Bulletin of the American Physical Society
2005 58th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 20–22, 2005; Chicago, IL
Session FG: CFD III |
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Chair: Oleg Vasilyev, University of Colorado Room: Hilton Chicago Williford A |
Monday, November 21, 2005 8:00AM - 8:13AM |
FG.00001: A unified approach for flow simulation of compressible and incompressible fluids Lars Pesch, Monika Polner, Jaap van der Vegt In many fluid dynamics problems one has to deal with flows which exhibit in some regions conditions under which the fluid is compressible, while in other regions it behaves essentially incompressible. To simulate such flows is a problem for many mathematical methods that are either tailored towards compressible or incompressible fluids. Also the inclusion of complicated equations of state, necessary to describe the physics of complex fluids, raises difficulties. A mathematical framework that can overcome these difficulties is provided by the use of the symmetrized compressible Navier-Stokes equations using entropy variables. In the presented method, these equations are solved with a Galerkin least-squares finite element method which provides the necessary flexibility to deal with a wide range of flow problems. A critical component in the algorithm is the use of a stabilization operator. Using dimensional analysis we have derived and analyzed a class of stabilization operators which is suitable for both compressible and incompressible flows, resulting in a unified mathematical framework. A key feature of the algorithm is its close relation to the underlying thermodynamics, which is incorporated by expressing the equations of state in terms of two material coefficients, volume expansivity and isothermal compressibility, which are available in analytical or tabulated form for relevant substances. The algorithm will be demonstrated with test cases for both fully incompressible as well as compressible fluids to underline the feasibility of the approach. [Preview Abstract] |
Monday, November 21, 2005 8:13AM - 8:26AM |
FG.00002: The Integro-Differential Scheme - A New Approach for Solving the Conservation Laws for Fluid Flow Gafar Elamin, Frederick Ferguson A new numerical scheme for solving the equations that govern fluid dynamics problems is developed. The new numerical innovation is based on a smart integration of the traditional finite volume and finite difference schemes and is so-called the Integro-Differential Scheme, (IDS). The strength of IDS rests on the implementation of the mean value theorem to the integral form of the conservation laws. This process transforms the integral equations into a finite difference scheme that lends itself to efficient numerical implementation. In this paper the new scheme is employed to solve the viscous flow over a flat plate problem and the shock/boundary layer interaction problem. In both cases, the results showed very good agreement with the physical expectation of the flow, the empirical formulas, and the experimental data. This agreement solidified the belief that the scheme is robust, efficient, and capable of solving a variety of complex fluid dynamics problems. [Preview Abstract] |
Monday, November 21, 2005 8:26AM - 8:39AM |
FG.00003: Dynamically Adaptive Wavelet Collocation Method for Shock Computations Jonathan Regele, Oleg Vasilyev Most explicit TVD schemes make use of artificial viscosity to reduce oscillations and avoid the stability requirements that an explicitly written dissipation term would require when solving hyperbolic conservation equations. In this talk an adaptive wavelet collocation method for shock computation is described. The method for determining a shock's location is similar to Harten's multiresolution algorithm, but its implementation is more continuous. The presence of wavelet coefficients on the finest level of resolution indicates that the maximum allowed resolution has been reached and localized artificial viscous terms should be added to smooth the solution. The localized viscosity is constructed by creating a mask of the wavelet coefficients on the finest level that are greater than a given threshold parameter. The mask is smoothed to reduce oscillations that can be induced due to spatial discontinuities in the second derivative. The main advantage of this technique are its generality and zero losses away from shocks. Since the viscosity is written explicitly, sonic points are no longer problematic and there is no need to track wind direction or introduce flux splitting. One- and two-dimensional examples are given and discussed. [Preview Abstract] |
Monday, November 21, 2005 8:39AM - 8:52AM |
FG.00004: A Modified MacCormack's Explicit Time Marching Scheme for Solving the Conservation equations Stephen Akwaboa, Frederick Ferguson In this study, the classical MacCormack's explicit unsteady scheme is modified and smartly programmed to serve as the basis for a Navier-Stokes solver. A two dimensional code capable of solving the perfect gas dynamic equations is developed. Geometry of particular interest used to solve the fluid flow problem are flow in parallel plates and rearward-facing step. The governing equations are programmed using FORTRAN to solve the 2D planar Navier-Stokes equations. The solution results are visualized using TECPLOT. Supersonic flow over the rearward facing step is strategically solved using a flat plate solution paradigm. Parametric studies performed for the rearward facing step flow indicate that the Mach number and the step height affect flow characteristics such as corner expansion, recirculation zone, and the base pressure which are of great importance in the design of SCRAMJET engine. [Preview Abstract] |
Monday, November 21, 2005 8:52AM - 9:05AM |
FG.00005: Implementation of WENO schemes in compressible multicomponent flow problems Eric Johnsen, Tim Colonius Shock-capturing schemes are capable of properly resolving discontinuities with correct wave speeds in single-fluid Riemann problems. However, when different fluids are present, oscillations develop at interfaces. A class of existing methods that suppress these oscillations is based on first- and second-order accurate reconstructions with Roe solvers. In this presentation, we extend these methods to high-order accurate Weighted Essentially Non-Oscillatory (WENO) schemes and Harten, Lax and van Leer (HLL) approximate Riemann solvers. In particular, we show that a finite volume scheme where the appropriately averaged primitive variables are reconstructed leads to oscillation-free solutions to multicomponent Riemann problems. We restrict our analysis to a stiffened equation of state, which can model interfaces in flows of gas and liquid components. Our method is high-order accurate, conservative, and positivity-preserving; these properties are verified by considering one-dimensional multicomponent Riemann problems and a two-dimensional shock-bubble interaction. [Preview Abstract] |
Monday, November 21, 2005 9:05AM - 9:18AM |
FG.00006: Improving Shock-Free Compressible RANS Solvers for LES on Unstructured Meshes Laurent Georges, Philippe Geuzaine The objective of this contribution is to describe modifications required by standard RANS-like second-order discretizations on unstructured meshes to perform equally well for LES applications. The major issue is the effect of the numerical dissipation introduced to stabilize the discretization of the convective fluxes. It is well-known that this dissipation competes and often overwhelms the effect of the subgrid scale (SGS) model. An easy way to circumvent this problem is to resort to kinetic-energy conserving central schemes. Following the work by Mahesh and al. (JCP, 2004), we have developed and implemented an extension to compressible shock-free flows of their kinetic-energy conserving scheme (initially developed for incompressible flow solvers on unstructured meshes). Yet stable, this scheme can be prone to large truncation errors (as any low order schemes). We have performed simulations to investigate whether the SGS model can prevent the formation of spurious oscillations and whether these oscillations can be avoided without the use of an additional numerical dissipation such as high order upwinding. To highlight the aforementioned issues and the proposed solutions to resolve them, we have successfully performed simulations of the flow past a sphere (DNS at \mbox{\textit{Re} = 300} and a LES with the WALE model at \mbox{\textit{Re} = 10,000}). [Preview Abstract] |
Monday, November 21, 2005 9:18AM - 9:31AM |
FG.00007: Taylor-Galerkin Residual Distribution Schemes with Applications to Astrophysical Flows James Rossmanith Residual distribution (RD) schemes are multidimensional extensions of the upwind method for solving hyperbolic PDEs. These schemes are most often implemented on unstructured triangular (tetrahedral) grids in 2D (3D). For steady-state flows, these methods have been shown to produce accurate and efficient results for several hyperbolic systems including the compressible Euler equations of gas dynamics and the ideal MHD equations of plasma physics. In particular, RD schemes have the capability to accurately approximate shock waves and contact discontinuities. In the last few years, research has focused on developing high-order versions of these methods for fully time-dependent flows. In this work we present some preliminary results on a Taylor-Galerkin residual distribution scheme for time-dependent flows. The work we present on RD schemes is part of a larger effort to develop more accurate and efficient computational methods for simulating astrophysical flows. Examples of phenomena where such tools are required are the accretion of matter onto black holes and the dynamics of pulsar wind nebulae. Under the assumption that the space-time metric remains fixed on the time scales of fluid motion, these flows are governed by the equations of relativistic hydrodynamics. In this work, in addition to a discussion of our work on Taylor- Galerkin residual distribution schemes in general, we will present some numerical examples from astrophysical fluid dynamics. [Preview Abstract] |
Monday, November 21, 2005 9:31AM - 9:44AM |
FG.00008: A numerical solution of hyperbolic equations using the PCMFS method Christoph Schmitt, Zvi Rusak, Suvranu De The numerical solution of the inviscid wave and Burgers equations using the point collocation-based method of finite spheres (PCMFS) [1] is developed. The PCMFS is a meshfree numerical technique for the solution of PDEs on complex domains which uses the moving least squares (MLS) method for spatial discretization. The burden of mesh generation and remeshing in solving propagating shock waves is therefore mitigated. Each MLS shape function is compactly supported on a ball of radius $r.$ Temporal discretization uses a first-order backward difference scheme. Solution accuracy is governed by the spatial step$\Delta x$, the Courant number$C=u_{^{\max }} \Delta t/\Delta x$, and the relative radius$\rho =r/\Delta x$. The solution of the linear wave equation shows that $\rho $ has no significant influence on solution accuracy. As time step$\Delta t$ decreases, errors in the form of dissipation reduce and relatively small errors in the form of dispersion appear. As $\Delta x$decreases, dispersion error is significantly reduced and dissipation error is unaffected. The solution of the Burgers equation shows that for$\rho >3$ dissipation errors dominate while for $\rho <3$ dispersion error is more significant. For $\rho \to 1$ the solution error is the least. For $C<1$ dispersion error dominates which is superseded by dissipation error for$C>1$. A significant observation is that for $C=1$ a shock wave propagates without dissipation or dispersion errors for$\rho \approx 1$. [1] De et al., \textit{Computers {\&} Structures} 83, (17-18), 1415-1425, 2005$.$ [Preview Abstract] |
Monday, November 21, 2005 9:44AM - 9:57AM |
FG.00009: An artificial nonlinear diffusivity method for supersonic reacting flows with shocks Benoit Fiorina, Sanjiva K. Lele A computational approach for modeling interactions between shocks waves, contact discontinuities and reactions zones with a high order compact scheme is investigated. To prevent the formation of spurious oscillations around shocks, artificial nonlinear viscosity [1], based on high-order derivative of the strain rate tensor is used. To capture temperature and species discontinuities a nonlinear diffusivity based on the entropy gradients is added. The damping of `wiggles' is controlled by the model constants and is largely independent of the mesh size and the shock strength. The same holds for the numerical shock thickness and allows a determination of the L2 error. In the shock tube problem, with fluids of different initial entropy separated by the diaphragm, an artificial diffusivity is required to accurately capture the contact surface. Finally, the method is applied to a CJ detonation wave and to multi-dimensional flows including 2-D oblique wave reflection and a jet in a supersonic cross-flow.\newline \newline [1] Cook {\&} Cabot, J. Comput. Phys. 203 (2005) [Preview Abstract] |
Monday, November 21, 2005 9:57AM - 10:10AM |
FG.00010: Supernova and Hydrodynamic Instabilities Srabasti Dutta We have developed a curved geometry front tracking algorithm for interface instabilities. The code has been verified by comparing simulations to analytical solutions and various experiments. Using this algorithm, we have proved that front tracking is an accurate and efficient algorithm in the sense that tracking an interface can reduce the error significantly. We have also conducted numerical simulations of Richtmyer-Meshkov instabilities in spherical geometry, and have demonstrated scaling invariant with respect to shock Mach number for fluid mixing statistics. Our results are validated by the convergence under both mesh refinement and statistical ensemble average. We also show that the spherical geometry converge to planar geometry when the number of modes of interface perturbation goes to infinity. We also present a tracked sharp flame numerical model for thermonuclear explosion of Chandrasekhar mass white dwarfs, also is called Type Ia supernova. Simulations for turbulent combustion in Type Ia supernova have been carried out by using this model. [Preview Abstract] |
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