http://lsec.cc.ac.cn/lcfd/DEWENO/paper/WENO_1996.pdf

Efficient Implementation of Weighted ENO Schemes

I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Numerical methods for conservation laws

Since the review of periodic flow phenomena by Berger & Wille (1972) in this journal, over twenty years ago, there has been a surge of activity regarding bluff body wakes. Many of the questions regarding wake vortex dynamics from the earlier review have now been answered in the literature, and perhaps an essential key to our new understandings (and indeed to new questions) has been the recent focus, over the past eight years, on the three-dimensional aspects of nominally two-dimensional wake flows. New techniques in experiment, using laser-induced fluorescence and PIV (Particle-Image-Velocimetry), are vigorously being applied to wakes, but interestingly, several of the new discoveries have come from careful use of classical methods. There is no question that strides forward in understanding of the wake problem are being made possible by ongoing three- dimensional direct numerical simulations, as well as by the surprisingly successful use of analytical modeling in these flows, and by secondary stability analyses. These new developments, and the discoveries of several new phenomena in wakes, are presented in this review.

Vortex Dynamics in the Cylinder Wake

Weighted essentially non-oscillatory schemes

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Direct numerical simulations are performed of gravity-current fronts in the lock-exchange configuration The case of small density differences is considered, where the Boussinesq approximations can be adopted The key objective of the investigation is a detailed analysis of the flow structure at the foremost part of the front, where no previous high-resolution data were available For the simulations, high-order numerical methods are used, based on spectral and spectral-element discretizations and compact finite differences A three-dimensional simulation is conducted of a front spreading along a no-slip boundary at a Reynolds number of about 750 The simulation exhibits all features typically observed in experimental flows near the gravity-current head, including the lobe-and-cleft structure at the leading edge The results reveal that the flow topology at the head differs from what has been assumed previously, in that the foremost point is not a stagnation point in a translating system Rather, the stagnation point is located below and slightly behind the foremost point in the vicinity of the wall The relevance of this finding for the mechanism behind the lobe-and-cleft instability is discussed In order to explore the high-Reynolds-number regime, and to assess potential Reynolds-number effects, two-dimensional simulations are conducted for Reynolds numbers up to about 30 000, for both no-slip and slip (ie shear-stress free) boundaries It is shown that although quantitative Reynolds-number effects persist over the whole range examined, no qualitative changes in the flow structure at the head can be observed A comparison of the two-dimensional results with laboratory data and the three-dimensional simulation provides evidence that a two-dimensional model is able to capture essential features of the flow at the head The simulations also show that for the free-slip case the shape of the head agrees closely with the classical inviscid theory of Benjamin

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7E6399B783CB5EA725725C5D2D49F997/S0022112000001221a.pdf/div-class-title-analysis-and-direct-numerical-simulation-of-the-flow-at-a-gravity-current-head-part-1-flow-topology-and-front-speed-for-slip-and-no-slip-boundaries-div.pdf

Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries

The article surveys the current state of our understanding of turbidity currents, with an emphasis on their fluid mechanics. It highlights the significant role these currents play within the global sediment cycle, and their importance in environmental processes and in the formation of hydrocarbon reservoirs. Events and mechanisms governing the initiation of turbidity currents are reviewed, along with experimental observations and findings from field studies regarding their internal velocity and density structure. As turbidity currents propagate over the seafloor, they can trigger the evolution of a host of topographical features through the processes of deposition and erosion, such as channels, levees, and sediment waves. Potential linear instability mechanisms are discussed that may determine the spatial scales of these features. Finally, the hierarchy of available theoretical models for analyzing the dynamics of turbidity currents is outlined, ranging from dimensional analysis and integral models to both depth-averaged and depth-resolving simulation approaches.

Turbidity Currents and Their Deposits

Vortex breakdown of nominally axisymmetric, swirling incompressible flows with jet- and wake-like axial velocity distributions issuing into a semi-infinite domain is studied by means of direct numerical simulations. By selecting a two-parametric velocity profile for which the steady axisymmetric breakdown is well-studied (Grabowski & Berger 1976), issues are addressed regarding the role of three-dimensionality and unsteadiness with respect to the existence, mode selection, and internal structure of vortex breakdown, in terms of the two governing parameters and the Reynolds number. Low Reynolds numbers are found to yield flow fields lacking breakdown bubbles or helical breakdown modes even for high swirl. In contrast, highly swirling flows at large Reynolds numbers exhibit bubble, helical or double-helical breakdown modes, where the axisymmetric mode is promoted by a jet-like axial velocity profile, while a wake-like profile renders the flow helically unstable and ultimately yields non-axisymmetric breakdown modes. It is shown that a transition from super- to subcritical flow, as defined by Benjamin (1962), accurately predicts the parameter combination yielding breakdown, if applied locally to flows with supercritical inflow profiles. Thus the basic form of breakdown is axisymmetric, and a transition to helical breakdown modes is shown to be caused by a sufficiently large pocket of absolute instability in the wake of the bubble, giving rise to a self-excited global mode. Two distinct eigenfunctions corresponding to azimuthal wavenumbers have been found to yield a helical or double-helical breakdown mode, respectively. Here the minus sign represents the fact that the winding sense of the spiral is opposite to that of the flow.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6AB7F50FBF65A5DED546C613E3B29895/S0022112003004749a.pdf/div-class-title-three-dimensional-vortex-breakdown-in-swirling-jets-and-wakes-direct-numerical-simulation-div.pdf

Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation

High-resolution simulations of particle-driven gravity currents

The evolution of the two- and three-dimensional structures in a temporally growing plane shear layer is numerically simulated with the discrete vortex dynamics method. We include two signs of vorticity and thus account for the effect of the weaker boundary layer leaving the splitter plate which is used to create a spatially developing mixing layer. The interaction between the two layers changes the symmetry properties seen in a single vorticity-layer calculation and results in closer agreement with experimental observations of the interface between the two streams. Our calculations show the formation of concentrated streamwise vortices in the braid region between the spanwise rollers, whereas the spanwise core instability is observed to grow only initially. Comparison with flow visualization experiments is given, and we find that the processes dominating the early stages of the mixing-layer development can be understood in terms of essentially inviscid vortex dynamics.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112088000928

Eckart Meiburg

Papers

Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries

Turbidity Currents and Their Deposits

Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation

High-resolution simulations of particle-driven gravity currents

Three-dimensional shear layers via vortex dynamics