Fotis Sotiropoulos

Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies

The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, Journal of Computational Physics 225 (2007) 1782-1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions unconditionally unstable iteration schemes result even when strong coupling FSI is employed. For such cases, however, combining the strong-coupling iteration with under-relaxation in conjunction with the Aitkens acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and an upper bound of the required for stability under-relaxation coefficient is derived.

Numerical investigation of the hydrodynamics of anguilliform swimming in the transitional and inertial flow regimes.

SUMMARY We employ numerical simulation to investigate the hydrodynamic performance of anguilliform locomotion and compare it with that of carangiform swimming as the Reynolds number (Re) and the tail-beat frequency (Strouhal number, St) are systematically varied. The virtual swimmer is a 3-D lamprey-like flexible body undulating with prescribed experimental kinematics of anguilliform type. Simulations are carried out for three Reynolds numbers spanning the transitional and inertial flow regimes, Re=300, 4000 (viscous flow), and ∞ (inviscid flow). The net mean force is found to be mainly dependent on the tail-beat frequency rather than the tail-beat amplitude. The critical Strouhal number, St*, at which the net mean force becomes zero (constant-speed self-propulsion) is, similar to carangiform swimming, a decreasing function of Re and approaches the range of St numbers at which most anguilliform swimmers swim in nature (St∼0.45) only as Re increases. The anguilliform swimmers force time series is characterized by significantly smaller fluctuations above the mean than that for carangiform swimmers. In stark contrast with carangiform swimmers, the propulsive efficiency of anguilliform swimmers at St* is not an increasing function of Re but instead is maximized in the transitional regime. Furthermore, the power required for anguilliform swimming is less than that for the carangiform swimmer at the same Re. We also show that the form drag decreases while viscous drag increases as St increases. Finally, our simulations reinforce our previous finding for carangiform swimmers that the 3-D wake structure depends primarily on the Strouhal number.

A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming

SUMMARY We carry out fluid–structure interaction simulations of self-propelled virtual swimmers to investigate the effects of body shape (form) and kinematics on the hydrodynamics of undulatory swimming. To separate the effects of form and kinematics, we employ four different virtual swimmers: a carangiform swimmer (i.e. a mackerel swimming like mackerel do in nature); an anguilliform swimmer (i.e. a lamprey swimming like lampreys do in nature); a hybrid swimmer with anguilliform kinematics but carangiform body shape (a mackerel swimming like a lamprey); and another hybrid swimmer with carangiform kinematics but anguilliform body shape (a lamprey swimming like a mackerel). By comparing the performance of swimmers with different kinematics but similar body shapes we study the effects of kinematics whereas by comparing swimmers with similar kinematics but different body shapes we study the effects of form. We show that the anguilliform kinematics not only reaches higher velocities but is also more efficient in the viscous (Re∼102) and transitional (Re∼103) regimes. However, in the inertial regime (Re=∞) carangiform kinematics achieves higher velocities and is also more efficient than the anguilliform kinematics. The mackerel body achieves higher swimming speeds in all cases but is more efficient in the inertial regime only whereas the lamprey body is more efficient in the transitional regime. We also show that form and kinematics have little overall effect on the 3-D structure of the wake (i.e. single vs double row vortex streets), which mainly depends on the Strouhal number. Nevertheless, body shape is found to somewhat affect the small-scale features and complexity of the vortex rings shed by the various swimmers.

Flow in prosthetic heart valves: State-of-the-art and future directions

Since the first successful implantation of a prosthetic heart valve four decades ago, over 50 different designs have been developed including both mechanical and bioprosthetic valves. Today, the most widely implanted design is the mechanical bileaflet, with over 170,000 implants worldwide each year. Several different mechanical valves are currently available and many of them have good bulk forward flow hemodynamics, with lower transvalvular pressure drops, larger effective orifice areas, and fewer regions of forward flow stasis than their earlier-generation counterparts such as the ball-and-cage and tilting-disc valves. However, mechanical valve implants suffer from complications resulting from thrombus deposition and patients implanted with these valves need to be under long-term anti-coagulant therapy. In general, blood thinners are not needed with bioprosthetic implants, but tissue valves suffer from structural failure with, an average life-time of 10–12 years, before replacement is needed. Flow-induced stresses on the formed elements in blood have been implicated in thrombus initiation within the mechanical valve prostheses. Regions of stress concentration on the leaflets during the complex motion of the leaflets have been implicated with structural failure of the leaflets with bioprosthetic valves. In vivo and in vitro experimental studies have yielded valuable information on the relationship between hemodynamic stresses and the problems associated with the implants. More recently, Computational Fluid Dynamics (CFD) has emerged as a promising tool, which, alongside experimentation, can yield insights of unprecedented detail into the hemodynamics of prosthetic heart valves. For CFD to realize its full potential, however, it must rely on numerical techniques that can handle the enormous geometrical complexities of prosthetic devices with spatial and temporal resolution sufficiently high to accurately capture all hemodynamically relevant scales of motion. Such algorithms do not exist today and their development should be a major research priority. For CFD to further gain the confidence of valve designers and medical practitioners it must also undergo comprehensive validation with experimental data. Such validation requires the use of high-resolution flow measuring tools and techniques and the integration of experimental studies with CFD modeling.

A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids

In the present note a general reconstruction algorithm for simulating incompressible flows with complex immersed boundaries on Cartesian grids is presented. In the proposed method an arbitrary three-dimensional solid surface immersed in the fluid is discretized using an unstructured, triangular mesh, and all the Cartesian grid nodes near the interface are identified. Then, the solution at these nodes is reconstructed via linear interpolation along the local normal to the body, in a way that the desired boundary conditions for both pressure and velocity fields are enforced. The overall accuracy of the resulting solver is second-order, as it is demonstrated in two test cases involving laminar flow past a sphere.

Vorticity dynamics of a bileaflet mechanical heart valve in an axisymmetric aorta

We present comprehensive particle image velocimetry measurements and direct numerical simulation (DNS) of physiological, pulsatile flow through a clinical quality bileaflet mechanical heart valve mounted in an idealized axisymmetric aorta geometry with a sudden expansion modeling the aortic sinus region. Instantaneous and ensemble-averaged velocity measurements as well as the associated statistics of leaflet kinematics are reported and analyzed in tandem to elucidate the structure of the velocity and vorticity fields of the ensuing flow-structure interaction. The measurements reveal that during the first half of the acceleration phase, the flow is laminar and repeatable from cycle to cycle. The valve housing shear layer rolls up into the sinus and begins to extract vorticity of opposite sign from the sinus wall. A start-up vortical structure is shed from the leaflets and is advected downstream as the leaflet shear layers become wavy and oscillatory. In the second half of flow acceleration the leaflet shea...

On the bimodal dynamics of the turbulent horseshoe vortex system in a wing-body junction

The turbulent boundary layer approaching a wall-mounted obstacle experiences a strong adverse pressure gradient and undergoes three-dimensional separation leading to the formation of a dynamically rich horseshoe vortex (HSV) system. In a pioneering experimental study, Devenport and Simpson [J. Fluid Mech. 210, 23 (1990)] showed that the HSV system forming at the leading edge region of a wing mounted on a flat plate at Re=1.15×105 exhibits bimodal, low-frequency oscillations, which away from the wall produce turbulent energy and stresses one order of magnitude higher than those produced by the conventional shear mechanism in the approaching turbulent boundary layer. We carry out numerical simulations for the experimental configuration of Devenport and Simpson using the detached-eddy-simulation (DES) approach. The DES length scale is adjusted for this flow to alleviate the well known shortcoming of DES; namely that of premature, laminar-like flow separation. The numerical simulations reproduce with good acc...

Physics-Driven CFD Modeling of Complex Anatomical Cardiovascular Flows—A TCPC Case Study

Recent developments in medical image acquisition combined with the latest advancements in numerical methods for solving the Navier-Stokes equations have created unprecedented opportunities for developing simple and reliable computational fluid dynamics (CFD) tools for meeting patient-specific surgical planning objectives. However, for CFD to reach its full potential and gain the trust and confidence of medical practitioners, physics-driven numerical modeling is required. This study reports on the experience gained from an ongoing integrated CFD modeling effort aimed at developing an advanced numerical simulation tool capable of accurately predicting flow characteristics in an anatomically correct total cavopulmonary connection (TCPC). An anatomical intra-atrial TCPC model is reconstructed from a stack of magnetic resonance (MR) images acquired in vivo. An exact replica of the computational geometry was built using transparent rapid prototyping. Following the same approach as in earlier studies on idealized models, flow structures, pressure drops, and energy losses were assessed both numerically and experimentally, then compared. Numerical studies were performed with both a first-order accurate commercial software and a recently developed, second-order accurate, in-house flow solver. The commercial CFD model could, with reasonable accuracy, capture global flow quantities of interest such as control volume power losses and pressure drops and time-averaged flow patterns. However, for steady inflow conditions, both flow visualization experiments and particle image velocimetry (PIV) measurements revealed unsteady, complex, and highly 3D flow structures, which could not be captured by this numerical model with the available computational resources and additional modeling efforts that are described. Preliminary time-accurate computations with the in-house flow solver were shown to capture for the first time these complex flow features and yielded solutions in good agreement with the experimental observations. Flow fields obtained were similar for the studied total cardiac output range (1–3 l/min); however hydrodynamic power loss increased dramatically with increasing cardiac output, suggesting significant energy demand at exercise conditions. The simulation of cardiovascular flows poses a formidable challenge to even the most advanced CFD tools currently available. A successful prediction requires a two-pronged, physics-based approach, which integrates high-resolution CFD tools and high-resolution laboratory measurements.

Vortex-induced vibrations of two cylinders in tandem arrangement in the proximity–wake interference region

We investigate numerically vortex-induced vibrations (VIV) of two identical two-dimensional elastically mounted cylinders in tandem in the proximity-wake interference regime at Reynolds number Re = 200 for systems having both one (transverse vibrations) and two (transverse and in-line) degrees of freedom (1-DOF and 2-DOF, respectively). For the 1-DOF system the computed results are in good qualitative agreement with available experiments at higher Reynolds numbers. Similar to these experiments our simulations reveal: (1) larger amplitudes of motion and a wider lock-in region for the tandem arrangement when compared with an isolated cylinder; (2) that at low reduced velocities the vibration amplitude of the front cylinder exceeds that of the rear cylinder; and (3) that above a threshold reduced velocity, large-amplitude VIV are excited for the rear cylinder with amplitudes significantly larger than those of the front cylinder. By analysing the simulated flow patterns we identify the VIV excitation mechanisms that lead to such complex responses and elucidate the near-wake vorticity dynamics and vortex-shedding modes excited in each case. We show that at low reduced velocities vortex shedding provides the initial excitation mechanism, which gives rise to a vertical separation between the two cylinders. When this vertical separation exceeds one cylinder diameter, however, a significant portion of the incoming flow is able to pass through the gap between the two cylinders and the gap-flow mechanism starts to dominate the VIV dynamics. The gap flow is able to periodically force either the top or the bottom shear layer of the front cylinder into the gap region, setting off a series of very complex vortex-to-vortex and vortex-to-cylinder interactions, which induces pressure gradients that result in a large oscillatory force in phase with the vortex shedding and lead to the experimentally observed larger vibration amplitudes. When the vortex shedding is the dominant mechanism the front cylinder vibration amplitude is larger than that of the rear cylinder. The reversing of this trend above a threshold reduced velocity is associated with the onset of the gap flow. The important role of the gap flow is further illustrated via a series of simulations for the 2-DOF system. We show that when the gap-flow mechanism is triggered, the 2-DOF system can develop and sustain large VIV amplitudes comparable to those observed in the corresponding (same reduced velocity) 1-DOF system. For sufficiently high reduced velocities, however, the two cylinders in the 2-DOF system approach each other, thus significantly reducing the size of the gap region. In such cases the gap flow is entirely eliminated, and the two cylinders vibrate together as a single body with vibration amplitudes up to 50% lower than the amplitudes of the corresponding 1-DOF in which the gap flow is active. Three-dimensional simulations are also carried out to examine the adequacy of two-dimensional simulations for describing the dynamic response of the tandem system at Re = 200. It is shown that even though the wake transitions to a weakly three-dimensional state when the gap flow is active, the three-dimensional modes are too weak to affect the dynamic response of the system, which is found to be identical to that obtained from the two-dimensional computations.