Enhancement of an industrial finite-volume code for large-eddy-type simulation of incompressible high Reynolds number flow using near-wall modelling

We present a validation strategy for enhancement of an unstructured industrial finite-volume solver designed for steady RANS problems for large-eddy-type simulation with near-wall modelling of incompressible high Reynolds number flow. Different parts of the projection-based discretisation are investigated to ensure LES capability of the numerical method. Turbulence model parameters are calibrated by using a minimisation of least-squares functionals for first and second order statistics of the basic benchmark problems decaying homogeneous turbulence and turbulent channel flow. Then the method is applied to the flow over a backward facing step at Re_h = 37,500. Of special interest is the role of the spatial and temporal discretisation error for low order schemes. For wall-bounded flows, present results confirm existing best practice guidelines for mesh design. For free-shear layers, a sensor to quantify the resolution quality of the LES based on the resolved turbulent kinetic energy is presented and applied to the flow over a backward facing step at Re_h = 37,500.     

The finite volume element method with quadratic basis functions

The paper presents a box scheme with quadratic basis functions for the discretisation of elliptic boundary value problems. The resulting discretisation matrix is non-symmetrical (and also not an M-matrix). The stability analysis is based on an elementwise estimation of the scalar product <A _h u _h ,u _h >. Sufficient conditions placed on the triangles of the triangulation lead to discrete ellipticity. Proof of anO(h ²) error estimate is given for these conditions.     

Transient fluid–structure interaction with non-matching spatial and temporal discretizations

This paper presents a review of spatial and temporal discretization schemes for unsteady flow interacting with structure. Two types of spatial coupling schemes are analyzed: (i) point-to-element projection and (ii) common-refinement based projection. It is shown that the point-to-element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh, leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution. The common-refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non-matching meshes. With respect to the temporal discretization, three coupling techniques are assessed: (i) conventional sequential staggered (CSS); (ii) generalized serial staggered (GSS) and (iii) combined interface boundary condition (CIBC). Traditionally, continuity of velocity and traction along interfaces are satisfied through algebraic interface conditions applied in a sequential fashion, which is often referred to as staggered computation. In existing partitioned staggered procedures, the interface conditions may undermine stability and accuracy of coupled fluid–structure simulations. By utilizing the CIBC technique on the velocity and traction boundary conditions, a staggered coupling procedure can be constructed with similar order of accuracy and stability as standalone computations. The effectiveness of spatial and temporal coupling schemes is investigated with the aid of simple 1D examples and new 2D subsonic flow-shell aeroelastic simulations.     

A conservative pressure-correction method for flow at all speeds

A new fully conservative Mach-uniform staggered scheme is discussed. With this scheme one can compute flow with a Mach number ranging from the incompressible limit M↓0 up to supersonic flow M>1, with nearly uniform efficiency and accuracy. Earlier methods are based on a nonconservative discretisation of the energy equation. This results in small discrepancies in the computed shock speed for the Euler equations. The new method has similar Mach-uniform properties as the earlier methods, but is found to converge to the correct weak solution.     

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.     

Application of Computer Algebra Systems for Stability Analysis of Difference Schemes on Curvilinear Grids

The paper deals with problems arising in the application of the computer algebra systems for the symbolic–numeric stability analysis of difference schemes and schemes of the finite-volume method approximating the two-dimensional Euler equations for compressible fluid flows on curvilinear spatial grids. We carry out a detailed comparison of the REDUCE 3.6 and Mathematica(Versions 2.2 and 3.0) from the point of view of their applicability to the solution of the above problems. We draw a conclusion that a preference should be given for Mathematica from the viewpoint of the execution of symbolic–numeric computations. We also describe in detail our new symbolic–numeric algorithm for stability investigation, which was implemented with the aid of Mathematica. The proposed method enables us to reduce the needed computer storage at the symbolic stages by a factor of about 20 as compared with the previous algorithms. A feature of the numerical stages is the use of the arithmetic of rational numbers, which enables us to avoid the accumulation of the roundoff errors. We present the examples of the application of the proposed symbolic–numeric method for stability analysis of very complex schemes of the finite-volume method on curvilinear grids, which are widely used in computational fluid dynamics.     

Runge–Kutta Residual Distribution Schemes

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.     

A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains

A finite volume scheme which is based on fourth order accurate central differences in the spatial directions and on a hybrid explicit/semi-implicit time stepping scheme was developed to solve the incompressible Navier-Stokes equations on cylindrical staggered grids. This includes a new fourth order accurate discretization of the velocity at the singularity of the cylindrical coordinate system and a new stability condition. The new method was applied in the direct numerical simulations (DNS) of the fully developed non-swirling turbulent flow through straight pipes with circular cross-section for the Reynolds number Re_τ = 360 based on the friction velocity u_τ and the pipe diameter. The obtained results are expressed in terms of statistical moments of the velocity components and are presented in comparison with those obtained with a second order accurate scheme and by measurements. It is shown that the fourth order spatial discretization leads to improved higher order statistical moments, while the first and the second order moments are more or less insensitive to the spatial discretization order.     

Difference schemes for the dispersive equation

In this paper a table of difference schemes for the dispersive equationu _i=au _xxx is presented. A collection of criterions for deriving stability conditions of difference schemes is given and applied to these difference schemes.     

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     

Efficient frequency response computation for low-order modelling of spatially distributed systems

ABSTRACT

Motivated by the challenges of designing feedback controllers for spatially distributed systems, we present an efficient approach to obtaining the frequency response of such systems, from which low-order models can be identified. This is achieved by combining the frequency responses of constituent lower-order subsystems in a way that exploits the interconnectivity arising from spatial discretisation. This approach extends to the singular subsystems that arise upon spatial discretisation of systems governed by PDAEs, with fluid flows being a prime example. The main result of this paper is a proof that the computational complexity of forming the overall frequency response is minimised if the subsystems are merged in a particular fashion. Doing so reduces the complexity by several orders of magnitude; a result demonstrated upon the numerical example of a spatially discretised wave-diffusion equation. By avoiding the construction, storage, or manipulation of large-scale system matrices, this modelling approach is well conditioned and computationally tractable for spatially distributed systems consisting of enormous numbers of subsystems, therefore bypassing many of the problems with conventional model reduction techniques.     

A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems

We give a space-time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (cG(p)), with a discontinuous Galerkin piecewise constant (dG(0)) or linear (dG(1)) approximation in time. A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong L_p-energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time).We also give upper bounds on the dG(0)cG(1) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.     

Particle simulation methods in plasma physics

A brief survey of particle simulation methods is presented. Section 2 outlines the particle-mesh (PM) method for collisionless phase fluids. Section 3 describes how PM is extended to make simulation studies of dense plasmas feasible: The particle-particle/particle-mesh (P³M) algorithms reduces the cost scaling to αN_p from the αN²_p for conventional methods, where N_p is the number of particles. Section 4 summarises particle modelling of fluids and magnetofluids. New algorithms (EPIC schemes) arise when the optimisation techniques of collisionless PM models are applied to the fluid and MHD equations. These are shown to outperform state-of-the-art finite difference schemes in both cost and performance.     

A staggered scheme for hyperbolic conservation laws applied to unsteady sheet cavitation

We demonstrate the advantages of discretizing on a staggered grid for the computation of solutions to hyperbolic systems of conservation laws arising from instationary flow of an inviscid fluid with an arbitrary equation of state. Results for a highly nonlinear, nonconvex equation of state obtained with the staggered discretisation are compared with those obtained with the Osher scheme for two different Riemann problems. The staggered approach is shown to be superior in simplicity and efficiency, without loss of accuracy. The method has been applied to simulate unsteady sheet cavitation on a NACA0012 hydrofoil. Results show good agreement with those obtained with a cavity interface tracking method. Received: 25 February 1999 / Accepted: 17 June 1999     

Combined interface boundary condition method for unsteady fluid–structure interaction

Traditionally, continuity of velocity and traction along interfaces are satisfied through algebraic interface conditions applied in a sequential or staggered fashion. In existing staggered procedures, the numerical treatment of the interface conditions can undermine the stability and accuracy of coupled fluid–structure simulations. This paper presents a new loosely-coupled partitioned procedure for modeling fluid–structure interaction called combined interface boundary condition (CIBC). The procedure relies on a higher-order treatment for improved accuracy and stability of fluid–structure coupling. By utilizing the CIBC technique on the velocity and momentum flux boundary conditions, a staggered coupling procedure can be constructed with similar order of accuracy and stability of standalone computations for either the fluids or structures. The new formulation involves a coupling parameter that adjusts the amount of interfacial traction in the form of acceleration correction, which plays a key role in the stability and accuracy of the coupled simulations. Introduced correction terms for velocity and traction transfer are explicitly added to the standard staggered time-stepping stencils based on the discretized coupling effects. The coupling scheme is demonstrated in the classical 1D closed- and open-domain elastic piston problems, but further work is needed to consider the analytical stability of these schemes, 3D problems and comparison to monolithic integration.     

The finite mass mesh method

The finite mass method is a purely Lagrangian scheme for the spatial discretisation of the macroscopic phenomenological laws that govern the flow of compressible fluids. In this article we investigate how to take into account long range gravitational forces in the framework of the finite mass method. This is achieved by incorporating an extra discrete potential energy of the gravitational field into the Lagrangian that underlies the finite mass method. The discretisation of the potential is done in an Eulerian fashion and employs an adaptive tensor product mesh fixed in space, hence the name finite mass mesh method for the new scheme. The transfer of information between the mass packets of the finite mass method and the discrete potential equation relies on numerical quadrature, for which different strategies will be proposed. The performance of the extended finite mass method for the simulation of two-dimensional gas pillars under self-gravity will be reported. Communicated by: G. Wittum     

Algebraic and graph theoretic characterizations of structured flowchart schemes

The paper concerns the relationship between graph theoretic and algebraic properties of structured flowchart schemes. For each of ten classes of flowchart schemes defined algebraically, a graph theoretic property is given which characterizes this class. The classes include the Dijkstra schemes, Elgot's CACI and $\text{G}$-schemes, the reducible schemes and Kosaraju's BJ_n-schemes. For two classes of schemes defined by a graph theoretic property, an equivalent algebraic characterization is found.     

Investigations into mixing of fluids in microchannels with lateral obstructions

This work presents a study of a passive micromixer with lateral obstructions along a microchannel. The mixing process is simulated by solving the continuity, momentum and diffusion equations. The mixing performance is quantified in terms of a parameter called ‘mixing efficiency’. A comparison of mixing efficiencies with and without obstructions clearly indicates the benefit of having obstructions along the microchannel. The numerical model was validated by comparing simulation results with experimental results for a micromixer. An extensive parametric study was carried out to investigate the influence of the geometrical and operational parameters in terms of the mixing efficiency and pressure drop, which are two important criteria for the design of micromixers. A very interesting observation reveals that there exists a critical Reynolds number (Re _cr  ~ 100) below which the mixing process is diffusion dominated and thus the mixing efficiency is reduced with increase in Re and above which the mixing process is advection dominated and mixing efficiency increases with increase in Re. Microchannels with symmetric and staggered protrusion arrangements were studied and compared. The mixing performance of the staggered arrangement was comparable with that of symmetric arrangement but the pressure drop was lower in the case of staggered arrangements making it more suitable.     

Broadcasting schemes for hypercubes with background traffic

Optimal broadcasting schemes for interconnection networks (INs) are most essential for the efficient interprocess communication amongst parallel computers. In this paper two novel broadcasting schemes are proposed for hypercube computers with bursty background traffic and a single-port mode of message passing communication. The schemes utilize a maximum entropy (ME) based queue-by-queue decomposition algorithm for arbitrary queueing network models (QNMs) [D.D. Kouvatsos, I. Awan, Perform. Eval. 51 (2003) 191] and are based on binomial trees and graph theoretic concepts. It is shown that the overall cost of the one-to-all broadcasting scheme is given by max{ω₁,ω₂,…,ω_2ⁿ/2}, where ω_i, i=1,2,…,2ⁿ/2 is the total weight at each leaf node of the binomial tree and n is the degree of the hypercube. Moreover, the upper bound of the total cost of the neighbourhood broadcasting scheme is determined by ∑_i=1^Fmax{ω_i}, where F is an upper bound of the number of steps and is equal to 1.33⌈log₂(n−1)⌉+1. Evidence based on empirical studies indicates the suitability of the schemes for achieving optimal broadcasting costs.