A least-squares finite element method for time-dependent incompressible flows with thermal convection

The time-dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity–pressure–vorticity–temperature–heat-flux ( u –P–ω–T– q ) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l²-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 10⁶, lid-driven cavity flow at Reynolds numbers up to 10⁴ and flow over a square obstacle at Reynolds number 200, are presented to validate the method.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows

This paper presents lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) model and incompressible LBGK model‐based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK‐based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 10⁵. However, for the pressure‐driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK‐based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd.     

A mixture differential quadrature method for solving two-dimensional incompressible Navier-Stokes equations

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Simulation of three-dimensional incompressible flows in generalized curvilinear coordinates using a high-order compact finite-difference lattice Boltzmann method

In the present study, a high-order compact finite-difference lattice Boltzmann method is applied for accurately computing 3-D incompressible flows in the generalized curvilinear coordinates to handle practical and realistic geometries with curved boundaries and nonuniform grids. The incompressible form of the 3-D nineteen discrete velocity lattice Boltzmann method is transformed into the generalized curvilinear coordinates. Herein, a fourth-order compact finite-difference scheme and a fourth-order Runge-Kutta scheme are used for the discretization of the spatial derivatives and the temporal term, respectively, in the resulting 3-D nineteen discrete velocity lattice Boltzmann equation to provide an accurate 3-D incompressible flow solver. A high-order spectral-type low-pass compact filtering technique is applied to have a stable solution. All boundary conditions are implemented based on the solution of the governing equations in the 3-D generalized curvilinear coordinates. Numerical solutions of different 3-D benchmark and practical incompressible flow problems are performed to demonstrate the accuracy and performance of the solution methodology presented. Herein, the 2-D cylindrical Couette flow, the decay of a 3-D double shear wave, the cubic lid-driven cavity flow with nonuniform grids, the flow through a square duct with 90° bend and the flow past a sphere at different flow conditions are considered for validating the present computations. Numerical results obtained show the accuracy and robustness of the present solution methodology based on the implementation of the high-order compact finite-difference lattice Boltzman method in the generalized curvilinear coordinates for solving 3-D incompressible flows over practical and realistic geometries.     

Development of an artificial compressibility methodology using flux vector splitting

An implicit, upwind arithmetic scheme that is efficient for the solution of laminar, steady, incompressible, two-dimensional flow fields in a generalised co-ordinate system is presented in this paper. The developed algorithm is based on the extended flux-vector-splitting (FVS) method for solving incompressible flow fields. As in the case of compressible flows, the FVS method consists of the decomposition of the convective fluxes into positive and negative parts that transmit information from the upstream and downstream flow field respectively. The extension of this method to the solution of incompressible flows is achieved by the method of artificial compressibility, whereby an artificial time derivative of the pressure is added to the continuity equation. In this way the incompressible equations take on a hyperbolic character with pseudopressure waves propagating with finite speed. In such problems the ‘information’ inside the field is transmitted along its characteristic curves. In this sense, we can use upwind schemes to represent the finite volume scheme of the problem's governing equations. For the representation of the problem variables at the cell faces, upwind schemes up to third order of accuracy are used, while for the development of a time-iterative procedure a first-order-accurate Euler backward-time difference scheme is used and a second-order central differencing for the shear stresses is presented. The discretized Navier–Stokes equations are solved by an implicit unfactored method using Newton iterations and Gauss–Siedel relaxation. To validate the derived arithmetical results against experimental data and other numerical solutions, various laminar flows with known behaviour from the literature are examined. © 1997 John Wiley & Sons, Ltd.     

Adaptive hybrid (prismatic–tetrahedral) grids for incompressible flows

Hybrid grids consisting of prisms and tetrahedra are employed for the solution of the 3-D Navier–Stokes equations of incompressible flow. A pressure correction scheme is employed with a finite volume–finite element spatial discretization. The traditional staggered grid formulation has been substituted with a collocated mesh approach which uses fourth-order artificial dissipation. The hybrid grid is refined adaptively in local regions of appreciable flow variations. The scheme operations are performed on an edge-wise basis which unifies treatment of both types of grid elements. The adaptive method is employed for incompressible flows in both single and multiply-connected domains. © 1998 John Wiley & Sons, Ltd.     

Finite volume computation of incompressible turbulent flows in general co-ordinates on staggered grids

A brief review of the computation of incompressible turbulent flow in complex geometries is given. A 2D finite volume method for the calculation of turbulent flow in general curvilinear co-ordinates is described. This method is based on a staggered grid arrangement and the contravariant flux componets are chosen as primitive variables. Turbulence is modelled either by the standard k–ε model or by a k–ε model based on RNG theory. Convection is approximated with central differences for the mean flow quantities and a TVD-type MUSCL scheme for the turbulence equations. The sensitivity of the method to the grid properties is investigated. An application of this method to a complex turbulent flow is presented. The results of computations are compared with experimental data and other numerical solutions and are found to be satisfactory.     

Direct numerical simulation of turbulent flow and heat transfer in a square duct with natural convection

In this paper, a direct numerical simulation of a fully developed turbulent flow and heat transfer are studied in a square duct with an imposed temperature difference between the vertical walls and the perfectly insulated horizontal walls. The natural convection is considered on the cross section in the duct. The numerical scheme employs a time-splitting method to integrate the three dimensional incompressible Navier-Stokes equation. The unsteady flow field was simulated at a Reynolds number of 400 based on the Mean friction velocity and the hydraulic diameter (Re _m = 6200), while the Prandtl number (Pr) is assumed 0.71. Four different Grashof numbers (Gr = 10⁴, 10⁵, 10⁶ and 10⁷) are considered. The results show that the secondary flow and turbulent characteristics are not affected obviously at lower Grashof number (Gr ≤ 10⁵) cases, while for the higher Grashof number cases, natural convection has an important effect, but the mean flow and mean temperature at the cross section are also affected strongly by Reynolds stresses. Compared with the laminar heat transfer at the same Grashof number, the intensity of the combined heat transfer is somewhat decreased.     

Investigation of solution of Navier–Stokes equations using a variational formulation

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.     

Modeling the flow past an oscillating airfoil by the method of viscous vortex domains

The Lagrangian vortex method for solving the Navier-Stokes equations is applied for numerically modeling the unsteady flow past a wing airfoil executing angular oscillations in a viscous incompressible flow. Formulas relating the unsteady forces on the airfoil and the vorticity field are derived. The calculated results are compared with the experimental data for the NACA-0012 airfoil executing harmonic oscillations in an air flow at the Reynolds number Re = 4.4 × 10⁴.     

A new general-purpose least-squares finite element model for steady incompressible low-viscosity laminar flow using isoparametric C1-continuous Hermite elements

This paper deals with a critical evaluation of various finite element models for low-viscosity laminar incompressible flow in geometrically complex domains. These models use Galerkin weighted residuals UVP, continuous penalty, discrete penalty and least-squares procedures. The model evaluations are based on the use of appropriate tensor product Lagrange and simplex quadratic triangular elements and a newly developed isoparametric Hermite element. All of the described models produce very accurate results for horizontal flows. In vertical flow domains, however, two different cases can be recognized. Downward flows, i.e. when the gravitational force is in the direction of the flow, usually do not present any special problem. In contrast, laminar flow of low-viscosity Newtonian fluids where the gravitational force is acting in the direction opposite to the flow presents a difficult case. We show that only by using the least-squares method in conjunction with C¹-continuous Hermite elements can this type of laminar flow be modelled accurately. The problem of smooth isoparametric mapping of C¹ Hermite elements, which is necessary in dealing with geometrically complicated domains, is tackled by means of an auxiliary optimization procedure. We conclude that the least-squares method in combination with isoparmetric Hermite elements offers a new general-purpose modelling technique which can accurately simulate all types of low-viscosity incompressible laminar flow in complex domains.     

A NUMERICAL STUDY OF THE FLOW OVER ELLIPSOIDAL OBJECTS INSIDE A CYLINDRICAL TUBE

A numerical scheme is developed to obtain the flow field around one, two and five ellipsoidal objects inside a cylindrical tube. The scheme uses the Galerkin finite element technique and the primitive variable(u–v–p) formulation. The two-dimensional incompressible Navier–Stokes equations are solved numerically by using the direct mixed interpolation method. A Picard iteration scheme is used for the solution of the resulting system of non-linear algebraic equations. The computer code is verified by checking with known analytical solutions for the flow past a sphere. Results for the shear stress distributions along the ellipsoids, forces and drag coefficients are obtained for different geometric ratios and Reynolds numbers. Some of the intermediate computational results on the velocity fields developed are also reported.     

不可压缩机翼绕流的有限谱法计算

结合有限谱QUICK格式求解不可压缩粘性流问题。这一格式用于模拟不同攻角下的NACA1200机翼绕流问题。利用体积力,提出了将流场速度从0加速到来流速度的方法。区别于传统的压力梯度为零的边界条件,推导出一个更精确的压力边界条件。为使速度散度保持为零,在泊松方程中给速度散度一个特殊的处理。这一成果说明了有限谱法不但具有很高的精度,而且能灵活地和其他格式一起构造出新的格式,从而成功地应用到复杂流场不可压缩流动的数值计算中。     

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

A high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

An efficient Legendre-Galerkin spectral method for the natural convection in two-dimensional cavities

An efficient numerical method is developed for solving the natural convection in two-dimensional cavities. The numerical scheme is proposed by using second-order projection scheme in time direction and Legendre-spectral in spatial variable of the incompressible flow. Finally, a series of numerical examples are presented to demonstrate the efficiency of our algorithm. The numerical strategies developed in this article could be readily applied to study other incompressible fluid problems.     

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

A vector-parallel scheme for Navier-Stokes computations at multi-gigaflop performance rates

A class of vector-parallel schemes for solution of steady compressible or incompressible viscous flow is developed and performance studies carried out. The algorithms employ an artificial transient treatment that permits rapid integration to a steady state. In the present work a four-stage explicit Runge-Kutta scheme employing variable local step size is utilized for the ODE system integration. The RK-4 scheme is restructured to allow vectorization and enhance concurrency in the calculation for a streamfunction-vorticity formulation of the flow problem. The parameters of the resulting RK scheme can be selected to accelerate convergence of the RK recursion. Four main procedures are considered which permit vector-parallel solution: a Jacobi update, a hybrid of the Jacobi and Gauss-Seidel method, red-black ordering and domain decomposition. Numerical performance studies are conducted with a representative viscous incompressible flow calculation. Results indicate that a scheme involving domain decomposition with a Gauss-Seidel type of update for the RK four-stage scheme is most effective and provides performance in excess of 8 Gflops on the Cray C-90.     

A method for predicting unsteady potential flow about an aerofoil

A new model is presented for the calculation of the incompressible, inviscid flow around an arbitrary aerofoil undergoing unsteady motion. The technique was developed from the steady flow algorithm of Leishman and Galbraith¹ in which use was made of a linear distribution of panel vorticity. The procedure is in the same class as that of Basu and Hancock² but, because of the particular approach to the manner of specifying the shed vorticity, only a set of linear simultaneous equations needs be solved, unlike the method of Reference 2, complicated by the necessary solution of a quadratic. A brief history of unsteady flow modelling is given in the introduction, followed by the mathematical details of the current method. Results are presented and discussed for a number of cases which clearly illustrate relevant characteristics of unsteady flow.     

A multigrid pseudo-spectral method for incompressible Navier–Stokes flows

A pseudo-spectral solver with multigrid acceleration for the numerical prediction of incompressible non-isothermal flows is presented. The spatial discretization is based on a Chebyshev collocation method on Gauss–Lobatto points and for the discretization in time the second-order backward differencing scheme (BDF2) is employed. The multigrid method is invoked at the level of algebraic system solving within a pressure-correction method. The approach combines the high accuracy of spectral methods with efficient solver properties of multigrid methods. The capabilities of the proposed scheme are illustrated by a buoyancy driven cavity flow as a standard benchmark case. To cite this article: K. Krastev, M. Schäfer, C. R. Mecanique 333 (2005).