Direct numerical simulation of compressible turbulent channel flows using the discontinuous Galerkin method

Direct numerical simulation of turbulent channel flows between isothermal walls have been carried out using discontinuous Galerkin method. Three Mach numbers are considered (0.2, 0.7, and 1.5) at a fixed Reynolds number ≈2800, based on the bulk velocity, bulk density, half channel width, and dynamic viscosity at the wall. Power law and log-law with the scaling of the mean streamwise velocity are considered to study their performance on compressible flows and their dependence on Mach numbers. It indicates that power law seems slightly better and less dependent on Mach number than the log-law in the overlap region. Mach number effects on the second-order (velocity, pressure, density, temperature, shear stress, and vorticity fluctuations) and higher-order (skewness and flatness of velocity, pressure, density, and temperature fluctuations) statistics are explored and discussed. Both inner (that is wall variables) and outer (that is global) scalings (with Mach number) are considered. It is found that for some second-order statistics (i.e. velocity, density, and temperature), the outer scaling collapses better than the inner scaling. It is also found that near-wall large-scale motions are affected by Mach number. The near-wall spanwise streak spacing increases with increasing Mach number. Iso-surfaces of the second invariant of the velocity gradient tensor are more sparsely distributed and elongated as Mach number increases, which is similar to the distribution of near-wall low speed streaks.     

Application of a discontinuous Galerkin finite element method to special relativistic hydrodynamic models

A Runge–Kutta discontinuous Galerkin (RKDG) finite element method is proposed for solving the special relativistic hydrodynamic (SRHD) equations and as a limiting case the ultra-relativistic hydrodynamic (URHD) equations. The latter model is obtained by ignoring the rest-mass energy when the internal energy of fluid particles is sufficiently large. Several test problems of SRHD and URHD models are carried out. For validation, the results of DG-method are compared with the staggered central scheme. The numerical results verify the accuracy of the proposed method qualitatively and quantitatively.     

Numerical solutions of unsteady flows with low inlet Mach numbers

This study deals with a numerical solution of a 2D unsteady flow of a compressible viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract and the flow can represent a model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds, and to the human vocal tract.     

Comparing the Adomian decomposition method and the Runge–Kutta method for solutions of the Stefan problem

The solution of the one-phase Stefan problem is presented. A Stefan's task is first approximated with a system of ordinary differential equations. A comparison between the Adomian decomposition method and the fourth-order Runge–Kutta method for solving this system is then presented.     

Convergence acceleration for computing steady-state compressible flow at low Mach numbers

A novel technique is introduced to accelerate the convergence of compressible low Mach number flow computations to the steady state. The stiffness due to the large disparity of flow velocity and acoustic wave speeds is bypassed by artificially reducing the speed of sound and thereby increasing the Mach number. This Mach number transformation is achieved by subtracting a constant value from the pressure in the entire flow field. Only the inviscid terms of the energy equation are influenced by that pressure decrease. The steady-state error is corrected by solving a scalar equation after each time step such that the steady-state solutions of the modified and non-modified schemes coincide. Thus, for each low Mach number simulation, one can obtain a convergence performance comparable to the corresponding simulation with a Mach number of about 0.4. This convergence acceleration is demonstrated for premixed laminar flames. If the present technique is implemented without time splitting, it corresponds to a novel low Mach number preconditioning.     

Implementation of the low Mach number method for calculating flows in the NOISEtte software package

This paper is devoted to the development of computational techniques for simulating low Mach number flows on unstructured meshes based on the Roe method and the edge-based vertexcentered higher accuracy schemes. The techniques are implemented in the in-house NOISEtte code. The results of predicting an inviscid compressible low Mach number flow over a NACA0012 airfoil are presented and analyzed. The computations are carried out on structured and unstructured triangular grids.     

Prediction of entrance length for low Reynolds number flow in pipe using neuro-fuzzy inference system

This paper proposes an adaptive network fuzzy inference system (ANFIS) for the prediction of entrance length in pipe for low Reynolds number flow. After using the computational fluid dynamics (CFD) technique to establish the basic database under various working conditions, an efficient rule database and optimal distribution of membership function is constructed from the hybrid-learning algorithm of ANFIS. An experimental data set is obtained with Reynolds number, diameter of the pipe, and inlet velocity as input parameters and entrance length as output parameter. The input-output data set is used for training and validation of the proposed techniques. After validation, they are forwarded for the prediction of entrance length. The entrance length estimation results obtained by the model are compared with existing predictive models and are presented. The model performed quite satisfactory results with the actual and predicted entrance length values. The model can also be used for estimating entrance length on-line but the accuracy of the model depends upon the proper training and selection of data points.     

The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M²) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM ² to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM ²= 0 in order to derive a solution with very small divergence can with good result be carried through withM ² as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatt · (i/(M x)–v/x ²) is in the stability region of the time stepping method, and asM ²=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.     

A high-resolution penalization method for large Mach number flows in the presence of obstacles

A penalization method is applied to model the interaction of large Mach number compressible flows with obstacles. A supplementary term is added to the compressible Navier-Stokes system, seeking to simulate the effect of the Brinkman-penalization technique used in incompressible flow simulations including obstacles. We present a computational study comparing numerical results obtained with this method to theoretical results and to simulations with Fluent software. Our work indicates that this technique can be very promising in applications to complex flows.     

The multistage homotopy analysis method: application to a biochemical reaction model of fractional order

In this paper, a new reliable algorithm called the multistage homotopy analysis method (MHAM) based on an adaptation of the standard homotopy analysis method (HAM) is presented to solve a time-fractional enzyme kinetics. This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The new algorithm is only a simple modification of the HAM, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. Numerical comparisons between the MHAM and the classical fourth-order Runge–Kutta method in the case of integer-order derivatives reveal that the new technique is a promising tool for nonlinear systems of integer and fractional order.     

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell’s equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space     

A new method of embedded fourth order with four stages to study raster CNN simulation

A new Runge-Kutta （PK） fourth order with four stages embedded method with error control is presentea m this paper for raster simulation in cellular neural network （CNN） environment. Through versatile algorithm, single layer/raster CNN array is implemented by incorporating the proposed technique. Simulation results have been obtained, and comparison has also been carried out to show the efficiency of the proposed numerical integration algorithm. The analytic expressions for local truncation error and global truncation error are derived. It is seen that the RK-embedded root mean square outperforms the RK-embedded Heronian mean and RK-embedded harmonic mean.     

Application of the method of manufactured solutions to the verification of a pressure-based finite-volume numerical scheme

The present study reports a numerical procedure based on a series of tests that make use of the method of manufactured solutions (MMS) and allow to evaluate the effective numerical performance with respect to the theoretical order of accuracy. The method is applied to a pressure-based finite volume numerical scheme suited to variable density flows representative of those encountered in combustion applications. The algorithm is based on a predictor-corrector time integration scheme that employs a projection method for the momentum equations. A physically consistent constraint is retained to ensure that the velocity field is solved correctly. The MMS application shows that the combination of this velocity constraint and the variable-coefficient Poisson solver is of fundamental importance to ensure both the numerical stability and the expected order of accuracy. Especially, the resort to an inner iteration procedure gives rise to undeniable improvements in terms of both the order of accuracy and error magnitude. The MMS applications confirm the interest of the method to conduct a preliminary check of the performance of any numerical algorithm applied to both fully incompressible and variable density flows. Finally, the analysis is ended by the application of the retained pressure-based finite-volume scheme to the numerical simulation of mixing layers featuring increasing values of the density contrast. The corresponding results shed some light onto the stability and robustness of the numerical scheme, important issues that are not addressed through MMS analyses.     

A multiphase compressible model for the simulation of multiphase flows

A compressible model able to manage incompressible two-phase flows as well as compressible motions is proposed. After a presentation of the multiphase compressible concept, the new model and related numerical methods are detailed on fixed structured grids. The presented model is a 1-fluid model with a reformulated mass conservation equation which takes into account the effects of compressibility. The coupling between pressure and flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The numerical model is then validated with four test cases involving the compression of an air bubble by water, the liquid injection in a closed cavity filled with air, a bubble subjected to an ultrasound field and finally the oscillations of a deformed air bubble in melted steel. The numerical results are compared with analytical results and convergence orders in space are provided.     

A Numerical Comparison of the Lax–Wendroff Discontinuous Galerkin Method Based on Different Numerical Fluxes

Discontinuous Galerkin (DG) method is a spatial discretization procedure, employing useful features from high-resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. In [(2005). Comput. Methods Appl. Mech. Eng. 194, 4528], we developed a Lax–Wendroff time discretization procedure for the DG method (LWDG), an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. In most of the DG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes, which could also be used. In this paper, we systematically investigate the performance of the LWDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., the second-order TVD fluxes and generalized Riemann solver, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two-dimensional systems.      

Optimal error estimates of the penalty method for the linearized viscoelastic flows

In this article, optimal error estimates of the penalty method for the linearized viscoelastic flows equations arising in the Oldroyd model are derived. Furthermore, error estimates for the backward Euler time discretization scheme in L ² and H ¹-norms are obtained.     

A parallel multilevel preconditioned iterative pressure Poisson solver for the large-eddy simulation of turbulent flow inside a duct

Turbulent Poiseuille flows inside the square duct are simulated by the large-eddy simulation based on the multilevel Schwarz preconditioned conjugate gradient pressure Poisson solver, which was developed on top of the Portable, Extensible Toolkit for Scientific Computation (PESTc). The impact of the five different matrix reordering techniques for an incomplete LU (ILU) decomposition as a subdomain solver on the overall performance of Schwarz-type preconditioners for the solution of the pressure Poisson equation are studied. The numerical results indicate that ILU of two-level fill-ins with the reverse Cuthill–McKee matrix ordering technique produces the best performance. Further investigation on the parallel performance of different multilevel methods was also conducted for two different problem sizes. It was observed that the computational cost saturates at around six-level for both the problem sizes explored. Also, though the one-level method is better for small problem size, for the larger problem size, the six-level method performs best in terms of scalability and compute time; hence, the benefit of a multilevel method is more obviously.     

A direct spectral domain decomposition method for the computation of rotating flows in a T-shape geometry

This paper presents a direct domain decomposition method, coupled with a Chebyshev collocation approximation, for solving the incompressible Navier-Stokes equations in the vorticity-streamfunction formulation. The method is based on the influence matrix technique used to treat the lack of vorticity boundary conditions on no-slip walls as well as to enforce the continuity conditions at the interfaces between adjacent subdomains. The multi-domain approach is proposed in order to extend the use of spectral approximations to non-rectangular geometries and singular solutions. It is applied to the computation of a four domain configuration, corresponding to a forced throughflow in a rotating channel-cavity system which is important in air cooling devices and cannot be modeled by single-domain spectral approximations.     

Local discontinuous Galerkin methods based on the multisymplectic formulation for two kinds of Hamiltonian PDEs

ABSTRACT

This paper examines the novel local discontinuous Galerkin (LDG) discretization for Hamiltonian PDEs based on its multisymplectic formulation. This new kind of LDG discretizations possess one major advantage over other standard LDG method, which, through specially chosen numerical fluxes, states the preservation of discrete conservation laws (i.e. energy), and also the multisymplectic structure while the symplectic time integration is adopted. Moreover, the corresponding local multisymplectic conservation law holds at the units of elements instead of each node. Taking the nonlinear Schrödinger equation and the KdV equation as the examples, we illustrate the derivations of discrete conservation laws and the corresponding numerical fluxes. Numerical experiments by using the modified LDG method are demonstrated for the sake of validating our theoretical results.     

Study of a Jacobian-free approach in the simulation of compressible fluid flows in porous media using a derivative-free spectral method

The development of Jacobian-free software for solving problems formulated by nonlinear partial differential equations is of increasing interest to simulate practical engineering processes. For the first time, this work uses the so-called derivative-free spectral algorithm for nonlinear equations in the simulation of flows in porous media. The model considered here is the one employed to describe the displacement of miscible compressible fluid in porous media with point sources and sinks, where the density of the fluid mixture varies exponentially with the pressure. This spectral algorithm is a modern method for solving large-scale nonlinear systems, which does not use any explicit information associated with the Jacobin matrix of the considered system, being a Jacobian-free approach. Two dimensional problems are presented, along with numerical results comparing the spectral algorithm to a well-developed Jacobian-free inexact Newton method. The results of this paper show that this modern spectral algorithm is a reliable and efficient method for simulation of compressible flows in porous media.