Simulations of Compressible Multiphase Flows Through a Tube of Varying Cross-Section

2012 ◽  
Author(s):  
Shaban A. Jolgam ◽  
Ahmed R. Ballil ◽  
Andrzej F. Nowakowski ◽  
Franck C. G. A. Nicolleau
Keyword(s):  
Boundary Conditions ◽  
Numerical Algorithm ◽  
Multiphase Flows ◽  
Incompressible Flows ◽  
Godunov Method ◽  
Relaxation Processes ◽  
Source Terms ◽  
Governing Equations ◽  
Pressure Relaxation ◽  
Strang Splitting

The simulation of multiphase compressible flows through high pressure nozzles is presented. The study uses the developed numerical approach. There are many important engineering applications which are concerned with multiphase flows and convergent-divergent nozzles. This work presents the developed extension of the model and numerical algorithm based on the so called parent model earlier introduced by Saurel and Abgrall [Saurel, R. and Abgrall, R., A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows, J. Comput. Phys. 150 (1999), 425–467]. This model which consists of conservation laws for each phase complemented with the volume fraction evolution equation is modified by adding a source term to simulate area variation. The model is strictly hyperbolic and non-conservative due to the existence of non-conservative terms. The model is able to deal with compressible and incompressible flows. Moreover, it can deal with mixtures and pure fluids, where each fluid has its own pressure and velocity. The presence of velocity and pressure relaxation terms in the governing equations has made the velocity and pressure relaxation processes essential to tackle the boundary conditions at the interface. The interface separating phases is considered as a numerical diffusion zone in this method. The model is solved using an efficient Eulerian numerical method. A second order Godunov-type scheme with approximate Riemann solver is used to enable capturing of a physical interface by the resolution of the Riemann problem. The solution is obtained by splitting the hyperbolic part and source terms parts in the numerical algorithm. The source terms, including relaxation parts of the model, are tackled in succession using Strang splitting technique. The governing equations are solved at each computational cell using the same numerical algorithm for the whole domain including the interface. The main aim of this work has been to study different flow regimes with respect to pressure boundary conditions through the numerical solutions of single and multiphase flows. The performance of the programme has been verified via well established benchmark test problems for multiphase flows.

Mass Conservation in the Immersed Boundary Method

2005 ◽  
Author(s):  
Frank Muldoon ◽  
Sumanta Acharya
Keyword(s):  
Boundary Conditions ◽  
Immersed Boundary Method ◽  
Incompressible Flows ◽  
Mass Conservation ◽  
Steady State Solution ◽  
Immersed Boundary ◽  
Governing Equations ◽  
Time Step ◽  
Backward Euler ◽  
Divergence Free

The immersed boundary approach for the modeling of complex geometries in incompressible flows is examined critically from the perspective of satisfying boundary conditions and mass conservation. The system of discretized equations for mass and momentum can be inconsistent if the real velocities are used in defining the forcing terms used to satisfy the boundary conditions. As a result, the velocity is generally not divergence free and the pressure at locations in the vicinity of the immersed boundary is not physical. However, the use of the pseudo velocities in defining the forcing (as frequently done when the governing equations are solved using a fractional step or projection method) combined with the use of the specified velocity on the immersed boundary is shown to result in a consistent set of equations which allows a divergence free velocity but, depending on the time step used to obtain a steady state solution, is shown to have an undesirable effect of allowing significant permeability of the immersed boundary. An improvement is shown if the pressure gradient is integrated in time using the Crank-Nicholson scheme instead of the backward Euler scheme. However, even with this improvement a significant reduction in the time step and hence increase in computational expense is still required for sufficient satisfaction of the boundary conditions.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

Application of Generalized Differential Quadrature Method to the Bending of Thick Laminated Plates with Various Boundary Conditions

2006 ◽  
Vol 5-6 ◽  
pp. 407-414 ◽  
Author(s):  
Mohammad Mohammadi Aghdam ◽  
M.R.N. Farahani ◽  
M. Dashty ◽  
S.M. Rezaei Niya
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Simple Procedure ◽  
Shear Deformation Theory ◽  
Laminated Plates ◽  
Differential Quadrature ◽  
Governing Equations ◽  
First Order ◽  
Various Boundary Conditions ◽  
Generalized Differential

Bending analysis of thick laminated rectangular plates with various boundary conditions is presented using Generalized Differential Quadrature (GDQ) method. Based on the Reissner first order shear deformation theory, the governing equations include a system of eight first order partial differential equations in terms of unknown displacements, forces and moments. Presence of all plate variables in the governing equations provide a simple procedure to satisfy different boundary condition during application of GDQ method to obtain accurate results with relatively small number of grid points even for plates with free edges .Illustrative examples including various combinations of clamped, simply supported and free boundary condition are given to demonstrate the accuracy and convergence of the presented GDQ technique. Results are compared with other analytical and finite element predictions and show reasonably good agreement.

Investigation of heat transfer in irregular porous cavity subjected to various boundary conditions

2019 ◽  
Vol 29 (1) ◽  
pp. 418-447 ◽  
Author(s):  
Irfan Anjum Badruddin
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Outer Boundary ◽  
Governing Equations ◽  
Content Type ◽  
Porous Cavity ◽  
Various Boundary Conditions ◽  
Heating Source ◽  
Transfer Behavior ◽  
Vertical Walls

Purpose The purpose of this paper is to investigate the heat transfer in an arbitrary cavity filled with porous medium. The geometry of the cavity is such that an isothermal heating source is placed centrally at the bottom of the cavity. The height and width of the heating source is varied to analyses its effect on the heat transfer characteristics. The investigation is carried out for three different cases of outer boundary conditions such as two outside vertical walls being maintained at cold temperature To, two vertical and top horizontal surface being heated to. To and the third case with top surface kept at To but other surfaces being adiabatic. Design/methodology/approach Finite element method is used to solve the governing equations. Findings It is observed that the cavity exhibits unique heat transfer behavior as compared to regular cavity. The cases of boundary conditions are found to affect the heat transfer rate in the porous cavity. Originality/value This is original work representing the heat transfer in irregular porous cavity with various boundary conditions. This work is neither being published nor under review in any other journal.

scholarly journals Visualization Techniques of Differentially Heated Lid-Driven Square Cavity

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Vekamulla Narayana
Keyword(s):  
Flow Field ◽  
Boundary Conditions ◽  
Computational Cost ◽  
Field Synergy ◽  
Governing Equations ◽  
Square Cavity ◽  
Field Synergy Principle ◽  
Visualization Techniques

In the present study, an attempt is made to explore the flow field inside the differentially heated lid-driven square cavity. The governing equations along with boundary conditions are solved numerically. The simulated results (100 ≤ Re ≤ 1000 and 0.001 ≤ Ri ≤ 10) are validated with previous results in the literature. The convection differencing schemes, namely, UPWIND, QUICK, SUPERBEE, and SFCD, are discussed and are used to simulate the flow using the MPI code. It is observed that the computational cost for all the differencing schemes get reduced tremendously when the MPI code is implemented. Plots demonstrate the influences of Re and Ri in terms of the contours of the fluid streamlines, isotherms, energy streamlines, and field synergy principle.

A New Cavitation-Induced-Momentum-Defect (CIMD) Correction Approach to Track Cavitation Events

2007 ◽  
Author(s):  
Vedanth Srinivasan ◽  
Abraham J. Salazar ◽  
Kozo Saito
Keyword(s):  
Liquid Phase ◽  
Transport Equation ◽  
Vapor Phase ◽  
Incompressible Flows ◽  
Vapor Transport ◽  
Source Terms ◽  
Local Pressure ◽  
Cloud Cavitation ◽  
Tracking Model ◽  
The Impact

A new unsteady cavitation event tracking model is developed for predicting vapor dynamics occurring in multi-dimensional incompressible flows. The procedure solves incompressible Navier-Stokes equations for the liquid phase with an additional vapor transport equation for the vapor phase. The model tracks regions of liquid vaporization and applies compressibility effects to compute the local variation in speed of sound using the Homogeneous Equilibrium Model (HEM) assumptions. The variation of local cell density as a function of local pressure is used to construct the source term in the vapor fraction transport equation. The novel Cavitation-Induced-Momentum-Defect (CIMD) correction methodology developed in this study serves to account for cavitation inception and collapse events as relevant momentum source terms in the liquid phase momentum equations. Effects of vapor phase accumulation and diffusion are incorporated by detailed relaxation models. A modified RNG K-ε model, including the effects of compressibility in the vapor regions, is employed for modeling turbulence effects. Turbulent kinetic energy and dissipation contributions from the vapor regions are integrated with the liquid phase turbulence using relevant source terms. Numerical simulations are carried out using a Finite Volume methodology available within the framework of commercial CFD software code Fluent v.6.2. Simulation results are in qualitative agreement with experiments for unsteady cloud cavitation behavior in planar nozzle flows. Multitude of mechanisms such as formation of vortex cavities, vapor cluster shedding and coalescence, cavity pinch off are sharply captured by the supplemented vapor transport equation. Our results concur with previously established theories concerning sheet and cloud cavitation such as the re-entrant jet motion, cavity closure and the impact of adverse pressure gradients on cavitation dynamics.

scholarly journals Vibration Characteristics of Piezoelectric Microbeams Based on the Modified Couple Stress Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
R. Ansari ◽  
M. A. Ashrafi ◽  
S. Hosseinzadeh
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Natural Frequencies ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Modified Couple Stress Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

The vibration behavior of piezoelectric microbeams is studied on the basis of the modified couple stress theory. The governing equations of motion and boundary conditions for the Euler-Bernoulli and Timoshenko beam models are derived using Hamilton’s principle. By the exact solution of the governing equations, an expression for natural frequencies of microbeams with simply supported boundary conditions is obtained. Numerical results for both beam models are presented and the effects of piezoelectricity and length scale parameter are illustrated. It is found that the influences of piezoelectricity and size effects are more prominent when the length of microbeams decreases. A comparison between two beam models also reveals that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams as compared to its Timoshenko counterpart.

Meshless Integral Method for Analysis of Elastoplastic Geotechnical Materials

2010 ◽  
Author(s):  
Jianfeng Ma ◽  
Joshua David Summers ◽  
Paul F. Joseph
Keyword(s):  
Boundary Conditions ◽  
Function Approximation ◽  
Integral Method ◽  
Weak Form ◽  
Constitutive Law ◽  
Governing Equations ◽  
Natural Boundary ◽  
Pressure Sensitive ◽  
Support Domain ◽  
Natural Boundary Conditions

The meshless integral method based on regularized boundary equation [1][2] is extended to analyze elastoplastic geotechnical materials. In this formulation, the problem domain is clouded with a node set using automatic node generation. The sub-domain and the support domain related to each node are also generated automatically using algorithms developed for this purpose. The governing integral equation is obtained from the weak form of elastoplasticity over a local sub-domain and the moving least-squares approximation is employed for meshless function approximation. The geotechnical materials are described by pressure-sensitive multi-surface Drucker-Prager/Cap plasticity constitutive law with hardening. A generalized collocation method is used to impose the essential boundary conditions and natural boundary conditions are incorporated in the system governing equations. A comparison of the meshless results with the FEM results shows that the meshless integral method is accurate and robust enough to solve geotechnical materials.

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