Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks.

Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments

Kinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support (finite elements, finite differences, finite volumes, spectral techniques, etc.). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function is related to a node that defines the associated degree of freedom. When the model involves high dimensional spaces (including physical and conformation spaces and time), standard discretization techniques fail due to an excessive computation time required to perform accurate numerical simulations. One appealing strategy that allows circumventing this limitation is based on the use of reduced approximation basis within an adaptive procedure making use of an efficient separation of variables. (c) 2006 Elsevier B.V. All rights reserved.

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A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids

Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves

The log conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] has been implemented in a FEM context using the DEVSS/DG formulation for viscoelastic fluid flow. We present a stability analysis in 1D and identify the failure of the numerical scheme to balance exponential growth as a possible source for numerical instabilities at high Weissenberg numbers. A different derivation of the log-based evolution equation than in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] is also presented. We show numerical results for the flow around a cylinder for an Oldroyd-B and a Giesekus model. With the log conformation representation, we are able to obtain solutions beyond the limiting Weissenberg numbers in the standard scheme. In particular, for the Giesekus model the improvement is rather dramatic: there does not seem to be a limit for the chosen model parameter (α = 0.01). However, it turns out that although in large parts of the flow the solution converges, we have not been able to obtain convergence in localized regions of the flow. Possible reasons include artefacts of the model and unresolved small scales. More work is necessary, including the use of more refined meshes and/or higher order schemes, before any conclusion can be made on the local convergence problems. © 2005 Elsevier B.V. All rights reserved.

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Flow of viscoelastic fluids past a cylinder at high Weissenberg number : stabilized simulations using matrix logarithms

Kinetic theory models described within the Fokker-Planck formalism involve high-dimensional spaces (including physical and conformation spaces and time). One appealing strategy for treating this kind of problems lies in the use of separated representations and tensor product approximations basis. This technique that was introduced in a former work [A. Ammar, B. Mokdad, E Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newtonian Fluid Mech. 139 (2006) 153-176] for treating steady state kinetic theory models is extended here for treating transient models. (c) 2007 Elsevier B.V. All rights reserved.

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A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids - Part II: Transient simulation using space-time separated representations

A finite volume method based on a velocity-only formulation is used to solve the flow field around a confined circular cylinder in a channel in order to investigate lateral wall proximity effects on stability, Strouhal number, hydrodynamic forces and wake structure behind the cylinder for a wide range of blockage ratios (0.1<β⩽0.9) and Reynolds numbers (0<Re⩽280). For blockage ratios less than approximately 0.85 a first critical Reynolds number is identified at which a supercritical Hopf bifurcation of the symmetric solution occurs. For blockage ratios greater than about 0.687 and at Reynolds numbers exceeding the first critical Reynolds number a second curve of neutral stability is seen, representing a pitchfork bifurcation of the steady symmetric solution to one of two possible steady asymmetric solutions. Either side of the neutral stability curve for the pitchfork bifurcation our linear stability analysis and direct numerical simulations demonstrate that although the flow is linearly stable it is unstable to finite two-dimensional perturbations. At blockage ratios larger than about 0.82 the steady asymmetric solutions also become unstable through a Hopf bifurcation. In contrast with the first Hopf bifurcation of the symmetric solution at lower Reynolds numbers numerical calculations of the lift coefficient reveal that the oscillations are no longer symmetric in the rising and falling parts of each cycle. Very strong vortices shed from the cylinder and the wall cause drastic increases in the amplitudes of the lift and drag coefficients. A co-dimension 2 point where pitchfork and Hopf bifurcations occur simultaneously has been located in parameter space. Altogether, four distinct regions in the parameter space (β,Re)∈(0,0.9]×(0,280] have been identified, each corresponding to a different class of flow: (i) Steady symmetric flow, (ii) symmetric vortex shedding, (iii) steady asymmetric flow, and (iv) asymmetric vortex shedding, where a periodic-in-time flow is classed as symmetric or asymmetric depending on whether the time-average over one cycle of the lift coefficient is zero or not. Numerical solutions are computed on meshes having up to 1.8 million degrees of freedom. Extensive comparisons are made with the results available in the literature.

A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder

A new constitutive equation for whole human blood is derived using ideas drawn from temporary polymer network theory to model the aggregation and disaggregation of erythrocytes in normal human blood at different shear rates. Each erythrocyte is represented by a dumbbell. The use of a linear spring law in the dumbbells leads to a multi-mode generalized Maxwell equation for the elastic stress and both the relaxation times and viscosities are functions of a time-dependent structure variable. An approximate constitutive equation is derived by choosing a single mode corresponding to the cell aggregate size where the largest number of cells are to be found. This size is identified in the case of steady flows. The model exhibits shear-thinning, viscoelasticity and thixotropy and these are clearly related to the microstructural properties of the fluid. Agreement with the experimental data of Bureau et al. [M. Bureau, J.C. Healy, D. Bourgoin, M. Joly, Rheological hysteresis of blood at low shear rate, Biorheology 17 (1980) 191–203] in the case of a simple triangular step shear rate flow is convincing.

A new microstructure-based constitutive model for human blood

A novel implicit cell-vertex finite volume method is described for the solution of the Navier-Stokes equations at high Reynolds numbers. The key idea is the elimination of the pressure term from the momentum equation by multiplying the momentum equation with the unit normal vector to a control volume boundary and integrating thereafter around this boundary. The resulting equations are expressed solely in terms of the velocity components. Thus any difficulties with pressure or vorticity boundary conditions are circumvented and the number of primary variables that need to be determined equals the number of space dimensions. The method is applied to both the steady and unsteady two-dimensional lid-driven cavity problem at Reynolds numbers up to 10000. Results are compared with those in the literature and show excellent agreement.

A novel fully implicit finite volume method applied to the lid‐driven cavity problem—Part I: High Reynolds number flow calculations

The smoothed particle hydrodynamics (SPH) method is extended and tested for the numerical simulation of transient viscoelastic free surface flows. The basic equations governing the free surface flow of an Oldroyd-B fluid are considered and approximated by SPH. In particular, a drop of an Oldroyd-B fluid impacting a rigid plate is simulated. Results for a Newtonian fluid are also presented for comparison. It is found that the original SPH method, which has been successfully applied to the simulation of transient viscoelastic flows in bounded domains (such as the start-up flow between parallel plates), is unable to simulate the viscoelastic free surface flow considered here because of the so-called tensile instability. This instability leads to unrealistic fracture and particle clustering in fluid stretching and may eventually result in complete blowup of the simulation. Recent works have shown that in simulations of elastic solids the tensile instability can be removed by an artificial stress. Here we show that the same idea also works for viscoelastic fluids provided that the constant parameter entering in the definition of the artificial stress is properly chosen. Numerical results obtained are in good agreement with those simulated by a finite difference technique.

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A numerical study of the SPH method for simulating transient viscoelastic free surface flows

Rheology is the science of the deformation and flow of materials. It deals with the theoretical concepts of kinematics, conservation laws and constitutive relations, describing the interrelation between force, deformation and flow. The experimental determination of the rheological behaviour of materials is called rheometry. The object of haemorheology is the application of rheology to the study of flow properties of blood and its formed elements, and the coupling of blood and the blood vessels in living organisms. This field involves the investigation of the macroscopic behaviour of blood determined in rheometric experiments, its microscopic properties in vitro and in vivo and studies of the interactions among blood cellular components and between these components and the endothelial cells that line blood vessels.

Robert G. Owens

Papers

A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder

A new microstructure-based constitutive model for human blood

A novel fully implicit finite volume method applied to the lid‐driven cavity problem—Part I: High Reynolds number flow calculations

A numerical study of the SPH method for simulating transient viscoelastic free surface flows

Rheological models for blood