Multifractals, free turbulence decay laws and the Scaling Gyroscopes Cascade model

<p>Turbulence being a dissipative system decays when being "free", i.e. without any force. The law of this decay has been intriguing for quite a while. Assuming that for vanishing viscosity, the whole spectrum is self-similar, as well as stationary for low wave numbers/large eddies  (E(k,t) ≈C<sub>S</sub> k<sup>S</sup>, k → 0) , it was shown [1] that the total energy of turbulence has a power-law decay: E(t) = ∫ E(k,t) dk ≈ t<sup>-a(s)</sup>: a(s) =2(s+1)/(s+3) . This was particularly thought to be relevant for s=4, C<sub>4</sub> being proportional to the Loitsianski integral, assumed to be time-invariant [2]. However, it was shown with the help of the eddy-damped quasi-normal Markovian (EDQNM) [3] that there is an energy backscatter term transferring energy from energy-containing eddies by nonlocal triads interactions to large eddies, which behaves like T<sub>NL</sub>≈ k<sup>4</sup> and therefore prevents the invariance of the Loitsianski integral. This implies that the theoretical exponent a(s) = 2(s+1)/(s+3)  is only valid for s<4 and that a(s) =a(4)=-(10-2γ)/7 for s≥ 4 with C<sub>4</sub>(t) ≈ t <sup>γ</sup>, γ>0. The turbulence decay is therefore slower than previously expected for s ≥ 4 due to the backscatter term that progressively stores energy in large eddies. <br>EDQNM provides the estimate γ ≈ 0.16. However, a strong limitation of EDQNM and similar models (e.g. Direct Interaction Approximation, Test Field Model) is that these models are not able to represent intermittency, which is a fundamental phenomenon of turbulence [4] and this could bring into questions the previous results. We, therefore, investigate this question with the Scaling Gyroscopes Cascade (SGC) model [5], which is based on nonlocal interactions and display multifractal intermittency [6]. We first theoretically argue that SGC confirms the existence of the backscatter term, but the turbulence decay is no longer smooth but occurs by puffs and we provide numerical evidence of this.</p><p>Keywords: Loitsianski integral; intermittency; infrared spectrum; SGC model; energy decay</p><p>[1]M. Lesieur and D. Schertzer, ‘‘Amortissement auto-similaire d’une turbulence a‘ grand nombre de Reynolds,’’ J. Mec. 17, 609 1978 .</p><p>[2]Davidson, P. A. (2000). Was Loitsyansky correct? A review of the arguments. <em>Journal of Turbulence</em>, <em>1</em>(1), 006-006.</p><p>[3]Frisch, U., Lesieur, M.,Schertzer, D. (1980). Comments on the quasi- normal Markovian approximation for fully-developed turbulence. Jour- nal of Fluid Mechanics, 97(1), 181-192.</p><p>[4]Morf, R. H., Orszag, S. A., Frisch, U. (1980). Spontaneous singularity in three-dimensional inviscid, incompressible ow. Physical Review Letters, 44(9), 572.</p><p>[5]Chigirinskaya, Y., Schertzer, D.,  Lovejoy, S. (1997). Scaling gyroscopes cascade: universal multifractal features of 2-D and 3-D turbulence. <em>Fractals and Chaos in Chemical Engineering. World Scientific, Singapore</em>, 371-384.</p><p>[6]Chigirinskaya, Y.,  Schertzer, D. (1997). Cascade of scaling gyroscopes: Lie structure, universal multifractals and self-organized criticality in turbulence. In <em>Stochastic Models in Geosystems</em> (pp. 57-81). Springer, New York, NY.</p>

Numerical Modeling of Freestream Turbulence Decay Using Different Commercial Computational Fluid Dynamics Codes

This work models the spatial decay of freestream turbulence using three different commercial computational fluid dynamics (CFD) codes: Fluent, star-ccm+, and cfx. The two-equation shear stress transport k–ω (SST-k–ω) steady Reynolds-averaged-Navier–Stokes (RANS) model was used, within each of these three different commercial codes, and the modeling variations were analyzed. Comparison of the results from the SST-k–ω model with experiments and large eddy simulation (LES) (carried out using star-ccm+) were also made, which reveal that all the commercial CFD codes demonstrate either a higher or slower rate of spatial turbulent kinetic energy (TKE) decay. Attempts were then made to unify the resultant modeling approach between these three CFD tools, by careful manipulation of the inlet boundary conditions and subsequent fine-tuning of the SST-k–ω model constant (β∞∗). The results obtained not only displayed uniformity among the three CFD codes but also demonstrated a much better agreement to the experiments and the LES results. Thereafter, the optimized model coefficient (β∞∗) was integrated with the three-equation k–kl–ω transition model to examine its applicability in modeling a turbulent boundary layer flow over a flat plate with low incoming turbulence. The results showed good agreement with the theoretical boundary layer correlations, with correct prediction of the transition location. The findings from this study can be used as a suitable modeling method to accurately model the effects of freestream turbulence on bluff-body and boundary layer flows.

Theory and verification of a new 3D RANS wake model

This paper details the background to the WakeBlaster model: a purpose-built, parabolic three-dimensional RANS solver, developed by ProPlanEn. WakeBlaster is a field model, rather than a single turbine model; it therefore eliminates the need for an empirical wake superposition model. It belongs to a class of very fast (a few core seconds, per flow case) mid-fidelity models, which are designed for industrial application in wind farm design, operation, and control. The domain is a three-dimensional structured grid, a node spacing of a tenth of a rotor diameter, by default. WakeBlaster uses eddy viscosity turbulence closure, which is parameterized by the local shear, time-lagged turbulence development, and stability corrections for ambient shear and turbulence decay. The model prescribes a profile at the end of the near wake, and the spatial variation of ambient flow, by using output from an external flow model.

An Experimental Study of Turbulent Mixing in Channel Flow Past a Grid

Grid turbulence is considered to be a canonical case of turbulent flow. In the presented paper, the flow structure is analyzed from the point of view of mixing properties, where vortical structures and their properties play a significant role. That is why the effect of various length-scales in turbulence is studied separately. The experimental study uses the Particle Image Velocimetry (PIV) method. The original method for spatial spectrum evaluation is applied. Results on vortex spatial spectrum and isotropy are presented. The scaling of turbulent kinetic energy (TKE) is measured; furthermore, the TKE is decomposed according to the length-scales of the fluctuations. By this method, we found that the decay of TKE associated with the smallest length-scales is more sensitive to the Reynolds number than that at larger length-scales. The TKE at the largest investigated length-scales decays more slowly. The turbulence decay-law is studied for various Reynolds numbers. The second and fourth statistical moments of vorticity are evaluated at various Reynolds numbers and distances from the grid. The isotropy is investigated in the sense of ratio of fluctuations in stream-wise to span-wise directions as the used data are captured using the planar PIV method. The full 3D fluctuation invariants were investigated in a representative position by means of the Stereo-PIV method.

Large Eddy Simulation of a Transonic Linear Cascade With Synthetic Inlet Turbulence

The aim of this work is to predict the boundary layer transition and the heat transfer on a highly loaded transonic turbine cascade using Large Eddy Simulations (LESs) with prescribed inlet synthetic turbulence. The numerical simulations were performed for the flow in a linear turbine cascade tested at the von Karman Institute for Fluid Dynamic (MUR test case). For the numerical case, two operating conditions with two different levels of free-stream turbulence intensity are evaluated. For the lower turbulence level case (Tu = 0.8%, MUR132) a laminar inflow is used for the LES simulations whereas for the higher one (Tu = 6%, MUR237) the inlet turbulence is prescribed by using the Synthetic Eddy Method (SEM) of Jarrin. The first part of this work deals with the LES setup. The standard Smagorinsky model was used as closure model. A value of the Smagorinsky constant CS = 0.05 was chosen whereas the turbulent viscosity was reduced in the region closest to the wall by changing the definition of the Smagorinsky length scale. To handle the strong fluctuations in the flow field the cell fluxes are computed using the WENO-P scheme. In the second part, precursor RANS and LES simulations are used to set the optimal values of the SEM parameters and to guarantee the correct level of turbulence at the blade leading edge. The turbulence decay of the synthetic turbulence is compared with the one of the RANS κ–ωSST model. Finally, a comparison between experimental and numerical results is done and the ability of LES to predict the boundary layer transition and the heat transfer on the blade surface is evaluated for the two different inflow conditions.

First characterization of a new perturbation system for gust generation: the chopper

We present a new system for the generation of rapid, strong flow perturbations in the aerodynamic wind tunnel at École Centrale de Nantes. The system is called the chopper, and it consists of a rotating bar cutting through the inlet of a wind tunnel test section, thus generating an inverse gust that travels downstream. The flow generated by the chopper is investigated with respect to the rotational frequency using an array equipped with hot-wires that is traversed downstream in the flow field. It is found that the gust can be described as a superposition of the mean gust velocity, an underlying gust shape, and additional turbulence. Following this approach, the evolution of the mean gust velocity and turbulence intensity are presented, and the evolution of the underlying inverse gust shape is explained. The turbulence is shown to be characterized by an integral length scale of approximately half the chopper blade width and a turbulence decay according to E(f)∝f-5/3.

Theory and Verification of a new 3D RANS Wake Model

This paper details the background to the WakeBlaster model: a purpose built, parabolic three-dimensional RANS solver, developed by ProPlanEn. WakeBlaster is a field model, rather than a single turbine model; it therefore eliminates the need for an empirical wake superposition model. It belongs to a class of very fast (a few core seconds, per flow case) mid-fidelity models, which are designed for industrial application in wind farm design, operation and control. The domain is a three-dimensional structured grid, with approximately 80 nodes covering the rotor disk, by default. WakeBlaster uses eddy viscosity turbulence closure, which is parameterized by the local shear, time-lagged turbulence development, and stability corrections for ambient shear and turbulence decay. The model prescribes a profile at the end of the near-wake, and the spatial variation of ambient flow, by using output from an external flow model. The WakeBlaster model is verified, calibrated and validated using a large volume of data from multiple onshore and offshore wind farms. This paper presents example simulations for one offshore wind farm.