Acceleration Methods for Coarse-Grained Numerical Solution of the Boltzmann Equation

2006 ◽  
Vol 129 (7) ◽  
pp. 908-912 ◽  
Author(s):  
Husain A. Al-Mohssen ◽  
Nicolas G. Hadjiconstantinou ◽  
Ioannis G. Kevrekidis
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Boltzmann Kinetic Equation ◽  
Coarse Grained ◽  
Navier Stokes ◽  
Explicit Integration ◽  
Simulation Tools ◽  
Efficient Simulation ◽  
Acceleration Methods ◽  
O 10

We present a coarse-grained steady-state solution framework for the Boltzmann kinetic equation based on a Newton-Broyden iteration. This approach is an extension of the equation-free framework proposed by Kevrekidis and coworkers, whose objective is the use of fine-scale simulation tools to directly extract coarse-grained, macroscopic information. Our current objective is the development of efficient simulation tools for modeling complex micro- and nanoscale flows. The iterative method proposed and used here consists of a short Boltzmann transient evolution step and a Newton-Broyden contraction mapping step based on the Boltzmann solution; the latter step only solves for the macroscopic field of interest (e.g., flow velocity). The predicted macroscopic field is then used as an initial condition for the Boltzmann solver for the next iteration. We have validated this approach for isothermal, one-dimensional flows in the low Knudsen number regime. We find that the Newton-Broyden iteration converges in O(10) iterations, starting from arbitrary guess solutions and a Navier-Stokes based initial Jacobian. This results in computational savings compared to time-explicit integration to steady states when the time to steady state is longer than O(40) mean collision times.

Acceleration Methods for Coarse-Grained Numerical Solution of the Boltzmann Equation

2006 ◽  
Author(s):  
Husain A. Al-Mohssen ◽  
Nicolas G. Hadjiconstantinou ◽  
Ioannis G. Kevrekidis
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Boltzmann Kinetic Equation ◽  
Coarse Grained ◽  
Navier Stokes ◽  
Explicit Integration ◽  
Simulation Tools ◽  
Efficient Simulation ◽  
Acceleration Methods ◽  
O 10

We present a coarse-grained steady state solution framework for the Boltzmann kinetic equation based on a Newton-Broyden iteration. This approach is an extension of the equation-free framework proposed by Kevrekidis and coworkers, whose objective is the use of fine-scale simulation tools to directly extract coarse-grained, macroscopic information. Our current objective is the development of efficient simulation tools for modeling complex micro/nanoscale flows. The iterative method proposed and used here consists of a short Boltzmann transient evolution step and a Newton-Broyden contraction mapping step based on the Boltzmann solution; the latter step only solves for the macroscopic field of interest (e.g. flow velocity). The predicted macroscopic field is then used as an initial condition for the Boltzmann solver for the next iteration. We have validated this approach for isothermal, one-dimensional flows in the low Knudsen number regime. We find that the Newton-Broyden iteration converges in O(10) iterations, starting from arbitrary guess solutions and a Navier-Stokes based initial Jacobian. This results in computational savings compared to time-explicit integration to steady states when the time to steady state is longer than O(40) mean collision times.

Numerical Solution of the Equations Governing the Steady State of a Thin Cylindrical Web Supported by an Air Cushion

1999 ◽  
Author(s):  
Sinan Müftü
Keyword(s):  
Steady State ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Steady State Solution ◽  
Iteration Scheme ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Air Cushion ◽  
Raphson Method ◽  
The Web

Abstract The numerical method used to solve the coupled nonlinear, partial differential equations (PDEs) representing the interaction between a thin, flexible, cylindrical web and an air cushion at steady state is analyzed. The web deflections are modeled by a cylindrical shell theory that allows moderately large deflections. The airflow is modeled in two-dimensions with a modified form of the Navier-Stokes and mass balance equations that have non-linear source terms. The coupled fluid/structure system is solved numerically in a stacked iteration scheme: The fluid equations are solved using pseudo-compressibility method with artificial viscosity; and the web equations are solved with a modified Newton-Raphson method. The convergence characteristics of the coupled system and the effects of the numerical parameters on the steady state solution are studied by numerical experiments.

On the Use of the Fully Compressible Navier-Stokes Equations for the Steady-State Solution of Natural Convection Problems in Closed Cavities

2006 ◽  
Vol 129 (3) ◽  
pp. 387-390 ◽  
Author(s):  
Sandip Mazumder
Keyword(s):  
Steady State ◽  
Stokes Equations ◽  
Pressure Rise ◽  
Steady State Solution ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Simple Strategy ◽  
Reference Pressure

The steady-state compressible form of the Navier-Stokes equations, along with no-slip boundary conditions on walls, represents a boundary value problem. In closed heated cavities, these equations are incapable of preserving the initial mass of the cavity and predicting the pressure rise. A simple strategy to adjust the reference pressure in the system is presented and demonstrated. The strategy is similar to solving the transient form of the governing equations, but completely eliminates truncation errors associated with temporal discretization of the transient terms. Results exhibit good agreement with previous reports. Additional results are shown to highlight differences between the fully compressible formulation and the Boussinesq approximation.

An airfoil theory of bifurcating laminar separation from thin obstacles

1990 ◽  
Vol 216 ◽  
pp. 255-284 ◽  
Author(s):  
C. J. Lee ◽  
H. K. Cheng
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Steady State ◽  
Steady State Solution ◽  
Equation System ◽  
Branch Points ◽  
Laminar Separation ◽  
Kutta Condition ◽  
Numerical Studies ◽  
Steady State Solutions

Global interaction of the boundary layer separating from an obstacle with resulting open/closed wakes is studied for a thin airfoil in a steady flow. Replacing the Kutta condition of the classical theory is the breakaway criterion of the laminar triple-deck interaction (Sychev 1972; Smith 1977), which, together with the assumption of a uniform wake/eddy pressure, leads to a nonlinear equation system for the breakaway location and wake shape. The solutions depend on a Reynolds numberReand an airfoil thickness ratio or incidence τ and, in the domain$Re^{\frac{1}{16}}\tau = O(1)$considered, the separation locations are found to be far removed from the classical Brillouin–Villat point for the breakaway from a smooth shape. Bifurcations of the steady-state solution are found among examples of symmetrical and asymmetrical flows, allowing open and closed wakes, as well as symmetry breaking in an otherwise symmetrical flow. Accordingly, the influence of thickness and incidence, as well as Reynolds number is critical in the vicinity of branch points and cut-off points where steady-state solutions can/must change branches/types. The study suggests a correspondence of this bifurcation feature with the lift hysteresis and other aerodynamic anomalies observed from wind-tunnel and numerical studies in subcritical and high-subcriticalReflows.

Dependency of Unsteady Time-Linearized Flutter Investigations on the Steady State Flow Field

2011 ◽  
Author(s):  
Michael Blocher ◽  
Markus May ◽  
Harald Schoenenborn
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Elastic Stability ◽  
Compressor Cascade ◽  
Steady State Flow ◽  
State Flow ◽  
Flow Parameters ◽  
Pressure Distributions ◽  
German Aerospace ◽  
École Polytechnique

The influence of the steady state flow solution on the aero-elastic stability behaviour of an annular compressor cascade shall be studied in order to determine sensitivities of the aero-dynamic damping with respect to characteristic flow parameters. In this context two different flow regimes — a subsonic and a transonic case — are subject to the analysis. The pressure distributions, steady as well as unsteady, on the blade surface of the NACA3506 profile are compared to experimental data that has been gained by the Institute of Aeroelasticity of the German Aerospace Center (DLR) during several wind tunnel tests at the annular compressor cascade facility RGP-400 of the Ecole Polytechnique Fe´de´rale de Lausanne (EPFL). Whereas a certain robustness of the unsteady CFD results can be stated for the subsonic flow regime, the transonic regime proves to be very sensitive with respect to the steady state solution.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

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