Quadratic Closure Research Articles

Numerical modeling and simulation of water hammer phenomena using the MacCormack method

ABSTRACT Water hammer refers to pressure fluctuations caused by changes in fluid speed in hydraulic systems. This research proposes a model using the MacCormack method to study and control water hammer, specifically focusing on managing valve closure to reduce shock wave pressure during transient flow. Numerical simulations compare linear and quadratic closure laws for both rapid and gradual valve closure, with the goal of identifying the optimal laws. The findings show that with faster closure times, the overpressure at the valve section decreases linearly in the second quarter of the return period (t4). The application of the quadratic law results in reduced pressures at the valve compared to the linear law, with a maximum pressure difference of 0.05 bar between the highest and lowest values. When searching for the optimal valve closure law to mitigate overpressure, it is found that the exponent ‘m’ falls within the range of 1.117–1.599. As the slow closing time increases gradually, the range of variation for ‘m’ decreases. Furthermore, in the case of underpressure prevention, the exponent ‘m’ ranges from 0.95 to 1.419, and the range of ‘m’ remains relatively constant with the increase in closing time.

On the influence of the fourth order orientation tensor on the dynamics of the second order one

Abstract The consequences of introducing the fourth order orientation tensor as an independent variable in addition to the second order one are investigated. In the first part consequences of the Second Law of Thermodynamics are exploited. The cases with the second order alignment tensor in the state space on one hand and with the second and fourth order alignment tensors on the other hand are analogous. In the latter case differential equations for the second and fourth order tensors result from linear force-flux relations with a coupling arising due to coupling terms in the free energy. In the second part the differential equations for the second order orientation tensor or the second and fourth order orientation tensors, respectively are given explicitly in the special case of a rotation symmetric orientation distribution. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. The fourth order parameter is eliminated later. The resulting equation for the second order parameter is a Duffing equation with a behavior of solutions completely different from the solutions of the Riccati equation.

Asymptotic fiber orientation states of the quadratically closed Folgar–Tucker equation and a subsequent closure improvement

Anisotropic fiber-reinforced composites are used in lightweight construction, which is of great industrial relevance. During mold filling of fiber suspensions, the microstructural evolution of the local fiber arrangement and orientation distribution is determined by the local velocity gradient. Based on the Folgar–Tucker equation, which describes the evolution of the second-order fiber orientation tensor in terms of the velocity gradient, the present study addresses selected states of deformation rates that can locally occur in complex flow fields. For such homogeneous flows, exact solutions for the asymptotic fiber orientation states are derived and discussed based on the quadratic closure. In contrast to the existing literature, the derived exact solutions take into account the fiber-fiber interaction. The analysis of the asymptotic solutions relying upon the common quadratic closure shows disadvantages with respect to the predicted material symmetry, namely, the anisotropy is overestimated for strong fiber-fiber interaction. This motivates us to suggest a novel normalized fully symmetric quadratic closure. Two versions of this new closure are derived regarding the prediction of anisotropic properties and the fiber orientation evolution. The fiber orientation states determined with the new closure approach show an improved prediction of anisotropy in both effective viscous and elastic composite behaviors. In addition, the symmetrized quadratic closure has a simple structure that reduces the effort in numerical implementation compared to more elaborated closure schemes.

Numerical simulation and modeling of the die swell for fiber suspension flows

A numerical model is developed to examine the die swell for fiber suspensions, which occurs in 3D printing extrusion processes. More specifically, it focuses on the fiber orientation distribution in a Newtonian suspending fluid and its effects on the shape of the free surface. In a first step, the flow in a 2D axisymmetric capillary die is explored in order to validate the numerical implementation for the fiber orientation model. It is found that the coupling effect flattens the velocity profile but has a small impact on the fiber orientation distribution in the suspension flow. In a second step, the modeling of die swell is considered and result from the literature for the final swelling height is obtained for an uncoupled flow. For fully coupled flows, different values for the coupling parameter and the fiber-fiber interaction coefficient have been tested in order to observe their effects on the free surface shape and on the final fiber orientation distribution. The increase of the fiber coupling parameter increases the die swell ratio when fibers are randomly distributed at the flow inlet, but the opposite behavior, i.e., a decrease of the swelling ratio, is noticed when a perfect fiber alignment condition is imposed at the entrance for low fiber interaction coefficients. The Tanner model, initially developed for viscoelastic fluids, is updated to estimate extrudate swell for fibers suspended in a Newtonian fluid in terms of the coupling parameter and the interaction coefficient. Although this analytical approach used a quadratic closure approximation, it gives qualitative results when compared with the numerical simulations.

CFD simulation of swirling flow induced by twist vanes in a rod bundle

Abstract Swirling flow induced by the twist vanes can enhance the heat transfer in the rod-bundle fuel assembly. In order to investigate accuracy of the k-e models in predicting the swirling flow in a rod bundle subchannel, the CFD simulations with the standard and realizable k-e model are benchmarked with In et al.’s experiment. The nonlinear closure models are utilized with the standard k-e model to account for the curved and rotational flow, while the curvature correction model is employed in the realizable k-e model. Comparison with the experiment indicates that the CFD simulations predict the relatively large core region of swirling flow. Utilization of the nonlinear closure models in the standard k-e model leads to the larger initial angular momentum. The curvature correction in the realizable k-e model reduces the initial angular momentum, but it does not affect the decay rate, which indicates that the curvature correction is effective near the twist vanes where the flow curvature is strong. The decay rate of swirling flow is generally overestimated by all the models. The cross-flow velocity profile in the gap is affected by prediction of swirling flow separation from rod surface. The cubic and quadratic closure model can improve turbulence modelling in the core of swirling flow, especially near the spacer, where the linear closure models significantly underestimate the production of turbulence. However, in the further downstream all the models underestimate the turbulence intensity in the swirling core, as a consequence of low-frequency large-scale flow structure in the core region. Underestimation of turbulence in the annular and wall region will result in the underestimation of wall heat transfer and inter-subchannel mixing.

Persistence, Turing Instability and Hopf Bifurcation in a Diffusive Plankton System with Delay and Quadratic Closure

A reaction–diffusion plankton system with delay and quadratic closure term is investigated to study the interactions between phytoplankton and zooplankton. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system. Our conclusions show that diffusion can induce Turing instability, delay can influence the stability of the positive equilibrium and induce Hopf bifurcations to occur. The computational formulas which determine the properties of bifurcating periodic solutions are given by calculating the normal form on the center manifold, and some numerical simulations are carried out for illustrating the theoretical results.

Numerical Simulation of Injection Molding using OpenFOAM

AbstractWe show how to use and extend OpenFOAM's incompressible two‐phase flow solvers for the simulation of injection molding with short fiber reinforced thermoplastics in a laminar flow regime. Second order fiber orientation tensors are computed using the Folgar‐Tucker equation (FTE) with quadratic closure. The FTE is coupled to the viscosity‐term of the Navier‐Stokes equations for the non‐Newtonian flow in a segregated manner. Phase dependent boundary conditions are implemented to simulate wall heat transfer, stickiness of the melt to the wall and to prevent air‐traps close to the wall. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Simulating microstructure evolution during passive mixing

The prediction of microstructure evolution during passive mixing is of major interest in order to qualify and quantify mixing devices as well as to predict the final morphology of the resulting blend. Direct numerical simulation fails because of the different characteristic lengths of the microstructure and the process itself. Micro-macro approaches could be a valuable alternative but the computational cost remains tremendous. For this reason many authors proposed the introduction of some microstructural variables able to qualify and quantify the mixing process at a mesoscale level. Some proposals considered only the effects induced by the flow kinematics, other introduced also the effects of shape relaxation due to the surface tension and coalescence. The most advanced integrate also the break-up process. However, the derivation of the evolution equations governing the evolution of such microstructural variables needs the introduction of some closure relations whose impact on the computed solution should be evaluated before applying it for simulating complex mixing flows. In this work we consider the Lee and Park’s model that considers the flow kinematics, the surface tension, the coalescence and the break-up mechanisms in the evolution of the area tensor. The accuracy of both a quadratic closure and an orthotropic relations will be analyzed in the first part of this work, and then the resulting closed model by using a quadratic closure will be used for simulating complex mixing flows.

Numerical simulation of flow-induced fiber orientation using normalization of second moment

A normalization scheme for the numerical solution of the moment approximation equation in fiber suspension flows is presented. Here, normalization refers to rescaling the trace of the second moment tensor to unity at each time step. The equivalence between the normalization scheme and the quadratic closure model is analytically proved. The performance of the scheme is investigated in simple shear flow with respect to the quadratic and hybrid closures, and a stochastic Monte-Carlo simulator that provides exact solution. The proposed scheme is a computationally efficient alternative to the quadratic closure: it performs equally well and is more efficient regarding computational time.

Modeling fiber interactions in semiconcentrated fiber suspensions

A set of rheological equations is developed for semiconcentrated suspensions of rigid fibers in a Newtonian fluid taking into account hydrodynamic and fiber-fiber interactions. The force generated by the fiber interactions is modeled using a linear hydrodynamic friction coefficient proportional to the relative velocity at the contact point, and weighted by the probability for contacts to occur. The equation of evolution of the second-order orientation tensor, containing advection and diffusion terms due to fiber interactions, is derived to predict fiber orientation under flow. The well known fourth-order orientation tensor, related to the hydrodynamic contribution, and a newly proposed fourth-order interaction tensor are used to evaluate the total stress in the composite. A linear and a quadratic closure approximation are proposed to describe the fourth-order interaction tensor. Results are presented using the quadratic form, which is found to be more accurate than the linear one. The model is shown to describe well simple shear data of suspensions of glass fibers in a Newtonian polybutene. Moreover, fiber orientation and the average number of contacts per fiber are predicted. The newly proposed interaction coefficient varies with fiber orientation, which appears to be realistic.

Holonomy Lie Algebras and the LCS Formula for Subarrangements of An

If X is the complement of a hypersurface in 8 n , then Kohno showed in [11] that the nilpotent completion of π 1 (X) is isomorphic to the nilpotent completion of the holonomy Lie algebra of X. When X is the complement of a hyperplane arrangement , the ranks ϕ k of the lower central series quotients of π 1 (X) are known in only two very special cases: if X is hypersolvable (in which case the quadratic closure of the cohomology ring is Koszul), or if the holonomy Lie algebra decomposes in degree 3 as a direct product of local components. In this paper, we use the holonomy Lie algebra to obtain a formula for ϕ k when is a subarrangement of A n . This extends Kohno's result [12] for braid arrangements, and provides the first instance of an LCS formula for arrangements that are not decomposable or hypersolvable.

A unified model for polystyrene–nanorod and polystyrene–nanoplatelet melt composites

We present a unified constitutive model capable of predicting the steady shear rheology of polystyrene (PS)–nanoparticle melt composites, where particles can be rods, platelets, or any geometry in between, as validated against experimental measurements. The composite model incorporates the rheological properties of the polymer matrix, the aspect ratio and characteristic length scale of the nanoparticles, the orientation of the nanoparticles, hydrodynamic particle–particle interactions, the interaction between the nanoparticles and the polymer, and flow conditions of melt processing. We demonstrate that our constitutive model predicts both the steady rheology of PS–carbon nanofiber composites and the steady rheology of PS–nanoclay composites. Along with presenting the model and validating it against experimental measurements, we evaluate three different closure approximations, an important constitutive assumption in a kinetic theory model, for both polymer–nanoparticle systems. Both composite systems are most accurately modeled with a quadratic closure approximation.

Limits of cubes

The celebrated Urysohn space is the completion of a countable universal homogeneous metric space which can itself be built as a direct limit of finite metric spaces. It is our purpose in this paper to give another example of a space constructed in this way, where the finite spaces are scaled cubes. The resulting countable space provides a context for a direct limit of finite symmetric groups with strictly diagonal embeddings, acting naturally on a module which additively is the “Nim field” (the quadratic closure of the field of order 2). Its completion is familiar in another guise: it is the set of Lebesgue-measurable subsets of the unit interval modulo null sets. We describe the isometry groups of these spaces and some interesting subgroups, and give some generalisations and speculations.

Numerical simulation of spin coating processes with carbon nanotubes suspensions

In this paper we address the problem of simulating the behaviour of Carbon nanotubes (CNTs) suspensions in spin coating processes. Spin coating is a procedure used to apply uniform thin films to flat substrates by means of a high rotating velocity and the subsequent centrifugal force. In this work we assume a suspension of chemically treated CNTs, such that they do not aggregate. In such a suspension, CNTs can be treated as rigid fibres whose orientation is dictated by the flow of the solvent. In order to treat the associated free-surface problem and to avoid the numerical problems associated to their FE solution, we have implemented a Natural Element strategy in an updated Lagrangian framework. The resulting model is thus composed by a description of the micro scale, related to the orientation of the carbon nanotubes and the assumption of a quadratic closure relation, together with the natural element approximation for the Navier-Stokes problem governing the macroscopic suspension kinematics.

THE EFFECTS OF CLOSURE MODEL OF FIBER ORIENTATION TENSOR ON THE INSTABILITY OF FIBER SUSPENSIONS IN THE TAYLOR–COUETTE FLOW

An analysis for the linear stability of fiber suspensions in the Taylor–Couette flow is conducted. The orientation state of fibers is determined using the linear, quadratic and composite HL-I closure model and the model of the fiber contribution to the total stress is established based on the slender-body theory. The linear stability equation is derived and solved with the Chebyshev spectral method. The results show that the suppression effect is most predominant for the composite HL-I model, while least predominant for the linear closure model. The longer fibers and suspensions with high volume fraction have a more predominant influence on suppressing the instability. The contribution of the suspending fluid to the change rate of the disturbance energy dominates the energy dissipation. While the contribution of the fiber shear stress increases monotonously with increasing fiber aspect ratio for the three closure models, the contribution based on the composite HL-I closure model is the largest, and that based on the linear closure model is the smallest. The effect of fiber aspect ratio on the change rate of the disturbance energy caused by the fiber shear stress is more predominant than that of the fiber volume fraction. The positive energy caused by the fiber shear stress consumes the gross energy of the fluid system and leads to a suppression of the flow instability.

Derivation of Frank-Ericksen elastic coefficients for polydomain nematics from mean-field molecular theory for anisotropic particles

The complete free energy density, including all eight Frank-Ericksen elastic coefficients and all anisotropic Ericksen-Leslie viscosities of nematic and discotic polydomain nematic liquid crystals are derived from the kinetic model of a spatially inhomogeneous system of uniaxial liquid crystal molecules with given shape. The authors take into account the known anisotropy of the translational diffusion tensor and its dependence on shape, rotational diffusion, and a macroscopic flow field for elongated particles (including disks). In this manuscript they release all of the previously made assumptions about closure relationships or the interrelationship between Frank elastic coefficients (such as a simple quadratic closure, or the one-constant approximation) in order to derive results which not only generalize or improve earlier results, but also apply to more general cases, and for arbitrary forms of the mean-field potential in terms of the scalar order parameter (or temperature). The kinetic model is shown to confirm all proposed inequalities between Frank-Ericksen-Leslie coefficients, i.e., satisfies the main result of the macroscopic approaches. They resolve quantitatively the effect of molecular shape, order parameters, and mean-field strength and form of the mean-field potential on all results, compare with experimental findings, theoretical predictions, and discuss some implications for various special cases of the general result derived in this work.

On the homotopy Lie algebra of an arrangement

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincar\'e polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras.

Validation of an extended hydrodynamic model for a submicron npn bipolar junction transistor

Transport phenomena in a submicron npn silicon bipolar junction transistor are described by using an 8-moment model for the electrons, combined with a solution of the drift–diffusion model for the holes. The validity of the constitutive equations for the fluxes and the production terms, obtained by means of the maximum entropy principle, and the hyperbolicity conditions are checked with direct simulation Monte Carlo. We verify numerically that the quadratic closure is more accurate with respect to the zero-order one, but some irregularities can appear in the solution due to the loss of hyperbolicity in some regions of the device.

New constitutive equations derived from a kinetic model for melts and concentrated solutions of linear polymers

In this paper, new constitutive equations for linear entangled polymer solutions and melts are derived from a recently proposed kinetic model (Fang et al. 2004) by using five closure approximations available in the literature. The simplest closure approximation considered is that due to Peterlin (1966). In this case, a mean-field-type Fokker-Planck equation underlying the evolution equation for an equilibrium averaged polymer segment orientation tensor is shown to be consistent with the fluctuation-dissipation theorem (Kubo et al. 1985). We compare the performance of the five new constitutive equations in their capacity to faithfully reproduce the predictions of the modified encapsulated FENE dumbbell model of Fang et al. (2004) for a number of shear and extensional flows. Comparisons are also made with the experimental data of Kahvand (1995) and Bhattacharjee et al. (2002, 2003). In the case of the Hinch-Leal and Bingham closures (Hinch and Leal 1976; Chaubal and Leal 1998) a combination with the quadratic closure of Doi (1981) is found to be necessary for stability in fast flows. The Hinch-Leal closure approximation, modified in this way, is found to outperform the other closures and its mathematical description is considerably simpler than that of the Bingham closure.

Numerical Analysis of the Rheology of Polymeric Liquid Crystals (3rd Report, Evaluation of a Second Order Tensor Type Constitutive Equation)

We have evaluated numerically a second order tensor type constitutive equation which includes both a molecular shape factor and a quadratic closure approximation, by means of comparing the results with the solutions obtained from the original Doi equation without closure approximations. The second order tensor type constitutive equation can simulate three types of director behaviors such as tumbling, wagging and aligning, by setting the molecular shape factor to be smaller than its inherent value based on the molecular aspect ratio. The shear stress and the first normal stress difference determined from the tensor type constitutive equation qualitatively agree with stresses obtained from the Doi equation, for low shear rate. When shear rate is high, however, even qualitative agreement is not obtained. Therefore, the constitutive equation is applicable to slow flows or situations where a velocity field is so simple that only an orientation field is a consideration.