Taylor-Reynolds Number Research Articles

Evolution of energy in flow driven by rising bubbles

We investigate by direct numerical simulations the flow that rising bubbles cause in an originally quiescent fluid. We employ the Eulerian-Lagrangian method with two-way coupling and periodic boundary conditions. In order to be able to treat up to 288000 bubbles, the following approximations and simplifications had to be introduced, as done before, e.g., by Climent and Magnaudet, Phys. Rev. Lett. 82, 4827 (1999). (i) The bubbles were treated as point particles, thus (ii) disregarding the near-field interactions among them, and (iii) effective force models for the lift and the drag forces were used. In particular, the lift coefficient was assumed to be 1/2, independent of the bubble Reynolds number and the local flow field. The results suggest that large-scale motions are generated, owing to an inverse energy cascade from the small to the large scales. However, as the Taylor-Reynolds number is only in the range of 1, the corresponding scaling of the energy spectrum with an exponent of -5/3 cannot develop over a pronounced range. In the long term, the property of local energy transfer, characteristic of real turbulence, is lost and the input of energy equals the viscous dissipation at all scales. Due to the lack of strong vortices, the bubbles spread rather uniformly in the flow. The mechanism for uniform spreading is as follows. Rising bubbles induce a velocity field behind them that acts on the following bubbles. Owing to the shear, those bubbles experience a lift force, which makes them spread to the left or right, thus preventing the formation of vertical bubble clusters and therefore of efficient forcing. Indeed, when the lift is artificially put to zero in the simulations, the flow is forced much more efficiently and a more pronounced energy that accumulation at large scales (due to the inverse energy cascade) is achieved.

Taylor Vortices–Induced Instability (Transition Flow) versus “Turbulence” in Concentric Cylinders Separated by Microscale Clearances

This paper is concerned with the relationship between the onset and the development of the Taylor instabilities and their treatment as turbulent flows in the most accepted turbulence models (Constantinescu (1); Ng-Pan (2); Hirs (3)) used with the Reynolds equation, in the range of 41.3√R/C < Re < 2000. The authors show that in between these limits there is a transition regime where the velocity and pressure profiles are fundamentally different from either a Couette flow or a fully developed turbulent flow. Thus the issue under consideration is whether the flow formations observed during Taylor instability regimes should be simulated using the widely accepted turbulence models as they presently are modeled in microscale clearance flows. We are considering the flow of light silicone oil in gaps varying from 3.302 mm (0.13 in.) to 0.127 mm (0.005 in.) between two concentric cylinders, with the inner cylinder rotating. The computational engine used in this study is a well-established and a tried software package: CFD-ACE+. It was found that the Taylor vortices (cells) begin to form at certain, but different, “critical” speeds, function of clearance size, and as the speed grows, the vortices become fully developed and evolve further into wavy vortices. Calculations show that both the 1st and 2nd critical Taylor numbers and Reynolds numbers are functions of the clearance size. The Taylor numbers decrease, while the Reynolds numbers increase with the decrease in clearance size. The onset of both instabilities is clearly characterized by the discontinuities in the Torque-√Ta (or Torque – Re) curve slope. The calculations presented here show that the slope changes in the above-mentioned graphs are due to the changes in the average velocity gradient on the outer cylinder and not to a change in the actual viscosity as it is implemented by the turbulence models mentioned above. Finally a comparison is made between present calculations and the data of Roberts (4), Cole (5), Walowit et al. (6), Weinstein (7), Koschmieder (8), and DiPrima (9).

Turbulent bubbly flow

The effect of bubbles on fully developed turbulent flow is investigated numerically and experimentally, summarizing the results of our previous papers (Mazzitelli et al., 2003, Physics of Fluids 15, L5. and Journal of Fluid Mechanics 488, 283; Rensen, J. et al. 2005, Journal of Fluid Mechanics 538, 153). On the numerical side, we simulate Navier–Stokes turbulence with a Taylor–Reynolds number of Re λ≈60, a large large-scale forcing, and periodic boundary conditions. The point-like bubbles follow their Lagrangian paths and act as point forces on the flow. As a consequence, the spectral slope is less steep as compared to the Kolmogorov case. The slope decrease is identified as a lift force effect. On the experimental side, we do hot-film anemometry in a turbulent water channel with Re λ ≈ 200 in which we have injected small bubbles up to a volume percentage of 3%. Here the challenge is to disentangle the bubble spikes from the hot-film velocity signal. To achieve this goal, we have developed a pattern recognition scheme. Furthermore, we injected microbubbles up to a volume percentage of 0.3%. Both in the counter flowing situation with small bubbles and in the co-flow situation with microbubbles, we obtain a less spectral slope, in agreement with the numerical result.

The relationship between the topological structures in turbulent flow and the distribution of a passive scalar with an imposed mean gradient

Data obtained from direct numerical simulations of isotropic homogenous turbulence and the diffusion of a scalar with an applied mean gradient is analysed using the topological techniques developed by Chong et al. (1990. Phys. Fluids 2(5), 765–777). Several simulations were run at various values of Taylor–Reynolds number ( Re λ ) and Schmidt number ( Sc). Comparing the numerical results obtained, relationships between the scalar characteristics and the topological features in the flow have been identified. Results from this work may be useful in the development of improved mixing processes by isolating the effect of the various topologies on the distribution of a passive scalar.

A parsimonious and universal description of turbulent velocity increments

This paper proposes a reformulation and extension of the concept of Extended Self-Similarity. In support of this new hypothesis, we discuss an analysis of the probability density function (pdf) of turbulent velocity increments based on the class of normal inverse Gaussian distributions. It allows for a parsimonious description of velocity increments that covers the whole range of amplitudes and all accessible scales from the finest resolution up to the integral scale. The analysis is performed for three different data sets obtained from a wind tunnel experiment, a free-jet experiment and an atmospheric boundary layer experiment with Taylor-Reynolds numbers $R_{\lambda} = 80,190,17000$ , respectively. The application of a time change in terms of the scale parameter $\delta$ of the normal inverse Gaussian distribution reveals some universal features that are inherent to the pdf of all three data sets.

Energy transfer in Rayleigh-Taylor instability.

The spatial structure and energy budget for Rayleigh-Taylor instability are examined using results from a 512 x 512 x 2040 point direct numerical simulation. The outer-scale Reynolds number of the flow follows a rough t(3) power law and reaches a final value of about 5500. Taylor microscales and Reynolds numbers are plotted to characterize anisotropy in the flow and document progress towards the mixing transition. A mixing parameter is defined which characterizes the relative rates of entrainment and mixing in the flow. The spectrum of each term in the kinetic energy equation is plotted, at regular time intervals, as a function of the inhomogeneous direction and the two-dimensional wave number for the homogeneous directions. The energy spectrum manifests the beginning of an inertial range by the latter stages of the simulation. The production and dissipation spectra become increasingly opposite and separate in wave space as the flow evolves. The transfer spectrum depends strongly on the inhomogeneous direction, with the net transfer being from large to small scales. Energy transfer at the bubble/spike fronts is strictly positive. Extensive cancellation occurs between the pressure and advection terms. The dilatation term produces negligible energy transfer, but its overall effect is to move energy from high to low density regions.

Two-point, two-time closures applied to forced isotropic turbulence

The direct interaction approximation and the local energy transfer theory were used to calculate freely decaying turbulence for a range of initial Taylor–Reynolds numbers 3⩽ R λ ( t=0)⩽129, and forced turbulence for evolved Taylor–Reynolds numbers R λ ( t evolved )≃88 and R λ ( t evolved )≃232. In the case of the freely decaying turbulence, the new feature of the work (as compared to McComb et al., J. Fluid Mech. 245 (1992) 279) was the ability to make comparisons with a direct numerical simulation at the same initial conditions. We were also able to obtain ‘experimental results’ from the simulation with an estimate of the experimental error. We present a small sample of results from an extensive investigation, sufficient to confirm that both closures agree with the simulation, to within the experimental error, in broad agreement with our earlier investigations. However, the main motivation of the work was to study the application of the closures to stationary turbulence and here we faced new problems such as how to ‘force’ the closures or the need to compute the time-history integrals of the closures over the long transient period during which the numerical simulation settles down. For this reason, our results should be seen as rather preliminary and tentative in nature. Essentially, we found that agreement between the closures and the simulation was not as good as for free decay for low-wavenumber behaviour but was quite good for high-wavenumber spectra and quantities determined by this region such as the Taylor microscale. We conclude that further work is needed to establish just what is a fair comparison between forced closure calculations and the analogous direct numerical simulation.

Transition stages of Rayleigh–Taylor instability between miscible fluids

Direct numerical simulations (DNS) are presented of three-dimensional, Rayleigh–Taylor instability (RTI) between two incompressible, miscible fluids, with a 3:1 density ratio. Periodic boundary conditions are imposed in the horizontal directions of a rectangular domain, with no-slip top and bottom walls. Solutions are obtained for the Navier–Stokes equations, augmented by a species transport-diffusion equation, with various initial perturbations. The DNS achieved outer-scale Reynolds numbers, based on mixing-zone height and its rate of growth, in excess of 3000. Initial growth is diffusive and independent of the initial perturbations. The onset of nonlinear growth is not predicted by available linear-stability theory. Following the diffusive-growth stage, growth rates are found to depend on the initial perturbations, up to the end of the simulations. Mixing is found to be even more sensitive to initial conditions than growth rates. Taylor microscales and Reynolds numbers are anisotropic throughout the simulations. Improved collapse of many statistics is achieved if the height of the mixing zone, rather than time, is used as the scaling or progress variable. Mixing has dynamical consequences for this flow, since it is driven by the action of the imposed acceleration field on local density differences.

A vortex-based model for the subgrid flux of a passive scalar

A model for the flux of a passive scalar by the subgrid motions in the large-eddy simulation of turbulent flow is proposed within the framework of the stretched-vortex subgrid stress model. The model is based on an analytical solution for the winding of a scalar field by an elemental subgrid vortex. This gives a tensor gradient-diffusion expression for the local flux of the scalar with subgrid turbulent diffusivity which depends upon the subgrid energy, the local cell size, and the vortex orientation in space. The scalar-flux subgrid model is tested by comparison of the results of 323 large-eddy simulation of passive-scalar transport by forced isotropic turbulence in the presence of a mean scalar gradient, with the direct-numerical simulation results of Overholt and Pope [Phys. Fluids 8, 2128 (1996)]. The present large-eddy simulation results predict that at large Taylor–Reynolds numbers, the ratio of the scalar variance to the squared product of the scalar gradient with the dissipation length of the turbulence, is asymptotic to a nearly constant value c′2/(α1 Lε)2≈0.36.

A Lagrangian Stochastic Model for the Dispersion and Deposition of Brownian Particles

A Lagrangian stochastic model for the dispersion and deposition of submicron-size particles is formulated and validated. The model satisfies the well-mixed condition, incorporates molecular diffusivity, and accounts for the effects of Reynolds number upon Lagrangian particle statistics. Reynolds number effects are found to be significant in the viscous sublayer and the buffer zone of a turbulent shear flow. The effects are due almost entirely to the change in the Lagrangian integral time scale. Sawford's correction to first-order Lagrangian stochastic models for the effects of Reynolds number is found to be appropriate for inhomogeneous turbulence even when the Taylor–Reynolds number Rλ ∼ O(0.1). The model predicts, in close accord with experiment and the results of direct numerical simulations, that the nondimensional particle deposition velocity K+ = 0.06Sc−2/3, where Sc is the Schmidt number. When Reynolds number effects are neglected, K+ is overpredicted by several orders of magnitude.

Non-equilibrium Monte Carlo simulation of decaying Navier–Stokes turbulence

Recently a master equation for the three-dimensional Navier–Stokes equation in k-space has been proposed. It has been shown, that the Hopf-equation can be derived from the time evolution of the stochastic process given by the master equation. Therefore it reproduces exactly the correct turbulence moment hierarchy. Here we present the results of a Monte Carlo simulation of turbulence in three space dimensions using the proposed master equation. We simulate the underlying stochastic process defined by the master equation by producing realizations and calculating averages. The results of the simulation at a Taylor–Reynolds number R λ of about 112 show a −5/3 scaling range for the energy spectrum and the Kolmogorov-constant is about 1.6.

Lagrangian acceleration measurements at large Reynolds numbers

We report experimental measurements of Lagrangian accelerations in a turbulent water flow between counter-rotating disks for Taylor–Reynolds numbers 900&amp;lt;Rλ&amp;lt;2000. Particle tracks were recorded by imaging tracer particles onto a position sensitive photodiode, and Lagrangian information was obtained from fits to the position versus time data. Several challenges associated with extracting Lagrangian statistical quantities from particle tracks are addressed. The acceleration variance is obtained as a function of Reynolds number and shows good agreement with Kolmogorov (1941) scaling. The Kolmogorov constant for the acceleration variance is found to be a0=7±3.

Different intermittency for longitudinal and transversal turbulent fluctuations

Scaling exponents of the longitudinal and transversal velocity structure functions in numerical Navier–Stokes turbulence simulations with Taylor–Reynolds numbers up to Reλ=110 are determined by the extended self similarity method. We find significant differences in the degree of intermittency: For the sixth moments the scaling corrections to the classical Kolmogorov expectations are δξ6L=−0.21±0.01 and δξ6T=−0.43±0.01, respectively, independent of Reλ. Also the generalized extended self similarity exponents ρp,q=δξp/δξq differ significantly for the longitudinal and transversal structure functions. Within the She–Leveque model this means that longitudinal and transversal fluctuations obey different types of hierarchies of the moments. Moreover, the She–Leveque model hierarchy parameters βL and βT show small but significant dependences on the order of the moment.

On longitudinal and lateral moment hierarchy in turbulence

A quantity of interest, β (degree of intermittency between scale hierarchy), which appears in a recent intermittency model by She and Lévêque [Phys. Rev. Lett. 72, 336 (1994)] is computed from turbulent simulations with Taylor-Reynolds number in the order of 100. It is found that the lateral and longitudinal β values are unequal. For most of the moment hierarchy order p, lateral βdeviates more from K-41 (i.e., smaller) than the longitudinal. β is also found to be p dependent.

Application of extended self-similarity in turbulence

From Navier-Stokes turbulence numerical simulations we show that for the extended self-similarity (ESS) method it is essential to take the third order structure function taken with the modulus and called D*3(r), rather than the standard third order structure function D3(r) itself. If this is done, we find ESS towards scales larger than order ~10 eta , where eta is the Kolmogorov scale. If D3(r) is used, there is no ESS. We also analyze ESS within the Batchelor parametrization of the second and third order longitudinal structure function and focus on the scaling of the transversal structure function. The Re-asymptotic inertial range scaling develops only beyond a Taylor-Reynolds number Re lambda >~ 500.

Reynolds number and scale dependence of turbulent eddy decorrelation

The temporal decay of the Lagrangian velocity correlation function of isotropic, homogeneous turbulence is calculated by following particle trajectories and via continued fraction representation. We report on the eddy size as well as on the Reynolds number dependence. Most remarkably, the exponential relaxation approximation for the correlation function turns out to be insufficient. Memory effects result in a considerable enhancement: the decay is more than an order of magnitude faster. The very high Reynolds number turbulent flow fields are generated by a reduced wave vector set approximation of the Navier-Stokes dynamics, but for smaller Re the conclusions are confirmed by direct numerical simulations. Two orders of magnitude of the Taylor-Reynolds number are covered, up to Reλ ˜ 104.

Anisotropy and scaling corrections in turbulence.

We analyze second-order turbulent velocity moments both in r and in p space. Finite size corrections induce dramatic differences between local r- and p-space scaling exponents. As analytically accessible examples we focus on two popular parametrizations: the Batchelor parametrization for the r-space structure function and a common parametrization for the energy spectrum, E(p)\ensuremath{\propto}${\mathit{p}}^{\mathrm{\ensuremath{-}}5/3}$exp(-p/${\mathit{p}}_{\mathit{d}}$). The spectral bottleneck energy pileup hidden in the Batchelor parametrization results in an extended r-space scaling range, comparable to experimental ones for the same Taylor-Reynolds number ${\mathrm{Re}}_{\ensuremath{\lambda}}$. Shear effects are discussed in terms of (global) apparent scaling correction \ensuremath{\delta}${\mathrm{\ensuremath{\zeta}}}^{\mathit{app}}$(${\mathrm{Re}}_{\ensuremath{\lambda}}$) to classical scaling, which again depend on whether looked at in r or in p space. The differences can be traced back to the subtleties of the crossovers in the velocity moments. Our observations emphasize the need for more experimental information on crossovers between different subranges. \textcopyright{} 1996 The American Physical Society.

On the Lundgren–Townsend model of turbulent fine scales

The strained-spiral vortex model of turbulent fines scales given by Lundgren [Phys. Fluids 25, 2193 (1982)] is used to calculate vorticity and velocity-derivative moments for homogeneous isotropic turbulence. A specific form of the relaxing spiral vortex is proposed modeled by a rolling-up vortex layer embedded in a background containing opposite signed vorticity and with zero total circulation at infinity. The numerical values of two dimensionless groups are fixed in order to give a Kolmogorov constant and skewness which are within the range of experiment. This gives the result that the ratio of the ensemble average hyperskewness S2p+1≡ (∂u/∂x)2p+1/[(∂u/∂x)2](2p+1)/2 to the hyperflatness F2p≡(∂u/∂x)2p/[(∂u/∂x)2] p, p=2,3,..., is constant independent of Taylor–Reynolds number Rλ, as is the ratio of the 2pth moment of one component of the vorticity Ω2p≡ω2px/(ω2x)p to F2p. A cutoff in a relevant time integration is then used to eliminate vortex-sheet-induced divergences in the integrals corresponding to ω2px, p=2,3,..., and an assumption is made that the lateral scale of the spiral vortex in the model is the geometric mean of the Taylor and the Kolmogorov microscales. This gives Ω2p=Ω̂2pRλp/2−3/4, F2p=F̂2pRλp/2−3/4 and S2p+1=Ŝ2p+1Rλp/2−3/4, p=2,3,..., with explicit calculation of the numbers Ω̂2p, F̂2p, and Ŝ2p+1. The results of the model are compared with experimental compilation of Van Atta and Antonia [Phys. Fluids 23, 252 (1980)] for F4 and with the isotropic turbulence calculations of Kerr [J. Fluid Mech. 153, 31 (1985)] and of Vincent and Meneguzzi [J. Fluid Mech. 225, 1 (1991)].

Rederivation and further assessment of the LET theory of isotropic turbulence, as applied to passive scalar convection

A simpler and more rigorous derivation is presented for the LET (Local Energy Transfer) theory, which generalizes the theory to the non-stationary case and which corrects some minor errors in the original formulation (McComb 1978), Previously, ad hoc generalizations of the LET theory (McComb & Shanmugasundaram 1984) gave good numerical results for the free decay of isotropic turbulence. The quantitative aspects of these previous computations are not significantly affected by the present corrections, although there are some important qualitative improvements. The revised LET theory is also extended to the problem of passive scalar convection, and numerical results have been obtained for freely decaying isotropic turbulence, with Taylor–Reynolds numbers in the range 5 [les ] R λ [les ] 1060, and for Prandtl numbers of 0.1, 0.5 and 1.0. At sufficiently high values of the Reynolds number, both velocity and scalar spectra are found to exhibit Kolmogorov-type power laws, with the Kolmogorov spectral constant taking the value α = 2.5 and the Corrsin–Oboukhov constant taking a value of β = 1.1.

Influence of High Rotational Speeds on the Heat Transfer and Discharge Coefficients in Labyrinth Seals

Heat transfer and leakage loss measurements were obtained for compressible flows in typical straight-through labyrinth seals with high rotational speeds. The experiments are an extension of our earlier measurements in a stationary test facility. In order to ensure direct comparisons to the original experiments, the principal dimensions of the test facility and gas dynamic parameters of the hot gas were kept similar. The new study encompasses a wide range of Taylor numbers, Reynolds numbers, and clearances between the rotating annular fins and the stationary shroud. Heat transfer coefficients are determined for the stator as well as for the rotor. Temperature measurements along the cooled rotor were performed utilizing a high-accuracy telemetric system. Continuous clearance control was achieved by employing specially designed gages. Detailed pressure and temperature measurements in the axial as well as in the circumferential direction were performed. Heat transfer coefficients and loss parameters are presented and compared with those obtained under steady-state conditions.