On the origin of spin-up processes in decaying two-dimensional turbulence

A remarkable feature of two-dimensional turbulence in a square container with no-slip walls is the spontaneous production of angular momentum due to flow-wall interactions, also known as spontaneous spin-up of the flow. In this paper we address the statistics of spin-up and discuss its likely origin. A signature of the spontaneous production of angular momentum is the development of a large-scale circulation cell. It is found that the global turnover time of the flow guides the spin-up process, which can be considered as a relaxation process of the macroscopic flow to an angular momentum containing state. The high turnover rate of the small-scale vortical structures emerging from the no-slip walls apparently has no significant effect on the spin-up rate. The presented data on the spin-up process strongly suggest that spin-up is not the net result of isolated vortex-wall interactions, with its associated pressure fluctuations on the domain boundaries, alone. The rapid spin-up of the flow clearly suggests the attraction to an angular momentum containing state.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

Study of a free surface in open-channel water flows in the regime from “weak” to “strong” turbulence

A liquid–air interface in an inclined open-channel water flows was studied experimentally as the flow changes from “weak” to “strong” turbulence. In this regime, the interface is highly agitated by bulk eddies and waves, but not broken. The surface deformation statistics were obtained under a variety of conditions, including different inclination angles and flow rates. The parameter space is described in terms of Reynolds, Froude, and Weber numbers. The surface-normal displacements were obtained via the time series of the fluctuating flow depth with an ultrasound transducer. Independently, the in-plane changes in surface structures were acquired with a high-speed camera. These structures are seen as surface cells. By applying a newly developed image processing technique, the cell celerity was found to agree well with the mean flow velocity. This suggests that the cells appear when a turbulent surface-renewal eddy interacts with the interface. As the flow changes to strong turbulence, the turbulence–interface interactions become dominant over the wave phenomena, and the turbulent structures at the surface become more 3D (similar to those in the bulk flow), compared to quasi-2D structures in the weak turbulence.     

Reynolds averaged simulation of flow and heat transfer in ribbed ducts

The accuracy of modern eddy-viscosity type turbulence models in predicting turbulent flows and heat transfer in complex passages is investigated. The particular geometries of interest here are those related to turbine blade cooling systems. This paper presents numerical data from the calculation of the turbulent flow field and heat transfer in two-dimensional (2D) cavities and three-dimensional (3D) ribbed ducts. It is found that heat transfer predictions obtained using the v²–f turbulence model for the 2D cavity are in good agreement with experimental data. However, there is only fair agreement with experimental data for the 3D ribbed duct. On the wall of the duct where ribs exist, predicted heat transfer agrees well with experimental data for all configurations (different streamwise rib spacing and the cavity depth) considered in this paper. But heat transfer predictions on the smooth-side wall do not concur with the experimental data. Evidence is provided that this is mainly due to the presence of strong secondary flow structures which might not be properly simulated with turbulence models based on eddy viscosity.     

2D numerical flow modeling in a macro‐rough channel

A 2D numerical flow model, developed at the Research unit of Hydrology, Applied Hydrodynamics and Hydraulic Constructions at ULg, has been applied to flows in a macro‐rough channel. The model solves the shallow water equations (SWE) with a two length scale, depth‐integrated k‐type approach for turbulence modeling. Data for the comparison have been provided by experiments conducted at the Laboratory of Hydraulic Constructions at EPFL. In the experiments with different non‐prismatic channel configurations, namely large‐scale cavities at the side walls, three different 2D flow characteristics could be observed in cavities. With the used numerical model features, especially regarding turbulence and friction modeling, a single set of bottom and side wall roughness could be found for a large range of discharges investigated in a prismatic channel. For the macro rough configurations, the numerical model gives an excellent agreement between experimental and numerical results regarding backwater curves and flow patterns if the side wall cavities have low aspect ratios. For configurations with high aspect ratios, the head loss generated by the preservation of important recirculation gyres in the cavities is slightly underestimated. The results of the computations reveal clearly that the separation of turbulence sources in the mathematical model is of great importance. Indeed, the turbulence related to 2D transverse shear effects and the 3D turbulence, generated by bed friction, can have very different amplitude. When separating these two effects in the numerical models, most of the flow features observed experimentally can be reproduced accurately. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical study of turbulent trailing-edge flows with base cavity effects using URANS

Turbulent flows over lifting surfaces exhibiting trailing-edge vortex shedding often cause adverse and complex phenomena, such as self-induced vibration and noise. In this paper, a numerical study on flow past a blunt-edged two-dimensional NACA 0015 section and the same section with various base cavity shapes and sizes at high Reynolds numbers has been performed using the unsteady Reynolds-averaged Navier–Stokes (URANS) approach with the realisable κ–ε turbulence model. The equations are solved using the control volume method of second-order accuracy in both spatial and time domains. The assessment of the application of URANS for periodic trailing-edge flow has shown that reasonable agreement is achieved for both the time-averaged and fluctuating parameters of interest, although some differences exist in the prediction of the near-wake streamwise velocity fluctuation magnitudes. The predicted Strouhal numbers of flows past the squared-off blunt configuration with varying degrees of bluntness agree well with published experimental measurements. It is found that the intensity of the vortex strengths at the trailing-edge is amplified when the degree of bluntness is increased, leading to an increase in the mean square pressure fluctuations. The numerical prediction shows that the presence of the base cavity at the trailing-edge does not change the inherent Strouhal number of the 2D section examined. However, it does have an apparent effect on the wake structure, local pressure fluctuations and the lift force fluctuations. It is observed that the size of the cavity has more influence on the periodic trailing-edge flow than its shape does.     

On Two-dimensional Predictions of Turbulent Cross-flow Induced Vibration: Forces on a Cylinder and Wake Interaction

In this paper a typical fluid-structure interaction scenario is investigated for a turbulent flow past a circular cylinder at a relatively low subcritical Reynolds number. Numerous experimental and numerical studies have been undertaken for a baseline Reynolds number of 4,000 involving a stationary cylinder to study in detail the near wake mean flow and turbulence characteristics. These studies conclusively show that the turbulent wake displays significant coherent periodic structures of large eddies that could be adequately and profitably resolved by “low order modelling” of turbulence. In this study, an unsteady numerical framework is employed for the simulations, incorporating an Arbitrary Lagrangian–Eulerian (ALE) method for the associated grid deformation to simulate the coupled motion of the circular cylinder with a single degree of freedom in the initial zone in a typical cylinder-flow response map or what is called “initial regime”. Particular attention is paid towards resolving the large scales of the fluid motion and the inherent coupling of the cylinder’s motion towards the associated evolution of the time averaged flow field. The flow-induced vibration effects regarding the kinetic energy exchange between the mean flow and the coherent periodic scales are investigated further. The predictions discussed and analyzed in detail in the paper display reasonable agreement with the chosen benchmark tests of the stationary cylinder and suggest that the conclusions outlined regarding the coupled flow-cylinder system potentially provides a valuable contribution to the state of the art.     

Planar velocity measurements of a two-dimensional compressible wake

The present study describes the application of particle image velocimetry (PIV) to investigate the compressible flow in the wake of a two-dimensional blunt base at a freestream Mach number M_X=2. The first part of the study addresses specific issues related to the application of PIV to supersonic wind tunnel flows, such as the seeding particle flow-tracing fidelity and the measurement spatial resolution. The seeding particle response is assessed through a planar oblique shock wave experiment. The measurement spatial resolution is enhanced by means of an advanced image-interrogation algorithm. In the second part, the experimental results are presented. The PIV measurements yield the spatial distribution of mean velocity and turbulence. The mean velocity distribution clearly reveals the main flow features such as expansion fans, separated shear layers, flow recirculation, reattachment, recompression and wake development. The turbulence distribution shows the growth of turbulent fluctuations in the separated shear layers up to the reattachment location. Increased velocity fluctuations are also present downstream of reattachment outside of the wake due to unsteady flow reattachment and recompression. The instantaneous velocity field is analyzed seeking coherent flow structures in the redeveloping wake. The instantaneous planar velocity and vorticity measurements return evidence of large-scale turbulent structures detected as spatially coherent vorticity fluctuations. The velocity pattern consistently shows large masses of fluid in vortical motion. The overall instantaneous wake flow is organized as a double row of counter-rotating structures. The single structures show vorticity contours of roughly elliptical shape in agreement with previous studies based on spatial correlation of planar light scattering. Peak vorticity is found to be five times higher than the mean vorticity value, suggesting that wake turbulence is dominated by the activity of large-scale structures. The unsteady behavior of the reattachment phenomenon is studied. Based on the instantaneous flow topology, the reattachment is observed to fluctuate mostly in the streamwise direction suggesting that the unsteady separation is dominated by a pumping-like motion.     

Gas-particle two-phase turbulent flow in a vertical duct

Two-phase gas-phase turbulent flows at various loadings between the two vertical parallel plates are analyzed. A thermodynamically consistent turbulent two-phase flow model that accounts for the phase fluctuation energy transport and interaction is used. The governing equation of the gas-phase is upgraded to a two-equation low Reynolds number turbulence closure model that can be integrated directly to the wall. A no-slip boundary condition for the gas-phase and slip-boundary condition for the particulate phase are used. The computational model is first applied to dilute gas-particle turbulent flow between two parallel vertical walls. The predicted mean velocity and turbulence intensity profiles are compared with the experimental data of Tsuji et al. (1984) for vertical pipe flows, and good agreement is observed. Examples of additional flow properties such as the phasic fluctuation energy, phasic fluctuation energy production and dissipation, as well as interaction momentum and energy supply terms are also presented and discussed.

Applications to the relatively dense gas-particle turbulent flows in a vertical channel are also studied. The model predictions are compared with the experimental data of Miller & Gidaspow and reasonable agreement is observed. It is shown that flow behavior is strongly affected by the phasic fluctuation energy, and the momentum and energy transfer between the particulate and the fluid constituents.     

The effect of boundary conditions by the side of the nozzle of a low Reynolds number jet

The near flow field of an axially symmetric water jet at Reynolds numbers from about 1,000 to 10,000 is investigated using laser-induced fluorescence (LIF), laser Doppler anemometry and particle tracking velocimetry. Spanwise and streamwise vortices are detected on the longitudinal plane and on cross-sections. Attention is focused onto the effects of rigid or free boundaries sideways to the nozzle outlet (no-slip or free-slip conditions), and particularly on the start up, growth and interaction of large vortical structures. On average, for the free-slip jet these structures develop more gradually and closer to the nozzle than for the no-slip jet; the local mixedness (derived from LIF measurements) is also higher for the free-slip case. Moreover, the measured velocity field decreases more slowly, with a longer potential core and a higher shear layer (momentum) thickness for the free-slip rather than for the no-slip conditions. The relation between spanwise and streamwise large-scale vortices is clarified by the observation that the ejection of fluid in cross-sections through streamwise vortices is coupled to the pairing of spanwise vortices on the longitudinal section.     

Development characteristics of fluctuating velocity field of drag-reducing surfactant solution flow in a duct

Development behavior of the fluctuating velocity of surfactant solution in a duct has been studied experimentally. The concentration of surfactants was kept constant at 1,000 ppm, mean velocity at 0.78 m/s and fluid temperature at 15 °C. Using laser Doppler velocimetry, the fluctuating streamwise velocity distributions at six cross sections, which ranged from 14 to 112 times of hydraulic diameter of the duct, were measured. From the results, the fluctuating structures of surfactant solution flow are observed to have structures different from that of turbulent water flow in the developing field. The wavelet analysis reveals that the high-level fluctuation of surfactant solution flow is characterized by periodicity rather than irregularity around the position where the fluctuation intensity takes a peak value and that the period and the scale of periodic flow structures are related to the relaxation times of the fluid. This indicates that the high-level fluctuation is deeply related to the elastic instability and has a different generation mechanism from that of turbulence observed in a Newtonian turbulent flow.     

A study of the energetic turbulence structures during stall delay

This study revealed the three-dimensional instantaneous topologies of the large-scale turbulence structures in the separated flow on the suction surface of wind turbine’s blade during stall delay. These structures are the major contributors to the first two POD (proper orthogonal decomposition) modes. The two kinds of instantaneous flow structures as major contributors to the first POD mode are: (1) extended regions of downwash flow with an upstream upward flow beside it and a compact vortex pair closer to the blade’s leading edge; (2) a large-scale clockwise vortex with strong induced flows. The two kinds of flow structures contributing significantly to the second POD mode are: (1) large counter-rotating vortices inducing strong upward velocities and a series of small vortices; (2) strong downwash flow coming from the leading-edge shear layer with a large and strong vortex on the left side and small vortices upstream. The statistical impacts of these large-scale and energetic structures on the turbulence have also been studied. It was observed that when these turbulence structures were removed from the flow, the peak values of some statistics were significantly reduced.     

Large Eddy Simulation of Turbulent Flow Through Submerged Vegetation

Large Eddy Simulations (LES) are performed for an open channel flow through idealized submerged vegetation with a water depth (h) to plant height (h _p) ratio of h/h _p = 1.5 according to the experimental configuration of Liu et al. (J Geophys Res Earth Sci, 2008). They used a 1D laser Doppler velocimeter (LDV) to measure longitudinal and vertical velocities as well as turbulence intensities along several verticals in the flow and the data are used for the validation of the present simulations. The code MGLET is used to solve the filtered Navier–Stokes equations on a Cartesian non-uniform grid. In order to represent solid objects in the flow, the immersed boundary method is employed. The computational domain is idealized with a box containing 16 submerged circular cylinders and periodic boundary conditions are applied in both longitudinal and transverse directions. The predicted streamwise as well as vertical mean velocities are in good agreement with the LDV measurements. Furthermore, fairly good agreement is found between calculated and measured streamwise and vertical turbulence intensities. Large-scale flow structures of different shapes are present in the form of vortex rolls above the vegetation tops as well as locally generated trailing and von- Karman-type vortices due to flow separation at the free end and the sides of the cylinders. In this paper, the flow field is analyzed statistically and evidence is provided for the existence of these structures based on the LES.     

Implementation Methods of Wall Functions in Cell-vertex Numerical Solvers

Two different implementation techniques of wall functions for cell-vertex based numerical methods are described and evaluated. The underlying wall model is based on the classical theory of the turbulent boundary layer. The present work focuses on the integration of this wall-model in a cell-vertex solver for large eddy simulations and its implications when applied to complex geometries, in particular domains with sudden expansions (more generally in presence of sharp edges). At corner nodes, the conjugation of law of the wall models using slip velocities on walls and of the cell-vertex approach leads to difficulties. Therefore, an alternative implementation of wall functions is introduced, which uses a no-slip condition at the wall. Both implementation methods are compared in a turbulent periodic channel flow, representing a typical validation case. The case of an injector for aero-engines is presented as an example for an industrial-scale application with a complex geometry.     

Stokes Plane-Parallel Vortex Systems in Channels

Plane-parallel vortex systems in a viscous incompressible fluid in channels with parallel walls and in a corner with no-slip conditions on the walls are investigated on the basis of exact solutions of the biharmonic equation. It is found that separation zones and paired (joined) vortices are formed when the fluid flows through these channels and the fluid flow path between these vortex formations is traced. The flow structures are considered in the region of intrusion of the Poiseuille or Hamel flow into the zone of predominance of the intense vortex formations and for outflow of the fluid from this zone.     

Near-Wake Turbulence Properties in the High Reynolds Number Incompressible Flow Around a Circular Cylinder Measured by Two- and Three-Component PIV

The main objective of the present experimental study is to analyse the turbulence properties in unsteady flows around bluff body wakes and to provide a database for improvement and validation of turbulence models, concerning the present class of non-equilibrium flows. The flow around a circular cylinder with a low aspect ratio () and a high blockage coefficient () is investigated. This confined environment is used in order to allow direct comparisons with realisable 3D Navier–Stokes computations avoiding ‘infinite’ conditions. The flow is investigated in the critical regime at Reynolds number 140,000. A cartography of the velocity fields in the near wake of the cylinder is obtained by PIV and Stereoscopic PIV techniques. Statistical means and phase-averaged quantities are determined. Furthermore, POD analysis is performed on the data set in order to extract coherent structures of the flow and to compare the results with those obtained by the conditional sampling technique. The Reynolds stresses, the strain-rate and vorticity fields as well as the turbulence production terms are determined.     

Three-dimensional temporally resolved measurements of turbulence–flame interactions using orthogonal-plane cinema-stereoscopic PIV

A new orthogonal-plane cinema-stereoscopic particle image velocimetry (OPCS-PIV) diagnostic has been used to measure the dynamics of three-dimensional turbulence–flame interactions. The diagnostic employed two orthogonal PIV planes, with one aligned perpendicular and one aligned parallel to the streamwise flow direction. In the plane normal to the flow, temporally resolved slices of the nine-component velocity gradient tensor were determined using Taylor’s hypothesis. Volumetric reconstruction of the 3D turbulence was performed using these slices. The PIV plane parallel to the streamwise flow direction was then used to measure the evolution of the turbulence; the path and strength of 3D turbulent structures as they interacted with the flame were determined from their image in this second plane. Structures of both vorticity and strain-rate magnitude were extracted from the flow. The geometry of these structures agreed well with predictions from direct numerical simulations. The interaction of turbulent structures with the flame also was observed. In three dimensions, these interactions had complex geometries that could not be reflected in either planar measurements or simple flame–vortex configurations.     

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.     

Modeling of Langmuir Circulation: Triple Decomposition of the Craik–Leibovich Model

Turbulent shear flows on shallow continental shelves (here shallow means that the interaction with the solid, no-slip bottom is important) are of great importance because of their role in vertical mixing as well as on the transport of sediment and bioactive material. The presence of a wavefield in these areas can lead to the appearance of Langmuir circulation which is known to strongly affect the dynamics of a turbulent flow. To investigate those dynamical effects within a RANS-type modeling framework, we apply a triple decompostion to the LES results of Langmuir circulation in order to further isolate the coherent structures from fluctuating velocity field. The results are compared to the classical double-decomposition. In contrast to the double-decomposition framework, the triple-decompostion more effectively educes the coherent structure field and quantifies the need to take into account the energy exchange between the coherent and random fluctuations as well as the overall impact of the coherent structures on the turbulence dynamics.     

Two-Dimensional Turbulence with Rigid Circular Walls

We report numerical computations of decaying two-dimensional Navier--Stokes turbulence inside a circular rigid boundary. We summarize previously reported calculations involving no-slip boundary conditions and present results with higher spatial resolution than achieved before (with, however, no qualitative changes in the observed behavior). We then report new results with stress-free boundary conditions (for a viscous fluid, but bounded by a perfectly slippery wall). The method used is spectral, involving expansions of the fields in orthonormal sets of functions which obey two boundary conditions (circular analogues of the Chandrasekhar–Reid functions). The computation takes place entirely in the spectral space. Large-scale Reynolds numbers are typically less than a thousand. Interest focuses on the role played by angular momentum, in determining the decay of the turbulence with no-slip boundary conditions, and the role of possible other ideal invariants in the stress-free case. Received 30 September 1996 and accepted 5 February 1997