Reynolds Stress Terms Research Articles

Numerical study on flow structure near two impermeable trapezoid submerged breakwaters on slop bottoms

Studying on hydrodynamic characteristics in water waves propagating over two impermeable trapezoid submerged breakwaters is of great significance to research design and construction of this type of submerged breakwaters. The turbulent characteristics are discussed with cnoidal waves in the wave flume with 1:20 smooth slope. The governing equations of the vertical 2D model are the Reynolds-averaged Navier-Stokes equations. The Reynolds stress terms are closed by a nonlinear (anisotropic) k - ɛ turbulence transportation model. Using a VOF method for tracking the free surface and source function wave generating method. Furthermore, the supplementary discussion is made by means of numerical results.

Mean flow deformation in a laminar separation bubble: separation and stability characteristics

The mutual interaction of laminar–turbulent transition and mean flow evolution is studied in a pressure-induced laminar separation bubble on a flat plate. The flat-plate boundary layer is subjected to a sufficiently strong adverse pressure gradient that a separation bubble develops. Upstream of the bubble a small-amplitude disturbance is introduced which causes transition. Downstream of transition, the mean flow strongly changes and, due to viscous–inviscid interaction, the overall pressure distribution is changed as well. As a consequence, the mean flow also changes upstream of the transition location. The difference in the mean flow between the forced and the unforced flows is denoted the mean flow deformation. Two different effects are caused by the mean flow deformation in the upstream, laminar part: a reduction of the size of the separation region and a stabilization of the flow with respect to small, linear perturbations. By carrying out numerical simulations based on the original base flow and the time-averaged deformed base flow, we are able to distinguish between direct and indirect nonlinear effects. Direct effects are caused by the quadratic nonlinearity of the Navier–Stokes equations, are associated with the generation of higher harmonics and are predominantly local. In contrast, the stabilization of the flow is an indirect effect, because it is independent of the Reynolds stress terms in the laminar region and is solely governed by the non-local alteration of the mean flow via the pressure.

A mixed large eddy simulation model based on the residual-based variational multiscale formulation

A mixed model based on the residual-based variational multiscale (RBVM) formulation is proposed for the large eddy simulation of turbulent flows. In this model the cross stresses are modeled by the RBVM term and the Reynolds stresses are represented by a Smagorinsky eddy viscosity. The mixed model is motivated by an a priori analysis of a turbulent flow field that indicates that the RBVM term correctly models the dissipation induced by the cross-stress term but underpredicts the contribution from the Reynolds stress term. The dynamic version of the mixed model is implemented in a Fourier-spectral code in order to predict the decay of incompressible, three-dimensional homogeneous isotropic turbulence. It is found that the mixed model yields better agreement with direct numerical simulation solution than the dynamic Smagorinsky and the RBVM models, which are its building blocks.

Numerical study on the characteristics of flow field and wave propagation near submerged breakwater on slope

In this study, characteristics of flow field and wave propagation near submerged breakwater on a sloping bed are investigated with numerical model. The governing equations of the vertical twodimensional model are Reynolds Averaged Navier Stokes equations. The Reynolds stress terms are closed by a nonlinear k − ɛ turbulence transportation model. The free surface is traced through the PILC-VOF method. The proposed numerical model is verified with experimental results. The numerical result shows that the wave profile may become more asymmetrical when wave propagates over breakwater. When wave crest propagates over breakwater, the anticlockwise vortex may generate. On the contrary, when wave hollow propagates over breakwater, the clockwise vortex may generate. Meanwhile, the influenced zone of vortex created by wave crest is larger than that created by wave hollow. All the maximum values of the turbulent kinetic energy, turbulent dissipation and eddy viscosity occur on the top of breakwater. Both the turbulent dissipation and eddy viscosity increase as the turbulent kinetic energy increases. Wave energy may rapidly decrease near the breakwater because turbulent dissipation increases and energy in lower harmonics is transferred into higher harmonics.

Evolution of flat-plate wakes in sink flow

Evolution of flat-plate wakes in sink flow has been studied both analytically and experimentally. For such wakes, a similarity solution is derived which considers simultaneous presence of both laminar and turbulent stresses inside the wake. This solution utilizes an additional Reynolds-stress term which represents the fluctuations similar to those in wall-bounded flows, accounting for the fluctuations originating from the plate boundary layer. In this solution, it is shown that the total stress, the sum of laminar and Reynolds shear stresses, becomes self-similar. To investigate the accuracy of the analytical results, the wake of a flat plate located at the centreline of a planar contraction is studied using hot-wire anemometry. Wakes of both tapered and blunt edges are considered. The length of the plates and the flow acceleration number K = 6.25 × 10−6 are chosen such that the boundary-layer profiles at the plate edge approach the self-similar laminar solution of Pohlhausen (Z. Angew. Math. Mech., vol. 1, 1921, p. 252). A short plate in which the boundary layer at the edge does not fully relaminarize is also considered. The development of the turbulent diffusivity used in the analysis is determined empirically for each experimental case. We have shown that the obtained similarity solutions, accounting also for the initial conditions in each case, generally agree well with the experimental results even in the near field. The results also show that the mean velocity of the transitional wake behind a tapered edge becomes self-similar almost immediately downstream of the edge.

Eddy Transport and Mixing in a Wind- and Buoyancy-Driven Jet on the Sphere

Abstract The baroclinically unstable wind- and buoyancy-driven flow in a zonally reentrant pie-shaped sector on a sphere is numerically modeled and then analyzed using the transformed Eulerian-mean (TEM) formalism. Mean fields are obtained by zonal and time averaging performed at fixed height. The very large latitudinal extent of the basin (50.7°S latitude to the equator) allows the latitude variation of the Coriolis parameter to strongly influence the flow. Persistent zonal jets are observed in the statistically steady state. Reynolds stress terms play an important role in redistributing zonal angular momentum: convergence of the lateral momentum flux gives rise to a strong eastward jet, with an adjacent westward jet equatorward and weaker multiple jets poleward. An equally prominent feature of the flow is a strong and persistent eddy that has the structure of a Kelvin cat’s eye and generally occupies the zonal width of the basin at latitudes 15°–10°S. A strongly mixed surface diabatic zone overlies the near-adiabatic interior, within which Ertel potential vorticity (but not thickness) is homogenized along the mean isopycnals everywhere in the basin where eddies have developed (and thus is not homogenized equatorward of the most energetic eastward jet). A region of low potential vorticity (PV) is formed adjacent to the strong baroclinic front associated with that jet and subsequently maintained by strong convective events. The eddy buoyancy flux is dominated by its skew component over large parts of the near-adiabatic interior, with cross-isopycnal components present only in the vicinity of the main jet and in the surface diabatic layer. Close to the main jet, the cross-isopycnal components are dominantly balanced by the triple correlation terms in the buoyancy variance budget, while the advection of buoyancy variance by the mean flow is not a dominant term in the eddy buoyancy variance budget. Along-isopycnal mixing in the near-adiabatic interior is estimated by applying the effective diffusivity diagnostic of Nakamura. The effective diffusivity is large at the flanks of the mean jet and beneath it and small in the jet core. The apparent horizontal diffusivity for buoyancy obtained from the flux–gradient relationship is the same magnitude as the effective diffusivity, but the structures are rather different. The diapycnal diffusivity is strongest in the surface layer and also in a convectively unstable region that extends to depths of hundreds of meters beneath the equatorward flank of the main jet.

Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids

Non-modal amplification of disturbances in streamwise-constant channel flows of Oldroyd-B fluids is studied from an input–output point of view by analysing the responses of the velocity components to spatio-temporal body forces. These inputs into the governing equations are assumed to be harmonic in the spanwise direction and stochastic in the wall-normal direction and in time. An explicit Reynolds number (Re) scaling of frequency responses from different forcing to different velocity components is developed, showing the sameRedependence as in Newtonian fluids. It is found that some of the frequency response components peak at non-zero temporal frequencies. This is in contrast to Newtonian fluids, where peaks are always observed at zero frequency, suggesting that viscoelastic effects introduce additional time scales and promote development of flow patterns with smaller time constants than in Newtonian fluids. The temporal frequencies, corresponding to the peaks in the components of frequency response, decrease with an increase in viscosity ratio (ratio of solvent viscosity to total viscosity) and show maxima for non-zero elasticity number. Our analysis of the Reynolds–Orr equation demonstrates that the energy-exchange term involving the streamwise/wall-normal polymer stress component τxyand the wall-normal gradient of the streamwise velocity ∂yubecomes increasingly important relative to the Reynolds-stress term as the elasticity number increases and is thus the main driving force for amplification in flows with strong viscoelastic effects.

Contribution of Reynolds stress distribution to the skin friction in compressible turbulent channel flows

An exact relationship for the local skin friction is derived for the compressible turbulent wall-bounded flow (channel, pipe, flat plate). This expression is an extension of the compressible case of that derived by Fukagata [Phys. Fluids 14, L73 (2002)] in the case of incompressible wall-bounded flows. This decomposition shows that the skin friction can be interpreted as the contribution of four physical processes, i.e., laminar, turbulent, compressible, and a fourth coming from the interaction between turbulence and compressibility. Compressible numerical simulations show that, even at Mach number M=2 , the main contribution comes from the turbulence, i.e., the Reynolds stress term.

Non‐normal growth in symmetric shear flow

AbstractAn analysis of symmetric instability from the perspective of Generalized Stability Theory is presented. For Richardson number smaller than one, the optimal growth exceeds that predicted by normal mode analysis yielding potentially a much faster generation of slantwise convection. In both normal and non‐normal evolution, the parcel trajectory remains close to the isentropes and energy growth results primarily from the vertical Reynolds stress term in the energy equation. The large non‐normal growth obtained results from the optimal perturbations having parcel trajectories in the mean shear plane, in the initial stage, that maximize the growth by Reynolds stress. This plane is perpendicular to the plane of the isentropes and the absolute momentum isolines, usually associated with slantwise convection. For Richardson number larger than unity, transient growth results primarily from the meridional heat flux term in the energy equation, however this growth is relatively small. Copyright © 2008 Royal Meteorological Society

Reconstructed 3D flame structures in noise-controlled swirl-stabilized combustor

Flame structures of turbulent premixed flames in a noise-controlled, swirl-stabilized combustor are investigated to clarify the mechanism of combustion noise reduction by the secondary fuel injection. Planar laser-induced fluorescence (PLIF) is conducted for several cases with different secondary fuel injection, and 3D flame structure is reconstructed from PLIF results on multiple planes. The secondary fuel injection suppresses the fluctuation of high-temperature gas in the recirculation zone and reduces Reynolds stress and entropy terms in the acoustic sound source. In the flame zone, effects of the injection frequency are discussed by introducing mean progress variable. The flame brush is very wide for the no control case, whereas it becomes thin and is confined to a narrow space for the secondary fuel injection cases. The investigated combustor gives minimum sound level at a relevant fuel injection frequency, which is very low compared with the natural acoustic mode of the combustor. The flame brush becomes very thin, and self-induced oscillations of the flame brush disappear at this relevant frequency. The oscillation of the flame brush represents large-scale fluctuation of the mean heat release rate. The relations between characteristics of flame brush and combustion noise are discussed by introducing instantaneous and dynamical effects of flame front on the entropy term of the sound source. The secondary fuel injection works for the control of the entropy term in the sound source because the thin flame brush represents suppression of the instantaneous and dynamical effects.

A New Model for Transition Flow of Thin Films in Long Journal Bearings

This paper investigates the flow of a viscous fluid in the narrow gap (C/R = 0.002 to 0.01) between two cylinders with low eccentricity ratios (ϵ ≤ 0.2), where the shaft radius is 25.4 mm (1.0 in.). The computational engine is provided by CFD-ACE+, a commercial multi-task software. Calculations show that during the Taylor vortex regime velocity profiles in the radial direction are sinusoidal, and that pressure does vary in the axial direction even for the case of the “long journal bearing” (L/D > 2). Both velocity and pressure profiles are axisymmetric and time-independent during the Taylor vortex regime for the concentric case (ϵ = 0). During the wavy vortex regime the velocities maintain their sinusoidal profiles, while pressure varies in both axial and circumferential directions. Both velocity and pressure profiles are non-axisymmetric and time-dependent. An order of magnitude analysis for the Navier-Stokes equation shows that the inertia, viscous terms, pressure, and the Reynolds stress terms are equally significant. Based on these findings, a new model for predicting the flow behavior in long journal bearing films in the transition regime (Taylor and wavy vortex regimes) is proposed and justified. The new model indicates that the velocity profiles are sinusoidal and depend on the local Reynolds number and the position in the axial direction. Unlike the modified turbulent viscosity of the most accepted models (Constantinescu, Ng-Pan, Hirs, and Gross et al.), the value of the dynamic viscosity used in the new model is kept at its laminar original value. Finally, a modified form for the Reynolds equation is proposed for the transition regime; a comparison is made between the results of this model and the four most accepted turbulence models mentioned above.

3-D NUMERICAL SIMULATION OF FLOW AND CLEAR WATER SCOUR BY INTERACTION BETWEEN BRIDGE PIERS

In this paper, the flow and local scour variation a round single pier and introducing interaction effect between bridge piers were studie d using 3D flow model. The model used a finite-volume method to solve the non-transi ent Navier-Stokes equations for three dimensions on a general non-orthogonal grid. The e k turbulence model is used to solve the Reynolds-stress term. The numeric al model solves the sediment continuity equation in conjunction with van Rijn’s bed-load sediment transport formula to simulate the bed evolution. The 3D flow model was verified through experimental study in the non cohesive bed material in an experimental flume. The different causes of local scour around the pier wer e simulated well, such as bow flow, down flow, horseshoe vortex, pressure variation and lee-wake vortex. It was found from this research study that the local scour depth by interaction process between bridge piers depends on the Froude number, the dist ance between piers and the diameter of piers. The maximum scour depth for double piers is higher than that for single pier. Furthermore, the effect of pier shape on the scour process was studied and it was found that the maximum scour depth for circu lar pier is less than that for rectangular one for both single and double piers ca ses. The results show good agreement between simulation and experimental results. Also, empirical equation was developed from the experimental data for computing the maximum scour depth due to the interaction between bridge piers.

CFD4NRS with a focus on experimental and CMFD investigations of bubbly flows

The paper discusses first current computational trends related to reactor safety problems and developments needed, as well as the need for new kinds of much more detailed experimental data for the validation of the new methods and codes. The numerous bubbly flow experiments conducted during the last decade at PSI are then reviewed and put in perspective with regards to the needs of the analysts. The latest experiments produced first-of-a-kind sets of experimental data from bubbly plumes and jets. Some of these data were conditionally ensemble or phase averaged to separate the effects of large-scale meandering and oscillations or coherent structures from the time-dependent small-scale effects; then the Reynolds stress terms could be properly identified. The large data sets that have been created can be further mined to extract additional information for the two-phase flow analyst. The companion analytical developments and simulations of bubbly flows with RANS and LES methods are discussed. LES methods are needed to overcome some of the basic limitations of RANS simulations.

A finite-difference convective model for Jupiter's equatorial jet

AbstractWe present results of a numerical model for studying the dynamics of Jupiter's equatorial jet. The computed domain is a piece of spherical shell around the equator. The bulk of the region is convective, with a thin radiative layer at the top. The shell is spinning fast, with a Coriolis number = ΩL/V on the order of 50. A prominent super-rotating equatorial jet is generated, and secondary alternating jets appear in the higher latitudes. The roles of terms in the zonal momentum equation are analyzed. Since both the Reynolds number and the Taylor number are large, the viscous terms are small. The zonal momentum balance is primarily between the Coriolis and the Reynolds stress terms.

Experimental study of turbulent cross-flow in a staggered tube bundle using particle image velocimetry

An experimental investigation was performed on turbulent cross-flow in a staggered tube bundle with transverse and longitudinal pitch-to-diameter ratios of 3.8 and 2.1, respectively. A particle image velocimetry technique was employed to obtain detailed measurements in the bundle at inlet-velocity-based Reynolds numbers of 4800, 9300 and 14,400. Quantities reported include mean velocities, turbulence intensities, Reynolds stresses, and various terms in the Reynolds-Averaged Navier–Stokes (RANS) equations. The results reveal higher shear rates in the wake regions. The streamwise and transverse turbulence fluctuations attained peak value of 85% and 105%, respectively, of the approach mean velocity in the wake regions. The flow evolves fairly rapidly and becomes spatially periodic in the streamwise direction after a relatively short distance. The flow exhibits strong Reynolds number dependence in the developing region but no significant Reynolds number effects are observed in the spatially periodic region. The pressure gradient terms in the streamwise and transverse RANS equations are nearly balanced by the Reynolds stress terms in the recirculation zones, and by the convective terms outside the recirculation region.

LES of Turbulence Flow in Bundled Rods and Analysis for the Driving Force of Secondary Flow

The mechanism of secondary flow was discussed based on the result of an LES for the turbulence flow parallel to bundled rods. The LES was carried out in the Cartesian grid system using an improved immersed boundary method developed for accurate simulation of complex wall boundary of industrial interest. The method was verified by comparing the LES with a DNS of turbulence flow in a pipe. The LES for bundled rods reasonably reproduced the secondary flow of the second kind. We proposed a theoretical method to analyze the mechanism of the secondary flow. The result of the analysis indicated that the main source of secondary flow was the non-conservative component of Reynolds stress term as was the case for a square duct. The theory proposed that non-conservative force corresponds to the difference between the pressure gradient along the wall and the Reynolds stress term decreasing rapidly near the wall. The present theory will unify the understandings for the secondary flow of the first and second kinds. The present LES technique and theory is useful for investigating the physics in turbulence flow in complex walls.

Reformulating and quantifying the generalized added mass in filtered gas-solid flow models

To account for mesoscale phenomena in coarse grid simulations, Reynolds stress terms appearing in the filtered gas-solid flow equations have to be modeled. A generalized added mass approach previously proposed by Zhang and VanderHeyden [Int. J. Multiphase Flow 28, 805 (2002)] to model the acoustic gas-solid interaction Reynolds stress term is analyzed. Theoretically, it is shown that a generalized added mass term appears directly from the filtered acoustic gas-solid interaction term and that it corresponds to a redistribution of the filtered gas phase pressure gradient over the phases. This direct contribution scales according to the mean square of the solid volume fraction fluctuations. Two-dimensional dynamic mesoscale simulations over a broad solid volume fraction range and for two domain sizes and two grid resolutions are carried out to calculate the magnitude of the generalized added mass effect. Calculated values of the mean square of the solid volume fraction fluctuations are qualitatively in agreement with the experimental observations of Zenit and Hunt [Int. J. Multiphase Flow 26, 763 (2000)]. A second, indirect contribution to the generalized added mass term from the filtered acoustic gas-solid interaction term is shown to be statistically significant, but one order of magnitude smaller than the direct contribution. A further quantification of the maximum generalized added mass effect as a function of the filter frequency is obtained from a mixture speed of sound test. The results show that a large generalized added mass coefficient, as previously reported by Zhang and VanderHeyden [Int. J. Multiphase Flow 28, 805 (2002)], is justified only in case the filter frequency is low (<20Hz), i.e., if the grid is, either spatially or temporally, sufficiently coarse.

A new hypothesis on meandering atmospheric flows in low wind speed conditions

Low wind speeds are often associated with high pollutant concentrations in the atmosphere. Dispersion modelling in such conditions is still an important challenge for scientists due to phenomena associated with low wind speeds, which are not well understood. One such phenomenon is the large horizontal oscillation of the atmosphere, which is called meandering. This study aims at providing a new hypothesis for the cause of meandering. Meandering is explained as an inherent property of atmospheric flows in low wind speed conditions, and generally no particular trigger mechanism is necessary to initiate meandering as discussed previously by several scientists (e.g. gravity waves). The hypothesis is verified by numerically and analytically solving the two-dimensional Reynolds averaged Navier–Stokes equations under the assumption of negligible Reynolds stress terms, Coriolis forces, and pressure gradients in low wind speed conditions. Meandering is shown to arise when the 2-D flow studied here is approaching or near approximate geostrophic balance, and is damped out and vanishes when the Reynolds stresses are larger. Further, the analytical solution provides an autocorrelation function for the horizontal velocity components, which was recently proposed by Anfossi et al. (Boundary Layer Meteorol. (2005), 114, 179–203) for use in low wind speed conditions. In addition, a new set of Langevin equations is proposed for simulating dispersion in low wind speed conditions.

427 エッジトーン現象における秩序構造の三成分分解・エネルギー輸送方程式からの考察(S35-2 乱流渦構造(2),S35 乱流渦構造)

The flow field with an edgetone phenomenon induced by interaction between a two dimensional jet and a flat plate was experimentally explored by applying phase averaging technique and triple decomposition analysis. We define a coherent structure as the region where the absolute phase averaging periodic vorticity is larger than 0.15. The coherent structure formed at nozzle exit develops into downstream and then impinges on a tip of the flat plate. Addressing on Reynolds stress term and turbulent energy production term, we consider the role of coherent structure in the process of momentum and energy transport. The periodic Reynolds stress and production have a large value at same place where the coherent structure passes through. Based on this result, we assume that the rotation of coherent structure transports momentum. This process may increase the production term of periodic component.

Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation

We perform a systematic multiscale analysis for the 2-D incompressible Eulerequation with rapidly oscillating initial data using a Lagrangian approach.The Lagrangian formulation enables us to capture the propagation of the multiscalesolution in a natural way. By making an appropriate multiscale expansion inthe vorticity-stream function formulation, we derive a well-posed homogenizedequation for the Euler equation. Based on the multiscale analysis in theLagrangian formulation, we also derive the corresponding multiscale analysisin the Eulerian formulation. Moreover, our multiscale analysis revealssome interesting structure for the Reynolds stress term, which provides atheoretical base for establishing systematic multiscale modeling of 2-Dincompressible flow.