Divergence-free Velocity Field Research Articles

Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach

The stability and accuracy of three methods which enforce either a divergence-free velocity field, density invariance, or their combination are tested here through the standard Taylor–Green and spin-down vortex problems. While various approaches to incompressible SPH (ISPH) have been proposed in the past decade, the present paper is restricted to the projection method for the pressure and velocity coupling. It is shown that the divergence-free ISPH method cannot maintain stability in certain situations although it is accurate before instability sets in. The density-invariant ISPH method is stable but inaccurate with random-noise like disturbances. The combined ISPH, combining advantages in divergence-free ISPH and density-invariant ISPH, can maintain accuracy and stability although at a higher computational cost. Redistribution of particles on a fixed uniform mesh is also shown to be effective but the attraction of a mesh-free method is lost. A new divergence-free ISPH approach is proposed here which maintains accuracy and stability while remaining mesh free without increasing computational cost by slightly shifting particles away from streamlines, although the necessary interpolation of hydrodynamic characteristics means the formulation ceases to be strictly conservative. This avoids the highly anisotropic particle spacing which eventually triggers instability. Importantly pressure fields are free from spurious oscillations, up to the highest Reynolds numbers tested.

Sensitivity analysis of initial condition parameters on the transitional temporal turbulent mixing layer

This paper aims at determining the most influential inlet turbulence parameters on the downstream transitional mixing region. To this end, a stochastic method is developed to generate a divergence-free random velocity field with a prescribed energy distribution in physical and wave-number space. In addition, predetermined integral length scales can be established. Ten direct numerical simulations of a temporally evolving transitional turbulent mixing layer are examined in detail. Simulation results show a large disparity in mean and instantaneous turbulent quantities, mainly effected by the energy spectrum, the integral length scale and the divergence freeness of the initial field.

Incompressible smoothed particle hydrodynamics

We present a smoothed particle hydrodynamic model for incompressible fluids. As opposed to solving a pressure Poisson equation in order to get a divergence-free velocity field, here incompressibility is achieved by requiring as a kinematic constraint that the volume of the fluid particles is constant. We use Lagrangian multipliers to enforce this restriction. These Lagrange multipliers play the role of non-thermodynamic pressures whose actual values are fixed through the kinematic restriction. We use the SHAKE methodology familiar in constrained molecular dynamics as an efficient method for finding the non-thermodynamic pressure satisfying the constraints. The model is tested for several flow configurations.

Simulation of incompressible flows around moving bodies using meshless finite differencing

A scheme using the mesh-free generalized finite differencing (GFD) on flows past moving solid bodies is proposed. The aim is to devise a method which can be applied to simulate flows past immersed moving bodies with reduced computational effort by eliminating the need to perform remeshing for every time step and the intense interpolation procedures. The generalized finite difference method (GFD) with weighted least squares approximation was used to solve the incompressible Navier Stokes equations under the arbitrary Lagrangian-Eulerian (ALE) formulation, on a stationary background set of Cartesian nodes and a cloud of nodes following each moving solid boundary to simulate flows near the solid-fluid interface. A combination of standard central-spaced finite differencing and GFD is applied to the background region and cloud region respectively as spatial discretization. The second-ordered Crank Nicolson time discretization is applied in the projection method, which computes an intermediate set of velocities and uses the pressure Poisson equation to obtain the divergence-free velocity field. Test cases were performed for workability and accuracy of the method, and satisfactory results were obtained for the driven cavity, the decaying vortex, the flow past circular cylinder and a flapping elliptic body in a square enclosed space.

Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry

In this paper we present and analyze different volume-of-fluid (VOF) reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three-dimensional space. The paper describes the improvement of the reconstruction when a least-square fit algorithm, which minimizes a distance functional, is applied. Its performance is tested for several smooth surfaces against other simple reconstruction methods. Then Eulerian, Lagrangian and mixed split advection schemes are presented and analyzed. In particular, one advection method is discussed that conserves mass exactly for a divergence-free velocity field, thus allowing computations to machine precision.

An adaptive phase field method for the mixture of two incompressible fluids

This paper develops an adaptive moving mesh method to solve a phase field model for the mixture of two incompressible fluids. The projection method is implemented on a half-staggered, moving quadrilateral mesh to keep the velocity field divergence-free, and the conjugate gradient or multigrid method is employed to solve the discrete Poisson equations. The current algorithm is composed by two independent parts: evolution of the governing equations and mesh-redistribution. In the first part, the incompressible Navier–Stokes equations are solved on a fixed half-staggered mesh by the rotational incremental pressure-correction scheme, and the Allen–Cahn type of phase equation is approximated by a conservative, second-order accurate central difference scheme, where the Lagrangian multiplier is used to preserve the mass-conservation of the phase field. The second part is an iteration procedure. During the mesh redistribution, the phase field is remapped onto the newly resulting meshes by the high-resolution conservative interpolation, while the non-conservative interpolation algorithm is applied to the velocity field. The projection technique is used to obtain a divergence-free velocity field at the end of this part. The resultant numerical scheme is stable, mass conservative, highly efficient and fast, and capable of handling variable density and viscosity. Several numerical experiments are presented to demonstrate the efficiency and robustness of the proposed algorithm.

The effect of non divergence-free velocity fields on field scale ground water solute transport

Solute plume subjected to field scale hydraulic conductivity heterogeneity shows a large dispersion/macrodispersion, which is the manifestation of existing fields scale heterogeneity on the solute plume. On the other hand, due to the scarcity of hydraulic conductivity measurements at field scale, hydraulic conductivity heterogeneity can only be defined statistically, which makes the hydraulic conductivity a random variable/function. Random hydraulic conductivity as a parameter in flow equation makes the pore flow velocity also random and the ground water solute transport equation is a stochastic differential equation now. In this study, the ensemble average of stochastic ground water solute transport equation is taken by the cumulant expansion method in order to upscale the laboratory scale transport equation to field scale by assuming pore flow velocity is a non stationary, non divergence-free and unsteady random function of space and time. Besides the stochastic explanation of macrodispersion and the velocity correction term obtained by Kavvas and Karakas (J Hydrol 179:321–351, 1996) before a new velocity correction term, which is a function of mean pore flow velocity divergence, is obtained in this study due to strict second order cumulant expansion (without omitting any term after the expansion) performed. The significance of the new velocity correction term is investigated on a one dimensional transport problem driven by a density dependent flow field.

Convergence Analysis of a Colocated Finite Volume Scheme for the Incompressible Navier–Stokes Equations on General 2D or 3D Meshes

We study a colocated cell‐centered finite volume method for the approximation of the incompressible Navier–Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressure, all of them colocated at the center of the cells of a unique mesh; such a configuration is known to lead to stability problems, hence the need for a stabilization technique, which we choose of the Brezzi–Pitkäranta type. The scheme features two essential properties: the discrete gradient is the transpose of the divergence terms, and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem and for the transient and the steady Navier–Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the $L^2$‐convergence of the components of the velocity, and, in the steady case, the weak $L^2$‐convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov’s theorem. The limit of this subsequence is then shown to be a weak solution of the Navier–Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and nonlinear cases.

Unconditionally stable discretizations of the immersed boundary equations

The immersed boundary (IB) method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method. Many implicit methods have been proposed to try to mitigate this timestep restriction, but none are known to be unconditionally stable, and the observed instability of even some of the fully implicit methods is not well understood. In this paper, we prove that particular backward Euler and Crank–Nicolson-like discretizations of the nonlinear immersed boundary terms of the IB equations in conjunction with unsteady Stokes Flow can yield unconditionally stable methods. We also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space. We further demonstrate through computational tests that approximate projection methods (which do not provide a discretely divergence-free velocity field) appear to have a stabilizing influence for these problems; and that the implicit methods of this paper, when used with the full Navier–Stokes equations, are no longer subject to such a strict timestep restriction and can be run up to the CFL constraint of the advection terms.

Effects of mixing and stirring on the critical behaviour

Stochastic dynamics of a nonconserved scalar order parameter near its critical point, subject to random stirring and mixing, is studied using the field theoretic renormalization group. The stirring and mixing are modelled by a random external Gaussian noise with the correlation function $\propto\delta(t-t') k^{4-d-y}$ and the divergence-free (due to incompressibility) velocity field, governed by the stochastic Navier--Stokes equation with a random Gaussian force with the correlation function $\propto\delta(t-t') k^{4-d-y'}$. Depending on the relations between the exponents $y$ and $y'$ and the space dimensionality $d$, the model reveals several types of scaling regimes. Some of them are well known (model A of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow), but there are three new nonequilibrium regimes (universality classes) associated with new nontrivial fixed points of the renormalization group equations. The corresponding critical dimensions are calculated in the two-loop approximation (second order of the triple expansion in $y$, $y'$ and $\epsilon=4-d$).

Ground water contaminant transport by nondivergence-free, unsteady and nonstationary velocity fields

Pore flow velocity is assumed to be a nondivergence-free, unsteady, and nonstationary random function of space and time for ground water contaminant transport in a heterogeneous medium. The laboratory-scale stochastic contaminant transport equation is up scaled to field scale by taking the ensemble average of the equation by using the cumulant expansion method. A new velocity correction, which is a function of mean pore flow velocity divergence, is obtained due to strict second order cumulant expansion (without omitting any term after the expansion). The field scale transport equations under the divergence-free pore flow velocity field assumption are also derived by simplifying the nondivergence-free field scale equation. The significance of the new velocity correction term is investigated on a two dimensional transport problem driven by a density dependent flow.

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier–Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier–Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray–Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.

The effect of non divergence-free velocity fields on field scale ground water solute transport

The effect of non divergence-free velocity fields on field scale ground water solute transport

Gemischte und nicht-standard Finite-Elemente-Methoden mit Anwendungen

Mixed finite element methods (MFEM) form a general mathematical framework for the spatial discretisation of partial differential equations, mainly applied to elliptic equations of second order. They become increasingly important for the solution of nonlinear problems. In contrast to standard finite element schemes the mixed finite element discretisation of problems in divergence form, i.e.\~$f +{\rm div} , \sigma = 0$ where $\sigma = A (\nabla , u)$, $\sigma \in L$ and $u \in H$, allows more flexibility in the design of the discrete approximation spaces contained in $L$ and $H$, i.e. in the spaces for the direct variables and the Lagrange multipliers. The workshop focuses on new developments in the field of mixed and non-standard finite element methods. The main points are \begin{itemize} \item The analysis of mixed FE formulations and of non-conforming methods, including, for instance, enhanced strain and discontinuous Galerkin methods, cf. (F.\~Armero), (D.\~Braess), (R.\~Dur\\'{a}n), (K.\~Garikipati), (P.\~Monk), (F.\~Radu), (D. Reddy), (S.\~Reese), (R.\~Sacco), (J.\~Sch\\"{o}berl), (W.\~Wagner), (R.\~Winther), (B.\~Wohlmuth). \pagebreak\item The effective and reliable error estimation of finite element methods as the basis for adaptive techniques and error control, cf. (M.\~Ainsworth), (P.\~Houston) and (R.\~Rannacher). \item Numerical techniques to handle strong and weak discontinuities predicted by the underlying boundary value problem, cf. (P.\~Steinmann). Cracks and material failure need to be simulated in accordance with the physical experiment, cf. (P.\~Hansbo), (U.\~Hoppe), (K.\~Weinberg). \item For a range of phenomena in structural mechanics the microscale of the material needs to be taken into account to obtain accurate numerical solutions, cf. (T.\~Arbogast). \end{itemize} The workshop aimed at bridging the gap between the computational engineering community and applied mathematicians and in consequence to unify the scientific language and foster later collaboration. Nonlinear mixed schemes were of particular concern for problems in elasticity and plasticity, but electromagnetics and mathematically related topics were also included. Mixed finite element methods for elliptic problems are based on a variational description in saddle-point form. Side conditions such as divergence free velocity fields in incompressible fluid dynamics are usually treated in this framework. The appearance of \`soft' side conditions is typical for structural mechanics as is the case with nearly incompressible materials or plates and shells with small thickness parameters. We also mention materials which almost satisfy the Kirchhoff condition, i.e. problems with a high but finite shear stiffness. In such cases, which are by no means \`soft' from the mathematical point of view, mixed methods lead to a more robust discretisation. The arising stability conditions and computational techniques cannot be understood fully by intuitive mechanical principles; however, from the mathematician's point of view their reasoning is natural, clear and insightful. Mixed and non-standard finite element methods gain increasing prominence in the prevention of locking phenomena. We highlight a topic which is currently actively investigated: the development of stable and efficient plate and shell elements with regard to shear locking, which is more intricate than volume locking. Here it is important to understand how techniques based on heuristic ideas are consistent with more modern mathematical methods. Availability of fast solvers is decisive for the competitiveness of numerical techniques. For a variety of applications, multigrid methods are crucial for the efficiency of the implementation. Methods have been proposed which do not appear plausible if one wants to deduce the algorithms directly from the physical model. The advanced methods depend on rigorous error estimators in order to guarantee that the numerical solutions represent the exact solutions of the physical model.

Superdiffusive Behaviour of a Passive Ornstein–Uhlenbeck Tracer in a Turbulent Shear Flow

In this paper, we are interested in the study of the diffusion of a passive particle with positive mass by a divergence free velocity field. We consider here the very simple turbulent shear flow case, in which we will prove the superdiffusive behaviour of the motion for large enough values of the energy spectrum of the velocity field. For small values, the proof of the diffusive behaviour of the model is also new, and it is shown that this diffusion is strictly greater than the one obtained with a non-massive particle. One interesting point to insist on is that we are able to obtain explicit hydrodynamic equations without even having the stationary measure of the studied processes

A conservative level set method for two phase flow

A conservative method of level set type for moving interfaces in divergence free velocity fields is presented. The interface is represented implicitly by the 0.5 level set of a function Φ being a smeared out Heaviside function, i.e., a function being zero on one side of the interface and one on the other. In a transition layer of finite, constant thickness Φ goes smoothly from zero to one. The interface is moved implicitly by the advection of Φ, which is split into two steps. First Φ is advected using a standard numerical method. Then an intermediate step is performed to make sure that the smooth profile of Φ and the thickness of the transition layer is preserved. Both these steps are performed using conservative second order approximations and thus conserving ∫ Φ. In this way good conservation of the area bounded by the 0.5 contour of Φ is obtained. Numerical tests shows up to second order accuracy and very good conservation of the area bounded by the interface. The method was also coupled to a Navier–Stokes solver for incompressible two phase flow with surface tension. Results with and without topological changes are presented.

On homogenization of nonlinear hyperbolic equations

In this paper we study homogenizationof nonlinear hyperbolic equations. The weak limit of thesolutions is investigatedby approximating the flux functions with piecewiselinear functions. We study mostly Riemann problemsfor layered velocity fields as well as forthe heterogeneous divergence free velocity fields.

Numerical solution of three-dimensional velocity-vorticity Navier-Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false-transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence-free velocity field. The vorticity transport equations in conservative form are solved using the second-order accurate Adams–Bashforth central difference scheme in order to assure divergence-free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri-diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri-diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three-dimensional problems, which otherwise requires enormous computational effort. For three-dimensional-driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100⩽Re⩽2000. Copyright © 2004 John Wiley & Sons, Ltd.

Trajectory structures and transport

The special problem of transport in two-dimensional divergence-free stochastic velocity fields is studied by developing a statistical approach, the nested subensemble method. The nonlinear process of trapping determined by such fields generates trajectory structures whose statistical characteristics are determined. These structures strongly influence the transport.

The repair paradigm and application to conservation laws

Repair is a conservative, post-processing procedure to be used in numerical methods for hyperbolic conservation laws in order to preserve certain qualitative characteristics of the numerical solution, such as positivity of density and internal energy, by means of redistribution of conserved quantities such as mass, momentum and total energy among the cells of the mesh. In this paper we describe the repair paradigm and prove several theorems which form a theoretical foundation for the repair procedures. We consider two applications of repair and present corresponding numerical results. The first application deals with improving properties of the remapping (conservative interpolation) stage of arbitrary Lagrangian–Eulerian (ALE) methods for the gas dynamics equations, in which the solution is conservatively transferred from one mesh to another. One requirement for remapping is that the interpolated density and internal energy on the new mesh have to stay positive. Another desirable property is that the remapping procedure should not create new extrema for the velocity field. For various reasons these properties may not be satisfied, especially for high-order methods. Repair plays a supplemental role by bringing gas dynamics quantities into physically justified bounds. Another application of repair is to improve the quality of numerical methods for advection of some scalar tracer field with prescribed divergence-free velocity field, in which case the advection equation can be written as a conservation law, and therefore the total amount of tracer is conserved. We show how the repair procedure allows us to reduce oscillations in a numerical solution obtained by a formally high-order method. Repair offers an alternative to more classical methods of reducing oscillations and maintaining positivity.