A dual-time method for two-dimensional unsteady incompressible flow calculations

A method for computing unsteady incompressible viscous flows on moving or deforming meshes is described. It uses a well-established time-marching finite-volume flow solver, developed for steady compressible flows past rigid bodies. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. Such methods can be extended to steady incompressible flows using an artificial compressibility scheme. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real-time discretization and a dual-time approach, which uses a technique similar to the artificial compressibility scheme. Results are presented for test cases on both fixed and deforming meshes. Experimental, numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved. © 1998 John Wiley & Sons, Ltd.     

Non‐conservative and conservative formulations of characteristics‐based numerical reconstructions for incompressible flows

An investigation of characteristics‐based (CB) schemes for solving the incompressible Navier–Stokes equations in conjunction with the artificial‐compressibility approach, is presented. Both non‐conservative and conservative CB numerical reconstructions are derived and their accuracy and convergence properties are assessed analytically and numerically. We demonstrate by means of eigenvalue analysis that there are differences in the spectral characteristics of these formulations that result in different convergence properties. Numerical tests for two‐ and three‐dimensional flows reveal that the two formulations provide similar accuracy but the non‐conservative formulation converges faster. Copyright © 2005 John Wiley & Sons, Ltd.     

An improved unsteady,unstructured, artificial compressibility,finite volume scheme for viscous incompressible flows: Part I. Theory and implementation

A robust, artificial compressibility scheme has been developed for modelling laminar steady state and transient, incompressible flows over a wide range of Reynolds and Rayleigh numbers. Artificial compressibility is applied in a consistant manner resulting in a system of preconditioned governing equations. A locally generalized preconditioner is introduced, designed to be robust and offer good convergence rates. Free artificial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artificial compressibility scheme. Memory efficiency is achieved through a multistage, pseudo‐time‐explicit time‐marching solution procedure. A node‐centred dual‐cell edge‐based finite volume discretization technique, suitable for unstructured grids, is used due to its computational efficiency and high‐resolution spatial accuracy. In the interest of computational efficiency and ease of implementation, stabilization is achieved via a scalar‐valued artificial dissipation scheme. Temporal accuracy is facilitated by employing a second‐order accurate, dual‐time‐stepping method. In this part of the paper the theory and implementation details are discussed. In Part II, the scheme will be applied to a number of example problems to solve flows over a wide range of Reynolds and Rayleigh numbers. Copyright © 2002 John Wiley & Sons, Ltd.     

A particle method for two‐phase flows with compressible air pocket

When using particle methods to simulate water–air flows with compressible air pockets, a major challenge is to deal with the large differences in physical properties (e.g., density and viscosity) between water and air. In addition, the accurate modeling of air compressibility is essential. To this end, a new two‐phase strategy is proposed to simulate incompressible and compressible fluids simultaneously without iterations between the solvers for incompressible and compressible flows. Water is modeled by the recently developed 2‐phase Consistent Particle Method for incompressible flows. For air modeling, a new compressible solver is proposed based on the ideal gas law and thermodynamics. The formulation avoids the problem of determining the actual sound speed that is dependent on the temperature and is therefore not necessarily constant. In addition, the compressible air solver is seamlessly integrated with the incompressible solver 2‐phase Consistent Particle Method because they both use the same predictor–corrector scheme to solve the governing equations. The performance of the proposed method is demonstrated by three benchmark problems as well as an experimental study of sloshing impact with entrapped air pockets in an oscillating tank. Copyright © 2016 John Wiley & Sons, Ltd.     

Performance of an analogy-based all-speed procedure without any explicit damping

 The performance of a numerical method which solves flow at all speeds, and does not use any explicit artificial viscosity or damping mechanism whatsoever, is investigated by testing a number of selected cases in compressible and incompressible flows. Contrary to existing methods, the momentum components are chosen as the dependent variables instead of the velocity components in order to provide a number of advantages. Among the motivations for this change is a flow analogy which permits incompressible methods to be used to solve compressible flows. The method is formulated within a control-volume-based finite-element approach using a collocated grid arrangement. The definition of two types of mass flux components at the control volume surfaces removes the possibility of velocity-pressure decoupling in the incompressible or Euler limits. In the absence of any dissipation mechanisms, the main concern of this work is to evaluate the performance of the method and the analogy for solving high speed compressible flows with shocks. The results and performance of the present work are compared with the exact and benchmark solutions and the results of other workers who use dissipation mechanisms to solve flow at all speeds.     

A matrix-form GSM–CFD solver for incompressible fluids and its application to hemodynamics

A GSM–CFD solver for incompressible flows is developed based on the gradient smoothing method (GSM). A matrix-form algorithm and corresponding data structure for GSM are devised to efficiently approximate the spatial gradients of field variables using the gradient smoothing operation. The calculated gradient values on various test fields show that the proposed GSM is capable of exactly reproducing linear field and of second order accuracy on all kinds of meshes. It is found that the GSM is much more robust to mesh deformation and therefore more suitable for problems with complicated geometries. Integrated with the artificial compressibility approach, the GSM is extended to solve the incompressible flows. As an example, the flow simulation of carotid bifurcation is carried out to show the effectiveness of the proposed GSM–CFD solver. The blood is modeled as incompressible Newtonian fluid and the vessel is treated as rigid wall in this paper.     

Effect of the Compressibility of the Fluid on the Parameters of a Hydraulic Gun

The effect of the compressibility of the fluid on the parameters of a hydraulic gun is evaluated. The quasionedimensional motion of an ideal compressible fluid is described by equations of nonstationary gas dynamics, which are solved numerically according to the algorithm proposed. The numerical solution for a compressible fluid is compared with an analytical solution for an incompressible fluid and with an experiment, which are performed by other authors. From an analysis of the results conclusions are drawn that the compressibility of the fluid can be disregarded. It is proposed that the Mach number the used as a criterion for assessing account for the compressibility of the fluid.     

A cut‐cell non‐conforming Cartesian mesh method for compressible and incompressible flow

This paper details a multigrid‐accelerated cut‐cell non‐conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub‐, trans‐, and supersonic flows. Cut‐cell technology is developed to furnish body‐fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge‐based vertex‐centred finite volume method. An alternative dual‐mesh construction strategy, similar to the cell‐centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch‐activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible–incompressible equation set on cut‐cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical simulation of separated boundary-layer flow

The numerical simulation of time-dependent, 2-D compressible boundary-layer flow containing a region of separation is studied. The separation is generated by the introduction of an adverse pressure gradient along the freestream boundary. In order to validate the numerical method, a low Mach-number laminar separation bubble flow is considered, which enables an extensive comparison with incompressible results. The generation of an adverse pressure gradient along the freestream boundary can be realized in various ways. An imposed decelerating flow boundary is compared with a suction technique. The effects of the strength of the pressure gradient and the presence of small upstream perturbations on the separation bubble are also investigated. The time-averaged characteristics of the flow are in good quantitative agreement with incompressible approximate theories predicting the condition for separation. The appearance of self-excited vortex shedding in unperturbed flows under a sufficiently strong adverse pressure gradient is consistent with incompressible flow simulations reported in the literature. The satisfactory result achieved in the calculation of the low-Mach-number flow encourages the application of the numerical method to flows with strong compressibility effects.     

An improved unsteady,unstructured, artificial compressibility,finite volume scheme for viscous incompressible flows: Part II. Application

In Part I of this paper, a preconditioned artificial compressibility scheme was developed for modelling laminar steady‐state and transient, incompressible flows for a wide range of Reynolds and Rayleigh numbers. In this part, several examples of laminar incompressible problems are solved and discussed. The influence of various AC parameters on robustness and convergence rates are assessed for a complex category of problems. It is shown that the scheme developed in Part I is an accurate, robust and easy to use method for solving incompressible laminar flow problems over a wide range of flow regimes. Copyright © 2002 John Wiley & Sons, Ltd.     

The least-squares finite element method for low-mach-number compressible viscous flows

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. The conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique, and finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such a difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: driven cavity flows at various Reynolds numbers and buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.     

A segregated CFD approach to pipe network analysis

The most popular pipe network algorithms fall into three categories depending on whether node, loop or element solving equations are considered. Although node methods have some advantages over the other two methods, some authors have found the node methods to be more unreliable than the other two classes of methods. Node methods are also used in Computational Fluid Dynamics (CFD) to solve the Navier–Stokes equations. Since significant progress has been made in this field in the recent past it was felt that this should have some bearing on the development of more reliable node methods for pipe network problems. In this paper the well-known SIMPLE algorrithm of Patankar and Spalding,¹ which is Known in CFD as a segregated method, is extended to deal with pipe network problems. The method can deal with both compressible and incompressible flows. Special attention is given to the solution of the pressure correction equation, the stability of the algorithm, sensitivity to initial conditions and convergence parameters. It is shown that the present method is not very sensitive to initial conditions. The method is very reliable and it deals more effectively with compressible flows than the conventional Newton–Raphson method for incompressible flows.     

Lagrangian finite element analysis of Newtonian fluid flows

A fully Lagrangian finite element method for the analysis of Newtonian flows is developed. The approach furnishes, in effect, a Lagrangian implementation of the compressible Navier–Stokes equations. As the flow proceeds, the mesh is maintained undistorted through continuous and adaptive remeshing of the fluid mass. The principal advantage of the present approach lies in the treatment of boundary conditions at material surfaces such as free boundaries, fluid/fluid or fluid/solid interfaces. In contrast to Eulerian approaches, boundary conditions are enforced at material surfaces ab initio and therefore require no special attention. Consistent tangents are obtained for Lagrangian implicit analysis of a Newtonian fluid flow which may exhibit compressibility effects. The accuracy of the approach is assessed by comparison of the solution for a sloshing problem with existing numerical results and its versatility demonstrated through a simulation of wave breaking. The finite element mesh is maintained undistorted throughout the computation by recourse to frequent and adaptive remeshing © 1998 John Wiley & Sons, Ltd.     

A fully explicit characteristic based split (CBS) scheme for viscoelastic flow calculations

In this paper, an explicit characteristic based split (CBS) scheme is proposed for the numerical solution of incompressible viscoelastic flow equations. The scheme proposed is free from simultaneous solution to the matrices arising from the finite element discretization of the governing equations. The experience gained from the solution of Newtonian fluid dynamics problems has been applied to the solution of viscoelastic flows. The Oldroyd‐B model has been employed to solve two benchmark problems of viscoelastic flow. They are viscoelastic flow past a circular cylinder and viscoelastic flow through planar contraction geometry. The results show that the solutions obtained are stable for the Weissenberg or Deborah number range studied in this paper. The solutions obtained at lower Weissenberg or Deborah numbers are accurate and agree excellently with the majority of available numerical data. However at higher Weissenberg or Deborah numbers, results show some sign of negative influence of the artificial dissipation added to the discrete constitutive equations. Copyright © 2004 John Wiley Sons, Ltd.     

A stream function implicit finite difference scheme for 2D incompressible flows of Newtonian fluids

The development of a new algorithm to solve the Navier–Stokes equations by an implicit formulation for the finite difference method is presented, that can be used to solve two‐dimensional incompressible flows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady fluid mechanics problems: cavity‐driven flow with Re=100, 400 and 1000 and flow in a sudden expansion with expansion ratio H/h=2 and Re=50, 100 and 200. The results obtained for the stream function are compared with values obtained by different available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

An artificial compressibility method for viscous incompressible and low Mach number flows

Within the framework of the pressure‐based algorithm, an artificial compressibility method is developed on a non‐orthogonal grid for incompressible and low Mach number fluid flow problems, using cell‐centered finite‐volume approximation. Resorting to the traditional pseudo‐compressibility concept, the continuity constraint is perturbed by the time derivative of pressure, the physical relevance of which is to invoke matrix preconditionings. The approach provokes density perturbations, assisting the transformation between primitive and conservative variables. A dual‐dissipation scheme for the pressure–velocity coupling is contrived, which has the expediences of greater flexibility and increased accuracy in a way similar to the monotone upstream‐centered schemes for conservation laws approach. To account for the flow directionality in the upwinding, a rotational matrix is introduced to evaluate the convective flux. Numerical experiments in reference to a few well‐documented laminar flows demonstrate that the entire contrivance expedites enhanced robustness and improved overall damping properties of the factored pseudo‐time integration procedure. Copyright © 2008 John Wiley & Sons, Ltd.     

An extension of Key's principle to nonlinear elasticity

A variational principle for finite isothermal deformations of anisotropic compressible and nearly incompressible hyperelastic materials is presented. It is equivalent to the nonlinear elastic field (Lagrangian) equations expressed in terms of the displacement field and a scalar function associated with the hydrostatic mean stress. The formulation for incompressible materials is recovered from the compressible one simply as a limit. The principle is particularly useful in the development of finite element analysis of nearly incompressible and of incompressible materials and is general in the sense that it uses a general form of constitutive equation. It can be considered as an extension of Key's principle to nonlinear elasticity. Various numerical implementations are used to illustrate the efficiency of the proposed formulation and to show the convergence behaviour for different types of elements. These numerical tests suggest that the formulation gives results which change smoothly as the material varies from compressible to incompressible.     

Shock-capturing method using characteristic-based dissipation filters in pressure-based algorithm

In this paper, an efficient blending procedure based on the pressure-based algorithm is presented to solve the compressible Euler equations on a non-orthogonal mesh with collocated finite volume formulation. The boundedness criteria for this procedure are determined from total variation diminishing (TVD) schemes with and without applying of artificial compression method (ACM) of Harten as a control switch of dissipation. The fluxes of the convected quantities including mass flow rate are approximated by using the characteristic-based TVD and TVD/ACM methods. The algorithm is tested for steady-state inviscid flows at different Mach numbers ranging from the transonic to the supersonic regime and the results are compared with the existing numerical solutions. The comparisons show that the ACM is a useful technique to modify standard high-resolution schemes, which prevents the smearing of discontinuities and improves the resolution of shocks in the pressure-based algorithm. Aside from having the ability of accurately capturing shocked flows, this approach also accelerates the convergence rate of the solution in the supersonic flows with only a maximum increase of 5% in the operations with respect to standard second-order TVD schemes.     

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

A novel Lagrangian gradient smoothing method (L‐GSM) is developed to solve “solid‐flow” (flow media with material strength) problems governed by Lagrangian form of Navier‐Stokes equations. It is a particle‐like method, similar to the smoothed particle hydrodynamics (SPH) method but without the so‐called tensile instability that exists in the SPH since its birth. The L‐GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no‐slip solid boundary, free‐surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L‐GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L‐GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear‐driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L‐GSM performance. It shows that the L‐GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testing results demonstrate that the proposed L‐GSM approach restores first‐order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L‐GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.