Volume 8, Issue 10, October 1996
Index of content:

The universal scaling exponents of anisotropy in turbulence and their measurement
View Description Hide DescriptionCorrelation functions of non‐scalar fields in isotropic hydrodynamicturbulence are characterized by a set of universal exponents. These exponents also characterize the rate of decay of the effects of anisotropic forcing in developed turbulence. These exponents are important for the general theory of turbulence, and for modelinganisotropic flows. We propose methods for measuring these exponents by designing new laboratory experiments.

Extensional deformation of Newtonian liquid bridges
View Description Hide DescriptionA numerical investigation is presented of axisymmetric, static and elongating, viscous Newtonian liquid bridges confined between identical circular disks. The time‐dependent interface shapes and applied forces on the end plates, which separate at a constant prescribed velocity, are calculated as functions of the capillary number, the viscosity ratio between the inner and outer fluids, and an initial bridge configuration characterized by the aspect ratio. The numerical simulations are in excellent agreement with available experimental data and provide useful insight into the different dynamical responses of extending liquid bridge configurations. In particular, liquid bridges surrounded by fluids of a relatively small viscosity deform in a fore‐aft symmetrical manner and undergo breakup sooner than in the case of relatively viscous outer fluids, which also require a greater applied force on the end plates to maintain the desired motion. Decreasing the capillary number (increasing interfacial tension) and the initial aspect ratio result in shorter bridge lengths prior to breakup and an increase in the applied forces on the end plates.

A theoretical model for centering of a thin viscous liquid shell in free and forced capillary oscillations
View Description Hide DescriptionIn previous numerical studies [Lee and Wang, J. Fluid Mech. 188, 411 (1988); Pelekasis et al., J. Fluid Mech. 230, 541 (1991)], it has been shown that when an inviscid and nonconcentric liquid shell undergoes a finite‐amplitude capillary oscillation, its enclosed bubble undertakes a slow translational oscillation relative to the shell. In the present work, we study the effects of viscosity on the slow motion, in both free and forced capillary oscillations. It is found that in a free oscillation, the shell cannot become concentric because the oscillations are damped by viscosity before centering occurs. In a forced oscillation which is sustained by an external source such as a modulated acoustic radiation pressure, centering does occur when the slow oscillations are damped. The predicted centering of the shell takes place more slowly than that observed in experiments [Wang et al., J. Colloid Interface Sci. 165, 19 (1994)]. However, it is noted that a comparison with experiments is not appropriate at this time, since the shell in the experiments had an uncontrolled rotation in the acoustic potential well.

Particle dispersion and deposition in direct numerical and large eddy simulations of vertical pipe flows
View Description Hide DescriptionThe motion of dense particles in a turbulent gas flow has been studied by means of numerical simulations. The single‐phase turbulent pipe flow was modelled using Direct Numerical Simulation and Large Eddy Simulation. At tube Reynolds numbers of 5300, 18300 and 42000 particles with dimensionless relaxation times ranging from 5 to 10^{4} were released. Assuming the system to be dilute, the characteristics of particle dispersion, deposition and concentration distribution were studied under various conditions of gravity and lift. This study shows that for small particles the deposition process is governed by the properties of the near‐wall layer where the wall‐normal turbulence intensity is low, while for large inertial particles turbulent dispersion determines the chances for particles to hit the tube wall. The motion of the latter particles appears to scale properly with the Lagrangian integral time scale of the turbulence. Furthermore we demonstrated the segregation of particles towards the wall, as a result of particle‐turbulence interaction.

Flow between a stationary and a rotating disk shrouded by a co‐rotating cylinder
View Description Hide DescriptionBoundary layers on stationary and rotating disks have received much attention since von Kármán’s [Z. Angew. Math. Mech. 1, 233 (1921)] and Bödewadt’s [Z. Angew. Math. Mech. 20, 241 (1940)] studies of the cases with disks of infinite radius. Theoretical treatments have focused on similarity treatments leading to conflicting ideas about existence and uniqueness, and where self‐similar solutions exist, whether they are physically realizable. The coupling between the boundary layer flows and the interior flow between them, while being of practical importance in a variety of situations such as turbomachinery and ocean circulations, is not well understood. Here, a numerical treatment of the axisymmetric Navier–Stokes equations, together with some experiments for the case of finite stationary and rotating disks bounded by a co‐rotating sidewall is presented. We show that in the long time limit, solutions are steady and essentially self‐similar. Yet the transients are not. In particular, axisymmetric waves propagate in the stationary disk boundary layer when the vortex lines entering the boundary layer develop inflection points, and there are subsequent eruptions of vortical flow out of the boundary layer deep into the interior at large Reynolds numbers.

Steady states and oscillatory instability of swirling flow in a cylinder with rotating top and bottom
View Description Hide DescriptionIn this study we present a numerical investigation of steady states, onset of oscillatory instability, and slightly supercritical oscillatory states of an axisymmetric swirling flow of a Newtonian incompressible fluid in a cylinder, with independently rotating top and bottom. The first part of the study is devoted to the influence of co‐ and counter‐rotation of the bottom on the steady vortex breakdown, which takes place in the well‐known problem of flow in a cylinder with a rotating top. It is shown that weak counter‐rotation of the bottom may suppress the vortex breakdown. Stronger counter‐rotation may induce a stable steady vortex breakdown at relatively large Reynolds numbers for which a vortex breakdown does not appear in the case of the stationary bottom. Weak corotation may promote the vortex breakdown at lower Reynolds numbers than in the cylinder with the stationary bottom. Stronger corotation leads to the detachment of the recirculation zone from the axis and the formation of an additional vortex ring. The second part of the study is devoted to the investigation of the onset of oscillatory instability of steady flows. It is shown that the oscillatory instability sets in due to a Hopf bifurcation. The critical Reynolds number and the critical frequency of oscillations were calculated as a function of the rotation ratio (ξ=Ω_{bottom}/Ω_{top}) for a fixed value of the aspect ratio γ (height/radius) of the cylinder γ=1.5. The stability analysis showed that there are several most unstable linear modes of the perturbation that become successively dominant with a continuous change of ξ. It is shown that the oscillatory instability may lead to an appearance and coexistence of more than one oscillating separation vortex bubble.

Coherent structures and dynamics in a neutrally stratified planetary boundary layer flow
View Description Hide DescriptionCoherent structures and the dynamics of a neutrally stratified planetary boundary layerflow are studied through a large eddy simulation, which includes surface roughness,Coriolis force, and a capping inversion. Quadrant analysis and flow visualization show that low‐speed negative momentum flux (ejection) is the dominant feature throughout most of the boundary layer. The initiation of vortical structures is observed to be associated with vorticity sheets and pressure maxima, which are formed dynamically when low‐speed negative momentum flux collides with either high‐speed negative momentum flux (sweep) or the mean flow. Four dimensional conditional averages are used to study the statistical behavior of ejections and sweeps. The shape, strength, lifetime, and origin of the conditionally sampled structures at three different heights are discussed. Near the surface, sweeps are observed to induce ejections when colliding with the surface. The evolution of sweep‐induced ejections near the wall is discussed.

Quantitative experimental and numerical investigation of a vortex ring impinging on a wall
View Description Hide DescriptionA joint experimental and computational methodology is developed and applied to investigate a vortex ring impinging normally on a wall. The method uses digital particle imagevelocimetry to make planar flow measurements, which are then used to initialize a second‐order finite difference calculation. The experiment and the simulation are compared at later times and agree extremely well. The ring undergoes two rebounds from the wall and continues to expand. During the approach to the wall, peak vorticity grows by 50% due to vortex stretching. Peak vorticity strengths of the secondary and tertiary vortices formed from the shedding boundary layer are 40% and 20% of the primary. In addition, a ring with a Gaussian core is simulated and compared to demonstrate the benefits of using realistic initial conditions.

On initial‐value and self‐similar solutions of the compressible Euler equations
View Description Hide DescriptionWe examine numerically the issue of convergence for initial‐value solutions and similarity solutions of the compressible Euler equations in two dimensions in the presence of vortex sheets (slip lines). We consider the problem of a normal shock wave impacting an inclined density discontinuity in the presence of a solid boundary. Two solution techniques are examined: the first solves the Euler equations by a Godunov method as an initial‐value problem and the second as a boundary value problem, after invoking self‐similarity. Our results indicate nonconvergence of the initial‐value calculation at fixed time, with increasing spatial‐temporal resolution. The similarity solution appears to converge to the weak ‘zero‐temperature’ solution of the Euler equations in the presence of the slip line. Some speculations on the geometric character of solutions of the initial‐value problem are presented.

Partition functions and equilibrium measures in two‐dimensional and quasi‐three‐dimensional turbulence
View Description Hide DescriptionAn attempt is made to construct numerically equilibrium measures for the Euler equations by first examining measures for discretized approximate systems and then searching on the computer for the limit of vanishing discretization. First the partition function is evaluated for two‐dimensional discretized incompressible fields with a hydrodynamical energy function and an infinite number of invariants; the behavior of the partition functions is examined as the discretization is refined. The results are contrasted with those of recent mean‐field theories, which are seen to be reasonable approximations only at moderate temperatures. The two‐dimensional vortex system has no phase transitions and no states invariant under refinement of the discretization, except at zero temperature. Finite‐temperature equilibrium measures may appear if a simple representation of vortex stretching is added to the system, in agreement with recent work on three‐dimensional turbulence, where these equilibrium measures are used as key building blocks.

Numerical determination of turbulent fractal dimensions
View Description Hide DescriptionThis paper focuses on the fractal dimension of a line embedded in a homogeneous turbulent field. Based on an intuitive picture motivated by fractal theory, the introduction provides a physical approach for analysing the fractal dimension. The analysis is first applied to the case of an isotropic turbulent field which is suddenly subjected to steady rotation. It is shown that for a non‐rotating field the fractal dimension D ^{′} of the line increases with time. When expressed as a function of t/t _{ d }, where t _{ d } is the time of decay, the fractal dimension depends on both the Reynolds number and the rotation rate. However, using a suitable combination of linear (rotation rate Ω) and non‐linear (t _{ d }) mechanisms to define a characteristic time t ^{*}=t _{ d }(1+αt _{ d }Ω), the dependence of the fractal dimension on rotation rate can be absorbed for Rossby numbers Ro≳1.54. The fractal dimension of a line immersed in periodic channel flow is also considered and the role of a no‐slip wall is analysed. It shows that the fractal dimension increases rapidly from unity on the solid surface, and that this sharp and substantial increase occurs with 15 wall units. In the cases considered, this region corresponds to zones where the rms value of the velocity fluctuation is maximum.

Two‐dimensional Gram–Charlier reconstruction of velocity correlations
View Description Hide DescriptionThe two‐point statistics obtained in a two‐dimensional mixing layer and a three‐dimensional wall jet are reconstructed from the summation of Hermite Polynomials. The use of Hermite Polynomials allows the rigorous and progressive decomposition of the statistical field into separate components, Gaussian and non‐Gaussian. The influence of individual terms can then be investigated. Two different schemes are used: a one‐dimensional temporal reconstruction of data from both experiments, which is capable of providing excellent agreement with the measurements, and a two‐dimensional scheme with the mixing layer data, which captures spatial and temporal characteristics of the velocity cross‐correlation. It is demonstrated that the technique can also recover information that may be lost or missing between two measuring points thereby providing a complementary method to linear stochastic estimation.

On velocity‐conditioned scalar mixing in homogeneous turbulence
View Description Hide DescriptionScalar mixing models are required to modelturbulent molecular mixing in full probability density function (pdf) simulations of turbulent reacting flows. Despite the existence of direct numerical simulation (DNS) data suggesting the contrary, most scalar mixing models assume that molecular mixing is independent of the instantaneous velocity, i.e., 〈D∇^{2}φV,ψ〉=〈D∇^{2}φψ〉. Since in a joint velocity, composition pdf calculation the velocity is known, this assumption is unnecessary and leads to a lack of local isotropy in the scalar field. Moreover, since velocity conditioning offers a numerically tractable approach for including the effects of local anisotropy and mean velocity gradients on scalar mixing, it should be of considerable interest for the numerical simulation of scalar mixing in inhomogeneous turbulent flows. An efficient numerical implementation of velocity‐conditioned scalar mixing for full pdf simulations is proposed and verified against DNS data for homogeneous turbulence (isotropic and shear flow) with a uniform mean scalar gradient. A second‐moment closure relating the velocity‐conditioned scalar dissipation to the scalar fluxes and Reynolds stresses that is exact in the limit of a joint Gaussian pdf is also derived for use with moment closure models.

Rapid distortion theory for compressible homogeneous turbulence under isotropic mean strain
View Description Hide DescriptionIsotropic compressible turbulence subjected to rapid isotropic compression is studied using inviscid rapid distortion theory (RDT) and direct numerical simulation. An exact solution to the rapid distortion problem is given. Comparisons are made between the simulation results and the RDT solution, as well as previously studied limiting cases of the RDT solution. The comparisons illustrate the range of applicability of the RDT solutions. Implications for the use of RDT results in modeling compressible turbulent flows are briefly discussed.

Coherent structures in two uniformly distorted plane turbulent wakes
View Description Hide DescriptionA pattern‐recognition analysis was performed on two uniformly distorted plane turbulent wakes in the far region: the first was a wake generated by a high solidity mesh strip and the second was a wake generated by a solid flat plate. We observed that the coherent structures in the porous wake are significantly different from those in the solid wake. The porous wake structures are characterized by ‘‘ejections’’ of turbulent fluid from the center of the wake. The solid wake has nearly two‐dimensional and quasi‐periodic structures with strong circulation patterns in the vertical plane. We interpret the results as showing that there is a deficit of spanwise vorticity, relative to lateral or streamwise vorticity, in the porous wake structures, as compared to the solid wake structures. It is shown that these differences have a marked effect on the mixing and entrainment in the two flows.

The near wake region of a high solidity mesh strip
View Description Hide DescriptionA pattern‐recognition analysis of the near region of a plane turbulent wake generated by a high solidity mesh strip reveals a markedly regular vortex street. In fact, the ratio of the ‘‘coherent’’ to ‘‘incoherent’’ (or ‘‘random’’) fluctuations is significantly larger than for many other plane turbulent wakes of similar Reynolds number. Since these incoherent fluctuations are so weak, the vortex street persists. This is in contrast to what is observed to happen in many other wakes, where secondary vortices, which are associated with the incoherent fluctuations, act to deform the main vortices of the vortex street into horseshoe‐like vortices very early in the development. Here, the secondary structures must be too weak in the near region to influence the vortex street to any significant effect.

Multiple scale modeling of turbulent nonequilibrium boundary layer flows
View Description Hide DescriptionMultiple scale models are investigated in order to improve modeling of turbulent nonequilibrium boundary layers. The model is first calibrated on the standard equilibrium boundary layer and then applied to several nonequilibrium flows. Applications include the flow around an airfoil, the shock–boundary layer interaction, the transonic bump flow, and a three‐dimensional bump. The multiple scale model predictions are compared with experimental results and with standard one‐scale models. Several improvements are obtained that make the multiple scale model an interesting tool for tackling the prediction of nonequilibrium turbulent flows.

Numerical simulation of particle interactions with wall turbulence
View Description Hide DescriptionThis paper presents the results of a numerical investigation of the effects of near‐neutral density solid particles on turbulent liquid flow in a channel. Interactions of particles, in a size range about the dissipative length scale, with wall turbulence have been simulated at low volume fractions (average volume fraction less than 4×10^{−4}). Fluid motion is calculated by directly solving the Navier‐Stokes equations by a pseudo‐spectral method and resolving all scales of motion. Particles are moved in a Lagrangian frame through the action of forces imposed by the fluid and gravity. Particle effects on fluid motion are fed back at each time step by calculating the velocity disturbance caused by the particles assuming the flow around them is locally Stokesian. Particle‐particle interactions are not considered. The slightly heavier‐than‐fluid particles of the size range considered are found to preferentially accumulate in the low‐speed streaks, as reported in several other investigations. It is also found that particles smaller than the dissipative length scale reduce turbulence intensities and Reynolds stress, whereas particles that are somewhat larger increase intensities and stress. By examining higher order turbulence statistics and doing a quadrant analysis of the Reynolds stress, it is found that the ejection‐sweep cycle is affected—primarily through suppression of sweeps by the smaller particles and enhancement of sweep activity by the larger particles. A preliminary assessment of the impact of these findings on scalar transfer is made, as enhancement of transfer rate is a motivation of the overall work on this subject. For the case investigated, comparison of the calculations with an existing experiment was possible, and shows good agreement.

Uniform shear flow in a binary mixture with general repulsive interactions
View Description Hide DescriptionA kinetic model for a binary mixture under uniform shear flow is exactly solved. The model incorporates a temperature dependence of the collision frequencies that allows the consideration of general repulsive interactions. The rheological properties of the mixture are obtained as functions of the shear rate, the parameters of the mixture (particle masses, concentrations, and force constants), and a parameter characterizing the interaction considered. In addition, the velocity distribution functions are explicitly obtained. While the transport coefficients are hardly sensitive to the interaction potential, the distribution functions are clearly influenced by the interaction parameter. In the tracer limit, a transition to an alternative state recently found in the context of Boltzmann equation is exactly identified in the case of Maxwell molecules. For non‐Maxwell molecules, preliminary results suggest that this transition is also present although the phenomenon is less significant. A comparison with previous results derived from the a Boltzmann equation for Maxwell molecules is also carried out.

Magnetohydrodynamic wave‐current system with constant vorticity
View Description Hide DescriptionThe basic linear theory is given for a perfectly conducting stratified wave‐current system in which there is an upper layer of incompressible fluid of constant vorticity propagating on a slightly heavier lower layer of stagnant fluid. An extraneous homogeneous horizontal magnetic fieldB _{0} is present, directed either longitudinally or transversely. Because of the magnetic field the basic governing equations for the system form a set of differential equations, in marked contrast to the single algebraic dispersion equation that would result were the magnetic field absent. The differential equations are solved, and the solutions are shown graphically for various cases of the magnetic field strength.