Immersed Boundary Research Articles

A high-order sharp-interface immersed boundary solver for high-speed flows

A sharp-interface immersed boundary (IB) solver for the three-dimensional compressible Navier-Stokes equations with shock discontinuities is presented. The IB method is based on high-order schemes and a ghost-cell method which uses Taylor polynomial interpolation and weighted least-squares error minimization to impose the boundary conditions on the immersed boundary. The results show that the method enables robust and accurate calculation of high-speed flow problems involving complex geometries, moving boundaries, and solid and membrane geometries. Third-order accuracy or higher is maintained even for supersonic conditions. The numerical framework is validated through several two-dimensional and three-dimensional benchmark problems at subsonic and supersonic speeds. Additionally, the method is demonstrated for airfoil dynamic stall and wing-store interaction investigations.

Assessment of immersed boundary methods for hypersonic flows with gas-surface interactions

The efficacy of immersed boundary (IB) methods with adaptive mesh refinement (AMR) techniques is assessed in the context of atmospheric entry applications, including effects of chemical nonequilibrium (CNE) and gas-surface interactions (GSI). We scrutinize a conservative cut-cell IB method and two non-conservative ghost-cell IB methods, comparing their results with analytical solutions, data from the literature, and results obtained with a reference solver that operates on body-fitted grids. All solvers employ the same external thermochemistry library, ensuring that all observed differences can be attributed solely to differences in the underlying numerical methodologies. We present results for eight benchmark cases. Four verification cases verify the implementation of chemistry, transport properties, catalytic boundary conditions, and shock capturing. Four validation cases encompass blunt geometries with adiabatic and isothermal, as well as inert, catalytic and ablative boundary conditions. Overall, the results obtained with the IB solvers are in very good agreement with the reference data. Discrepancies arise in cases with large temperature or concentration gradients at the wall, and these are linked to conservation errors inherent to ghost-cell methods. Only a strictly conservative cut-cell IB method is on par with body-fitted grid methods.

BiGlobal stability analysis for flow in complex geometry based on immersed boundary method

A numerical framework of global stability analysis based on the immersed boundary (IB) method is established for predicting the unsteadiness onset of flow system including complex geometry. The solid wall is considered as the boundary force term and the stability analysis is applied on the orthogonal Cartesian grid, which facilitate the application of spectral collocation method in the stability analysis. To convert the linearized N-S equation with source term to homogeneous, the boundary force can be viewed as an unknown quantity, and then the no-slip boundary condition is introduced into the governing equation to close the system. The method is first tested on two canonical cases, the flow past a stationary cylinder and an isolated airfoil. The relevant results are consistent with phenomena observed in the numerical calculations and experiments, verifying the effectiveness of the method. Applying the spectral collocation method contributes to the accuracy of stability analysis and reduces the grid nodes, making it more feasible to consider complex geometry. Then, to highlight the superiority of treating the flow system with complex geometry, such as including multi-objects, the method is applied to the stability analysis of the flow past a four-square-arranged cylinder cascade. The abundant instability perturbation modes induced by the interaction of multi-objects and the corresponding relationship with the whole system are captured, which is also verified by the unsteady calculation.

Gravitational settling of two impermeable semi-torus particles

This research aimed to numerically investigate the dynamics of two semi-torus particles using the Immersed Boundary (IB) method. Various initial configurations were considered: vertically aligned along the centre of the fluid channel (Case A), vertically aligned in an asymmetric location (Case B), and horizontally aligned symmetrically to the channel centreline (Case C). Additionally, the impact of different starting alignments on particle dynamics was explored. When the particles were dropped along the middle of the channel, their dynamics depended on the fluid viscosity. Higher viscosities led to drafting and kissing phases, while lower viscosities resulted in multiple cycles of drafting, kissing, and tumbling (DKT). Similarly, the dynamics of particles dropped from an asymmetric location were also influenced by viscosity. For horizontally aligned particles along the centreline, the obtained results agreed well with previous studies. Notably, the initial orientation of the particles had a significant effect on their behaviour, particularly in less viscous fluid conditions.

Efficient GPU parallelization of adaptive mesh refinement technique for high-order compressible solver with immersed boundary

A new, highly parallelized, adaptive mesh refinement (AMR) library, equipped with an accurate immersed boundary (IB) method for solving the compressible Navier–Stokes system is presented. The library, named ADAM, is designed to efficiently exploit modern exascale GPU-accelerated supercomputers and it is implemented with a highly modular structure in order to make easy to leverage it for a wide range of CFD applications. The structured Cartesian grids at the basis of (octree) AMR approach allows to implement very high order accurate models retaining a low computational cost and high level of parallelization. The accurate IB method coupled with efficient AMR technique enables the simulation flows with complex (possibly moving and deforming) geometries. The library is applied to the simulation of a strong shock diffraction over a solid sphere and a detailed discussion concerning the physical results and the parallel performance obtained is presented.

A new approach to aircraft ditching analysis by coupling free surface lattice Boltzmann and immersed boundary method incorporating surface tension effects

This study primarily addresses the phenomenon of aircraft ditching. This phenomenon presentspresenting a significant challenge for numerical simulations due to its highly nonlinear and multiphase-coupled water entry impact problem involving fluid-structure interaction. Traditional numerical methods have difficulties and deficiencies in free surface evolution and accuracy. An improved iterative velocity correction scheme has been presented by coupling the single-phase free surface-lattice Boltzmann method (FS-LBM) based on the volume of fluid method with the immersed boundary (IB) method to simulate water entry impact issues. Notably, this model incorporates the effect of surface tension, enhancing its ability to accurately simulate various phenomena, such as rolling, breaking, splashing, and fusion with the free surface in gas-liquid-solid multi-physical field conditions. Through the correction of the boundary force of the interface lattice and the gas lattice and the improved scheme, the non-slip boundary is satisfied, and the particle penetration is prevented. Furthermore, several benchmark tests have been conducted to verify the rationality and feasibility of the FS-LBM-IB method in simulating aircraft ditching. A scaled model of an aircraft has been developed to investigate the dynamic response characteristics of aircraft ditching at a specific attitude angle, with the simulation results confirming the accuracy of the FS-LBM-IB method in predicting the secondary head-up of the aircraft under suction force. This research highlights the efficiency of the FS-LBM-IB method as a novel approach for simulating aircraft ditching behavior.

A semi-implicit semi-Lagrangian method for simulating immersed boundary motion under high inertia and elasticity

In this paper, we present an efficient and stable fractional-step 2D immersed boundary (IB) method for solving the interaction problems between bulk fluid and elastic interface, in particular, when the fluid inertia and the interfacial elasticity are the significant factors affecting its dynamics. In myriads of real-world applications, the effects of high inertia and elasticity are dominant. So the complex fluid dynamics under such harsh conditions is an important topic in computational physics and is inherently challenging due to high computational complexity. In turn, it requires to solve the governing equations of elastic interfacial motion in an implicit manner so that more stable simulations can be performed by relaxing the Courant-Friedrichs-Lewy (CFL) condition. The contributions of the proposed approach are three folds. First, an iteration-free semi-Lagrangian method is employed in Navier-Stokes (NS) equations. Second, the elastic force acting along the interface is treated semi-implicitly in IB formulations. Both approaches improve the numerical stability associated with the high fluidic inertia and interfacial elasticity. Finally, to solve the resulting linear system, our novel idea is to transform the original 3-by-3 block matrix system into a reduced 2-by-2 block matrix system using a discrete projection operator in a staggered grid, and then explicitly represent the exact solution via the Schur complement of the Helmholtz operator. Owing to this feature, we refer to this proposed approach as reduced immersed boundary method (rIBM). We show that the two systems are equivalent in theory, whereas the conventional immersed boundary projection method (IBPM) modifies the discrete momentum equation in the original system. A series of numerical tests is conducted to confirm the stability of the rIBM using relatively larger time-step sizes, specifically with Reynolds number and inverse capillary number equal to or larger than approximately 1000. By estimating the computational time, the numerical efficiency of the proposed method is further verified in comparison with the conventional IBPM and the Crank-Nicolson scheme-based IB method. In conclusion, the proposed approach not only improves the numerical stability, but also increases the computational speed, suitable for solving more realistic problems.

A computational method for complex-shaped hydraulic turbomachinery flow based on the immersed boundary method

Traditional numerical simulation techniques, such as sliding mesh, dynamic mesh, and others, have many limitations in dealing with flow simulation with the large-scale movement of solid boundaries, which is the case for simulating the flow of complex-shaped hydraulic turbomachinery such as propellers, pumps, and turbines. The immersed boundary (IB) method provides a new approach to solve the above-mentioned limitations. Therefore, this study proposes a sharp-interface IB method based on the level-set function that is suitable for simulating the flow through turbomachinery with complex geometries. This method is applied to actual three-dimensional numerical simulations of high-Reynolds number propellers using an in-house computational fluid dynamics solver. The results show that the proposed method can provide comparatively accurate predictions of unsteady load coefficients within the propeller flow passage and capture the correct propeller wake characteristics as well as the interaction between the propeller wake and free surface. This study is aimed at providing a theoretical basis and engineering reference for the application of the IB method in engineering numerical simulations.

A flexible forcing immersed boundary scheme-based one-step simplified lattice Boltzmann method for two-dimensional fluid-solid interaction problems

In this study, we coupled the one-step simplified lattice Boltzmann method (OSLBM) [1] with flexible forcing immersed boundary (FFIB) scheme [2] and proposed a computationally efficient FFIB-OSLBM solver for complex two-dimensional fluid-solid interaction problems. The adopted OSLBM algorithm has the benefits of lower virtual memory requirement and easy implementation of physical boundary conditions over the traditional lattice Boltzmann (LBM) solver and is also computationally more efficient than the simplified lattice Boltzmann method (SLBM) [3]. Unlike in SLBM, the macroscopic variable is updated in one-step in OSLBM, thus avoiding the predictor and corrector steps and reducing the computational time. Furthermore, the adopted FFIB algorithm uses an implicit approach to calculate the body force density, which accurately satisfies the no-slip velocity condition at the boundary of the immersed solid. Moreover, this method requires less computational effort since it discards the time-consuming, complex matrix inversion process associated with the conventional immersed boundary (IB) method. We have conducted the numerical accuracy test and noticed that the proposed FFIB-OSLBM solver is nearly second-order accurate in space, which is higher than the typical coupled IB-LBM solvers. In addition, the applicability of the present solver is tested through a set of benchmark problems, and the results obtained from the present solver show an excellent agreement with the data available in the references. Compared to the FFIB-LBM [2], the present solver consumes at least 20% less computational memory and CPU time for all the validation problems investigated in this study.

Comparison of Immersed Boundary Simulations of Heart Valve Hemodynamics Against In Vitro 4D Flow MRI Data.

The immersed boundary (IB) method is a mathematical framework for fluid-structure interaction problems (FSI) that was originally developed to simulate flows around heart valves. Direct comparison of FSI simulations around heart valves against experimental data is challenging, however, due to the difficulty of performing robust and effective simulations, the complications of modeling a specific physical experiment, and the need to acquire experimental data that is directly comparable to simulation data. Such comparators are a necessary precursor for further formal validation studies of FSI simulations involving heart valves. In this work, we performed physical experiments of flow through a pulmonary valve in an in vitro pulse duplicator, and measured the corresponding velocity field using 4D flow MRI (4-dimensional flow magnetic resonance imaging). We constructed a computer model of this pulmonary artery setup, including modeling valve geometry and material properties via a technique called design-based elasticity, and simulated flow through it with the IB method. The simulated flow fields showed excellent qualitative agreement with experiments, excellent agreement on integral metrics, and reasonable relative error in the entire flow domain and on slices of interest. These results illustrate how to construct a computational model of a physical experiment for use as a comparator.

Applying a Cartesian method to moving boundaries

The paper describes the development of a Cartesian method able of simulating compressible and viscous flows around moving/deforming objects as well as fluid–structure interactions (FSI). Besides, the research addresses the issues related to the partitioned coupling of aerodynamic and structural tools and suggests the use of proper drivers based on open-source libraries.The flow solver is based on dynamic immersed boundaries (IB) which take into account the motion of Lagrangian bodies through an inertial Cartesian mesh. This means having cells without volume changes and/or roto-translations. In particular, the major difficulties arise from the correct evaluation of flow quantities at cells emerging from the solid areas. Indeed, no flow-state is associated with cells inside the body. As a consequence, a ‘field-extension’ procedure is applied to estimate the flow state vector where needed. The use of fluid and wall values makes the procedure consistent. A proper IB model accounts for the body motion into a stationary mesh that are dynamically adapted to the flow. Both rigid and deforming motions are allowed by means of proper BCs based on the local wall-velocities.The paper discusses the accuracy and stability constraints of the method when applied to well-known benchmarks available in literature. In particular, the ability to analyze rigid body-motions is assessed by simulating the viscous flows around an oscillating circular cylinder and a moving sphere. Both cases focus on the laminar regime which represents an effective starting-point for validating the dynamic IB-method.Part of the research activities is devoted to developing an interface for coupling the present method with a structural solver in the framework of a partitioned process. A shared platform is set-up for driving the solution sequence and analyzing the accuracy of the results. A literature case is discussed which consists of a flexible cantilever mounted in the laminar wake of a rigid square cylinder.The final aim is to make affordable the study of complex flows whose characteristics strongly depend on the structural dynamics of immersed objects. In particular, the use of an IB method based on a fast and automatic meshing process is expected to simplify and speed-up the simulation of fluid–structure interaction (FSI) problems.

The volume-filtering immersed boundary method

We present a novel framework to deal with static and moving immersed boundaries (IB) based on volume-filtering. In this strategy, called Volume-Filtering Immersed Boundary (VFIB) method, transport equations are derived by filtering the Navier-Stokes equations and accounting for stresses at the solid-fluid interface. The result is that boundary conditions that normally apply on the solid-fluid interface are transformed into bodyforces that apply on the right-hand side of the filtered transport equations. In this method, the filter width acts as a parameter that controls the level of resolution. The IB is considered well-resolved if the filter width is much smaller than the characteristic corrugation scale of the interface. There are several innovations in this IB method. First, it sheds light on the role of the internal flow which arises when the transport equations are solved inside the IB. We show that, it is essential to separate stresses due to the external and internal fluids in order to get accurate forces, and provide a method to do so. Second, we show that the volumes associated with Lagrangian forcing points on the boundary depend on the local topology of the surface. We provide a straightforward way to compute these volumes using a triangle tessellation of the interface and the surface density function. Third, we provide an efficient procedure to compute the solid volume fraction, thus, enabling tagging interior/exterior cells. This volume fraction is also involved in the procedure to separate stresses due to the external fluid from the total stresses. Fourth, we show a path forward to extend the VFIB method to Large Eddy Simulations involving IBs. Lastly, we apply the VFIB in several numerical tests involving two- and three-dimensional static and moving IBs. We show greatly improved results compared to prior IB methods. Further, we test several filter kernels and show that, for well-resolved IBs, the choice of the kernel plays little role.

A sharp interface Lagrangian-Eulerian method for flexible-body fluid-structure interaction

This paper introduces a sharp-interface approach to simulating fluid-structure interaction (FSI) involving flexible bodies described by general nonlinear material models and across a broad range of mass density ratios. This new flexible-body immersed Lagrangian-Eulerian (ILE) scheme extends our prior work on integrating partitioned and immersed approaches to rigid-body FSI. Our numerical approach incorporates the geometrical and domain solution flexibility of the immersed boundary (IB) method with an accuracy comparable to body-fitted approaches that sharply resolve flows and stresses up to the fluid-structure interface. Unlike many IB methods, our ILE formulation uses distinct momentum equations for the fluid and solid subregions with a Dirichlet-Neumann coupling strategy that connects fluid and solid subproblems through simple interface conditions. As in earlier work, we use approximate Lagrange multiplier forces to treat the kinematic interface conditions along the fluid-structure interface. This penalty approach simplifies the linear solvers needed by our formulation by introducing two representations of the fluid-structure interface, one that moves with the fluid and another that moves with the structure, that are connected by stiff springs. This approach also enables the use of multi-rate time stepping, which allows us to use different time step sizes for the fluid and structure subproblems. Our fluid solver relies on an immersed interface method (IIM) for discrete surfaces to impose stress jump conditions along complex interfaces while enabling the use of fast structured-grid solvers for the incompressible Navier-Stokes equations. The dynamics of the volumetric structural mesh are determined using a standard finite element approach to large-deformation nonlinear elasticity via a nearly incompressible solid mechanics formulation. This formulation also readily accommodates compressible structures with a constant total volume, and it can handle fully compressible solid structures for cases in which at least part of the solid boundary does not contact the incompressible fluid. Selected grid convergence studies demonstrate second-order convergence in volume conservation and in the pointwise discrepancies between corresponding positions of the two interface representations as well as between first and second-order convergence in the structural displacements. The time stepping scheme is also demonstrated to yield second-order convergence. To assess and validate the robustness and accuracy of the new algorithm, comparisons are made with computational and experimental FSI benchmarks. Test cases include both smooth and sharp geometries in various flow conditions. We also demonstrate the capabilities of this methodology by applying it to model the transport and capture of a geometrically realistic, deformable blood clot in an inferior vena cava filter.

The immersed boundary method: A SIMPLE approach

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.

An Unfitted Method with Elastic Bed Boundary Conditions for the Analysis of Heterogeneous Arterial Sections

This manuscript presents a novel formulation for a linear elastic model of a heterogeneous arterial section undergoing uniform pressure in a quasi-static regime. The novelties are twofold. First, an elastic bed support on the external boundary (elastic bed boundary condition) replaces the classical Dirichlet boundary condition (i.e., blocking displacements at arbitrarily selected nodes) for elastic solids to ensure a solvable problem. In addition, this modeling approach can be used to effectively account for the effect of the surrounding material on the vessel. Secondly, to study many geometrical configurations corresponding to different patients, we devise an unfitted strategy based on the Immersed Boundary (IB) framework. It allows using the same (background) mesh for all possible configurations both to describe the geometrical features of the cross-section (using level sets) and to compute the solution of the mechanical problem. Results on coronary arterial sections from realistic segmented images demonstrate that the proposed unfitted IB-based approach provides results equivalent to the standard finite elements (FE) for the same number of active degrees of freedom with an average difference in the displacement field of less than 0.5%. However, the proposed methodology does not require the use of a different mesh for every configuration. Thus, it is paving the way for dimensionality reduction.

Enhancement of Rotor Loading and Suppression of Stator Separation through Reduction of the Blade–Row Gap

An immersed boundary (IB) method is applied to study the effect of the blade–row gap in a low-speed single-stage compressor. The advantage of using an IB method is that the rotor/stator interface can be eliminated and, thus, the blade–row interaction can be considered at an extremely small gap. The IB method was modified to internal-flow problems, and the adaptive mesh refinement (AMR) technique, together with a wall model, used to facilitate the simulations for high Reynolds-number flows. The results showed that both the pressure rise and the efficiency were observed to be higher in the smaller-gap cases. Comparisons between the results of two gaps, 35%ca and 3.5%ca, are highlighted and further analysis at a specific flow coefficient showed that the increase of the stage performance was contributed to by the enhancement of rotor loading and the suppression to the flow separation of the stator. Correspondingly, the increases of the total pressure rise on the rotor and the stator outlets were observed to be 0.5% and 4.3%, respectively. Although the increase on the rotor outlet is much lower than that on the stator outlet, its significance is that a higher level of static pressure is formed near the hub of the gap, which, thus, reduces the adverse pressure gradient of this region in the stator passage. This improvement suppresses the flow separation near the hub of the stator and, thereby, results in a considerable increase to the pressure rise on the stator outlet as a consequence. The effect of the gap on unsteady pressure fluctuation is also presented.

A 3D Fully Non-Hydrostatic Model for Free-Surface Flows with Complex Immersed Boundaries

A fully non-hydrostatic hydrodynamic model is developed to simulate a three-dimensional, incompressible, and viscous free-surface flow passing downstream rigid rectangular and circular cylinders. A direct numerical simulation (DNS) based on the volume of fluid (VOF) and immersed boundary (IB) method is presented for solving the Navier–Stokes equations. The numerical scheme provides accurate solutions with high efficiency using the novel computational procedure to model severe surface deformations. A staggered finite difference method with a Cartesian mesh coordinate system is used to discretize the governing equations with the complexity of the deformed free-surface flow, for which the numerical schemes include a free-surface tracking technique based on the VOF and a VOS-based IB method to simulate 3D dam-break flows passing the slender objects. Additionally, the case studies demonstrate the accuracy and flexibility of the proposed model to predict the impact forces of the surface flow against the different configurations of structures. The results reveal that the temporal variation of the impact force acted on the rectangular obstacle is dominated by the aspect ratio. The force increases with the increase in the shape parameter. The resistance caused by a thin obstacle is considerably less than the blunt shape.

Lattice Boltzmann simulation of complex thermal flows via a simplified immersed boundary method

A simplified immersed boundary (IB) lattice Boltzmann method (LBM) is developed in this study for simulating incompressible flows with heat transfer. The no-slip and constant temperature boundary conditions are both enforced by correcting the distribution functions. Similar to the previous distribution function correction (DFC)-IB method (S. Tao et al., 2021 [59]), the kinetic nature of LBM has been retained in the present scheme, which directly modifies the distribution functions other than models the interface as a source of IB force in the boundary treatment. However, this study presents two improvements. First, we reduce the number of distribution functions that need to be modified, and hence the computation load of the method can be reduced significantly. Furthermore, the correction term can be determined directly and explicitly, replacing the calculation of a local position-related adjustment parameter in the previous DFC-IB method. As a result, the computed velocity and temperature at the interface are consistent with their desired counterparts, and hence the Dirichlet boundary conditions can be accurately specified. The present simplified DFC IB method is validated in the simulations with both stationary and moving boundaries, including the flow past a hot cylinder, mixed convection in a square lid-driven cavity, and sedimentation of a particle in a long channel. The results demonstrate the accuracy and efficiency of the method.

Aerodynamics of a foil undergoing the coexisting of fast and slow pitching

The aerodynamic characteristics of the foil undergoing the coexisting of fast and slow pitching have been studied via Immersed Boundary (IB) method. With an adjustable parameter [Formula: see text], the foil kinematics is defined by making the period corresponding to the positive amplitude faster than that corresponding to the negative amplitude, and vice versa. An opposite scenario is observed at [Formula: see text] and [Formula: see text], where the positive lift is improved via decreasing [Formula: see text] at [Formula: see text] but the negative lift enlarges with increasing [Formula: see text] at [Formula: see text], which is of benefit to generate up-burst and dive of the pitching foil, respectively. The lift generation is investigated with the time history of the lift coefficient and the time-mean streamwise velocity, which renders a physical basis for understanding the behavior of flying animals.

A non-equilibrium slip wall model for large-eddy simulation with an immersed boundary method

A non-equilibrium wall model for large-eddy simulation with the immersed boundary (IB) method is proposed to reduce the required number of grid points in simulating wall-bounded turbulence. The proposed wall model is presented as an appropriate slip velocity on the wall. The slip velocity is constructed by integrating the simplified turbulent boundary layer (TBL) equation along the wall-normal direction, which enhances the integral momentum balance near the wall on a coarse grid. The effect of pressure gradient on the near wall flow is taken into account by retaining the pressure gradient term in the simplified TBL equation. The proposed model is implemented in the form of a direct-forcing IB method with moving-least-square reconstruction near the wall. The benchmarks of plane channel turbulence and the flows over a backward-facing step are used for validation. The proposed model improves the wall stresses and velocity profiles in the region where the pressure gradient dominates the near wall flows.