Reduced Order Models of Low Mach Number Isothermal Flows in Microchannels

2013 ◽  
Author(s):  
Leila Issa ◽  
Issam Lakkis
Keyword(s):  
Mach Number ◽  
Analog Circuit ◽  
Stokes Equations ◽  
Momentum Equation ◽  
Ideal Gas ◽  
Reduced Order Models ◽  
Two Dimensional ◽  
Reduced Order ◽  
Low Mach Number ◽  
Circuit Components

We present reduced order models of unsteady low Mach number isothermal ideal gas flows in two-dimensional rectangular microchannels subject to first order slip boundary conditions. The Navier-Stokes equations are simplified using Low Mach Number expansions of the pressure and velocity fields. This approximation allows decoupling the density from spatial pressure variations, thus simplifying the momentum equation. The resulting diffusion equation and the subsequent pressure-flow-rate relationship enables modeling the flow using analog circuit components. The accuracy of the proposed models is investigated for steady and unsteady flows in a two-dimensional channel for different values of Reynolds and Knudsen numbers.

Reduced-Order Modeling of Low Mach Number Unsteady Microchannel Flows

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Leila Issa ◽  
Issam Lakkis
Keyword(s):  
Mach Number ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Solid Boundary ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Reduced Order ◽  
Low Mach Number ◽  
First Order ◽  
Slip Boundary

We present reduced-order models of unsteady low-Mach-number ideal gas flows in two-dimensional rectangular microchannels subject to first-order slip-boundary conditions. The pressure and density are related by a polytropic process, allowing for isothermal or isentropic flow assumptions. The Navier–Stokes equations are simplified using low-Mach-number expansions of the pressure and velocity fields. Up to first order, this approximation results in a system that is subject to no-slip condition at the solid boundary. The second-order system satisfies the slip-boundary conditions. The resulting equations and the subsequent pressure-flow-rate relationships enable modeling the flow using analog circuit components. The accuracy of the proposed models is investigated for steady and unsteady flows in a two-dimensional channel for different values of Mach and Knudsen numbers.

Artificial Compressibility Method and Preconditioning Method for Solving Two Dimensional Incompressibile Flow

2011 ◽  
Author(s):  
Hyungro Lee ◽  
Einkeun Kwak ◽  
Seungsoo Lee
Keyword(s):  
Mach Number ◽  
Turbulent Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Artificial Compressibility ◽  
Artificial Compressibility Method ◽  
Preconditioning Method

In this study, two commonly used numerical methods for the analysis of incompressible flows (or low Mach number flows), Chorins’ artificial compressibility method and Wiess and Smith’s preconditioning method are compared. Also, the convergence characteristics of two methods are numerically investigated for two-dimensional laminar and turbulent flows. Although the two methods have similar governing equations, the eigensystems and other details are very different. The eigensystems of the artificial compressibility method and the preconditioning method are analytically examined. An artificial compressibility code that solves the incompressible RANS (Reynolds Averaged Navier-Stokes) equations is newly developed for the study. An artificial compressibility code and a well-verified existing low Mach number code uses Roe’s approximate Riemann solver in conjunction with a cell centered finite volume method. Using MUSCL extrapolation with nonlinear limiters, 2nd order spatial accuracy is achieved while maintaining TVD (total variation diminishing) property. AF-ADI (approximate factorization-alternate direction implicit) method is used to get the steady solution for both codes. Menter’s k–ω SST turbulence model is used for the analysis of turbulent flows. Navier-Stokes equations and the turbulence model equations are solved in a loosely coupled manner.

A Numerical Study of Unsteady Natural Convection in a Rectangular Enclosure: The Effect of Compressibility

2004 ◽  
Author(s):  
K. M. Akyuzlu ◽  
Y. Pavri ◽  
A. Antoniou
Keyword(s):  
Mathematical Model ◽  
Stokes Equations ◽  
Numerical Study ◽  
Vertical Wall ◽  
Ideal Gas ◽  
Working Fluid ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Rectangular Enclosure ◽  
Circulation Patterns

A two-dimensional, mathematical model is adopted to investigate the development of buoyancy driven circulation patterns and temperature contours inside a rectangular enclosure filled with a compressible fluid (Pr=1.0). One of the vertical walls of the enclosure is kept at a higher temperature then the opposing vertical wall. The top and the bottom of the enclosure are assumed insulated. The physics based mathematical model for this problem consists of conservation of mass, momentum (two-dimensional Navier-Stokes equations) and energy equations for the enclosed fluid subjected to appropriate boundary conditions. The working fluid is assumed to be compressible through a simple ideal gas relation. The governing equations are discretized using second order accurate central differencing for spatial derivatives and first order forward finite differencing for time derivatives where the computation domain is represented by a uniform orthogonal mesh. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns (primitive variables) of the problem. A numerical experiment is carried out for a benchmark case (driven cavity flow) to verify the accuracy of the proposed solution procedure. Numerical experiments are then carried out using the proposed compressible flow model to simulate the development of the buoyancy driven circulation patterns for Rayleigh numbers between 103 and 105. Finally, an attempt is made to determine the effect of compressibility of the working fluid by comparing the results of the proposed model to that of models that use incompressible flow assumptions together with Boussinesq approximation.

Grid-Free Vortex Methods for Natural Convection in Two-Dimensional Domains

2012 ◽  
Author(s):  
Issam Lakkis
Keyword(s):  
Natural Convection ◽  
Nonlinear Diffusion ◽  
Reacting Flows ◽  
Ideal Gas ◽  
Two Dimensional ◽  
Vortex Methods ◽  
Free Vortex ◽  
Low Mach Number ◽  
Buoyancy Driven Flows ◽  
Vorticity Generation

Vortex methods for simulating natural convection of an ideal gas in unbounded two-dimensional domains are presented. In particular, the redistribution method for diffusion is extended to enable simulation of nonlinear diffusion of an ideal gas in isobaric conditions encountered in unbounded low-Mach number flows. We also address the problem of handling source terms in grid-free vortex methods and propose a fast, accurate, and physically motivated method for solving the associated inverse problems. Examples include generation of baroclinic vorticity in non-reacting buoyancy driven flows, and in addition, generation of internal energy and species in buoyant reacting flows. Accuracy and speed of the proposed algorithms for nonlinear diffusion and vorticity generation are investigated separately. Simulations of natural convection of a “thermal patch” for Grashof number ranging from to 1562.5 to 25000 are presented.

scholarly journals Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

2020 ◽  
Author(s):  
David Maltese ◽  
Antonín Novotný
Keyword(s):  
Mach Number ◽  
Initial Data ◽  
Error Estimate ◽  
Strong Solution ◽  
Stokes Equations ◽  
Main Tool ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Mac Scheme

Abstract We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

Mach Number Influence on Reduced-Order Models of Inviscid Potential Flows in Turbomachinery

2002 ◽  
Vol 124 (4) ◽  
pp. 977-987 ◽  
Author(s):  
Bogdan I. Epureanu ◽  
Earl H. Dowell ◽  
Kenneth C. Hall
Keyword(s):  
Mach Number ◽  
Steady Flow ◽  
Inviscid Flow ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Spatial Gradients ◽  
Potential Flows ◽  
Proper Orthogonal Decomposition Technique ◽  
Flow Through ◽  
Phase Angles

An unsteady inviscid flow through a cascade of oscillating airfoils is investigated. An inviscid nonlinear subsonic and transonic model is used to compute the steady flow solution. Then a small amplitude motion of the airfoils about their steady flow configuration is considered. The unsteady flow is linearized about the nonlinear steady response based on the observation that in many practical cases the unsteadiness in the flow has a substantially smaller magnitude than the steady component. Several reduced-order modal models are constructed in the frequency domain using the proper orthogonal decomposition technique. The dependency of the required number of aerodynamic modes in a reduced-order model on the far-field upstream Mach number is investigated. It is shown that the transonic reduced-order models require a larger number of modes than the subsonic models for a similar geometry, range of reduced frequencies and interblade phase angles. The increased number of modes may be due to the increased Mach number per se, or the presence of the strong spatial gradients in the region of the shock. These two possible causes are investigated. Also, the geometry of the cascade is shown to influence strongly the shape of the aerodynamic modes, but only weakly the required dimension of the reduced-order models.

Inverse cascades in two‐dimensional compressible turbulence. I. Incompressible forcing at low Mach number

1990 ◽  
Vol 2 (8) ◽  
pp. 1481-1486 ◽  
Author(s):  
J. P. Dahlburg ◽  
R. B. Dahlburg ◽  
J. H. Gardner ◽  
J. M. Picone
Keyword(s):  
Mach Number ◽  
Compressible Turbulence ◽  
Two Dimensional ◽  
Low Mach Number
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