Large-scale Reynolds Number Research Articles

Non-universality and dissipative anomaly in compressible magnetohydrodynamic turbulence

We systematically study the dissipative anomaly in compressible magnetohydrodynamic (MHD) turbulence using direct numerical simulations, and show that the total dissipation remains finite as viscosity diminishes. The dimensionless dissipation rate $\mathcal {C}_{\varepsilon }$ fits well with the model $\mathcal {C}_{\varepsilon } = \mathcal {C}_{\varepsilon,\infty } + \mathcal {D}/R_L^-$ for all levels of flow compressibility considered here, where $R_L^-$ is the generalized large-scale Reynolds number. The asymptotic value $\mathcal {C}_{\varepsilon,\infty }$ describes the total energy transfer flux, and decreases with increase of the flow compressibility, indicating non-universality of the dimensionless dissipation rate in compressible MHD turbulence. After introducing an empirically modified dissipation rate, the data from compressible cases collapse to a form similar to the incompressible MHD case depending only on the modified Reynolds number.

Forecasting small-scale dynamics of fluid turbulence using deep neural networks

Turbulence in fluid flows is characterized by a wide range of interacting scales. Since the scale range increases as some power of the flow Reynolds number, a faithful simulation of the entire scale range is prohibitively expensive at high Reynolds numbers. The most expensive aspect concerns the small-scale motions; thus, major emphasis is placed on understanding and modeling them, taking advantage of their putative universality. In this work, using physics-informed deep learning methods, we present a modeling framework to capture and predict the small-scale dynamics of turbulence, via the velocity gradient tensor. The model is based on obtaining functional closures for the pressure Hessian and viscous Laplacian contributions as functions of velocity gradient tensor. This task is accomplished using deep neural networks that are consistent with physical constraints and explicitly incorporate Reynolds number dependence to account for small-scale intermittency. We then utilize a massive direct numerical simulation database, spanning two orders of magnitude in the large-scale Reynolds number, for training and validation. The model learns from low to moderate Reynolds numbers and successfully predicts velocity gradient statistics at both seen and higher (unseen) Reynolds numbers. The success of our present approach demonstrates the viability of deep learning over traditional modeling approaches in capturing and predicting small-scale features of turbulence.

Universality of local dissipation scales in turbulent boundary layer flows with and without free-stream turbulence

Measurements of the small-scale dissipation statistics of turbulent boundary layer flows with and without free-stream turbulence are reported for Reτ ≈ 1000 (Reθ ≈ 2000). The scaling of the dissipation scale distribution is examined in these two boundary conditions. Results demonstrated that the local large-scale Reynolds number based on the measured longitudinal integral length scale fails to properly normalize the dissipation scale distribution near the wall in these two free-stream conditions due to the imperfect characterization of the upper bound of the inertial cascade by the integral length scale. A surrogate found from turbulent kinetic energy and mean dissipation rate only moderately improved the scaling of the dissipation scales, relative to the measured integral length scale. When a length scale based on the distance from the wall [as suggested by Bailey and Witte, “On the universality of local dissipation scales in turbulent channel flow,” J. Fluid Mech. 786, 234–252 (2015)] was utilized to scale the dissipation scale distribution, in the region near the wall, there was a noticeable improvement in the collapse of the normalized distribution of dissipation scales. In addition, unlike in channel flows, in the outer layer of the turbulent boundary layer, the normalized distributions of the local dissipation scales were observed to be dependent on the wall-normal position. This was found to be attributable to the presence of external intermittency in the outer layer as the presence of free-stream turbulence was found to restore the scaling behavior by replacing the intermittent laminar flow with turbulent flow.

Generating and controlling homogeneous air turbulence using random jet arrays

The use of random jet arrays, already employed in water tank facilities to generate zero-mean-flow homogeneous turbulence, is extended to air as a working fluid. A novel facility is introduced that uses two facing arrays of individually controlled jets (256 in total) to force steady homogeneous turbulence with negligible mean flow, shear, and strain. Quasi-synthetic jet pumps are created by expanding pressurized air through small straight nozzles and are actuated by fast-response low-voltage solenoid valves. Velocity fields, two-point correlations, energy spectra, and second-order structure functions are obtained from 2D PIV and are used to characterize the turbulence from the integral-to-the Kolmogorov scales. Several metrics are defined to quantify how well zero-mean-flow homogeneous turbulence is approximated for a wide range of forcing and geometric parameters. With increasing jet firing time duration, both the velocity fluctuations and the integral length scales are augmented and therefore the Reynolds number is increased. We reach a Taylor-microscale Reynolds number of 470, a large-scale Reynolds number of 74,000, and an integral-to-Kolmogorov length scale ratio of 680. The volume of the present homogeneous turbulence, the largest reported to date in a zero-mean-flow facility, is much larger than the integral length scale, allowing for the natural development of the energy cascade. The turbulence is found to be anisotropic irrespective of the distance between the jet arrays. Fine grids placed in front of the jets are effective at modulating the turbulence, reducing both velocity fluctuations and integral scales. Varying the jet-to-jet spacing within each array has no effect on the integral length scale, suggesting that this is dictated by the length scale of the jets.

On the universality of local dissipation scales in turbulent channel flow

Well-resolved measurements of the small-scale dissipation statistics within turbulent channel flow are reported for a range of Reynolds numbers from $Re_{{\it\tau}}\approx 500$ to 4000. In this flow, the local large-scale Reynolds number based on the longitudinal integral length scale is found to poorly describe the Reynolds number dependence of the small-scale statistics. When a length scale based on Townsend’s attached-eddy hypothesis is used to define the local large-scale Reynolds number, the Reynolds number scaling behaviour was found to be more consistent with that observed in homogeneous, isotropic turbulence. The Reynolds number scaling of the dissipation moments up to the sixth moment was examined and the results were found to be in good agreement with predicted scaling behaviour (Schumacher et al., Proc. Natl Acad. Sci. USA, vol. 111, 2014, pp. 10961–10965). The probability density functions of the local dissipation scales (Yakhot, Physica D, vol. 215 (2), 2006, pp. 166–174) were also determined and, when the revised local large-scale Reynolds number is used for normalization, provide support for the existence of a universal distribution which scales differently for inner and outer regions.

Effect of Periodic Bottom Plate Heating on Large Scale Flow in Turbulent Rayleigh-Bénard Convection

We report analysis of temperature fluctuations measured inside a Rayleigh-Benard convection cell of aspect ratio unity with steady and time-varying heating of the bottom boundary. The working fluid is cryogenic helium gas at Rayleigh number Ra = 10 13 , for which a large scale coherent flow (mean wind) exists. We observe the wind to occasionally vanish under both steady and time-varying heating conditions. However, with applied periodic modulation of the lower boundary temperature at the frequency of the wind, we can observe destruction of the periodic temperature variation at the cell mid-plane, due to relative changes in phase between the mean wind and the propagating temperature wave. The wind speed is not observed to change with modulation, demonstrating that it is set by mean properties of the system. Finally we propose that the frequency of the mean wind – and hence the large scale Reynolds number – could be better resolved by finding resonance with applied forcing.

Dissipation-scale fluctuations and mixing transition in turbulent flows

A small separation between reactants, not exceeding 10−8 − 10−7 cm, is the necessary condition for various chemical reactions. It is shown that random advection and stretching by turbulence leads to the formation of scalar-enriched sheets of strongly fluctuating thickness ηc. The molecular-level mixing is achieved by diffusion across these sheets (interfaces) separating the reactigants. Since the diffusion time scale is $\tau_{d}\,{\propto}\,\eta_{c}^{2}$, knowledge of the probability density Q(ηc, Re) is crucial for evaluation of mixing times and chemical reaction rates. According to Kolmogorov–Batchelor phenomenology, the stretching time τeddy ≈ L/urms = O(1) is independent of large-scale Reynolds number Re = urmsL/ν and the diffusion time $\tau_{d}\,{\approx}\,\tau_{\it eddy}/\sqrt{{\it Re}}\,{\ll}\, \tau_{\it eddy}$ is very small. Therefore, in previous studies, molecular diffusion was frequently neglected as being too fast to contribute substantially to the reaction rates. In this paper, taking into account strong intermittent fluctuations of the scalar dissipation scales, this conclusion is re-examined. We derive the probability density Q(ηc, Re, Sc), calculate the mean scalar dissipation scale and predict transition in the reaction rate behaviour from ${\cal R}\,{\propto}\,\sqrt{Re}$ ($Re\,{\leq}\, 10^{3}-10^{4})$ to the high-Re asymptotics ${\cal R}\,{\propto}\, {\it Re}^{0}$. These conclusions are compared with known experimental and numerical data.

Probability densities in strong turbulence

In this work we, using Mellin’s transform combined with the Gaussian large-scale boundary condition, calculate probability densities (PDFs) of velocity increments P ( δ r u , r ) , velocity derivatives P ( u ′ , r ) and the PDF of the fluctuating dissipation scales Q ( η , Re ) , where Re is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF P L ( δ r u , r ) often quoted in the literature. It is shown that the probability density of the small-scale velocity fluctuations includes information about the large (integral) scale dynamics which is responsible for the deviation of P ( δ r u , r ) from P L ( δ r u , r ) . An expression for the function D ( h ) of the multifractal theory, free from spurious logarithms recently discussed in [U. Frisch, M. Martins Afonso, A. Mazzino, V. Yakhot, J. Fluid Mech. 542 (2005) 97] is also obtained.

Anomalous Scaling of Structure Functions and Dynamic Constraints on Turbulence Simulations

The connection between anomalous scaling of structure functions (intermittency) and numerical methods for turbulence simulations is discussed. It is argued that the computational work for direct numerical simulations (DNS) of fully developed turbulence increases as $Re^{4}$, and not as $Re^{3}$ expected from Kolmogorov's theory, where $Re$ is a large-scale Reynolds number. Various relations for the moments of acceleration and velocity derivatives are derived. An infinite set of exact constraints on dynamically consistent subgrid models for Large Eddy Simulations (LES) is derived from the Navier-Stokes equations, and some problems of principle associated with existing LES models are highlighted.

The Regularized Dia Closure for Two-Dimensional Turbulence

A regularized version of the direct interaction approximation closure (RDIA) is compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large-scale Reynolds numbers ranging between low (≈ 50) and high (≈ 4000). The regularization localizes transfer by removing the interaction between large-scale and small-scale eddies depending on a specified cut-off ratio α. It thus eliminates spurious convection effects of small-scale eddies by large-scale eddies in the Eulerian direct interaction approximation (DIA) that causes the underestimation of small-scale kinetic energy by the DIA. Cumulant update versions of the RDIA closure that have comparable performance but are much more efficient computationally have also been analyzed. Both the closures and DNS use discrete wavenumber representations relevant to flows on a doubly periodic domain. This means that any differences between them are intrinsic and not partly due to using continuous wavenumber formulation for the closures. Comparisons between the regularized closures and DNS have focused on evolved kinetic energy and palinstrophy spectra and as well on enstrophy flux spectra and on the evolution of skewness which depends sensitively on small-scale differences. All of these diagnostics compare quite well when α = 6. And this is the case for runs started from each of three initial spectra, for the range of evolved large-scale Reynolds numbers ranging from ≈ 50 to ≈ 4000 and for regularized DIA closures with, and particularly without, cumulant update restarts. The performance of the RDIA compared with quasi-Lagrangian closure models is discussed.

Pressure–velocity correlations and scaling exponents in turbulence

It is shown that each structure function being the large-scale Reynolds number. This result has implications for the resolution requirements of direct simulations of turbulence. A new rigorous dynamic constraint relating scaling exponents of the structure functions to the codimension of the most singular features of turbulence is derived. A modification of the model by Gotoh & Nakano (2003) for the pressure–velocity correlations, based on the Bernoulli equation, is proposed. This proposal leads to an analytic expression for the scaling exponents of velocity structure functions.

Dynamics and spectra of cumulant update closures for two-dimensional turbulence

Non-Markovian closure theories are compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large scale Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures, as well as DNS, are formulated for discrete wave numbers relevant to flows on the doubly periodic domain and are compared with the results of continuous wave number closures. The direct interaction approximation (DIA), self-consistent field theory (SCFT) and local energy-transfer theory (LET) closures are also compared with cumulant update versions of these closures (CUDIA, CUSCFT, CULET). The cumulant update closures are shown to have comparable performance to the standard closures but are much more efficient allowing long time integrations. The discrete wave number closures perform considerably better than continuous wave number closures as far as evolved energy and transfer spectra and skewness are concerned. The discrete wave number closures are in reasonable agreement with DNS in the energy containing range of the large scales for Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures tend to underestimate the enstrophy flux to high wave numbers, increasingly so with increasing Reynolds number, resulting in underestimation of small-scale kinetic energy.

Two-Dimensional Turbulence with Rigid Circular Walls

We report numerical computations of decaying two-dimensional Navier--Stokes turbulence inside a circular rigid boundary. We summarize previously reported calculations involving no-slip boundary conditions and present results with higher spatial resolution than achieved before (with, however, no qualitative changes in the observed behavior). We then report new results with stress-free boundary conditions (for a viscous fluid, but bounded by a perfectly slippery wall). The method used is spectral, involving expansions of the fields in orthonormal sets of functions which obey two boundary conditions (circular analogues of the Chandrasekhar–Reid functions). The computation takes place entirely in the spectral space. Large-scale Reynolds numbers are typically less than a thousand. Interest focuses on the role played by angular momentum, in determining the decay of the turbulence with no-slip boundary conditions, and the role of possible other ideal invariants in the stress-free case.

Relaxation in two dimensions and the ‘‘sinh-Poisson’’ equation

Long-time states of a turbulent, decaying, two-dimensional, Navier–Stokes flow are shown numerically to relax toward maximum-entropy configurations, as defined by the ‘‘sinh-Poisson’’ equation. The large-scale Reynolds number is about 14 000, the spatial resolution is (512)2, the boundary conditions are spatially periodic, and the evolution takes place over nearly 400 large-scale eddy-turnover times.

On the motion of small particles in a homogeneous isotropic turbulent flow

The motion of small spherical solid particles are simulated numerically in a decaying homogeneous isotropic turbulent gas flow field generated by the large eddy simulation. By comparing with the previous experimental and theoretical studies, the present method is found to be a successful tool to generate the properties of the particle motion involving the second-order statistics, such as the mean-square displacement, the dispersion coefficient, and the root-mean-square velocity fluctuation. The present results are complementary to the experimental data and include a detailed study of the effects of the flow turbulence, the particle’s inertia, and the particle’s free-fall velocity in a still fluid on the particle dispersion and turbulence intensity. By performing particle simulation in the flow fields generated with different values of the coefficient in the subgrid model and with different sizes of the calculation domain, it is found that the particle motion is indeed controlled mainly by the large eddies, and there are only minor contributions of the high wave number components of the flow field to the particle motion. Also included in the results is the effect of the turbulence decay on the long time particle dispersion. It is found that the peak value of the dispersion coefficient during the turbulence decay can be correlated with the large-scale Reynolds number and the velocity ratio between the particle’s free-fall velocity and the root-mean-square velocity fluctuation of the fluid. Simulation also shows that the particle’s settling velocity in turbulence is greater than that in a still fluid.