Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

High-order ghost-point immersed boundary method for viscous compressible flows based on summation-by-parts operators

A high-order immersed boundary method is devised for the compressible Navier-Stokes equations by employing high-order summation-by-parts difference operators. The immersed boundaries are treated as sharp interfaces by enforcing the solid wall boundary conditions via flow variables at ghost points. Two different interpolation schemes are tested to compute values at the ghost points and a hybrid treatment is used. The first method provides the bilinearly interpolated flow variables at the image points of the corresponding ghost points and the second method applies the boundary condition at the immersed boundary by using the weighted least squares method with high-order polynomials. The approach is verified and validated for compressible flow past a circular cylinder at moderate Reynolds numbers. The tonal sound generated by vortex shedding from a circular cylinder is also investigated. In order to demonstrate the capability of the solver to handle complex geometries in practical cases, flow in a cross-section of a human upper airway is simulated.     

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical implementation of Aristov–Pukhnachev's formulation for axisymmetric viscous incompressible flows

In the present work a finite‐difference technique is developed for the implementation of a new method proposed by Aristov and Pukhnachev (Doklady Phys. 2004; 49 (2):112–115) for modeling of the axisymmetric viscous incompressible fluid flows. A new function is introduced that is related to the pressure and a system similar to the vorticity/stream function formulation is derived for the cross‐flow. This system is coupled to an equation for the azimuthal velocity component. The scheme and the algorithm treat the equations for the cross‐flow as an inextricably coupled system, which allows one to satisfy two conditions for the stream function with no condition on the auxiliary function. The issue of singularity of the matrix is tackled by adding a small parameter in the boundary conditions. The scheme is thoroughly validated on grids with different resolutions. The new numerical tool is applied to the Taylor flow between concentric rotating cylinders when the upper and lower lids are allowed to rotate independently from the inner cylinder, while the outer cylinder is held at rest. The phenomenology of this flow is adequately represented by the numerical model, including the hysteresis that takes place near certain specific values of the Reynolds number. Thus, the present results can be construed to demonstrate the viability of the new model. The success can be attributed to the adequate physical nature of the auxiliary function. The proposed technique can be used in the future for in‐depth investigations of the bifurcation phenomena in rotating flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Characteristic‐based inlet and outlet boundary conditions for incompressible flows

Determining boundary conditions (BCs) for incompressible flows is such a delicate matter that affects the accuracy of the results. In this research, a new characteristic‐based BC for incompressible Navier‐Stokes equations is introduced. Discretization of equations has been done via finite volume. Additionally, artificial compressibility correction has been employed to deal with equations. Ordinary extrapolation from inner cells of a domain was used as a traditional way to estimate pressure and velocities on solid wall and inlet/outlet boundaries. Here, this method was substituted by the newly proposed BCs based on the characteristics of artificial compressibility equations. To follow this purpose, a computer code has been developed to carry out series of numerical tests for a flow over a backward‐facing step and was applied to a wide range of Reynolds numbers and grid combinations. Calculation of convective and viscous fluxes was done using Jameson's averaging scheme. Employing the characteristic‐based method for determining BCs has shown an improved convergence rate and reduced calculation time comparing with those of traditional ones. Furthermore, with the reduction of domain and computational cells, a similar accuracy was achieved for the results in comparison with the ones obtained from the traditional extrapolation method, and these results were in good agreement with the ones in the literature.     

Dispersion and stability analyses of the linearized two‐dimensional shallow water equations in boundary‐fitted co‐ordinates

In the present investigation, a Fourier analysis is used to study the phase and group speeds of a linearized, two‐dimensional shallow water equations, in a non‐orthogonal boundary‐fitted co‐ordinate system. The phase and group speeds for the spatially discretized equations, using the second‐order scheme in an Arakawa C grid, are calculated for grids with varying degrees of non‐orthogonality and compared with those obtained from the continuous case. The spatially discrete system is seen to be slightly dispersive, with the degree of dispersivity increasing with an decrease in the grid non‐orthogonality angle or decrease in grid resolution and this is in agreement with the conclusions reached by Sankaranarayanan and Spaulding (J. Comput. Phys., 2003; 184 : 299–320). The stability condition for the non‐orthogonal case is satisfied even when the grid non‐orthogonality angle, is as low as 30° for the Crank Nicolson and three‐time level schemes. A two‐dimensional wave deformation analysis, based on complex propagation factor developed by Leendertse (Report RM‐5294‐PR, The Rand Corp., Santa Monica, CA, 1967), is used to estimate the amplitude and phase errors of the two‐time level Crank–Nicolson scheme. There is no dissipation in the amplitude of the solution. However, the phase error is found to increase, as the grid angle decreases for a constant Courant number, and increases as Courant number increases. Copyright © 2003 John Wiley & Sons, Ltd.     

Volume preserving immersed boundary methods for two‐phase fluid flows

In this article, we propose a simple area‐preserving correction scheme for two‐phase immiscible incompressible flows with an immersed boundary method (IBM). The IBM was originally developed to model blood flow in the heart and has been widely applied to biofluid dynamics problems with complex geometries and immersed elastic membranes. The main idea of the IBM is to use a regular Eulerian computational grid for the fluid mechanics along with a Lagrangian representation of the immersed boundary. Using the discrete Dirac delta function and the indicator function, we can include the surface tension force, variable viscosity and mass density, and gravitational force effects. The principal advantage of the IBM for two‐phase fluid flows is its inherent accuracy due in part to its ability to use a large number of interfacial marker points on the interface. However, because the interface between two fluids is moved in a discrete manner, this can result in a lack of volume conservation. The idea of an area preserving correction scheme is to correct the interface location normally to the interface so that the area remains constant. Various numerical experiments are presented to illustrate the efficiency and accuracy of the proposed conservative IBM for two‐phase fluid flows. Copyright © 2011 John Wiley & Sons, Ltd.     

The prediction of spatially periodic flows using a finite‐volume model

A periodic boundary condition has been developed that can be used in conjunction with a specified flow rate to produce accurate results in spatially periodic geometries. This condition is useful in situations where the flow rate is known, or more importantly, in cases where the pressure gradient is not known a priori, such as in countercurrent flows. Using the present condition, the flow rate is imposed at the inlet in terms of a bulk velocity, but the velocity field evolves as part of the solution. The condition is formulated to be suitable for both fixed and moving periodic domains. For the case of a moving domain, a correction is introduced to account for changes in the instantaneous velocity through the periodic edges. Under periodic conditions, these corrections integrate to zero over a complete (temporal) period. The new periodic condition is shown to produce accurate results for flat and wavy‐walled channels under both induced flow and countercurrent conditions. Copyright © 2003 John Wiley & Sons, Ltd.     

Momentum‐based approximation of incompressible multiphase fluid flows

We introduce a time stepping technique using the momentum as dependent variable to solve incompressible multiphase problems. The main advantage of this approach is that the mass matrix is time‐independent making this technique suitable for spectral methods. A level set method is applied to reconstruct the fluid properties such as density. We also introduce a stabilization method using an entropy‐viscosity technique and a compression technique to limit the flattening of the level set function. We extend our algorithm to immiscible conducting fluids by coupling the incompressible Navier‐Stokes and the Maxwell equations. We validate the proposed algorithm against analytical and manufactured solutions. Results on test cases such as Newton's bucket problem and a variation thereof are provided. Surface tension effects are tested on benchmark problems involving bubbles. A numerical simulation of a phenomenon related to the industrial production of aluminium is presented at the end of the paper.     

Boundary‐fitted non‐linear dispersive wave model for regions of arbitrary geometry

A vertically integrated non‐linear dispersive wave model is expressed in non‐orthogonal curvilinear co‐ordinate system for simulating shallow or deep water wave motions in regions of arbitrary geometry. Both dependent and independent variables are transformed so that an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Thus, the wall condition for enclosures surrounding a typical physical domain, such as a channel, port or harbor, is satisfied accurately and easily. The numerical scheme is based on staggered grid finite‐difference approximations, which result in implicit formulations for the momentum equations and semi‐explicit formulation for the continuity equation. Test cases of linear wave propagation in converging, diverging and circular channels are performed to check the reliability of model simulations against the analytical solutions. Cnoidal waves of different steepness values in a circular channel are also considered as examples to non‐linear wave propagation within curved walls. In closing, remarks concerning versatility and practical uses of the numerical model are made. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Parallelization of a vorticity formulation for the analysis of incompressible viscous fluid flows

A parallel computer implementation of a vorticity formulation for the analysis of incompressible viscous fluid flow problems is presented. The vorticity formulation involves a three‐step process, two kinematic steps followed by a kinetic step. The first kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition. The vortex sheet strengths are related to the vorticity flux boundary conditions. The second kinematic step determines the interior velocity field from the regular form of the generalized Helmholtz decomposition. The third kinetic step solves the vorticity equation using a Galerkin finite element method with boundary conditions determined in the first step and velocities determined in the second step. The accuracy of the numerical algorithm is demonstrated through the driven‐cavity problem and the 2‐D cylinder in a free‐stream problem, which represent both internal and external flows. Each of the three steps requires a unique parallelization effort, which are evaluated in terms of parallel efficiency. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of two‐dimensional laminar incompressible offset jet flows

Two‐dimensional transient laminar incompressible offset jet is simulated numerically to gain insight into convective recirculation and flow processes induced by an offset jet. The behaviour of the jet with respect to offset ratio (OR) and Reynolds number (Re) are described in detail. The transient development of the velocity is simulated for various regions: recirculation, impingement and wall jet development. It is found that the reattachment length is dependent on both Re and OR for the range considered. Simulations are made to show the effect of entrainment on recirculation eddy. A detailed study of u velocity decay is reported. The decay rate of horizontal velocity component (u) is linear in impingement region. It is found that at high OR, velocity decay depends on Re only. Velocity profile in the wall jet region shows good agreement with experimental as well as similarity solutions. It is found that the effect of Re and OR are significant to bottom wall vorticity up to impingement region. Far downstream bottom wall vorticity is independent of OR. Copyright © 2005 John Wiley & Sons, Ltd.     

Coupling finite difference method with finite particle method for modeling viscous incompressible flows

A hybrid approach to couple finite difference method (FDM) with finite particle method (FPM) (ie, FDM-FPM) is developed to simulate viscous incompressible flows. FDM is a grid-based method that is convenient for implementing multiple or adaptive resolutions and is computationally efficient. FPM is an improved smoothed particle hydrodynamics (SPH), which is widely used in modeling fluid flows with free surfaces and complex boundaries. The proposed FDM-FPM leverages their advantages and is appealing in modeling viscous incompressible flows to balance accuracy and efficiency. In order to exchange the interface information between FDM and FPM for achieving consistency, stability, and convergence, a transition region is created in the particle region to maintain the stability of the interface between two methods. The mass flux algorithm is defined to control the particle creation and deletion. The mass is updated by N-S equations instead of the interpolation. In order to allow information exchange, an overlapping zone is defined near the interface. The information of overlapping zone is obtained by an FPM-type interpolation. Taylor-Green vortices and lid-driven shear cavity flows are simulated to test the accuracy and the conservation of the FDM-FPM hybrid approach. The standing waves and flows around NACA airfoils are further simulated to test the ability to deal with free surfaces and complex boundaries. The results show that FDM-FPM retains not only the high efficiency of FDM with multiple resolutions but also the ability of FPM in modeling free surfaces and complex boundaries.     

A new modification of the immersed‐boundary method for simulating flows with complex moving boundaries

In this paper, a new immersed‐boundary method for simulating flows over complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. The present method is based on a finite‐difference approach on a staggered mesh together with a fractional‐step method. It must be noted that the immersed boundary is generally not coincident with the position of the solution variables on the grid, therefore, an appropriate strategy is needed to construct a relationship between the curved boundary and the grid points nearby. Furthermore, a momentum forcing is added on the body boundaries and also inside the body to satisfy the no‐slip boundary condition. The immersed boundary is represented by a series of interfacial markers, and the markers are also used as Lagrangian forcing points. A linear interpolation is then used to scale the Lagrangian forcing from the interfacial markers to the corresponding grid points nearby. This treatment of the immersed‐boundary is used to simulate several problems, which have been validated with previous experimental results in the open literature, verifying the accuracy of the present method. Copyright © 2006 John Wiley & Sons, Ltd.     

Further experiences with computing non‐hydrostatic free‐surface flows involving water waves

A semi‐implicit, staggered finite volume technique for non‐hydrostatic, free‐surface flow governed by the incompressible Euler equations is presented that has a proper balance between accuracy, robustness and computing time. The procedure is intended to be used for predicting wave propagation in coastal areas. The splitting of the pressure into hydrostatic and non‐hydrostatic components is utilized. To ease the task of discretization and to enhance the accuracy of the scheme, a vertical boundary‐fitted co‐ordinate system is employed, permitting more resolution near the bottom as well as near the free surface. The issue of the implementation of boundary conditions is addressed. As recently proposed by the present authors, the Keller‐box scheme for accurate approximation of frequency wave dispersion requiring a limited vertical resolution is incorporated. The both locally and globally mass conserved solution is achieved with the aid of a projection method in the discrete sense. An efficient preconditioned Krylov subspace technique to solve the discretized Poisson equation for pressure correction with an unsymmetric matrix is treated. Some numerical experiments to show the accuracy, robustness and efficiency of the proposed method are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

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.