Velocity Derivative Skewness Research Articles

Laws of turbulence decay from direct numerical simulations.

Inspection of available data on the decay exponent for the kinetic energy of homogeneous and isotropic turbulence (HIT) shows that it varies by as much as 100%. Measurements and simulations often show no correspondence with theoretical arguments, which are themselves varied. This situation is unsatisfactory given that HIT is a building block of turbulence theory and modelling. We take recourse to a large base of direct numerical simulations and study decaying HIT for a variety of initial conditions. We show that the Kolmogorov decay exponent and the Birkhoff-Saffman decay are both observed, albeit approximately, for long periods of time if the initial conditions are appropriately arranged. We also present, for both cases, other turbulent statistics such as the velocity derivative skewness, energy spectra and dissipation, and show that the decay and growth laws are approximately as expected theoretically, though the wavenumber spectrum near the origin begins to change relatively quickly, suggesting that the invariants do not strictly exist. We comment briefly on why the decay exponent has varied so widely in past experiments and simulations. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Effect of spatial dimension on a model of fluid turbulence

A numerical study of the $d$ -dimensional eddy damped quasi-normal Markovian equations is performed to investigate the dependence on spatial dimension of homogeneous isotropic fluid turbulence. Relationships between structure functions and energy and transfer spectra are derived for the $d$ -dimensional case. Additionally, an equation for the $d$ -dimensional enstrophy analogue is derived and related to the velocity derivative skewness. Comparisons are made to recent four-dimensional direct numerical simulation results. Measured energy spectra show a magnified bottleneck effect which grows with dimension whilst transfer spectra show a varying peak in the nonlinear energy transfer as the dimension is increased. These results are consistent with an increased forward energy transfer at higher dimensions, further evidenced by measurements of a larger asymptotic dissipation rate with growing dimension. The enstrophy production term, related to the velocity derivative skewness, is seen to reach a maximum at around five dimensions and may reach zero in the limit of infinite dimensions, raising interesting questions about the nature of turbulence in this limit.

Homogeneous isotropic turbulence in four spatial dimensions

Direct numerical simulation is performed for the forced Navier–Stokes equation in four spatial dimensions. Well equilibrated, long time runs at sufficient resolution were obtained to reliably measure spectral quantities, the velocity derivative skewness, and the dimensionless dissipation rate. Comparisons to corresponding two- and three-dimensional results are made. Energy fluctuations are measured, and the results show a clear reduction moving from three to four dimensions. The dynamics show simplifications in four dimensions with a picture of increased forward energy transfer resulting in an extended inertial range with a smaller Kolmogorov scale. This enhanced forward transfer is linked to our finding of increased dissipative anomaly and velocity derivative skewness.

Velocity derivative skewness and its budget in non-equilibrium time-reversed turbulence

The velocity derivative skewness and its budget equation are analysed in non-equilibrium time-reversed turbulence. We show that the positive skewness is related to the behaviors of backward energy transfer and enstrophy divergence during the non-equilibrium procedure. A dynamic equilibrium state of the turbulence corresponds to a balanced budget of the skewness, whereas the balance is destroyed when the turbulence is perturbed and with a non-equilibrium phenomenon. Production and pressure terms are dominant to the evolution of the skewness compared to the viscosity term. Furthermore, based on this budget equation, a theoretical value of the skewness is derived for sufficiently high-Reynolds turbulent flows. The budget presented here might inspire future investigations for the modeling of non-equilibrium turbulence.

Inertial range skewness of the longitudinal velocity derivative in locally isotropic turbulence

A velocity derivative skewness akin to inertial-scale motions of turbulence is introduced and studied analytically, numerically, and experimentally. As an alternative to a conventional fine-scale parameter, it has an advantage in measurability while retaining a connection to vorticity dynamics.

Improved vortex method for large-eddy simulation inflow generation

The generation of turbulent inflow conditions in large-eddy simulation (LES) is a key ingredient for general applications of LES in both academic turbulent flows and industrial designs with complicated engineering flows. This is because accurate predictions of the fluid behaviour are strongly dependent on the inflow conditions, particularly in turbulent flows at high turbulent Reynolds numbers. This paper aims at improving the vortex method (VM) of Sergent that demands long adaptive distances (12 times the half channel height, for a channel flow at Reτ=395) to achieve high quality turbulence, and evaluating the equilibrium of the flow field obtained in terms of both the equilibrium of the mean flow and that of the turbulence (inter-scale turbulent energy transfer). The mean flow equilibrium is checked with classic criteria such as the friction velocity. In order to assess the equilibrium of turbulence, we propose using the velocity-derivative skewness, because it associates with the balance of energy transfer between large- and small-scale fluid motions. Numerical tests with the optimised set of model parameters reveal that the IVM is very efficient, in terms of adaptive distance, in generating high-quality synthetic turbulent fluctuations over a moderate distance: 6h for channel flow and 21δ for flat-plate boundary layer, with h and δ being respectively the half channel height and the nominal boundary layer thickness.

Boundedness of the velocity derivative skewness in various turbulent flows

The variation of$S$, the velocity derivative skewness, with the Taylor microscale Reynolds number$\mathit{Re}_{{\it\lambda}}$is examined for different turbulent flows by considering the locally isotropic form of the transport equation for the mean energy dissipation rate$\overline{{\it\epsilon}}_{iso}$. In each flow, the equation can be expressed in the form$S+2G/\mathit{Re}_{{\it\lambda}}=C/\mathit{Re}_{{\it\lambda}}$, where$G$is a non-dimensional rate of destruction of$\overline{{\it\epsilon}}_{iso}$and$C$is a flow-dependent constant. Since$2G/\mathit{Re}_{{\it\lambda}}$is found to be very nearly constant for$\mathit{Re}_{{\it\lambda}}\geqslant 70$,$S$should approach a universal constant when$\mathit{Re}_{{\it\lambda}}$is sufficiently large, but the way this constant is approached is flow dependent. For example, the approach is slow in grid turbulence and rapid along the axis of a round jet. For all the flows considered, the approach is reasonably well supported by experimental and numerical data. The constancy of$S$at large$\mathit{Re}_{{\it\lambda}}$has obvious ramifications for small-scale turbulence research since it violates the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) but is consistent with the original similarity hypothesis (Kolmogorov,Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303).

Velocity derivative skewness in fractal-generated, non-equilibrium grid turbulence

The evolution of the velocity derivative skewness, S(∂u/∂x), is investigated along two streamwise axes and four transverse positions in the wake of a square-fractal-element grid. In the near-field, the produced turbulence exhibits non-equilibrium characteristics including Cϵ∼ReMα/ReLβ. In the far-field, the turbulence agrees with canonical grid turbulence results and Cϵ is approximately constant. It is found that in the non-equilibrium region, the value of −S(∂u/∂x) is dependent on both streamwise and transverse positions, but after a sufficient decay period, it takes on a near constant value in the far-field. It is demonstrated that the evolution Cϵ approximately corresponds to that of −S(∂u/∂x), which is suggestive that some of the non-equilibrium properties are likely a result of residual strain from the turbulence generating conditions.

Energy spectra power laws and structures

Direct numerical simulations (DNS) of two inviscid flows, the Taylor–Green flow and two orthogonal interacting Lamb dipoles, together with the DNS of forced isotropic turbulence, were performed to generate data for a comparative study. The isotropic turbulent field was considered after the transient and, in particular, when the velocity derivative skewness oscillates around −0.5. At this time, Rλ ≈ 257 and a one decade wide k−5/3 range was present in the energy spectrum. For the inviscid flows the fields were considered when a wide k−3 range was achieved. This power law spectral decay corresponds to infinite enstrophy and is considered one of the requirements to demonstrate that the Euler equations lead to a finite time singularity (FTS). Flow visualizations and statistics of the strain rate tensor and vorticity components in the principal axes of the strain rate tensor (λ, λ) were used to classify structures. The key role of the intermediate component 2 is demonstrated by its good correlation with enstrophy production. Filtering of the fields shows that the slope of the power law is directly connected to self-similar structures, whose radius of curvature is smaller the steeper the spectrum.

Velocity derivative skewness in isotropic turbulence and its measurement with hot wires

We investigate the effect of the hot wire resolution on the measurement of the velocity derivative skewness in homogeneous isotropic turbulence. Single- and cross-wire configurations (with different lengths and separations of the wires, and temporal sampling resolution) are considered. Predictions of the attenuation on the basis of a model for the energy spectrum are compared to experimental and numerical data in grid and box turbulence, respectively. It is shown that the model-based correction is accurate for the single wire but not for the cross-wire. In the latter case, the effect of the separation between the wires is opposite to that found in the experiments and simulations. Moreover, the attenuation predicted by the numerical data is in good agreement with that observed in the experiment. For both probe configurations, the sampling resolution has a sizeable attenuation effect, but, for the X-probe, the impact of the separation between the wires is more important. In both cases, the length of the wires has only a minor effect, in the non-dimensional range of wire length investigated. Finally, the present experimental data support the conclusion that the skewness is constant with the Reynolds number, in agreement with Kolmogorov’s 41 theory.

Inertial consistent subgrid model for large-eddy simulation based on the lattice Boltzmann method

The recently introduced inertial-range (IR) consistent Smagorinsky model and the classical Smagorinsky model are applied to the large-eddy simulation (LES) of decaying homogeneous isotropic turbulence based on the lattice Boltzmann method (LBM), which is implemented using the 19-velocity D3Q19 lattice model. The objectives of this study are to examine the effectiveness of the LES-LBM technique for study of turbulence and to extend and validate the efficiency of the inertial-range consistent Smagorinsky model for lattice Boltzmann fluid dynamics. The LES-LBM results are compared with the direct numerical simulation data as well as experimental data. The time evolution of the kinetic energy and the decay exponents of the dissipation rate, the velocity derivative skewness, and instantaneous energy spectra are analyzed. The dependency of behavior of the model coefficients on the ratio of grid width Δ and the Kolmogorov scale η is examined numerically. The results demonstrate that the LES-LBM in conjunction with the IR consistent Smagorinsky model can be used to simulate turbulence more satisfactorily than the standard Smagorinsky model.

Refined vorticity statistics of decaying rotating three-dimensional turbulence

The influence of background rotation on all nontrivial third-order vorticity correlations is studied for an unbounded incompressible homogeneous turbulent flow, using pseudo-spectral direct numerical simulation. The behaviour of third-order vorticity correlations is found to be consistent with exact theoretical predictions presented herein for axisymmetric turbulence without mirror symmetry. Particular attention is given to the vertical vorticity skewness S ω3. Its dependence on the viscosity, the initial value of the velocity gradient skewness, and the background rotation rate has been thoroughly investigated. The initial growth rate of S ω3 provides evidence for a power-law behaviour proportional to t 0.75 ± 0.1 for all considered cases, in agreement with recent experimental results by Morize et al. [Phys. Fluids 17 (2005) 095105]. It is also found that higher background rotation rates – implying more linearity – result in lower final values of while lower viscosities and lower initial (absolute) values of the velocity derivative skewness both yield higher final values of S ω3.

Decaying grid-generated turbulence in a rotating tank

The decay of initially three-dimensional homogeneous turbulence in a rotating frame is experimentally investigated. Turbulence is generated by rapidly towing a grid in a rotating water tank, and the velocity field in a plane perpendicular to the rotation axis is measured by means of particle image velocimetry. During the decay, strong cyclonic coherent vortices emerge, as the result of enhanced stretching of the cyclonic vorticity by the background rotation, and the selective instability of the anticyclonic vorticity by the Coriolis force. This asymmetry towards cyclonic vorticity grows on a time scale Ω−1 (Ω is the rotation rate), until the friction from the Ekman layers becomes dominant. The energy spectrum perpendicular to the rotation axis becomes steeper as the instantaneous Rossby number Roω=ω′∕2Ω decreases below the value 2±0.5 (ω′ is the root-mean square of the vertical vorticity). The spectral exponent increases in time from its classical Kolmogorov value 5∕3 up to values larger than 2. Below the threshold Roω<2, the velocity derivative skewness decreases as ∣S∣∝Roω, reflecting the inhibition of the energy transfers by the background rotation, with a net inverse energy cascade that develops at large scales.

On the evolution of decaying isotropic turbulence

A direct numerical simulation is performed on 2563 grids for decaying isotropic turbulence. The total kinematic energy, Taylor micro-scale, Taylor micro-scale Reynolds number and the velocity derivative skewness are calculated. The snapshots of energy spectra and energy transfer spectra are plotted. These measurements verify the DIA predictions: decaying isotropic turbulence has the energy propagation and occupies the final decay periods. The skewness remains to some level with small variation even in the final decay period.

An equality about the velocity derivative skewness in turbulence

We study velocity derivative skewness S of incompressible homogeneous isotropic turbulence. By using exact relations of isotropic turbulence and various typical models of second-order structure function DLL(r) and energy spectrum E(k), it is found that −S=C(kc/kd)2 when Taylor-microscale Reynolds number Rλ is high. Here, C is a coefficient, kc is the center wavenumber of energy dissipation spectrum, and kd is the Kolmogorov wavenumber. Therefore, the problem of Reynolds number dependence of S becomes the problem of Reynolds number dependence of kc/kd. In the inertial range, we have scaling DLL(r)∼rζ2 and E(k)∼k−(ζ2+1), ζ2 is the second-order inertial-range scaling exponent. Equality −S=C(kc/kd)2 is valid in the case of ζ2>2/3 (intermittency models of Kolmogorov’s 1962 theory) as well as in the case of ζ2=2/3 (Kolmogorov’s 1941 theory).

An analytical model of velocity-derivative skewness of rotating homogeneous turbulence

The velocity-derivative skewness of rotating homogeneous turbulence is analytically derived from the turbulent kinetic energy dissipation rate equation by assuming a simple energy spectrum function. It predicts the cut-off of spectral energy at large rotation rate, and the predicted reduction of skewness due to rotation agrees well with the results of direct numerical simulation and eddy-damped quasi-normal Markovian closure.

The approach to self-preservation of scalar fluctuations decay in isotropic turbulence

An analytical study of isotropic scalar fluctuations decay in isotropic turbulence is undertaken. From a fixed-point analysis, the existence of two complete self-preserving regimes is demonstrated. One of them relates to the final period of decay whereas the other one corresponds to the decay at large Reynolds and Péclet numbers. In both cases, the scalar-to-velocity timescale ratio is constant. In the final period, its asymptotic value is determined by the values of the coefficients of velocity and scalar enstrophy destruction. At large Reynolds and Péclet numbers, it depends on the latter as well as on the values of the mixed-derivative and velocity derivative skewness coefficients. A model for the approach to full self-preservation of scalar fluctuations decay is proposed. This model accounts for non-equilibrium resulting from initial unbalanced vortex stretching. In the conditions of experiments on temperature fluctuations decay in grid-turbulence, it displays a transient regime, extending over the region of measurements, during which the decay exponent is not constant but slowly varying.

Development of enstrophy and spectra in numerical turbulence

Decaying isotropic turbulence with initial Taylor microscale Reynolds number (Rλ≤258) is studied via direct numerical simulations (DNS), with spectral resolution ≤2563. DNS results are compared with two-point closure, in the form of the direct interaction approximation (DIA) and the test field model (TFM). The goals of this study are to understand the time-dependence of enstrophy and spectra as they evolve from random initial conditions, and to assess and interpret differences between DNS and closure. Two time scales are identified in the DNS. The first is that for the development of normalized enstrophy production (velocity derivative skewness) and is independent of Rλ. The second is that for the saturation of the enstrophy which follows after a longer period of near exponential growth and is strongly Rλ dependent. For ν≠0, the DIA represents the time development of both integral quantities, such as enstrophy and spectra, with surprising accuracy in spite of its lack of invariance to random large-scale sweeping. The TFM has significant energy range errors, which we attribute to its Markovianization assumption.

Cloud-Environment Interface Instability: Part II: Extension to Three Spatial Dimensions

Abstract Three-dimensional numerical experiments were performed with thermals rising in a stably stratified environment to study the cloud-environment boundary instability. This work extends that reported in Part I. It is shown that the analytical theory developed in Part I, which describes the evolution of the laminar interface between the thermal and its environment, applies to the three-dimensional case with only minor modifications. As in the two-dimensional case, the scale selection and growth rate of the unstable modes appear to depend upon the depth and velocity change across the shear layer near the interface, which is in rough agreement with classical linear theory developed for the case of planar geometry. Analysis is presented that indicates further evolution of the three-dimensional eddies results in a transition to turbulence. A decrease of the Taylor-microscale Reynolds number and leveling off of the average enstrophy and velocity-derivative skewness is observed in the numerical experiments,...

Simulation of spatially evolving turbulence and the applicability of Taylor’s hypothesis in compressible flow

For the numerical simulation of inhomogeneous turbulent flows, a method is developed for generating stochastic inflow boundary conditions with a prescribed power spectrum. Turbulence statistics from spatial simulations using this method with a low fluctuation Mach number are in excellent agreement with the experimental data, which validates the procedure. Turbulence statistics from spatial simulations are also compared to those from temporal simulations using Taylor’s hypothesis. Statistics such as turbulence intensity, vorticity, and velocity derivative skewness compare favorably with the temporal simulation. However, the statistics of dilatation show a significant departure from those obtained in the temporal simulation. To directly check the applicability of Taylor’s hypothesis, space-time correlations of fluctuations in velocity, vorticity, and dilatation are investigated. Convection velocities based on vorticity and velocity fluctuations are computed as functions of the spatial and temporal separations. The profile of the space-time correlation of dilatation fluctuations is explained via a wave propagation model.