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Semi-implicit direct forcing immersed boundary method for incompressible viscous thermal flow problems: A Schur complement approach
Abstract
An extended immersed boundary method utilizing a semi-implicit direct forcing approach for the simulation of confined incompressible viscous thermal flow problems is presented. The method utilizes a Schur complement approach to enforce the kinematic constraints of no-slip and the corresponding thermal boundary conditions for immersed surfaces. The developed methodology can be straightforwardly adapted to any existing incompressible time marching solver based on a segregated pressure-velocity coupling. The method accurately meets the thermal and the no-slip boundary conditions on the surfaces of immersed bodies for the entire range of Rayleigh numbers . Strategies for further increasing the computational efficiency of the developed approach are discussed. The method has been extensively verified by applying it for the simulation of a number of representative fully 3D confined natural convection steady and periodic flows. Complex dynamic phenomena typical of this kind of flow including vortical structures and convection cells and instability characteristics, were simulated and visualized and the results were found to compare favorably with results known from literature.
... Remarkably, to date, the capabilities of the direct forcing IB approach in ETHD flows have not been fully exploited, apart from the study by [26], which used the IB-lattice Boltzmann method for analyzing ETHD flows in 2D configurations. The currently developed methodology, which constitutes an extension of our recently developed framework [35] to electro-driven flows, bridges the above gap. To do this, we couple the incompressible Navier-Stokes and energy equations with the Poisson-Nernst-Planck (PNP) and the electrostatic system of equations governing the transport of charged species and the spatial distribution of electric potential, respectively. ...
... The boundary conditions on the surface of the sphere are enforced by utilizing the direct forcing IB method [35,39]. Following the formalism of the direct forcing IB method, the problem is discretized on two independent grids, namely, the Eulerian and the Lagrangian grids. ...
... All the additional unknowns were treated implicitly with the fields governed by the corresponding transport equations; these additional unknowns, in fact, appeared as a result of applying the IB method and due to the regularized volumetric sources. The strategy, which is similar to that applied in our previous study [35], is as follows. Each pair of Eqs. ...
... A slightly different semi-implicit formulation coupling Lagrangian forces and heat fluxes with a predicted (i.e. non-divergence free) velocity field was proposed in [31], and subsequently it was used for the investigation of 3D moving boundary flows [32] and natural convection confined flows developing around hot and cold cylinders [33,34]. ...
... As a novel aspect, the developed formulation is reduced to the solution of a Poisson-body forces system of equations constituting a regularized saddle point, that can be conveniently transformed into equivalent positive definite system [36]. We then propose a physically justified approximation of the system allowing for immense decrease in memory consumption compared to the previously developed numerical methodologies [31,32]. Additionally, we put an emphasis on the portability of the developed methodology to provide its convenient embedding into any available solvers of the Laplace and Helmholtz equations. ...
... the Thomas solver [2]. The latter, which is a direct solver, is superior to the commonly used iterative BiCgStab [1] algorithm, when used for obtaining the matrix-vector product of the inverse Helmholtz operator; it has been extensively verified in our previous studies [41,31,32]. With emphasis on the portability of the developed methodology, the solver is utilized in this study in a black-box manner by modifying only the RHS of Eq. (4). ...
A novel formulation of the direct forcing immersed boundary (IB) method, that treats it as an integral part of a SIMPLE method is presented for the simulation of incompressible flows. The incompressibility and no-slip kinematic constraints are treated implicitly as distributed Lagrange multipliers and are fully coupled with each other by combining them into a single Poisson-body forces system of equations that constitutes a regularized
saddle point problem. A physically justified approximation of the system, resembling a technique typical of the penalty method, is carried out to facilitate the solution. By utilizing the Schur complement approach, the approximated system is decomposed into two separate systems of equations, allowing us to compute the values for the volumetric force and pressure corrections. The first system, which is characterized by a small (only O(10)) value of the condition number, is conveniently solved by the BiCGStab method [1], converging within 2-3 iterations, while the second system is addressed by the direct TPF solver [2], characterized by O(N4/3) complexity. The entire methodology is designed to
be highly portable, which facilitates the use of any available solver that was designed to simulate incompressible flows governed by the Helmholtz and Laplace operators but could not benefit from the immersed boundary formalism. The capabilities of the developed methodology applied to the simulation of representative shear- and buoyancy-driven confined flows developing in the presence of stationary immersed bodies are demonstrated.
A further application of the developed approach to moving boundary and two-way coupled fluid-structure interaction problems is discussed.
... Interest in this field has been motivated by its addressing a broad spectrum of engineering applications based on gas-solid heat exchangers and a fundamental understanding of the instability of highly separated confined flows. It is worth mentioning in this context the works of [54], [55], [56], [57] and the studies of [58], [59], [60], [61], [62], which addressed natural and Natarajan [64]. The authors developed a diffuse immersed boundary approach for thermally driven non-Boussinesq flows, which, however, relies on a quasi-incompressible formulation of the governing equations and therefore cannot be considered to be a fully compressible approach. ...
... The results obtained in the current study were performed for a wide range of governing parameters: ∈ {10 3 , 10 4 , 10 5 , 10 6 [72], [58] obtained by employing the Boussinesq approximation. with the lowest number do not contain secondary convective cells, and the temperature distribution is close to linear along the radial direction from the hot cylinder to the cold cavity walls, as expected from systems for which conduction constitutes the major heat transfer mechanism. ...
... were compared with the corresponding independently reported data obtained by employing the Boussinesq approximation. In particular, we focus on the studies in [72] and [58] that simulate the natural convection flow from a hot cylinder placed within a three-dimensional cavity. The geometry and the boundary conditions of this specific configuration are shown in Fig. 4.18. ...
The goal of this report is to present the final project conducted in order to fulfill the requirements of the M.Sc. degree at the Department of Mechanical Engineering, Ben-Gurion University (BGU) of the Negev. The project comprises theoretical research investigating natural convection compressible flow with high temperature differences and with complex geometries. The research motivation comes from long-term research investigating and simulating the steady state and transient multiphase flow regimes existing in the reactor core, that was established by the Soreq Nuclear Center. The main objective of this project is to develop a comprehensive numerical methodology that is capable of theoretical modeling of natural convection compressible flow with high temperature differences and with complex geometries, using standard techniques of computational fluid dynamics (CFD) - pressure-based solution algorithms and immersed boundary methods. This report contains: - A comprehensive literature review surveying methods for the simulation of natural convection flow and immersed boundary methods. - An extended outline of the objectives of the performed research. - A comprehensive physical model, including the governing equations, definitions, constitutive laws, and dimensional analysis. - A verification study by favorable comparison with corresponding independent numerical data available in the literature for incompressible, and non-Bossinesq compressible flows, without complex geometry. - A comparison between results obtained in the present study and results from previous studies for configurations with low temperature difference and complex geometry. - A solution and analysis of the configurations with high temperature differences and complex geometry. - A summary, conclusions , and recommendations for possible future work.
... Interest in this field was generated by its relevance to a broad spectrum of engineering applications based on gas-solid heat exchangers and a need to investigate the instability characteristics of highly separated confined flows. Worth mentioning in this context are the works [24][25][26][27] and the studies [28][29][30][31][32] that addressed natural convection confined flows in the presence of bodies of complex two-and three-dimensional geometries, respectively. ...
... The Nusselt number Nu was calculated as is detailed below in subsection 5.2. The Nusselt numbers obtained for the whole range of Ra and for the lowest value of the temperature-difference parameter (ε = 0.005) were compared with the corresponding results available in literature [28], [43] calculated by employing the Boussinesq approximation. show that for the lowest Rayleigh number, Ra =10 3 , there were no significant differences between the spatial distributions of the path lines and the temperature fields, respectively, obtained for the lowest (ε=0.005) and the highest (ε=0.6) ...
... The results of the current study obtained for the value of ε = 0.005 were compared with those from corresponding studies of natural convection flow from a hot cylinder placed Figure 38: Physical model of a hot cylinder inside a cold tube adapted from [28] and [43]. ...
... Interest in this field has been motivated by its addressing a broad spectrum of engineering applications based on gas-solid heat exchangers and a fundamental understanding of the instability of highly separated confined flows. It is worth mentioning in this context the works of [54], [55], [56], [57] and the studies of [58], [59], [60], [61], [62], which addressed natural and Natarajan [64]. The authors developed a diffuse immersed boundary approach for thermally driven non-Boussinesq flows, which, however, relies on a quasi-incompressible formulation of the governing equations and therefore cannot be considered to be a fully compressible approach. ...
... The results obtained in the current study were performed for a wide range of governing parameters: ∈ {10 3 , 10 4 , 10 5 , 10 6 [72], [58] obtained by employing the Boussinesq approximation. with the lowest number do not contain secondary convective cells, and the temperature distribution is close to linear along the radial direction from the hot cylinder to the cold cavity walls, as expected from systems for which conduction constitutes the major heat transfer mechanism. ...
... were compared with the corresponding independently reported data obtained by employing the Boussinesq approximation. In particular, we focus on the studies in [72] and [58] that simulate the natural convection flow from a hot cylinder placed within a three-dimensional cavity. The geometry and the boundary conditions of this specific configuration are shown in Fig. 4.18. ...
... Unfortunately, a purely implicit implementation of the IBM typically involves substantial modification of the original solvers, which are not initially equipped with the IBM capability. For this reason, a semi-implicit implementation of the IBM, in which the Lagrangian forces are implicitly coupled with a non-solenoidal velocity field subsequently projected onto a divergence free subspace has attracted increasing interest in recent years [52][53][54]. The semi-implicit formulation of the IBM can be straight forwardly embedded into the whole family of pressure-velocity segregated NS solvers based on projection or fractional step algorithms, while maintaining the accuracy of the imposed constraints of no-slip bounded by the discretization error of the numerical scheme [54]. ...
... For this reason, a semi-implicit implementation of the IBM, in which the Lagrangian forces are implicitly coupled with a non-solenoidal velocity field subsequently projected onto a divergence free subspace has attracted increasing interest in recent years [52][53][54]. The semi-implicit formulation of the IBM can be straight forwardly embedded into the whole family of pressure-velocity segregated NS solvers based on projection or fractional step algorithms, while maintaining the accuracy of the imposed constraints of no-slip bounded by the discretization error of the numerical scheme [54]. ...
... The key idea underlying the semi-implicit implementation is to analytically decompose the operator coupling the NS equations with the constraints of no-slip and to pre-compute the contribution of the latter at the beginning of the computational process. It was shown in our previous study [54] that, after the pre-computing is completed, the efficiency of the time integration performed by the algorithm based on the semi-implicit implementation of the IBM is comparable with that of its explicit counterpart in the case of stationary immersed bodies. The present paper extends the previously developed methodology to flow simulations in the presence of periodically moving immersed bodies whose kinematics is governed by periodic functions and can therefore be split into a finite number of discrete states. ...
An extended immersed boundary methodology utilizing a semi-implicit direct forcing approach was formulated for the simulation of incompressible flows in the presence of periodically moving immersed bodies. The methodology utilizes a Schur complement approach to enforce no-slip kinematic constraints for immersed surfaces. The methodology is split into an “embarrassingly” parallel pre-computing stage and a time integration stage, both of which take advantage of the general parallel file system (GPFS) for efficient writing and reading of large amounts of data. The methodology can be embedded straight forwardly into the whole family of pressure–velocity segregated solvers of incompressible Navier–Stokes equations based on projection or fractional step approaches. The methodology accurately meets the no-slip kinematic constraints on the surfaces of immersed oscillating bodies. In this study, it was extensively verified by applying it for the simulation of a number of representative flows developing in the presence of an oscillating sphere. The capabilities of the methodology for the simulation of incompressible flow generated by a number of bodies whose motion is governed by general periodic kinematics were demonstrated by simulation of the flow developing in the presence of two out-of-phase oscillating spheres. The physical characteristics of the generated flows in terms of the time evolutions of the total drag coefficients were presented as a function of Reynolds values. The vortical structures inherent in the generated flows were visualized by presenting the isosurfaces of theλ2 criterion.
... A deeper insight into the impact of the separation phenomenon on the characteristics of thermally driven flows has been acquired from the investigation of flow regimes developing within tall vertical channels [4,6,11] with horizontal fins and within square and cubic cold cavities containing immersed obstacles in the form of one [12][13][14][15] or more [16][17][18][19][20] hot horizontal cylinders. The investigation of natural convection flow within cold cubic enclosures containing horizontal hot cylinders is, specifically, of theoretical interest, as it is possible to obtain highly separated periodic flows at a Ra value as small as ̃10 5 ÷ 10 6 (see e.g. ...
... Eqs. (1)-(3) are solved by utilizing the semi-implicit direct forcing IB method recently developed in Ref. [15], in accordance with which, the volumetric f and q sources introduced into the governing equations (1)-(3) are solved implicitly. The detailed description of the method, applied to the simulation of the natural convection flows falling within the scope of the present study, is presented in our previous paper [21] and is therefore not repeated here for the sake of brevity. ...
... The standard second-order finite volume method [27] is utilized for discretization of all the spatial derivatives, while a second-order backward difference scheme is utilized for the time discretization. The algorithm consists of precomputing and time integration steps [15]. All the results reported here were calculated on both 200 3 and 300 3 grids with the aim to prove the grid independence of the obtained results. ...
The study numerically investigated the instability characteristics of a 3D highly separated natural convection flow developing within a cold cubic enclosure in the presence of a tandem of hot and cold horizontally aligned cylinders. The immersed boundary (IB) method was utilized to enforce kinematic no-slip and thermal boundary conditions on the surfaces of the two cylinders. The obtained results were based on the analysis of slightly supercritical flows simulated for three different distances between the cylinders and for Rayleigh numbers Ra∝O(10^6). It was found that the transition to unsteadiness of the flow sets in via the first Hopf bifurcation, which preserves two types of reflectional symmetry with respect to the central cross-section of the cubic enclosure. The impact of the closeness of the cavity boundaries to the cylinder surfaces on the quantitative and
qualitative characteristics of the observed instabilities was extensively investigated. The study elucidated the fundamental instability characteristics typical of highly separated thermally driven flows in confined containers in terms of the bifurcation characteristics and the impact of the object orientation and closeness to the container boundaries on preserving the spatio-temporal symmetries of the slightly supercritical flow.
... Nowadays, however, with the increase of available computational power and the development of advanced numerical techniques, this analysis has become trivial for 2D flow configurations and possible for some 3D flow configurations [30]. In particular, the present study is performed by applying a recently developed and thoroughly verified methodology comprising the semi-implicit direct forcing immersed boundary (IB) method based on the Schur complement approach [31]. ...
... Eqs. (1)-(3) are solved by utilizing the semi-implicit direct forcing IB method recently developed in [31]. For the sake of completeness, a brief description of the method follows. ...
... introduced by Roma et al. in [34] and verified in a number of studies [31,[35][36][37][38]. Here, Dr is the cell width in the r direction. ...
... introduced by Roma et al. [20] and verified elsewhere [21][22][23][24][25][26]. Here r is the cell width in the r direction. ...
We report a generic theoretical framework for accurate simulation of the temporal and spatial evolution of fused fiber-optic components, fabricated by the "heat and pull" technique. The methodology is based on the solution of quasi-3D incompressible Navier-Stokes equations formulated for immiscible two-phase flow. The two-phase interface is resolved by employing an interface tracking approach combined with the immersed boundary method. The model facilitates accurate spatiotemporal prediction of the evolution of both the external shape of the optical component and the internal dopant concentration during fabrication. Validation of the model was obtained by extensive comparison to experimental results. The model was found to be a convenient theoretical tool that may reliably facilitate the design and fabrication process of a wide spectrum of optic components.
... Recently, natural convection in a three-dimensional cubic enclosure immersed with cylinder has received much attention (Feldman 2018b;Lee et al. 2016;Seo et al. 2016;Spizzichino et al. 2019b). Comparing with the twodimensional calculations, significant difference exists for the thermal and flow fields (Choi et al. 2015b;Pandey et al. 2019) as well as the surface-averaged Nusselt number (Choi et al. 2015a) in the three-dimensional simulations, even the overall computational parameters are similar. ...
Multiphase flows with momentum, heat, and mass transfer exist widely in a variety of industrial applications. With the rapid development of numerical algorithms and computer capacity, advanced numerical simulation has become a promising tool in investigating multiphase transport problems. Immersed boundary (IB) method has recently emerged as such a popular interface capturing method for efficient simulations of multiphase flows, and significant achievements have been obtained. In this review, we attempt to give an overview of recent progresses on IB method for multiphase transport phenomena. Firstly, the governing equations, the basic ideas, and different boundary conditions for the IB methods are introduced. This is followed by numerical strategies, from which the IB methods are classified into two types, namely the artificial boundary method and the authentic boundary method. Discussions on the implementation of various boundary conditions at the interphase surface with momentum, heat, and mass transfer for different IB methods are then presented, together with a summary. Then, the state-of-the-art applications of IB methods to multiphase flows, including the isothermal flows, the heat transfer flows, and the mass transfer problems are outlined, with particular emphasis on the latter two topics. Finally, the conclusions and future challenges are identified.
... The first goal will make it possible to simulate mixing flow phenomena typical for classical dam break flow, for example, and will upgrade the overall flexibility of the developed approach bringing it closer to functionalities of VOF and LS methods. The second goal will be achieved by utilizing segregated pressure-velocity coupling, such as the SIMPLE algorithm (or its derivatives) combined with the Schur complement method, as detailed in our recent work [54] addressing natural convection flow in the presence of immersed bodies of an arbitrary geometry. ...
In this paper we present an extended formulation of the immersed boundary (IB) method that facilitates simulation of incompressible immiscible two-phase flows. In the developed formulation the pressure field and the surface tension forces associated with interface curvature are implicitly introduced in the form of distributed Lagrange multipliers. The approach provides for impermeability between both phases and exhibits accurate mass conservation without the need for additional correction procedures. Further, we present a grid independence study and extensive verification of the developed method for representative 2D two-phase flows dominated by buoyancy, shear stress, and surface tension forces.
Purpose
This paper aims to present an unconditionally energy-stable scheme for approximating the convective heat transfer equation.
Design/methodology/approach
The scheme stems from the generalized positive auxiliary variable (gPAV) idea and exploits a special treatment for the convection term. The original convection term is replaced by its linear approximation plus a correction term, which is under the control of an auxiliary variable. The scheme entails the computation of two temperature fields within each time step, and the linear algebraic system resulting from the discretization involves a coefficient matrix that is updated periodically. This auxiliary variable is given by a well-defined explicit formula that guarantees the positivity of its computed value.
Findings
Compared with the semi-implicit scheme and the gPAV-based scheme without the treatment on the convection term, the current scheme can provide an expanded accuracy range and achieve more accurate simulations at large (or fairly large) time step sizes. Extensive numerical experiments have been presented to demonstrate the accuracy and stability performance of the scheme developed herein.
Originality/value
This study shows the unconditional discrete energy stability property of the current scheme, irrespective of the time step sizes.
A three-dimensional direct simulation was carried out to examine the natural convective fluid flow and heat transfer characteristics in a long enclosure with a twisted cylinder. The effects of the Rayleigh number, dimensionless mean radius, and wavelength (or wavenumber) for a twisted cylinder were investigated. In addition to the heat transfer characteristics at cylinder and enclosure, the distributions of temperature and fluid flow are also presented by the iso-surfaces of temperature and vortical structure formed inside the enclosure. An artificial neural network was applied to predict the characteristics of natural convective flow and heat transfer by considering various simulation parameters for the elliptical and twisted cylinders. The input parameters considered in this study significantly affect on the heat transfer characteristics at cylinder and enclosure.
In this study, a three-dimensional numerical simulation of the natural convection in a long rectangular enclosure containing a sinusoidal cylinder was conducted. The sinusoidal cylinder was examined to improve the total heat transfer performance in comparison to that of a circular cylinder. The effects of simulation parameters of the sinusoidal cylinder were compared with the results of a circular cylinder in the Rayleigh number range of 10⁴≤Ra≤10⁶. The key simulation parameters were the mean radius of the cylinder (0.1L≤Rmean≤0.4 L) and its wavelength (or wavenumber (1≤λ≤3)). The heat transfer characteristics at the surfaces of the cylinder and walls of the enclosure were investigated. The effect of the cylinder shape on the heat transfer performance was notable at Rmean=0.4 L, and the performance improved by up to 27%. An artificial neural network (ANN) model was also used for the natural convection heat transfer from the sinusoidal cylinder. The impact of the parameters on the heat transfer performance was quantitatively evaluated. The results show that the Rayleigh number and the mean radius of the cylinder have a great influence on the total heat transfer performance at the cylinder surface and enclosure walls.
Conventional rectangular busbars employed in electric power systems suffer from several drawbacks, such as containing excess conductor material and exhibiting low current densities and poor heat dissipation. Therefore, the goal of this paper is to develop an irregular-shaped high-current (2000-A) busbar based on a coupled multiphysics analysis approach in which the influences of the proximity and the skin effects were considered. First, 3-D models based on the finite-element method (FEM) are established for a conventional rectangular busbar and the proposed irregular-shaped busbar, and the current densities of the two are compared. Second, a double coupling method is employed to couple the fluid–thermal field calculations with the electromagnetic power losses, which are considered to be heat sources, to determine the proportion of convective and radiative heat transferred from the surfaces of the busbar conductors. Third, based on the electromagnetic field analysis, the dynamic and thermal stabilities of the irregular-shaped busbar are verified with a new method. The time at which the short-circuit current reached a maximum is determined accurately. The electromagnetic force, temperature, and thermal stress distributions of the irregular-shaped busbar under short-circuit conditions are obtained. Finally, the accuracy of the FEM simulations is verified by experiments, which indicate that the multiphysics field-coupled analytical approach is valid.
Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor S 2 + a 2 ; here S and 0 are respectively the symmetric and antisymmetric parts of the velocity gradient tensor Vu. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier-Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of V u or the complex eigenvalues of Vu, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.
The numerical simulation of spherical and ellipsoidal bubbles in purified fluids requires the imposition of the free-slip boundary condition at the bubble surface. This paper describes a numerical method for the implementation of free-slip boundary conditions in the context of immersed boundary methods. In contrast to other numerical approaches for multiphase flows, the realization is not straightforward. The reason is that the immersed boundary method treats the liquid as well as the gas phase as a field of constant density and viscosity with a fictitious fluid inside the bubble. The motion of the disperse phase is computed explicitly by solving the momentum balance for each of its elements and is coupled to the continuous phase via additional source terms in the Navier-Stokes equations. The paper starts with illustrating that an ad hoc method is unsuccessful. On this basis, a new method is proposed employing appropriate direct forcing at the bubble surface. A central finding is that with common ratios between the step size of the grid and the bubble diameter, curvature terms need to be accounted for to obtain satisfactory results. The new method is first developed for spherical objects and then extended to generally curved interfaces. This is done by introducing a local coordinate system which approximates the surface in the vicinity of a Lagrangian marker with the help of the two principal curvatures of the surface at this point. The numerical scheme is then validated for spherical and ellipsoidal objects with or without prescribed constant angular velocity. It is shown that the proposed method achieves similar convergence behavior as the method for no-slip boundaries. The results are compared to analytical solutions for creeping flow around a sphere and to numerical reference data obtained on a body-fitted grid. The numerical tests confirm the excellent performance of the proposed method.
We develop an immersed boundary (IB) method for modeling flows around fixed
or moving rigid bodies that is suitable for a broad range of Reynolds numbers,
including steady Stokes flow. The spatio-temporal discretization of the fluid
equations is based on a standard staggered-grid approach. Fluid-body
interaction is handled using Peskin's IB method; however, unlike existing IB
approaches to such problems, we do not rely on penalty or fractional-step
formulations. Instead, we use an unsplit scheme that ensures the no-slip
constraint on the surface and interior of the body is enforced exactly.
Fractional-step approaches, by contrast, can impose such constraints only
approximately. Imposing these constraints exactly requires the solution of a
large linear system that includes the fluid velocity and pressure as well as
Lagrange multiplier forces that impose the motion of the body. To solve this
system efficiently, we develop a preconditioner for the constrained IB
formulation that is based on an analytical approximation to the Schur
complement. This approach is enabled by the near translational and rotational
invariance of Peskin's IB method. We demonstrate that only a few cycles of a
geometric multigrid method for the fluid equations are required in each
application of the preconditioner, and we demonstrate robust convergence of the
overall Krylov solver despite the approximations made in the preconditioner. We
apply the method to a number of test problems at zero and finite Reynolds
numbers, and we demonstrate first-order convergence of the method to several
analytical solutions and benchmark computations.
A strong coupling algorithm is presented for simulating the dynamic interactions between incompressible viscous flows and rigid-body systems in both two- and three-dimensional problems. In this work, the Navier–Stokes equations for incompressible flow are solved on a uniform Cartesian grid by the vorticity-based immersed boundary projection method of Colonius and Taira. Dynamical equations for arbitrary rigid-body systems are also developed. The proposed coupling method attempts to unify the treatment of constraints in the fluid and structure—the incompressibility of the fluid, the linkages in the rigid-body system, and the conditions at the interface—through the use of Lagrange multipliers. The resulting partitioned system of equations is solved with a simple relaxation scheme, based on an identification of virtual inertia from the fluid. The scheme achieves convergence in only 2 to 5 iterations per time step for a wide variety of mass ratios. The formulation requires that only a subset of the discrete fluid equations be solved in each iteration. Several two- and three-dimensional numerical tests are conducted to validate and demonstrate the method, including a falling cylinder, flapping of flexible wings, self-excited oscillations of a system of many linked plates in a free stream, and passive pivoting of a finite aspect ratio plate under the influence of gravity in a free stream. The results from the current method are compared with previous experimental and numerical results and good agreement is achieved.
An efficient approach for the simulation of finite-size particles with interface resolution was presented by Uhlmann [M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys. 209 (2005) 448–476.]. The present paper proposes several enhancements of this method which considerably improve the results and extend the range of applicability. An important step is a simple low-cost iterative procedure for the Euler–Lagrange coupling yielding a substantially better imposition of boundary conditions at the interface, even for large time steps. Furthermore, it is known that the basic method is restricted to ratios of particle density and fluid density larger than some critical value above 1, hence excluding, for example, non-buoyant particles. This can be remedied by an efficient integration step for the artificial flow field inside the particles to extend the accessible density range down to 0.3. This paper also shows that the basic scheme is inconsistent when moving surfaces are allowed to approach closer than twice the step size. A remedy is developed based on excluding from the force computation all surface markers whose stencil overlaps with the stencil of a marker located on the surface of a collision partner. The resulting algorithm is throughly validated and is demonstrated to substantially improve upon the original method.
Center for Turbulence Research, Mechanical Engineering Department, Stanford University, Stanford, California 94305; email: jops@ctr.stanford.edu
The recursive zonal equal area (EQ) sphere partitioning algorithm is a practical algorithm for partitioning higher dimensional spheres into regions of equal area and small diameter. This paper describes the partition algorithm and its implementation in Matlab, provides numerical results and gives a sketch of the proof of the bounds on the diameter of regions. A companion paper [13] gives details of the proof.
A second-order accurate, highly efficient method is developed for simulating unsteady three-dimensional incompressible flows in complex geometries. This is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid. The governing equations, therefore, can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures. Two different forcings are tested showing that while the quality of the results is essentially the same in both cases, the efficiency of the calculation strongly depends on the particular expression. A major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme; this ranges from zeroth-order for the most commonly used interpolations up to second-order for an ad hoc velocity interpolation. The present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature. Finally in the last example we show the flow inside an IC piston/cylinder assembly at high Reynolds number; to our knowledge this is the first example in which the immersed boundary technique is applied to a full three-dimensional complex flow with moving boundaries and with a Reynolds number high enough to require a subgrid-scale turbulence model.
Factorization of the incompressible Stokes operator linking pressure and
velocity is revisited. The main purpose is to use the inverse of the Stokes
operator with a large time step as a preconditioner for Newton and Arnoldi
iterations applied to computation of steady three-dimensional flows and to
study of their stability. It is shown that the Stokes operator can be inversed
within an acceptable computational effort. This inverse includes fast direct
inverses of several Helmholtz operators and iterative inverse of the pressure
matrix. It is shown, additionally, that fast direct solvers can be attractive
for the inverse of the Helmholtz and Laplace operators on fine grids and at
large Reynolds numbers, as well as for other problems where convergence of
iterative methods slows down. Implementation of the Stokes operator inverse to
time-stepping-based formulation of the Newton and Arnoldi iterations is
discussed.
Computational models are developed to predict the natural convection heat transfer and buoyancy for a Montgolfiere under conditions relevant to the Titan atmosphere. Idealized single- and double-walled balloon geometries are simulated using algorithms suitable for both laminar and (averaged) turbulent convection. Steady-state performance results are compared with existing heat transfer coefficient correlations. The laminar results, in particular, are used to test the validity of the correlations in the absence of uncertainties associated with turbulence modeling. Some discrepancies are observed, which appear to be primarily associated with temperature nonuniformity on the balloon surface. The predicted buoyancy for both the single- and double-walled balloons in the turbulent convection regime, predicted with standard two-equation turbulence models, showed trends similar to those with the empirical correlations. There was also good agreement with recently conducted experiments in a cryogenic facility designed to simulate the Titan atmosphere.
The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid-structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least
C
1
and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. The method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.
The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor. In a previous work we introduced the Immersed Boundary Smooth Extension (IBSE) method, a variation of the IB method that achieves high-order accuracy for elliptic PDE by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations. In this work, we extend the IBSE method to allow for the imposition of a divergence constraint, and demonstrate high-order convergence for the Stokes and incompressible Navier-Stokes equations: up to third-order pointwise convergence for the velocity field, and second-order pointwise convergence for all elements of the stress tensor. The method is flexible to the underlying discretization: we demonstrate solutions produced using both a Fourier spectral discretization and a standard second-order finite-difference discretization.
An extended formulation of the immersed boundary method, which facilitates simulation of incompressible isothermal and natural convection flows around immersed bodies and which may be applied for linear stability analysis of the flows, is presented. The Lagrangian forces and heat sources are distributed on the fluid — structure interface. The method treats pressure, the Lagrangian forces, and heat sources as distributed Lagrange multipliers, thereby implicitly providing the kinematic constraints of no-slip and the corresponding thermal boundary conditions for immersed surfaces. Extensive verification of the developed method for both isothermal and natural convection 2D flows is provided. Strategies for adapting the developed approach to realistic 3D configurations are discussed.
A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3,700 are used to verify the accuracy and physical fidelity of the formulation.
Three-dimensional numerical simulations were conducted for the natural convection phenomena around a hot inner circular cylinder positioned in a cold cubic enclosure in the Rayleigh number range of 103 ≤ Ra ≤ 106 at the Prandtl number of Pr = 0.7. The immersed boundary method (IBM) was used to capture the virtual wall boundary of the inner cylinder, based on the finite volume method (FVM). In this study, the transition of the flow regime from the steady state to the unsteady state and consequent three-dimensionality in the system induced by the increase in the flow instability were investigated. Detailed three-dimensional vortical structures of the convection cells at a relatively high Rayleigh number of Ra = 106 were analyzed using the visualization technique, and the heat transfer characteristics in the system resulting from the change in the vortical structures were addressed.
A fluid between two spheres, concentric or not, at different temperatures
will flow in the presence of a constant gravitational force. Although there is
no possible hydrostatic state, energy transport is dominated by diffusion if
temperature difference between the spheres is small enough. In this conductive
regime the average Nusselt number remains approximately constant for all
Rayleigh numbers below some critical value. Above the critical Rayleigh number,
plumes appear and thermal convection takes place. We study this phenomenon, in
particular the case where the inner sphere is displaced from the centre, using
a two-component thermal lattice Boltzmann method to characterize the convective
instability, the evolution of the flow patterns and the dependence of the
Nusselt number on the Rayleigh number beyond the transition.
Laminar natural convection flow inside multi-layered spherical shells with internal hot and external cold boundaries was investigated. Direct numerical simulations (DNS), which were performed by utilizing the immersed boundary method, addressed the fully 3D natural convection flow inside spherical shells with concentric, eccentric, equi-spaced and non-equi-spaced zero-thickness internal baffles. The insulation efficiency of the spherical shell was studied for up to four equi-spaced concentric internal layers. A unified functional dependency correlating modified Nu à and Ra à numbers was derived for spherical shells with up to four equi-spaced concentric internal layers. The effects of both vertical and horizontal eccentricity of the internal layers and of the width variation of concentric layers on the overall insulating performance of the spherical shell were investigated and quantified in terms of the Nu–Ra functionality.
The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet it only achieves first-order
spatial accuracy near embedded boundaries. In this paper, we introduce a new
high-order numerical method which we call the Immersed Boundary Smooth
Extension (IBSE) method. The IBSE method achieves high-order accuracy by
smoothly extending the unknown solution of the PDE from a given smooth domain
to a larger computational domain, enabling the use of simple Cartesian-grid
discretizations (e.g. Fourier spectral methods). The method preserves much of
the flexibility and robustness of the original IB method. In particular, it
requires minimal geometric information to describe the boundary and relies only
on convolution with regularized delta-functions to communicate information
between the computational grid and the boundary. We present a fast algorithm
for solving elliptic equations, which forms the basis for simple, high-order
implicit-time methods for parabolic PDE and implicit-explicit methods for
related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat,
Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise
convergence for Dirichlet problems and third-order pointwise convergence for
Neumann problems.
We investigate the forces and unsteady flow structures associated with harmonic oscillations of an airfoil in the streamwise (surging) and transverse (plunging) directions in two-dimensional simulations at low Reynolds number. For the surging case, we show that there are specific frequencies where the wake instability synchronizes with the unsteady motion of the airfoil, leading to significant changes in the mean forces. Resonant behaviour of the time-averaged forces is observed near the vortex shedding frequency and its subharmonic; the behaviour is reminiscent of the dynamics of the generic nonlinear oscillator known as the Arnol’d tongue or the resonance horn. Below the wake instability frequency, there are two regimes where the fluctuating forces are amplified and attenuated, respectively. A detailed study of the flow structures associated with leading-edge vortex (LEV) growth and detachment are used to relate this behaviour with the LEV acting either in phase with the quasi-steady component of the forces for the amplification case, or out of phase for the attenuation case. Comparisons with wind tunnel measurements show that phenomenologically similar dynamics occur at higher Reynolds number. Finally, we show that qualitatively similar phenomena occur during both surging and plunging.
Laminar and turbulent natural convection inside concentric spherical shells with isothermal cold and hot boundaries is numerically investigated up to Rayleigh number values Ra <= 10(12) and Pr = 0.71. The study utilizes direct numerical simulation (DNS), large eddy simulation (LES) and Reynolds averaged Navier-Stokes (RANS) approaches for investigation of the laminar, transitional and fully developed turbulent flow regimes, respectively. Three-dimensional flow patterns for slightly supercritical oscillatory flow regime inside the shell, with internal/external diameter ratio equal to D-i/D-o = 0.714 are presented and may be potentially useful for verification of the future linear stability analysis results. Particular attention has been given to the complex, fully three-dimensional unsteady flows occurring in narrow shell geometries characterized by 0.85 <= D-i/D-o <= 0.95. For this geometry considerable deviations in predicted heat flux rate through the shell boundaries are observed when compared with existing heat transfer correlations for the entire range of Ra numbers. The deviations tend to increase for transitional and fully turbulent flows. A new correlation for the heat transfer rate is suggested for laminar and transitional flow regimes.
A boundary condition implemented-immersed boundary method (IBM) involving velocity correction and heat flux correction is presented in this paper. In the framework of IBM, the velocity correction is made with Dirichlet condition (non-slip), and the temperature correction is made with Neumann (heat flux) condition. The main feature of present approach is to accurately satisfy the governing equations and boundary conditions through velocity and heat flux correction, which is performed by introducing a forcing term in the momentum equation and a heat source/sink term in the energy equation to consider the effect of the immersed boundary. The forcing term and heat source/sink are determined in such a way that the physical boundary conditions for velocity and temperature can be accurately satisfied. Numerical experiments for both forced convection and natural convection problems are conducted to validate the capability and efficiency of present method. Good agreements with available data in the literature have been achieved.
A linear stability analysis has been performed of the buoyancy-driven flow of a Boussinesq fluid in a spherical gap where the inner shell is warmer than the outer one (T1>T2T1>T2) for the radii ratio η=R1R2=0.714. We show that the two-dimensional axisymmetric basic flow becomes unstable with respect to non-axisymmetric perturbations. The stability diagrams, critical Grashof number GrcGrc and wave number mcmc versus the Prandtl number Pr are presented for both the steady and the oscillatory instabilities. In this way, the results bridge the gap to recent three-dimensional simulations performed in Scurtu et al. (2010), Feldman and Colonius (2013) for air in this ηη. Furthermore we investigate the energy exchange between the basic flow and perturbations in terms of a Reynolds–Orr-equation.
Natural convection in a spherical geometry is considered for prediction of the buoyancy of single- and double-walled
balloons in a cryogenic environment such as Titan’s atmosphere. The steady-state flow characteristics
obtained by solving the Reynolds-averaged Navier–Stokes equations with a standard turbulence model are used to
determine the net buoyancy as a function of heat input. Thermal radiation effects are shown to have a minor impact
on the buoyancy, as would be expected at cryogenic conditions. The predicted buoyancy and temperature fields
compare favorably with experiments preformed on a 1-m-diameter Montgolfiere prototype in a cryogenic facility. In
addition, both numerical and experimental results were compared with correlations for the heat transfer coefficients
for free convection internal and external to the balloon as well as in the concentric gap of the double-walled balloons.
Finally, scaling issues related to inferring the performance of the full-scale Montgolfiere from the model-scale results
are examined.
A boundary condition-enforced immersed boundary method is presented in this paper for simulation of free and forced convection problems with Dirichlet-type boundary condition. The heat source/sink is introduced into the energy equation to model the effect of immersed boundary. Different from previous works, in this paper, the heat source/sink is not pre-calculated, but determined implicitly in such a way that temperature at the immersed boundary interpolated from the corrected temperature field accurately satisfies the thermal condition. The main advantage of the proposed method lies in its simple concept, easy implementation and robustness in stability. Another important contribution of the paper is that it presents two efficient ways to calculate the average Nusselt number. They are based on temperature correction at Eulerian points and heat flux at Lagrangian points, in which no approximation for temperature gradients is needed. Numerical experiments for both forced convection and natural convection problems have been conducted to validate the capability and efficiency of the present method and proposed two ways to calculate the average Nusselt number. Good agreements with available data in the literature have been achieved.
Accurate description of particle–fluid interaction is one of the big challenges in the community of multiphase flows. Toward this direction, the combined multi-direct forcing and immersed boundary method were presented to simulate flows laden with finite-size moving particles with full-scale solutions. In the approach, the hydrodynamic interactions between moving rigid boundary and fluid were calculated using the multi-direct forcing scheme. The no-slip boundary conditions at the immersed boundaries can be satisfied well in this way. Direct numerical simulations of particle sedimentation under various conditions were performed based on the multi-direct forcing scheme, the immersed boundary method and the high-order finite difference. It is proved that this approach can successfully simulate the interactions between fluid and particle, the interactions between particle and particle as well as the interactions between particles and wall. The hitting and rebounding process of the single particle sedimentation, the drafting–kissing–tumbling of two settling particles and many particles sedimentation were observed. The quantitative comparisons against other studies were also conducted to validate the present approach.
Numerical calculations are carried out for the three-dimensional natural convection induced by a temperature difference between a cold outer cubic enclosure and a hot inner sphere. The immersed-boundary method (IBM) to model a sphere based on the finite volume method is used to study a three-dimensional natural convection for different Rayleigh numbers varying in the range of 103–106. This study investigates the effect of the inner sphere location on the heat transfer and fluid flow. The flow and thermal fields eventually reach the steady state for all Rayleigh numbers regardless of the sphere location. For Rayleigh numbers of 105 and 106, the variation of local Nusselt number of the sphere along the circumferential direction is large, showing the strong three dimensionality of the natural convection in the enclosure unlike to the cases of lower Rayleigh numbers of 103 and 104. For the highest Rayleigh number, the local peaks of the Nusselt number on the top wall of the enclosure shows the sinusoidal distribution along the circumferential direction. The flow and thermal fields, and the local and surface-averaged Nusselt numbers on the sphere and the enclosure are highlighted in detail.
Buoyancy-induced turbulent flow and natural convection heat transfer between two differentially heated concentric isothermal spheres is studied numerically. The low-Reynolds-number k–ω model is used for turbulence modeling. The two-dimensional governing equations are discretized using control volume method and solved by employing the alternating direction implicit scheme. Results are presented in the form of streamline and temperature patterns, and local and average Nusselt numbers, over the heated and cooled boundaries for a wide range of Rayleigh numbers (102–1010), extending the previous studies to the turbulent flow regime and for the radius ratio of 2. The results of the flow pattern and average Nusselt numbers were compared with the previously published experimental and numerical investigations and very good agreements were observed. For low values of Rayleigh numbers, regions with conduction-dominated flow pattern accompanied with low values of Nusselt numbers were observed, while for higher Rayleigh numbers, the flow pattern was changed to the convection dominated boundary layer type flow, resulting in an increase in the rate of heat transfer and flow velocities adjacent to both inner and outer boundaries. The average Nusselt numbers were correlated against Rayleigh number and a 1/4 power dependence of Ra in both laminar and turbulent regimes is obtained.
In this paper, the sedimentation of a sphere and its radial migration in a Poiseuille flow in a vertical tube filled with a Newtonian fluid are simulated with a finite-difference-based distributed Lagrange multiplier (DLM) method. The flow features, the settling velocities, the trajectories and the angular velocities of the spheres sedimenting in a tube at different Reynolds numbers are presented. The results show that at relatively low Reynolds numbers, the sphere approaches the tube axis monotonically, whereas in a high-Reynolds-number regime where shedding of vortices takes place, the sphere takes up a spiral trajectory that is closer to the tube wall than the tube axis. The rotation motion and the lateral motion of the sphere are highly correlated through the Magnus effect, which is verified to be an important (but not the only) driving force for the lateral migration of the sphere at relatively high Reynolds numbers. The standard vortex structures in the wake of a sphere, for Reynolds number higher than 400, are composed of a loop mainly located in a plane perpendicular to the streamwise direction and two streamwise vortex pairs. When moving downstream, the legs of the hairpin vortex retract and at the same time a streamwise vortex pair with rotation opposite to that of the legs forms between the loops. For Reynolds number around 400, the wake structures shed during the impact of the sphere on the wall typically form into streamwise vortex structures or else into hairpin vortices when the sphere spirals down. The radial, angular and axial velocities of both neutrally buoyant and non-neutrally buoyant spheres in a circular Poiseuille flow are reported. The results are in remarkably good agreement with the available experimental data. It is shown that suppresion of the sphere rotation produces significant large additional lift forces pointing towards the tube axis on the spheres in the neutrally buoyant and more-dense-downflow cases, whereas it has a negligible effect on the migration of the more dense sphere in upflow.
Natural convective fluid motions in the gap between two concentric non-rotating spheres are numerically studied. The case of homogeneously heated inner sphere and cooled outer sphere is considered for the radial aspect ratio π = 0.714 and Prandtl number Pr = 0.7. The natural convection problem is characterized by the Rayleigh number associated with the heat transfer within the fluid. For small values of the Rayleigh number an axisymmetric single vortex of crescent shape is formed as the basic flow. By increasing the Rayleigh number, flow patterns with banana-type cells, oriented in north-south direction and aligned in the azimuthal direction, are formed. Various characteristics of these flows as well as their transient evolution are investigated.
In this article we discuss a methodology that allows the direct numerical simula-tion of incompressible viscous fluid flow past moving rigid bodies. The simulation methods rest essentially on the combination of: (a) Lagrange-multiplier-based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region. (b) Finite element approximations of the Navier–Stokes equations occurring in the global model. (c) Time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model. The above methodology is particularly well suited to the direct numerical simulation of particulate flow, such as the flow of mixtures of rigid solid particles and incom-pressible viscous fluids, possibly non-Newtonian. We conclude this article with the presentation of the results of various numerical experiments, including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions.
The tensor product method was presented in [SJ and this paper contains an elaboration of the application of this method to the direct solution of partial difference equations. Since the results reported in [5] were obtained, we have found that EGERVARY [12] applied tensor products to the analysis of the fivepoint approximation of Poisson's equation. His computational scheme is not feasible for large problems. We describe a method for the direct and explicit solution of the partial difference equations arising from a large class of problems. This class includes all second order separable elliptic partial difference equations. The approach is a natural and classic one. If the problem is separable, then the solution can be expressed in terms of tensor products (direct products) of solutions of lower dimensional (and hence much simpler) problems. This implies that the matrices involved in the corresponding partial difference equation can also be expressed in terms of tensor products of lower order matrices. In the simplest cases the difference equations can be written in the form (1.t) (I| + B| u=g where A, B and I are matrices and u and g are vectors. If A and B are n n matrices then the matrix I| +B| is n2 n 2. We show how this n2 n 2 matrix can be explicitly inverted in terms of the eigenvectors and eigenvalues of the matrices A and B. It is, of course, no surprise that this can be done. We merely obtain the solution of (t .t) by the analog of the usual method involving the Green's function. The surprise is that this exact solution may be evaluated numerically with [ewer operations than required by the common overrelaxation iterative schemes. There are no numerical instabilities in this evaluation. This approach also leads to a simple and direct method for the analysis of alternating direction implicit schemes for solving (t.1). This analysis is presented elsewhere [6]. Section 2 contains a brief resume of pertinent results concerning tensor products and matrices. In the next section general second order separable partial difference equations are defined, analyzed, and an explicit expression for the solution of boundary value problems is obtained. Finally, it is shown how to efficiently evaluate the explicit form of the solution (one does not, as proposed in [12], compute the inverse of the matrix in (IA) even though this inverse is explicitly available). Several types of possible generalizations are investigated in
In this paper we present a new implementation of the distributed Lagrange multiplier/fictitious domain (DLM) method by making some modifications over the original algorithm for the Newtonian case developed by Glowinski et al. [Int. J. Multiphase Flow 25 (1999) 755], and its extended version for the viscoelastic case by Singh et al. [J. Non-Newtonian Fluid Mech. 91 (2000) 165]. The key modification is to replace a finite-element triangulation for the velocity and a “staggered” (twice coarser) triangulation for the pressure with a rectangular discretization for the velocity and the pressure. The sedimentation of a single circular particle in a Newtonian fluid at different Reynolds numbers, sedimentation of particles in the Oldroyd-B fluid, and lateral migration of a single particle in a Poiseuille flow of a Newtonian fluid are numerically simulated with our code. The results show that the new implementation can give a more accurate prediction of the motion of particles compared to the previous DLM codes and even the boundary-fitted methods in some cases. The centering of a particle and the well-organized Karman vortex street are observed at high Reynolds numbers in our simulation of a particle sedimenting in a Newtonian fluid. Both results obtained using the DLM method and the spectral element method reveal that the direct contribution of the viscoelastic normal stress to the force on a particle in the Oldroyd-B fluid is very important.
The transient laminar natural convection heat transfer of fluids between two concentric isothermal spheres is investigated theoretically. The fluid is initially at rest and then the inner wall is subjected to a step change of temperature. The stream function/vorticity formulation is employed for the analysis because of the symmetry of the problem. The transient behavior of the flow field and its subsequent effect on the temperature distribution for different Rayleigh numbers and radius ratios are analyzed by finite difference methods. Forward differences are used for the time derivatives and second-order central differences for the space derivatives. The alternating direction implicit method is used for solution of the discretization equations. The results show that the Rayleigh number and radius ratio have a profound influence on the temperature and flow fields. The results of average Nusselt numbers are also compared with those of previous experimental and numerical investigations. Excellent agreement is obtained.
In this study, we extended the distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) formulation of Glowinski et al. [Int. J. Multiphase Flow 25 (1999) 755] for the fluid/rigid-body interactions to deal with the fluid/flexible-body interactions by replacing Newton’s equations of motion for the rigid body with the continuum equations for the general solid material. Similar to the rigid-body case where the DLM is introduced as a pseudo body force to enforce the constraint of rigid-body motion of the fictitious fluid in the solid domain, the Lagrange multiplier in our formulation is to enforce the fictitious fluid to move at the same velocity as the solid. For our computational scheme, a first-order accurate fractional step scheme is employed to decouple the entire system into three sub-systems: a fluid problem, a solid problem and a Lagrange multiplier problem; the flow problem is solved with the projection method on half-staggered grids; the solid problem is solved with the Lagrangian finite element method and the Newton iterative method; and the incompressibility of the material is implemented with the penalty function method. The proposed method is applied to two typical fluid–structure interaction problems: the flow-driven oscillation of a flexible plate along the flow direction and the self-sustained oscillation across the flow direction. Both results compare favorably with previously reported numerical and experimental results, and show that our method is suited to the simulation of the motion of an incompressible non-linear elastic material in a fluid.
We consider the solution of both symmetric and unsymmetric systems of sparse linear equations. A new parallel distributed memory multifrontal approach is described. To handle numerical pivoting efficiently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been developed. We discuss some of the main algorithmic choices and compare both implementation issues and the performance of the LDLT and LU factorizations. Performance analysis on an IBM SP2 shows the efficiency and the potential of the method. The test problems used are from the Rutherford–Boeing collection and from the PARASOL end users.
The subject of this paper is the flow of a viscous incompressible fluid in a region containing immersed boundaries which move with the fluid and exert forces on the fluid. An example of such a boundary is the flexible leaflet of a human heart valve. It is the main achievement of the present paper that a method for solving the Navier-Stokes equations on a rectangular domain can now be applied to a problem involving this type of immersed boundary. This is accomplished by replacing the boundary by a field of force which is defined on the mesh points of the rectangular domain and which is calculated from the configuration of the boundary. In order to link the representations of the boundary and fluid, since boundary points and mesh points need not coincide, a semi-discrete analog of the δ function is introduced. Because the boundary forces are of order h−1, and because they are sensitive to small changes in boundary configuration, they tend to produce numerical instability. This difficulty is overcome by an implicit method for calculating the boundary forces, a method which takes into account the displacements that will be produced by the boundary forces themselves. The numerical scheme is applied to the two-dimensional simulation of flow around the natural mitral valve.
A computational setting for the Immersed Boundary Method employing an adaptive mesh refinement is presented. Enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested, progressively finer rectangular grid patches which dynamically follow the immersed boundary motion. The set of equations describing the interaction between a non-stationary, viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method, specially designed for locally refined meshes, to an implicit formulation of the Immersed Boundary Method. The main contributions of this work concern the formulation and the implementation of a multilevel self-adaptive version of the Immersed Boundary Method on locally refined meshes. This approach is tested for a particular two-dimensional model problem, for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary, and on the associated uniform mesh, built with the resolution of the finest level.
A new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion. Like previous methods, a boundary force is applied at the immersed surface to satisfy the no-slip constraint. This extra constraint can be added to the incompressible Navier–Stokes equations by introducing regularization and interpolation operators. The current method gives prominence to the role of the boundary force acting as a Lagrange multiplier to satisfy the no-slip condition. This role is analogous to the effect of pressure on the momentum equation to satisfy the divergence-free constraint. The current immersed boundary method removes slip and non-divergence-free components of the velocity field through a projection. The boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger CFL numbers compared to some past methods. Symmetry and positive-definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field. Examples show that the current formulation achieves second-order temporal accuracy and better than first-order spatial accuracy in L2-norms for one- and two-dimensional test problems. Results from two-dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies.
In this article we discuss the application of a distributed Lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modelled by the Navier-Stokes equations around moving bodies, we suppose the rigid bodies motion known a priori. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. Numerical experiment results for two-dimensional flow around a moving disk are presented.
We present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid. The main idea is to incorporate Peskin’s regularized delta function approach [Acta Numerica 11 (2002) 1] into a direct formulation of the fluid–solid interaction force in order to allow for a smooth transfer between Eulerian and Lagrangian representations while at the same time avoiding strong restrictions of the time step. This technique was implemented in a finite-difference and fractional-step context. A variety of two- and three-dimensional simulations are presented, ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles. The accuracy and efficiency of the current method are clearly demonstrated.
We present a method for solving the incompressible Navier–Stokes equations in irregular domains. These equations are discretized using finite difference method in a uniform Cartesian grid. Stationary rigid boundaries are embedded in the Cartesian grid and singular forces are applied at the rigid boundaries to impose the no-slip conditions. The singular forces are then distributed to the nearby Cartesian grid points using the immersed boundary method. In the present work, the singular forces are computed implicitly by solving a small system of equations at each time step. This system of equations is derived from a second order projection method. The main advantage of this method is that it imposes the no-slip boundary condition exactly and avoids the need for small time step to maintain stability. The ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2-dimensional flows past a circular cylinder, multiple rigid obstacles and 3-dimensional flow past a sphere.
A general, numerical, marching procedure is presented for the calculation of the transport processes in three-dimensional flows characterised by the presence of one coordinate in which physical influences are exerted in only one direction. Such flows give rise to parabolic differential equations and so can be called three-dimensional parabolic flows. The procedure can be regarded as a boundary-layer method, provided it is recognised that, unlike earlier published methods with this name, it takes full account of the cross-stream diffusion of momentum, etc., and of the pressure variation in the cross-stream plane. The pressure field is determined by: first calculating an intermediate velocity field based on an estimated pressure field; and then obtaining appropriate correction so as to satisfy the continuity equation. To illustrate the procedure, calculations are presented for the developing laminar flow and heat transfer in a square duct with a laterally-moving wall.
In this paper, we analyze the main features and discuss tuning of a solver for the direct solution of sparse linear equations on distributed memory computers, developed by the authors in the context of the PARASOL Project (ESPRIT IV LTR Project (No 20160)). To cover a large class of test problems (symmetric positive definite, general symmetric, unsymmetric, rank deficient matrices), we have designed a very general purpose method based on a multifrontal approach. Dynamic distributed task scheduling has been developed to accommodate numerical pivoting due to partial pivoting and to allow the migration of tasks to lightly loaded processors. Fully asynchronous communications have been used to efficiently overlap communication with computation. We illustrate our design choices by runs on test matrices from the Rutherford-Boeing sparse collection and from industrial partners in the PARASOL project. 1 Introduction We consider the direct solution of large sparse linear systems on d...
An implicit-forcing immersed boundary method for simulating viscous flows in irregular domains
- S W Su
- M C Lai
- C A Lin
S. W. Su, M. C. Lai, C. A. Lin, An implicit-forcing immersed boundary method for simulating viscous flows in irregular domains, Comput.
Fluid. 36 (2007) 313-324.
Multifrontal parallel distributed symmetric and unsymmetric solvers
- P Amestoy
- I Duff
- J Léxcellent
- J Koster
P. Amestoy, I. Duff, J. LÉxcellent, J. Koster, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl.
Mech. Engrg. 184 (1998) 501-520.