Axisymmetric Turbulent Wake Research Articles

Spatial distribution of coherent structures in a self-similar axisymmetric turbulent wake

Spatial distribution of coherent structures in a self-similar axisymmetric turbulent wake

Interaction of two axisymmetric turbulent wakes

The interaction between turbulent axisymmetric wakes plays an important role in many industrial applications, notably in the modeling of wind farms. This work proposes an experimental study on the interaction between two bluff bodies. We find that a simple mathematical expression for the wake interaction length based on nonequilibrium turbulence scalings can be used to collapse the streamwise developments of the second, third, and fourth moments of the streamwise fluctuating velocity.

The impact of swirl and wake strength on turbulent axisymmetric wake evolution

An experimental investigation of swirl and wake strength influence on axisymmetric turbulent wake evolution was conducted. A novel wake generator design wire mounted in a wind tunnel test section with low free-stream turbulence produced wake Reynolds numbers based on momentum thickness and free-stream velocity in excess of 14 000 and swirl numbers up to 0.4 with minimal blockage. Steady-state blade element momentum simulations of reference wind turbine designs indicated that wind turbines operate in the flow regimes studied, indicating the practical aspects of this work. Stereoscopic particle image velocimetry was used to acquire three components of velocity in the swirling wake at locations up to approximately ten diameters downstream. Quantitative measures of wake growth and decay were deduced using available equilibrium similarity scaling for the swirling wake. The results show an increase above 50% in growth and axial velocity decay rate constants over the range of swirl strength studied compared to those of the non-swirling wake. Tangential velocity decay constants were shown to decrease with swirl strength over the range of conditions studied. Notably, changes in wake strength have little influence on growth and decay rates when compared to changes in swirl strength for the flow regimes studied in this work.

Application of the Townsend-George wake theory to field measurements of wind turbine wakes

As wind turbines are usually clustered in wind farms, knowledge of the evolution of the wind turbine wakes is important because downstream turbines will be exposed to them, causing higher loads and maintenance times. For that reason, wind turbine wakes have been studied intensively and different engineering wake models were derived. However, none of them is constructed from a few basic and robust assumptions, while such a formalism already exists for the axisymmetric turbulent wake. Therefore, we will apply these models to data obtained in a wind farm using a scanning LiDAR. The wakes of two wind turbines are analyzed in four different wind directions chosen so that their wakes will have different degrees of interaction. The axisymmetric wake models are found to perform better than the Jensen wake model, and the main source of improvement is shown to be the presence of a virtual origin. Indeed, a virtual origin is shown to also improve the Jensen wake model significantly. Overall, our results indicate that a view from classical axisymmetric wake theory can help to improve the understanding of the evolution of wind turbine wakes in the atmospheric boundary layer.

Non-Kolmogorov scaling and dissipation laws in planar turbulent plume

We explore theoretically non-Kolmogorov scaling and dissipation laws by employing continuous symmetry group transformations on statistical model equations for planar turbulent plume. The dissipation laws for mean turbulent K. E. (k¯) and mean thermal intensity (kθ¯) at infinite Reynolds number limit are obtained as ϵ∼(k¯)3/2/(δuReδum), ϵθ∼(k¯)1/2(kθ¯)/(δuReδum), where the exponent m=3a2/(a1+a2) preferably belongs to [−3/2, 1], a1, a2 the dilation symmetry group parameters. Here Reδu is the local Reynolds number based on plume velocity width δu. The Kolmogorov dissipation law will hold only when k¯∼u¯2∼u′v′¯, which implies the same streamwise variations of Reynolds stresses, while non-Kolmogorov dissipation can hold only when k¯∼u¯2≁u′v′¯ indicating different streamwise variations of Reynolds stresses. When m = −3/2, the plume scales as a non-equilibrium exponential function of streamwise distance x, while for m ∈ (−3/2, 0) ∪ (0, 1], there exist multiple non-equilibrium power law scalings and classical equilibrium scaling corresponds to m = 0. The turbulent kinetic energy dissipation law for m ≈ 1 is reported both experimentally and theoretically in axisymmetric turbulent wake and planar jet. When m = 1, the plume velocity and temperature widths (δu, δT) grow as ∼(x + a)2/5, centerline mean velocity grows as u¯c∼(x+a)1/5, and mean temperature decays as θc ∼ (x + a)−3/5, with a being a virtual origin. The non-equilibrium dissipation scaling instigates a variation of plume entrainment coefficient with streamwise distance. In particular, the region where dissipation scaling with exponent m = 1 holds, the entrainment coefficient in planar plume varies as ∼(x + a)−3/5, whereas the entrainment coefficient reported recently in planar jet varies as ∼(x + a)−1/3.

Multitude scaling laws in axisymmetric turbulent wake

We establish theoretically multitude scaling laws of a self-similar (statistical) axisymmetric turbulent wake. At infinite Reynolds number limit, the flow evolves as general power law and a new exponential law of streamwise distance, consistent with the criterion of equilibrium similarity hypothesis. We found power law scalings for components of the homogeneous dissipation rate (ϵ) obeying the non-Richardson-Kolmogorov cascade as ϵu∼ku3/2/(lRelm), ϵv∼kv3/2/l, kv∼ku/Rel2m, 0 < m < 3. Here ku and kv are the components of the Reynolds normal stress, l is the local length scale, and Rel is the Reynolds number. The Richardson-Kolmogorov cascade corresponds to m = 0. For m ≈ 1, the power law agrees with non-equilibrium scaling laws observed in recent experiments of the axisymmetric wake. On the contrary, the exponential scaling law follows the above dissipation law with different regions of existence for power index m = 3. At finite Reynolds number with kinematic viscosity ν, scalings obey the dissipation laws ...

Related self-similar statistics of the turbulent/non-turbulent interface and the turbulence dissipation

The scalings of the local entrainment velocity$v_{n}$of the turbulent/non-turbulent interface and of the turbulence dissipation rate are closely related to each other in an axisymmetric and self-similar turbulent wake. The turbulence dissipation scaling implied by the Kolmogorov equilibrium cascade phenomenology is consistent with a Kolmogorov scaling of$v_{n}$whereas the non-equilibrium dissipation scaling reported for various turbulent flows in Vassilicos (Annu. Rev. Fluid Mech., vol. 47, 2015, pp. 95–114), Dairayet al.(J. Fluid Mech., vol. 781, 2015, pp. 166–195), Goto &amp; Vassilicos (Phys. Lett. A, vol. 379 (16), 2015, pp. 1144–1148) and Obligadoet al.(Phys. Rev. Fluids, vol. 1 (4), 2016, 044409) is consistent with a different scaling of $v_{n}$. We present results from a direct numerical simulation of a spatially developing axisymmetric and self-similar turbulent wake which supports this conclusion and the assumptions that it is based on.

Numerical modeling of the final stage of axisymmetric turbulent wake decay

Using the modified e ~ e turbulence model the numerical simulation of the final stage of viscose stage of turbulent wake decay behind axisymmetric bodies was performed.

Direct numerical simulation of a turbulent wake: The non-equilibrium dissipation law

A Direct Numerical Simulation (DNS) study of an axisymmetric turbulent wake generated by a square plate placed normal to the incoming flow is presented. It is shown that the new axisymmetric turbulent wake scalings obtained recently for a fractal-like wake generator (Dairay et al., 2015), specifically a plate with irregular multiscale periphery placed normal to the incoming flow, are also present in an axisymmetric turbulent wake generated by a regular square plate. These new scalings are therefore not caused by the multiscale nature of the wake generator but have more general validity.

Non-equilibrium scaling laws in axisymmetric turbulent wakes

We present a combined direct numerical simulation and hot-wire anemometry study of an axisymmetric turbulent wake. The data lead to a revised theory of axisymmetric turbulent wakes which relies on the mean streamwise momentum and turbulent kinetic energy equations, self-similarity of the mean flow, turbulent kinetic energy, Reynolds shear stress and turbulent dissipation profiles, non-equilibrium dissipation scalings and an assumption of constant anisotropy. This theory is supported by the present data up to a distance of 100 times the wake generator’s size, which is as far as these data extend.

Axisymmetric Turbulent Wakes with New Nonequilibrium Similarity Scalings

The recently discovered nonequilibrium turbulence dissipation law implies the existence of axisymmetric turbulent wake regions where the mean flow velocity deficit decays as the inverse of the distance from the wake-generating body and the wake width grows as the square root of that distance. This behavior is different from any documented boundary-free turbulent shear flow to date. Its existence is confirmed in wind tunnel experiments of wakes generated by plates with irregular edges placed normal to an incoming free stream. The wake characteristics of irregular bodies such as buildings, bridges, mountains, trees, coral reefs, and wind turbines are critical in many areas of environmental engineering and fluid mechanics.

Numerical simulation of turbulent wakes with variable total excess momentum

Numerical simulation of the dynamics of plane and axisymmetric turbulent wakes in a homogeneous fluid is performed using the modified e ∼ ɛ model. The calculation results are in good agreement with the known experimental data. Asymptotic decay of axisymmetric turbulent wakes with small nonzero total excess momentum is numerically simulated.

On the universality of turbulent axisymmetric wakes

AbstractDirect numerical simulations (DNS) of two time-dependent, axially homogeneous, axisymmetric turbulent wakes having very different initial conditions are presented in order to assess whether they reach a universal self-similar state as classically hypothesized by Townsend. It is shown that an extensive early-time period exists during which the two wakes are individually self-similar with wake widths growing like$\delta \propto {t}^{1/ 3} $, as predicted by classical dimensional analysis, but have very different growth rates and are thus not universal. Subsequently, however, the turbulence adjusts to yield, eventually, wakes that are structurally identical and have the same growth rate (also with$\delta \propto {t}^{1/ 3} $) so provide clear evidence of a universal, self-similar state. The former non-universal but self-similar state extends, in terms of a spatially equivalent flow behind a spherical body of diameter$d$, to a distance of$O(3000d)$whereas the final universal state does not appear before$O(5000d)$(and exists despite relatively low values of the Reynolds number and no evidence of a spectral${\kappa }^{\ensuremath{-} 5/ 3} $inertial subrange). Universal wake evolution is therefore likely to be rare in practice. Despite its low Reynolds number, the flow does not exhibit the sometime-suggested alternative self-similar behaviour with$\delta \propto {t}^{1/ 2} $(as for the genuinely laminar case) at large times (or, equivalently, distances), since the eddy viscosity remains large compared to the molecular viscosity and its temporal variations are not negligible.

Three-Component Vectorial Proper Orthogonal Decomposition of Axisymmetric Wake Behind a Disk

One-component scalar, two-component vectorial, and three-component vectorial proper orthogonal decompositions of the axisymmetric turbulent wake have been studied to investigate possible effects of the number of components comprising the two-point correlation tensor forming the kernel of the proper orthogonal decompositions integral equation. A wind-tunnel experiment has been performed 50 diameters downstream of the wake generator, which was a disk of 20 mm in diameter. The Reynolds number based on the disk diameter was 20,400. Twelve cross hot wires were used to obtain the simultaneous multipoint measurements. Six of the probes were located on a fixed rake and the other six probes were located on a movable rake, which was traversed from 10 to 180 deg in 10 deg increments. Two experiments were carried out to obtain first the streamwise and azimuthal components of the velocity, and second to obtain the streamwise and radial components of the velocity. Seven out of nine components of the two-point correlation tensor were computed using measured velocities and the remaining two two-point correlations were extracted from the data using the continuity equations. The one-component scalar, and two- and three-component vectorial decomposition results were essentially in agreement. Eigenvalues for the full vector proper orthogonal decompositions integrated over frequency showed that Fourier mode 2 was the largest in both the streamwise and azimuthal velocity. Azimuthal mode 2 peaks at the near-zero frequency, whereas azimuthal mode 1 peaks at a frequency which corresponds to the vortex shedding frequency. Similar features in turbulence kinetic energy distributions were also observed in the modally decomposed two-point cross-spectra and two-point cross-correlations. Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Integral Solution for the Mean Flow Profiles of Turbulent Jets, Plumes, and Wakes

Integral methods are used to derive similarity solutions for several quantities of interest including the cross-stream velocity, Reynolds stress, the dominant turbulent kinetic energy production term, and eddy diffusivities of momentum and heat for axisymmetric and planar turbulent jets, plumes, and wakes. A universal constant is evaluated for axisymmetric and planar plumes. The cross-stream velocity profiles show that jets and axisymmetric plumes experience an outflow near the axis and an inflow far away from it. The outflow is attributed to the decay of the centerline velocity with downstream distance, and the extent and magnitude of outflow correlates with the streamwise decay of the centerline velocity. It is also shown that the entrainment velocity should not in general be equated to the product of the entrainment coefficient and the centerline velocity. It is found that, due to similar governing equations, profiles for jets and plumes are qualitatively similar. Our results show that the derived quantities are strong functions of streamwise and cross-stream positions, in contrast to previous approaches that assumed constant (in the cross-stream direction) eddy viscosity and thermal diffusivity. The turbulent Prandtl number is approximately equal to unity which matches the value quoted in the literature.

Equilibrium similarity, effects of initial conditions and local Reynolds number on the axisymmetric wake

Equilibrium similarity considerations are applied to the axisymmetric turbulent wake, without the arbitrary assumptions of earlier theoretical studies. Two solutions for the turbulent flow are found: one for infinite local Reynolds number which grows spatially as x1/3; and another for small local Reynolds number which grows as x1/2. Both solutions can be dependent on the upstream conditions. Also, the local Reynolds number diminishes with increasing downstream distance, so that even when the initial Reynolds number is large, the flow evolves downstream from one state to the other. Most of the available experimental data are at too low an initial Reynolds number and/or are measured too near the wake generator to provide evidence for the x1/3 solution. New results, however, from a laboratory experiment on a disk wake and direct numerical simulations (DNS) are in excellent agreement with this solution, once the flow has had large enough downstream distance to evolve. Beyond this the ratio of turbulence intensity to centerline velocity deficit is constant, until the flow unlocks itself from this behavior when the local Reynolds number goes below about 500 and the viscous terms become important. When this happens the turbulence intensity ratio falls slowly until the x1/2 region is reached. No experimental data are available far enough downstream to provide unambiguous evidence for the x1/2 solution. The prediction that the flow should evolve into such a state, however, is confirmed by recent DNS results which reach the x1/2 solution at about 200 000 momentum thicknesses downstream. After this the turbulence intensity ratio is again constant, until box-size affects the calculation and the energy decays exponentially.

Proper orthogonal decomposition of an axisymmetric turbulent wake behind a disk

A proper orthogonal decomposition (POD) study of the axisymmetric turbulent wake behind a disk has been performed using multipoint hot-wire data. The Reynolds number based on the free stream velocity and disk diameter was kept constant at 28 000. The investigated region spanned from 10 to 50 disk diameters downstream. The hot-wire data were obtained using two rakes: a seven wire fixed array and a six wire array azimuthally traversable to span the cross section of the flow in increments of 15°. The instantaneous streamwise velocity component data were Fourier transformed in time and decomposed in Fourier series in the azimuthal direction to form the kernel for the POD. For all downstream positions, two distinct peaks were found in the first eigenspectrum: one at azimuthal mode 2 at near zero frequency, and another at azimuthal mode 1 at a fixed Strouhal number (fd/U∞) of 0.126. Both peaks decrease in magnitude as the flow evolves downstream, but the peak at the Strouhal number 0.126 decreases more rapidly than the one at near-zero frequency, leaving the latter to eventually dominate. Because of this evolution, the eigenvalues integrated over frequency show an azimuthal mode-1 dominance at x/D=10 and a mode-2 dominance by x/D=50. The results are compared to those recently obtained in the axisymmetric far jet, and the results of previous wake investigations.

Turbulence, similarity scaling and vortex geometry in the wake of a towed sphere in a stably stratified fluid

Late wakes (Nt&gt; 20) of towed spheres in a stably stratified fluid were analysed in a plane using a reliable, customized DPIV technique that provides sufficient spatial and temporal resolution to cover all important scales of motion in this freely decaying geophysical flow. Systematic experiments were conducted with independent variation ofRe∈ [103, 104] andF∈ [1, 10] (F ≡ 2U/NDis an internal Froude number based on the buoyancy frequency,N, and the sphere radius,D/2), and for selected {Re, F} pairs above this range.The normalized wake width grows at approximately the same rate as in a three-dimensional unstratified wake, but it becomes narrower, not wider, with decreasingF(i.e. as stratification effects become more important). The centreline defect velocity, on the other hand, reaches values an order of magnitude above those measured for three-dimensional unstratified wakes at equivalent downstream locations. Both observations are argued to be consequences of the very high degree of order and coherence that emerge in the late-wake vortex structures.Streamwise-averaged turbulence quantities, such as the velocity fluctuation magnitude, and mean-square enstrophy, show similar power law behaviour for allRe≤ 5 × 103, with exponents equal to those expected in three-dimensional axisym-metric turbulent wakes. There is no obvious physical reason why three-dimensional arguments are so successful in such a flow, and at such long evolution times. The scaling collapses none of the cases forRebelow 4 – 5 × 103, appearing to establish a minimumRefor a class of self-similar stratified wake flows that evolve from fully turbulent initial conditions.Individual vortex cross-sections appear to be well approximated by Gaussian distributions at allRe, F andNtstudied here. The scaling behaviour of individual vortices mimics that of the statistical, wake-averaged quantities, and differs measurably from a simple two-dimensional viscous diffusion model. The importance of formulating a realistic three-dimensional model is discussed, and some limited steps in this direction point to future useful experiments and modelling efforts.

Self-similar solutions for an axisymmetric turbulent wake

Self-similar solutions for an axisymmetric turbulent wake

Temperature fluctuation balance in an axisymmetric turbulent wake

Results of experiments conducted in the thermal wake behind an axisymmetric body are used to evaluate the radial profiles of different terms of the balance equation for the mean square of temperature fluctuations.