An improved immersed boundary method with direct forcing for the simulation of particle laden flows

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.     

On the use of immersed boundary methods for shock/obstacle interactions

This paper describes the implementation of immersed boundary method using the direct-forcing concept to investigate complex shock–obstacle interactions. An interpolation algorithm is developed for more stable boundary conditions with easier implementation procedure. The values of the fluid variables at the embedded ghost-cells are obtained using a local quadratic scheme which involves the neighboring fluid nodes. Detailed discussions of the method are presented on the interpolation of flow variables, direct-forcing of ghost cells, resolution of immersed-boundary points and internal treatment. The method is then applied to a high-order WENO scheme to simulate the complex fluid–solid interactions. The developed solver is first validated against the theoretical solutions of supersonic flow past triangular prism and circular cylinder. Simulated results for test cases with moving shocks are further compared with the previous experimental results of literature in terms of triple-point trajectory and vortex evolution. Excellent agreement is obtained showing the accuracy and the capability of the proposed method for solving complex strong-shock/obstacle interactions for both stationary and moving shock waves.     

Towards oscillation-free implementation of the immersed boundary method with spectral-like methods

It is known that, when the immersed boundary method (IBM) is implemented within spectral-like methods, the Gibbs oscillation seriously deteriorates the calculation of derivatives near the body surface. In this paper, a radial basis function (RBF) based smoothing technique is proposed with the intention of eliminating or efficiently reducing the Gibbs oscillation without affecting the flow field outside the body. Based on this technique, a combined IBM/spectral scheme is developed to solve the incompressible Navier–Stokes equations. Numerical simulations of flow through a periodic lattice of cylinders of various cross sections are performed. The results demonstrate that the proposed methodology is able to give accurate and nearly oscillation-free numerical solutions of incompressible viscous flows.     

A note on pressure accuracy in immersed boundary method for Stokes flow

In this short note, we provide a simplified one-dimensional analysis and two-dimensional numerical experiments to predict that the overall accuracy for the pressure or indicator function in immersed boundary calculations is first-order accurate in L₁ norm, half-order accurate in L₂ norm, but has O(1) error in L_∞ norm. Despite the pressure has O(1) error near the interface, the velocity field still has the first-order convergence in immersed boundary calculations.     

On computational issues of immersed finite element methods

The objective of this paper is to provide a review of recent finite element formulations for immersed methods. In these finite element formulations, independent Lagrangian solid meshes are introduced to move on top of a background Eulerian fluid mesh. This key feature allows the handling, without excessive fluid mesh adaptation, multiple deformable solids immersed in viscous fluid. In particular, pros and cons of both explicit and implicit approaches are illustrated along with subtle differences between incompressible and slightly compressible models.     

Numerical simulations of two-dimensional foam by the immersed boundary method

In this paper, we present an immersed boundary (IB) method to simulate a dry foam, i.e., a foam in which most of the volume is attributed to its gas phase. Dry foam dynamics involves the interaction between a gas and a collection of thin liquid-film internal boundaries that partition the gas into discrete cells or bubbles. The liquid-film boundaries are flexible, contract under the influence of surface tension, and are permeable to the gas, which moves across them by diffusion at a rate proportional to the local pressure difference across the boundary. Such problems are conventionally studied by assuming that the pressure is uniform within each bubble. Here, we introduce instead an IB method that takes into account the non-equilibrium fluid mechanics of the gas. To model gas diffusion across the internal liquid-film boundaries, we allow normal slip between the boundary and the gas at a velocity proportional to the (normal) force generated by the boundary surface tension. We implement this method in the two-dimensional case, and test it by verifying the von Neumann relation, which governs the coarsening of a two-dimensional dry foam. The method is further validated by a convergence study, which confirms its first-order accuracy.     

Prediction of wall-pressure fluctuation in turbulent flows with an immersed boundary method

The objective of this paper is to assess the accuracy and efficiency of the immersed boundary (IB) method to predict the wall pressure fluctuations in turbulent flows, where the flow dynamics in the near-wall region is fundamental to correctly predict the overall flow. The present approach achieves sufficient accuracy at the immersed boundary and overcomes deficiencies in previous IB methods by introducing additional constraints – a compatibility for the interpolated velocity boundary condition related to mass conservation and the formal decoupling of the pressure on this surfaces. The immersed boundary-approximated domain method (IB-ADM) developed in the present study satisfies these conditions with an inexpensive computational overhead. The IB-ADM correctly predicts the near-wall velocity, pressure and scalar fields in several example problems, including flows around a very thin solid object for which incorrect results were obtained with previous IB methods. In order to have sufficient near-wall mesh resolution for LES and DNS computations, the present approach uses local mesh refinement. The present method has been also successfully applied to computation of the wall-pressure space–time correlation in DNS of turbulent channel flow on grids not aligned with the boundaries. When applied to a turbulent flow around an airfoil, the computed flow statistics – the mean/RMS flow field and power spectra of the wall pressure – are in good agreement with experiment.     

Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method

In this paper, we develop an immersed boundary (IB) method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid. In order to take into account the inextensibility constraint of the vesicle, the penalty immersed boundary (pIB) method is used to virtually decouple the fluid and vesicle dynamics. As numerical tests of our current pIB method, the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored, and compared with the previous literature. The method is also validated by a series of convergence study, which confirms its consistent first-order accuracy on the velocity field, the vesicle configuration, the vesicle area and the perimeter errors. In addition, the method is also applied to study a binary-component vesicle problem.     

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations

The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.     

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin’s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system’s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix–vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.     

基于浸入边界-多松弛时间格子玻尔兹曼通量求解法的流固耦合算法研究

针对流固耦合问题,发展了基于浸入边界-多松弛时间格子玻尔兹曼通量求解法(immersed boundary method multi-relaxation-time lattice Boltzmann flux solver,IB-MRT-LBFS)的弱耦合算法.依据多尺度Chapman-Enskog展开,建立不可压宏观方程状态变量和通量与格子玻尔兹曼方程中粒子密度分布函数之间的关系;采用强制浸入边界法处理流固界面使固壁表面满足无滑移边界条件,根据修正的速度求解动量方程力源项;结构运动方程采用四阶龙格-库塔法求解.格子模型与浸入边界法的引入使流固耦合计算可以在笛卡尔网格下进行,无需生成贴体网格及运用动网格技术,简化了计算过程.数值模拟了单圆柱横向涡激振动、单圆柱及串列双圆柱双自由度涡激振动问题.结果表明,IB-MRT-LBFS能够准确预测圆柱涡激振动的锁定区间、振动响应、受力情况以及捕捉尾流场结构形态,验证了该算法在求解流固耦合问题的有效性和可行性.     

DNS of buoyancy-dominated turbulent flows on a bluff body using the immersed boundary method

A novel immersed boundary (IB) method has been developed for simulating multi-material heat transfer problem – a cylinder in a channel heated from below with mixed convection. The method is based on a second-order velocity/scalar reconstruction near the IB. A novel algorithm has been developed for the IB method to handle conjugate heat transfer. The fluid–solid interface is constructed as a collection of disjoint faces of control volumes associated to different material zones. Coupling conditions for the material zones have been developed such that continuity and conservation of the scalar flux are satisfied by a second-order interpolation. Predictions of the local Nusselt number on the cylinder surface show good agreement with the experimental data. The effect of the Boussinesq approximation on this problem was also investigated. Comparison with the variable density formulation suggests that, in spite of a small thermal expansion coefficient of water, the variable density formulation in a transitional flow with mixed convection is preferable.     

The slip boundary condition in the dynamics of solid particles immersed in Stokesian flows

This paper reviews the classical theories of the dynamics of solid particles in a viscous flow via traditional Navier–Stokes/Stokes equations with slip or nonslip boundary conditions, and also studies a corresponding dynamical equation. The emphasis of this paper is on the explicit expressions of the slip effect at the solid-fluid boundary in the dynamics of solid particles in Stokesian fluids.     

Large-eddy simulation of flows past a flapping airfoil using immersed boundary method

The numerical simulation of flows past flapping foils at moderate Reynolds numbers presents two challenges to computational fluid dynamics: turbulent flows and moving boundaries. The direct forcing immersed boundary(IB) method has been developed to simulate laminar flows. However,its performance in simulating turbulent flows and transitional flows with moving boundaries has not been fully evaluated. In the present work,we use the IB method to simulate fully developed turbulent channel flows and transitional flows past a stationary/plunging SD7003 airfoil. To suppress the non-physical force oscillations in the plunging case,we use the smoothed discrete delta function for interpolation in the IB method. The results of the present work demonstrate that the IB method can be used to simulate turbulent flows and transitional flows with moving boundaries.     

Influence of the vertical profile of Saharan dust on the visible direct radiative forcing

The vertical profile of Saharan dust in the atmosphere is generally characterized by a large aerosol concentration in the mid troposphere, differently from the climatological distribution of other types of particles, that show a peak at the surface and a rapid decrease with height. Saharan dust is also characterized by particles of relatively large size of irregular shape, and variable values of the single scattering albedo (the ratio between radiation scattering and extinction). The dust's peculiar vertical distribution is expected to produce an effect on the calculation of the direct aerosol radiative forcing at the surface and at the top of the atmosphere. This effect is investigated by comparing estimates of aerosol direct visible radiative forcing at the surface and at the top of the atmosphere for dust vertical profiles measured in the Mediterranean, and for the climatological profile. The radiative forcing is estimated by means of an accurate radiative transfer model, and for the ocean surface. The sensitivity of the results on the solar zenith angle, aerosol optical depth, and aerosol absorption is also investigated. The aerosol radiative forcing at the surface shows a very small dependency on the aerosol vertical profile. At the top of the atmosphere, the radiative forcing is weakly dependent on the vertical profile (up to 10% variation on the daily average forcing) for low absorbing particles; conversely, it shows a strong dependency (the daily radiative forcing may vary up to 100%) for absorbing particles. The top of the atmosphere visible radiative forcing efficiency produced by dust having single scattering albedo <0.7 is higher by 4 W m⁻² when the observed vertical profile instead of the standard profile is used in the calculations (i.e. it produces a lower cooling). For values of the single scattering albedo around 0.67, the sign of the forcing depends on the vertical profile. The influence of the vertical distribution on the radiative forcing is largest at small values of the solar zenith angle, and at short wavelengths.     

Three-dimensional simulation of elastic capsules in shear flow by the penalty immersed boundary method

An improved penalty immersed boundary method (pIBM) has been proposed for simulation of flow-induced deformation of three-dimensional (3D) elastic capsules. The motion of the capsule membrane is described in the Lagrangian coordinates. The membrane deformation takes account of the bending and twisting effects as well as the stretching and shearing effects. The method of subdivision surfaces is adopted to generate the mesh of membrane and the corresponding shape functions, which are required to be C¹ continuous. The membrane motion is then solved by the subdivision-surface based finite element method on the triangular unstructured mesh. On the other hand, the fluid motion is defined on the Eulerian domain, and is advanced by the fractional step method on a staggered Cartesian grid. Coupling of the fluid motion and the membrane motion is realized in the framework of the pIBM. Using the proposed method, deformation of 3D elastic capsules in a linear shear flow is studied in detail, and validations are examined by comparing with previous studies. Both the neo-Hookean membrane and the Skalak membrane are tested. For an initially spherical capsule the tank-treading motion is formed under various dimensionless shear rates and reduced bending moduli. It is found that buckling occurs near the equator of the capsule for small shear rates but near the tips for large shear rates, which is suppressed by including the bending rigidity of the membrane. Effects of the Reynolds number and the membrane density are investigated for an initially spherical capsule. For a non-spherical capsule, with the initial shape of the oblate spheroid or the biconcave circular disk as a model of the red blood cell, the swinging motion is observed due to the shape memory effect. By decreasing the dimensionless shear rate or increasing the reduced bending modulus, the swinging motion is transited into the tumbling motion.     

On the accuracy of covariance matrix: Hessian versus Gauss-Newton methods in atmospheric remote sensing with infrared spectroscopy

Most retrieval schemes use a linear approximation of the radiative transfer function within each iteration as well as for error analysis. Like most standard methods, the improved Hessian method relies on a quadratic form of the cost function and linear approximation in the error analysis. Often, there is no robust criterion in determining step size that can be used to calculate covariance matrix by discrete perturbation of the cost function in the Hessian approach. The Hessian method improved recently, however, overcomes this problem by employing adaptive algorithm which uses small step sizes in steep directions and large step sizes in flat directions of the cost function. The results of retrievals of atmospheric trace gases from simulated limb emission spectra show that Gauss-Newton algorithm and the improved Hessian generally give nearly identical volume mixing ratios and error covariance matrices in the original state vector space. Due to interlevel correlations, however, the agreement in the uncertainities in the original state vector coordinate system is partly lost in a space in which the elements of state vector are independent after orthogonal coordinate transformation. The significant discrepancies between the estimated uncertainities by the two methods are found to be related with elements of state vector that are dominantly controlled by flattest eigenvector directions of the inverse covariance matrix. The improved Hessian method determines the uncertainities in those shallowest directions with better accuracy than Gauss-Newton approach. The performance of the Hessian method is also found to be better in resolving structures related to the shallowest eigenvector directions as revealed by better vertical resolutions in the retrieved profiles of the trace species.     

Analysis of an immersed boundary method for three-dimensional flows in vorticity formulation

This article presents numerical analysis and practical considerations for three-dimensional flow computation using an implicit immersed boundary method. The Euler equations, or half a step of the Navier–Stokes equations when using fractional step algorithms, are investigated in their vorticity formulation. The context of flow computation around an arbitrarily shaped body is especially investigated.In conventional immersed boundary methods using vorticity, singular vortex are dispatched over the body surface. In the present study, one prefers using sources of potential velocity field, dispatched on the body, whose nature is not vorticity. Such a formulation is compatible to the Euler equations. In practice, these sources of potential flow produce a velocity through this surface, aiming in practice at cancelling a flow-through velocity.This article focuses on the use of the source-to-flow-through linear application, its properties being the key points for fast convergence. Its self-adjointness, or lack thereof, conditioning and preconditioning aspects are investigated. It follows that computing a velocity field with no-flow-through conditions in complex geometry, when using the source-to-flow-through linear application, can be achieved for 4/3 of the computational cost of standard Poisson equation in a Cartesian box.The robustness of immersed boundaries is especially interesting when used together with vortex-in-cell methods, well known for their robustness in time and their ability to compute accurately convective effects. A few examples, based on real-world geometries, illustrate the method capabilities.