Incompressibility Condition Research Articles

A new meshless approach in simulating free surface flows using continuous MLS shape functions and Voronoi diagram

PurposeThe purpose of this paper is to develop an efficient and accurate mesh-less method to simulate free flows with continuous deformation in boundary positions.Design/methodology/approachA two-step pressure projection method in a Lagrangian form is used to solve the governing equations of mass and momentum conservation. In the first step, velocity field is calculated in which incompressibility is not enforced. In the second step, a pressure Poisson equation is applied to satisfy incompressibility conditions. The numerical proposed method is used for spatial discretization of the governing equations. Three benchmark-free surface problems, namely, dam break, solitary wave propagation and evolution of an elliptical bubble with available experimental results and analytical solutions, are used to test the accuracy of the proposed method. The results prove the accuracy of the method in simulating free surface problems.FindingsThe Voronoi diagram instead of kernel function summation can be used to estimate the particle or nodal volume concept in particle-based (mesh-less) methods for function approximation. This idea probably works well especially for highly irregular node distributions.Originality/valueThe continuous moving least squares shape functions are applied for function approximation, and the Voronoi diagram concept is also used to estimate region influence of computational nodal points or particle volumes. Combinations of these two concepts and finite differences formulation for first derivatives gives an accurate numerical model for Laplacian operator in the proposed method.

A path-dependent necking instability analysis of the thin substrate composite plates considering nonlinear reinforced layer effects

In this paper, an analytical path-dependent instability model is proposed for the thin substrate composite plates, considering the nonlinear reinforced layer effects. The model is introduced by extending a modified maximum force criterion (MMFC) and the vertex criteria to predict diffuse and localized necking, respectively, for the thin substrate-supported plates. The MMFC considers the strain hardening on the diffuse necking as well as the loading conditions. The vertex criteria presented by St $$ \ddot{\mathrm{o}}\mathrm{re} $$ n and Rice are usually based on the J 2 deformation theory of classical plasticity, which explores the localized necking through the rate discontinuity assumption at the necking band. Both models will be combined with the strain path effect through a linear adoption of an equivalent strain. It will be investigated by applying a pre-strain in the major and minor directions. Moreover, a dependent to yield criterion (DYC) angle is used for prediction of the necking band angle in the vertex theory. Also, a nonlinear neo-Hookean reinforced layer is considered to make a sensible delay at plastic deformation. Since the reinforcing layers are the materials with high stretchability, the effects of material properties and thickness values will be studied for supporting metal layers. Also, it is assumed that the incompressibility condition remains for corresponded elastomer and metal layer strains. Finally, the quadratic Hill criterion is used to investigate the anisotropy effect. The model is verified by experimental and other theoretical results.

Numerical modeling of flood waves in a bumpy channel with the different boundary conditions

In this paper, the Incompressible Smoothed Particle Hydrodynamics (ISPH) method is presented to simulate flood waves in uneven beds. The SPH method is a mesh free particle modeling approach that is capable of tracking the large deformation of free surfaces in an easy and accurate manner. Wave breaking is one of the phenomena that its free surface is complicated. Therefore, ISPH method is robust tool for the modeling of this kind of free surface. The basic equations are the incompressible mass conservation and Navier–Stokes equations that are solved using a two-step fractional method. In the first step, these equations are solved to compute velocity components by omitting the pressure term and in the absence of incompressible condition. In the second step, the continuity constraint is satisfied and the Poisson equation is solved to calculate pressure terms. In the present model, a new technique is applied to allocate density of the particles for the calculations. By employing this technique, ISPH method is stabled. The validation by comparison with laboratory data is conducted for bumpy channel with various boundary conditions. The numerical results showed good agreement with available experimental data. Also relative error is calculated for two numerical cases.

Application of CUPID for subchannel-scale thermal–hydraulic analysis of pressurized water reactor core under single-phase conditions

Abstract There have been recent efforts to establish methods for high-fidelity and multi-physics simulation with coupled thermal–hydraulic (T/H) and neutronics codes for the entire core of a light water reactor under accident conditions. Considering the computing power necessary for a pin-by-pin analysis of the entire core, subchannel-scale T/H analysis is considered appropriate to achieve acceptable accuracy in an optimal computational time. In the present study, the applicability of in-house code CUPID of the Korea Atomic Energy Research Institute was extended to the subchannel-scale T/H analysis. CUPID is a component-scale T/H analysis code, which uses three-dimensional two-fluid models with various closure models and incorporates a highly parallelized numerical solver. In this study, key models required for a subchannel-scale T/H analysis were implemented in CUPID. Afterward, the code was validated against four subchannel experiments under unheated and heated single-phase incompressible flow conditions. Thereafter, a subchannel-scale T/H analysis of the entire core for an Advanced Power Reactor 1400 reactor core was carried out. For the high-fidelity simulation, detailed geometrical features and individual rod power distributions were considered in this demonstration. In this study, CUPID shows its capability of reproducing key phenomena in a subchannel and dealing with the subchannel-scale whole core T/H analysis.

Gradient Dynamics and Entropy Production Maximization

AbstractWe compare two methods for modeling dissipative processes, namely gradient dynamics and entropy production maximization. Both methods require similar physical inputs–-how energy (or entropy) is stored and how it is dissipated. Gradient dynamics describes irreversible evolution by means of dissipation potential and entropy, it automatically satisfies Onsager reciprocal relations as well as their nonlinear generalization (Maxwell–Onsager relations), and it has statistical interpretation. Entropy production maximization is based on knowledge of free energy (or another thermodynamic potential) and entropy production. It also leads to the linear Onsager reciprocal relations and it has proven successful in thermodynamics of complex materials. Both methods are thermodynamically sound as they ensure approach to equilibrium, and we compare them and discuss their advantages and shortcomings. In particular, conditions under which the two approaches coincide and are capable of providing the same constitutive relations are identified. Besides, a commonly used but not often mentioned step in the entropy production maximization is pinpointed and the condition of incompressibility is incorporated into gradient dynamics.

An arbitrary Lagrangian‐Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem

SummaryAn arbitrary Lagrangian‐Eulerian framework, which combines the advantages of both Lagrangian and Eulerian methods, is presented to solve incompressible multiphase flow problems. The incompressible Navier‐Stokes equations are discretized using the side‐centered unstructured finite volume method, where the velocity vector components are defined at the midpoint of each cell face, while the pressure term is defined at element centroids. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. The surface tension term at the interface is treated as a force tangent to the interface and computed with several different approaches including the use of Legendre polynomials. In addition, the several different discretizations of interface kinematic boundary conditions are investigated. For the application of the interface kinematic boundary condition, a special attention is given to satisfy both local and global discrete geometric conservation law to conserve the total mass of both species at machine precision. The mesh vertices are deformed by solving the linear elasticity equations due to the normal displacement of interface. The resulting algebraic equations are solved in a fully coupled manner, and a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned subdomain is used for the resulting fully coupled system. The method is validated by simulating the classical benchmark problem of a single rising bubble in a viscous fluid due to buoyancy. The results of numerical simulations are found out to be in an excellent agreement with the earlier results in the literature. The mass of the bubble is conserved, and discontinuous pressure field is obtained to avoid errors due to the incompressibility condition in the vicinity of the interface, where the density and viscosity jumps occur.

Database Approach for Laminar-Turbulent Transition Prediction: Navier–Stokes Compatible Reformulation

This paper presents a transition prediction model that adds additional equations to the Reynolds-averaged Navier–Stokes set. It is based on the so-called -factor method, which is valid for natural transition on wings due to the hypothesis that it relies on (modal instabilities disturbing the boundary layer). It consists of an integration of the disturbance growth rates along the edge streamline. Herein, the disturbance growth rates are obtained via a database approach: ONERA—The French Aerospace Lab’s so-called parabolas method. Numerically, integration of the growth rates along the edge streamline is performed through additional transport equations, thus removing the need for any topology specification from the end user. Implementation caveats are illustrated on the ONERA-D airfoil in incompressible conditions. This method is then successfully applied for transition prediction on a two-dimensional industrial laminar airfoil at transonic conditions, with varying upstream conditions and angles of attack. Satisfactory agreements with both the experimental data and the conventional transition predictions (local, linear stability results) are obtained, which illustrate the accuracy of the predicted transition location.

A discussion on the mechanics of lipid membranes: Lagrange multipliers and a singular substrate

We present rigorous analysis of Lagrange multipliers arising in the variational computations of a membrane’s energy functional. Through the convergence analysis, it is found that the local Lagrange multiplier is sensitive to the properties of membranes. Unlike the local Lagrange multiplier, the bulk liquid incompressibility condition via the global Lagrangian field acts as an additional boundary input rather than a constraint, in the perspective of the surrounding membrane. With the enhanced understanding of Lagrange multiplier fields, a membrane–substrate interaction problem is considered where singularities are present at the cross section of the substrate. A viable analytical solution is obtained under the modified boundary conditions. In addition, a complete analysis of necessary boundary condition in the case of flat surfaces is presented within the prescription of superposed incremental deformations.

Estimates of the evolution of small perturbations in the radial spread (drain) of a viscous ring

This paper studies the evolution of small perturbations in the kinematic and dynamic characteristics of the radial flow of a flat ring filled with a homogeneous Newtonian fluid or an ideal incompressible fluid. When the flow rate is specified as a function of time, the main motion is completely determined by the incompressibility condition regardless of the properties of the medium. A biparabolic equation for the stream function with four homogeneous boundary conditions which simulate adhesion to the expanding (contracting) walls of the ring is derived. Upper bounds for the perturbation are obtained using the method of integral relations for quadratic functionals. The case of an exponential decay of initial perturbations is considered in a finite or infinite time interval. The admissibility of the inviscid limit in this problem is proved, and upper and lower bounds for this limit are estimated.

Theoretical Investigation on Particle Brownian Motion on Micro-air-bubble Characteristic in H2O Solvent

Micro-air-bubble has a high potential contribution in waste water, farming, and fishery treatment. In this research, submicron scale of micro-air-bubble was observed to determine its stability in H2O solvent. By increasing its stability, it can be used for several applications, such as bio-preservative for medical and food transport. The micro-air-bubble was assumed in spherical shape that in incompressible gas boundary condition. So, the random motion of particle (Brownian motion) can be solved by using Stokes-Einstein approximation. But, Hadamard and Rybczynski equation is promoted to solve for larger bubble (micro scale). While, the effect of physical properties (e.g. diffusion coefficient, density, and flow rate) have taken important role in its characteristics in water. According to the theoretical investigation that have been done, decreasing of bubble velocity indicates that the bubble dissolves away or shrinking to the surface. To obtain longevity bubble in pure water medium, it is recomended to apply some surfactant molecules (e.g. NaCl) in micro-air-bubble medium.

A new two-step projection method in an ISPH model for free surface flow computations

In this study, we propose a new two-step projection method in connection with an ISPH model. Compared to the traditional ISPH model that uses the future time step pressure to calculate the intermediate velocity, in the present study the pressure at both current time step and future time step is employed. In addition, the velocity and pressure obtained at the end step of the projection method are re-projected onto the updated particle locations using an interpolation method. While the use of current time step pressure provides a better energy conserving property of the numerical scheme, the interpolation smooths the velocity field and ensures a more stringent constraint for incompressibility condition. The accuracy of the model is further enhanced by implementing C2 consistency kernel approximation for variable and derivative calculation. The model is first calibrated by using the still water and liquid sloshing tests. It is then validated against a series of benchmark tests of linear wave and solitary wave propagation in constant water depth, followed by the case of nonlinear wave transformation over a submerged breakwater. The numerical results are compared to available theories, experimental data, and previously published simulation results. It is shown that the present model can achieve a much better energy conservation than the traditional ISPH model, even with the use of a much larger time step.

Magneto-Rayleigh–Taylor instability in compressible Z-pinch liner plasmas**Project supported by the National Natural Science Foundation of China (Grant Nos. 11475027, 11274051, 11105017, and 11275030) and the National Basic Research Program of China (Grants No. 2013CB834100).

In this paper, the characteristics of magneto-Rayleigh–Taylor (MRT) instability of liner plasmas in MagLIF is theoretically investigated. A three-region slab model, based on ideal MHD equations, is used to derive the dispersion relation of MRT instability. The effect of compressibility on the development of MRT instability is specially examined. It is shown that the growth rate of MRT instability in compressible condition is generally lower than that in incompressible condition in the presence of magnetic field. In the case of zero magnetic field, the growth rate in compressible assumption is approximately the same as that in incompressible assumption. Generally, MRT instability in (x, y) plane can be remarkably mitigated due to the presence of magnetic field especially for short-wavelength perturbations. Perturbations may be nearly completely mitigated when the magnetic field is increased to over 1000 T during liner implosions. The feedthrough of MRT instability in liner outer surface on inner surface is also discussed.

Creep strain rates analysis in cylinder under temperature gradient materials by using Seth’s theory

PurposeThe purpose of this paper is to present study of creep strain rates in a circular cylinder under temperature gradient materials by using Seth’s transition theory.Design/methodology/approachSeth’s transition theory is applied to the problem of creep stresses and strain rates in a cylinder under temperature gradient materials by finite deformation. Neither the yield criterion nor the associated flow rule is assumed here. The results obtained here are applicable to compressible materials. If the additional condition of incompressibility is imposed, then the expression for stresses corresponds to those arising from Tresca yield condition.FindingsThermal effect increases the values of axial stress at the external surface of a circular cylinder for incompressible material as compared to compressible materials. With the introduction of thermal effects, the maximum value of strain rates occurs at the external surface for incompressible material as compared to the compressible materials.Originality/valueThe model proposed in this paper is used commonly either as pressure vessels intended for storage industrial gases or media transportation of high pressurized fluids and the design of turbine rotors.

Microscopic statistical description of incompressible Navier-Stokes granular fluids

Based on the recently established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic 1-body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the same 1-body PDF asymptotically relaxes to a stationary and spatially uniform Maxwellian PDF.

An Eulerian model for the motion of granular material with a large Stokes number in fluid flow

This study introduced a novel Euler–Euler approach for modeling granular multiphase flow. The motion of particles with a large Stokes number was investigated assuming that granular material has unilateral compressibility. Solid pressure in the momentum equations for granular multiphase flow was determined so that the unilateral incompressibility condition was satisfied. Using the continuity condition of the granular phase, the equation was rewritten in the optimal form to calculate the solid pressure. A discrete formulation of smoothed particle hydrodynamics was applied for the convective terms so that the discrete matrix was positive semidefinite for the convergence and the discretization for an unstructured mesh was allowed. Frictional stress was then determined from solid pressure and, by using the solid pressure and frictional stress, momentum equations for the granular phase were solved. The method was incorporated into ANSYS FLUENT by a UDF (user defined function). Model validation was performed through a comparison with two previous results, and efficacy of the proposed model was confirmed.

On the use of Laplace's equation for pressure and a mesh-free method for 3D simulation of nonlinear sloshing in tanks; Reply to the discussion

In this reply, we re-emphasize on the fact that in the simulation of inviscid incompressible fluid flows using Lagrangian description, under some conditions, the divergence operator and the material derivative operator commute and thus the Laplace of the pressure is applicable in practical engineering problems as it has already been used and verified in numerous studies by many researchers. We show that the incompressibility condition suggested by the author of the discussion (discusser), as the replacement for divergence of the velocity field in using finite time increments, is insufficient and wrong. It will be shown that the concept has totally been misinterpreted by the discusser.

On elastic growth modelling of straight hair

Certain macroscopic similarities of the nail and hair elongation mechanisms enable the mass-growth activity that takes place within a straight hair follicle to be modelled through a suitable extension of a relevant small-strain pseudo-elasticity model of human nail growth. Basic differences which are taken into consideration are the facts that straight hair (a) resembles the form of a cylindrical rod, rather than a plate, while (b) its material constitution seems microscopically transversely isotropic, rather than isotropic. A complete analytical solution of the obtained governing differential equations is detailed for the case where incompressible mass-growth conditions prevail within the hair matrix. In addition to estimating displacement and stress distributions that develop within the growing matrix, and the resulting hair elongation, that solution enables prediction of a clinically observed zone of hair-fibre hardening that lies between the matrix soft tissue and the hard keratinous hair shaft. It also predicts that the longitudinal dimension of the hair matrix and that of the hair-fibre hardening zone depend on the material properties of the soft tissue of the follicle. Consideration of more advanced micro- or macro-scopic features of the hair follicle, such as layered structure or curved form, can be handled mathematically in a similar manner at the expense of analytical simplicity.

Study of the Shock Wave–Turbulent Boundary Layer Interaction Using a 3D von Kármán Length Scale

Abstract Traditional turbulence models are initially formulated and calibrated under incompressible conditions. Thus, these models are always of low fidelity when extended to high speed, complex and compressible flows. In this work, a compressible von Kármán length scale is proposed for compressible flows considering the variable densities. The length scale is the ratio between the new vorticity and its gradient. The new length scale is actually based on phenomenological theory, which is then integrated into the KDO (turbulence Kinetic energy Dependent Only) turbulence model, arriving at a compressible model called CKDO (Compressible KDO). In the CKDO turbulence model, all the extra terms produced by compressibility are modeled as dissipation. Compression corners of 8, 16, 20 and 24 angles are studied within SST, SA, KDO and CKDO. These test cases are known as the typical shock wave–boundary layer interactions. The results show that the new length scale in CKDO is able to well capture the surface pressure and skin friction distributions. Besides, compared with the standard von Kármán length scale, the new length scale in CKDO can better capture the size and position of the separation bubble. With the increase of the corner angle, CKDO shows more prominent potential for describing compressible flows.

Evaluation of aerodynamic characteristics of a coupled fluid-structure system using generalized Bernoulli's principle: An application to vocal folds vibration.

In this work we explore the aerodynamics flow characteristics of a coupled fluid-structure interaction system using a generalized Bernoulli equation derived directly from the Cauchy momentum equations. Unlike the conventional Bernoulli equation where incompressible, inviscid, and steady flow conditions are assumed, this generalized Bernoulli equation includes the contributions from compressibility, viscous, and unsteadiness, which could be essential in defining aerodynamic characteristics. The application of the derived Bernoulli's principle is on a fully-coupled fluid-structure interaction simulation of the vocal folds vibration. The coupled system is simulated using the immersed finite element method where compressible Navier-Stokes equations are used to describe the air and an elastic pliable structure to describe the vocal fold. The vibration of the vocal fold works to open and close the glottal flow. The aerodynamics flow characteristics are evaluated using the derived Bernoulli's principles for a vibration cycle in a carefully partitioned control volume based on the moving structure. The results agree very well to experimental observations, which validate the strategy and its use in other types of flow characteristics that involve coupled fluid-structure interactions.

Eulerian solid-fluid coupling

We present a new method that achieves a two-way coupling between deformable solids and an incompressible fluid where the underlying geometric representation is entirely Eulerian. Using the recently developed Eulerian Solids approach [Levin et al. 2011], we are able to simulate multiple solids undergoing complex, frictional contact while simultaneously interacting with a fluid. The complexity of the scenarios we are able to simulate surpasses those that we have seen from any previous method. Eulerian Solids have previously been integrated using explicit schemes, but we develop an implicit scheme that allows large time steps to be taken. The in-compressibility condition is satisfied in both the solid and the fluid, which has the added benefit of simplifying collision handling.