Vorticity Thickness Research Articles

Scaling of the turbulent/non-turbulent interface in boundary layers

AbstractScaling of the interface that demarcates a turbulent boundary layer from the non-turbulent free stream is sought using theoretical reasoning and experimental evidence in a zero-pressure-gradient boundary layer. The data-analysis, utilising particle image velocimetry (PIV) measurements at four different Reynolds numbers ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\delta u_{\tau }/\nu =1200\mbox{--}14\, 500$), indicates the presence of a viscosity dominated interface at all Reynolds numbers. It is found that the mean normal velocity across the interface and the tangential velocity jump scale with the skin-friction velocity$u_{\tau }$and are approximately$u_{\tau }/10$and$u_{\tau }$, respectively. The width of the superlayer is characterised by the local vorticity thickness$\delta _{\omega }$and scales with the viscous length scale$\nu /u_{\tau }$. An order of magnitude analysis of the tangential momentum balance within the superlayer suggests that the turbulent motions also scale with inner velocity and length scales$u_{\tau }$and$\nu /u_{\tau }$, respectively. The influence of the wall on the dynamics in the superlayer is considered via Townsend’s similarity hypothesis, which can be extended to account for the viscous influence at the turbulent/non-turbulent interface. Similar to a turbulent far-wake the turbulent motions in the superlayer are of the same order as the mean velocity deficit, which lends to a physical explanation for the emergence of the wake profile in the outer part of the boundary layer.

LES Study of 3D Incompressible Temporal Mixing Layer Using Different Well-Known Subgrid Scale (SGS) Models

In this paper, 3D incompressible temporal mixing layer at three different Reynolds numbers has been investigated using large eddy simulation (LES) turbulence model. To consider the effects of the subgrid scales, six different well-known SGS models, Smagorinsky, dynamic Smagorinsky, Yoshizava one equation, Spalart–Allmaras one equation, Lagrangian dynamic Smagorinsky two equation (LDS) and six-equations Reynolds stress (RSM) were applied. The computed LES results for vorticity thickness, average total kinetic energy, and Reynolds shear stress components of stress tensor revealed that the LDS and RSM models agree well with DNS data. Different steps of forming of coherent flow structures at different times were also discussed.

On the influence of internal density variations on the linear stability characteristics of planar shear layers

Planar shear layers are known to be sensitive to density variation within the vorticity region. The current study will expand on the available literature on the stability characteristics of shear layers with density variation through consideration of new regimes in terms of the velocity and density profiles. Both spatial and spatio-temporal linear analyses were considered. In addition to the maximum density change within the shear layer, the effect of density to vorticity thickness ratio, and transverse shifting of the density profile relative to the velocity profile were considered. For a shear layer experiencing combustion, these effects relate to the relative position and thickness of the flame in the shear layer. Thick density profiles have a stabilizing affect that imposes essentially a low-pass attenuator modulation on the spatial stability. The shear layer is destabilized through the presence of an intense yet thin density reduction that overlaps with the peak vorticity profile, a condition that may be present near the shear layer origination point. The velocity ratio at which the flow transitions between absolute and convective instability is dependent on the magnitude of the density reduction in the shear layer as well as the relative density profile thickness and transverse location. The results generally indicate that the instability of the planar shear layer is strongly influenced by the characteristics of the density-weighted vorticity distribution within the shear layer.

Large eddy simulation of flow development and noise generation of free and swirling jets

Large eddy simulation is performed for investigating the local and far-field behaviors of free and swirling jets at moderate Reynolds number. By solving compressible boundary layer equations, the inflow profiles with different swirl number are calculated, and then their stability characteristics are analyzed based on linear stability theory. The amplification rates of swirling jets are higher than the free one, particularly for higher or negative azimuthal wavenumber modes. Multiple unstable modes are superimposed to construct inflow forcing. The quantities of flow and acoustic are presented and compared against the results of existed experiments and other computations, besides, the comparisons are also made among themselves. For swirling jets, the spreadings of jet half-width and vorticity thickness at the initial and transition stage are enhanced, but they are surpassed by the free jet at turbulent mixing stage. In all cases, the development of mixing layer initially is greatly influenced by frequencies f0 and f0/2 associated with upstream forcing. As the swirl intensity is increased, the growth rates of fluctuation quantities on the centerline and the nozzle lip line are raised, but the peak levels on the centerline are reduced substantially. In swirling jets, the strength of vortex pairing is decreased, and the pairing noise is weakened correspondingly. The overall sound pressure levels are lower than that of the free jet at all observation angles (ϕ), and about 3 decibels (dB) is reduced at ϕ ≃ 30° in the strong swirling case at a distance of 60 radii. The Fourier analyses of pressure and acoustic sources show that the modes are varied greatly, which suggests that the noise reduction should be corresponding to the change of instability waves.

Instability regimes in the primary breakup region of planar coflowing sheets

AbstractThis article investigates the appearance of instabilities in two planar coflowing fluid sheets with different densities and viscosities via experiments, numerical simulation and linear stability analysis. At low dynamic pressure ratios a convective instability is shown to appear for which the frequency of the waves in the primary atomization region is influenced by both liquid and gas velocities. For large dynamic pressure ratios an asymptotic regime is obtained in which frequency is solely controlled by gas velocity and the instability becomes absolute. The transition from convective to absolute is shown to be influenced by the velocity defect induced by the presence of the separator plate. We show that in this regime the splitter plate thickness can also affect the nature of the instability if it is larger than the gas vorticity thickness. Computational and experimental results are in agreement with the predictions of a spatio-temporal stability analysis.

Effects of Noncircular Inlet on the Flow Structures in Turbulent Jets

Turbulent jet flows with noncircular nozzle inlet are investigated by using a Reynolds Stress Model. In order to analyze the effects of noncircular inlet, the cross section of inlet are selected as circular, square, and equilateral triangular shape. The jet half-width, vorticity thickness, and developments of the secondary flow are presented. From the result, it is confirmed that the secondary flows of square and equilateral triangular nozzle are more vigorous than that of the circular jet. This development of secondary flows is closely related to the variations of vortical motions in axial and azimuthal directions.

Newly Developed Empirical Correlations for Simulation of Transitional Flows

An effective new formulation is developed for simulation of transitional flows. This formulation is based on modifications made to the latest numerical model utilizing vorticity and momentum thickness Reynolds numbers concepts. In this respect, rigorous experiments were conducted in a wind tunnel to modify the existing formulation to a more reliable form suitable for modeling of transitional flows. Test model was a linear cascade of axial compressor blades. Wind tunnel tests consisted of measurements of surface pressure distributions and velocity profiles utilizing hot film anemometry. Different freestream turbulence intensities, flow incidences, and Reynolds numbers were examined. New correlations were imposed to a commercial numerical flow solver;applying them to some standard objects produced more reliable results than those obtained from other formulations, presented so far. This attribution is more emphasized especially while dealing with modeling laminar separation bubbles, where transition occurs within the free shear layer.

An eddy-driven abyssal circulation in a bowl-shaped basin due to deep water formation

Numerical experiments were performed to investigate the effects of eddies generated in deep water formation processes on an abyssal circulation in a closed bowl-shaped basin. Two sets of experiments were performed. One set was eddy-restricted experiments in which only a volume-driven (upwelling-driven) circulation was simulated and the other set was eddy-permitted experiments in which both a volume-driven circulation and an eddy-driven circulation were simulated. In the two layer experiment where the lower layer water is formed, a mean along-slope current is formed in the lower layer for both the eddy-restricted and eddy-permitted experiments. The direction of the current was not unique in the eddy-restricted experiment, but it was cyclonic in the eddy-permitted experiment. In the three layer experiments where water of the intermediate layer is formed, the mean along-slope current in the lowest layer is negligibly small in the eddy-restricted experiment, while it is large and cyclonic in the eddy-permitted experiment. The driving forcings of the eddy-driven circulation are quantified in terms of eddy fluxes of relative vorticity (Reynolds stress) and layer thickness (bolus velocity). These terms increase as the volume of the newly formed water increases, but they do not change greatly with the slope height. The magnitude of these terms changes with the slope width, but the sum of these terms does not vary greatly. As a result, the intensity of the eddy-driven circulation depends primarily on the volume of newly formed water. These dependences of eddy fluxes were interpreted using downgradient diffusion of potential vorticity.

Time resolved flow-field measurements of a turbulent mixing layer over a rectangular cavity

High Reynolds number, low Mach number, turbulent shear flow past a rectangular, shallow cavity has been experimentally investigated with the use of dual-camera cinematographic particle image velocimetry (CPIV). The CPIV had a 3 kHz sampling rate, which was sufficient to monitor the time evolution of large-scale vortices as they formed, evolved downstream and impinged on the downstream cavity wall. The time-averaged flow properties (velocity and vorticity fields, streamwise velocity profiles and momentum and vorticity thickness) were in agreement with previous cavity flow studies under similar operating conditions. The time-resolved results show that the separated shear layer quickly rolled-up and formed eddies immediately downstream of the separation point. The vortices convect downstream at approximately half the free-stream speed. Vorticity strength intermittency as the structures approach the downstream edge suggests an increase in the three-dimensionality of the flow. Time-resolved correlations reveal that the in-plane coherence of the vortices decays within 2–3 structure diameters, and quasi-periodic flow features are present with a vortex passage frequency of ~1 kHz. The power spectra of the vertical velocity fluctuations within the shear layer revealed a peak at a non-dimensional frequency corresponding to that predicted using linear, inviscid instability theory.

Three-dimensional simulations of cellular non-premixed jet flames

The formation, dynamics and structure of cellular flames in circular non-premixed jets are examined with three-dimensional numerical simulations incorporating detailed descriptions of chemistry and transport. Similar to past experiments reported in the literature, CO 2-diluted hydrogen in diluted or pure oxygen co-flowing streams in the proximity of the extinction limit are considered. As in the experiments, several preferred cellular states are found to co-exist with the particular state realized depending on initial conditions as well as on the jet characteristics. The simulations provide additionally the temporal transitions to different stationary or rotating cellular flames, their detailed structure, and the dependence of the scaling of the realized number of cells with the vorticity thickness.

Investigation of Turbulent-Mixing-Layer Flow with Polymer Additives and Bubble Injection by Particle Image Velocimetry

Turbulent-mixing-layer flows in a vertical channel with polymer additives and bubble injection were experimentally studied by particle image velocimetry (PIV). The transparent test section was 1.0 m long with a 0.04 m × 0.11 m cross section, and the main stream was split into two substreams by a specially designed splitter. The velocities on the low- and high-speed sides are 0.36 m/s and 1.4 m/s, respectively. The Reynolds number based on the velocity difference between the two streams and the vorticity thickness δ at the trailing edge of the splitter was 13,500 for the pure water mixing-layer flow. Aquesous (200 ppm) Polyacrylamide (PAM) solutions and pure water were used as the liquid phase, and gas bubbles with a 0.5% void fraction were injected into the mixing-layer containing polymer additives, from the center of the splitter end, to check the influence of polymer additives and bubbles on the turbulent structure and properties. The distribution of the turbulence properties changed more significantly for the mixing-layer in the polymer additive case than in the pure water case. In the latter case, the Reynolds shear stress (RSS) peak was less than that for the pure water case at the same downstream cross section, but the vorticity peak in the polymer additive case was larger than that in the pure water case in the region the mixing-layer developed (x < 5δ). However, upon bubble injection, the effect of the polymer additives was diminished.

Conditional statistics of inert droplet effects on turbulent combustion in reacting mixing layers

Direct numerical simulation (DNS) of turbulent reacting mixing layers laden with evaporating inert droplets is used to assess the droplet effects in the context of the conditional moment closure (CMC) for multiphase turbulent combustion. The temporally developing mixing layer has an initial Reynolds number of 1000 based on the vorticity thickness with more than 16 million Lagrangian droplets traced. An equivalent mixture fraction incorporating the inert vapour mass fractions clearly demonstrates the effects of vapour dilution on the mixture. Instantaneous fields and conditional statistics, such as the singly conditioned scalar dissipation rate, the gas temperature ⟨ T g|η ⟩, conditional variance of the gas temperature ⟨ T g ”2|η ⟩ and conditional covariance between the fuel mass fraction and gas temperature ⟨ Y f ” T g ”|η ⟩ show considerable droplet effects. Comparison between the droplet-free and droplet-laden reacting mixing layer cases suggests significant extinction in the latter case. The resulting large conditional fluctuations around the conditional means contradict the basic assumption behind the first-order singly conditioned CMC. More sophisticated CMC approaches, such as doubly conditioned or second-order CMCs are, in principle, better able to incorporate the effects of evaporating droplets, but significant modelling challenges exist. The scalar dissipation rate doubly conditioned on the mixture fraction and a normalized gas temperature ⟨ χ | η, ζ ⟩ exemplifies the modelling complexity in the CMC of multiphase combustion.

Evolution of Three-Dimensional Temporally Evolving Plane Mixing Layers Under Strong Vortex Disturbances

The evolution of three-dimensional temporally evolving plane mixing layers under strong vortex disturbances is investigated by DNS method. The convective Mach number and the Reynolds number used is set to be 0.8 and 200. The isentropic vortex is chosen as the model for vortex disturbances. Three different directions of the vortex axis (streamwise, spanwise, and normal directions) are considered in the simulations. For comparison, the computation with linear unstable oblique waves as disturbances is also executed. The results show that in the case of the streamwise vortex disturbance, significant increase in vorticity thickness can be found only in the early stage of development. In the case of the normal vortex disturbance, a high growth rate of the plane mixing layer is kept all the time, while in the linear wave case saturation occurs in the later stage of the simulation and the vorticity thickness stops increasing further. In the case of the spanwise vortex disturbance, the flow is quite quiet and the vorticity thickness attains the lowest growth rate.

Characteristics of Annular Mixing Layer in High Subsonic Jet

Experimental studies were conducted on an annular mixing layer which evolved from freely expanding adiabatic circular jet produced by a conical nozzle. The mixing layer region shrouding the potential core was explored at four different jet-exit Mach numbers - 0.5, 0.6, 0.7 and 0.8. Pressure measurements were carried out by means of a miniature Pitot probe attached to an automated traverse. Mean velocity profiles were obtained from the radial distribution of pressures across the annular mixing layer thickness at various axial locations. The iso-velocity curves, plotted in longitudinal plane, exhibited that the velocity in the line of the nozzle lip was always about 60% of the nozzle exit velocity for the entire range of Mach numbers studied. The mixing layer thickness, assessed in terms of vorticity thickness, was found to grow linearly with the longitudinal distance downstream of the nozzle. The mixing layer growth rate was found to decline with increasing Mach number. However, the mixing layer was found ...

Direct Numerical Simulation of Inert Droplet Effects on Scalar Dissipation Rate in Turbulent Reacting and Non-Reacting Shear Layers

Three-dimensional direct numerical simulation has been performed to investigate the effects of inert evaporating droplets on scalar dissipation rate χ in temporally-developing turbulent reacting and non-reacting mixing layers with the Reynolds number based on the vorticity thickness up to 8000 and the number of traced Lagrangian droplets up to 107. The detailed instantaneous field analysis and ensemble-averaged statistics reveal complex interactions among combustion, droplet dynamics and evaporation, all of which have a considerable influence on χ. The presence of inert evaporating droplets promotes χ in both non-reacting and reacting mixing layers. In the latter, combustion reduces χ, so when combustion is suppressed by evaporating droplets, χ is enhanced. The transport equation of χ has been analyzed to investigate the various effects on χ in detail. The terms in the equation contain explicitly the evaporation rate and its spatial derivative, acting as a sink and a source for χ, respectively. On the whole, the net effect of the evaporation-rate terms is to promote χ. However, the production and dissipation terms are the dominant source and sink terms, respectively.

Optical Characterization of a Simulated Weakly Compressible Shear Layer: Unforced and Forced

This paper uses a discrete-vortex code to examine a shear layer’s response to forcing at its origin and to develop a relationship between a shear layer’s optical characteristics and the commonly used characteristic growth length, vorticity thickness. The code and its thermodynamic overlay have been used in previous studies to predict the optically-aberrating characteristics of relatively-high-Mach-number, subsonic shear layers that can be classified as weakly compressible. A weighted average natural frequency is introduced and used to characterize the unforced shear layer in terms of an optical characteristic length referred to as optical coherence length. It is shown that optical coherence length is related to vorticity thickness by a factor of approximately 3.18. The study also shows that the use of singlefrequency forcing produces a regularized shear layer for distances preceding the point where the unforced shear layer’s natural frequency matches the forcing frequency. In the case of the forced shear layer, a greater thickness is produced closer to its point of origin until collapsing * Graduate Research Assistant, Aerospace and Mechanical Engineering Department, Fitzpatrick Hall, Notre Dame, IN 46556, DEPS Fellow, AIAA Student Member. ** Research Assistant Professor, Aerospace and Mechanical Engineering Department, Hessert Center, Notre Dame, IN 46556, AIAA Member. § Professor, Aerospace and Mechanical Engineering Department, Hessert Center, Notre Dame, IN 46556, AIAA Fellow.

On the origin of the inflectional instability of a laminar separation bubble

This is an experimental and theoretical study of a laminar separation bubble and the associated linear stability mechanisms. Experiments were performed over a flat plate kept in a wind tunnel, with an imposed pressure gradient typical of an aerofoil that would involve a laminar separation bubble. The separation bubble was characterized by measurement of surface-pressure distribution and streamwise velocity using hot-wire anemometry. Single component hot-wire anemometry was also used for a detailed study of the transition dynamics. It was found that the so-called dead-air region in the front portion of the bubble corresponded to a region of small disturbance amplitudes, with the amplitude reaching a maximum value close to the reattachment point. An exponential growth rate of the disturbance was seen in the region upstream of the mean maximum height of the bubble, and this was indicative of a linear instability mechanism at work. An infinitesimal disturbance was impulsively introduced into the boundary layer upstream of separation location, and the wave packet was tracked (in an ensemble-averaged sense) while it was getting advected downstream. The disturbance was found to be convective in nature. Linear stability analyses (both the Orr–Sommerfeld and Rayleigh calculations) were performed for mean velocity profiles, starting from an attached adverse-pressure-gradient boundary layer all the way up to the front portion of the separation-bubble region (i.e. up to the end of the dead-air region in which linear evolution of the disturbance could be expected). The conclusion from the present work is that the primary instability mechanism in a separation bubble is inflectional in nature, and its origin can be traced back to upstream of the separation location. In other words, the inviscid inflectional instability of the separated shear layer should be logically seen as an extension of the instability of the upstream attached adverse-pressure-gradient boundary layer. This modifies the traditional view that pegs the origin of the instability in a separation bubble to the detached shear layer outside the bubble, with its associated Kelvin–Helmholtz mechanism. We contend that only when the separated shear layer has moved considerably away from the wall (and this happens near the maximum-height location of the mean bubble), a description by the Kelvin–Helmholtz instability paradigm, with its associated scaling principles, could become relevant. We also propose a new scaling for the most amplified frequency for a wall-bounded shear layer in terms of the inflection-point height and the vorticity thickness and show it to be universal.

Gradient trajectory analysis of a scalar field with external intermittency

The passive scalar field of a temporally evolving shear layer is investigated using gradient trajectories as a means to analyse the scalar probability density function and the conditional scalar dissipation rate in the presence of external intermittency. These results are of significance for turbulent combustion, where improved predictions of the statistics of the conditional dissipation rate are needed in several models. First, the variation of the conventional first and second moments of the conditional dissipation rate across the layer is quantitatively documented in detail. A strong dependence of the conditional dissipation rate on the lateral position and on the conditioning value of the scalar is observed. The dependence on the transverse distance to the centre-plane partially explains the double-hump profile usually reported when this dependence is ignored. The variation with the scalar observed in the ratio between the second and first moments would invalidate certain assumptions commonly done in turbulent combustion. It is also seen that conditioning on the scalar does not reduce the fluctuation of the dissipation rate with respect to unconditional values. Next, the role of external intermittency in these results is investigated. For that purpose, the flow is partitioned into different zones based on different types of gradient trajectories passing through each point, thereby introducing non-local information in comparison with the standard turbulent/non-turbulent separation based on the conventional intermittency function. In addition to the homogeneous outer regions, three zones are identified: a turbulent zone, a turbulence interface and quasi-laminar diffusion layers. The relative contribution from each of these zones to the conventional intermittency factor is reported. The statistics are then conditioned on each of these zones, and the spatial variation of the scalar distribution and of the conditional scalar dissipation rate is explained in terms of the observed zonal statistics. For the Reynolds numbers of the present simulation, between 1500 and 3000 based on the vorticity thickness and the velocity difference, and a Schmidt number equal to 1, it results that the major contribution to both statistics is due to the turbulence interfaces. At the same time, the turbulent zone shows a distinct behaviour, being approximately homogeneous but anisotropic.

Scalar Concentration Profiles in the Canopy and Roughness Sublayer

The mean flow and scalar concentration profiles within and above a tall canopy are well known to violate the standard boundary-layer flux-gradient relationships. We present a theory for the scalar concentration profile that is comprised of a canopy exchange model coupled to a modified surface-layer model. The coupling between the two components and the modifications to the surface-layer profiles are formulated through the mixing-layer analogy for the flow at canopy top. This analogy provides an additional length scale—the vorticity thickness—upon which the profiles depend and a set of criteria that allows a reduction in the empiricism associated with earlier forms in the literature. Predictions of the mean scalar concentration profiles are shown to match observations over a wide range of diabatic stabilities for both potential temperature and water vapour.

On the turbulence structure in inert and reacting compressible mixing layers

Direct numerical simulation is used to investigate effects of heat release and compressibility on mixing-layer turbulence during a period of self-similarity. Temporally evolving mixing layers are analysed at convective Mach numbers between 0.15 and 1.1 and in a Reynolds number range of 15000 to 35000 based on vorticity thickness. The turbulence inhibiting effects of heat release are traced back to mean density variations using an analysis of the fluctuating pressure field based on a Green's function.