Bulletin of the American Physical Society
62nd Annual Meeting of the APS Division of Fluid Dynamics
Volume 54, Number 19
Sunday–Tuesday, November 22–24, 2009; Minneapolis, Minnesota
Session BL: CFD II: Methods |
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Chair: Carlos Pantano, University of Illinois at Urbana-Champaign Room: 200A |
Sunday, November 22, 2009 10:30AM - 10:43AM |
BL.00001: A finite-volume contact-capturing scheme Arpit Tiwari, Ratnesh Shukla, Carlos Pantano, Jonathan Freund Finite-volume schemes have been remarkably successful at capturing shock waves. Here, the local characteristics, of course, move into the shock, so in a sense these shock capturing schemes have the task of adding sufficient dissipation so that the captured shock remains relatively sharp but also sufficiently resolved to be compatible with the underlying discretization. In a similar spirit, we have developed a scheme for capturing contact discontinuities, with our particular interest being phase or material boundaries. These are fundamentally different than shocks because the characteristics are parallel to the contact, so a capturing scheme needs to counter the numerical diffusion (as needed for shocks) that thickens the contact in time. In analogy to the well-placed dissipation in shock capturing, terms are added to the equations that also capture these contacts over long times as sharp near-jumps, similar to captured shocks. These terms preserve both sharp mass and material jumps and are compatible with various equations of state, including interfaces between perfect gas and Mie-Gr{\"u}neison materials. The algorithm is demonstrated in various bubble and pore collapse scenarios. [Preview Abstract] |
Sunday, November 22, 2009 10:43AM - 10:56AM |
BL.00002: Analysis of Explicit Algorithms for Fluctuating Hydrodynamics A.C. Donev, E. Vanden-Eijnden, A. Garcia, J. Bell We describe the development and analysis of finite-volume methods for the Landau-Lifshitz Navier-Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white-noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. We introduce a systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatio-temporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order Runge Kutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations. Together with a novel random-direction method for evaluating the stochastic fluxes in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit. [Preview Abstract] |
Sunday, November 22, 2009 10:56AM - 11:09AM |
BL.00003: High-Order Finite-Difference Solution of the Poisson Equation with Interface Jump Conditions Alexandre Marques, Jean-Christophe Nave, Rodolfo Rosales The Poisson equation with jumps in function value and normal derivative across an interface is of central importance in the numerical study of multi-phase flows. In this presentation we introduce a method to obtain a high-order solution to such problem. The method is based on the construction of corrector functions that provide accurate extensions of the jump conditions around the interface. The accuracy of the method results from the combination of Hermite interpolants and a high-order representation of the interface using the gradient-augmented level-set technique. These corrector functions can be easily incorporated in standard finite-difference discretization schemes, only generating additional terms to the right-hand side of the system. As a result, computational cost is not significantly affected when compared to the first order accurate ghost fluid method. [Preview Abstract] |
Sunday, November 22, 2009 11:09AM - 11:22AM |
BL.00004: A new Ghost Fluid approach on non-graded adaptive cartesian grid for solving the Poisson equation with jump conditions enforced on an irregular interface Asdis Helgadottir, Frederic Gibou A poisson solver on a non-graded adaptive Cartesian grid, for the poisson equation with jump enforced at an irregular interface, is presented. A Ghost Fluid method, similar to that presented by Liu \emph{et al.} (in JCP:160(2000), 151 - 178) is used, with two main differences: 1) the uniform grid is replaced with optimum quad tree (2D) and octree (3D) structures, which significantly saves computational time and memory usage, 2) the jump in the normal derivative is enforced analytically (or more so by a numerical integration of the analytical term) instead of being approximated with finite difference as done by Liu \emph{et al.}. The method is simple and results in a positive definite matrix that can easily be solved with black box solvers. [Preview Abstract] |
Sunday, November 22, 2009 11:22AM - 11:35AM |
BL.00005: A new approach to computing open-boundary flows with SPH S. Majid Hosseini, James Feng In Smoothed Particle Hydrodynamics (SPH) methods, incompressibility is typically imposed by a projection method. This entails an artificial Neumann boundary condition for the pressure Poisson equation, which is often inconsistent with physical conditions at inflow and outflow boundaries. For this reason open-boundary flows have rarely been computed using SPH. In this work, we demonstrate that the classical pressure boundary condition produces a numerical boundary layer that compromises the solution near the boundaries. We resolve this problem by utilizing a ``rotational incremental pressure-correction scheme'' with a consistent pressure boundary condition. We show that this scheme computes the pressure and velocity accurately near open boundaries, and extends the scope of SPH simulation beyond the usual closed and periodic boundary conditions. [Preview Abstract] |
Sunday, November 22, 2009 11:35AM - 11:48AM |
BL.00006: Exponential Time Integrator for Solving the Lattice Boltzmann Equations Based on a Spectral-Element Discontinuous Galerkin Approach Kalu Chibueze Uga, Misun Min, Taehun Lee, Paul Fischer I'll present a high-order time integration method for solving lattice Boltzmann equation (LBE). We use high-order spectral-element discretization in space based on discontinuous Galerkin approach and apply a Krylov subspace approximation for time-advancing. The semi-discrete form of the spectral-element discontinuous Galerkin (SEDG) method on the LBE brings us to an ODE of the form $\raise0.7ex\hbox{${\partial U}$} \!\mathord{\left/ {\vphantom {{\partial U} {\partial t}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\partial t}$}\,\,=\,\,-AU$ with initial condition$U(0)=U_0 $ where $U$is a solution vector. A is a large sparse matrix based on a polynomial approximation order N. The solution of the equation is $U(t)=U_0 e^{-At}$. The explicit one-step method is based on the computation of matrix functions of the type $U(t+\delta )=U(t)e^{-A\delta }$. We project the matrix exponential and the solution vector onto a finite dimensional Krylov subspace $K_m $ of order $m$. We use the Arnoldi algorithm to generate an orthogonal basis $V_m $ and an Hessenberg matrix$H_m $ for approximating $e^{-A\delta }U(t)\,\,\approx \,V_m e^{-H_m \delta }V_m^T U(t)$. We will study convergence of the exponential time integrator for possible use of larger time step with high-order $m. $We will demonstrate its efficiency and accuracy compared to the Runge-Kutta time-stepping methods. [Preview Abstract] |
Sunday, November 22, 2009 11:48AM - 12:01PM |
BL.00007: A Lattice-based Numerical Solution Of Collisional Boltzmann Equation Boe Green, Prakash Vedula Accurate prediction capabilities of nonequilibrium flow behavior are important for efficient design of a wide range of emerging technologies ranging from micro-scale flow devices and hypersonic re-entry vehicles. Challenges for nonequilibrium flow predictions are mainly due to the breakdown of the continuum flow field hypothesis and the corresponding classical continuum field equations (e.g. Navier-Stokes). To address these challenges we propose an efficient lattice-based computational method (called Collisional Lattice Boltzmann Method or cLBM) for description of non-equilibrium flows, based on fundamental Boltzmann kinetic theory and the underlying full collision operator, without the use of any equilibrium-based approximations. In this moment method, which is applicable for a wide range of Knudsen numbers, the contributions of the full collision operator to the evolution of moments are computed via multinomial expansions and analytical integrals using a discrete quadrature on the lattice while spatial transport is accounted for through a streaming process on the lattice. The underlying conservation laws and invariants of collisional relaxation are also preserved in our approach, along with rates of evolution of selected low order moments. Prediction capabilities of cLBM are demonstrated via good agreement between results obtained from cLBM and other approaches using Couette, Poiseulle and lid-driven cavity flows. [Preview Abstract] |
Sunday, November 22, 2009 12:01PM - 12:14PM |
BL.00008: Semi-implicit Unstructured Finite Element Lattice Boltzmann Equation Method for Incompressible Binary Fluids Taehun Lee A semi-implicit finite element lattice Boltzmann equation method for incompressible binary fluids with large density and viscosity differences is proposed. The collision is treated implicitly and the intermolecular forcing terms are treated explicitly in order to achieve stability at high Reynolds number and avoid implicit treatment of the non-linear forcing terms. The characteristic Galerkin finite element approximation is adopted for the solution of the streaming, which provides geometric flexibility while retaining high-order accuracy. Unstructured body-fitted mesh enables mass conservation at the solid/liquid/gas triple contact line. The equilibrium contact angle is naturally imposed by the surface integral of the free energy. The proposed method is applied to several benchmark cases including drop sliding on patterned superhydrophobic surfaces. [Preview Abstract] |
Sunday, November 22, 2009 12:14PM - 12:27PM |
BL.00009: Gas-kinetic BGK Schemes for 3D Viscous Flow Jin Jiang, YueHong Qian Gas-kinetic BGK scheme developed as an Euler and Navier-Stokes solver is dated back to the early 1990s. There are now numerous literatures on the method. Here we focused on extending this approach to 3D viscous flow. Firstly, to validate the code, some test cases are carried out, including 1D Sod problem, interaction between shock and boundary layer. Then to improve its computational efficiency, two main convergence acceleration techniques, which are local time-stepping and implicit residual smoothing, have adopted and tested. The results indicate that the speed-up to convergence steady state is significant. The last is to incorporate turbulence model into current code with the increasing Reynolds number. As a proof of accuracy, the transonic flow over ONERA M6 wing and pressure distributions at various selected span-wise directions have been tested. The results are in good agreement with experimental data, which implies the extension to turbulent flow is very encouraging and of good help for further development. [Preview Abstract] |
Sunday, November 22, 2009 12:27PM - 12:40PM |
BL.00010: A New Weighted-Integral Based High-Order Method Li-Jun Xuan, Jie-Zhi Wu A weighted-integral based scheme (WIBS) has been constructed where the integrals of the function weighted by test functions are recorded as degree of freedoms (DOFs). The time evolution of DOFs is by stable Runge-Kutta method from the weak form of the original equation. At the boundary of every two cells, the function values are interpolated from the DOFs of the neighboring cells to calculate flux and volumetric integral in the weak form. The basic idea is to increase the order of interpolation by increasing both interpolating cells and DOFs simultaneously. The interpolation on more cells permits the use of WENO to capture discontinuity, while more DOFs can shrink the size of the interpolating stencil. The compactness of the reconstruction can increase the accuracy (especially for short waves) and fully retain it at boundary. Many existing schemes (e.g., FV, FE, FD, DG, SV, SD, Hermiter, etc.) can be viewed as special subclasses of WIBS. For 1-D hyperbolic conservation law systems, a high stability is found and the order of accuracy is perfectly held. A WENO-WIBS scheme has also been constructed to capture the discontinuity successfully. The results of WIBS for various benchmark problems are compared with those of 5th-order WENO. [Preview Abstract] |
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