Characterization of plunging liquid jets: A combined experimental and numerical investigation

This paper presents a combined experimental and numerical study of the flow characteristics of round vertical liquid jets plunging into a cylindrical liquid bath. The main objective of the experimental work consists in determining the plunging jet flow patterns, entrained air bubble sizes and the influence of the jet velocity and variations of jet falling lengths on the jet penetration depth. The instability of the jet influenced by the jet velocity and falling length is also probed. On the numerical side, two different approaches were used, namely the mixture model approach and interface-tracking approach using the level-set technique with the standard two-equation turbulence model. The numerical results are contrasted with the experimental data. Good agreements were found between experiments and the two modelling approaches on the jet penetration depth and entraining flow characteristics, with interface tracking rendering better predictions. However, visible differences are observed as to the jet instability, free surface deformation and subsequent air bubble entrainment, where interface tracking is seen to be more accurate. The CFD results support the notion that the jet with the higher flow rate thus more susceptible to surface instabilities, entrains more bubbles, reflecting in turn a smaller penetration depth as a result of momentum diffusion due to bubble concentration and generated fluctuations. The liquid average velocity field and air concentration under tank water surface were compared to existing semi-analytical correlations. Noticeable differences were revealed as to the maximum velocity at the jet centreline and associated bubble concentration. The mixture model predicts a higher velocity than the level-set and the theory at the early stage of jet penetration, due to a higher concentration of air that cannot rise to the surface and remain trapped around the jet head. The location of the maximum air content and the peak value of air holdup are also predicted differently.     

A numerical study of a rectangular jet along a shear free boundary: Surface jet

A numerical simulation of a rectangular surface jet is performed at a Reynolds number of ${\mathit{Re}}_j=4400$. The global parameters of the jet e.g. maximum velocity decay, jet surface normal and lateral spread rates, entrainment, jet momentum flux and turbulent momentum flux are in agreement with several other studies reported in the literature. It is shown that the mean velocity and Reynolds stress profiles scale with the maximum local streamwise velocity and jet half width in the surface normal and lateral directions. The current simulation provides balance, explicitly calculated budgets for the turbulence kinetic energy, Reynolds normal and shear stresses. The surface jet develops a thin layer of fast moving fluid in the lateral direction near the surface. This layer is called the ‘surface current’. It has been suggested that the surface current arises due to the Reynolds stress anisotropy in the near surface region. The current study shows that this explanation is incomplete. The turbulence production for the Reynolds stress in the lateral direction is negative, which can drive the mean flow in the lateral direction. The higher level of negative production in the near surface region is responsible for the development of the surface current.     

极低湍流度下翼型近场尾流雷诺切应力的分布

简述了在西北工业大学低湍流度风洞中，用热线风速仪测量翼型边界层及近场尾流中速度型、湍流强度、雷诺正应力及切应力等的简况；文中着重分析了极低湍流度下（０．０２％）试验所得近场尾流中雷诺切应力的分布规律。结果表明，在尾流宽度方向及流动方向均有明显的规律性，且与常规风洞湍流度下结果相比，其有关特征值明显偏低。     

Compact finite difference schemes on non‐uniform meshes. Application to direct numerical simulations of compressible flows

In this paper, the development of a fourth‐ (respectively third‐) order compact scheme for the approximation of first (respectively second) derivatives on non‐uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi‐ and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical method for the determination of weakly non‐hydrostatic non‐linear free surface wave propagation

A numerical method is described that may be used to determine the propagation characteristics of weakly non‐hydrostatic non‐linear free surface waves over a general, bottom topography. In shallow water of constant undisturbed depth, such waves are equivalent to the familiar cnoidal waves characterized by sharp crests and relatively flat troughs. For a certain range of parameters, these propagate without change of form by virtue of the weakly non‐hydrostatic balance in the vertical momentum equation. Effectively, this counters the tendency for the non‐linearity in a purely hydrostatic theory to lead to a continuously deforming surface wave profile. The realistic representation furnished by cnoidal wave theory of free surface waves in the shallow near‐shore zone has led to its utilization in evaluating their propagation characteristics. Nonetheless, the classic analytical theory is inapplicable to the case of wave propagation over a sloping beach or off‐shore sand bar topography. Under these conditions, a local change in form of the surface wave profile is anticipated before the waves break and knowing this is required in order to evaluate fully the propagation process. The efficacy of the numerical method is first demonstrated by comparing the solution for water of constant depth with the evaluation of the analytical solution expressed in terms of the Jacobian elliptic function cn. The general method described in the paper is then illustrated by experiments to determine the change in profile of weakly non‐hydrostatic non‐linear surface waves propagating over bed forms representative of those found in shallow coastal seas. Copyright © 2001 John Wiley & Sons, Ltd.     

Three‐dimensional numerical simulation of compound meandering open channel flow by the Reynolds stress model

Turbulent flow in a compound meandering open channel with seminatural cross sections is one of the most complicated turbulent flows as the flow pattern is influenced by the combined action of various forces, such as centrifugal force, pressure, and shear stresses. In this paper, a three‐dimensional (3D) Reynolds stress model (RSM) is adopted to simulate the compound meandering channel flows. Governing equations of the flow are solved numerically with finite‐volume method. The velocity fields, wall shear stresses, and Reynolds stresses are calculated for a range of input conditions. Good agreement between the simulated results and measurements indicates that RSM can successfully predict the complicated flow phenomenon. Copyright © 2008 John Wiley & Sons, Ltd.     

Flow modulation of a planar free shear layer with large bubbles—direct numerical simulations

The flow of a planar free shear layer with cylindrical bubbles is simulated using a finite difference/front tracking scheme. This approach allows direct numerical simulation of the multiphase flow by wholly incorporating the local bubble flow field in conjunction with the large scale vortical structures of the liquid. The role of large bubbles in modifying low Reynolds number ( 250) shear flow structures is investigated, specifically for bubbles whose diameter approaches the scale of the largest liquid eddies. The results indicate that duration of eddy crossing is the main mechanism for flow modulation, which is typically characterized by decreased vortex coherency and size, modified fluctuation statistics and significant variations in pairing/merging phenomena. The comparison of fluctuating statistics and flow field visualization also allowed qualitative discrimination between the modulation of the non-linear eddy dynamics and fluctuations due simply to the random bubble induced perturbations.     

GPU‐accelerated direct numerical simulations of decaying compressible turbulence employing a GKM‐based solver

Gas Kinetic Method‐based flow solvers have become popular in recent years owing to their robustness in simulating high Mach number compressible flows. We evaluate the performance of the newly developed analytical gas kinetic method (AGKM) by Xuan et al. in performing direct numerical simulation of canonical compressible turbulent flow on graphical processing unit (GPU)s. We find that for a range of turbulent Mach numbers, AGKM results shows excellent agreement with high order accurate results obtained with traditional Navier–Stokes solvers in terms of key turbulence statistics. Further, AGKM is found to be more efficient as compared with the traditional gas kinetic method for GPU implementation. We present a brief overview of the optimizations performed on NVIDIA K20 GPU and show that GPU optimizations boost the speedup up‐to 40x as compared with single core CPU computations. Hence, AGKM can be used as an efficient method for performing fast and accurate direct numerical simulations of compressible turbulent flows on simple GPU‐based workstations. Copyright © 2016 John Wiley & Sons, Ltd.     

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

A numerical study of turbulent flow over a two‐dimensional hill

Calculations of mean velocities and Reynolds stresses are reported for the recirculating flow established in the wake of two‐dimensional polynomial‐shaped obstacles that are symmetrical about a vertical axis and mounted in the water channel downstream of a fully developed channel flow for Re=6×10⁴. The study involves calculations of mean and fluctuating flow properties in the streamwise and spanwise directions and include comparisons with experimental data [Almeida GP, Durão DFG, Heitor MV. Wake flows behind two‐dimensional model hills. Experimental Thermal and Fluid Science 1993; 7: 87–101] for flow around a single obstacle with data resulting from the interaction of consecutive obstacles, using two versions of the low‐Reynolds number differential second‐moment (DSM) closure model. The results include analysis of the turbulent stresses in local flow co‐ordinates and reveal flow structure qualitatively similar to that found in other turbulent flows with a reattachment zone. It is found that the standard isotropization of production model (IPM), based on that proposed by Gibson and Launder [Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics 1978; 86(3): 191–511], with the incorporation of the wall reflection model of Craft and Launder [New wall‐reflection model applied to the turbulent impinging jet. AIAA Journal 1992; 32(12): 2970–2972] predicts the mean velocities quite well, but underestimates the size of the recirculation region and turbulent quantities in the shear layer. These inadequacies are circumvented by adopting a new cubic Reynolds stress closure scheme based on that more recently developed by Craft and Launder [A Reynolds stress closure designed for complex geometries. International Journal of Heat and Fluid Flow 1996; 17: 245–254] which satisfies the two component limit (TCL) of turbulence. In this model the geometry‐specific quantities, such as the wall‐normal vector or wall distance, are replaced by invariant dimensionless gradient indicators. Also, the model captures the diverse behaviour of the different components of the stress dissipation, ε_ij, near the wall and uses a novel decomposition for the fluctuating pressure terms. Copyright © 2001 John Wiley & Sons, Ltd.     

A local‐analytic‐based discretization procedure for the numerical solution of incompressible flows

An Erratum has been published for this article in International Journal for Numerical Methods in Fluids 2005, 49(8): 933. We present a local‐analytic‐based discretization procedure for the numerical solution of viscous fluid flows governed by the incompressible Navier–Stokes equations. The general procedure consists of building local interpolants obtained from local analytic solutions of the linear multi‐dimensional advection–diffusion equation, prototypical of the linearized momentum equations. In view of the local analytic behaviour, the resulting computational stencil and coefficient values are functions of the local flow conditions. The velocity–pressure coupling is achieved by a discrete projection method. Numerical examples in the form of well‐established verification and validation benchmarks are presented to demonstrate the capabilities of the formulation. The discretization procedure is implemented alongside the ability to treat embedded and non‐matching grids with relative motion. Of interest are flows at high Reynolds number, ??(10⁵)–??(10⁷), for which the formulation is found to be robust. Applications include flow past a circular cylinder undergoing vortex‐induced vibrations (VIV) at high Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical and experimental investigation of flow over a semicircular weir

The water flow over a semicircular weir is investigated numerically and experimentally in this paper. The numerical model solves the Reynolds equation for a mean flow field with thek-ε-turbulent model. To trace the motion of the free surface, the COF method with geometric reconstruction is employed. The velocity of the flow is measured by means of LDV technique. Four types of flow patterns, the position of the separation and reattachment point, the distribution of shear stress on the bed at downstream of the weir are presented and discussed. The numerical results agree well with the experiment data.     

A projection method for the spectral solution of non‐homogeneous and incompressible Navier–Stokes equations

This paper is devoted to the development of a parallel, spectral and second‐order time‐accurate method for solving the incompressible and variable density Navier–Stokes equations. The method is well suited for finite thickness density layers and is very efficient, especially for three‐dimensional computations. It is based on an exact projection technique. To enforce incompressibility, for a non‐homogeneous fluid, the pressure is computed using an iterative algorithm. A complete study of the convergence properties of this algorithm is done for different density variations. Numerical simulations showing, qualitatively, the capabilities of the developed Navier–Stokes solver for many realistic problems are presented. The numerical procedure is also validated quantitatively by reproducing growth rates from the linear instability theory in a three‐dimensional direct numerical simulation of an unstable, non‐homogeneous, flow configuration. It is also shown that, even in a turbulent flow, the spectral accuracy is recovered. Copyright © 2012 John Wiley & Sons, Ltd.     

Dynamics of stress wave propagation in a chain of photoelastic discs impacted by a planar shock wave; Part I, experimental investigation

The propagation of stress waves through a chain of discs has been studied experimentally. Optically transparent 20-mm diameter discs, made of epoxy, were loaded dynamically by head-on collision with an incident planar shock wave. The loading was done in a vertical shock tube. The head-on collision between the punch-plate, placed on top of the chain of discs, and the incident shock wave resulted in a head-on reflected shock wave inducing behind it a fairly uniform step-wise pressure pulse having duration of about 6 ms. The recorded fringe patterns of the stress field, in the discs-chain, show that the input pressure pulse was broken into several oscillating cycles. The back and forth bouncing of stress waves gave rise to two different modes of the contact stress oscillations, which continued until the overall stress reaches equilibrium with the input conditions. The registered propagation velocity of the stress wave was significantly lower than the appropriate speed of sound in the material from which the discs were made.      

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

We develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low‐order non‐conforming finite element approximations (the so‐called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L² stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L² norm, second‐order space convergence for the velocity and first‐order space convergence for the pressure are observed. Copyright © 2010 John Wiley & Sons, Ltd.     

A numerical investigation of CO_2 dilution on the thermochemical characteristics of a swirl stabilized diffusion flame

The turbulent combustion flow modeling is performed to study the effects of CO₂ addition to the fuel and oxidizer streams on the thermochemical characteristics of a swirl stabilized diffusion flame. A flamelet approach along with three well-known turbulence models is utilized to model the turbulent combustion flow field. The k-ω shear stress transport(SST) model shows the best agreement with the experimental measurements compared with other models. Therefore, the k-ω SST model is used...     

An attempt to improve accuracy of higher‐order statistics and spectra in direct numerical simulation of incompressible wall turbulence by using the compact schemes for viscous terms

We attempt to improve accuracy in the high‐wavenumber region in DNS of incompressible wall turbulence such as found in fully developed turbulent channel flow. In particular, it is shown that the improvement of accuracy of viscous terms in the Navier–Stokes equations leads to the improvement of accuracy of higher‐order statistics and various spectra. It is emphasized that increase in required computational cost will not be crucial when incompressible flow is simulated, because the introduction of a higher‐order scheme into the viscous terms does not increase computational cost for solving the Poisson equation. We introduced fourth‐order and eighth‐order central compact schemes for discretizing the viscous terms in DNS of a fully developed turbulent channel flow. The results are compared with those using second‐order and fourth‐order central‐difference schemes applied to the viscous terms and those obtained by the spectral method. The results show that accuracy improvement of the viscous terms improve accuracy of higher‐order statistics (i.e., skewness and flatness factors of streamwise velocity fluctuation) and various spectra of velocity and pressure fluctuations in the high‐wavenumber region. Copyright © 2013 John Wiley & Sons, Ltd.     

A simulation of free surface waves for incompressible two‐phase flows using a curvilinear level set formulation

A level set formulation in a generalized curvilinear coordinate is developed to simulate the free surface waves generated by moving bodies or the sloshing of fluid in a container. The Reynolds‐averaged Navier–Stokes (RANS) equations are modified to account for variable density and viscosity in two‐phase (i.e. water–air) fluid flow systems. A local level set method is used to update the level set function and a least square technique adopted to re‐initialize it at each time step. To assess the developed algorithm and its versatility, a selection of different fluid–structure interaction problems are examined, i.e. an oscillating flow in a two‐dimensional square tank, a breaking dam involving different density fluids, sloshing in a two‐dimensional rectangular tank and a Wigley ship hull travelling in calm water. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast non‐Fickian particle‐tracking diffusion simulator and the effect of shear on the pollutant diffusion process

This paper presents a fast method for the generation of non‐Fickian particle paths within a particle‐tracking pollutant diffusion model based on a Fourier spectral representation of fractional Brownian motion (fBm), a generalization of ordinary Brownian motion. Correlated diffusive components in a particle‐tracking algorithm are modelled using fBm increments that have long‐range correlations over numerous spatial and/or temporal scales; hence producing non‐Fickian diffusion. A fast algorithm to generate fBm and its increment by using its power spectral density S(f) in a fast Fourier transform algorithm is given. A general equation for the scaling of fBm within a velocity flow field with simple linear shear is presented. An initial numerical study of the nature of fBm shear dispersion has been conducted by incorporating fBm increments into a non‐Fickian particle‐tracking algorithm. It is shown that the effect of simple (i.e. linear) shear on the diffusion process is to produce enhanced diffusive phenomena with the longitudinal spreading of the plume scaling with exponent ∼1+H, where H is the Hurst exponent used to describe fBm. Finally, a more complex shear zone at the entrance of a coastal bay model is investigated using both a traditional particle‐tracking method and the fBm‐based method. Copyright © 2000 John Wiley & Sons, Ltd.