Volume 8, Issue 5, May 1996
Index of content:

Nonlinear critical Reynolds number of a stratified plane Poiseuille flow
View Description Hide DescriptionThe minimum Reynolds number,Re ^{ N } _{ c }, below which all the two‐dimensional and monochromatic wave disturbances decay, is studied numerically for a thermally stratified plane Poiseuille flow. The Re ^{ N } _{ c } is found to be sensitive to the change of Prandtl number P. Under unstable stratification with Rayleigh number just below critical, Re ^{ N } _{ c }≂1000 is found for P≤10^{−3}.

Distribution of particles suspended in convective flow in differentially heated cavity
View Description Hide DescriptionOur aim is to explore, both experimentally and theoretically, the cumulative effects of small particle–liquid density difference, where the particles are used as tracers in recirculating flow. As an example we take a flow field generated in a differentially heated cavity. The main flow structure in such a cavity consists in one or two spiraling motions. Long‐term observations of such structures with the help of tracers (small particles) indicated that accumulation of the particles may set in at some flow regions. For theoretical insight into the phenomenon, a simple analytical model of recirculating (rotating) flow was studied. It was assumed that particles are spherical and rigid, and their presence does not affect the flow field. The particle Reynolds number is negligibly small, hence only the effects of particle–liquid density difference are of importance. Besides buoyancy, the effects of Saffman’s force and the inertial forces are also taken into account when calculating particle trajectories. Both cases were analyzed, particles with density slightly higher and lower than the fluid. It was found that in our case the inertial forces are egligible. In the numerical experiment trajectories of particles were investigated. The particles were allocated at random in the flow field obtained by numerical solution of the natural convection in the differentially heated cavity. In the experimental part, behavior of a dilute particle suspension in the convective cell was explored. In the model‐analytical study of a simple spiraling motion, it was found that due to the interaction of the recirculating convective flow field and the gravity‐buoyancy force, the particles may be trapped in some flow regions, whereas the rest of the flow field becomes particle‐free. This prediction agrees fairly well with the numerical and experimental findings.

Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at finite Prandtl number
View Description Hide DescriptionWe examine thermal convection in a rotating spherical shell with central gravity and a spatially non‐uniformly heated outer surface at two values of the Prandtl number: P _{ r }=7.0, appropriate for water at room temperature, and P _{ r }=0.7, appropriate for air at standard temperature and pressure, by numerical calculation. Four calculations are performed in a sequence: the onset of convection with homogeneous temperature boundary condition, nonlinear boundary‐forced steady convection, stability of the forced steady convection to infinitesimal disturbances, and time stepping of subsequent secondary convection. Unlike our previous study of the infinite Prandtl number limit [J. Fluid Mech. 250, 209 (1993)] inertial terms in the equation of motion for moderate Prandtl numbers play a key role in the dynamics. The effects of an inhomogeneous temperature boundary condition on nonlinear convection are illustrated by varying the wavelength and strength of the imposed boundary temperature. It is shown that even a slight inhomogeneity in the thermal boundary condition can lock azimuthally drifting convection and make it stationary, or modify the normal drifting convection rolls to a vacillating structure. In the infinite Prandtl number case, when inertial forces are absent from the equation of motion, resonance occurs when the wavelengths of boundary forcing and natural convection coincide. Fluid inertia destroys this resonance for finite Prandtl number fluids. The same effect reduces in size the stability region where steady convection is locked to the boundary, and steady convection becomes unstable to time‐dependent convection. The period of the secondary convection is close to that obtained with uniform temperature boundaries but the spatial structure is dramatically changed, exhibiting vacillations between the wavelength of the boundary temperature and that of the natural convection.

Three‐dimensional disturbances; considering starting profiles and optimal profiles in Couette and Poiseuille flow
View Description Hide DescriptionThe influence of the velocity profile on transient (algebraic) growth of three‐dimensional disturbances in channel flow is investigated. Only streamwise‐independent perturbations are considered and the effect of a single forcing Orr–Sommerfeld mode is studied. This restriction is motivated by previous investigations, which show that the largest possible (global optimal) energy growth of a disturbance is generally obtained for streamwise‐independent perturbations. Analytic solutions for the energy growth on starting profiles for plane Couette and plane Poiseuille flow are deduced and the findings reveal that the most favorable profiles for transient growth are indeed the fully developed ones. An isoperimetric method to compute optimal profiles is then presented with the purpose to increase the energy growth compared to the fully developed profiles. These profiles are dependent on the spanwise wave number, as well as the perturbation growth time. A numerical investigation show that significant energy magnification can take place on such a profile, even at short growth times of the perturbation. It is further established that these profiles are stable only for low Reynolds number.

Phase‐locked eduction of vortex shedding in flow past an inclined flat plate
View Description Hide DescriptionFlow past an inclined flat plate at an angle of attack of 30° and a Reynolds number of 30 000 is investigated experimentally. The velocity field in the wake is measured with a laser doppler anemometer (LDA) in the region from one plate breadth downstream to three and a half‐plate breadths downstream. Controlled forcing is applied to the wake by vibrating the plate in the across‐wind direction at a frequency in the middle of the lock‐in range. The forcing serves to enhance the regularity and two‐dimensionality of vortex shedding from the plate. It also facilitates phase‐locked averaging of the LDA data. The LDA bursts are sorted according to their arrival instants relative to a particular phase of the vortex shedding cycles. The phase‐averaged velocity results reveal large‐scale vortical structures in the wake. Dynamical properties of these structures such as coherent vorticity and Reynolds stress production are discussed. The wake is found to be strongly asymmetric. The flow dynamics in the wake are dominated by a train of counterclockwise vortices shed from the trailing edge of the plate. The development, shedding and subsequent convection of these vortices are studied by following the consecutive phases of the shedding cycle.

Lévy stable distributions for velocity and velocity difference in systems of vortex elements
View Description Hide DescriptionThe probability density functions(PDFs) of the velocity and the velocity difference field induced by a distribution of a large number of discrete vortex elements are investigated numerically and analytically. Tails of PDFs of the velocity and velocity difference induced by a single vortex element are found. Treating velocities induced by different vortex elements as independent random variables, PDFs of the velocity and velocity difference induced by all vortex elements are found using limit distribution theorems for stable distributions. Our results generalize and extend the analysis by Takayasu [Prog. Theor. Phys. 72, 471 (1984)]. In particular, we are able to treat general distributions of vorticity, and obtain results for velocity differences and velocity derivatives of arbitrary order. The PDF for velocity differences of a system of singular vortex elements is shown to be Cauchy in the case of small separation r, both in 2 and 3 dimensions. A similar type of analysis is also applied to non‐singular vortex blobs. We perform numerical simulations of the system of vortex elements in two dimensions, and find that the results compare favorably with the theory based on the independence assumption. These results are related to the experimental and numerical measurements of velocity and velocity difference statistics in the literature. In particular, the appearance of the Cauchy distribution for the velocity difference can be used to explain the experimental observations of Tong and Goldburg [Phys. Lett. A 127, 147 (1988); Phys. Rev. A 37, 2125, (1988); Phys. Fluids 31, 2841 (1988)] for turbulent flows. In addition, for intermediate values of the separation distance, near exponential tails are found.

Transverse and longitudinal scaling laws in non‐homogeneous low Re turbulence
View Description Hide DescriptionThe statistical properties of the velocity fluctuation components at moderate Re _{λ} are studied in homogeneous and non‐homogeneous turbulence by an experimental technique. Measurements are performed by means of a hot‐wire anemometer with X‐probe downstream of a screen for several positions, x/M=9 up to 109, where M is the screen mesh size, with Re _{λ} ranging from 37 to 82. The homogeneity of the flow is analyzed by means of velocity measurements in different transverse positions and a direct evaluation of the local isotropy of the flow is performed by means of velocity spectra. The scaling properties of the statistical moments of the structure functions up to the order of six, are investigated in the extreme positions by means of the extended self‐similarity (ESS) method and the intermittency exponents are detected for both homogeneous and non‐homogeneous flow conditions. A comparison of the longitudinal and transverse intermittency exponents as functions of the position is then performed and discussed in addition to the analysis of the transition from the anomalous to a regular scaling for small spatial separation.

Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging–diverging channel flows
View Description Hide DescriptionA study of Lagrangianchaos, Eulerian chaos, and mixing enhancement in converging–diverging channel flows, using spectral element direct numerical simulations, is presented. The time‐dependent, incompressible Navier–Stokes and continuity equations are solved for laminar, transitional, and chaotic flow regimes for 100≤Re≤850. Classical fluiddynamics representations and dynamical system techniques characterize Eulerian flows, whereas Lagrangian trajectories and finite‐time Lagrangian Lyapunov exponents identify Lagrangian chaotic flow regimes and quantify mixing enhancement. Classical representations demonstrate that the flow evolution to an aperiodic chaotic regime occurs through a sequence of instabilities, leading to three successive supercritical Hopf bifurcations. Poincaré sections and Eulerian Lyapunov exponent evaluations verify the first Hopf bifurcation at 125<Re<150 and the onset of Eulerian chaos at Re≊550. Lagrangian trajectories and finite‐time Lagrangian Lyapunov exponents reveal the onset of Lagrangianchaos, its relation with the appearance of the first Hopf bifurcation, the interplay between Lagrangian and Eulerian chaos, and the coexistence of Lagrangian chaotic flows with Eulerian nonchaotic velocity fields. Last, Lagrangian and Eulerian Lyapunov exponents are used to demonstrate that the onset of Eulerian chaos coincides with the spreading of a strong Lagrangian chaotic regime from the vortex region to the whole fluid domain.

Large eddy simulation of particle‐laden turbulent channel flow
View Description Hide DescriptionParticle transport in fully‐developed turbulent channel flow has been investigated using large eddy simulation(LES) of the incompressible Navier–Stokes equations. Calculations were performed at channel flowReynolds numbers, Re_{τ}, of 180 and 644 (based on friction velocity and channel half width); subgrid‐scale stresses were parametrized using the Lagrangian dynamic eddyviscosity model. Particle motion was governed by both drag and gravitational forces and the volume fraction of the dispersed phase was small enough such that particle collisions were negligible and properties of the carrier flow were not modified. Material properties of the particles used in the simulations were identical to those in the DNS calculations of Rouson and Eaton [Proceedings of the 7th Workshop on Two‐Phase Flow Predictions (1994)] and experimental measurements of Kulick et al. [J. Fluid Mech. 277, 109 (1994)]. Statistical properties of the dispersed phase in the channel flow at Re_{τ}=180 are in good agreement with the DNS; reasonable agreement is obtained between the LES at Re_{τ}=644 and experimental measurements. It is shown that the LES correctly predicts the greater streamwise particle fluctuation level relative to the fluid and increasing anisotropy of velocity fluctuations in the dispersed phase with increasing values of the particle time constant. Analysis of particle fluctuation levels demonstrates the importance of production by mean gradients in the particle velocity as well as the fluid‐particle velocity correlation. Preferential concentration of particles by turbulence is also investigated. Visualizations of the particle number density field near the wall and along the channel centerline are similar to those observed in DNS and the experiments of Fessler et al. [Phys. Fluids 6, 3742 (1994)]. Quantitative measures of preferential concentration are also in good agreement with Fessler et al. [Phys. Fluids 6, 3742 (1994)].

Analysis and modeling of subgrid scalar mixing using numerical data
View Description Hide DescriptionDirect numerical simulations (DNS) of passive scalar mixing in isotropic turbulence is used to study, analyze, and, subsequently, model the role of small (subgrid) scales in the mixing process. In particular, we attempt to model the dissipation of the large‐scale (supergrid) scalar fluctuations caused by the subgrid scales by decomposing it into two parts: (i) the effect due to the interaction among the subgrid scales, E^{≫} _{φ}; and, (ii) the effect due to interaction between the supergrid and the subgrid scales, E^{≳<} _{φ}. Model comparison with DNS data shows good agreement.

Dynamics of coherent structures and transition to turbulence in free square jets
View Description Hide DescriptionWe report results of time‐dependent numerical simulation of spatially developing free square jets initialized with a thin square vortex‐sheet with slightly rounded corner‐regions. The studies focus on the near field of jets with Mach number 0.3–0.6 and moderately high Reynolds numbers. A monotonically‐integrated large‐eddy‐simulation approach is used, based on the solution of the unfiltered inviscid equations and appropriate inflow/outflow open boundary conditions. The simulations show that the initial development of the square jet is characterized by the dynamics of vortex rings and braid vortices. Farther downstream, strong vortex interactions lead to the breakdown of the vortices, and to a more disorganized flow regime characterized by smaller scale elongated vortices and spectral content consistent with that of the Kolmogorov (K41) inertial subrange. Entrainment rates significantly larger than those for round jets are directly related to the enhanced fluid and momentum transport between jet and surroundings determined by the vortex dynamics underlying the axis‐rotation of the jet cross‐section. The first axis‐rotation of the jet cross‐section can be directly correlated with self‐induced vortex‐ring deformation. However, subsequent jet axis‐rotations are the result of strong interactions between ring and braid vortices, rather than being correlated with successive self‐induced vortex‐ring deformations, as previously conjectured based on laboratory observations. The interaction between braid and ring vortices has the effect of inhibiting the periodic self‐induced axis‐rotations observed in the case of isolated square vortex rings.

On the nature of vortex interactions and models in unforced nearly‐inviscid two‐dimensional turbulence
View Description Hide DescriptionA powerful feature‐tracking tool is applied to several high‐resolution, very long‐duration, regularized contour dynamics (contour surgery) simulations of unforced nearly‐inviscid two‐dimensional turbulence (2DT) on the surface of a sphere. Particularly low density gases of vortices (i.e. on average, very widely separated vortices) are examined to ascertain the nature of their interactions. The simplest (minimal) model system is studied, namely a set of vortex patches of just two vorticity values, ±ω_{0}, whose total circulation is zero. The areas of the patches are selected initially from a pre‐assigned, stable (nearly time invariant) power‐law distribution. When the vorticity occupation fraction f≳0.01, often more than three vortices are found relatively close together at the onset of a strong interaction. But, when f≲0.01, all such interactions involve only three nearby vortex patches, not all having the same sign of vorticity. This is related to the well‐known collapse of three singular (point) vortices. Thus, under these conditions, isolated two‐vortex interactions, which have figured in recent ad hoc theories and models for decaying 2DT, cannot occur. Taking into account these results, we propose an asymptotically‐motivated and computationally‐efficient ‘‘reduced’’ model.

Turbulence suppression by active control
View Description Hide DescriptionIt has recently been recognized that the non‐normality of the dynamical operator obtained by the linearization of the equations of motion about the strongly sheared background flow plays a central role in the dynamics of both fully developed turbulence and laminar/turbulent transition. This advance has led to the development of a deterministic theory for the role of coherent structures in shear turbulence as well as a stochastic theory for the maintenance of the turbulent state. In this work the theory of stochastically forced non‐normal dynamical systems is extended to explore the possibility of controlling the transition process and of suppressing fully developed shear turbulence.Modelingturbulence as a stochastically forced non‐normal dynamical system allows a great variety of control strategies to be explored and their physical mechanism understood. Two distinct active control mechanisms have been found to produce suppression of turbulent energy by up to 70%. A physical explanation of these effective control mechanisms is given and possible applications are discussed.

Numerical simulations of Richtmyer–Meshkov instabilities in finite‐thickness fluid layers
View Description Hide DescriptionDirect numerical simulations of Richtmyer–Meshkov instabilities in shocked fluid layers are reported and compared with analytic theory. To investigate new phenomena such as freeze‐out, interface coupling, and feedthrough, several new configurations are simulated on a two‐dimensional hydrocode. The basic system is an A/B/A combination, where A is air and B is a finite‐thickness layer of freon, SF_{6}, or helium. The middle layer B has perturbations either on its upstream or downstream side, or on both sides, in which case the perturbations may be in phase (sinuous) or out of phase (varicose). The evolution of such perturbations under a Mach 1.5 shock is calculated, including the effect of a reshock. Recently reported gas curtain experiments [J. M. Budzinski et al., Phys. Fluids 6, 3510 (1994)] are also simulated and the code results are found to agree very well with the experiments. A new gas curtain configuration is also considered, involving an initially sinuous SF_{6} or helium layer and a new pattern, opposite mushrooms, is predicted to emerge. Upon reshock a relatively simple sinuous gas curtain is found to evolve into a highly complex pattern of nested mushrooms.

A threshold line dissociation model for the direct simulation Monte Carlo method
View Description Hide DescriptionA new dissociation model for the direct simulation Monte Carlo (DSMC) method is formulated using the threshold line concept. This model considers in some detail the coupling between the vibrational energy distribution of molecules in a gas and the rate of dissociation. For a particular reaction, the new threshold line model only requires determination of a leading multiplicative constant and is therefore significantly less deterministic than existing DSMC chemistry models. The new model is evaluated in a heat bath under equilibrium and nonequilibrium conditions. It is also employed to calculate hypersonic shock waves of nitrogen and air. Comparison of the results generated with the new model is made with existing experimental data and with previous computational results. The more detailed DSMC implementation reveals properties of the threshold line dissociation model not apparent in the previous continuum implementation.

On subgrid scale modeling in large eddy simulations of compressible fluid flow
View Description Hide DescriptionThe purpose of this study is to theoretically and numerically investigate the behavior of different subgrid scale (SGS) models for large eddy simulations(LES) of compressible fluids in engineering applications. Various models have been analyzed including eddyviscosity, scale similarity, mixed and dynamic models and finally the concept of monotone integrated large eddy simulation (MILES). The theoretical investigation concerns primarily the demonstration and motivation of the frame indifference constraint and the application of this physical requirement on the various SGS models. LES’ utilizing the presented models have been carried out in a configuration corresponding to a bluff body flow in a rectilinear duct. Experimental data, obtained by LDV, are used for investigating the simulation results. It is found that the results from the numerical simulations are fairly resilient to the SGS models selected when the computational grid is fine enough. However, the different SGS models reproduce the energy dissipation or the interscale energy transfer −B⋅D̃ and the SGS fluxes div B in highly dissimilar ways.

On the turbulent transport of a passive scalar by anisotropic turbulence
View Description Hide DescriptionThe effect of anisotropicturbulence on the vertical transport of a tracer in a fluid layer is investigated numerically. It is found that for a fixed vertical turbulence spectrum an increase in the intensity of horizontal turbulence reduces the efficiency of the vertical transport. The reduction factor is inversely proportional to the anisotropy (defined as the ratio of horizontal to vertical rms velocities) for large Peclet numbers. This effect had been overlooked in studies of turbulent particle transport in stratified media, where the reduction in vertical transport coefficients was entirely attributed to vertical mode suppression by stratification. Our study is based on Gaussian velocity fields.

Kinetic study of the effects of boundary geometry on rarefied vapor flow
View Description Hide DescriptionThe effect of boundary geometry on the properties of a rarefied vapor flow emerging from a narrow and infinitely long source was studied. The boundary consisted of walls, along both sides of the source which were parallel to the bulk velocity direction and of a collimator which plane was perpendicular to this direction. Two‐dimensional numerical solutions for the flow field were obtained based on the Bhatnagar–Gross–Krook model. It was found that the rarefied flow is affected mainly by the size of the collimator aperture and by the walls separation. The streamwise length of the walls and the location of the collimator have a minor influence on the far field density.

Transition from spiral‐ to bubble‐type vortex breakdown
View Description Hide DescriptionNumerical calculations employing the fully three‐dimensional, time‐dependent Navier–Stokes equations have been utilized to compute a transition from spiral‐ to bubble‐type vortex breakdown for an unconfined longitudinal vortex. The transition was initiated through a small increase in the magnitude of the free‐stream axial velocity deceleration. The resulting bubble structure, which consisted of a single toroidal recirculation cell, ultimately grew unstable. Transition back to a stronger spiral breakdown followed. Results are in qualitative agreement with experimental observations.

Conditional sampling of dissipation in moderate Reynolds number grid turbulence
View Description Hide DescriptionMulti‐point hot‐wire measurements in grid turbulence have been used to conditionally average the dissipation over inertial‐scale features of the turbulent field. Velocities were measured at 5 separate spatial points obtaining six simultaneous velocity components, at turbulent Rey‐ nolds numbers Re _{λ} of 160 and 230. The dissipation was approximated with pseudo‐ or surro‐ gate dissipation ε_{ r } ^{′} based on one or two velocity gradients. The conditioned dissipation ε_{ r } ^{′} inside the most intense large‐scale rotational regions of the flow is more than twice as large as the expected value of ε_{ r } ^{′} over the whole domain. Conditional averaging was also carried out over strong large‐scale fronts showing smaller excess in dissipation.