Low-Mach-number asymptotics for two-phase flows of granular materials

2011 ◽  
Vol 669 ◽  
pp. 472-497 ◽  
Author(s):  
C. VARSAKELIS ◽  
M. V. PAPALEXANDRIS
Keyword(s):  
Mach Number ◽  
Granular Materials ◽  
Perturbation Parameter ◽  
Constant Density ◽  
Two Phase ◽  
Low Mach Number ◽  
Equilibrium Limit ◽  
Governing System ◽  
Special Cases ◽  
Multi Phase

In this paper, we generalize the concept of low-Mach-number approximation to multi-phase flows and apply it to the two-phase flow model of Papalexandris (J. Fluid Mech., vol. 517, 2004, p. 103) for granular materials. In our approach, the governing system of equations is first non-dimensionalized with values that correspond to a reference thermodynamic state of the phase with the smaller speed of sound. By doing so, the Mach number based on this reference state emerges as a perturbation parameter of the equations in hand. Subsequently, we expand each variable in power series of this parameter and apply singular perturbation techniques to derive the low-Mach-number equations. As expected, the resulting equations are considerably simpler than the unperturbed compressible equations. Our methodology is quite general and can be directly applied for the systematic reduction of continuum models for granular materials and for many different types of multi-phase flows. The structure of the low-Mach-number equations for two special cases of particular interest, namely, constant-density flows and the equilibrium limit is also discussed and analysed. The paper concludes with some proposals for experimental validation of the equations.

Double-Inclusion Model and Overall Moduli of Multi-Phase Composites

1994 ◽  
Vol 116 (3) ◽  
pp. 305-309 ◽  
Author(s):  
Muneo Hori ◽  
Sia Nemat-Nasser
Keyword(s):  
Ellipsoidal Inclusion ◽  
The Self ◽  
Average Field ◽  
Two Phase ◽  
Homogeneous Domain ◽  
Special Cases ◽  
Self Consistent ◽  
Inclusion Model ◽  
Exact Bounds ◽  
Multi Phase

The double-inclusion model consists of an ellipsoidal inclusion of arbitrary elasticity, containing another ellipsoidal heterogeneity of arbitrary elasticity, size, and orientation, which are embedded in an infinitely extended homogeneous domain of yet another arbitrary elasticity. Average field quantities for the double inclusion are obtained analytically, and used to estimate the overall moduli of two-phase composites. The technique includes the self-consistent and other related methods as special cases. Furthermore, exact bounds for the overall moduli are obtained on the basis of the double-inclusion model. The double-inclusion model has been generalized (Nemat-Nasser and Hori, 1993) to a multi-inclusion model, where, again, all the average field quantities are estimated analytically. The application of the multiinclusion model includes a composite containing inclusions with multi-layer coatings.

scholarly journals Modified Gamma-Based Model for Solving Low Mach Number Compressible Two Phase Flows

2013 ◽  
Vol 8 (1) ◽  
pp. 75-89 ◽  
Author(s):  
Junichi OOIDA ◽  
Kota NAKANO ◽  
Kohei NAGANE
Keyword(s):  
Mach Number ◽  
Two Phase Flows ◽  
Two Phase ◽  
Low Mach Number

Projection method for high-order compact schemes for low Mach number flows in enclosures

2014 ◽  
Vol 24 (5) ◽  
pp. 1141-1174 ◽  
Author(s):  
Artur Tyliszczak
Keyword(s):  
Mach Number ◽  
Boundary Conditions ◽  
Projection Method ◽  
High Order ◽  
Staggered Grid ◽  
Constant Density ◽  
Computational Domain ◽  
Content Type ◽  
Low Mach Number ◽  
Large Density

Purpose – Variable density flows play an important role in many technological devices and natural phenomena. The purpose of this paper is to develop a robust and accurate method for low Mach number flows with large density and temperature variations. Design/methodology/approach – Low Mach number approximation approach is used in the paper combined with a predictor-corrector method and accurate compact scheme of fourth and sixth order. A novel algorithm is formulated for the projection method in which the boundary conditions for the pressure are implemented in such a way that the continuity equation is fulfilled everywhere in the computational domain, including the boundary nodes. Findings – It is shown that proposed implementation of the boundary conditions considerably improves a solution accuracy. Assessment of the accuracy was performed based on the constant density Burggraf flow and for two benchmark cases for the natural convection problems: steady flow in a square cavity and unsteady flow in a tall cavity. In all the cases the results agree very well with exemplary solutions. Originality/value – A staggered or half-staggered grid arrangement is usually used for the projection method for both constant and low Mach number flows. The staggered approach ensures stability and strong pressure-velocity coupling. In the paper a high-order compact method has been implemented in the framework of low Mach number approximation on collocated meshes. The resulting algorithm is accurate, robust for large density variations and is almost free from the pressure oscillations.

scholarly journals A homogeneous relaxation low Mach number model

2021 ◽  
Author(s):  
Gloria Faccanoni ◽  
Bérénice Grec ◽  
Yohan Penel
Keyword(s):  
Fluid Flow ◽  
Mach Number ◽  
Equation Of State ◽  
Specific Model ◽  
Relaxation Model ◽  
Two Phase ◽  
Asymptotic Preserving ◽  
Low Mach Number ◽  
Asymptotic Preserving Scheme ◽  
Phase Fluid

In the present paper, we investigate a new homogeneous relaxation model describing the behaviour of a two-phase fluid flow in a low Mach number regime, which can be obtained as a low Mach number approximation of the well-known HRM. For this specific model, we derive an equation of state to describe the thermodynamics of the two-phase fluid. We prove some theoretical properties satisfied by the solutions of the model, and provide a well-balanced scheme. To go further, we investigate the instantaneous relaxation regime, and prove the formal convergence of this model towards the low Mach number approximation of the well-known HEM. An asymptotic-preserving scheme is introduced to allow numerical simulations of the coupling between spatial regions with different relaxation characteristic times.

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