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A semi-implicit direct forcing immersed boundary method for periodically moving immersed bodies: A Schur complement approach
Abstract
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 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.
... [34]). The problem can be remedied by utilizing semi-implicit formulation of the immersed boundary method, which imposes kinematic constraints of no-slip on the predicted non-solenoidal velocity field up to machine zero precession [48]. The second challenge is an extension of the currently presented method to simulate fully 3D natural convection flows. ...
In this work, based on the moving-least-squares immersed boundary method, we proposed a new technique to improve the calculation of the volume force representing the body boundary. For boundary with simple geometry, we theoretically analyze the error between the desired volume force at boundary and the actual force given by the original method. The ratio between the two forces is very close to a constant and exhibits a very narrow distribution. A spatially uniform coefficient is then introduced to correct the force and can be fixed by the least-square method over all boundary markers. Such method is explicit and non-iterative, and is easy to implement into the existing scheme. Several test cases have been simulated with stationary and moving boundaries. Our new method can reduce the residual boundary velocity to the level comparable to that given by the iterative method, but requires much less computing time. Preliminary test suggests that the current method is also effective for nonuniform Eulerian grid. Moreover, the new method can be readily combined with the iterative method and further reduces the residual boundary velocity.
In this work, based on the moving-least-squares immersed boundary method, we proposed a new technique to improve the calculation of the volume force representing the body boundary. For boundary with simple geometry, we theoretically analyse the error between the desired volume force at boundary and the actual force given by the original method. The ratio between the two forces is very close to a constant. Numerical experiments reveal that for complex geometry, this ratio exhibits very narrow distribution around certain value. A spatially uniform coefficient is then introduced to correct the force and fixed by the least-square method over all boundary markers. Such method is explicit and non-iterative, and can be easily implemented into the existing scheme. Several test cases have been simulated with stationary and moving boundaries. Our new method can reduce the residual boundary velocity to the level comparable to that given by the iterative method, but requires much less computing time. Moreover, the new method can be readily combined with the iterative method and further reduces the residual boundary velocity.
A computational framework is designed to accurately predict the elastic response of thin shells undergoing large displacements induced by local hydrodynamic forces, as well as to resolve the complex fluid pattern arising from its interaction with an incompressible fluid. Within the context of partitioned algorithms, two different approaches are employed for the fluid and structural domain. The fluid motion is resolved with a pressure projection method on a Cartesian structured grid. The immersed shell is modeled by means of a NURBS surface, and the elastic response is obtained from a displacement-based isogeometric analysis relying on the Kirchhoff–Love theory. The two solvers exchange data through a direct-forcing immersed-boundary approach, where the interpolation/spreading of the variables between Lagrangian and Eulerian grids is implemented with a Moving Least Squares approximation, which has proven to be very effective for moving boundaries. In this scenario, the isoparametric paradigm is exploited to perform an adaptive collocation of the Lagrangian markers, decoupling the local grid density of fluid and shell domains and reducing the computational expense. The accuracy of the method is verified by refinement analyses, segregating the Eulerian/Lagrangian refinement, which confirm the expected scheme accuracy in space and time. The effectiveness of the method is then validated against different test-cases of engineering and biologic inspiration, involving fundamentally different physical and numerical conditions, namely: (i) a flapping flag, (ii) an inverted flag, (iii) a clamped plate, (iv) a buoyant seaweed in a free stream. Both strong and loose coupling approaches are implemented to handle different fluid-to-structure density ratios, providing accurate results.
Solid tumor dissemination from the primary site to the sites of metastasis involves tumor cell transport through the blood or lymph circulation systems. Once the tumor cells enter the bloodstream, they encounter a new hostile microenvironment. The cells must withstand hemodynamic forces and overcome the effects of fluid shear. The cells are exposed to immunological signaling insults from leukocytes, to collisions with erythrocytes, and to interactions with platelets or macrophages. Finally, the cells need to attach to the blood vessel walls and extravasate to the surrounding stroma to form tumor metastases. Although only a small fraction of invasive cells is able to complete the metastatic process, most cancer-related deaths are the result of tumor metastasis. Thus, investigating the intracellular properties of circulating tumor cells and the extracellular conditions that allow the tumor cells to survive and thrive in this microenvironment is of vital interest. In this chapter, we discuss the intravascular microenvironment that the circulating tumor cells must endure. We summarize the current experimental and computational literature on tumor cells in the circulation system. We also illustrate various aspects of the intravascular transport of circulating tumor cells using a mathematical model based on immersed boundary principles.
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.
This paper presents a simple, efficient, and high-order accurate sliding-mesh interface approach to the spectral difference (SD) method. We demonstrate the approach by solving the two-dimensional compressible Navier–Stokes equations on quadrilateral grids. This approach is an extension of the straight mortar method originally designed for stationary domains [7] and [8]. Our sliding method creates curved dynamic mortars on sliding-mesh interfaces to couple rotating and stationary domains. On the nonconforming sliding-mesh interfaces, the related variables are first projected from cell faces to mortars to compute common fluxes, and then the common fluxes are projected back from the mortars to the cell faces to ensure conservation. To verify the spatial order of accuracy of the sliding-mesh spectral difference (SSD) method, both inviscid and viscous flow cases are tested. It is shown that the SSD method preserves the high-order accuracy of the SD method. Meanwhile, the SSD method is found to be very efficient in terms of computational cost. This novel sliding-mesh interface method is very suitable for parallel processing with domain decomposition. It can be applied to a wide range of problems, such as the hydrodynamics of marine propellers, the aerodynamics of rotorcraft, wind turbines, and oscillating wing power generators, etc.
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 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.
In the present study, we apply a robust Incompressible Smoothed Particle Hydrodynamics method improved by advanced second-order discretization in order to examine active micromixers equipped with a rotating or an oscillating stir-bar. Since it has a Lagrangian nature and it is based on particles, Smoothed Particle Hydrodynamics is an appropriate and convenient method for the moving boundary and for one-way coupled fluid–solid interactions. However, this method has not been applied to simulate active micromixers. The present study thus introduces a convenient Smoothed Particle Hydrodynamics method for modelling active micromixers. In addition, mixing flow is investigated by considering the vorticity field. Cross-shaped and straight stir-bars are also examined and evaluated. The presented results show that the cross-shaped stir-bar is more efficient than the straight stir-bar. In addition, the rotating cross-shaped stir-bar can be introduced as an appropriate stirrer, while the rotating straight stir-bar demonstrates weak performance.
In this paper the principles of the FOAM C++ class library for continuum mechanics are outlined. The intention is to make it as easy as possible to develop reliable and efficient computational continuum mechanics (CCM) codes: this is achieved by making the top level syntax of the code as close as possible to conventional mathematical notation for tensors and partial differential equations. Object orientation techniques enable the creation of data types which closely mimic those of continuum mechanics, and the operator overloading possible in C++ allows normal mathematical symbols to be used for the basic operations. As an example, the implementation of various types of turbulence modelling in a FOAM computational uid-dynamics (CFD) code is discussed, and calculations performed on a standard test case, that of flow around an square prism, are presented. To demonstrate the exibility of the FOAM library, codes for solving structures and magneto-hydrodynamics are also presented w...
Fluid–structure interaction (FSI) problems commonly encountered in engineering and biological applications involve geometrically complex flexible or rigid bodies undergoing large deformations. Immersed boundary (IB) methods have emerged as a powerful simulation tool for tackling such flows due to their inherent ability to handle arbitrarily complex bodies without the need for expensive and cumbersome dynamic re-meshing strategies. Depending on the approach such methods adopt to satisfy boundary conditions on solid surfaces they can be broadly classified as diffused and sharp interface methods. In this review, we present an overview of the fundamentals of both classes of methods with emphasis on solution algorithms for simulating FSI problems. We summarize and juxtapose different IB approaches for imposing boundary conditions, efficient iterative algorithms for solving the incompressible Navier–Stokes equations in the presence of dynamic immersed boundaries, and strong and loose coupling FSI strategies. We also present recent results from the application of such methods to study a wide range of problems, including vortex-induced vibrations, aquatic swimming, insect flying, human walking and renewable energy. Limitations of such methods and the need for future research to mitigate them are also discussed.
We present the development of a sliding mesh capability for an unsteady high order (order ⩾ 3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier–Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular–quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian–Eulerian form of the incompressible Navier–Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x–y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier–Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier–Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.
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 immersed boundary method is a methodology for dealing with boundary conditions at fluid–fluid and fluid–solid interfaces. The immersed boundary method has been attracting growing attention in the recent years due to its simplicity in mesh processing. Great effort has been made to develop its new features and promote its applications in new areas. This review is focused on assessing the immersed boundary method fundamentals and the latest progresses especially the strategies to address the challenges and the applications of the immersed boundary method. Various numerical examples are also presented for demonstrating the capability of the immersed boundary method, including blood flow and blood cells, flapping flag, flow around a hoverfly, turbulence flow over a wavy boundary, shock wave-induced vibration, and acoustic waves scattered by a cylinder and a sphere. The major challenges and several open issues in this field are highlighted.
In this review, we introduce immersed boundary (IB) methods for fluid-structure interactions (FSIs) of rigid and elastic bodies. IB methods impose momentum forcing on an Eulerian mesh to satisfy boundary conditions on the interface between fluid and structure, which enables us to use a non-body conforming grid system for complex-shaped moving bodies. Imposition of the momentum forcing is performed directly through discrete delta function or indirectly through velocity reconstruction, by which IB methods have their own strengths and weaknesses to FSI problems of rigid and elastic bodies. To deal with FSI, IB methods using monolithic and partitioned (strong and weak coupling) approaches with different stability and cost have been suggested. Nevertheless, two important problems in FSI, cases of low density ratio of solid to fluid and high Reynolds number, have not been completely overcome by current IB methods in terms of the stability, accuracy and cost. These aspects are examined in this review.
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.
An efficient immersed boundary (IB) method is presented for the direct numerical simulation of fluid flow past a pair of circular cylinders and rigid particulate flows. The cylinders are settled in either a tandem or side-by-side arrangement. The grid applied in this paper involves an uneven Cartesian grid, and local differential quadrature (LDQ) is employed to discretize the governing equations. Specifically, a solid in a target region is embedded into the Cartesian grid, and linear interpolation is adopted to calculate the IB and the virtual force on the cell center. The virtual force is substituted into the governing equations to determine the effect of the solid in the IB. The parameters for numerical calculation are the Reynolds number (10–200) and the Prandtl number of air (0.7). The spacing intervals of the tandem arrangement and side-by-side arrangement are g* = 1.5–4 and s* = 1.5–4, respectively. According to the aforementioned conditions, the local Nusselt number and average Nusselt number of the cylinder surface in each arrangement are calculated to investigate how dissimilar flow conditions and spacing intervals between the cylinders influence the effect of heat transfer enhancement. In addition, the present model is applied to simulate the sedimentation of one particle and two particles in a box. Related changes in the flow field of fluid-particles interaction are also discussed.
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.
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.
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.
This paper presents a new sliding mesh technique for the computation of unsteady viscous flows in the presence of rotating bodies. The compressible Euler and incompressible Navier-Stokes equations are solved using a higher-order (> 2) finite volume method on unstructured grids. A sliding mesh approach is employed at the interface between computational grids in relative motion. In order to prevent loss of accuracy, two distinct families of higher-order sliding mesh interfaces are developed. These approaches fit naturally
in a high-order finite volume framework. To this end, Moving Least Squares (MLS) approximants are used for the transmission of the information for one grid to another. A particular attention is paid for the study of the
accuracy and conservation properties of the numerical scheme for static and rotating grids. The capabilities of the present solver to compute complex unsteady vortical flow motions created by rotating geometries are illustrated on a cross-flow configuration.
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.
In this paper, we develop a geometrically flexible technique for computational fluid–structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term “immersogeometric analysis” to identify this paradigm.
Recent experiments showed that standing acoustic waves could be exploited to
induce self-propulsion of rigid metallic particles in the direction
perpendicular to the acoustic wave. We propose in this paper a physical
mechanism for these observations based on the interplay between inertial forces
in the fluid and the geometrical asymmetry of the particle shape. We consider
an axisymmetric rigid near-sphere oscillating in a quiescent fluid along a
direction perpendicular to its symmetry axis. The kinematics of oscillations
can be either prescribed or can result dynamically from the presence of an
external oscillating velocity field. Steady streaming in the fluid, the
inertial rectification of the time-periodic oscillating flow, generates steady
stresses on the particle which, in general, do not average to zero, resulting
in a finite propulsion speed along the axis of the symmetry of the particle and
perpendicular to the oscillation direction. Our derivation of the propulsion
speed is obtained at leading order in the Reynolds number and the deviation of
the shape from that of a sphere. The results of our model are consistent with
the experimental measurements, and more generally explains how time periodic
forcing from an acoustic field can be harnessed to generate autonomous motion.
The interactions between incompressible fluid flows and immersed struc-tures are nonlinear multi-physics phenomena that have applications to a wide range of scientific and engineering disciplines. In this article, we review representative numeri-cal methods based on conforming and non-conforming meshes that are currently avail-able for computing fluid-structure interaction problems, with an emphasis on some of the recent developments in the field. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study in fluid-structure interactions.
The present study investigates the natural convection induced by a temperature difference between a cold outer square enclosure and two hot inner circular cylinders. A two-dimensional solution for natural convection in an enclosure with inner cylinders is obtained using an accurate and efficient immersed boundary method. The immersed boundary method based on the finite volume method is used to handle inner cylinders located at different vertical centerline positions of the enclosure for different Rayleigh numbers in the range 103⩽Ra⩽106103⩽Ra⩽106. The results in the case of two cylinders are compared with those in the case of the single cylinder in order to see the effect of the interaction between the adjacent two inner hot cylinders in addition to the interaction between two hot inner cylinders and the cold walls of the enclosure. The distribution of isotherms and streamlines eventually reaches a steady state or changes its state from steady to unsteady, depending on the values of the Rayleigh number and the cylinder position in the enclosure. The distribution of the local and surface-averaged Nusselt numbers is obtained for different Rayleigh numbers and the cylinder positions.
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.
The advent of isogeometric analysis (IGA) using the same basis functions for design and analysis constitutes a milestone in the unification of geometric modeling and numerical simulation. However, an important class of geometric models based on the CSG (Constructive Solid Geometry) concept such as trimmed NURBS surfaces do not fully support the isogeometric paradigm, since basis functions do not explicitly represent the boundary. The finite cell method (FCM) is a high-order fictitious domain method, which offers simple meshing of potentially complex domains into a structured grid of cuboid cells, while still achieving exponential rates of convergence for smooth problems. In the present paper, we first discuss the possibility to directly couple the finite cell method to CSG, without any necessity for meshing the three-dimensional domain, and then explore a combination of the best of the two approaches IGA and FCM, closely following ideas of the recently introduced shell FCM. The resulting finite cell extension to isogeometric analysis achieves a truly straightforward transfer of a trimmed NURBS surface into an analysis suitable NURBS basis, while benefiting from the favorable properties of the high-order and high-continuity basis functions. Accuracy and efficiency of the new approach are demonstrated by a numerical benchmark, and its versatility is outlined by the analysis of different trimmed design variants of a brake disk.
We explore hierarchical refinement of NURBS as a basis for adaptive isogeometric and immersed boundary analysis. We use the principle of B-spline subdivision to derive a local refinement procedure, which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions. We test hierarchical refinement of NURBS for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases. Using the B-spline version of the finite cell method, we illustrate the potential of immersed boundary methods as a seamless isogeometric design-through-analysis procedure for complex engineering parts defined by T-spline CAD surfaces, specifically a ship propeller and an automobile wheel. We show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features.
An immersed boundary method (IBM) with second-order spatial accuracy is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method is based on the computationally efficient direct-forcing method of Uhlmann [M. Uhlmann, An immersed boundary method with direct forcing for simulation of particulate flows, J. Comput. Phys. 209 (2005) 448–476] that is embedded in a finite-volume/pressure-correction method. The IBM consists of two grids: a fixed uniform Eulerian grid for the fluid phase and a uniform Lagrangian grid attached to and moving with the particles. A regularized delta function is used to communicate between the two grids and proved to be effective in suppressing grid locking. Without significant loss of efficiency, the original method is improved by: (1) a better approximation of the no-slip/no-penetration (ns/np) condition on the surface of the particles by a multidirect forcing scheme, (2) a correction for the excess in the effective particle diameter by a slight retraction of the Lagrangian grid from the surface towards the interior of the particles with a fraction of the Eulerian grid spacing, and (3) an enhancement of the numerical stability for particle–fluid mass density ratios near unity by a direct account of the inertia of the fluid contained within the particles. The new IBM contains two new parameters: the number of iterations Ns of the multidirect forcing scheme and the retraction distance rd. The effect of Ns and rd on the accuracy is investigated for five different flows. The results show that rd has a strong influence on the effective particle diameter and little influence on the error in the ns/np condition, while exactly the opposite holds for Ns. A novel finding of this study is the demonstration that rd has a strong influence on the order of grid convergence. It is found that for spheres the choice of rd = 0.3Δx yields second-order accuracy compared to first-order accuracy of the original method that corresponds to rd = 0. Finally, Ns = 2 appears optimal for reducing the error in the ns/np condition and maintaining the computational efficiency of the method.
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.
Unsteady flow due to an oscillating sphere with a velocity U0cosωt’, in which U0 and ω are the amplitude and frequency of the oscillation and t’ is time, is investigated at finite Reynolds number. The methods used are: (i) Fourier mode expansion in the frequency domain; (ii) a time-dependent finite difference technique in the time domain; and (iii) a matched asymptotic expansion for high-frequency oscillation. The flow fields of the steady streaming component, the second and third harmonic components are obtained with the fundamental component. The dependence of the unsteady drag on ω is examined at small and finite Reynolds numbers. For large Stokes number, ε = (ωa2/2v)½ [dbl greater-than sign] 1, in which a is the radius of the sphere and v is the kinematic viscosity, the numerical result for the unsteady drag agrees well with the high-frequency asymptotic solution; and the Stokes (1851) solution is valid for finite Re at ε [dbl greater-than sign] 1. For small Strouhal number, St = ωa/U0 [double less-than sign] 1, the imaginary component of the unsteady drag (Scaled by 6πU0pfva, in which Pf is the fluid density) behaves as Dml [similar] (h0Stlog St–h1St), m = 1,3,5… This is in direct contrast to an earlier result obtained for an unsteady flow over a stationary sphere with a small-amplitude oscillation in the free-stream velocity (hereinafter referred to as the SA case) in which D1[similar] –h1 St (Mei, Lawrence & Adrian 1991). Computations for flow over a sphere with a free-stream velocity U0(1–α1+α1cosωt’) at Re = U02a/v = 0.2 and St [double less-than sign] 1 show that h0 for the first mode varies from 0 (at α1 = 0) to around 0.5 (at α1 = 1) and that the SA case is a degenerated case in which the logarithmic dependence of the drag in St is suppressed by the strong mean uniform flow.
This review describes recent developments in self-propelling nano- and micro-scale swimming devices. The ability of these devices to transport nano-scale components in a fluidic environment is demonstrated. Furthermore, the adaptations needed for these devices to meet biological transport challenges such as targeted drug delivery are highlighted. Particular emphasis is placed on describing autonomously powered devices driven by asymmetrical chemical reactions. Methods to control the speed and direction of such swimming devices using external fields are described, and contrasted to recent demonstrations of statistical autonomous migrations and organisations driven by chemical gradients, inter swimmer interactions and external photo-stimulus. Finally the challenges and advantages of converting other nature inspired swimming mechanisms into realistic artificial self-powered devices are considered.
A direct-forcing pressure correction method is developed to simulate fluid–particle interaction problems. In this paper, the sedimentation flow is investigated. This method uses a pressure correction method to solve incompressible flow fields. A direct-forcing method is introduced to capture the particle motions. It is found that the direct-forcing method can also be served as a wall-boundary condition. By applying Gauss's divergence theorem, the formulas for computing the hydrodynamic force and torque acting on the particle from flows are derived from the volume integral of the particle instead of the particle surface. The order of accuracy of the present method is demonstrated by the errors of velocity, pressure, and wall stress. To demonstrate the efficiency and capability of the present method, sedimentations of many spherical particles in an enclosure are simulated. Copyright © 2010 John Wiley & Sons, Ltd.
Heat or mass transfer from spherical particles in oscillatory flow has important applications in combustion and spray drying. This work provides a parametric investigation of drag forces experienced by, and transport of a passive scalar from, an isolated rigid fixed sphere in steady and oscillatory axisymmetric flows. At Schmidt Prandtl number of 1, oscillatory flows with Reynolds numbers in the range 1–100 and oscillation amplitudes in the range 0.05–5 sphere diameters are investigated using numerical simulation. Scalar concentration is uniform on the surface of the sphere and zero in the far field. Coefficients of peak drag for steady and oscillatory flows are presented and compared to values obtained from Basset's analytical solution for Stokes flow, and the relative contributions of the added mass, Stokes drag, and Basset history terms are examined. At the higher Reynolds numbers and amplitudes, it is found that the time-average mass transfer rate can be more than double that for diffusion in quiescent fluid, or in Stokes flow. Time-average Sherwood Nusselt numbers for oscillatory flows asymptote to the Stokes limit at low oscillation amplitude, regardless of Reynolds number. An unexpected result is that at intermediate Reynolds numbers and oscillation amplitudes, it is possible to depress the time-average mass-transfer coefficient slightly below that for Stokes flow. Within the Reynolds number range considered, Sherwood–Nusselt numbers in steady flow are found to be always higher than for an oscillatory flow of the same root-mean-square rms velocity. © 2002 American Institute of Physics.
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.
A novel b-spline based immersed finite element method is introduced for the computation of geometrically and topologically complex problems. The geometry description and the finite element analysis rely on a block structured logically Cartesian mesh which encloses the domain of interest. A signed distance function is used for representing the domain on the Cartesian mesh, whereby the domain boundary is the zeroth level set of the signed distance function. Away from the domain boundaries, the standard b-spline basis functions are used for the finite element interpolation. Close to domain boundaries, a new approach has been developed for modifying the b-spline basis functions so that they locally interpolate the Dirichlet boundary conditions. The efficiency and robustness of the proposed approach is demonstrated with a number of one-, two- and three-dimensional linear boundary value problems.
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.
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.