Flux difference splitting for open-channel flows

A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional shallow-water equations in open channels, together with an extension to two-dimensional flows. A linearized problem, analogous to that of Riemann for gas dynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearized problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. The scheme is applied to a one-dimensional dam-break problem, and to a problem of flow in a river whose geometry induces a region of supercritical flow. The scheme is also applied to a two-dimensional dam-break problem. The numerical results are compared with the exact solution, or other numerical results, where available.     

Vortex structures in geophysical convection

An intriguing variety of vortex structures arise during buoyant convection, especially in the presence of background stratification and rotation. These vortices play an important role in environmental fluid motions, bearing upon small-scale turbulence to planetary-scale circulation. A brief review of vortex motions associated with buoyant convection is presented in this paper, emphasizing the sources of vorticity, evolution of vortex structures and their role in oceanic and atmospheric dynamics. The genesis of a variety of vortices, for example, mushroom vortices, geostrophic and ageostrophic vortices, dipolar structures and hetons in buoyant convection flows is described, and parameterizations to represent their properties are discussed. New laboratory and numerical simulation results on vortex-related phenomena in stratified and rotating fluids and their implications in geophysical convective flows are also presented.     

Concurrent use of finite element and finite volume methods for shallow water flows in locally 1‐D channel networks

A numerical model is developed for shallow water equation in locally 1‐D channel networks. The model concurrently uses the standard Galerkin finite element method for the continuity equation and the finite volume method with an upwind scheme for the momentum equation. The surface gradient method is consistently employed. A minimum treatment is given for channel junctions so that application to multiply connected channels do not require any special consideration The model is capable of computing different types of transcritical flows, wet and dry flows, and flows with complex source terms. Standardized test problems and laboratory experimental data are used for verifying the model. Applicability of the models is validated in a multiply connected channel network draining hydromorphic farmlands located in a West African savanna, and Manning's roughness coefficient is identified, so that the steady solution is consistent with field observations. Unsteady simulation demonstrates that the model is capable of stably reproducing shifts of hydraulic jumps in flows of sub‐millimeter water depths. Copyright © 2011 John Wiley & Sons, Ltd.     

Direct Numerical Simulation of a Square Jet Ejected Transversely into an Accelerating, Laminar Main Flow

A numerical simulation of a square jet ejected transversely into a laminar boundary-layer flow was performed at a jet-to-main-flow velocity ratio of 9.78 and jet Reynolds number of 6330. The jet consisted of a single pulse with a duration equal to the time required for the jet fluid to travel 173 jet widths. A strongly-favourable streamwise pressure gradient was applied to the boundary layer and produced a freestream acceleration that is above the typical threshold required for relaminarization. The results of the simulation illustrate the effect of the favourable streamwise pressure gradient on the flowfield created by the transverse jet. Notably, the horseshoe vortex system created upwind of the jet remains steady in time and does not induce noticeable fluctuations in the jet flow. The upwind and downwind shear layers of the jet roll-up through a Kelvin–Helmholtz-like instability into discrete shear-layer vortices. Jet vorticity in the upwind and downwind shear layers accumulates near the corners of the jet and produces two sets of vortex pairs, the former of which couple with the shear-layer vortices to produce large, counter-rotating vortices in the freestream, while the latter are unstable and periodically produce hairpin vortices in the main-flow boundary layer and elongated vortices in the freestream behind the jet. The departure of the jet flowfield from the vortical structures typically observed in transverse jets illustrates the substantive effect of the favourable streamwise pressure gradient on the flowfield created by the jet.     

Mixing and coherent vortices in turbulent coaxial jets

Direct numerical simulations associated with mixing in constant-density round coaxial jets are performed. They are validated by comparison against laboratory experiments. The mixing process is studied by seeding a passive tracer first in the outer annular jet, then in the inner jet. We demonstrate the important role played by coherent vortices in the mixing mechanisms. The turbulent mixing exhibits an intermittent character as a consequence of fluid ejections caused by the counter-rotating streamwise vortices. We quantify also the domination of the outer jet and show that the fluid issuing from the central jet remains confined. To cite this article: G. Balarac, M. Si-Ameur, C. R. Mecanique 333 (2005).     

Least-squares finite-element method for shallow-water equations with source terms

Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

Study on the influences of interaction behaviors between multiple combustion-gas jets on expansion characteristics of Taylor cavities

The purpose of this study is to investigate means of controlling the interior ballistic stability of a bulk-loaded propellant gun(BLPG).Experiments on the interaction of twin combustion gas jets and liquid medium in a cylindrical stepped-wall combustion chamber are conducted in detail to obtain time series processes of jet expansion,and a numerical simulation under the same working conditions is also conducted to verify the reliability of the numerical method by comparing numerical results and experimental results.From this,numerical simulations on mutual interference and expansion characteristics of multiple combustion gas jets(four,six,and eight jets) in liquid medium are carried out,and the distribution characteristic of pressure,velocity,temperature,and evolutionary processes of Taylor cavities and streamlines of jet flow Held are obtained in detail.The results of numerical simulations show that when different numbers of combustion gas jets expand in liquid medium,there are two different types of vortices in the jet flow field,including corner vortices of liquid phase near the step and backflow vortices of gas phase within Taylor cavities.Because of these two types of vortices,the radial expansion characteristic of the jets is increased,while changing numbers of combustion gas jets can restrain Kelvin-Helmholtz instability to a certain degree in jet expansion processes,which can at last realize the goal of controlling the interior ballistic stability of a BLPG.The optimum method for both suppressing Kelvin-Helmholtz instability and promoting radial expansion of Taylor cavities can be determined by analyzing the change of characteristic parameters in a jet flow field.     

Solution of shallow-water equations using least-squares finite-element method

A least-squares finite-element method （LSFEM） for the non-conservative shallow-water equations is presented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercritical flows, and flows with smooth and sharp gradient changes. Advantages of the model include： （1） sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; （2） upwind scheme is no needed; （3） a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi （LBB） condition; and （4） the resulting system of equations is symmetric and positive-definite （SPD） which can be solved efficiently with the preconditioned conjugate gradient method. The model is verified with flow over a bump, tide induced flow, and dam-break. Computed results are compared with analytic solutions or other numerical results, and show the model is conservative and accurate. The model is then used to simulate flow past a circular cylinder. Important flow charac-teristics, such as variation of water surface around the cylinder and vortex shedding behind the cylinder are investigated. Computed results compare well with experiment data and other numerical results.     

Computation of axisymmetric confined jets in a diffuser

The paper reports on a numerical study of turbulent confined jets in a conical duct with a 5° divergence. The flow has a large ratio of jet to ambient velocities at the entrance so that it gives rise to strong recirculation. The calculations are carried out with a general finite volume method designed for calculating incompressible elliptic flows with complex boundaries. Turbulence is simulated by the standard κ–? model. The sensitivity of the solution to numerical discretization errors is examined using three convection schemes, i.e. hybrid central/upwind differencing, QUICK and SOUCUP, on two grids consisting of 68 × 50 and 102 × 82 points respectively. An examination is also made of the influence of inlet boundary conditions on the predicted flow field. The computed results are compared with experimental data for mean axial velocity, turbulent shear stress and turbulent kinetic energy profiles. It is shown that the calculations reproduce the essential features of the flow observed in the experiments.     

Numerical study of unsteady flow and heat transfer of circular tangential direction jets flowing over the inner cylinder surface in the annular chamber

In order to understand the dynamics of vortices on heat transfer, the unsteady flow field of tangential direction jets flowing in the annular chamber is numerically investigated by scale-adaptive simulation (SAS). The jet Reynolds number is 332,000 based on the jet’s diameter and inflow velocity for a specific geometric model. The analogy theory is used to obtain the convective heat transfer coefficient distribution on the hub surface. Spectral analysis via fast Fourier transform (FFT) is used to analyze frequency information that flows inside the chamber. The proper orthogonal decomposition (POD) method is performed on the velocity field in the chamber and the convective heat transfer coefficient on the hub surface using a snapshot method. The fast Fourier transform helps find the dominant frequency of the unsteady flow in the chamber. The time sequence of velocity fields on the radial plane shows the presence of cyclic flapping of the jet. The proper orthogonal decomposition analysis indicates that the unsteady periodic flow phenomenon in the chamber and unsteady heat transfer on the hub surface are mainly related to the dynamics of the counter-rotating vortices caused by the jet.     

Numerical simulations of vortices in mixing layers and plane jets

Direct numerical simulations of two-dimensional mixing layers and jets are presented first within the temporal approximation, and then with spatial calculations. The evolution of a temperature field as a passive scalar is investigated simultaneously. Besides usual statistical quantities, the results are presented through visualisations of coloured vorticity- and temperature-contours in order to describe the evolutions of the fields with time. Concentrated vortices develop and the evolution of the layer is dominated by the interactions between these eddies which can be viewed as the coherent structures observed in laboratory experiments.     

Characteristics of strongly-forced turbulent jets and non-premixed jet flames

Previous researchers have demonstrated that strong pulsations of the fuel flow rate can significantly reduce the flame length and luminosity of laminar/transitional non-premixed jet flames. The physical mechanisms responsible for these changes are investigated experimentally in acoustically-forced jet flows where the peak velocity fluctuations are up to eight times the mean flow velocity. Both reacting and non-reacting flows were studied and Reynolds numbers, based on the mean flow properties, ranged from 800 to 10,000 (corresponding to peak Reynolds numbers of 1,450–23,000), and forcing frequencies ranged from 290 to 1,140 Hz. Both the first and second organ-pipe resonance modes of the fuel delivery tube were excited to obtain these frequencies. An analysis of the acoustic forcing characteristics within the resonance tube is provided in order to understand the source of the high amplitude forcing. Flow visualization of jets with first resonant forcing confirms the presence of large-scale coherent vortices and strong reverse flow near the exit of the fuel tube. With second-resonant forcing, however, vortices are not emitted from the tube as they are drawn back into the fuel tube before they can fully form. Increased fine-scale turbulence is associated with both resonant cases, but particularly at second resonance. The power spectra of the velocity fluctuations for a resonantly pulsed jet show the presence of an inertial subrange indicating that the flow becomes fully turbulent even for mean-Reynolds-number jets that are nominally laminar. It is shown that these pulsed jet flows exhibit strong similarities to synthetic jets and that the Strouhal number, based on the maximum velocity at the fuel tube exit, is the dominant parameter for scaling these flows. The Strouhal number determines the downstream location where the coherent vortices breakdown, and is found to provide better collapse of flame length data (both current and previous) than other parameters that have been used in the literature.     

COMPUTATIONAL STUDIES OF IMPINGING JETS USING K-ϵ TURBULENCE MODELS

This paper reports numerical modelling of impinging jet flows using Rodi and Malin corrections to the k–ϵ turbulence model, carried out using the PHOENICS finite volume code. Axisymmetric calculations were performed on single round free jets and impinging jets and the effects of pressure ratio, height and nozzle exit velocity profile were investigated numerically. It was found that both the Rodi and Malin corrections tend to improve the prediction of the hydrodynamic field of free and impinging jets but still leave significant errors in the predicted wall jet growth. These numerical experiments suggest that conditions before impingement significantly affect radial wall jet development, primarily by changing the wall jet's initial thickness.     

Semi-implicit Skew Upwind Method for problems in environmental hydraulics

A new computational method is presented for reducing numerical diffusion in environmental fluid problems. This method, which is referred to as the Semi-Implicit Skew Upwind Method (SISUM), is a robust solution procedure for the conditional convergence of the discretized transport equations. The method retains the advantage of the low numerical diffusion of the conventional skew upwind schemes but does not suffer from over- or under-shooting often found in these methods due to the improved interpolation schemes. The effectiveness of SISUM is demonstrated in several examples. The comparison of the results of a hybrid scheme and SISUM with field observations of convection-dominated pollutant transport in strongly curvilinear river flow shows that SISUM successfully eliminates the high numerical diffusion produced by the hybrid scheme. The robustness of the method was tested by solving the hydrodynamics of a circular clarifier model with a large density gravity source term in the vertical-momentum equation.     

Averaging over Fast Gravity Waves for Geophysical Flows with Unbalanced Initial Data

Various facets of recent mathematical theories for averaging over fast gravity waves on advective time scales for geophysical flows with unbalanced initial data are presented here including nonlinear Rossby adjustment and simplified reduced dynamics. This work is presented within the context of simplified geophysical models involving the rotating shallow-water equations and the rotating stably stratified Boussinesq equations. Novel mechanisms for enhanced gravity wave dissipation through the catalytic interaction with potential vortical modes are also developed here within the context of the rotating shallow-water equations. Received 2 May 1997 and accepted 20 August 1997     

PIV study of large-scale flow organisation in slot jets

The paper reports on particle image velocimetry (PIV) measurements in turbulent slot jets bounded by two solid walls with the separation distance smaller than the jet width (5–40%). In the far-field such jets are known to manifest features of quasi-two dimensional, two component turbulence. Stereoscopic and tomographic PIV systems were used to analyse local flows. Proper orthogonal decomposition (POD) was applied to extract coherent modes of the velocity fluctuations. The measurements were performed both in the initial region close to the nozzle exit and in the far fields of the developed turbulent slot jets for Re ⩾ 10,000. A POD analysis in the initial region indicates a correlation between quasi-2D vortices rolled-up in the shear layer and local flows in cross-stream planes. While the near-field turbulence shows full 3D features, the wall-normal velocity fluctuations day out gradually due to strong wall-damping resulting in an almost two-component turbulence. On the other hand, the longitudinal vortex rolls take over to act as the main agents in wall-normal and spanwise mixing and momentum transfer. The quantitative analysis indicates that the jet meandering amplitude was aperiodically modulated when arrangement of the large-scale quasi-2D vortices changed between asymmetric and symmetric pattern relatively to the jet axis. The paper shows that the dynamics of turbulent slot jets are more complex than those of 2D, plane and rectangular 3D jets. In particular, the detected secondary longitudinal vortex filaments and meandering modulation is expected to be important for turbulent transport and mixing in slot jets. This issue requires further investigations.     

A comparative study of sound generation by laminar,combusting and non-combusting jet flows

Sound production by two-dimensional, laminar jet flows with and without combustion is studied numerically and theoretically. The compressible Navier–Stokes, energy and progress variable equations are solved by resolving both the near field and the acoustics. The combusting jet flows are compared to non-combusting jets of the same jet Mach number, with the non-combusting, non-isothermal jets having the same steady temperature difference as the combusting jets. This infers that the magnitude of entropic and density disturbances is similar in some of the combusting and non-combusting cases. The flows are perturbed by a sinusoidal inlet velocity fluctuation at different Strouhal numbers. The computational domain is resolved to the far field in all cases, allowing direct examination of the sound radiated and its sources. Lighthill’s acoustic analogy is then solved numerically using Green’s functions. The radiated sound calculated using Lighthill’s equation is in good agreement with that from the simulations for all cases, validating the numerical solution of Lighthill’s equation. The contribution of the source terms in Dowling’s reformulation of Lighthill’s equation is then investigated. It is shown that the source term relating to changes in the momentum of density inhomogeneities is the dominant source term for all non-reacting, non-isothermal cases. Further, this source term has similar magnitude in the combusting cases and is one of the several source terms that have similar magnitude to the source term involving fluctuations in the heat release rate.     

A Numerical Study of an Operator Splitting Method for Rotating Flows with Large Ageostrophic Initial Data

We propose an operator splitting method which is especially suitable for long-time integration of geophysical equations characterized by the presence of multiple-time scales and weak-operator splitting. The method is illustrated on the classical rotating shallow-water equations on a periodic domain with large ageostrophic (unprepared) initial data. The asymptotic splitting decomposes the solution into a first part which solves the quasigeostrophic equation; a second one which is the “slow” ageostrophic component of the flow; and a corrector. The particular decomposition we use ensures that the corrector is small for large rotation. By considering only the “slow” ageostrophic and quasigeostrophic components a numerical approximation to the shallow-water equations is derived that effectively removes the time-step restrictions caused by the presence of fast waves. The splitting is exact in the asymptotic limit of large rotation and includes the nonlinearity of the equations. Numerical examples are included. These examples demonstrate a significant reduction in the computational cost over direct numerical approximations of the shallow-water equations. We conclude with an outline of a general operator splitting method for more general primitive geophysical equations. Received 1 July 1998 and accepted 1 December 1998