General Two-dimensional Flow Research Articles

EPIC: The Elliptical Parcel-In-Cell method

We present a novel approach to simulating general two-dimensional flows, which could also be applied to other areas of continuum mechanics. The approach generalises the Particle-In-Cell (PIC) method, originally used to model two-dimensional hydrodynamics, by representing fluid elements by elliptical parcels. The rotation and deformation of these parcels are calculated, and parcels split beyond a critical aspect ratio. Conversely, small parcels are eliminated by merging them with larger ones. The elliptical parcels well represent the flow deformation and have excellent conservation properties. In contrast to earlier work that combined PIC with elliptical parcels that split and merge, a vorticity-based framework is used, and accurate integration over ellipses is performed efficiently by two-point Gaussian quadrature. The small-scale mixing associated with parcel splitting and merging is shown to be strongly convergent with grid resolution. The robustness, versatility, accuracy and efficiency of the new Elliptical Parcel-In-Cell (EPIC) method is demonstrated for a variety of standard test cases, and compared with a standard pseudo-spectral method. The results indicate that EPIC is a promising, Lagrangian-based alternative to grid-based methods.

The Tensor Diffusion approach for simulating viscoelastic fluids

In this paper, the novel Tensor Diffusion approach for the numerical simulation of viscoelastic fluids is introduced based on the idea, that the extra-stress tensor in the momentum equation of the flow model can be replaced by the product of the strain-rate tensor and a (nonsymmetric) tensor-valued viscosity. As potential advantage (which can be demonstrated for fully developed channel, resp., pipe flows), this approach offers the possibility to reduce the full nonlinear viscoelastic three-field model to a generalized Tensor Stokes problem, avoiding the need of considering a separate stress tensor in the solution process. Moreover, an (artificial) diffusive operator is introduced into the momentum equation in the case of “no solvent” viscoelastic fluids, which is related to the physics of the problem leading to an improved numerical behaviour w.r.t. approximation and convergence properties of the iterative solvers. After validating the Tensor Diffusion approach for fully developed channel flow configurations, it is evaluated in the context of more general two-dimensional flow settings of benchmarking character, too, taking into account a symmetrized formulation. Numerical simulations for several prototypical flow configurations illustrate the mathematical and numerical properties of this new approach and can be viewed as proof-of-concept for further development.

Numerical study of the 3D-shape of a drop immersed in a fluid under an elongational flow with vorticity

This work focuses on a three-dimensional analysis of the deformation of a drop — immersed in a Newtonian fluid— generated by a 2D elongational flow with vorticity. The study of steady-state deformations of the cross-section of the drop shows a prevalent non-circular shape. The axisymmetric idealization of the ellipsoid is not observed nor the linear dependency between capillary number and deformation of the drop, as Taylor and Cox theory predicted. Our numerical results are consistent with experiments and other numerical simulations. However, in the latter cases, measurements of the cross section of the drop are few while a limited class of flows is applied. In this work, deformations induced by general two-dimensional flows upon the 3D drop shape are presented with special emphasis about the length scale along the third axis —perpendicular to the plane of the applied flow field.

Shear-induced particle migration: Predictions from experimental evaluation of the particle stress tensor

This paper addresses the modeling of the phenomenon of particle migration in the flow of monodispersed non-colloidal suspensions at neglected inertia using the Suspension Balance Model (SBM). The SBM describes the migration flux of particles as the divergence of the particle stress tensor. It is selected in this work because of its parameters that can be measured experimentally and its capability to quantify well the shear-induced migration phenomenon. A recent experiment [10,11] reported measurements of the different parameters in the SBM, which are used in this work to study their effects on the prediction of the particle migration phenomenon. For that purpose, a two-dimensional solver capable of solving the set of conservation equations of the SBM using the finite volume method is developed within the “OpenFOAM®” CFD toolbox [34]. The code is validated by simulating the suspension flows in a channel of rectangular cross-section, and in a wide gap Couette cell. Solutions are generated using the newly measured SBM parameters, and results are compared to similar ones obtained using the old SBM parameters. It is found that the new measured parameters have no significant influence on prediction of particle migration as compared to those proposed in the literature. Finally, the SBM is extended to general two-dimensional flows through a frame-invariant formulation that takes into account the local kinematics of the suspension including buoyancy effects. The frame-invariant model is applied to the resuspension and mixing of a monodispersed suspension in a horizontal Couette cell. The predicted results are found to be in good agreement with experimental measurements.

Texture optimization for example-based synthesis

We present a novel technique for texture synthesis using optimization. We define a Markov Random Field (MRF)-based similarity metric for measuring the quality of synthesized texture with respect to a given input sample. This allows us to formulate the synthesis problem as minimization of an energy function, which is optimized using an Expectation Maximization (EM)-like algorithm. In contrast to most example-based techniques that do region-growing, ours is a joint optimization approach that progressively refines the entire texture. Additionally, our approach is ideally suited to allow for controllable synthesis of textures. Specifically, we demonstrate controllability by animating image textures using flow fields. We allow for general two-dimensional flow fields that may dynamically change over time. Applications of this technique include dynamic texturing of fluid animations and texture-based flow visualization.

Three-dimensional centrifugal-type instabilities of two-dimensional flows in rotating systems

This paper deals with the stability of incompressible inviscid planar basic flows in a rotating frame. We give a sufficient condition for such flows to undergo three-dimensional shortwave centrifugal-type instabilities. This criterion reduces to the Bradshaw–Richardson (1969) or Pedley (1969) criterion in the specific case of parallel shear flows subject to rotation, to Rayleigh’s centrifugal criterion (1916) in the case of axisymmetric vortices in inertial frames, to the Kloosterziel and van Heijst (1991) criterion in the case of axisymmetric vortices subject to rotation and to Bayly’s criterion (1988) in the case of general two-dimensional flows in inertial frames. The criterion states that a steady 2D basic flow subject to rotation Ω is unstable if there exists a streamline for which at each point 2(V/R+Ω)(W+2Ω)<0 where W is the vorticity of the streamline, ℛ is the local algebraic radius of curvature of the streamline and V is the local norm of the velocity. If this condition is satisfied then the flow is unstable according to the geometrical optics method introduced by Lifschitz and Hameiri (1991), which consists in following wave packets along the flow trajectories using a Wentzel–Kramers–Brillouin formalism. When the streamlines are closed, it is further shown that a localized unstable normal mode can be constructed in the vicinity of a streamline. As an application, this new criterion is used to study the centrifugal-type instabilities in the Stuart vortices, which is a family of exact solutions describing a row of periodic co-rotating eddies. For each solution of that family and for each rotation parameter f=2Ω, we give the unstable streamline interval, according to the criterion of instability. This criterion gives only a sufficient condition of centrifugal instability. The equations of the geometrical optics method are therefore numerically solved to obtain the true centrifugally unstable streamline intervals. It turns out that our criterion gives excellent results for highly concentrated vortices, i.e., the two approaches yield the same unstable streamline intervals. In less concentrated vortices, some streamlines undergo centrifugal instability although our criterion is not fulfilled. From these numerical results, another criterion of centrifugal instability for a flow with closed streamlines is conjectured which reduces to the change of sign of the absolute vorticity W+2Ω somewhere in the flow.

Magnetohydrodynamic equilibria of a cylindrical plasma with poloidal mass flow and arbitrary cross sectional shape

The equilibrium of a cylindrical plasma with purely poloidal mass flow and cross section of arbitrary shape is investigated within the framework of the ideal MHD theory. For the system under consideration it is shown that only incompressible flows are possible and, consequently, the general two-dimensional flow equilibrium equations reduce to a single second-order quasilinear partial differential equation for the poloidal magnetic flux function , in which four profile functionals of appear. Apart from a singularity occurring when the modulus of Mach number associated with the Alfvén velocity for the poloidal magnetic field is unity, this equation is always elliptic and permits the construction of several classes of analytic solutions. Specific exact equilibria for a plasma confined within a perfectly conducting circular cylindrical boundary and having (i) a flat current density and (ii) a peaked current density are obtained and studied.

Numerical simulation of particle migration in concentrated suspensions by a finite volume method

This paper is concerned with the numerical modelling of particle migration in concentrated suspensions by the finite volume method. The constitutive equation for the particle flux, originally proposed by Phillips et al., Phys. Fluids A, 4 (1992) 30–40, in one dimension, is implemented in general two-dimensional flows with arbitrary geometry and boundary conditions. The numerical implementation is benchmarked against the exact solution in circular Couette flow, including the transient case where a closed-form solution is not available. The numerical predictions in an eccentric circular geometry, where the inner cylinder is placed off the axis of the outer cylinder, are given, showing that particles migrate towards the outer cylinder, but at different rates depending on their azimuthal positions.

Strong flows of dilute suspensions of microstructure

We consider dilute suspensions that have a microstructure that may be characterized by an axial state vector. Examples include axisymmetric particles, line elements of the fluid itself, or, as an approximation, droplets of fluid or polymer molecules. Past studies, in which sufficient conditions for stretch or coherent orientation of the microstructure are obtained for steady flows with homogeneous velocity gradient tensors are shown not to apply to the general situation. Instead, a careful analysis of the microdynamical equations reveals that stretching and orientation of the microstructure by the flow must be analyzed over a time interval. Using techniques from the theory of dynamical systems, a quantitative measure is developed to determine orientations and/or stretched lengths of the microstructure, that are robust and attractive to nearby states. This leads to a strong flow criterion for unsteady flows with inhomogeneous velocity gradient tensors in which the effects of history dependence are apparent. A particular model system is treated in the case of general two-dimensional flow. The sensitivity of the results to changes in the modeling assumptions is investigated.

The classification of dilatant flow types

The kinematic vorticity number W has been used to classify the geometry of two-dimensional, non-dilatant flow as a function of flow vorticity. A similar kinematic dilatancy number A can be defined, which expresses flow geometry as a function of dilatancy rate. General two-dimensional flow geometry can be described in a simple, non-ambiguous way by just W and A. Mohr circles for flow can be used to illustrate this classification method.

Finite volume methods for two-dimensional incompressible flows with complex boundaries

The paper discusses possible ways of developing accurate and efficient finite volume methods for complex, incompressible flows. The issues of boundary-fitted grid generation, choice of coupled or uncoupled procedures, Cartesian or grid-aligned velocity components, staggered or non-staggered variable arrangement, discretization methods, solution algorithms and convergence acceleration by multigrid methods are addressed. An example of a finite volume method for general two-dimensional flows is described in some detail and applications of this method to a variety of flows are presented.

The behaviour of macromolecules in the flow through a sudden planar contraction

Analytic steady-state results for FENE-P model macromolecules, in the nearly coiled-up and nearly stretched state respectively, in general two-dimensional flow fields are derived. These results are utilized in the flow through a sudden planar contraction. Special emphasis is devoted to the structure tensor 〈R R〉, which furnishes, among other things, the mean square extension and the average orientation of the macromolecules.

General two-dimensional magnetohydrodynamic equilibria with mass flow

A set of reduced ideal MHD equations is derived to investigate equilibria of plasmas with mass flow in general two-dimensional geometry. These equations provide a means of investigating the effects of flow on self-consistent equilibria in a number of new two-dimensional configurations, such as helically symmetric configurations with helical axis, which are relevant to stellarators, as well as axisymmetric configurations. It is found that, as in the axisymmetric case, general two-dimensional flow equilibria are governed by a second-order quasilinear partial differential equation for a magnetic flux function, which is coupled to a Bernoulli-type equation for the density. The equation for the magnetic flux function becomes hyperbolic at certain critical flow speeds which follow from its characteristic equation. When the equation is hyperbolic, shock phenomena may exist. As a particular example, unidirectional flow along the lines of symmetry is considered. In this case, the equation mentioned above is always elliptic. An exact solution for the case of helically symmetric unidirectional flow is found and studied to determine flow effects on the magnetic topology.

The effects of conformation-dependent friction and internal viscosity on the dynamics of the nonlinear dumbbell model for a dilute polymer solution

The effects of a nonlinear spring, variable hydrodynamic friction and internal viscosity are examined in detail for the dumbbell model of dilute polymer solutions in a general two-dimensional flow. Both steady and transient startup flows are investigated and the response of the bulk stress is calculated. The existence of a hysteresis in the steady end-to-end length of the dumbbell, which arises from the variable friction factor, is found to depend strongly on both the flow type (i.e. the ratio of vorticity to the rate of strain) and the moleculecular weight. The influence of internal viscosity is examined in detail using two methods of solution for the problems of transient and steady two-dimensional flows. The first method uses a perturbation expansion valid for small values of the internal viscosity and leads largely to analytical results. The second technique uses the pre-averaging approximation of Cerf and is capable of predicting the model response when the internal viscosity becomes large. The most striking result for large values of the internal viscosity and a linear spring dumbbell is the appearance of multiple steady states in the dumbbell length for flows between simple shear and pure straining, and the removal of the well known singularity in end-to-end length which otherwise appears in the linear dumbbell model at a critical value of the strain rate.

Finite element analysis of viscous, incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique. A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.

Streaming Birefringence of Rigid Macromolecules in General Two-Dimensional Laminar Flow

Quantitative expressions for the direction of the angle of isocline and for the amount of birefringence due to a dilute solution of rigid macromolecules in a general two-dimensional laminar flow are derived. If E is the principal strain rate and Λ0 is the angle between the streamline direction and the direction of the principal strain rate axis, the angle of isocline, measured from the first principal strain rate axis is χ=−(Esin2Λ0/6D){1−(E2/27D2)[sin22Λ0+(24b2/35)]+···],where b=[(a12—a22) / (a12+a22)] is a shape factor for an ellipsoid of revolution of semimajor axis a1 and semiminor axis a2 and D is the rotary diffusion constant for this ellipsoid. The amount of birefringence is Δn=(4π/15)(cGEb/nD) {1−(E2/18D2)[sin22Λ0+(6b2/35)]+···},where n is the mean index of refraction of the solution, c the volume concentration of the macromolecules, and G=g1—g2 is the optical anisotropy of the ellipsoids. It is seen that if the principal strain rate is not at 45° to the streamline at the point of observation, this will make itself felt in the position of the angle of isocline before it influences the amount of birefringence. Detailed expressions for the effect of polydispersity show that there is a simple relationship between the birefringence and angle of isocline measured in Couette flow and these quantities measured in general two-dimensional flow only if (a) Λ0=45° in the general flow or (b) the birefringence and angle of isocline values are linear with strain rate.