Penalty Immersed Boundary Method Research Articles

A self-propelled flexible plate with a keel-like structure

Caudal keels, a pair of lateral keel-like structures in tunas along the caudal peduncle, are a remarkable specialization. Although various hypotheses about the function of caudal keels have been proposed, our understanding of their underlying hydrodynamic mechanism is still limited. The penalty immersed boundary method was used to explore the hydrodynamics of a self-propelled flexible plate with a keel-like structure on the leading edge of the plate in an effort to understand the role of the caudal keel in nature. The clamped leading edge of the flexible plate was forced into a prescribed harmonic oscillation in the vertical direction but was free to move in the horizontal direction. For comparison, simulations without a keel were also performed. Vortical structures and pressure distributions were visualized to characterize the hydrodynamic benefits of the keel. The keel generates streamwise vortices that result in negative pressure and enhance the average cruising speed and thrust. The underlying propulsion mechanism was analyzed in detail by examining the phase of the heaving stroke. The average cruising speed and the propulsion efficiency are increased by more than 11.0% and 6.7%, respectively, by the presence of the keel. A parametric study was performed to determine the set of parameters of the keel that maximizes the propulsion efficiency η as a function of the reduced length (l/L) and the reduced height (h/L) of the keel.

The Nonlinear Coupling of Oscillating Bubble and Floating Body with Circular Hole

The fluid-structure interaction of the oscillating bubble and floating body with circular hole is essentially the nonlinear coupling problem among the incomplete movable boundary, free surface and bubble. This problem is particularly complicated in bubble dynamics. Combined with the volume of fluid method, the Eulerian finite element method is employed to deal with the fluid movement. Based on the improved penalty immersed boundary method, the transient axisymmetric numerical model is established in this paper, considering the fluid-structure interaction effect. The results of simulation are consistent with those of the electric discharge bubble experiment and explosion experiment. Subsequently, considering the influence of the hole size, floating body density, explosive location, and buoyancy, this complex fluid-structure interaction problem is analyzed systematically. Through numerical simulation, we get some new conclusions. When the radius of the hole Rh less than the maximum radius of the oscillating bubble, the changes in the whole system are incredibly intense, and the free surface crushing will emerge. The energy of the bubble acts more on the radial direction of the floating body, when the explosive location parameter is small. When the floating body has the same density as the water, the multiple spike skirt is displayed vividly. And the buoyancy of fluid can produce a lifting effect on the floating body.

Heat transfer enhancement in a poiseuille channel flow by using multiple wall-mounted flexible flags

The flapping dynamics of multiple wall-mounted flexible flags vertically clamped in a heated channel and their effects on heat transfer performance were explored numerically. The fluid-structure-thermal interaction between the flapping flexible flags and the surrounding fluid was modeled with the penalty immersed boundary method. The vortex mixing process was examined to analyze their effects on heat transfer enhancement. The effects of varying the gap distance G/d, the reduced distance d/L, the bending rigidity γ, the channel height H/L and the Reynolds number Re on heat transfer enhancement were scrutinized to optimize cooling, where the flag length is L. The thermal enhancement factor was determined to assess both the heat transfer enhancement benefit and the pressure drop penalty. The asymmetric flag configuration G/d = 0.5 results in better cooling performance than the symmetric configuration G/d = 0 because of the resulting thermal mixing of the cold mainstream and the near-wall hot fluid. More flags can be clamped inside the channel by varying the reduced distance d/L. Multiple flags with γ = 0.04 provide the maximum convective heat transfer at Re = 600. Increasing the blockage ratio, i.e. the ratio of the cross-sectional height occupied by the flags to the channel height, enhances the friction factor.

Vortex induced vibration of a circular cylinder with a filament by using penalty immersed boundary method

In this paper, vortex induced vibration of a circular cylinder with an attached filament is numerically studied by a Penalty Immersed boundary method (PIBM). In the current work, we study the effects of a single filament on the flow wake of a fixed cylinder, as well as its dynamic response when it is free to move transversely. Two key parameters, the bending coefficient (Kb) and the length (L) of the filament are considered. We report that for a static cylinder, by increasing the length or the rigidity of the filament, both the mean drag force and the fluctuation amplitude of the lift force on the cylinder can be reduced. When the cylinder is free to move, it is found that the existence of a filament actually enhances the vibration of the cylinder by comparing to that of a bare cylinder, differentiating from the prediction of lift fluctuation on a fixed cylinder. In the meanwhile, it leads to a broader resonance region on the vibration response of the cylinder with respect to the reduced velocity. Moreover, various flow patterns are identified in the wake of cylinder for different filament lengths.

An improved penalty immersed boundary method for multiphase flow simulation

SummaryAn improved version of the penalty immersed boundary method is proposed for simulating multiphase flows with nonuniform density and moving interface. The background fluid with a uniform density is solved on the fixed Eulerian grid, while the excessive mass is modelled by the Lagrangian points. Both the positive and negative density differences are taken into account, and the surface tension is included in the formulation. For validation of the present method, the Rayleigh‐Taylor instability problem and deformation of a droplet in a simple shear flow are computed. The numerical results are compared with the previous studies and the spatial accuracy is between the first and second order. Convergence of the computational results as the penalty coefficient increases is obtained. Furthermore, interaction of the fluid‐fluid interface with the rigid surface is also considered. To improve the numerical stability, the reinitialization operation is introduced to guarantee the quality of the Lagrangian mesh. As a demonstrating example, a droplet bypassing an obstacle in a channel flow is tested.

A penalty immersed boundary method for viscoelastic particulate flows

We extend the penalty immersed boundary (pIB) method to investigate the interaction between circular rigid particles and a surrounding viscoelastic Oldroyd-B fluid. The basic idea of the pIB method is the splitting of an immersed boundary, which here is a rigid body, notionally into two Lagrangian components: one is a massive component carrying all mass of the rigid body, and the other is massless. These two components are connected by a system of stiff springs with zero rest length. The massive component has no direct interaction with the surrounding fluid and behaves as though in a vacuum, following the dynamics of a rigid body, in which the acting forces and torques are generated from the system of stiff springs that connects the two Lagrangian components. The massless component interacts with the surrounding Oldroyd-B fluid: it moves at the local fluid velocity and exerts force locally on the fluid. We verify the pIB method combined with Oldroyd-B fluid model by investigating the effects of the wall and elasticity of the fluid on the lateral position of a circular particle falling under the influence of gravity and by studying convergence of the numerical solutions. We also simulate the interaction between multiple circular particles and the surrounding Oldroyd-B fluid and compare the dynamics of the particles in various flow conditions.

Heat transfer enhancement by asymmetrically clamped flexible flags in a channel flow

Two flexible flags clamped in a heated channel were numerically modeled to investigate the dynamics of the flexible flags and their effects on heat transfer enhancement. The penalty immersed boundary method was adopted to analyze the fluid–structure–thermal interaction between the surrounding fluid and the flexible flags. A system comprising the thermally conductive flags in an asymmetric configuration (FAC) with respect to the channel centerline is described for the first time in the present study. The effect of the resulting vortices on heat transfer enhancement was investigated. The FAC generated a reverse Kármán vortex street that encouraged a greater degree of thermal mixing in the wake compared to the vortical structures generated by the flags in a symmetric configuration (FSC). The ratio of FAC occupying a cross-section to the channel height decreased, resulting in a decrease in the pressure drop compared to FSC. The FAC significantly improved the thermal efficiency compared to the FSC. The effects of the gap distance between FAC (G/L) and the ratio of the channel height to the flag length (H/L) on the thermal enhancement were characterized to identify the parameters that optimized the thermal efficiency. The relationship between the flapping dynamics and the heat transfer properties was examined in detail. The presence of the FAC with the optimal parameters increased convective heat transfer by 207% and the thermal efficiency factor by 135% compared to the baseline (open channel) flow. The thermal efficiency factor obtained in the present study was compared with that obtained in the previous studies.

An immersed boundary method for fluid–structure interaction with compressible multiphase flows

This paper presents a two-dimensional immersed boundary method for fluid–structure interaction with compressible multiphase flows involving large structure deformations. This method involves three important parts: flow solver, structure solver and fluid–structure interaction coupling. In the flow solver, the compressible multiphase Navier–Stokes equations for ideal gases are solved by a finite difference method based on a staggered Cartesian mesh, where a fifth-order accuracy Weighted Essentially Non-Oscillation (WENO) scheme is used to handle spatial discretization of the convective term, a fourth-order central difference scheme is employed to discretize the viscous term, the third-order TVD Runge–Kutta scheme is used to discretize the temporal term, and the level-set method is adopted to capture the multi-material interface. In this work, the structure considered is a geometrically non-linear beam which is solved by using a finite element method based on the absolute nodal coordinate formulation (ANCF). The fluid dynamics and the structure motion are coupled in a partitioned iterative manner with a feedback penalty immersed boundary method where the flow dynamics is defined on a fixed Lagrangian grid and the structure dynamics is described on a global coordinate. We perform several validation cases (including fluid over a cylinder, structure dynamics, flow induced vibration of a flexible plate, deformation of a flexible panel induced by shock waves in a shock tube, an inclined flexible plate in a hypersonic flow, and shock-induced collapse of a cylindrical helium cavity in the air), and compare the results with experimental and other numerical data. The present results agree well with the published data and the current experiment. Finally, we further demonstrate the versatility of the present method by applying it to a flexible plate interacting with multiphase flows.

Heat transfer enhancement by flexible flags clamped vertically in a Poiseuille channel flow

A pair of flexible flags clamped vertically in a heated channel was numerically modeled to investigate the dynamics of the flexible flags and their effects on heat transfer enhancement. The penalty immersed boundary method was adopted to analyze the fluid–structure–thermal interaction between the surrounding fluid and the flexible flags. The flexible flags displayed three distinct movement modes: a flapping mode, a fully deflected mode, and an irregular mode that depended on the relationship between the hydrodynamic force and the restoring force. In the flapping mode, vortices shed from flexible flags merged and increased in magnitude. The merged vortical structures swept out the thermal boundary layer and enhanced thermal mixing between the fluid near the heated wall and the channel core flow. Compared to rigid flags, the flexible flags significantly improved the thermal efficiency. The effects of the bending rigidity, channel height, and Reynolds number on the thermal efficiency were observed, and an optimal parameter set was obtained. The presence of the flexible flags with the optimal parameter set resulted in an increase of up to 185% in the net heat flux and 106% in the thermal efficiency factor, compared to the baseline flow. The correlation between the vorticity and the temperature field was examined in detail using the dynamic mode decomposition (DMD) method.

Self-propelled heaving and pitching flexible fin in a quiescent flow

A self-propelled flexible fin with heaving and pitching motions in a quiescent flow has been simulated by using the penalty immersed boundary method. The flexible fin can move freely in the horizontal direction and the body of the flexible fin moves passively along with the active head motion. The position of the head of the fin was described as a harmonic heaving oscillation in the vertical direction, while the inclination angle of the head was prescribed as a harmonic oscillation with a moving clamped condition for the heaving and pitching fin. The cruising speed and the swimming efficiency of the self-propelled fin were determined as functions of the bending stiffness (γ), the heaving amplitude (Ah), the pitching amplitude (Ap), the flapping frequency (f) and the phase difference (Δϕ) between Ah and Ap. We optimized the cruising speed and the swimming efficiency with respect to γ, f, Ah, Ap and Δϕ. For a certain range of Ap, the swimming efficiency of the heaving and pitching fin is larger than that of a heaving-only fin.

Vortex interaction between two tandem flexible propulsors with a paddling-based locomotion

Schooling behaviours among self-propelled animals can benefit propulsion. Inspired by the schooling behaviours of swimming jellyfish, flexible bodies that self-propel through a paddling-based motion were modelled in a tandem configuration. This present study explored the hydrodynamic patterns generated by the interactions between two flexible bodies and the surrounding fluid in the framework of the penalty immersed boundary method. The hydrodynamic patterns produced in the wake revealed flow-mediated interactions between two tandem propulsors, including vortex–vortex and vortex–body interactions. Two tandem flexible propulsors paddling with identical amplitude and frequency produced stable configurations as a result of the flow-mediated interactions. Both the upstream and downstream propulsors benefited from the tandem configuration in terms of the locomotion velocity and the cost, compared with an isolated propulsion system. The interactions were examined as a function of the initial gap distance and the phase difference in the paddling frequency. The equilibrium gap distance between two propulsors remained constant, regardless of the initial gap distance, although it did depend on the phase difference in the paddling frequency.

A penalty immersed boundary method for a rigid body in fluid

We extend the penalty immersed boundary (pIB) method to the interaction between a rigid body and a surrounding fluid. The pIB method is based on the idea of splitting an immersed boundary, which here is a rigid body, notionally into two Lagrangian components: one is a massive component carrying all mass of the rigid body and the other is massless. These two components are connected by a system of stiff springs with 0 rest length. The massless component interacts with the surrounding fluid: it moves at the local fluid velocity and exerts force locally on the fluid. The massive component has no direct interaction with the surrounding fluid and behaves as though in a vacuum, following the dynamics of a rigid body, in which the acting forces and torques are generated from the system of stiff springs that connects the two Lagrangian components. We verify the pIB method by computing the drag coefficients of a cylinder and ball descending though a fluid under the influence of gravity and also by studying the interaction of two such descending cylinders and likewise the interaction of two such descending balls. The computational results are quite comparable to those in the literature. As a further example of an application, we include a freely falling maple seed with autorotation.

A further study of inverted hydrodynamic drafting for flow past two flexible filaments in tandem arrangement

Two tandem filaments in a uniform flow are simulated by using a penalty immersed boundary method. First, we investigate two identical filaments, duplicating the 'inverted drafting'. We then fix the structural parameters of the leader while vary that of the follower in kb2 and ρs2. It is found as the follower is stiffer or lighter than the leader, the follower flaps in larger amplitude than the leader. While the drag force acting on the follower is lower than that of the leader, which is in conflict with the so-called 'inverted drafting'. However, if we compare the dynamics of the follower to that of an isolated filament with the same structural properties, the 'inverted drafting' holds again. Furthermore, we investigate the characteristic of energy transfer between filaments and their surrounding fluid. The correlation parameter Clv indicates the follower extracts more energy than the leader except for the cases with ρs2 ≤ 0.2.

Inertial migration of a 3D elastic capsule in a plane Poiseuille flow

The inertial migration of a three-dimensional (3D) elastic capsule in a plane Poiseuille flow has been simulated by using the penalty immersed boundary method. An improved front-tracking method was developed to take into account the difference between the viscosities of the interior and exterior fluids. The calculation domain was confined to the capsule boundary by employing the ray casting algorithm. The dynamic motions and the lateral equilibrium positions of the elastic capsules in moderate Reynolds number (Re) flows were examined as functions of the initial lateral position (y0), Re, the aspect ratio (ε), the viscosity ratio (λ), the membrane stretching coefficient (ϕ), the bending coefficient (γ) and the initial roll angle (θx). Three dynamical motions were observed: tank-treading, swinging and tumbling. The initial behavior of the capsule is influenced by the initial lateral position (y0), but the equilibrium position of the capsule is not affected by variation in y0. The equilibrium position is determined by the balance between the wall effect and the shear gradient effect.

Cross-type optical separation of elastic oblate capsules in a uniform flow

The dynamic behavior of an elastic capsule with an initially oblate spheroidal shape during cross-type optical separation was numerically investigated. The penalty immersed boundary method was adopted for the fluid-membrane interaction, and the optical force calculation was conducted by using the ray optics method including the ray-surface intersection algorithm. The oblate elastic capsule of b/a = 0.5 with different surface Young's moduli and different initial inclination angles was considered. The oblate capsule with higher surface Young's moduli was less deformed, and was more migrated for each initial inclination angle. Unlike the oblate rigid particle, the initially inclined capsules with moderate inclination angles were similarly migrated since the oblate elastic capsule was deformed during rotation near the laser beam axis. The oblate capsules can be separated according to the surface Young's modulus, except for nearly non-inclined capsules. As the fluid velocity decreased, the migration distance increased. The maximum deformation parameter was insensitive to the fluid velocity. Furthermore, a new dimensionless number (Sec) was introduced to predict the migration distance of the oblate elastic capsule.

Migration of Elastic Capsules by an Optical Force in a Uniform flow

The behavior of an elastic capsule by an optical force in a uniform flow is examined by using the penalty immersed boundary method. The elastic capsule is subjected to the laser beam with Gaussian distribution in the perpendicular direction to the fluid flow. The elastic capsule migrated by the optical force along the direction of the laser beam propagation, and the migration distance is dependent on its properties. The oblate capsule with b/a = 0.5 obeying the neo-Hookean constitutive law is first considered, and the effects of the surface Young's modulus and the initial inclination angle on the migration distance are studied. The migration distance of the oblate capsule is increased as the surface Young's modulus increases, and the non-inclined oblate capsule is more migrated than the differently inclined capsules. Then the spherical, oblate, and biconcave capsules obeying the Skalak constitutive law are considered. A comparison of the trajectories of the capsules indicates that the migration of the spherical capsule is the largest. Unlike the oblate capsule, the non-inclined biconcave capsule is less migrated than other inclination angles due to its initial shape.

An Unconditionally Energy Stable Penalty Immersed Boundary Method for Simulating the Dynamics of an Inextensible Interface Interacting with a Solid Particle

In this paper, a novel penalty method based on the immersed boundary formulation is proposed for simulating the transient Stokes flow with an inextensible interface enclosing a suspended solid particle. The main idea of this approach relies on the penalty techniques by modifying the constitutive equation of Stokes flow to weaken the incompressibility condition, relating the surface divergence to the elastic tension $$\sigma $$? to relax the interface's inextensibility, and connecting the particle surface-velocity with the particle surface force $${\varvec{F}}$$F to regularize the particle's rigid motion. The advantage of these regularized governing equations is that when they are discretized by the standard centered difference scheme on a staggered grid, the resulting linear system can easily be reduced by eliminating the unknowns $$p_h, \sigma _h$$ph,?h and $${\varvec{F}}_h$$Fh directly, so that we just need to solve a smaller linear system of the velocity approximation $${\varvec{u}}_h$$uh. This advantage is preserved and even enhanced when such approach is applied to the transient Stokes flow with multiple compound vesicles. Moreover, this smaller linear system is symmetric and negative-definite, which enables us to use efficient linear solvers. Another important feature of the proposed method is that the discretization scheme is unconditionally stable in the sense that an appropriately defined energy functional associated with the discrete system is decreasing and hence bounded in time. We numerically test the accuracy and stability of the immersed boundary discretization scheme. The tank-treading and tumbling motions of inextensible interface with a suspended solid particle in the simple shear flow will be studied extensively. The simulation of the motion of multiple compound vesicles will be performed as well. Numerical results illustrate the superior performance of the proposed penalty method.

Optical separation of ellipsoidal particles in a uniform flow

The behavior of an ellipsoidal particle subjected to a vertical optical force by a loosely focused laser beam in a uniform flow was studied numerically. The fluid flow and the particle motion were separately solved and coupled using the penalty immersed boundary method, and the optical force was calculated using the dynamic ray tracing method. The optical force and optically induced torque on the ellipsoidal particle varied according to the aspect ratio and initial inclination angle. The ellipsoidal particle, whose major axis was initially aligned with the laser beam axis, was more migrated as the aspect ratio increased. The migration distance also depended on the initial inclination angle, even for a given ellipsoidal particle shape. As the laser beam power increased and the flow velocity decreased, the effect of the initial inclination angle increased. The ellipsoidal particles with different aspect ratios could be effectively separated if the rotation along the spanwise direction was suppressed. Moreover, the migration distance could be predicted analytically by introducing a new dimensionless number Sc to represent the ratio of the optical force to the viscous force for the ellipsoidal particles.

Simulation of swimming oblate jellyfish with a paddling-based locomotion

AbstractThe hydrodynamics of a swimming jellyfish depends on the morphology of the species. For example, oblate jellyfish appear to generate wide vortex structures near the bell margin. The vortex structures affect both the propulsion system and the feeding structure because the swimming and prey capturing activities are interrelated processes in these taxa. A three-dimensional computational model was established for an oblate jellyfish to analyse how the vortex structures present in the wake affect the swimming mechanism and the propulsion efficiency, which is defined as the ratio of power output (thrust multiplied by centre velocity) to power input (energy rate required for bell contraction). An improved penalty immersed boundary method was adopted to consider the interactions between the swimming jellyfish and the ambient fluid. The vortex structures in the wake of the swimming jellyfish were investigated in detail. The vortices generated by the contraction and expansion of the jellyfish bell interact with the vortex structures generated by the forward-moving behaviour of the jellyfish. The resulting vortex structures not only transfer momentum to the swimming jellyfish via the fluid, thereby providing the main source of thrust, but also have an implication for feeding. The effects of the elastic properties of the jellyfish on the propulsion were examined. The propulsion efficiency reaches its optimum value at particular elastic properties. We also investigated the effect of the swimming pattern of jellyfish on the propulsion efficiency. The efficiency increases with the flapping frequency and force duration.

Lateral migration of an elastic capsule by optical force in a uniform flow

The lateral migration of an elastic capsule under an optical force in a uniform flow was studied to show the separation characteristics according to the elastic properties in the cross-type optical separator. The initially spherical capsule was moved through the fluid flow using a laser beam with a Gaussian distribution focused along the direction normal to the flow device surface. To simulate such a system, a penalty immersed boundary method was adopted to enable fluid-membrane coupling, and a dynamic ray tracing method was applied to the optical force calculation. The effects of the elastic properties of the capsule membrane (the surface Young's modulus and the bending modulus) on the lateral migration were studied. By increasing the surface Young's modulus, the capsule deformed less and the migration distance increased; however, buckling occurred in the capsule with a high surface Young's modulus. Buckling could be suppressed by increasing the bending rigidity. The effects of the flow velocity and the laser beam power were also examined. In the simulation, the S number, i.e., the ratio of the optical force to the viscous force, was adjusted by decreasing the flow velocity or increasing the laser beam power. The migration distance increased as the S number increased, and a constant lateral migration distance was obtained for a rigid particle for a given S number. An elastic capsule under conditions intermediate between a fixed flow velocity and a fixed laser beam power, however, did not yield a constant lateral migration distance due to the extent of the deformation in the different situations. To predict the lateral migration distance of an elastic capsule, a nondimensional parameter, S_{e}, was defined to include the effects of the optical force, the elastic force, and the fluid viscous force. A unified tendency of the lateral migration distance with S_{e} was obtained for capsules with intermediate elasticity, by varying either the flow velocity or the laser beam power.