Discrete Delta Functions Research Articles

An efficient direct‐forcing immersed boundary method for flow around a pair of spheres

AbstractThe numerical study of flow around a pair of spheres and a square array of spheres is investigated by using a direct‐forcing immersed boundary method. Using high resolution three‐dimensional computations, we analyzed the flow around several configurations: a sphere, a pair of spheres in a tandem arrangement with center‐to‐center streamwise ratio L/D ranging from 1 to 6, and a square array with 9 spheres in a uniform arrangement. In the latter case, we explore the ratio of array diameter (DG) to sphere diameter (D) at 4, 5, 6 and 7. The center‐to‐center streamwise and transverse pitch is the same, varied from L/D = 1.5, 2, 2.5 to 3, and they were arranged in a square periodic array to allow uniform distribution within the array. Based on the effective direct‐forcing immersed boundary projection method, the fractional time marching methodology is applied for solving four field variables involving three velocities and one pressure component. The pressure Poisson equation is advanced in space by using the fast Fourier transform (FFT) and a tridiagonal matrix algorithm (TDMA), effectively solving for the diagonally dominant tridiagonal matrix equations. A direct‐forcing immersed boundary method is involved to treat the interfacial terms by adding the appropriate sources as force function at the boundary, separating the phases. Geometries featuring the stationary solid obstacles in the flow are embedded in the Cartesian grid with special discretizations near the embedded boundary using a discrete Dirac delta function to ensure the accuracy of the solution in the cut cells. An important characteristic of flow over the multiple spheres is devised by comparing with the drag and lift coefficients, as well as vortex shedding.

An immersed boundary method‐discrete unified gas kinetic scheme simulation of particle‐laden turbulent channel flow on a nonuniform orthogonal mesh

AbstractParticle‐resolved simulations of turbulent particle‐laden flows provide a powerful research tool to explore detailed flow physics at all scales. However, efficient particle‐resolved simulations for wall‐bounded turbulent particle‐laden flows remain a challenging task. In this article, we develop a simulation approach for a turbulent channel flow laden with finite‐size particles on a nonuniform mesh by combining the discrete unified gas kinetic scheme (DUGKS) and the immersed boundary method (IBM). The standard discrete delta function was modified according to reproducible kernel particle method to take into account mesh non‐uniformity and correctly conserve force moments. Simulation results based on uniform and nonuniform meshes are compared to validate and examine the accuracy of the nonuniform mesh DUGKS‐IBM. Finally, the computational performance of the nonuniform mesh DUGKS‐IBM is discussed.

Analysis of the immersed boundary method for turbulent fluid-structure interaction with Lattice Boltzmann method

An efficient and new methodology to deal with fluid structure interaction at high Reynolds number flows is presented in this article. It relies on the coupling of the lattice Boltzmann method and the immersed boundary method with a second order predictor corrector model for the structure. The effect of the lagrangian weight of the immersed boundary method is also analyzed in the context of fluid structure interaction. Both curved and moving boundaries are considered, and several methods to calculate the lagrangian weight are compared on relevant test-cases: the laminar flow around a cylinder at Reynolds number Re=100, the Poiseuille flow in a 2D channel and a 3D fluid-structure interaction test case: a deformable flapping flag immersed in a laminar flow. A convergence study in space and time is performed and the three following parameters are investigated: the number of lagrangian markers along the boundary, the value of the lagrangian weight and the shape of the discrete delta function used in the immersed boundary. Finally, the novelty of the paper is two-fold: a new expression of the lagrangian weight which is found to reduce the error by 20% in the case of fluid-structure interaction, and the coupling of a turbulence model with a second order predictor corrector model for improved stability and accuracy. This numerical methodology is found here to be accurate for challenging cases with high added mass effect and high Reynolds number flows.

An immersed boundary/multi-relaxation time lattice Boltzmann method on adaptive octree grids for the particle-resolved simulation of particle-laden flows

We present an immersed boundary/multi-relaxation time lattice Boltzmann method for the particle-resolved simulation of particle-laden flows. The no-slip boundary condition is handled by an explicit feedback immersed boundary method for fixed and moving particles with arbitrary shape. Two special treatments are applied to improve the simulation stability and accuracy: (i) a smoothed discrete delta function (Yang et al. (2009) [47]) is adopted to suppress the spurious force oscillations and (ii) a multi-relaxation time collision operator is utilized to fix the viscosity-dependent numerical slip error of the immersed boundaries. To further reduce the computational effort, we extend the method from fixed and uniform Cartesian grids to adaptive quadtree/octree grids by implementing it in the open-source software Basilisk. A Lax-Wendroff streaming operator is adopted in our method to retain second-order accuracy in both space and time on non-uniform grids, as well as to maintain a unified time scale over the entire computational domain. Consequently, one collision-streaming operation only is required per time step at all grid levels. We test our proposed method on a set of validation cases corresponding to assorted incompressible flows with fixed and moving rigid particles in both 2D and 3D. The accuracy and robustness of our method are demonstrated via thorough comparison with the analytical, experimental and numerical data provided in the literature.

Numerical Study of an Indicator Function for Front-Tracking Methods

In this paper, we present a detailed derivation and numerical investigation of an indicator function for front-tracking methods. We use the discrete Dirac delta function to construct an indicator function from a set of Lagrangian points and solve the resulting discrete Poisson equation with the zero Dirichlet boundary condition using an iterative method. We present several computational tests to investigate the effect of parameters such as distance between points, uniformity of the distance, and types of the Dirac delta functions on the indicator function.

An immersed boundary method with implicit body force for compressible viscous flow

We present a new immersed boundary method for the simulation of compressible viscous flow. The method is based on the singular source approach where the immersed boundary adds surface singularities to the governing equations. The strengths of the surface singularities are treated as algebraic variables under the framework of differential-algebraic equations and are implicitly calculated to enforce the corresponding boundary conditions. Discrete delta functions are used to approximate the surface singularities in the Eulerian grid and interpolate variables back to the surface mesh. A half-explicit Runge-Kutta method is used to achieve desired temporal accuracy. The method has been generalized from the well-known no-slip and isothermal condition to stress and heat flux conditions. The latter necessitates the introduction of further singularities in the stresses and heat flux, which is a new feature of the formulation. Furthermore, the approach has been extended to porous surfaces. The method is also applicable to flow-structure interaction problems. Two- and three-dimensional numerical tests are carried out to demonstrate accuracy. The results are compared with previous numerical and experimental studies or analytical solutions. Finally, a fully three-dimensional fluid-structure interaction simulation of a subsonic parachute is showcased to demonstrate applicability to a real problem.

Remarks on the numerical approximation of Dirac delta functions

We investigate the convergence rate of the solutions of one and two-dimensional Poisson-type PDEs where the Dirac delta function, representing the forcing term, is approximated by several expressions.The goal is to see if the solution to a Poisson’s equation converges when solved by a numerical method, with a source or sink approximated using a delta proposed in the literature. We will look at two parameters, how fast it converges, by estimating the order of convergence, and how well, by calculating the error between the analytical form and the numerical result. We investigate smoothed discrete delta functions based on the Immersed boundary (IB) approach, and we revisit their definitions, as in level set methods, by expanding their support for assessing higher-order of convergence in PDE solutions. We developed a Python package utilizing FiPy, a PDE solver based on the finite volume (FV) technique, and accelerated the solver with the AmgX package, a GPU solution. We have observed that when the support is wider, better results may be achieved. Moreover, the overall trends of error and the convergence rate in the 2D configuration differ from the 1D problems.

A method of immersed layers on Cartesian grids, with application to incompressible flows

The immersed boundary method (IBM) of Peskin (J. Comput. Phys., 1977), and derived forms such as the projection method of Taira and Colonius (J. Comput. Phys., 2007), have been useful for simulating flow physics in problems with moving interfaces on stationary grids. However, in their interface treatment, these methods do not distinguish one side from the other, but rather, apply the motion constraint to both sides, and the associated interface force is an inseparable mix of contributions from each side. In this work, we define a discrete Heaviside function, a natural companion to the familiar discrete Dirac delta function (DDF), to define a masked version of each field on the grid which, to within the error of the DDF, takes the intended value of the field on the respective sides of the interface. From this foundation we develop discrete operators and identities that are uniformly applicable to any surface geometry. We use these to develop extended forms of prototypical partial differential equations, including Poisson, convection-diffusion, and incompressible Navier-Stokes, that govern the discrete masked fields. These equations contain the familiar forcing term of the IBM, but also additional terms that regularize the jumps in field quantities onto the grid and enable us to individually specify the constraints on field behavior on each side of the interface. Drawing the connection between these terms and the layer potentials in elliptic problems, we refer to them generically as immersed layers. We demonstrate the application of the method to several representative problems, including two-dimensional incompressible flows inside a rotating cylinder and external to a rotating square.

The local tangential lifting method for moving interface problems on surfaces with applications

In this paper, a new numerical computational frame is presented for solving moving interface problems modeled by parabolic PDEs on static and evolving surfaces. The surface PDEs can have Dirac delta source distributions and discontinuous coefficients. One application is for thermally driven moving interfaces on surfaces such as Stefan problems and dendritic solidification phenomena on solid surfaces. One novelty of the new method is the local tangential lifting method to construct discrete delta functions on surfaces. The idea of the local tangential lifting method is to transform a local surface problem to a local two dimensional one on the tangent planes of surfaces at some selected surface nodes. Moreover, a surface version of the front tracking method is developed to track moving interfaces on surfaces. Strategies have been developed for computing geodesic curvatures of interfaces on surfaces. Various numerical examples are presented to demonstrate the accuracy of the new methods. It is also interesting to see the comparison of the dendritic solidification processes in two dimensional spaces and on surfaces.

Discrete weierstrass transform in discrete hermitian clifford analysis

The classical Weierstrass transform is an isometric operator mapping elements of the weighted L2−space L2(R,exp(−x2/2)) to the Fock space. It has numereous applications in physics and applied mathematics. In this paper, we define an analogue version of this transform in discrete Hermitian Clifford analysis, where functions are defined on a grid rather than the continuous space. This new transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. Furthermore, a discrete Weierstrass space with appropriate inner product is constructed, for which the discrete Hermite polynomials form a basis. In this setting, we also investigate the behaviour of the discrete delta functions and check if they are elements of this newly defined Weierstrass space.

An iterative divergence-free immersed boundary method in the finite element framework for moving bodies

A novel iterative direct-forcing immersed boundary (IB) method, which meets the divergence-free and no-slip conditions simultaneously, is proposed in the finite element framework for incompressible flow problems. The coupled velocity and the pressure of the fluid field are solved by the Characteristic-based Split scheme. The formulation for the extra body force is derived according to the no-slip boundary condition. The novelty of the proposed IB method lies in that the calculation of the pressure, the iterations of the velocity, the pressure and the extra body force are accomplished in a loop at the same time. Therefore, the divergence-free and no-slip conditions are ensured fully at the same time. The accuracy of the current IB method is verified by several cases, including static and moving boundaries. Meanwhile, the contributing factors of the spurious force oscillations for moving boundaries are studied. A brand-new viewpoint is proposed, which is different with the common standpoint. The major source of spurious force oscillations pointed out in this paper is that the change of interpolation coefficients when boundaries sweep within an element, instead of the change of interpolation points when boundaries move across adjacent elements, as discussed in existing references. The oscillations which are caused by the inherent properties of the immersed boundary method cannot be avoided, while effective methods can be employed to suppress them, including refining the meshes and applying a proper discrete delta function.

Sharp Computational Images From Diffuse Beams: Factorization of the Discrete Delta Function

Discrete delta functions define the limits of attainable spatial resolution for all imaging systems. Here we construct broad, multi-dimensional discrete functions that replicate closely the action of a Dirac delta function under aperiodic convolution. These arrays spread the energy of a sharp probe beam to simultaneously sample multiple points across the volume of a large object, without losing image sharpness. A diffuse point-spread function applied in any imaging system can reveal the underlying structure of objects less intrusively and with equal or better signal-to-noise ratio. These multi-dimensional arrays are related to previously known, but relatively rarely employed, one-dimensional integer Huffman sequences. Practical point-spread functions can now be made sufficiently large to span the size of the object under measure. Such large arrays can be applied to ghost imaging, which has demonstrated potential to greatly improve signal-to-noise ratios and reduce the total dose required for tomographic imaging. The discrete arrays built here parallel the continuum self-adjoint or Hermitian functions that underpin wave theory and quantum mechanics.

A relaxed multi-direct-forcing immersed boundary-cascaded lattice Boltzmann method accelerated on GPU

A new relaxed multi-direct-forcing immersed boundary-cascaded lattice Boltzmann method (MDF IB-CLBM) is proposed in this paper. This new technique improves the efficiency and accuracy of implementing no-slip boundaries on a single graphics processing unit (GPU). The traditional MDF-IBM method essentially solves a linear system iteratively with a relaxation parameter 1.0. By introducing an estimated optimal relaxation parameter, no-slip boundary is fulfilled after only one iteration in a typical case of flow around a fixed cylinder. This is a 7-fold speed-up over the traditional method. Since it is computationally expensive to calculate the estimated optimal relaxation parameter for moving boundaries in every time step, the impact of many factors on the value of the estimated optimal relaxation parameter is investigated. The studied parameters include the discrete delta function, shape of no-slip boundary, the relative distance between several no-slip boundaries, the spacing size between the Lagrangian points, and the relative movement of boundaries over underlying lattices. It is found that the first three factors have a more pronounced influence. Therefore, by considering these three key factors, an appropriate constant relaxation parameter can be determined before starting the simulations. Moreover, the relaxed method provides significant improvement to the traditional MDF-IBM with only a few modifications. The effectiveness of the method is demonstrated by dealing with the flow around a fixed cylinder, the flow past several obstacles, a particle moving in a linear shear flow, and the sedimentation of multiple particles in an enclosure.

Is Lagrangian weight crucial in the direct forcing immersed boundary method?

Particle resolved direct numerical simulation (PR-DNS) is one of the most powerful research tools for particle laden flows. Among a few most popular PR-DNS methods, the direct forcing immersed boundary method (DF-IBM) has obtained great success and has been adopted in various simulations of rigid particulate flows. Within DF-IBM, Eulerian and Lagrangian frameworks are used to depict the continuum and dispersed phases, respectively. Interpolation between the two frameworks is accomplished through a discrete delta function. It is generally believed that a Lagrangian weight attached to each Lagrangian marker, which is distributed on a particle’s surface, needs to be carefully chosen. To be more specific, the Lagrangian weight is supposed to match the local Eulerian cell. The matching requirement is not trivial for non-uniform Eulerian mesh or irregular shaped particles. There are various methods developed to calculate the Lagrangian weight. Here, the Lagrangian weights in a few testing cases have been calculated following two intuitively “straightforward” methods. It turns out there are substantial discrepancies in the Lagrangian weights obtained from different methods. However, further numerical examples demonstrate that such discrepancies have negligible effects on the flow dynamics. So a natural question is raised: Is Lagrangian weight crucial in the direct forcing immersed boundary method? A negative answer to this question is suggested. More detailed analysis is provided in a forthcoming paper.

A Moving-Least-Square Immersed Boundary Method for Rigid and Deformable Boundaries in Viscous Flow

AbstractWe present a moving-least-square immersed boundary method for solving viscous incompressible flow involving deformable and rigid boundaries on a uniform Cartesian grid. For rigid boundaries, noslip conditions at the rigid interfaces are enforced using the immersed-boundary direct-forcing method. We propose a reconstruction approach that utilizes moving least squares (MLS) method to reconstruct the velocity at the forcing points in the vicinity of the rigid boundaries. For deformable boundaries, MLS method is employed to construct the interpolation and distribution operators for the immersed boundary points in the vicinity of the rigid boundaries instead of using discrete delta functions. The MLS approach allows us to avoid distributing the Lagrangian forces into the solid domains as well as to avoid using the velocity of points inside the solid domains to compute the velocity of the deformable boundaries. The present numerical technique has been validated by several examples including a Poiseuille flow in a tube, deformations of elastic capsules in shear flow and dynamics of red-blood cell in microfluidic devices.

Conservative modified Crank–Nicolson and time-splitting wavelet methods for modeling Bose–Einstein condensates in delta potentials

This paper explores two wavelet-based energy-conserving algorithms for the Gross–Pitaevskii equation with delta potentials in Bose–Einstein condensates, named modified Crank–Nicolson wavelet method and time-splitting wavelet method, respectively. Both proposed methods can preserve the intrinsic properties of original problems as much as possible. Meanwhile, the rigorous error estimates and some conservative properties are investigated. They are proved to preserve the charge conservation exactly. The global energy conservation laws can be preserved under several conditions. In practical computations, to avoid a large drift in energy values caused by discontinuous potential well, an improved discrete delta function is implemented. Numerical experiments for attractive and repulsive cases are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

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 3700 are used to verify the accuracy and physical fidelity of the formulation.

Statistical Analysis of the Reaction Progress Variable and Mixture Fraction Gradients in Flames Propagating into Droplet Mist: A Direct Numerical Simulation Analysis

ABSTRACTThe statistics of reaction progress variable, , and mixture fraction, , and their gradients (i.e., and ) in flames propagating in droplet mist, where the fuel was supplied in the form of monodisperse droplets, have been analyzed for different values of turbulent velocity fluctuations (), droplet equivalence ratios (, and droplet diameters ( based on three-dimensional direct numerical simulations (DNS) in a canonical configuration under decaying turbulence. The combustion process in the gaseous phase has been found to take place predominantly in fuel-lean mode, even for . The probability of finding fuel-lean mixture increases with increasing initial droplet diameter due to slower evaporation of larger droplets. It has been shown that the joint probability density function (i.e., joint PDF) of and (i.e., ), cannot be approximated in terms of discrete delta functions throughout the flame brush for the cases considered here. Furthermore, the magnitude of cannot be adequately approximated by the product of marginal PDFs of , and variable, (i.e., ). The statistical properties of the Favre probability density functions (Favre-PDFs) of the mixture fraction, , and oxidizer-based reaction progress variable, , have been analyzed at several locations across the flame brush and a -function distribution has been found to capture the Favre-PDFs of and obtained from the DNS data. Furthermore, a log-normal distribution has been shown to capture the qualitative behaviors of the PDFs of the gradient of the mixture fraction and the gradient of the reaction progress variable, and , respectively, but discrepancies between the log-normal distribution and the DNS data were observed at the tails of PDFs. In addition, the interrelation between and was examined in terms of the PDFs of the cosine of the angle between them (i.e., and it was observed that most droplet cases exhibited much greater likelihood of positive values of than negative values. Finally, the joint PDF of and , , has been compared with that of P( (i.e., assuming statistical independence of and ) and a good level of agreement has been obtained. The bivariate log-normal distribution has been considered both assuming correlation between and and assuming no correlation for the purpose of modeling , and the variant with no correlation has been found to be more successful in capturing qualitative behavior of although quantitative discrepancies have been observed due to inaccuracies involved in parameterizing P( and P(by log-normal distributions.

A Gaussian-like immersed-boundary kernel with three continuous derivatives and improved translational invariance

The immersed boundary (IB) method is a general mathematical framework for studying problems involving fluid–structure interactions in which an elastic structure is immersed in a viscous incompressible fluid. In the IB formulation, the fluid described by Eulerian variables is coupled with the immersed structure described by Lagrangian variables via the use of the Dirac delta function. From a numerical standpoint, the Lagrangian force spreading and the Eulerian velocity interpolation are carried out by a regularized, compactly supported discrete delta function, which is assumed to be a tensor product of a single-variable immersed-boundary kernel. IB kernels are derived from a set of postulates designed to achieve approximate grid translational invariance, interpolation accuracy and computational efficiency. In this note, we present a new 6-point immersed-boundary kernel that is C3 and yields a substantially improved translational invariance compared to other common IB kernels.

Direct-forcing immersed boundary lattice Boltzmann simulation of particle/fluid interactions for spherical and non-spherical particles

The lattice Boltzmann method (LBM) is a useful technique for simulating multiphase flows and modeling complex physics. Specifically, we use LBM combined with a direct-forcing (DF) immersed boundary (IB) method to simulate fluid–particle interactions in two-phase particulate flows. Two grids are used in the simulation: a fixed uniform Eulerian grid for the fluid phase and a Lagrangian grid that is attached to and moves with the immersed particles. Forces are calculated at each Lagrangian point. To exchange numerical information between the two grids, discrete delta functions are used. The resulting DF IB-LBM approach is then successfully applied to a variety of reference flows, namely the sedimentation of one and two circular particles in a vertical channel, the sedimentation of one or two spheres in an enclosure, and a neutrally buoyant prolate spheroid in a Couette flow. This last application proves that the developed approach can be used also for non-spherical particles. The three forcing schemes and the different factors affecting the simulation (added mass effect, corrected radius) are also discussed.