polynomial of arbitrary degree is used to represent the variation of the intensity of the source distribution over each element. The effects of the order of the distribution, the number of elements, the normalization of the body coordinates, the fineness ratio and the geometry of the profile on the performance of the method have been studied in detail by using a number of test cases of known solutions. With appropriate choice of these parameters, this approach for both the direct and inverse axisymmetric problems can be as accurate as the surface singularity approach even for simple bodies with inflection points. However, the present scheme has the advantage of being much simpler and faster. A new technique has been developed for the calculation of the body radius in the authors' iterative inverse problem scheme. This technique proved to be essential for velocity distributions representing bodies with inflection points. Such bodies are of great interest in the design of low drag shapes.     

Analytical Study of Laminar Boundary Layers near Blunted Bodies

Mathematical Models and Computer Simulations - In high-speed flows, blunt body elements having an irregular shape due to which gas dynamic parameters undergo significant changes are, as a rule, the...     

Comparative study of behaviour of some axisymmetric finite elements under static and dynamic loadings

Axisymmetric finite element analysis can be applied to several structures of importance, e.g. reactor buildings, pressure vessels, domes, fluid containers, cooling towers, chimneys, etc. This paper presents a comparative study of some axisymmetric finite elements to judge their efficacy in solving axisymmetric thin as well as thick shells. The elements studied are (i) paralinear, (ii) clement with relative displacement degrees of freedom, (iii) cubilinear, (iv) element with incompatible modes, (v) parabolic, (vi) Lagrangian and (vii) Ahmad's shell element. To enable comparison of the results from these elements, a uniform circular cylindrical structure has been chosen for the analysis. A parametric study involving wall thickness and height of the cylinder as the parameters has been carried out to ascertain the relative competence of these elements under static as well as earthquake loadings. The effect of reduced order of integration has also been examined. The range of suitability of each element has been discussed. It has been concluded that the element with relative displacement degrees of freedom can be used over the entire range of parameters considered whereas the other elements have limited use and should be used with caution.     

Liquid and gas flows in microchannels of varying cross section: a comparative analysis of the flow dynamics and design perspectives

This paper presents a comparative study of the flow of liquid and gases in microchannels of converging and diverging cross sections. Towards this, the static pressure across the microchannels is measured for different flow rates of the two fluids. The study includes both experimental and numerical investigations, thus providing several useful insights into the local information of flow parameters as well. Three different microchannels of varying angles of convergence/divergence (4°, 8° and 12°) are studied to understand the effect of the angle on flow properties such as pressure drop, Poiseuille number and diodicity. A comparison of the forces involved in liquid and gas flows shows their relative significance and effect on the flow structure. A diodic effect corresponds to a difference in the flow resistance in a microchannel of varying cross section, when the flow is subjected alternatively to converging and diverging orientations. In the present experiments, the diodic effect is observed for both liquid and gas as working fluids. The effect of governing parameters—Reynolds number and Knudsen number, on the diodicity is analysed. Based on these results, a comparison of design perspectives that may be useful in the design of converging/diverging microchannels for liquid and gas flows is provided.     

Exact natural frequencies for out-of-plane motion of plane structures composed of curved beam members

Exact finite elements form the basis of a new and convenient procedure for converging with certainty upon any required natural frequency of out-of-plane motion of any plane structure composed of slender elastic curved members. Solution of the inherent transcendental eigenvalue problem is achieved through a variation on the powerful Wittrick-Williams algorithm. Two illustrative examples are included.     

A solution of Fredholm integral equation by means of the spline fit approximation

This paper includes a calculation for two-dimensional and axisymmetric potential flow about an arbitrary body shape by means of the singularity. Vortices are distributed continuously over the body surface using a spline fit function, and the distribution is computed as a solution of Fredholm integral equation. The accuracy of the solution can be found easily by examining whether velocity interior to the boundary surface is regarded as vanishing or not, even if no exact solution exists. The method was applied to potential flows about a circular cylinder, turbine cascades and axisymmetric entrances, and compare with exact analyses or experiments.     

Exact solution for wind tunnel interference using the panel method

Wind tunnel wall interference effects on lifting and non-lifting bodies are computed here for incompressible two-dimensional flows. The flows around bodies under consideration are computed by a panel method using linearly varying vortex distributions. The method is regarded as an exact numerical method. Also to exactly compensate for the wind tunnel wall, we use the method of images of the complete body in the wind tunnel. The image system consists of two cascades which extend infinitely on both sides of the wind tunnel. Thus, the exact wind tunnel wall interference effects are obtained for circular and elliptic cylinders, NACA 23012, NACA 64A010 and NACA 0010 airfoils. The velocity distributions and lift coefficient variations are presented for different blockages. The effects of airfoil incidences and the relative proximity of the two walls are also investigated. The present analysis has given rise to some results which otherwise would not have been possible by earlier existing very approximate methods. Wind tunnel walls are found to change the circulation around the body in the wind tunnel. This effect is further accentuated if we either change the incidence or change the relative proximity of the wind tunnel walls.     

Dynamic finite element analysis of nonaxisymmetric structures

A dynamic finite element method of analysis is developed for structural configurations which are derived from axisymmetric geometries but contain definite nonaxisymmetric features in the circumferential direction. The purpose of the analysis is to develop a method which will take into consideration the fact that the stress and strain conditions in these geometries will be related to the corresponding axisymmetrie solution. This analysis is an extension of previously published work in which a similar approach was developed for static structural problems. The analysis is developed in terms of a cylindrical coordinate system r, θ and z. As the first step of the analysis, the geometry is divided into several segments in the r-θ plane. Each segment is then divided into a set of quadrilateral elements in the r-z plane. The axisymmetric displacements are obtained for each segment by solving a related axisymmetric configuration. A perturbation analysis is then performed to match the solutions at certain points between the segments, and obtain the perturbation displacements for the total structure. The total displacement is then the axisymmetric displacement plus the perturbation displacement. The analysis allows for elastic-plastic materials with orthotropic yield criterion based on Hill's yield function and kinematic strain hardening. The finite element dynamic equations are solved by finite differences by dividing the time domain into incremental steps. The solution has been programmed on a computer and applied to a number of examples.     

Special short wave elements for flow acoustics

A numerical formulation based on the Partition of Unity Method (PUM) is proposed for modelling the propagation of short acoustic waves on irrotational mean flows. The method seeks to reduce the pollution error which exists in conventional FE schemes at high frequencies by using a local basis which is enriched by plane wave solutions of the convected Helmholtz equation. Initially the method is demonstrated with reference to a one dimensional model consisting of a variable area converging–diverging duct with mean flow. Next a simple two-dimensional model of a straight duct with uniform flow, is considered. In both cases the accuracy and the conditioning of the numerical solution is investigated for ranges of frequency and Mach number characteristic of aero-engine bypass ducts.     

Prediction of viscous flow around a fully submerged appended body

This paper describes the first stages in the development and application of a finite-difference calculation procedure employing a two-equation turbulence model (k, ε), to the prediction of the flow around an appended, submerged body.Initial studies were conducted to predict the steady flow field over and behind an appendage protruding through a plane boundary layer. The procedures developed to calculate both the two- and three-dimensional wake fields are presented. For the two-dimensional case the method employed is partially-parabolic and thus takes account of lateral pressure variation but cannot predict flow separation and recirculation. The three-dimensional flow field is also non-parabolic; however, by using the two-dimensional partially-parabolic solution it is possible to decouple the pressure in the momentum equations and thus render the solution parabolic. This considerably simplifies the computation procedure compared with that of a three-dimensional partially-parabolic solution. Comparison between calculated velocities and measured data demonstrate that the technique has the potential to warrant extension to consideration of the flow around an appended body.The application of the partially-parabolic calculation procedure to an unappended axisymmetric body is described and comparison again made between predicted and measured parameters. The calculated axial variation of skin friction and boundary layer thickness are shown to be in close agreement with experimental values but the static pressure gradient and total velocity are both over predicted in the tail region.The implications of the above comparison on the application of the prediction method to submerged bodies is discussed and proposals made for further work.     

Higher order numerical solution of the integral equation for the two-dimensional neumann problem

Interest in the problem of two-dimensional potential flow in arbitrary multiply-connected domains has been stimulated by the need to calculate flow about multiple airfoil configurations consisting of slats and flaps detached from the main airfoil. General methods of solution are based on the use of a singularity distribution over the boundary. The distribution is obtained as the solution of an integral equation over the boundary. In implementing this solution various investigators approximate the boundary by an inscribed polygon, whose faces are small flat surface elements. The singularity on each element is taken as constant by some investigators and linearly varying by others. This paper systematically investigates the effectiveness of higher order approximations of the integral equation, including use of curved surface elements and parabolically-varying singularity. It is found that the approach using flat elements with constant singularity is mathematically consistent as is the next higher-order approach with parabolic elements and linearly varying singularity. The popular approach based on flat elements with linearly varying singularity is shown to be mathematically inconsistent, and examples are presented for which the effect of element curvature is greater than that of the singularity derivative. A number of examples are presented to show that: (1) the higher order solutions give very little increase in accuracy for the important case of exterior flow about a convex body: (2) for bodies with substantial concave regions and for interior flows in ducts, the use of parabolic elements and linearly varying singularity can give a dramatic increase in accuracy; and (3) the use of still higher order solutions leads to a rather small additional gain in accuracy.