Time-Derivative Preconditioning Methods for Multicomponent Flows—Part II: Two-Dimensional Applications

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
Jeffrey A. Housman ◽  
Cetin C. Kiris ◽  
Mohamed M. Hafez
Keyword(s):  
Numerical Methods ◽  
Riemann Problem ◽  
Hybrid Method ◽  
Time Derivative ◽  
Hybrid Approach ◽  
Implicit Time ◽  
Two Dimensional ◽  
First Case ◽  
Multicomponent Flows ◽  
Time Marching

A time-derivative preconditioned system of equations suitable for the numerical simulation of multicomponent/multiphase inviscid flows at all speeds was described in Part I of this paper. The system was shown to be hyperbolic in time and remain well conditioned in the incompressible limit, allowing time marching numerical methods to remain an efficient solution strategy. Application of conservative numerical methods to multicomponent flows containing sharp fluid interfaces was shown to generate nonphysical pressure and velocity oscillations across the contact surface, which separates the fluid components. It was demonstrated using the one-dimensional Riemann problem that these oscillations may lead to stability problems when the interface separates fluids with large density ratios, such as water and air. The effect of which leads to the requirement of small physical time steps and slow subiteration convergence for the implicit time marching numerical method. Alternatively, the nonconservative and hybrid formulations developed by the present authors were shown to eliminate this nonphysical behavior. While the nonconservative method did not converge to the correct weak solution for flow containing shocks, the hybrid method was able to capture the physically correct entropy solution and converge to the exact solution of the Riemann problem as the grid is refined. In Part II of this paper, the conservative, nonconservative, and hybrid formulations described in Part I are implemented within a two-dimensional structured body-fitted overset grid solver, and a study of two unsteady flow applications is reported. In the first application, a multiphase cavitating flow around a NACA0015 hydrofoil contained in a channel is solved, and sensitivity to the cavitation number and the spatial order of accuracy of the discretization are discussed. Next, the interaction of a shock moving in air with a cylindrical bubble of another fluid is analyzed. In the first case, the cylindrical bubble is filled with helium gas, and both the conservative and hybrid approaches perform similarly. In the second case, the bubble is filled with water and the conservative method fails to maintain numerical stability. The performance of the hybrid method is shown to be unchanged when the gas is replaced with a liquid, demonstrating the robustness and accuracy of the hybrid approach.

Time-Derivative Preconditioning Methods for Multicomponent Flows—Part I: Riemann Problems

2009 ◽  
Vol 76 (2) ◽  
Author(s):  
Jeffrey A. Housman ◽  
Cetin C. Kiris ◽  
Mohamed M. Hafez
Keyword(s):  
Numerical Methods ◽  
Time Derivative ◽  
Density Ratio ◽  
Stability And Convergence ◽  
Implicit Time ◽  
Fluid Interfaces ◽  
Incompressible Limit ◽  
Riemann Problems ◽  
Multicomponent Flows ◽  
Time Marching

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid multicomponent and multiphase flows at all speeds is described. The system is shown to be hyperbolic in time and remains well conditioned in the incompressible limit, allowing time marching numerical methods to remain an efficient solution strategy. It is well known that the application of conservative numerical methods to multicomponent flows containing sharp fluid interfaces will generate nonphysical pressure and velocity oscillations across the component interface. These oscillations may lead to stability problems when the interface separates fluids with large density ratio, such as water and air. The effect of which may lead to the requirement of small physical time steps and slow subiteration convergence for implicit time marching numerical methods. At low speeds the use of nonconservative methods may be considered. In this paper a characteristic-based preconditioned nonconservative method is described. This method preserves pressure and velocity equilibrium across fluid interfaces, obtains density ratio independent stability and convergence, and remains well conditioned in the incompressible limit of the equations. To extend the method to transonic and supersonic flows containing shocks, a hybrid formulation is described, which combines a conservative preconditioned Roe method with the nonconservative preconditioned characteristic-based method. The hybrid method retains the pressure and velocity equilibrium at component interfaces and converges to the physically correct weak solution. To demonstrate the effectiveness of the nonconservative and hybrid approaches, a series of one-dimensional multicomponent Riemann problems is solved with each of the methods. The solutions are compared with the exact solution to the Riemann problem, and stability of the numerical methods are discussed.

Numerical Modeling of Radiative and Reactive Flow Field Around a Blunt Body at Hypersonic Regimes

2010 ◽  
Author(s):  
Morteza Rahmanpour ◽  
Reza Ebrahimi ◽  
Mehrzad Shams
Keyword(s):  
Flow Field ◽  
Differential Equations ◽  
Heat Fluxes ◽  
Transitional Flow ◽  
Navier Stokes ◽  
Implicit Time ◽  
Discrete Ordinates ◽  
Two Dimensional ◽  
Fully Implicit ◽  
Time Marching

A numerical method for calculation of strong radiation for two-dimensional reactive air flow field is developed. The governing equations are taken to be two dimensional, compressible Reynolds-average Navier-Stokes and species transport equations. Also, radiation heat flux in energy equation is evaluated using a model of discrete ordinate method. The model used S4 approximation to reduce the governing system of integro-differential equations to coupled set of partial differential equations. A multiband model is used to construct absorption coefficients. Tangent slab approximation is assumed to determine the characteristic parameters needed in the Discrete Ordinates Method. The turbulent diffusion and heat fluxes are modeled by Baldwin and Lomax method. The flow solution is obtained with a fully implicit time marching method. A thermochemical nonequilibrium formulation appropriate to hypersonic transitional flow of air is presented. The method is compared with existing experimental results and good agreement is observed.

scholarly journals A Parallel Adaptive Newton-Krylov-Schwarz Method for 3D Compressible Inviscid Flow Simulations

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Marzio Sala ◽  
Pénélope Leyland ◽  
Angelo Casagrande
Keyword(s):  
Compressible Flows ◽  
Time Derivative ◽  
Inviscid Flow ◽  
Implicit Time ◽  
Residual Distribution ◽  
Preconditioning Techniques ◽  
Flow Simulations ◽  
Efficient Data ◽  
Time Marching ◽  
2D And 3D

A parallel adaptive pseudo transient Newton-Krylov-Schwarz (αΨNKS) method for the solution of compressible flows is presented. Multidimensional upwind residual distribution schemes are used for space discretisation, while an implicit time-marching scheme is employed for the discretisation of the (pseudo)time derivative. The linear system arising from the Newton method applied to the resulting nonlinear system is solved by the means of Krylov iterations with Schwarz-type preconditioners. A scalable and efficient data structure for theαΨNKS procedure is presented. The main computational kernels are considered, and an extensive analysis is reported to compare the Krylov accelerators, the preconditioning techniques. Results, obtained on a distributed memory computer, are presented for 2D and 3D problems of aeronautical interest on unstructured grids.

scholarly journals Coupled Numerical Methods to Analyze Interacting Acoustic-Dynamic Models by Multidomain Decomposition Techniques

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Delfim Soares
Keyword(s):  
Numerical Methods ◽  
Dynamic Models ◽  
Coupled Model ◽  
Implicit Time ◽  
Decomposition Techniques ◽  
Iterative Coupling ◽  
Coupling Approach ◽  
Time Marching ◽  
Coupling Algorithms ◽  
Element Method

In this work, coupled numerical analysis of interacting acoustic and dynamic models is focused. In this context, several numerical methods, such as the finite difference method, the finite element method, the boundary element method, meshless methods, and so forth, are considered to model each subdomain of the coupled model, and multidomain decomposition techniques are applied to deal with the coupling relations. Two basic coupling algorithms are discussed here, namely the explicit direct coupling approach and the implicit iterative coupling approach, which are formulated based on explicit/implicit time-marching techniques. Completely independent spatial and temporal discretizations among the interacting subdomains are permitted, allowing optimal discretization for each sub-domain of the model to be considered. At the end of the paper, numerical results are presented, illustrating the performance and potentialities of the discussed methodologies.

scholarly journals Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

2021 ◽  
Vol 11 (8) ◽  
pp. 3421
Author(s):  
Cheng-Yu Ku ◽  
Li-Dan Hong ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang
Keyword(s):  
Porous Media ◽  
Time Domain ◽  
Numerical Solutions ◽  
Space Time ◽  
Two Dimensional ◽  
Transient Flows ◽  
Layered Porous Media ◽  
Time Marching ◽  
Time Basis ◽  
Marching Scheme

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

scholarly journals Bra-ket wormholes in gravitationally prepared states

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Yiming Chen ◽  
Victor Gorbenko ◽  
Juan Maldacena
Keyword(s):  
Hyperbolic Space ◽  
Path Integral ◽  
Flat Space ◽  
Low Energy ◽  
Two Dimensional ◽  
De Sitter ◽  
Euclidean Case ◽  
Fine Grained ◽  
Entropy Formula ◽  
First Case

Abstract We consider two dimensional CFT states that are produced by a gravitational path integral.As a first case, we consider a state produced by Euclidean AdS2 evolution followed by flat space evolution. We use the fine grained entropy formula to explore the nature of the state. We find that the naive hyperbolic space geometry leads to a paradox. This is solved if we include a geometry that connects the bra with the ket, a bra-ket wormhole. The semiclassical Lorentzian interpretation leads to CFT state entangled with an expanding and collapsing Friedmann cosmology.As a second case, we consider a state produced by Lorentzian dS2 evolution, again followed by flat space evolution. The most naive geometry also leads to a similar paradox. We explore several possible bra-ket wormholes. The most obvious one leads to a badly divergent temperature. The most promising one also leads to a divergent temperature but by making a projection onto low energy states we find that it has features that look similar to the previous Euclidean case. In particular, the maximum entropy of an interval in the future is set by the de Sitter entropy.

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