Luigino Zovatto

Flow about a circular cylinder between parallel walls

The flow about a body placed inside a channel diers from its unbounded counterpart because of the eects of wall connement, shear in the incoming velocity prole, and separation of vorticity from the channel walls. The case of a circular cylinder placed between two parallel walls is here studied numerically with a nite element method based on the vorticity{streamfunction formulation for values of the Reynolds number consistent with a two-dimensional assumption. The transition from steady flow to a periodic vortex shedding regime has been analysed: transition is delayed as the body approaches one wall because the interaction between the cylinder wake and the wall boundary layer vorticity constrains the separating shear layer, reducing its oscillations. The results conrm previous observations of the inhibition of vortex shedding for a body placed near one wall. The unsteady vortex shedding regime changes, from a pattern similar to the von K arm an street (with some dierences) when the body is in about the centre of the channel, to a single row of same-sign vortices as the body approaches one wall. The separated vortex dynamics leading to this topological modication is presented. The mean drag coecients, once they have been normalized properly, are comparable when the cylinder is placed at dierent distances from one wall, down to gaps less than one cylinder diameter. At smaller gaps the body behaves similarly to a surface-mounted obstacle. The lift force is given by a repulsive component plus an attractive one. The former, well known from literature, is given by the deviation of the wake behind the body. Evidence of the latter, which is a consequence of the shear in front of the body, is given.

Birth of three-dimensionality in a pulsed jet through a circular orifice

The pulsed flow about a circular non-centred orifice inside a cylindrical duct is analysed to obtain insight into the basic three-dimensional vortex dynamics that may be expected behind natural cardiac valves. The problem is approached by a highresolution numerical simulation. Results show how a small finite eccentricity generates af ully three-dimensional vortex wake that evolves quite differently from that found under an axisymmetric approximation. The flow field is analysed in terms of velocity and vorticity fields. The vortex wake structure is that of an initially quasi-axisymmetric vortex ring that progressively deforms into a three-dimensional structure, and whose vortex lines tend to reconnect with the boundary-layer-induced vorticity. The vortex structure explains secondary circulation and the presence of a diastolic backflow jet localized behind the longer orifice edge, in agreement with previous experimental observations. Possible relevance of the results to flows of cardiovascular interest is discussed.

A new meshless approach for subject-specific strain prediction in long bones: Evaluation of accuracy.

BACKGROUND The Finite Element Method is at present the method of choice for strain prediction in bones from Computed Tomography data. However, accurate methods rely on the correct topological representation of the bone surface, which requires a massive operator effort, thus restricting their applicability to clinical practice. Meshless methods, which do not rely on a pre-defined topological discretisation of the domain, might greatly improve the numerical process automation, but currently their application to biomechanics is negligible. METHODS A meshless implementation of an innovative numerical approach based on a direct discrete formulation of physical laws, the Cell Method, was developed to predict strains in a cadaver femur from Computed Tomography data. The model accuracy was estimated by comparing the predicted strains with those experimentally measured on the same specimen in a previous study. As a reference, the results were compared to those obtained with a state-of-the-art finite element model. FINDINGS The Cell Method meshless model predicted strains highly correlated with the experimental measurements (R2=0.85) with a good global accuracy (RMSE=15.6%). The model performed slightly worse than the finite element one, but this was probably due to the need to sub-sample the original data, and the lower order of the interpolation used (linear vs parabolic). INTERPRETATION Although there is surely room for improvement, the accuracy already obtained with this meshless implementation of the Cell Method makes it a good candidate for some clinical applications, especially considering the full automation of the method, which does not require any data pre-processing.

Flow on the symmetry plane of a total cavo-pulmonary connection

The flow inside a total cavo-pulmonary connection, a bypass operation of the right heart adopted in the presence of congenital malformation, is here studied for a specific geometry which has been recently introduced in clinics. The analysis has been performed by preliminary experimental observation and a novel Navier-Stokes formulation on the symmetry plane. This method, once some basic hypotheses are verified, allows to reproduce the flow on the symmetry plane of a three-dimensional field by using an extension of the two-dimensional approach. The analysis has confirmed the existence of a central vortex showing that it is not a real vortex (i.e. a place with accumulation of vorticity) but, rather, a weakly dissipative recirculating zone. It is surrounded by a shear layer that becomes spontaneously unsteady at moderately high Reynolds number. The topological changes and energy dissipation have been analysed in both cases of unbalanced and of balanced pulmonary artery and caval flows.

Extension of the Meshless Approach for the Cell Method to Three-Dimensional Numerical Integration of Discrete Conservation Laws

In this work, an extension to the three-dimensional space of the meshless approach proposed by the authors in a preceding paper is presented. The Cell Method (CM) for the numerical integration of discrete conservation laws is adopted. As in the two-dimensional case, for every node a local mesh is generated, formed by all tetrahedra whose vertices coincide with the node itself and its neighbors. Conservation equations are then written directly in discrete form on the tributary region (dual cell) made up of the polyhedron whose faces are represented by the planes passing through the circumcenters and/or the barycenters of the local mesh, the circumcenters and/or barycenters of the faces of the tetrahedra, and the midpoints of their sides. Such an approach avoids the construction of the global mesh, and is particularly efficient for non-linear problems whose iterative solution would require large CPU resources (problems with time-varying domain, or mixed Lagrangian-Eulerian formulations are examples). Moreover, the flexibility of the method allows the run-time addition or deletion of nodes (in order to increase solution accuracy, especially at critical zones of the domain) which, for a global mesh in 3D domains, would result computationally very intensive. The paper describes some algorithms (applied to Laplace equation) in order to compare their convergence order as a function of the number of neighbor nodes and the type of boundary conditions.

THE MESHLESS APPROACH FOR THE CELL METHOD: A NEW WAY FOR THE NUMERICAL SOLUTION OF DISCRETE CONSERVATION LAWS

A new methodology for the solution of discrete conservation laws, based on a point by point approach, is presented. For each node, a local mesh is firstly generated, made up of all triangles whose vertices coincide with the node itself and its neighbours. The solution is then determined through mass, energy and momentum balances directly written in a discrete form over a tributary region, represented by the polygon whose vertices are the barycenters and/or the circumcenters of the triangles belonging to the local mesh. This approach avoids global mesh generation (computationally much more expensive), and is particularly efficient for non-linear problems, such as fracture mechanics. In the paper, the numerical method is described in detail for Laplace equation, together with the convergence order as a function of the number of nodes and the type of boundary conditions. Finally, in order to further simplify the procedure, it is proposed to consider the tributary area formed by the circle with center in the generic node and radius equal to the average of the distances between the node and its neighbours. This results in a considerable ease in writing the discrete form of the governing equations, while maintaining the same accuracy and order of convergence than the approach based on local triangles.

Pulsatile Flow Inside Moderately Elastic Arteries, Its Modelling and Effects of Elasticity

Pulsatile flow inside a moderately elastic circular conduit with a smooth expansion is studied as a model to understand the influence of wall elasticity in artery flow. The solution of the simultaneous fluid-wall evolution is evaluated by a perturbative method, where the zeroth order solution is represented by the flow in a rigid vessel; the first order correction gives the wall motion and induced flow modification without the need to solve the difficult coupled problem. Such an approach essentially assumes a locally infinite celerity, therefore it represent a good approximation for the fluid-wall interaction in sites of limited extent (branches, stenosis, aneurism, etc.), which include typical situations associated with vascular diseases. The problem is solved numerically in the axisymmetric approximation; the influence of wall elasticity on the flow and on the unsteady wall shear stress is studied in correspondence of parameters taken from realistic artery flow. Attention is posed to the role of phase difference between the incoming pressure and flow pulses.

Improving the Convergence Order of the Meshless Approach for the Cell Method for Numerical Integration of Discrete Conservation Laws

In this work, the problem of increasing the convergence order of the integral meshless method already proposed by the same authors is addressed. Solutions are determined through equations directly written in discrete form over a tributary region represented by the circle with center in the generic node and radius given by the average of the distances between the node itself and its neighbors, thus allowing a considerable ease in writing the discrete form of the governing equations. The proposed approach, besides avoiding global mesh generation, adopts interpolating polynomials, which exactly reproduce nodal values of field variables, and eliminates some problems typically encountered when posing Dirichlet and Neumann boundary conditions with the Finite Element Method. Several numerical schemes adopting extended or compact computational cells are proposed and tested for the Laplace equation, in line with the previous papers. Results show that, when using interpolating polynomials that satisfy also the differential operator in some nodes, compact computational cells characterized by the fifth-order of convergence may be constructed.

ACCURACY OF A CELL-BASED MESHLESS APPROACH FOR CT-BASED MODELLING OF HUMAN BONES

The Finite Element Method is the most widely used tool for strain prediction in bones from CT data. Accurate FE modelling procedures have been devised, but they rely on the correct topological representation of bone surface and hence their clinical applicability is limited by the massive operator effort required. One alternative approach would be the adoption of meshless methods, which could significantly improve the process automation not requiring the domain discretization. However, their application to biomechanics is at present negligible. The aim of this study was twofold: i) to present a first implementation of a cell-based meshless method for the prediction of strains in bones starting from CT data; ii) to validate it against strain-gauge measurements on a cadaver femur, comparing the accuracy obtained with a state-of-the-art parabolic tetrahedral FE model [Taddei, 2006].