Smooth Flow Regions Research Articles

Shock capturing with entropy-based artificial viscosity for staggered grid discontinuous spectral element method

This paper presents a shock capturing technique for a staggered grid discontinuous spectral element method (DSEM), which adds localized and smooth artificial viscosity to systems of nonlinear conservation laws. The artificial diffusivity model used in this work is a modified form of the entropy viscosity (EV) presented by Guermond et al. (2011). We extend the application of this method to high-order discontinuous schemes for the simulation of high speed flows with discontinuities on staggered grids. Direct implementation of the entropy viscosity method in DSEM leads to a non-smooth artificial viscosity, which in turn leads to oscillations and instability of the solution. To smoothen the artificial viscosity, the EV method is coupled with a spectral filter and an interface treatment technique. The resulting artificial viscosity is locally large near discontinuities and transitions smoothly to zero in smooth flow regions. The method enables using elements with orders higher than unity while avoiding adaptive mesh refinement and preserving the locality and compactness of the discontinuous Galerkin (DG) scheme. The method is implemented for the inviscid compressible Euler equations in two space dimensions and its effectiveness is demonstrated through its application to a series of benchmark problems.

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two- and three-dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two-level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two-level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two- and three-dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two- to three-dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non-symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.

A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics

A new method based on a continuous, piece-wise linear approximation of the equations for Lagrangian shock hydrodynamics is presented. Numerical instabilities are controlled by a stabilizing operator derived using the paradigm of the variational multiscale analysis. Encouraging numerical comparisons with existing methods in the case of quadrilateral and hexahedral elements indicate that the proposed method is capable of preventing hourglass patterns in the solution, while maintaining accuracy in regions of smooth flow. The proposed approach satisfies Galilean invariance properties and hinges upon the interpretation of the Lagrangian shock hydrodynamics equations as a system of nonlinear wave equations. A specific implementation in terms of a predictor/multi-corrector version of the mid-point time integrator guarantees global conservation of mass, momentum, and total energy for each iterate. Stability and formal order of accuracy are investigated applying the von Neumann analysis to the linearized shock hydrodynamics equations in one dimension. This approach yields tight bounds for stable time-step estimation, formal second-order accuracy of the method in time and space, and valuable indications on the choice of the most appropriate values for the stabilization parameters present in the formulation.

An h-Adaptive Finite-Element Technique for Constructing 3D Wind Fields

Abstract An h-adaptive, mass-consistent finite-element model (FEM) has been developed for constructing 3D wind fields over irregular terrain utilizing sparse meteorological tower data. The element size in the computational domain is dynamically controlled by an a posteriori error estimator based on the L2 norm. In the h-adaptive FEM algorithm, large element sizes are typically associated with smooth flow regions and small errors; small element sizes are attributed to fast-changing flow regions and large errors. The adaptive procedure employed in this model uses mesh refinement–unrefinement to satisfy error criteria. Results are presented for wind fields using sparse data obtained from two regions within Nevada: 1) the Nevada Test Site, located approximately 65 mi (1 mi ≈ 1.6 km) northwest of Las Vegas, and 2) the central region of Nevada, about 100 mi southeast of Reno.

Flow characteristics of a small moving bed downcomer with an orifice under negative pressure gradient

Behaviour of gas–solid mixtures in a small moving bed downcomer operating under negative pressure gradient has been investigated experimentally, because flow characteristics in a downcomer are important factors in powder handling processes, such as a multi-stage spouted bed or a circulating fluidized bed. In this study, a small downcomer in which the moving bed height could be controlled was employed with the intention of using a downcomer as a connecting tube in a multi-stage spouted bed. Based on experimental observation, the moving bed flow changed from smooth flow (Region(I)) to slip-stick flow (Region(II)) and then changed from slip-stick flow to intermittent flow (Regime(III)), as the gas pressure gradient in the downcomer was increased. In this study, the critical point for steady-state (Region(I)) operation was also examined experimentally and theoretically, by using eight kinds of solid particles (Geldart B, D and A powders) and six different orifice sizes which were used to restrict solids flow at the base of the downcomer. As a result, a correlation of solids flow rate in the stable moving bed downcomer under negative pressure gradient was derived and the critical point between stable and unstable flow was explained by the proposed model.

Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes

A simple and efficient localized artificial diffusivity scheme is developed for the purpose of capturing discontinuities on curvilinear and anisotropic meshes using a high-order compact differencing scheme. The artificial diffusivity is dynamically localized in space to capture different types of discontinuities such as a shock wave, contact surface or material discontinuity. The method is intended for use with large-eddy simulation of compressible transitional and turbulent flows. The method captures the discontinuities on curvilinear and anisotropic meshes with minimum impact on the smooth flow regions. The amplitude of wiggles near a discontinuity and the number of grid points used to capture the discontinuity do not depend on the mesh size. The comparisons between the proposed method and high-order shock-capturing schemes illustrate the advantage of the method for the simulation of flows involving shocks, turbulence and their interactions. The multi-dimensional formulation is tested on a variety of 1D and 2D, steady and unsteady, different types of discontinuity-related problems on curvilinear and anisotropic meshes. A simplification of the method which reduces the computational cost does not show any major detrimental effect on the discontinuity capturing under the conditions examined.

Improvements in speed for explicit, transient compressible flow solvers

AbstractSeveral explicit high‐resolution schemes for transient compressible flows with moving shocks are combined in such a way so as to achieve the highest possible speed without compromising accuracy. The main algorithmic changes considered comprise the following: replacing limiting and approximate Riemann solvers by simpler schemes during the initial stages of Runge–Kutta solvers, and only using limiting and approximate Riemann solvers for the last stage; automatically switching to simpler schemes for smooth flow regions; automatic deactivation of quiescent regions; and unstructured grids with cartesian cores or embedded cartesian grids. The results from several examples demonstrate that speedup factors of 1:4 are attainable without compromising the accuracy of the traditional FCT schemes. Copyright © 2007 John Wiley & Sons, Ltd.

Runge–Kutta time‐stepping schemes with TVD central differencing for the water hammer equations

AbstractIn the present study, Runge–Kutta schemes are used to simulate unsteady flow in elastic pipes due to sudden valve closure. The spatial derivatives are discretized using a central difference scheme. Second‐order dissipative terms are added in regions of high gradients while they are switched off in smooth flow regions using a total variation diminishing (TVD) switch. The method is applied to both one‐ and two‐dimensional water hammer formulations. Both laminar and turbulent flow cases are simulated. Different turbulence models are tested including the Baldwin–Lomax and Cebeci–Smith models. The results of the present method are in good agreement with analytical results and with experimental data available in the literature. The two‐dimensional model is shown to predict more accurately the frictional damping of the pressure transient. Moreover, through order of magnitude and dimensional analysis, a non‐dimensional parameter is identified that controls the damping of pressure transients in elastic pipes. Copyright © 2006 John Wiley & Sons, Ltd.

Smooth transonic flow in an array of counter-rotating vortices

Numerical solutions to the steady two-dimensional compressible Euler equations corresponding to a compressible analogue of the Mallier & Maslowe (Phys. Fluids, vol. A 5, 1993, p. 1074) vortex are presented. The steady compressible Euler equations are derived for homentropic flow and solved using a spectral method. A solution branch is parameterized by the inverse of the sound speed at infinity, . Along these branches numerical evidence for the existence of extensive regions of smooth steady transonic flow, with local Mach numbers as large as 1.276, is presented.

Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks

We develop a tuned center-difference (TCD) scheme optimized for large-eddy simulations (LES) using a method proposed by Ghosal. For LES of weakly compressible decaying turbulence, these optimized stencils are shown to provide superior performance when compared to higher-order centered schemes with the same stencil width. A hybrid method combining the TCD stencil with a weighted essentially non-oscillatory (WENO) method is then constructed for use in the LES of strongly compressible, shock-driven flows. The user-specified, optimum WENO weights are chosen to match those of the TCD scheme. It is expected that these weights will be achieved automatically in regions of smooth flow away from shocks, but in practice a switch is found to be necessary. The hybrid TCD–WENO scheme is shown to work well for unsteady gas-dynamic flows in one and two dimensions.

A theoretical Taylor–Galerkin model for first‐order hyperbolic equation

AbstractThis paper presents a class of Taylor–Galerkin (TG) finite‐element models for solving the first‐order hyperbolic equation which admits discontinuities. Five parameters are introduced for purposes of controlling stability, monotonicity and accuracy. In this paper, the total variation diminishing concept and the theory of M‐matrix are applied to construct a monotonic TG model for capturing discontinuities. To avoid making the scheme overly diffusive, we apply a flux‐corrected transport (FCT) technique of Boris and Book to overcome the difficulty with anti‐diffusive flux. In smooth flow regions, our strategyof developing the temporal and spatial high‐order TG finite‐element model is based on modified equation analysis. In regions where discontinuity is encountered, we resort to two dispersively more accurate models to make the prediction accuracy as high as that obtained in smooth cases. These models are developed using the entropy‐increasing principle and the theory of group velocity. Guided by this theory, a slower group velocity should be used ahead of the shock. To avoid a train of post‐shocks, free parameters should be chosen properly to obtain a group velocity which takes on a larger value than the exact phase velocity. In this paper, we also apply the entropy‐increasing principle to determine free parameters introduced in the finite‐element model. Under the entropy‐increasing requirement, it is mandatory that coefficients of the even and odd derivative terms shown in the modified equation should change signs alternatively in order to avoid non‐physical wiggles. Several benchmark problems have been investigated to confirm the integrity of these proposed characteristic models. Copyright © 2003 John Wiley & Sons, Ltd.

Wave structure in the contact line region during high speed droplet impact on a surface: Solution of the Riemann problem for the stiffened gas equation of state

Upon impact of a liquid droplet on a rigid substrate at sufficiently high velocity, a wave front adjacent to the impacted surface is created, separating the compressed area from the undisturbed bulk of the liquid. At a certain point in the evolution of the phenomenon, the free surface at the vicinity of the contact line yields and an intense lateral jetting phenomenon occurs. The often employed assumption that the wave front is composed of a single shock wave across its entire surface leads to the emergence of an anomaly, prior to the jetting eruption. This anomaly can be removed by the proposition of the existence of a multiple wave structure at the contact line region. In this article, the wave structure at the contact line region is explored analytically. The exact one-dimensional Riemann problem solution for the stiffened gas equation of state is presented, under the assumption of an isentropic flow in the smooth flow region (i.e., in the region with no contact discontinuities). For the case of impact velocity V=500 m/s and a water droplet with radius R=0.1 mm, we show that in addition to the single shock structure, solutions consisting of an expansion fan and shock wave are physically admissible. In this scenario, the pressure created at the contact line decreases compared to the pressure of the single shock structure. It is demonstrated that the liquid particle velocity in the intermediate region between the waves is lower than the velocity in the bulk of compressed liquid. An algorithm for estimating the intermediate region state, when the left and right states are known, is also presented.

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

Numerical Entropy Production on Shocks and Smooth Transitions

In this work, we propose an algorithm to measure the numerical entropy production of several high order central schemes. The results obtained on a numerical test indicate that this quantity permits to estimate the local error, in regions of smooth flow, and to detect and locate shocks.

A Finite Volume Central Differencing Scheme for Simulation of the Shut Down Procedure of a Hydraulic System

AbstractA new central differencing finite volume scheme is investigated for solution of unsteady hydraulic problems as water hammer in pipe systems. Special time stepping procedure similar to Runge-Kutta algorithm is used to stabilize this second order scheme. It is monotonized by adding dissipative terms including second and fourth derivatives of the conserved variables, with coefficients proportional to derivatives of pressure or volumetric flow, which keeps the second order of accuracy in smooth flow regions. The one-dimensional unsteady incompressible equations are solved for a water hammer situation, and results are compared to existing analytical solutions. Results are also compared with numerical results of classical characteristic method, which is proved to be fairly accurate. The scheme could easily be generalized to two-dimensional case. Finally this procedure is used for analysis of the shut down procedure of a hydraulic system. Components of the system are modeled and effects of important para...

Riemann Solvers in Relativistic Hydrodynamics: Basics and Astrophysical Applications

My contribution to these proceedings summarizes a general overview on High Resolution Shock Capturing methods (HRSC) in the field of relativistic hydrodynamics with special emphasis on Riemann solvers. HRSC techniques achieve highly accurate numerical approximations (formally second order or better) in smooth regions of the flow, and capture the motion of unresolved steep gradients without creating spurious oscillations. In the first part I will show how these techniques have been extended to relativistic hydrodynamics, making it possible to explore some challenging astrophysical scenarios. I will review recent literature concerning the main properties of different special relativistic Riemann solvers, and discuss several 1D and 2D test problems which are commonly used to evaluate the performance of numerical methods in relativistic hydrodynamics. In the second part I will illustrate the use of HRSC methods in several astrophysical applications where special and general relativistic hydrodynamical processes play a crucial role.

A proof of existence of perturbed steady transonic shocks via a free boundary problem

We prove the existence of a solution of a free boundary problem for the transonic small-disturbance equation. The free boundary is the position of a transonic shock dividing two regions of smooth flow. Assuming inviscid, irrotational flow, as modeled by the transonic small-disturbance equation, the equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. To study the stability of a uniform planar transonic shock, we consider perturbation by a steady C1+ϵ upstream disturbance. If the upstream disturbance is small in a C1 sense, then there is a steady solution in which the downstream flow and the transonic shock are Holder-continuous perturbations of the uniform configuration. This result provides a new use of inviscid perturbation techniques to demonstrate, in two dimensions, the stability of transonic shock waves of the type that appear, for example, over the wing of an airplane, along an airfoil, or as bow shocks in a flow with a supersonic free-stream velocity. © 2000 John Wiley & Sons, Inc.

A proof of existence of perturbed steady transonic shocks via a free boundary problem

We prove the existence of a solution of a free boundary problem for the transonic small-disturbance equation. The free boundary is the position of a transonic shock dividing two regions of smooth flow. Assuming inviscid, irrotational flow, as modeled by the transonic small-disturbance equation, the equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. To study the stability of a uniform planar transonic shock, we consider perturbation by a steady C1+ϵ upstream disturbance. If the upstream disturbance is small in a C1 sense, then there is a steady solution in which the downstream flow and the transonic shock are Hölder-continuous perturbations of the uniform configuration. This result provides a new use of inviscid perturbation techniques to demonstrate, in two dimensions, the stability of transonic shock waves of the type that appear, for example, over the wing of an airplane, along an airfoil, or as bow shocks in a flow with a supersonic free-stream velocity. © 2000 John Wiley & Sons, Inc.

Grid influence on upwind schemes for the Euler and Navier-Stokes equations

The grid influence on the standard Roe's upwind scheme in two space dimensions is investigated. It is shown that inaccurate solutions can be obtained even in smooth flow regions and even with regularly structured triangular meshes. Some numerical analyses and various inviscid and viscous computations are conducted to provide a better understanding of this behavior. High-order reconstructions together with the standard scheme are shown to give only limited improvements. A grid-independent upwind scheme based on a five-wave model has been tested and has shown some potential in improving the situation.

A Shock-Adaptive Godunov Scheme Based on the Generalised Lagrangian Formulation

Application of the Godunov scheme to the Euler equations of gas dynamics based on the Eulerian formulation of flow smears discontinuities, sliplines especially, over several computational cells, while the accuracy in the smooth flow region is of the order O(h), where h is the cell width. Based on the generalised Lagrangian formulation (GLF) of Hui et al., the Godunov scheme yields superior accuracy. By the use of coordinate streamlines in the GLF, the slipline—itself a streamline—is resolved crisply. Infinite shock resolution is achieved through the splitting of shock-cells. An improved entropy-conservation formulation of the governing equations is also proposed for computations in smooth flow regions. Finally, the use of the GLF substantially simplifies the programming logic resulting in a very robust, accurate, and efficient scheme.