Immersed Boundary Simulations Research Articles

Code verification of immersed boundary techniques using the method of manufactured solutions

Code verification plays a crucial role for all finite element applications, especially for non-standard ones, such as immersed boundary approaches, which are typically based on novel algorithms and often error-prone in-house implementations. Instead of relying on rarely available analytical solutions or overkill FEM simulations, in this article, the capabilities of the method of manufactured solutions (MoMS) are explored, enabling an easy and straightforward derivation of closed-form reference solutions. The focus is kept on immersed problems, in particular, on the finite cell method (FCM), and manufactured solutions are derived for 2D and 3D problems involving voids and single/multiple inclusions. We propose several approaches for the construction of the manufactured solutions, where zero traction conditions for void regions and continuous normal stresses along material interfaces are directly fulfilled. Thus, no weak boundary conditions are required for reproducing the manufactured solution via FCM. This not only enables code verification for FCM implementations that lack the option of applying weak boundary conditions, but also keeps the simulation complexity low, when testing other relevant features, e.g., different integration schemes or the implementation of enrichment functions. The flexibility and wide application range of the MoMS in the context of immersed boundary simulations is demonstrated using static, quasi-static, and transient problems in the context of linear elasticity. Finally, the analytical derivations of the manufactured solutions used in this paper are provided as supplementary material.

Code verification of non‐linear immersed boundary simulations using the method of manufactured solutions

AbstractNon‐standard finite element technologies, such as immersed boundary approaches, are typically based on novel algorithms and advanced methods, which require reliable testing of the implemented code. For this purpose, the method of manufactured solutions (MoMS) offers a great framework, enabling an easy and straightforward derivation of closed‐form reference solutions. In this contribution, the focus is kept on non‐linear analysis via the finite cell method (FCM), which is typically based on an unfitted geometry discretization and higher‐order shape functions. The code verification via MoMS generally requires the application of boundary conditions to all boundaries of the simulation domain, which need to be enforced in a weak sense on the immersed boundaries. To avoid this, we propose a novel way of deriving manufactured solutions, for which the necessary constraints on the embedded boundaries are directly fulfilled. Thus, weak boundary conditions can be eliminated from the FCM simulation, and the simulation complexity is reduced when testing other relevant features of the immersed code. In particular, we focus on finite strain analysis of 3D structures with a Neo‐Hookean material model, and show that the proposed technique enables a reliable code verification approach for all load steps throughout the deformation process.

Time-domain immersed-boundary simulation of acoustic propagation between two spherical gas bubblesa).

Acoustic fields and resonance responses from two spherical gas bubbles are investigated using a time-domain simulation. Interior acoustic fields are obtained simultaneously from the simulation of the entire acoustic field propagation with an immersed-boundary method. The linear resonance responses are obtained and discussed for each of the bubbles with respect to the respective interior gas velocities. Also, these are analyzed in the frequency domain in terms of the coupled interactions. Unlike previous numerical and analytical solutions, the method allows for two bubbles of different sizes and shapes to be in contact with each other, which is representative of applicable underwater scattering targets.

Rheology of non-Brownian particle suspensions in viscoelastic solutions. Part 1: Effect of the polymer concentration

We study the effect of varying polymer concentration, measured by the dimensionless polymer viscosity partition function β, on the steady shear rheology of rigid particle suspensions using direct numerical simulation of the Oldroyd-B model. We compare the bulk rheology using immersed boundary simulations at Φ=2.5% and 5% to body-fitted single-particle simulations and find that the per-particle viscosity and first normal stress difference coefficient are always shear-thickening at all values of β considered. However, as β decreases, the polymer stress transforms the flow field near each particle from closed concentric streamlines to helical streamlines that advect stretched polymers away from the particle surface. At low β, the polymer stress is diffuse, where the distribution of the particle induced fluid stress (PIFS) caused by the stretched polymers is spread out in the simulation domain rather than concentrated near the particle surface. Therefore in multiparticle simulations, the polymer stress can be significantly affected by particle-particle interactions. The stress generated by a given particle is disrupted by the presence of particles in its vicinity, leading to a significantly lower PIFS than that of the single-particle simulation. In addition, at increased volume fractions and low values of β, the polymer stress distribution on the particle surface shifts so as to increase the magnitude of the polymer stress moments, resulting in a shear-thickening stresslet contribution to the viscosity that is not seen in single particle or high β simulations. This result indicates that for suspensions in highly viscoelastic suspending fluids that are characterized by a low β parameter, hydrodynamic interactions are significant even at modest particle concentrations and fully resolved multiparticle simulations are necessary to understand the rheological behavior.

Rheology of non-Brownian particle suspensions in viscoelastic solutions. Part II: Effect of a shear thinning suspending fluid

The shear rheology of particle suspensions in shear-thinning polymeric fluids is studied experimentally using parallel plate measurements and numerically using fully resolved, 3D finite volume simulations with the Giesekus fluid model. We show in our experiments that the steady shear viscosity and first normal stress difference coefficient of the suspension evolve from shear-thickening to substantially shear-thinning as the degree of shear-thinning of the suspending fluid increases. Moreover, in highly shear-thinning fluids, the suspension exhibits greater shear-thinning of the viscosity than the suspending fluid itself. Our dilute body-fitted simulations show that in the absence of hydrodynamic interactions, shear-thinning can arise from the particle-induced fluid stress (PIFS), which ceases to grow with increasing shear rate at low values of β (solvent viscosity ratio) and finite values of α (the Giesekus drag coefficient). In a Giesekus suspending fluid, the polymers surrounding the suspended particle are unable to stretch sufficiently at high Weissenberg numbers (Wi) and the reduced polymer stress results in a lower PIFS. When coupled with the shear-thinning stresslet, this effect creates an overall shear-thinning of the viscosity. We then explore the effects of particle-particle interactions on the suspension rheology using immersed boundary simulations. We show that multiparticle simulations are necessary to obtain the shear-thinning behavior of the per-particle viscosity of suspensions in shear-thinning fluids at moderate values of β. Particle-particle interactions lead to a substantial decrease in the PIFS and an enhancement of the shear-thinning of the stresslet compared to the single particle simulations. This combination leads to the shear-thinning of the per-particle viscosity seen in experiments. We also find that very low values of β and finite values of α have opposing effects on the per-particle viscosity that can lead to a nonmonotonic per-particle viscosity versus shear rate in a highly shear-thinning fluid. Overall, the addition of rigid particles to highly shear-thinning fluids, such as joint synovial fluid, leads to increased viscosity and also increased shear-thinning at high shear rates.

Twirling, whirling, and overwhirling revisited: Viscous dynamics of rotating filaments and ribbons

Filaments rotating in fluids arise throughout nature and human engineering, from the thin helices that drive bacterial swimming to methods for laying deep sea cables. Previous work examined the instability that occurs when a filament of circular cross-section rotates in a viscous fluid. Here we extend that analysis to probe the dynamics beyond onset of the instability and how the behavior is affected by cross-sectional shape. Our analysis reveals a rich phase space and elucidates the past discrepancy between immersed boundary simulations and other analytic and computational approaches.

Hydrogen bubble growth in alkaline water electrolysis: An immersed boundary simulation study

Enhancing the efficiency of industrial water electrolysis for hydrogen production is important for the energy transition. In electrolysis, hydrogen is produced at the cathode, which forms bubbles due to the diffusion of dissolved hydrogen in the surrounding supersaturated electrolyte. Hydrogen (and oxygen) bubbles play an important role in the achievable electrolysis efficiency. The growth of the bubbles is determined by diffusive and convective mass transfer. In turn, the presence and the growth of the hydrogen bubbles affect the electrolysis process at the cathode.In the present study, we simulate the growth of a single hydrogen bubble attached to a vertical cathode in a 30 wt% KOH solution in a cathodic compartment represented by a narrow channel. We solve the Navier–Stokes equations, mass transport equations and potential equation for a tertiary current distribution. A sharp interface immersed boundary method with an artificial compressibility method for the pressure is employed. To verify the numerical accuracy of the method, we performed a grid refinement study and checked the global momentum and hydrogen mass balances. We investigate the effects of flow rate and operation pressure upon bubble growth behaviour, species concentrations, potential and current density. We compare different cases in two ways: for the same time and for the same bubble radius. We observe that increasing the flow velocity leads to a small increase in efficiency. Increasing the operation pressure causes higher hydrogen density which slows down the bubble growth. For a given bubble radius, increasing the pressure leads to a small decrease in efficiency.

Immersed boundary simulations of cell-cell interactions in whole blood

We present a new method for the geometric reconstruction of elastic surfaces simulated by the immersed boundary method with the goal of simulating the motion and interactions of cells in whole blood. Our method uses parameter-free radial basis functions for high-order meshless parametric reconstruction of point clouds and the elastic force computations required by the immersed boundary method. This numerical framework allows us to consider the effect of endothelial geometry and red blood cell motion on the motion of platelets. We find red blood cells to be crucial for understanding the motion of platelets, to the point that the geometry of the vessel wall has a negligible effect in the presence of RBCs. We describe certain interactions that force the platelets to remain near the endothelium for extended periods, including a novel platelet motion that can be seen only in 3-dimensional simulations that we term “unicycling.” We also observe red blood cell-mediated interactions between platelets and the endothelium for which the platelet has reduced speed. We suggest that these behaviors serve as mechanisms that allow platelets to better maintain vascular integrity.

The role of breakup and coalescence in fine-scale bubble-induced turbulence. I. Dynamics

We study the effect of bubble breakup and coalescence on fine-scale dynamics of bubbly turbulent flows using direct numerical simulations. We perform two different simulations of dilute bubbly flows of void fraction 0.5%: one with bubbles breaking up and coalescing and the other without these physical processes. The volume of the fluid method is used for simulating bubbles undergoing breakup and coalescence while the bubbles are treated as rigid spheres in the immersed boundary method simulation. The energy spectrum in both types of simulation, consistent with previous studies, exhibits a −3 slope. We follow a single infinitesimal fluid element as it evolves to understand velocity gradient dynamics using conditional mean trajectories. We note finite-time divergence when the fluid element evolves under the action of inertial and pressure dynamics. The inertial, pressure, and viscous velocity gradient dynamics, when considered individually, produce the same results for bubble-induced turbulence (BIT) as with the classical homogeneous isotropic turbulence (HIT). Yet when the overall velocity gradient dynamics is considered, BIT results in non-cyclic trajectories moving toward stable node and unstable saddle while classical HIT shows cyclic behavior in their trajectories that move toward the origin. Interestingly, both the volume of fluid and immersed boundary simulations produce similar results. Therefore, there are two main takeaways from this research. First, new velocity gradient models are needed for BIT as their velocity gradients behavior is entirely different from the HIT. Second, we can neglect the bubble topology, breakup, and coalescence while studying or modeling the fine-scale dynamics of BIT.

The Presence of a Substrate Strengthens The Jet Generated by Upside-Down Jellyfish

Upside-down jellyfish, Cassiopea, are prevalent in warm and shallow parts of the oceans throughout the world. They are unique among jellyfish in that they rest upside down against the substrate and extend their oral arms upwards. This configuration allows them to continually pull water along the substrate, through their oral arms, and up into the water column for feeding, nutrient and gas exchange, and waste removal. Although the hydrodynamics of the pulsation of jellyfish bells has been studied in many contexts, it is not clear how the presence or absence of the substrate alters the bulk flow patterns generated by Cassiopea medusae. In this paper, we use three-dimensional (3D) particle tracking velocimetry and 3D immersed boundary simulations to characterize the flow generated by upside-down jellyfish. In both cases, the oral arms are removed, which allows us to isolate the effect of the substrate. The experimental results are used to validate numerical simulations, and the numerical simulations show that the presence of the substrate enhances the generation of vortices, which in turn augments the upward velocities of the resulting jets. Furthermore, the presence of the substrate creates a flow pattern where the water volume within the bell is ejected with each pulse cycle. These results suggest that the positioning of the upside-down jellyfish such that its bell is pressed against the ocean floor is beneficial for augmenting vertical flow and increasing the volume of water sampled during each pulse.

Immersed boundary simulations of flows driven by moving thin membranes

Immersed boundary methods are extensively used for simulations of dynamic solid objects interacting with fluids due to their computational efficiency and modeling flexibility compared to body-fitted grid methods. However, thin geometries, such as shells and membranes, cause a violation of the boundary conditions across the surface for many immersed boundary projection algorithms. Using a one-dimensional analytical derivation and multi-dimensional numerical simulations, this manuscript shows that adjustment of the Poisson matrix itself is require to avoid large velocity, pressure, and force prediction errors when the pressure jump across the interface is substantial and that these errors increase with Reynolds number. A new minimal thickness modification is developed for the Boundary Data Immersion Method (BDIM-σ), which avoids these issues while still enabling the use of efficient projection algorithms for high-speed immersed surface simulations.

A Counter-Extrapolation Approach for the Boundary Velocity Calculation in Immersed Boundary Simulations

The immersed boundary method (IBM) is often adopted to simulate various flow systems. The two key steps in IBM, namely the force distribution and velocity interpolation steps, are both performed in the immersed boundary layer region. As a result, the flow velocity in this area becomes less reliable, and the boundary velocity calculated from the local flow is less accurate. To enhance the boundary velocity accuracy, here we propose the counter-extrapolation method (CEM), which estimates the boundary velocity using the flow velocity outside of the immersed boundary layer, and thus effectively improved the accuracy in the calculated velocity. Several benchmark tests, which involve flat and curved boundaries and steady and unsteady flows, are conducted; and CEM is compared to other IBM variants recently proposed in the literature. Our results show remarkable accuracy improvements in flow velocity and flow-structure interaction from CEM compared to other methods.

Similar but Distinct Roles of Membrane and Interior Fluid Viscosities in Capsule Dynamics in Shear Flows.

The dynamics of biological capsules and red blood cells in shear flows has been studied extensively with experimental, analytical, and numerical methods. In particular, the effects of various parameters, including the shear rate or shear stress, membrane elasticity, capsule shape, and interior fluid viscosity, have been investigated carefully. The role of the membrane viscosity for capsule deformation dynamics has not been examined adequately. In previous studies, the so-called energy dissipation ratio has been used to account for the membrane viscosity effect by increasing the interior viscosity; however, the applicability and accuracy of this treatment have not been evaluated carefully. In this study, using the recently developed finite-difference scheme for immersed boundary simulations of viscoelastic membranes, we conduct comprehensive numerical simulations of the deformation processes of an originally spherical capsule in shear flows with various combinations of membrane and interior fluid viscosities. Our results show that the membrane and interior fluid viscosity have similar however different effects on the capsule deformation dynamics. While the capsule deformation decreases with both membrane and interior fluid viscosities, a typical decrease-then-increase variation is observed for the inclination angle as the membrane viscosity increases, instead of the monotonic decrease in the inclination angle with the interior fluid viscosity increase. Also, although both large membrane and interior fluid viscosity values can introduce oscillations in the capsule deformation and inclination, larger aptitudes and slow decay processes are noticed at larger membrane viscosities. The variations of other dynamic parameters of the capsule, including the circumference, average membrane velocity, and rotation frequency, are also analyzed, and an intuitive mechanism is proposed to relate the membrane velocity and rotation frequency to the capsule deformation and inclination angle. The simple mechanism is then applied to explain the spoon-like variation patterns for membrane velocity and rotation frequency observed in our results. Furthermore, we examine the validity of the energy dissipation ratio approach based on the mathematical functional dependence. Our results and analysis show that the dissipation ratio is a system and process dependent variable and it cannot be treated as a constant even for the same capsule. This research is valuable for a better understanding of the complex capsule dynamics in flows and also suggests that the membrane viscosity needs to be considered explicitly for accurate and reliable results in future studies.

Immersed boundary simulations of fluid shear-induced deformation of a cantilever beam

A biofilm is a colony of microorganisms that adheres and grows on a surface typically in contact with a stagnant or flowing fluid environment. The hydrodynamic interaction between the fluid and the film structure plays a key role in the various phases of the biofilm life-cycle — formation, growth, detachment, and reestablishment. The fluid-flow conditions determine the shear stresses exerted upon the film structure causing it to deform up to a critical point, beyond which the biofilm is uprooted from the surface into a planktonic state in the flowing fluid. To develop a fundamental understanding of the effects of the fluid-flow parameters on the mechanics of the biofilm, this work considered an ideal 2D model scenario of ‘a rectangular cantilever beam immersed in a channel filled with viscous, incompressible fluid, where one end of the beam is fixed to the channel wall, bending in response to a shear flow.’ The Immersed boundary (IB) method was employed to simulate numerically the fluid–structure interaction. The effect of physical parameters of the problem — stiffness and height of the cantilever and flow velocity, along with the numerical parameters on the equilibrium structure of the elastic beam was investigated and the simulation results were qualitatively compared with respect to linear beam theory. A careful analysis of the “corner effects” — irregularities in beam shape near the free and fixed ends, has been presented. It was shown that this issue can be effectively eliminated by smoothing out the corners with a “fillet” or rounded shape. Finally, the IB formulation was extended to include porosity in the beam to investigate the effect of the resultant porous flow on the beam deflection.

Immersed boundary simulation of drop stability

The stability of the quasi-two-dimensional droplet flow is of great importance in microfluidic devices. We check the drop’s stability in the square box using the immersed boundary and lattice Boltzmann methods. We implement two-dimensional equations within the immersed boundary approach in the Palabos programming platform. We check the influence of the boundaries on the drop movement. We estimate fluctuations in the quantities while applying different initial conditions of the linear and angular velocities. We found that the level of fluctuations depends on the symmetrical displacement of drop at the initial state. The effect is connected with the hydrodynamic interaction of drop with the walls.

Finite-difference and integral schemes for Maxwell viscous stress calculation in immersed boundary simulations of viscoelastic membranes.

The immersed boundary method (IBM) has been frequently utilized to simulate the motion and deformation of biological cells and capsules in various flow situations. Despite the convenience in dealing with flow-membrane interaction, direct implementation of membrane viscosity in IBM suffers severe numerical instability. It has been shown that adding an artificial elastic element in series to the viscous component in the membrane mechanics can efficiently improve the numerical stability in IBM membrane simulations. Recently Li and Zhang (Int J Numer Methods Biomed Eng 35:e3200, 2019) proposed a finite-difference method for calculating membrane viscous stress. In the present paper, two new schemes are developed based on the convolution integral expression of the Maxwell viscoelastic element. We then conduct several tests for the accuracy, stability, and efficiency performances of these three viscous stress schemes. By studying the behavior of a one-dimensional Maxwell element under sinusoidal deformation, we find that a good accuracy can be achieved by selecting an appropriate relaxation time. The twisting sphere tests confirm that, compared to the numerical errors induced by other components in capsule simulations, such as the finite element method for membrane discretization and IBM for flow-membrane interaction, the errors from the viscous stress calculation are negligible. Moreover, extensive simulations are conducted for the dynamic deformation of a spherical capsule in shear flow, using different numerical schemes and various combinations of the artificial spring constants and calculation frequency for the membrane viscous stress calculation. No difference is observed among the results from the three schemes; and these viscous stress schemes require very little extra computation time compared to other components in IBM simulations. The simulation results converge gradually with the increase in the artificial spring stiffness; however, a threshold value exists for the spring stiffness to maintain the numerical stability. The viscous stress calculation frequency has noapparent influence on the calculation results, but a large frequency number can cause the simulation to collapse. We therefore suggest to calculate the membrane viscous stress at each simulation time step, such that a better numerical stabilitycan be achieved. The three numerical schemes have nearly identical performances in all aspects, and they can all be utilized in future IBM simulations of viscoelastic membranes.

Coarse graining the dynamics of immersed and driven fiber assemblies

An important class of fluid-structure problems involve the dynamics of ordered arrays of immersed, flexible fibers. While specialized numerical methods have been developed to study fluid-fiber systems, they become infeasible when there are many, rather than a few, fibers present, nor do these methods lend themselves to analytical calculation. Here, we introduce a coarse-grained continuum model, based on local-slender body theory, for elastic fibers immersed in a viscous Newtonian fluid. It takes the form of an anisotropic Brinkman equation whose skeletal drag is coupled to elastic forces. This model has two significant benefits: (1) the density effects of the fibers in a suspension become analytically manifest, and (2) it allows for the rapid simulation of dense suspensions of fibers in regimes inaccessible to standard methods. As a first validation, without fitting parameters, we achieve very reasonable agreement with 3D Immersed Boundary simulations of a bed of anchored fibers bent by a shear flow. Secondly, we characterize the effect of density on the relaxation time of fiber beds under oscillatory shear, and find close agreement to results from full numerical simulations. We then study buckling instabilities in beds of fibers, using our model both numerically and analytically to understand the role of fiber density and the structure of buckling transitions. We next apply our model to study the flow-induced bending of inclined fibers in a channel, as has been recently studied as a flow rectifier, examining the nature of the internal flows within the bed, and the emergence of inhomogeneous permeability. Finally, we extend the method to study a simple model of metachronal waves on beds of actuated fibers, as a model for ciliary beds. Our simulations reproduce qualitatively the pumping action of coordinated waves of compression through the bed.

Passive vortical flows enhance mass transport in the interior of a coral colony

Corals exchange nutrients and dissolved gases with the surrounding environment for metabolic purposes. A recent study demonstrated that corals can actively stir quiescent water columns and produce vortical flows that enhance mass transfer rates by up to 400%. Here, three-dimensional immersed-boundary simulations of the flow through a Pocillopora meandrina colony demonstrate that the passive geometric features of the branching colony produce highly vortical internal flows that enhance mass transfer at the interior of the colony, compensating almost exactly for flows speed reductions there of up to 64% so that the advection time scale remains roughly constant throughout the colony.

A finite difference method with subsampling for immersed boundary simulations of the capsule dynamics with viscoelastic membranes.

The membrane or interfacial viscosity is an important property in many multiphase and biofluidic situations, such as the red blood cell dynamics and emulsion stability. The immersed boundary method (IBM), which incorporates the dynamic flow-membrane interaction via force distribution and velocity interpolation, has been extensively employed in simulations of such systems. Unfortunately, direct implementation of membrane viscosity in IBM suffers severe numerical instability, which causes an IBM calculation to break down before generating any useful results. Few attempts have been recently reported; however, several concerns exist in these attempts, such as the inconsistency to the classical definition of membrane viscosity, the inability to model the shear and dilatational viscosities separately, the unjustified mathematical formulations, and the complicated algorithms and computation. To overcome these concerns, in this paper, we propose a finite difference approach for implementing membrane viscosity in immersed boundary simulations. The viscous stress is obtained via finite difference approximations to the differential strain-stress relationship, with the help of a subsampling scheme to reduce the numerical noise in the calculated strain rates. This simple method has also avoided the complicated matrix calculations in previous attempts, and hence, a better computational efficiency is expected. Detailed mathematical description of the method and key steps for its implementation in immersed boundary programs are provided. Validation and illustration calculations are performed, and our results are compared with analytical solutions and previous publications with satisfactory agreement. The influences of membrane mesh resolution and simulation time step are also examined; and the results show no indication that our finite difference method has downgraded the general IBM accuracy. Based on these simulations and analysis, we believe that our method would be a better choice for future IBM simulations of capsule dynamics with viscoelastic membranes.

On the viscosity of adhesive hard sphere dispersions: Critical scaling and the role of rigid contacts

The viscosity of colloidal dispersions is a highly sensitive function of the microstructure of the dispersion and the nature of the interactions among the particles. While the viscosity of solutions of colloidal hard spheres is well understood, the viscosity in a broader array of colloidal solutions remains poorly studied. Here, we use immersed boundary simulations to investigate the viscosity of a prototypical colloidal solution, adhesive hard spheres, which under the right conditions can percolate to form sample spanning networks. We examine how rigidity of the particle–particle contacts in these dispersions can enhance the dispersion viscosity. By rigidly constraining a fraction of the adhered particles, we establish a reaction coordinate, the extent of rigidity, along which we can track the development of a gelled state characterized by a diverging viscosity. The fraction of rigid bonds is analogous to the extent of reaction for cross-linking in chemical gels, which suggests a close connection between chemical gelation with polymers and physical gelation with adhesive hard spheres. A critical gel point is identified when particles connected by rigid bonds begin to percolate the sample. In the vicinity of critical gel point, we observe a diverging viscosity and critical scaling behavior in agreement with previous studies on adhesive hard spheres. We use these results to formulate empirical equations for the viscosity across a broad range of volume fractions as a function of the extent of rigidity. Most importantly, through these simulations, we show that rigid constraints among the bonded particles are essential to producing the large viscosities measured experimentally in dispersions of adhesive hard spheres. Hydrodynamic interactions alone are insufficient for reproducing experimental observations.