Immersed Boundary – Thermal Lattice Boltzmann Methods for Non-Newtonian Flows Over a Heated Cylinder: A Comparative Study

In this study, we compare different diffuse and sharp interface schemes of direct-forcing immersed boundary — thermal lattice Boltzmann method (IB-TLBM) for non-Newtonian flow over a heated circular cylinder. Both effects of the discrete lattice and the body force on the momentum and energy equations are considered, by applying the split-forcing Lattice Boltzmann equations. A new technique based on predetermined parameters of direct forcing IB-TLBM is presented for computing the Nusselt number. The study covers both steady and unsteady regimes (20相似文献   

Semiclassical Lattice Boltzmann Simulations of Rarefied Circular Pipe Flows

Computations of microscopic circular pipe flow in a rarefied quantum gas are presented using a semiclassical axisymmetric lattice Boltzmann method. The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al. [Phys. Rev. E., 64 (2001), 011208] is adopted by adding forcing terms into the resulting microdynamic evolution equation. The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form. The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion. The velocity profiles and the mass flow rates of pipe flows with several Knudsen numbers covering different flow regimes are presented. It is found the Knudsen minimum can be captured in all three statistics studied. The results also indicate distinct characteristics of the effects of quantum statistics.     

Elements of the Lattice Boltzmann Method I: Linear Advection Equation

This paper opens a series of papers aimed at finalizing the development of the lattice Boltzmann method for complex hydrodynamic systems. The lattice Boltzmann method is introduced at the elementary level of the linear advection equation. Details are provided on lifting the target macroscopic equations to a kinetic equation, and, after that, to the fully discrete lattice Boltzmann scheme. The over-relaxation method is put forward as a cornerstone of the second-order temporal discretization, and its enhancement with the use of the entropy estimate is explained in detail. A new asymptotic expansion of the entropy estimate is derived, and implemented in the sample code. It is shown that the lattice Boltzmann method provides a computationally efficient way of numerically solving the advection equation with a controlled amount of numerical dissipation, while retaining positivity.     

Lattice BGK Model for Incompressible Axisymmetric Flows

In this paper, a lattice Boltzmann BGK (LBGK) model is proposed for simulating incompressible axisymmetric flows. Unlike other existing axisymmetric lattice Boltzmann models, the present LBGK model can eliminate the compressible effects only with the small Mach number limit. Furthermore, the source terms of the model are simple and contain no velocity gradients. Through the Chapman-Enskog expansion, the macroscopic equations for incompressible axisymmetric flows can be exactly recovered from the present LBGK model. Numerical simulations of the Hagen-Poiseuille flow, the pulsatile Womersley flow, the flow over a sphere, and the swirling flow in a closed cylindrical cavity are performed. The results agree well with the analytic solutions and the existing numerical or experimental data reported in some previous studies.     

Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGK Model

This paper aims to study the numerical features of a coupling scheme between the immersed boundary (IB) method and the lattice Boltzmann BGK (LBGK) model by four typical test problems: the relaxation of a circular membrane, the shearing flow induced by a moving fiber in the middle of a channel, the shearing flow near a non-slip rigid wall, and the circular Couette flow between two inversely rotating cylinders. The accuracy and robustness of the IB-LBGK coupling scheme, the performances of different discrete Dirac delta functions, the effect of iteration on the coupling scheme, the importance of the external forcing term treatment, the sensitivity of the coupling scheme to flow and boundary parameters, the velocity slip near non-slip rigid wall, and the origination of numerical instabilities are investigated in detail via the four test cases. It is found that the iteration in the coupling cycle can effectively improve stability, the introduction of a second-order forcing term in LBGK model is crucial, the discrete fiber segment length and the orientation of the fiber boundary obviously affect accuracy and stability, and the emergence of both temporal and spatial fluctuations of boundary parameters seems to be the indication of numerical instability. These elaborate results shed light on the nature of the coupling scheme and may benefit those who wish to use or improve the method.     

A Robust Immersed Boundary-Lattice Boltzmann Method for Simulation of Fluid-Structure Interaction Problems

A robust immersed boundary-lattice Boltzmann method (IB-LBM) is proposed to simulate fluid-structure interaction (FSI) problems in this work. Compared with the conventional IB-LBM, the current method employs the fractional step technique to solve the lattice Boltzmann equation (LBE) with a forcing term. Consequently, the non-physical oscillation of body force calculation, which is frequently encountered in the traditional IB-LBM, is suppressed greatly. It is of importance for the simulation of FSI problems. In the meanwhile, the no-slip boundary condition is strictly satisfied by using the velocity correction scheme. Moreover, based on the relationship between the velocity correction and forcing term, the boundary force can be calculated accurately and easily. A few test cases are first performed to validate the current method. Subsequently, a series of FSI problems, including the vortex-induced vibration of a circular cylinder, an elastic filament flapping in the wake of a fixed cylinder and sedimentation of particles, are simulated. Based on the good agreement between the current results and those in the literature, it is demonstrated that the proposed IB-LBM has the capability to handle various FSI problems effectively.     

Development of a High-Resolution Scheme for Solving the PNP-NS Equations in Curved Channels

A high-order finite difference scheme has been developed to approximate the spatial derivative terms present in the unsteady Poisson-Nernst-Planck (PNP) equations and incompressible Navier-Stokes (NS) equations. Near the wall the sharp solution profiles are resolved by using the combined compact difference (CCD) scheme developed in five-point stencil. This CCD scheme has a sixth-order accuracy for the second-order derivative terms while a seventh-order accuracy for the first-order derivative terms. PNP-NS equations have been also transformed to the curvilinear coordinate system to study the effects of channel shapes on the development of electroosmotic flow. In this study, the developed scheme has been analyzed rigorously through the modified equation analysis. In addition, the developed method has been computationally verified through four problems which are amenable to their own exact solutions. The electroosmotic flow details in planar and wavy channels have been explored with the emphasis on the formation of Coulomb force. Significance of different forces resulting from the pressure gradient, diffusion and Coulomb origins on the convective electroosmotic flow motion is also investigated in detail.     

A Localized Mass-Conserving Lattice Boltzmann Approach for Non-Newtonian Fluid Flows

A mass-conserving lattice Boltzmann model based on the Bhatnagar-Gross-Krook (BGK) model is proposed for non-Newtonian fluid flows. The equilibrium distribution function includes the local shear rate related with the viscosity and a variable parameter changing with the shear rate. With the additional parameter, the relaxation time in the collision can be fixed invariable to the viscosity. Through the Chapman-Enskog analysis, the macroscopic equations can be recovered from the present mass-conserving model. Two flow problems are simulated to validate the present model with a local computing scheme for the shear rate, and good agreement with analytical solutions and/or other published results are obtained. The results also indicate that the present modified model is more applicable to practical non-Newtonian fluid flows owing to its better accuracy and more robustness than previous methods.     

Lattice Boltzmann Modeling of Advection-Diffusion-Reaction Equations: Pattern Formation Under Uniform Differential Advection

A lattice Boltzmann model for the study of advection-diffusion-reaction (ADR) problems is proposed. Via multiscale expansion analysis, we derive from the LB model the resulting macroscopic equations. It is shown that a linear equilibrium distribution is sufficient to produce ADR equations within error terms of the order of the Mach number squared. Furthermore, we study spatially varying structures arising from the interaction of advective transport with a cubic autocatalytic reaction-diffusion process under an imposed uniform flow. While advecting all the present species leads to trivial translation of the Turing patterns, differential advection leads to flow induced instability characterized with traveling stripes with a velocity dependent wave vector parallel to the flow direction. Predictions from a linear stability analysis of the model equations are found to be in line with these observations.     

In vivo flexor tendon forces increase with finger and wrist flexion during active finger flexion and extension.

The effects of different hand motions and positions used during early protected motion rehabilitation on tendon forces are not well understood. The goal of this study was to determine in vivo forces in human flexor digitorum profundus (FDP) and flexor digitorum superficialis (FDS) tendons of the index finger during active unresisted finger flexion and extension. During open carpal tunnel surgery (n = 12), flexor tendon forces were acquired with buckle force transducers, and finger positions were recorded on video while subjects actively flexed and extended the fingers at two different wrist angles. Mean in vivo FDP tendon forces varied between 1.3N +/- 0.9 N and 4.0 N +/- 2.9 N while mean FDS tendon forces ranged from 1.3N +/- 0.5 N to 8.5 N +/- 10.7 N. FDP force increased with active finger flexion at both wrist angles of 0 degrees or 30 degrees flexion. FDS force increased with finger flexion when the wrist was in 30 degrees flexion, but was unchanged when the wrist was in 0 degrees of flexion. Tendon forces were similar regardless of whether the fingers were moving in the flexion or extension direction. Active finger flexion and extension with the wrist at 0 degrees and 30 degrees flexion may be used during early rehabilitation protocols with limited risk of repair rupture. This risk can be further decreased for a FDS tendon repair by reducing wrist flexion angle.     

Force calibration for an endovascular robotic system with proximal force measurement

Surgeons, while performing manual endovascular procedures with conventional surgical tools (catheters and guidewires), experience forces on the tool outside the patient's body that are proximal to the point of actuation. Currently, most of the robotic systems for endovascular procedures use active catheters to navigate vasculature and to measure the contact forces at the distal end (tool tip). These tools are more expensive than the conventional surgical tools used in endovascular procedures. To avoid dependence on specialized devices like active catheters, we have developed a novel endovascular robotic system (ERS) that uses conventional surgical tools. Our robot can indirectly measure proximal forces and provide haptic feedback to surgeons. This paper discusses the theory, methodology, and calibration of indirect proximal force measurement. This new calibration technique is presented as a nested optimization problem that is solved using bi‐level optimization. The results of experimental validation of the new force calibration methodology are also discussed. The results show that unbiasing of the indirect force measurement by means of force calibration will allow the use of conventional tools in robotic endovascular procedures.     

Patellofemoral and tibiofemoral joint loading during a single-leg forward hop following ACL reconstruction

Altered biomechanics are frequently observed following anterior cruciate ligament reconstruction (ACLR). Yet, little is known about knee-joint loading, particularly in the patellofemoral-joint, despite patellofemoral-joint osteoarthritis commonly occurring post-ACLR. This study compared knee-joint reaction forces and impulses during the landing phase of a single-leg forward hop in the reconstructed knee of people 12-24 months post-ACLR and uninjured controls. Experimental marker data and ground forces for 66 participants with ACLR (28 ± 6 years, 78 ± 15 kg) and 33 uninjured controls (26 ± 5 years, 70 ± 12 kg) were input into scaled-generic musculoskeletal models to calculate joint angles, joint moments, muscle forces, and the knee-joint reaction forces and impulses. The ACLR group exhibited a lower peak knee flexion angle (mean difference: ?6°; 95% confidence interval: [?10°, ?2°]), internal knee extension moment (?3.63 [?5.29, ?1.97] percentage of body weight × participant height (body weight [BW] × HT), external knee adduction moment (-1.36 [?2.16, ?0.56]% BW × HT) and quadriceps force (?2.02 [?2.95, ?1.09] BW). The ACLR group also exhibited a lower peak patellofemoral-joint compressive force (?2.24 [?3.31, ?1.18] BW), net tibiofemoral-joint compressive force (?0.74 [?1.20, 0.28] BW), and medial compartment force (?0.76 [?1.08, ?0.44] BW). Finally, only the impulse of the patellofemoral-joint compressive force was lower in the ACLR group (?0.13 [?0.23, ?0.03] body weight-seconds). Lower compressive forces are evident in the patellofemoral- and tibiofemoral-joints of ACLR knees compared to uninjured controls during a single-leg forward hop-landing task. Our findings may have implications for understanding the contributing factors for incidence and progression of knee osteoarthritis after ACLR surgery.     

Dimension-Reduced Hyperbolic Moment Method for the Boltzmann Equation with BGK-Type Collision

We develop the dimension-reduced hyperbolic moment method for the Boltzmann equation, to improve solution efficiency using a numerical regularized moment method for problems with low-dimensional macroscopic variables and high-dimensional microscopic variables. In the present work, we deduce the globally hyperbolic moment equations for the dimension-reduced Boltzmann equation based on the Hermite expansion and a globally hyperbolic regularization. The numbers of Maxwell boundary condition required for well-posedness are studied. The numerical scheme is then developed and an improved projection algorithm between two different Hermite expansion spaces is developed. By solving several benchmark problems, we validate the method developed and demonstrate the significant efficiency improvement by dimension-reduction.     

A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

The objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number (Kn); thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation. For aerodynamic flows, it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.     

Discrete mechanics and optimal control for constrained systems

The equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange‐d'Alembert principle. This paper derives a structure‐preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two‐link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Muscles that do not cross the knee contribute to the knee adduction moment and tibiofemoral compartment loading during gait

The aims of this study were to evaluate and explain the individual muscle contributions to the medial and lateral knee compartment forces during gait, and to determine whether these quantities could be inferred from their contributions to the external knee adduction moment. Gait data from eight healthy male subjects were used to compute each individual muscle contribution to the external knee adduction moment, the net tibiofemoral joint reaction force, and reaction moment. The individual muscle contributions to the medial and lateral compartment forces were then found using a least‐squares approach. While knee‐spanning muscles were the primary contributors, non‐knee‐spanning muscles (e.g., the gluteus medius) also contributed substantially to the medial compartment compressive force. Furthermore, knee‐spanning muscles tended to compress both compartments, while most non‐knee‐spanning muscles tended to compress the medial compartment but unload the lateral compartment. Muscle contributions to the external knee adduction moment, particularly those from knee‐spanning muscles, did not accurately reflect their tendencies to compress or unload the medial compartment. This finding may further explain why gait modifications may reduce the knee adduction moment without necessarily decreasing the medial compartment force. © 2012 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 30:1586–1595, 2012     

A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested to demonstrate the accuracy and the stability of the new scheme. The results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibria from a highly unbalanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.     

New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations

We present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.     

Interface corrective force measurements in Boston brace treatment

Brace application has been reported to be effective in treating idiopathic adolescent scoliosis. The exact working mechanism of a thoracolumbo spinal orthosis is a result of different mechanisms and is not completely understood. One of the supposed working mechanisms is a direct compressive force working through the brace upon the body and thereby correcting the scoliotic deformity, achieving optimal fit of the individual orthosis. In this study we measured these direct forces exerted by the pads in a Boston brace in 16 patients with idiopathic adolescent scoliosis, using the electronic PEDAR measuring device (Novel, Munich, Germany). This is designed as an in-shoe measuring system consisting of two shoe insoles (size 8 1/2), wired to a computer, recording static and dynamic pressure distribution under the plantar surface of the foot. After positioning the inserts between the lumbar and thoracic pads and the body, we measured the forces acting upon the body in eight different postures. In all positions the mean corrective force through the lumbar brace pad was larger than the mean corrective force over the thoracic brace pad. Some changes in body posture resulted in statistically significant alterations in the exerted forces. There was no significant correlation between the magnitude of the compressive force over the lumbar and thoracic brace-pad and the degree of correction of the major curve. Comparing the corrective forces in a relatively new (<6 months) and old (>6 months) brace, there was no statistically relevant difference, although the corrective force was slightly larger in the new braces. We think that the use of this pressure measurement device is practicable and of value for studies of the working mechanism of brace treatment, and in the future it might be of help in achieving optimal fit of the individual orthosis.     

Twist and its effect on ACL graft forces.

Graft tension is a controversial topic in anterior cruciate ligament (ACL) surgery. Evidence suggests a narrow range of graft tensions, which allow the graft to remodel to a stable and mature neoligament. In previous cadaver experiments, we showed that twisting the graft could modulate the graft forces. In this study we hypothesized that the same phenomena would be found in patients, and that twisting the graft intraoperatively can reduce peak forces in the graft. The effects of twist on graft forces in bone-patellar tendon-bone grafts were measured during anterior cruciate ligament surgery on 15 consecutive patients using a custom-made tension-measurement device. Variations in surgical procedure that could potentially affect the graft force patterns were quantified. Graft force as a function of knee-flexion angle was measured with the graft in the neutral, untwisted position and repeated with the tibial bone block rotated externally or internally by 180 degrees. In eight of the 15 knees, external twisting of the graft reduced the maximal graft force to 50%. However, in five knees the forces in extension increased by twisting to a maximum of 300%. Of the surgical variables, only the femoral tunnel position appeared to have a consistent effect on the graft force pattern. Due to the unpredictable effect of graft twisting, a general recommendation on whether the graft should be twisted, and if so, in which direction, cannot be given. Intraoperatively, graft twisting may however be considered in every individual knee to modulate the graft force as a function of flexion angle.