Incompressibility Condition Research Articles

An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

A new mixed variational formulation for the Navier--Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier--Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additi...

A Method of Calculating Critical Depth of Burial of Explosive Charges to Generate Bulging and Cratering in Rock

For underground explosions, a thin to medium thickness layer near the cavity of an explosion can be considered a theoretical shell structure. Detonation products transmit the effective energy of explosives to this shell which can expand thus leading to irreversible deformation of the surrounding medium. Based on mass conservation, incompressible conditions, and boundary conditions, the possible kinematic velocity fields in the plastic zone are established. Based on limit equilibrium theory, this work built equations of material resistance corresponding to different possible kinematic velocity fields. Combined with initial conditions and boundary conditions, equations of motion and material resistance are solved, respectively. It is found that critical depth of burial is positively related to a dimensionless impact factor, which reflects the characteristics of the explosives and the surrounding medium. Finally, an example is given, which suggests that this method is capable of calculating the critical depth of burial and the calculated results are consistent with empirical results.

Numerical Investigation on Effect of Immersed Blade Depth on the Performance of Undershot Water Turbines

Energy, especially electricity, plays a vital role in global social and economic development. High annual rain rate in Malaysia seems a good potential for electricity generation especially through small hydro powers. Undershot water turbines are one of the hydropower turbines used for many years. However, the effect of blade depth immersed in the flowing water is not fully investigated. Therefore, the purpose of this paper is to study the effect of immersed blade depth for straight blade undershot water turbine in power generation by using Computational Fluid Dynamics (CFD) method. ANSYS CFX 15.0 was used to perform three dimensional analysis under steady state, incompressible, and non-isothermal conditions. The water wheel with number of blades of 6 and four different immersed depth was applied for each simulation. There are four different immersed depth was applied to each simulation, which are 20 mm, 40 mm, 60 mm and 80 mm. From the simulation result, it was found that the optimum immersed depth is 40 mm where the torque load and power generated were 0.264 N.m and 1.318 Watt respectively.

Density-based solver for all Mach number flows

A density-based solver for turbomachinery application is developed based on the central-upwind scheme of Kurganov using the open source CFD-library OpenFOAM. Preconditioning of Weiss and Smith is utilised to extend the applicability down to the incompressibility limit. Implicit residual averaging, bulk viscosity damping and local time stepping are employed to speed up the simulations. A low-storage four-stage Runge-Kutta scheme and dual time-stepping are used for time integration. The presented solver is validated using measurement data from three well-known reference cases. The inviscid performance is investigated using the circular bump test cases at incompressible, subsonic, transonic and supersonic conditions. The laminar flow around a circular cylinder is analysed and compared to analytical and experimental data. Finally, a low speed centrifugal compressor is simulated. All three cases show a very good agreement with the reference data.

Steady thermal stresses in solid disk under heat generation subjected to variable density

Seth's transition theory is applied to the problem of stresses in a solid rotating disk under heat generation subjected to variable density by infinitesimal 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. It has been seen that circumferential stress are maximum at the outer surface for incompressible material as compare to disk made compressible materials. Density variation parameter increases the value of circumferential as well as radial stress at the outer surface of solid disk for compressible and incompressible materials. The present solution is illustrated by numerical results and is compared with heat generation case.

Determine variation of poisson ratios and thermal creep stresses and strain rates in an isotropic disc

Seth's transition theory is applied to the problem of thermal creep transition stresses and strain rates in a thin rotating disc with shaft having variable density 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. Thermal effect decreased value of radial stress at the internal surface of the rotating isotropic disc made of compressible material as well as incompressible material and this value of radial stress further much increases with the increase in angular speed. With the introduction of thermal effects, the maximum value of strain rates further increases at the internal surface for compressible materials as compare to incompressible material.

Kinematics of the deformation zone in plugless rolling of oil pipes

This article describes the process of plugless rolling of oil pipes. Correlations between caliber parameters and swaging were obtained. The mathematical model of the process of plugless rolling of pipes was made. Main parameters of the deformation zone were determined. A set of equations to determine the surface of the contact area of the round billet and the tool for two-roll and three-roll calibers was obtained. The velocity field of metal billets was presented. From the assumption that the axial component of the velocity vector changes according to the quadratic law and the metal entrance velocity in the deformation zone is connected with the exit velocity from the deformation zone by the reduction ratio, the dependence for the determination of this component of the velocity vector was obtained. The linear function of the radial component of the velocity vector was justified. From the relation of axial and radial components of the velocity vector by the tangent of the angle between the considered section and the vertical symmetry plane of the roll a dependence for the determination of the radial component of the velocity vector was defined. The third component of the velocity vector was determined subject to the incompressibility condition, which was defined for three-roll and two-roll calibers. Based on the determined components of the velocity vector, the components of the strain velocity tensor were defined for three-roll and two-roll calibers. The components of the strain velocity tensor were calculated for the case of rolling tubes in a three-roll reduction mill. The components of the strain velocity tensor were numerically estimated, and the components of the strain velocity tensor that may be neglected in engineering calculations were determined.

Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm

In this paper, the shallow water problem is discussed. By treating the incompressible condition as the constraint, a constrained Hamilton variational principle is presented for the shallow water problem. Based on the constrained Hamilton variational principle, a shallow water equation based on displacement and pressure (SWE-DP) is developed. A hybrid numerical method combining the finite element method for spatial discretization and the Zu-class method for time integration is created for the SWEDP. The correctness of the proposed SWE-DP is verified by numerical comparisons with two existing shallow water equations (SWEs). The effectiveness of the hybrid numerical method proposed for the SWE-DP is also verified by numerical experiments. Moreover, the numerical experiments demonstrate that the Zu-class method shows excellent performance with respect to simulating the long time evolution of the shallow water.

Global-wellposedness of 2D Boussinesq equations with mixed partial temperature-dependent viscosity and thermal diffusivity

In this paper, we study the global well-posedness of 2D anisotropic nonlinear Boussinesq equations with horizontal temperature-dependent viscosity and vertical thermal diffusivity in the whole space. Due to lacking vertical viscosity and horizontal thermal diffusivity, there is no smooth effect in those directions. Besides, the nonlinearity of temperature-dependent viscosity gives rise to new difficulties. To solve it, we make full use of the incompressible condition and anisotropic inequalities to obtain the H1 estimates of velocity field and H1+s estimates of temperature for any s∈(0,1/2]. In the end, we build up a uniqueness criterion which together with the a priori estimates admits a unique global solution without any smallness assumptions.

An unconditionally stable semi-implicit FSI finite element method

Abstract The paper addresses the numerical solution of fluid–structure interaction (FSI) problems involving incompressible viscous Newtonian fluid and hyperelastic material. A well known challenge in computing FSI systems is to provide an effective time-marching algorithm, which avoids numerical instabilities due to the loose coupling of fluid and structure motion on the FSI interface. In this work, we introduce a semi-implicit finite element scheme for an Arbitrary Lagrangian–Eulerian formulation of the fluid–structure interaction problem. The approach strongly enforces the coupling conditions on the fluid–structure interface, but requires only a linear problem to be solved on each time step. Further, we prove that the numerical solution to the fully discrete problem satisfies the correct energy balance, and the stability estimate follows without any extra model simplifications or assumptions on the time step. The analysis covers the cases of Saint Venant–Kirchhoff compressible and incompressible neo-Hookean materials. Results of several numerical experiments are included to illustrate the properties of the method and its applicability for the simulation of certain hemodynamic flows. We also experiment with the enforcement of material incompressibility condition in the finite element method via an integral constraint or alternatively letting the Poisson ratio in the compressible model to be close to 1 2 . From these experiments conclusions are drawn concerning the accuracy of flow statistics prediction for incompressible vs. nearly incompressible structure models.

Analysis of vertical rolling using double parabolic model and stream function velocity field

Vertical rolling has been widely used in the roughing stand of hot strip rolling to improve the precision of slab width which influences the finished product quality significantly in actual production. Double parabolic dog-bone function model and corresponding velocity and strain rate fields are firstly proposed based on the incompressibility condition and stream function. They are successfully applied to three-dimensional vertical rolling. Using the first variation principle of rigid-plastic material, an analytical solution of slab total power functional in vertical rolling is obtained. Then, the shape parameters and the rolling force are received by minimizing the power functional. The error of shape and power parameters is within 4 % compared with finite element method (FEM) simulation’s result and less than 9.5 % compared with other models’ result.

Parameter-free method for computing the turbulent flow in a plane channel in a wide range of Reynolds numbers

Near-wall turbulence in a plane channel at high Reynolds numbers is studied by applying direct numerical simulation. A new numerical algorithm is proposed in which convective fluxes are explicitly approximated using the CABARET scheme and two grid elliptic equations are solved in order to satisfy the incompressibility condition. The equations are of high dimension and are solved by employing a fast direct method that can be efficiently parallelized. In contrast to most methods, including spectral ones, the CABARET scheme does not involve any tuning parameters. It has a compact stencil, which simplifies setting boundary conditions and improves the efficiency of parallelization in computations on multiprocessor computer systems. Numerical results are obtained in a wide range of Reynolds numbers and are compared with experimental data and numerical results of other authors. The error in the computed drag coefficient for turbulent flow is examined as a function of grid characteristics.

A Total Lagrangian ANCF Liquid Sloshing Approach for Multibody System Applications

The objective of this investigation is to develop a total Lagrangian nonincremental liquid sloshing solution procedure based on the finite element (FE) absolute nodal coordinate formulation (ANCF). The proposed liquid sloshing modeling approach can be used to avoid the difficulties of integrating most of fluid dynamics formulations, which are based on the Eulerian approach, with multibody system (MBS) dynamics formulations, which are based on a total Lagrangian approach. The proposed total Lagrangian FE fluid dynamics formulation, which can be systematically integrated with computational MBS algorithms, differs significantly from the conventional FE or finite volume methods which are based on an Eulerian representation that employs the velocity field of a fixed control volume in the region of interest. The ANCF fluid equations are expressed in terms of displacement and gradient coordinates of material points, allowing for straightforward implementation of kinematic constraint equations and for the systematic modeling of the interaction of the fluid with the external environment or with rigid and flexible bodies. The fluid incompressibility conditions and surface traction forces are considered and derived directly from the Navier–Stokes equations. Two ANCF brick elements, one is obtained using an incomplete polynomial representation and the other is obtained from a B-spline volume representation, are used. The new approach ensures the continuity of the displacement gradients at the nodal points and allows for imposing higher degree of continuity across the element interface by applying algebraic constraint equations that can be used to eliminate dependent variables and reduce the model dimensionality. Regardless of the magnitude of the fluid displacement, the fluid has a constant mass matrix, leading to zero Coriolis and centrifugal forces. The analysis presented in this paper demonstrates the feasibility of developing an efficient nonincremental total Lagrangian approach for modeling sloshing problems in MBS system applications in which the bodies can experience large displacements including finite rotations. Several examples are presented in order to shed light on the potential of using the ANCF liquid sloshing formulation developed in this study.

A stream function solver for liquid simulations

This paper presents a liquid simulation technique that enforces the incompressibility condition using a stream function solve instead of a pressure projection. Previous methods have used stream function techniques for the simulation of detailed single-phase flows, but a formulation for liquid simulation has proved elusive in part due to the free surface boundary conditions. In this paper, we introduce a stream function approach to liquid simulations with novel boundary conditions for free surfaces, solid obstacles, and solid-fluid coupling. Although our approach increases the dimension of the linear system necessary to enforce incompressibility, it provides interesting and surprising benefits. First, the resulting flow is guaranteed to be divergence-free regardless of the accuracy of the solve. Second, our free-surface boundary conditions guarantee divergence-free motion even in the un-simulated air phase, which enables two-phase flow simulation by only computing a single phase. We implemented this method using a variant of FLIP simulation which only samples particles within a narrow band of the liquid surface, and we illustrate the effectiveness of our method for detailed two-phase flow simulations with complex boundaries, detailed bubble interactions, and two-way solid-fluid coupling.

A combined finite volume - finite element scheme for a dispersive shallow water system

We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation. The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces. The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition. Several numerical experiments are performed to confirm the relevance of our approach.

Rate dependent plastic deformation analysis of creeping short fiber composites using the virtual fiber method in the non-reinforced regions

In this research paper, analysis of time-dependent plastic deformation in non-reinforced regions of creeping short fiber composites is carried out under axial tensile stress using the virtual fiber method. As an important application of the present method, shuttles and spaceships, turbine blades and discs are usually subjected to the creep effects. So, analysis of the creep phenomenon is necessary and important in various industries. The nature of the stress distributions is investigated using the equilibrium and constitutive equations, compatibility relations, and incompressibility condition with various boundary conditions, and geometric relations. Analytical and finite element methods are used to predict the stress distributions in the steady state creep of the short fiber composites. Also, the creep analysis (nodal solution) is done by FEM to predict the possible creep rupture, in which, debonding at the interface of the matrix and fiber may happen due to the maximum tensile axial stress values at the interface. In addition to similarities of the present method and experimental results [34], a good agreement of the stress distributions between the present analytical solutions and the FEM results is found. As a result, the short fiber composites with large non-reinforced regions in the axial direction are proper because it increases the factor of safety for design.

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

SummaryA semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

SummaryThis paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity, and pressure, for the Brinkman problem with nonstandard boundary conditions. The approach is based on the introduction of Galerkin least‐squares terms arising from the constitutive equation relating the aforementioned unknowns and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic, which, thanks to the Lax–Milgram theorem, and besides proving the well‐posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart–Thomas elements of any order for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle‐bubble and edge‐bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators and showing the behavior of the associated adaptive algorithms, are reported. Copyright © 2015 John Wiley & Sons, Ltd.

Derivation of Relations and Analysis of Tube Bending Processes Using Discontinuous Fields of Plastic Strains

The generalized strain scheme in bending metal tubes at bending machines with the use of a mandrel presented in Śloderbach (1999; 2002; 2013<sub>1,2</sub>; 2014) satisfies initial and boundary kinematic conditions of bending, conditions of continuity and inseparability of strains. This paper introduces three formal simplifications gradually imposed into forms of principal components of the generalized strain model giving suitable simplifications of the 1st, 2nd and 3rd types. Such mathematical simplifications cause that the obtained strain fields do not satisfy the condition of consistency of displacements and strain continuity. The simplified methods determine safer values of the wall thickness than those from the generalized continuous strain scheme. The condition of plastic incompressibility was used for the derivation of an expression for distribution of wall thickness of the bent elbow in the layers subjected to tension and compression for three examples of discontinuous kinematic strain fields.

Upper bound analysis of rolling force and dog-bone shape via sine function model in vertical rolling

The control of the slab width in actual production is realized by the widely use of vertical rolling (edge rolling). According to the incompressibility condition, the sine function dog-bone model is firstly proposed in this paper for steady state deformation in vertical rolling with flat rolls combined with the actual conditions. Based on the first variation principle of rigid-plastic material, the variable upper bound integration method is used to integrate plastic deformation, shear and friction power terms. The upper bound solutions of rolling force and dog-bone shape are solved numerically, and this process is carried out using Matlab Optimization Toolbox by minimizing the total power functional. Results show that the rolling force increases and the dog-bone shape size become larger while engineering strain, initial thickness, or roll radius increases. The results obtained from sine function model are compared with those of experimental data in reference and FEM simulation, and a good agreement is found. The comparison shows that it is possible to determine the required optimum rolling force and dog-bone shape by using sine function model.