Showing papers in "International Journal for Numerical Methods in Fluids in 1983"
TL;DR: In this paper, the authors used mesh refnement and extrapolation to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls.
Abstract: Details are given of the computational method used to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls. Second-order, central difference approximations were used. Mesh refnement and extrapolation led to solutions for 103⩽Ra⩽10 6 which are believed to be accurate to better than 1 per cent at the highest Rayleigh number and down to one-tenth of that at the lowest value.
2,529 citations
TL;DR: In this article, a number of contributed solutions to the problem of laminar natural convection in a square cavity have been compared with what is regarded as a solution of high accuracy, and the purposes of this exercise have been to confirm the accuracy of the bench mark solution and to provide a basis for the assessment of the various methods and computer codes used to obtain the contributed solutions.
Abstract: A number of contributed solutions to the problem of laminar natural convection in a square cavity have been compared with what is regarded as a solution of high accuracy. The purposes of this exercise have been to confirm the accuracy of the bench mark solution and to provide a basis for the assessment of the various methods and computer codes used to obtain the contributed solutions.
678 citations
TL;DR: In this article, the two-dimensional Burgers' equation is used as a model equation for comparing the accuracy of different computational algorithms, and the authors present a more detailed review of the two dimensions of the Burgers's equation.
Abstract: Burgers’ equation is well suited to modelling fluid flows as it incorporates directly the interaction between the non-linear convection processes and the diffusive viscous processes. In one dimension the Cole-Hopf procedure transforms Burgers’ equation into the linear heat conduction equation. As a result many exact solutions of Burgers’ equation are available in the literature. Thus Burgers’ equation has often been used as a model equation for comparing the accuracy of different computational algorithms. This aspect of Burgers’ equation is reviewed by Fletcher.’ The two-dimensional Burgers’ equations
175 citations
TL;DR: In this article, a boundary element method for the solution of non-zero Reynolds number incompressible flows in which the non-linear terms are lumped together to form a forcing function is presented.
Abstract: A formulation of the boundary element method for the solution of non-zero Reynolds number incompressible flows in which the non-linear terms are lumped together to form a forcing function is presented. Solutions can be obtained at low to moderate Reynolds numbers. The method was tested using the flow of a fluid in a two-dimensional converging channel (Hamel flow) for which an exact solution is available. An axisymmetric formulation is demonstrated by examining the drag experienced by a sphere held stationary in uniform flow. Performance of the method was satisfactory. New results for an axisymmetric free jet at zero Reynolds number obtained using the boundary element method are also included. The method is ideal for this type of free-surface problem.
94 citations
TL;DR: In this article, a finite element model for the calculation of near and far field acoustical radiation from axisymmetric turbofan inlets is presented, where the far field is represented by infinite elements or wave envelope elements.
Abstract: Finite element models are presented for the calculation of near and far field acoustical radiation. These models are applied to the specific problem of fan noise radiation from axisymmetric turbofan inlets. In all cases conventional acoustic finite elements are used within an inner region close to the inlet. The far field is represented by infinite elements or wave envelope elements. Theory and results are presented for the case with zero mean flow. Comparisons of computed data with analytic solutions and measured values establish the utility of both the infinite element and wave envelope element schemes in determining the near field values of acoustical pressure. The wave envelope scheme is shown to be effective also in the far field. Both schemes use meshes an order of magnitude more sparse that would be required in conventional numerical discretizations, and may consequently be applied at modest computational cost.
88 citations
TL;DR: In this paper, the problem of wind induced flow in a rectangular basin is examined in detail using the Galerkin method and a no-slip bottom boundary condition with a vertically varying eddy viscosity distribution is employed in the calculation.
Abstract: The three dimensional linear hydrodynamic equations which describe wind induced flow in a sea are solved using the Galerkin method. A basis set of eigenfunctions is used in the calculation. These eigenfunctions are determined numerically using an expansion of B-splines.
Using the Galerkin method the problem of wind induced flow in a rectangular basin is examined in detail. A no-slip bottom boundary condition with a vertically varying eddy viscosity distribution is employed in the calculation. With a low (of order 1 cm2/s) value of viscosity at the sea bed there is high current shear in this region. Viscosities of the order of 1 cm2/s) value of viscosity at the sea bed there is high current shear in this region. Viscosities of the order of 1 cm2/s near the sea bed together with high current shear in this region are physically realistic and have been observed in the sea.
In order to accurately compute the eigenfunctions associated with large (of order 2000 cm2/s at the sea surface to 1 cm2/s at the sea bed) vertical variation of viscosity, an expansion of the order of thirty-five B-splines has to be used. The spline functions are distributed through the vertical so as to give the maximum resolution in the high shear region near the sea bed.
Calculations show that in the case of a no-slip bottom boundary condition, with an associated region of high current shear near the sea bed, the Galerkin method with a basis set of the order of ten eigenfunctions (a Galerkin-eigenfunction method) yields an accurate solution of the hydrodynamic equations. However, solving the same problem using the Galerkin method with a basis set of B-splines, requires an expansion of the order of thirty-five spline functions in order to obtain the same accuracy.
Comparisons of current profiles and time series of sea surface elevation computed using a model with a slip bottom boundary condition and a model with a no-slip boundary condition have been made. These comparisions show that consistent solutions are obtained from the two models when a physically relistic coefficient of bottom friction is used in the slip model, and a physically realistic bottom roughness length and thickness of the bottom boundary layer are employed in the no-slip model.
78 citations
TL;DR: In this paper, a finite element method for the analysis of viscous flow of fluid at high Reynolds numbers is presented, which is based on the explicit numerical integration scheme in time and uses three node triangular finite elements.
Abstract: This paper presents the finite element method for the analysis of unsteady viscous flow of fluid at high Reynolds numbers. The method is based on the explicit numerical integration scheme in time and uses three node triangular finite elements. For the convenience of the formulation, slight compressibility is considered. For the explicit scheme, the selective lumping two step scheme has been successfully employed. Vortex shedding behind a cylinder has been computed and compared with the conventional experimental results. The results agree favourably when both schemes are compared.
66 citations
TL;DR: In this paper, two numerical methods for solving the problem of solute transport in unsteady flow in unconfined aquifers are studied: the method of characteristics (MOC) based on the finite difference method (FDM), and the finite element method(FEM).
Abstract: Two numerical methods for solving the problem of solute transport in unsteady flow in unconfined aquifers are studied. They are the method of characteristics (MOC) based on the finite difference method (FDM), and the finite element method (FEM). The FEM is further subdivided into four schemes: moving mesh, pseudo-Lagrangian (FEM1); stationary mesh, pseudo-Lagrangian (FEM2); pseudo saturated-unsaturated, Eulerian (FEM3); and non-stationary element, Eulerian (FEM4).
Experiments on a one-dimensional flow case are performed to illustrate the schemes and to determine the effect of discretization on accuracy. In two-dimensional flow the above methods are compared with experimental results from a sand box model. Results indicate that for a similar degree of accuracy, the FEM requires less computational effort than the MOC. Among the four FEM schemes, FEM4 appears to be most attractive as it is the most efficient and most convenient to apply.
61 citations
TL;DR: In this paper, two methods are evaluated to reduce the noise level of finite element solutions to the shallow water equations, and these results are compared with published results for equal-order interpolations.
Abstract: Numerical noise has been a problem with finite element solutions to the shallow water equations. Two methods used to reduce the noise level are evaluated, and these results are compared with published results for equal-order interpolations. The two methods are mixed-interpolation (quadratic interpolation for velocity and linear interpolation for sea level) and a spectral form of the wave equation. Whereas mixed interpolation removes the troublesome sea level mode, it can still have considerable noise in velocity. The spectral wave equation is efficient and does not contain the spurious eigenmodes which contribute to high noise levels.
56 citations
TL;DR: In this article, a methodology for solving a particular class of optimum design problems in Fluid Mechanics, namely optimum design problem for aerofoils when the corresponding fluid flow is potential, is described.
Abstract: Described in this paper is a methodology for solving a particular class of optimum design problems in Fluid Mechanics, namely optimum design problems for aerofoils when the corresponding fluid flow is potential. The methods described in this paper operate directly in the physical space, and take advantage of the variational formulation of the partial differential equation modelling the flow. The techniques of optimal control, optimization and the finite element method are used. Numerical examples are also given.
38 citations
TL;DR: In this paper, a successful and economical fractional step algorithm for the convection-dispersion-reaction equation is described, where exact solutions are adopted for the reaction and convection steps, the latter by the introduction of a moving co-ordinate system.
Abstract: A successful and economical fractional step algorithm for the convection-dispersion-reaction equation is described. Exact solutions are adopted for the reaction and convection steps, the latter by the introduction of a moving co-ordinate system. The dispersion step uses an optimized finite difference algorithm which specifically accommodates the grid non-uniformity. The excellent performance of the algorithm is confirmed by numerical experiments together with computations of the Fourier response and integrated square error characteristics.
TL;DR: In this paper, a simple numerical model based on the shallow water equations in radial symmetry is used to simulate both instantaneous and continuous spills of liquefied natural gas (LNG) onto a water surface.
Abstract: A simple numerical model based on the shallow water equations in radial symmetry is used to simulate both instantaneous and continuous spills of liquefied natural gas (LNG) onto a water surface. Using the computed results, a study is made of the similarities and differences in the pool structure resulting from the two types of spills. For instantaneous spills a relation linear on a logarithmic plot is suggested between the maximum pool size and the spill volume. The effects of shear forces and surface cohesivity on the evolution of the spill are also examined.
TL;DR: In this paper, a finite element model to tackle the moving boundary problem of a wave run-up on moderately steep slopes is developed, where the element near the boundary is allowed to change its dimension so that the fluid boundary is closely followed.
Abstract: A finite element model to tackle the moving boundary problem of wave run-up on moderately steep slopes is developed. The special aspects considered in this study are (1) the modification of shallow water equations to accommodate the effect of vertical accelerations and (2) the use of Lagrangian acceleration coupled with an element that adapts itself to the moving boundary closely. The pressure term in the one-dimensional momentum equation is derived using the Eulerian equation in the vertical direction. This takes care of the vertical accelerations which are significant during the motion of a wave on moderately steep slopes. The element near the boundary is allowed to change its dimension so that the fluid boundary is closely followed. Such a flexible element precludes the need for approximation of the variables with regard to the indefinite position of the boundary. This element is split into two when its dimension becomes unduly large compared to the unchanging elements. The need for such a splitting is shown by an examination of the entries in the global matrix. Results of water profile as a wave runs up a structure are given. A brief history of the work on similar problems is outlined.
TL;DR: In this paper, a mixed finite element method for the analysis of 2D Navier-Stokes flow is presented, in which the deviatoric-stress, pressure, and velocity are discretized in each element, and the convective acceleration is treated by the conventional Galerkin approach, and all integrations are performed by necessary order quadrature rules.
Abstract: A formulation of a mixed finite element method for the analysis of unsteady, convective, incompressible viscous flow is presented in which: (i) the deviatoric-stress, pressure, and velocity are discretized in each element, (ii) the deviatoric stress and pressure are subject to the constraint of the homogeneous momentum balance condition in each element, a priori, (iii) the convective acceleration is treated by the conventional Galerkin approach, (iv) the finite element system of equations involves only the constant term of the pressure field (which can otherwise be an arbitrary polynomial) in each element, in addition to the nodal velocities, and (v) all integrations are performed by the necessary order quadrature rules. A fundamental analysis of the stability of the numerical scheme is presented. The method is easily applicable to 3-dimensional problems. However, solutions to several problems of 2-dimensional Navier-Stokes' flow, and their comparisons with available solutions in terms of accuracy and efficiency, are discussed in detail in Part II of this paper.
TL;DR: In this article, shallow water equations in radial symmetry are solved numerically to simulate the collapse of a cylindrical liquid column into an area surrounded by a concentric dike.
Abstract: The shallow-water equations in radial symmetry are solved numerically to simulate the collapse of a cylindrical liquid column into an area surrounded by a concentric dike. The following three subcases of this problem are considered: a liquid column collapsing onto a layer of the same liquid, a liquid column collapsing onto a solid surface, and a column of lighter liquid collapsing onto a heavier liquid (i.e. liquefied natural gas (LNG) spilled onto water). The results for the three categories are compared and the differences and similarities between them are analysed.
TL;DR: In this paper, a survey of finite difference schemes for the steady-state convectiondiffusion equation in one and two dimensions is presented, showing that none of these schemes is universally second order.
Abstract: SUMMARY This paper presents a survey of several finite difference schemes for the steady-state convectiondiffusion equation in one and two dimensions. Most difference schemes have O(h2) truncation error. The behaviour of these schemes on a one-dimensional model problem is analysed in detail, especially for the case when convection dominates diffusion. It is concluded that none of these schemes is universally second order. One recently proposed scheme is found to yield highly inaccurate solutions for the case of practical interest, i.e. when convection dominates diffusion. Extensions to two and three dimensions are also discussed.
TL;DR: In this paper, a finite element stream function formulation is presented for the solution to the two-dimensional double-glazing problem, and a restricted variational principle is used, in conjunction with a triangular finite element of C1 continuity, to discretize the two coupled governing partial differential equations (4th order in stream function and second order in temperature).
Abstract: A finite element stream function formulation is presented for the solution to the two-dimensional double-glazing problem. Laminar flow with constant properties is considered and the Boussinesq approximation used. A restricted variational principle is used, in conjunction with a triangular finite element of C1 continuity, to discretize the two coupled governing partial differential equations (4th order in stream function and second order in temperature). The resulting non-linear system of equations is solved in a segregated (decoupled) manner by the Newton-Raphson linearizing technique.
Results are produced for the standard test case of an upright square cavity. These are for Rayleigh numbers in the range 103−105, with a Prandtl number of 0.71. Comparisons are made with benchmark results presented at the 1981 International Comparison study in Venice. In the discussion of results, emphasis is placed on the variation of local Nusselt number along the isothermal walls, particularly near the corner. This reveals a noticeable source of error in the evaluation of the maximum Nusselt number by lower order discretization methods.
TL;DR: Finite difference methods for solving the linear model describing unsteady state flow in pipelines are considered and these methods are compared with each other in order to determine the best one, which meets the criteria of accuracy and relatively small computation time.
Abstract: SUMMARY Finite difference methods for solving the linear model describing unsteady state flow in pipelines are considered in the present paper. These methods are compared with each other in order to determine the best one, which meets the criteria of accuracy and relatively small computation time. Gas plays an extremely significant role in the fuel-energetic balance of most industrialized countries of the world. High calorific value combined with the facility of transport places it in the group of most valuable row materials. For that very reason its economic utilization is a problem of major importance. It should be dealt with by the optimization (with regard to a given criterion) of both the process of on-line control of gas transport system and the design of new or the reconstruction of existing networks. One cannot properly realize any of the enumerated tasks without first solving problems raised by network simulation. In the process of system control the simulation supplies us with the information on the values of pressures and flows indispensable in the selection of suitable parameters both for compressor stations and reduction stations. In the process of design the simulation allows us to correctly select network configurations, geometrical dimensions of pipelines, as well as the sites of both compressor and reduction stations for given parameters of gas supply and demand. Two kinds of simulation are commonly differentiated: the static one and the dynamic one. The present article deals with the dynamic simulation, i.e. with the case in which the parameters characterizing the gas supply of the system and its load are functions of time (in the static simulation they are independent of time). Correct simulation of dynamic properties necessitates the selection of the suitable mathematical model and the suitable numerical method enabling us to solve this model. Finite difference methods for solving the model elaborated in Reference 1 are considered in the present paper. These methods are compared with each other in order to determine the best one, which meets the criteria of accuracy and relatively small computation time. The investigations described have been undertaken chiefly because in many professional publications (e.g. References 2-4) various numerical schemes had been advanced, whereas the criteria for their selection had not been presented. 0271-2091/83/020125- 11$01.10 0 1983 by John Wiley & Sons, Ltd.
TL;DR: In this paper, a numerical approximation is taken to the solution of the complex flows existing in gas turbine engines with transonic blading and a quasi-3D approach decouples the problem into through-flow and blade-toblade solutions.
Abstract: SUMMARY A numerical approximation is taken to the solution of the complex flows existing in gas turbine engines with transonic blading. The quasi-3D approach decouples the problem into through-flow and blade-toblade solutions. An industrially practical finite element through-flow solution is developed and for blade-to-blade solutions a transonic finite areas method is utilized. The finite element code developed is capable of operating in an analysis or a design mode. In both modes a dynamic relaxation factor is employed and considerable reduction in solution time can be achieved. Comparisons to streamline curvature methods are carried out for simple analytical and complex industrial problems.
TL;DR: In this paper, a variational formulation for the solution of two dimensional, incompressible viscous flows has been developed by one of the authors, and the main objective of the present paper is to demonstrate the applicability of this approach for practical problems and in particular to investigate the introduction of boundary conditions to the Navier-Stokes equations through a VAE formulation.
Abstract: A variational formulation for the solution of two dimensional, incompressible viscous flows has been developed by one of the authors.1 The main objective of the present paper is to demonstrate the applicability of this approach for the solution of practical problems and in particular to investigate the introduction of boundary conditions to the Navier-Stokes equations through a variational formulation. The application of boundary conditions for typical internal and external flow problems is presented. Sample cases include flow around a cylinder and flow through a stepped channel.
Quadrilateral, bilinear isoparametric elements are utilized in the formulation. A single-step, implicit, and fully coupled numerical integration scheme based on the variational principle is employed. Presented results include sample cases with different Reynolds numbers for laminar and turbulent flows. Turbulence is modelled using a simple mixing length model. Numerical results show good agreement with existing solutions.
TL;DR: In this paper, a multi-vortex model of the vortex sheets shed from the sharp leading edges of slender wings is considered, which is designed to deal with those situations in which more than one centre of rotation is formed on the wing, for example on a slender wing with lengthwise camber or with a strake.
Abstract: SUMMARY A multi-vortex model of the vortex sheets shed from the sharp leading edges of slender wings is considered. The method, which is developed within the framework of slender-body theory, is designed to deal with those situations in which more than one centre of rotation is formed on the wing, for example on a slender wing with lengthwise camber or with a strake. Numerical results are presented, firstly for situations where comparison can be made with a vortex sheet model and secondly for cases, such as those described above, where a vortex sheet model is unable to describe the flow. Where comparison is available, agreement is good and in the cases where more than one vortex system is present interesting interactions are obtained. KEY Woms Low-aspect Ratio Wings Leading-edge Separation Vortex Shedding
TL;DR: In this paper, a finite-difference solution of the full axisymmetric, non-linear governing equations was used to obtain the flow field, and a special implicit technique for the Coriolis terms which maintains geostrophy was developed.
Abstract: The flow of an incompressible fluid in a rapidly rotating right circular cylinder is considered. A source/sink mass distribution at the lateral wall, which is azimuthally uniform and symmetric across the midplane, causes a deviation from wheel flow. The container is only partially full and the inner free surface is allowed to deviate slightly from the vertical. A finite-difference solution of the full axisymmetric, non-linear governing equations was used to obtain the flow field. A special implicit technique for the Coriolis terms which maintains geostrophy was developed and is described. The results obtained for a low Rossby number flow compare quite favourably with the linearized solution. Results are also presented for a case wherein the non-linear terms are important.
TL;DR: In this paper, the spatially third-order accurate QUICK finite difference technique is applied to the solution of the depth-integrated equations of motion for steady, subcritical, free surface flow in a wide, shallow, rectangular channel with and without an abrupt expansion.
Abstract: The spatially third-order accurate QUICK finite difference technique is applied to the solution of the depth-integrated equations of motion for steady, subcritical, free surface flow in a wide, shallow, rectangular channel with and without an abrupt expansion. The conservative, control-volume discretization of the equations of motion and the use of QUICK in approximating required cell and cell face average quantities is discussed. Results presented show that it is possible to obtain stable solutions for advective free surface flows without resorting to implicit numerical smoothing.
TL;DR: In this article, the bbundary integral formulation and boundary element method are extended to include lifting flow problems, which involves inclusion of a branch cut in the flow field and imposition of a Kutta condition to determine the circulation, Γ additional boundary integral contributions arise from the cut surface.
Abstract: The bbundary integral formulation and boundary element method are extended to include lifting flow problems. This involves inclusion of a branch cut in the flow field and imposition of a Kutta condition to determine the circulation, Γ Additional boundary integral contributions arise from the cut surface. Techniques for calculating Γ are developed and we treat, in particular, a superposition procedure which permits very efficient computation. Numerical results are presented for an NACA0012 aerofoil at several angles of attack.
TL;DR: In this paper, a new method for fluid mass balance evaluation based on the equivalent nodal source (E.N.S.) concept allows one to express the balance in conservative terms, and interpret finite element equations as nodal balance equations.
Abstract: SUMMARY The quasi-three-dimensional equations controlling the groundwater flow in heterogeneous and interconnected aquifer systems are discretized by finite elements, considering also the aquifer branching. A new method for fluid mass balance evaluation based on the equivalent nodal source (E.N.S.) concept allows one to express the balance in conservative terms, and interpret finite element equations as nodal balance equations. The solution of the system is based on the frontal method. Use of substructures limits the frontal increase in correspondence to the aquifer branching. In the steady state, the frontal method is integrated with an iterative solution technique to eliminate the frontal increase caused by the presence of aquitards. It converges very rapidly, using a forcing technique with an automatic parameter definition. In the unsteady case the same scope is achieved using a predictor-corrector procedure which employs the Crank-Nicolson method in the corrector phase. This very stable procedure permits use of fairly long time-steps and concerns the case of source terms depending on piezometry (problem of interaction between water table and river). This method has been tested with several fairly complex cases.
TL;DR: In this paper, heat transfer associated with forced convection between bundles of cylindrical fuel rods is analyzed using the finite element method. And the solid and fluid zones are analyzed, for temperature distribution, as a single domain.
Abstract: Heat transfer associated with forced convection between bundles of cylindrical fuel rods is analysed using the finite element method. A subchannel technique is employed and the numerical results are compared with previous experimental and numerical values. The solid and fluid zones are analysed, for temperature distribution, as a single domain.
TL;DR: In this article, a numerical model was developed to study the interactions of multiple angled jet flows in the inlet port plane of the Detroit Diesel 6V-92 two-stroke engine cylinder.
Abstract: SUMMARY A numerical model is developed to study the interactions of multiple angled jet flows in the inlet port plane of the Detroit Diesel 6V-92 two-stroke engine cylinder. The random vortex method is used in two dimensions. Results show axisymmetric swirl initially. As flow develops, the centre of the swirl moves to the mid-radius region and begins to precess about the cylinder centre. The flow becomes progressively more chaotic as time progresses.