No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
SUPG stabilized finite element resolution of the Navier–Stokes equations
Abstract
In this paper an analysis of the viscous incompressible flow has been carried out, from the very definition of the governing equations, up to the resolution of some practical problems, passing through the comprehensive study of the stabilized finite element techniques used in their resolution. As a consequence of this analysis, a code based upon a realistic interpretation of the forces has been written, which allows for the modelling of the open channel flow, with optimum results in the resolution of some benchmark and real flow problems related with the wastewater industry.
... Finally, S t is the turbulent term. Due to the simplifications made in the deduction process, several slightly different formulations for this term can be found in literature [2,9,20,21,22]. The expression used in the present work has been deduced in [19] and it has been compared [19,22] with other expressions, achieving good results. The turbulent term, then, is expressed as ...
... To check the effectiveness of introducing the turbulent term in the 2D-SWE, as well as to compare two discretizations of it (that will be described in section 6), the cavity flow problem has been chosen. It is a classical benchmark for the 2D Navier-Stokes equations with a maximum Reynolds number of 10000, over which value the steady solution becomes unstable [21]. In the Cavity Flow, the form of the streamlines, the u-velocity at the central section and the position of the circulation center depend completely on the imposed value of the viscosity. ...
... This test is not common in the shallow water literature and experimental results have not been found as it might be expected. However there are many numerical results and comparisons between them [21,24,25,26,27]. ...
A first-order finite volume model for the resolution of the 2D shallow water equations with turbulent term is presented. An upwind discretization of the equations that include the turbulent term is carried out. A method to reduce the excess of numerical viscosity (or diffusion) produced by the upwinding of the flux term is proposed. Two different discretizations of the turbulent term are compared, and results for uniform distributions of the viscosity are presented. Finally, two discretizations of the time derivative which are more efficient than Euler's are proposed and compared. Copyright © 2008 John Wiley & Sons, Ltd.
... An alternative computational method based on compressibility was originally proposed by Chorin [4]. The basic idea is the inclusion of additional terms to the equation of continuity of incompressible fluid flows, which is further explored by Temam [29], Malan et al. [25], Vellando et al. [38], Madsen and Schäffer [24], Drikakis [5], etc. This type of computational method is called the artificial compressibility method. ...
... Additional stabilization terms are used, e.g., the PSPG term, which is based on the original idea by Brezzi and Pitkäranta [2], and the LSIC term, the original idea of which comes from the GLS method [7,9,36], etc. Those are fully discussed by Hughes et al. [11][12][13][14][15][16], etc., Tezduyar et al. [22,[31][32][33][34][35], etc., and others [1,23,38], etc. ...
There is discontinuity between compressible and incompressible states in fluid flows. If we subtract the thermal effect from compressible fluid flows, we obtain adiabatic fluid flows, from which incompressible fluid flows are obtained if we let the acoustic velocity tend to infinity. Thus, we employ the idea of adiabatic fluid flows to solve incompressible flows. In the computation, the physical value of the acoustic velocity is employed. This idea corresponds to an extension of artificial compressibility under physical considerations. We present the new SUPG formulation of adiabatic fluid flows, by which not only the effect of SUPG but also those of PSPG and LSIC of incompressible fluid flows are derived. After the numerical verifications, three-dimensional solitary wave propagations are computed. Close agreement between computed and experimental values is obtained.
... A rigorous analysis of SUPG is presented in [17,18]. A great amount of SUPG applications can be found in advection-diffusion problems [19,20] and Navier-Stokes problems [21,22]. The main issue of SUPG is choosing the stabilization parameter. ...
In this work, various high-accuracy numerical schemes for transport problems in fractured media are further developed and compared. Specifically, to capture sharp gradients and abrupt changes in time, schemes with low order of accuracy are not always sufficient. To this end, discontinuous Galerkin up to order two, Streamline Upwind Petrov-Galerkin, and finite differences, are formulated. The resulting schemes are solved with sparse direct numerical solvers. Moreover, time discontinuous Galerkin methods of order one and two are solved monolithically and in a decoupled fashion, respectively, employing finite elements in space on locally refined meshes. Our algorithmic developments are substantiated with one regular fracture network and several further configurations in fractured media with large parameter contrasts on small length scales. Therein, the evaluation of the numerical schemes and implementations focuses on three key aspects, namely accuracy, monotonicity, and computational costs.
... The Navier-Stokes equations are a set of nonlinear time-dependent partial differential equations That have a wide range of applications in aerospace, ocean engineering, vehicle engineering, civil engineering, etc. [1][2][3]. With advances in computing power, many numerical methods have been applied to simulate two-dimensional incompressible flows. ...
The backward substitution method is a newly developed meshless method that has been used for the simulation of many problems in science and engineering with high accuracy and efficiency. In this paper, we explore the feasibility of employing the backward substitution method for simulating two-dimensional incompressible flows. Two non-increment pressure correction projection methods are considered to decompose the original velocity and pressure coupling system into two boundary value problems of intermediate velocity and pressure. Then, two boundary value problems are solved by the backward substitution method in each time iteration step. Five numerical examples are provided to demonstrate the accuracy, computational efficiency and convergence of the method. Comparisons with some existing meshless methods verify the method's advantages and potential applications to engineering.
... Among other examples of topics related to the water industry, we only cite here, for the sake of completeness, the following: two-phase problems (see [49,50]), suspension flows, partial phase separation and cavitation (see [51]), fluid-structure interaction (see [52,53]), 3-D problems for higher Reynolds number fluid flows (turbines, etc.) (see [54]), modelling of water treatment installations (see [55]), etc. ...
The approximate solution of imbibition phenomenon governed by non-linear partial differential equation is discussed in the present paper. Primary oil recovery process due to natural soil pressure, but in the secondary oil recovery process water flooding plays an important role. When water is injected in the injection well for recovering the reaming oil after primary oil recovery process, it comes to contact with the native oil and at that time the imbibition phenomenon occurs due to different viscosity. For the mathematical modelling, we consider the homogeneous porous medium and optimal homotopy analysis method has been used to solve the partial differential equation governed by it. The graphical representation of the solution is given by MATHEMATICA and physically interpreted.
... In this way the difference between the u values on both sides of the interface, which is responsible for the numerical viscosity added, is reduced (see Figure 3). The streamlines obtained using the second order scheme are shown in Figure 4 and we see that they agree very well with the reference streamlines taken from Vellando et al. [4] ( Figure 5). ...
A second order finite volume model for the resolution of the two dimensional shallow
water equationswith turbulent term is presented. It is shown that, if a first order
upwind method is used to discretize the hydrodynamic equations, a considerable
amount of numerical viscosity (or diffusion) is produced. For this reason a second
order method has been developed, which makes use of the mean gradient of
the variables in a cell. To compare the first and second order methods, the Cavity
Flow problem is used. Then a backward step problem is solved, using the k − ε
turbulence model to calculate the turbulent viscosity at every point. The results are
compared with experimental measures and they confirm the good behavior of the
model.
Keywords: finite volumes, shallow water equations, numerical viscosity, turbulent
term, gradient mean values.
... Computational modeling nowadays is widely used in studies of water treatment plants, for instance in chlorine contact tanks [40,51,45,34,21,19], settling tanks [22], clarification basins [44], distribution system [47], bacterial inactivation [39], and storage tanks [27,28,49,3]. Computational models are also used for the chlorine concentration decay [43,15]. ...
In this paper we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. The mathematical model contains several ingredients particularly designed for this problem, namely, a boundary condition to model falling jets on free surfaces, an arbitrary Lagrangian-Eulerian formulation to account for the motion of the free surface due to demand and supply of water, and a coupling of the hydrodynamics with a convection-diffusion-reaction equation modeling the time evolution of chlorine. From the numerical point of view, the equations resulting from the mathematical model are approximated using a finite element formulation, with linear continuous interpolations on tetrahedra for all the unknowns. To make it possible, and also to be able to deal with convection dominated flows, a stabilized formulation is used. In order to capture the sharp gradients present in the chlorine concentration, particularly near the injection zone, a discontinuity capturing technique is employed. This article is protected by copyright. All rights reserved.
... The 2D study of viscous fluids with Reynolds numbers below 10.000 can be very accurately solved by the utilization of constant values for the viscosity in the turbulent term [2], but in most cases of practical interest it is neccesary to calculate the turbulent viscosity at every point and a turbulence model is therefore needed. The depth-averaged k − ε model has been used to obtain the turbulent viscosity by many researchers [3] and it has been chosed for our work. ...
The behaviour of a fluid in 3D may be described by the Navier-Stokes equations which are a hy-perbolic system of non linear conservation laws. Their complexity has led to the development of the two-dimensional shallow water equations (2D-SWE) based on some simplifying hypothesis, the most important of which is the hydrostatic pressure distribution [1]. This set of equations describes remark-ably well the fluid behavior when the ratio of the depth to the horizontal dimensions is small and the magnitude of the vertical velocity component is much smaller than the magnitude of the horizontal velocity components at the space and time scales of interest for the resolution of a given problem. This situation can be found, for instance, in the flow in channels and rivers or tidal flows. 2D-SWE take into account the effects of turbulence both through the frictional terms and the diffusion-like term, that involves second derivatives. Frictional terms quantify the turbulence effects in the verti-cal, while the second derivatives term quantifies the turbulent losses produced by the horizontal mixing of momentum. This last term may not be significant in many practical problems when we only need an estimate of energy losses and 2D-SWE are frequently used without considering the second derivatives term, what is a reasonable simplification in many cases but not always. Thus, in the simulation of flows in which recirculation zones play a significant role, the inclusion of this turbulent term may become very important.
... Among other examples of topics related to the water industry, we only cite here, for the sake of completeness, the following: two-phase problems (see [49,50]), suspension flows, partial phase separation and cavitation (see [51]), fluid-structure interaction (see [52,53]), 3-D problems for higher Reynolds number fluid flows (turbines, etc.) (see [54]), modelling of water treatment installations (see [55]), etc. ...
The study field of water comprises a large variety of activities and interests, and therefore, areas of work. These areas face real engineering problems and, as a consequence, the contributions by some techniques from applied mathematics are really important. On the one hand, it is necessary to have analysis tools that allow us to carry out reliable simulations of the different models by analyzing different configurations, operation modes, load conditions, etc., in order to study existing installations from their basic characteristic data. They are determinist processes whose mathematical expressions consist of coupled systems of different types of equations, algebraic, ordinary differential, and partial differential equations, typically nonlinear, for which specific numerical techniques are necessary. In addition, given the uncertainty of many of the data (especially in existing configurations), it is frequently necessary to solve important inverse problems where other techniques (statistical, minimum quadratic, etc.) are also highly interesting. On the other hand, design is necessary in order to carry out new configurations. Frequently, the absence of initial data and the need of fulfilling different types of restrictions (some of them prone to subjectivity) turn design processes into real optimization problems where the classical methods frequently fail and for which the most current techniques based on neural networks, genetic algorithms, fuzzy theory, chaos theory, etc. are indispensable. This document describes the most important mathematical aspects needed in some of the stages of the integral cycle of water, with special emphasis on the most current topics.
... To obtain the real behavior of the flow, the resolution of the all-term-including Navier-Stokes equations is needed, and the numerical resolution of these equations involve tough stability problems that do not arise for the former simplified formulae. The code has been previously applied to the resolution of some problems related to environmental engineering, with very interesting results (Vellando et al., 2002). The particulars regarding the numerical resolution of the equations in the code HYDRAFEM can be found further in the Numerical Formulation section. ...
This work shows improvements made in mixing operations at water treatment plants, as a result of the hydrodynamic analysis of the mixing processes carried out by the use of a Finite Element Model. The code, developed in the Civil Engineering Department of the University of La Coruña, Spain, solves the Navier-Stokes equations that rule viscous incompressible flow by using a Streamline Upwind/Petrov-Galerkin (SUPG) stabilization technique. The incorporation of the SUPG formulation leads to obtaining stable solutions for Reynolds numbers of a moderate order in connection with meshes that are not very refined. Some water treatment units present significant deficiencies in their design. The numerical evaluation of the flow avoids the high expenses of the trial-and-error processes involved in installing and removing the mixing mechanisms and those derived from the need to halt the water treatment processes. As a result, an optimum design of the treatment plant is obtained at a low cost.
The present work consists in the implementation of a second order Finite Volume Method for the numerical resolution of the Two-Dimensional Shallow Water Equations (2D-SWE) with turbulent term. The model achieves second order accuracy in space by making use of the mean gradient of the variables in a cell and second order in time by using the second order Henn Method. The eddy viscosity values are obtained from the Rodi's depth-averaged κ - ε method. The model has been applied to the Cavity Flow Problem (for different Reynolds numbers) and to a real problem to compare the numerical results with some experimental measurements.
Started with the concept and fundamental of Subgrid Scale stabilized (SGS) method, an improved SSS finite element formulation is proposed to obtain time independent solutions by solving the unsteady incompressible Navier-Stokes equations. Based on some rational approximation and simplification, a new stabilization parameter which has relation with the selective time step size is deduced. As a result, the numerical instability resulting from high Reynolds and/or small time step size is eliminated. Combined with standard k-ε turbulence model and wall function method to estimate turbulent viscosity, the improved SGS method is applied to solve incompressible turbulent flows. The finite element implementation of wall function along with solution of turbulence transport equations is discussed in detail and the positivity preserving limiter for turbulence variables is introduced. A segregated algorithm for time stepping to solve incompressible turbulent flows is developed. The numerical examples of laminar and turbulent flows over a backward facing step are presented in order to show the possibility, stability and high accuracy of the proposed approach for simulating steady incompressible viscous flows.
Purpose
– A new coupled finite element model has been developed for the joint resolution of both the shallow water equations, that governs the free surface flow, and the groundwater flow equation that governs the motion of water through a porous media. The paper aims to discuss these issues.
Design/methodology/approach
– The model is based upon two different modules (surface and ground water) previously developed by the authors, that have been validated separately.
Findings
– The newly developed software allows for the assessment of the fluid flow in natural watersheds taking into account both the surface and the underground flow in the way it really takes place in nature.
Originality/value
– The main achievement of this work has dealt with the coupling of both models, allowing for a proper moving interface treatment that simulates the actual interaction that takes place between surface and groundwater in natural watersheds.
This work shows the achievements obtained in the improvement of the mixing operations in water treatment plants, thanks to the hydrodynamic analysis of the mixing processes carried out by the use of a SUPG stabilized FE model.
The interaction between the free surface and the ground water flows in a basin is a fundamental issue in the study of several environmental problems that require a comprehensive knowledge of the balance and quality of the water flows. Nowadays there is a good number of numerical models in the market, that allow for an evaluation of the free surface hydrodynamics, in one side, and for the groundwater flow in the other side. Still, there is a lack of model skills to evaluate the free surface and ground hydrodynamics in a joint way, and moreover including water quality criteria. The Ingeniería del Agua y del Medio Ambiente (Water and Environmental Engineering Group) group has been working in these fields for some years and has developed a software that allows for the numerical evaluation of this kind of problems. In this work it is shown the application of the developed codes to the working out of a model that evaluates the joined surface and ground water flows in an opencast mine to be restored as a lake. The so-predicted results are presented as a conclusion to shortly elaborate other alternatives in order to restore the opencast mine as a lake. Meirama opencast mine The mining activities that have been taking place in the Meirama opencast lignite mine are to be abandoned at the end of 2007. The firm LIMEISA (LIgnitos de MEIrama Sociedad Anónima) has been exploiting Meirama mine since 1980. The lignite deposit is situated in the province of La Coruña, north-western Spain, 25 km from the city of La Coruña. A 550 Mw power station, fuelled by the lignites of Meirama, was built nearby to produce energy from Meirama coals. At the moment of its closure in 2007 the mine will have produced some 87 million tons of lignite. Mine tailings of some 160 million m3 were moved to a nearby storage tip. The mine has been producing from 4 to 10 million m 3 per year during the first years of the exploitation and some 3 million m 3
The equations governing unsteady flows in secondary settling tanks, a component of the wastewater treatment process, are analysed using the finite element method. The model corresponding to such liquid–solid flows is highly non-linear and coupled, and incorporates the effects of turbulence. The results of numerical simulations are compared against experimental results from tests on full-scale settling tanks, and against results obtained from a finite difference code based on an idealized one-dimensional flux theory. The results compare well with the test results, over the range of applicability of the model. Copyright © 2005 John Wiley & Sons, Ltd.
En el presente trabajo se exponen los resultados de la aplicación de una formulación numérica propuesta por los autores, en la resolución de varios problemas de flujo relacionados con el tratamiento de aguas residuales. La formulación expuesta está basada en el Método de los Elementos Finitos, y resuelve las ecuaciones de Navier-Stokes que gobiernan el flujo viscoso incompresible. El desarrollo de este código permite modelar de manera adecuada el flujo viscoso incompresible y es capaz de evaluar el comportamiento del agua en depósitos y canales de las estaciones de tratamiento de aguas, permitiendo así conseguir un funcionamiento óptimo de éstas, gracias a la modificación de los parámetros hidráulicos y geométricos de estas plantas.
Congreso de Métodos Numéricos en Ingeniería 2005, Granada, Spain Se presenta un modelo en volúmenes finitos para la resolución numérica de las ecuaciones de aguas someras con término turbulento. Tras mostrar el modo en que se discretizan las ecuaciones, se valida el modelo simplificado, es decir sin inclusión del término turbulento. A continuación se obtienen resultados tomando en consideración este término, para distribuciones uniformes de la viscosidad. Se describe luego el modo en que se ha reducido, por medio de un coeficiente, la viscosidad numérica producida por la discretización descentrada. Por último se comprueba la influencia del tamaño de malla en el valor admisible de este coeficiente.
The conventional finite-element formulation of the equations of motion (written in pressure-velocity variables) requires that the order of interpolation for pressure be one less than that used for the velocity components. This constraint is inconvenient and can be argued to be physically inconsistent when inertial effects are dominant. The origins of the constraint are discussed and three new finite-element formulations are advanced that permit equal order representation of pressure and velocity. Of these, the velocity correction scheme, similar to that commonly used in finite-difference procedures, offers superior performance for the examples examined in this paper.
Laser-Doppler measurements of velocity distribution and reattachment length are reported downstream of a single backward-facing step mounted in a two-dimensional channel. Results are presented for laminar, transitional and turbulent flow of air in a Reynolds-number range of 70 < Re < 8000. The experimental results show that the various flow regimes are characterized by typical variations of the separation length with Reynolds number. The reported laser-Doppler measurements do not only yield the expected primary zone of recirculating flow attached to the backward-facing step but also show additional regions of flow separation downstream of the step and on both sides of the channel test section. These additional separation regions have not been previously reported in the literature.
A third-order upwind finite element scheme based on the Petrov-Galerkin formulation is presented for highly accurate solutions of convection dominated flows. A new modified weighting function which is expressed by the sum of a standard weighting function and its second and third derivatives is applied to the formulation. The discrete forms for artificial dissipation terms in the upwind scheme can be rewritten by expressions of fourth and fifth spatial derivatives of an unknown function by applying the Taylor-series expansion. The third-order upwinding technique is first applied to the one-dimensional advection-diffusion equation so that the structures of the upwind scheme are explained in detail. Next the upwind scheme is extended to the incompressible Navier-Stokes equations in multidimensions. Numerical results for a driven cavity flow in a two-dimensional square region are presented to demonstrate the effectiveness and applicability of the upwind finite element scheme proposed in this paper.
Penalty functions are used to modify variational principles used in finite element analysis to enforce constraints. The procedure
is found useful in imposing constraints implicit in the functional or to impose required interelement continuity. While the
process is approximate, quite good practical results can be obtained if sufficient accuracy is available in the computer.
The procedure is illustrated on several problems of interest in elasticity and fluid mechanics.
A Galerkin/Least-Squares formulation using equal order serendipity velocity-pressure elements is used in conjunction with a low-Reynolds number k–ϵ model to solve a turbulent Poiseuille type flow at Re=5000. Predictions are compared to available experimental results. The pressure solution obtained using the GLS formulation is in close agreement with the measurements. This work illustrates that the GLS formulation can be extended successfully to predict the pressure solution for the case of incompressible turbulent flows.
A finite-element method has been developed that combines the segregated velocity-pressure equal-order formulation of the Navier-Stokes equation originated from the SIMPLE algorithm and the streamline upwind Petrov-Galerkin weighted residual method. To verify the proposed finite-element method, driven cavity flow and backward-facing step flow have been considered. The present results are compared with existing experimental results using laser Doppler velocimetry and numerical results using the finite-difference method and the velocity-pressure integrated, mixed-order interpolation method. It has been shown that the present method gives accurate results with less memory and execution time than the conventional finite-element method.
The author gives necessary and sufficient conditions for existence and uniqueness of a class of problems of ″saddle point″ type which are often encountered in applying the method of Lagrangian multipliers. A study of the approximation of such problems by means of ″discrete problems″ (with or without numerical integration) is also given, and sufficient conditions for the convergence and error bounds are obtained.
In applications of the finite element method to complicated nonlinear problems, it is proposed to simplify the operator in each element, typically by a linerization process. Local approximations can then be more easily constructed in the usual manner, and a proper assembly can be made. Such an approach would lead to an approximate global variational statement; or if the variational principle exists but is hard to handle, it should suggest an iterative scheme with a built-in physical basis, hence a better chance to converge. The case of high subsonic flow over a circular cylinder is studied in detail by this method. With linear triangular elements, numerical results are obtained for 100 nodal unknowns in each quadrant. The convergence is extremely rapid up to a free stream Mach number of 0.42, slightly above critical. The accuracy is excellent as compared against Imai's analytical results.
The weighted residual method of finite elements is applied to one- and
two-dimensional problems of convective heat transport in formulations
using nonlinear and linear shape functions and weighting functions which
are not chosen equal to the shape functions. In a one-dimensional
problem with constant coefficients and parabolic shape functions, an
optimum weighting function is found so that the exact solution at the
nodal points is obtained. Two-dimensional weighting functions are
constructed by taking products of one-dimensional functions, giving
nine- and eight-node elements. The upwinding parameters can vary linear
or quadratically from one side to the other side of the element. The
performance of some of these elements is compared with solutions
obtained by using regular eight-node element Galerkin schemes with a
very fine mesh, and the ability of upwinding to reduce oscillations is
clearly demonstrated.
A new Petrov-Galerkin method for the incompressible Navier-Stokes equations is presented. The use of the so-called ‘optimal upwind’ parameter in multidimensions is justified by a time-scale analysis of the relevant physical processes. The resulting procedure circumvents the Babuška-Brezzi condition and allows equal order interpolation for velocity and pressure to be used.
A numerical procedure for solving the time-dependent, incompressible Navier–Stokes and energy equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a time-splitting procedure which has unique features in its formulation as well as in its evaluation from the viewpoint of efficiency and accuracy. The applicability of the proposed formulation was verified by computations and comparison with known results.
The paper is concerned with the finite element solution of the
two-dimensional convective-diffusion equation. The numerical
difficulties, related to the essentially elliptic and parabolic nature
of the two terms, have earlier been overcome, with loss of accuracy, in
the finite difference approximation, by considering a central difference
approximation for the first term and backward upwind differences for the
second term. Since Galerkin type finite element forms result in
expressions similar to those of central differences, this paper seeks a
finite element procedure equivalent to the upwind difference procedure.
This is accomplished through the use of weighting functions of
nonsymmetric forms different from those originally used as shape
functions. By suitable choice of such functions, stability can be
achieved with accuracy loss that is inherent in upwinding in the finite
differences. All other advantages of finite elements are preserved.
A comparison between different stabilized formulations and the standard Galerkin formulation, with and without penalty function approach, is presented for the lid-driven cavity problem. Computations are done for Re = 400, 1000, 5000 and 10 000, employing different grid resolutions. Results obtained are presented as velocity profiles, pressure profiles, velocity vectors and pressure contours and are compared with the benchmark solution. The Galerkin/Least-squares formulation using biquadratic (serendipity) elements exhibits better stability properties compared to the bilinear interpolation case. The SUPG formulation with constant pressure elements seems less accurate than the GLS formulations.
A quantitative evaluation for the penalty function finite element method for two-dimensional viscous incompressible flow using primitive variables is made, using bilinear and biquadratic elements. We conclude that this procedure, more efficient than full velocity pressure formulation, can achieve excellent accuracy using rather coarse meshes, that biquadratic elements produce improved accuracy over the more commonly used bilinears, and that the penalty method effectively imposes the continuity constraint. We point out the dangers of modelling problems containing singularities, as is the case of the driven cavity flow, with finite element formulations using primitive variables and how to overcome these problems. The driven cavity flow for Reynolds number up to 400 is solved and the answers compared with the best available solutions, the Jeffery-Hamel flow in a convergent control channel is solved for Reynolds number up to 61 and the results compared with the analytical solution. Finally extensions of the method are indicated via an example of natural convection in a square cavity for Rayleigh number up to 106, and the advantage of the method discussed.
A finite element procedure is presented for the solution of the Navier-Stokes equations, adopting a segregated velocity-pressure formulation. The procedure is based on the derivation of an approximate pressure equation, which allows the uncoupling of pressure and velocity solutions. This equation is derived from the discretized continuity equation by considering the relationships established between velocity and pressure in the discretized momentum equations. The numerical performance of the proposed method is demonstrated by investigating several examples. The results are compared with finite difference predictions.
This paper describes the application of a recently developed universal adaptive finite element algorithm to the simulation of several turbulent flows. The objective of the present work is to show how the controlled accuracy of adaptive methods provides the means to perform careful quantitative comparisons of two-equation models. The formulation uses the logarithmic form of turbulence variables, which naturally leads to a simple algorithm applicable to all two-equation turbulence models. The new methodology is free of ad-hoc stability enhancement measures such as clipping and limiters which may often differ from one model to the other. Such techniques limit the predictive capability of a turbulence model and cloud the issues of a comparison study. The present procedure results in one adaptive solver applicable to all two-equation models. The approach is demonstrated by comparing three popular turbulence models on a few non-trivial compressible and incompressible flows. We have chosen the following models: the standard k−ϵ model, the k−τ model of Speziale and the k−ω model of Wilcox. Results show that accurate solutions can be obtained for all models, and that systematic comparison of turbulence models can be made.
A new finite element formulation for convection dominated flows is developed. The basis of the formulation is the streamline upwind concept, which provides an accurate multidimensional generalization of optimal one-dimensional upwind schemes. When implemented as a consistent Petrov-Galerkin weighted residual method, it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods.The accuracy of the streamline upwind/Petrov-Galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples. The formulation is extended to the incompressible Navier-Stokes equations. An efficient implicit pressure/explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step. The effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a Reynolds number of 100.
A finite element method is presented for the numerical simulation of time-dependent incompressible viscous flows. The method is based on a fractional step approach to the time integration of the Navier-Stokes equations in which only the incompressibility condition is treated implicitly. This leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost-effective solutions of large-scale problems. Numerical results indicate the versatility and effectiveness of the proposed method.
A review of recent work and new developments are presented for the penalty-function/finite element formulation of incompressible viscous flows. Basic features of the penalty method are described in the context of the steady and unsteady Navier-Stokes equations. Galerkin and “upwind” treatments of convection terms are discussed. Numerical results indicate the versatility and effectiveness of the new methods.
A velocity—pressure integrated, mixed interpolation, Galerkin finite element computation of the Navier-Stokes equations using fine grids, is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions: and the pressure was interpolated using linear shape functions defined on a triangular element, which is contained inside the quadratic element for velocity variables. Comprehensive computational results for a cavity flow for Reynolds number of 400 through 10,000 and a laminar backward-facing step flow for Reynolds number of 100 through 900 are presented in this paper. Many high Reynolds number flows involve convection dominated motion as well as diffusion dominated motion (such as the fluid motion inside the subtle pressure driven recirculation zones where the local Reynolds number may become vanishingly small) in the flow domain. The computational results for both of the fluid motions compared favorably with the high accuracy finite difference computational results and/or experimental data available.
We present in this study the Taylor–Galerkin finite-element model to stimulate shallow water equations for bore wave propagation in a domain of two dimensions. To provide the necessary precision for the prediction of a sharply varying solution profile, the generalized Taylor–Galerkin finite-element model is constructed through introduction of four parameters. This paper also presents the fundamental theory behind the choice of free parameters. One set of parameters is theoretically determined to obtain the high-order accurate Taylor– Galerkin finite-element model. The other set of free parameters is determined using the underlying discrete maximum principle to obtain the low-order monotonic Taylor–Galerkin finite-element model. Theoretical study reveals that the higher-order scheme exhibits dispersive errors near the discontinuity while lower-order scheme dissipates the discontinuity. A scheme which has a high-resolution shock-capturing ability as a built-in feature is, thus, needed in the present study. Notice that lumping of the mass matrix equations is invoked in the low-order scheme to allow simulation of the hydraulic problem with discontinuities. We check the prediction accuracy against suitable test problems, preferably ones for which exact solutions are available. Based on numerical results, it is concluded that the Taylor–Galerkin-flux-corrected transport (TG-FCT) finite-element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles.
A segregated finite element algorithm for the solution of the SUPG formulation of the incompressible steady-state Navier-Stokes equations is investigated in this paper. The method features equal order interpolation for all the flow variables. The SIMPLEST algorithm is employed which results in symmetric coefficient matrices for the momentum equations. The same iterative linear equation solver can therefore be employed for the solution of the momentum and pressure equations. The effect of applying the SUPG weighting functions only to the convective terms is compared with the effect of applying the functions to all the terms in the momentum equations. The effect of using an iterative linear equation solver as opposed to a direct linear equation solver is also discussed.
Stabilized methods are proposed and analyzed for a linearized form of the incompressible Navier-Stokes equations. The methods are extended and tested for the nonlinear model. The methods combine the good features of stabilized methods already proposed for the Stokes flow and for advective-diffusive flows. These methods also generalize previous works restricted to low-order interpolations, thus allowing any combination of velocity and continuous pressure interpolations. A careful design of the stability parameters is suggested which considerably simplifies these generalizations.
The vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10,000 and meshes consisting of as many as 257 × 257 points. For Re = 1000, the (129 × 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional grid-clustering coordinate transformations.
The finite element discretisation technique is used to effect a solution of the Navier- Stokes equations. Two methods of formulation are presented, and a comparison of the effeciency of the methods, associated with the solution of particular problems, is made. The first uses velocity and pressure as field variables and the second stream function and vorticity. It appears that, for contained flow problems the first formulation has some advantages over previous approaches using the finite elemental method[1,2].
We consider the numerical solution on unstructured dynamic meshes of the averaged Navier–Stokes equations equipped with the k–ε turbulence model and a wall function. We discuss discretization issues pertaining to conservation laws, moving grids, and numerical dissipation. We also present a robust spring analogy method for constructing dynamic meshes. We validate our implementation of this two-equation turbulence model and justify its usage for a class of vortex shedding problems by correlating our computational results with experimental data obtained for a flow past a square cylinder. We also apply our solution methodology to the two-dimensional aerodynamic stability analysis of the Tacoma Narrows Bridge, and report numerical results that are in good agreement with observed data.
(REF CONTD) Publ. Co., 1977, x + 500pp.
Galerkin/least-squares finite element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of SUPG, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented.
Finite element vorticity-based methods are applied to the analysis of viscous flows.
Thesis (Ph. D.)--University of Texas at Austin. Vita. Bibliography: ℓ. 102-103.
If your new or interested in the field of CFD this book with introduction to cfd by john d anderson should be in your personal library.
On the existence uniqueness and approximation of saddle point problems arising from Lagrange multipliers. Rev. Franc ßaise Automatique Informatique Reserche Operationnelle, Ser. Rouge Anal. Num e er
- F Brezzi
F. Brezzi, On the existence uniqueness and approximation of saddle point problems arising from Lagrange multipliers. Rev. Franc ßaise Automatique Informatique Reserche Operationnelle, Ser. Rouge Anal. Num e er. 8 (R2) (1974) 129–151.
On the resolution of the Navier–Stokes equations by the finite element method using a SUPG stabilization technique, Application to some wastewater treatment problems
- P Vellando
P. Vellando, On the resolution of the Navier–Stokes equations by the finite element method using a SUPG stabilization technique, Application to some wastewater treatment problems, Doctoral thesis, Universidad de La Coru~ n na (Spain), Marzo, 2001.
Cálculos hidrodinámicos y de transporte de sustancias solubles para flujos turbulentos en lámina libre, Tesis doctoral
- J Bonillo
J. Bonillo, C a alculos hidrodin a amicos y de transporte de sustancias solubles para flujos turbulentos en l a amina libre, Tesis doctoral, Universidad de La Coru~ n na, 2000.
On the existence uniqueness and approximation of saddle point problems arising from Lagrange multipliers. Rev. Française Automatique Informatique Reserche Operationnelle
- Brezzi
On the resolution of the Navier-Stokes equations by the finite element method using a SUPG stabilization technique, Application to some wastewater treatment problems, Doctoral thesis
- P Vellando
Review of finite element analysis of incompressible viscous flow by the penalty function formulation
- Hughes