Experimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes
Margherita Dotti, Rasmus Korslund Schlander, Preben Buchhave, Clara Marika Velte
mmyjournal manuscript No. (will be inserted by the editor)
Experimental investigation of the turbulent cascadedevelopment by injection of single large-scale Fourier modes
Margherita Dotti · Rasmus K. Schlander · Preben Buchhave · Clara M.Velte
Received: date / Accepted: date
Abstract
The current work presents an experimen-tal investigation of the dynamic interactions betweenflow scales caused by repeated actions of the nonlinearterm of the Navier-Stokes equation. Injecting a narrowband oscillation, representing a single Fourier mode,into a round jet flow allows the measurement of thedownstream generation and development of higher har-monic spectral components and to measure when thesecomponents are eventually absorbed into fully devel-oped turbulence. Furthermore, the dynamic evolutionof the measured power spectra observed corresponds
Financial support from the Poul Due Jensen Foundation(Grundfos Foundation) for this research is gratefully acknowl-edged. Grant number 2018-039.M. DottiDepartment of Mechanical Engineering, Technical Universityof Denmark, Nils Koppels All´e, Bldg. 403, 2800 Kgs. Lyngby,Denmark.E-mail: [email protected]
Present address:
Department ofChemical and Biochemical Engineering, Technical Universityof Denmark, Søltofts Plads 228A, 2800 Kgs. Lyngby, Den-markR.K. SchlanderDepartment of Mechanical Engineering, Technical Universityof Denmark, Nils Koppels All´e, Bldg. 403, 2800 Kgs. Lyngby,Denmark.E-mail: [email protected]
Present address:
Depart-ment of Aeronautics, Imperial College, London SW7 2AZ,United KingdomP. BuchhaveIntarsia Optics, Sønderskovvej 3, 3460 Birkerød, Denmark.E-mail: [email protected]. VelteDepartment of Mechanical Engineering, Technical Universityof Denmark, Nils Koppels All´e, Bldg. 403, 2800 Kgs. Lyngby,Denmark.ORCID: 0000-0002-8657-0383E-mail: [email protected] well to the measured cascaded delays reported by oth-ers. Closely matching spectral development and cas-cade delays have also been derived directly from a one-dimensional solution of the Navier-Stokes equation de-scribed in a companion paper. The results in the cur-rent work provide vital information about how initialconditions influence development of the shape of thespectrum and about the extent of the time scales inthe triad interaction process, which should be of signif-icance to turbulence modelers.
Keywords
Turbulence cascade · Triad interactions · Laser Doppler Anemometry · Hot-wire Anemometry · Velocity Power Spectrum · Turbulence modelingparameters
Most of the 20th century knowledge of turbulence isfounded on the equilibrium gas dynamics analogy tosmall and intermediate scales of turbulence, as put forthby Kolmogorov [1,2]. Indeed, almost all turbulence the-ories and models are in one way or another based on theKolmogorov theory of turbulence (which the authorswill hereafter refer to as the K41 theory). In particular,many approaches rely on the existence of a continu-ous exchange of turbulent energy from small to largewave numbers, the Richardson cascade, where predom-inantly local interactions between scales are assumed tooccur [2,3,4].By taking the Fourier transform of the non-linearadvection term in the Navier-Stokes equation, one canimmediately observe the possibilities of energy trans-fer between different wavenumbers of the velocity field.However, being able to infer the restriction of localityof the triad interactions, as postulated by Richardson, a r X i v : . [ phy s i c s . f l u - dyn ] A p r Margherita Dotti et al.
Fig. 1
Generation and interaction of non-local scales [5]. directly from the governing equations appears to haveso far eluded the turbulence community. On the otherhand, accumulating evidence witnesses that the actualunderlying processes of energy exchange between scalesmay in fact be quite different from the ideas of K41.An example of non-local energy exchange, in direct vi-olation of the Richardson cascade concept, is shownin Figure 1 [5]. This figure zooms into a shear layer,from a Direct Numerical Simulation (DNS) computa-tion, where structures of small size appear and directlyexchange energy with significantly larger structures.Historically, much (but not all) of the published ex-perimental evidence supported the K41 theory. How-ever, George [6], inspired by recent developments of hisand others, argued that the K41 only applies to equilib-rium flows, while failing to reproduce results from flowsout of equilibrium.Recently, interesting dynamic effects in the trans-fer of turbulent energy by the triad interaction pro-cess have been reported. For instance, the data of thehot-wire anemometer analysis conducted by Josserand et al. [7] reveals both a lack of time reversal symme-try and a delay in the triad interactions. This delayseems to depend on the separation of the k-vectors en-tering the process. Other relevant studies concern thecomparison between the strength of non-local interac-tions against the local ones [8,9] and the persistenceof the initial large-scale structures during the furtherturbulence development [10]. As Zhou [8] pointed out,and several investigations concluded, the local energytransfer resulted from non-local interactions [11,12,13],which contrasts with the classical Kolmogorov picture.Indeed, direct coupling between scales of quite dif-ferent size seems to be possible [14], as is also shownin Figure 1. And even though the net effect of the en-ergy transport is towards higher wavenumbers, the en-ergy may move in the direction of both smaller andlarger scales. Other outstanding questions concern thepersistence [15] of the initial conditions into the de- veloped turbulence and the precise form of the finaldecay of the turbulent fluctuations [16]. Moreover, thefact that many papers report deviations from the − / The Navier-Stokes equation is a three-dimensional de-scription of the time evolution of fluid flow. The equa-tion is essentially the application of Newton’s secondlaw to the motion of fluid passing through an infinites-imal fluid control volume, as shown in Figure 2.The development of the flow structures and the in-terchange of energy between them is primarily causedby the non-linear term in the Navier-Stokes equation.This nonlinear term causes a variation in velocity, dueto the change in convection of momentum through acontrol volume, CV = dA · ds , where dA is the cross-sectional area and ds is the thickness of the CV . Theviscous diffusion term causes a loss of velocity by diffu-sion of momentum to the fluid surrounding the controlvolume.The pressure gradient influences the velocity by fluc-tuating (dynamic) pressure caused by velocity fluctu-ations elsewhere in the fluid and propagating instan- xperimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes 3 Fig. 2
Fluid control volume and instantaneous velocity u and a component u i at a point in space-time. The red lineindicates a streak line. taneously to the control volume in an incompressiblefluid. This term must be computed by a full solution tothe Laplace equation including pressure terms at theboundaries or inferred from theoretical models. Thus,the equation is local, describing the conditions at a‘point’, non-local effects only entering though pressureeffects created by velocity fluctuations or through bound-ary conditions elsewhere in the fluid, transmitted to thepoint of interest.The velocity field may be decomposed in severalways, for example by analyzing the triad interactionsbetween the Fourier coefficients or wave vector compo-nents of the flow. If one includes time in the decom-position [6,20], it can be seen that the phase matchcondition is extended beyond the classical one betweenthree wavenumber vectors, k = k + k , to one thatincludes also frequency and time, [ k − ( k + k )] · r − [ ω − ( ω + ω )] t = 0. A mismatch in the wavenumbers, ∆ k = k − ( k + k ), can therefore be compensated bya mismatch in the frequencies, ∆ω = ω − ( ω + ω ).This can give rise to time delays in the interactions in-troducing dynamics into the classical triad interactionsanalysis [9].Several experiments have been designed to explorethe underlying wave interaction dynamics: In the firstexperiment, an actual narrow-band oscillating veloc-ity, that resembles a single Fourier mode, is injectedinto a fully developed turbulent jet flow. By conduct-ing measurements downstream of the injection position,the development in time of the power spectrum is tracedthrough laser Doppler anemometry (LDA) and hot-wireanemometry (HWA) measurements. The downstreamposition is interpreted in relation to the Navier-Stokesequation as a time interval in which the equations op-erate on the flow passing a series of control volumes,as shown in Figure 2. Tests of zither-like grid gener-ated modes were performed as early as 1980 [23], anddetailed studies of full and narrow-band velocity corre-lations in wind tunnel flow was made as early as 1971. However, the clear evolution of the spectrum that wereport here has not been reported before.A ‘full’ solution to the Navier-Stokes equation re-quires a four-dimensional numerical solution with a highspatial and temporal resolution (DNS). However thesenumerical solutions do not necessarily provide a phys-ical understanding, i.e., the comprehension of the flowbased on the governing equations, which is the primaryinterest in the current work. Therefore, to understandthe inner workings of the interactions between Fouriercomponents (the triad interactions), a one-dimensionalmodel has been developed and implemented [21]. Themethod involves projection of the forces acting on thefluid in the control volume onto the instantaneous ve-locity direction. The time sampling interval ∆t is thenconverted to the convection sampling interval ∆s = u∆t [24]. This method, obviously, does not allow to seethe full 4-dimensional motion of the fluid. However, theforces acting in the direction of the velocity will changethe momentum and allow computation of kinetic energyand spatial velocity structures. The main problem is theinability of computing the pressure gradient, since thisrequires the solution of Laplace’s equation for the wholefluid volume at the present instant in time. To includepressure, it will be necessary to invoke a model or useseparate information about the fluctuating pressure. The experiments described in the following are designedto inject a single Fourier mode into a well-defined flowand by measurement of the time trace along the down-stream evolution follow the time development of thevelocity field. As Fourier modes are essentially planewaves [26], the attempt was to inject a two-dimensionaloscillating wave front developing along the mean flowdirection.The axisymmetric turbulent round jet can be con-sidered ideal to investigate the turbulent cascade. In-deed, the round turbulent jet was the first flow for whichthe theory of Kolmogorov was supported by showingthe − / Margherita Dotti et al.
Fig. 3
Jet generator and oscillating airfoil. intensity turbulent round jet flow. Secondly, measure-ments in jet flows of different Reynolds numbers fromthe same jet facility were carried out behind a largerairfoil with less vigorous oscillations using a hot-wireanemometer. Finally, the same hot-wire anemometersystem was used to measure vortex shedding in a lowintensity turbulent jet of significantly larger jet exit di-ameter, employed as an open wind tunnel. This thirdexperiment was performed in the laminar jet core usinga sharp-edged rod to create a distinct vortex sheddingfrequency.3.1 Oscillations in turbulent jet – small airfoil (LDA)A jet generator, which is the one used in several projectsof the Turbulence Research Laboratory of DTU [31,32,33], produces an axisymmetric fully developed turbu-lent flow 30 jet exit diameters downstream of the nozzleexit. The jet nozzle, shown in Figure 3, has an exit di-ameter of D = 10 mm , a contraction ratio of 3 . Fig. 4
Flow visualization airfoil experiments. flow perturbed by the airfoil. Initially, the airfoil spanexperiment was conducted by flapping in an oscillatingmanner a 50 × mm airfoil with a thick leading edgeand a sharp trailing edge.The airfoil was actuated by a small motor, creatingan oscillation frequency of 10 Hz . The streamwise ve-locity component was measured using an in-house state-of-the-art side scatter LDA system [32]. Since the sidescattering optics of the LDA system was highly sensi-tive to misalignments, it was decided to traverse the jetinstead of the LDA system. The measuring distance was300 mm and the size of the measurement volume was,due to the side scattering configuration, nearly spheri-cal with a diameter of 200 µm . When performing LDAmeasurements, the ambient air was seeded with glycerinparticles, injected by means of pressurized air, so thata nearly uniform spatial seeding density is achieved.The size of the scattering particles ( ∼ − µm ) hasbeen shown to be small enough to faithfully track theflow and observed to be large enough to scatter enoughlight for a to produce a satisfactory signal quality [35].The Doppler signal at each downstream position wasmeasured in 400 records, each of 2 s , with a samplingrate of 25 M Hz . The Reynolds number at the jet exitwas Re D = U D/ν = 2 . · . The LDA measurementvolume was then placed at 1D, 2D, 4D, 6D, 8D, 10Djet diameters downstream from the trailing edge of theairfoil, as depicted in Figure 5.The corresponding measured temporal streamwisevelocity power spectra are displayed in Figure 6. Thesequence of plots shows interesting dynamics in the en-ergy exchange process between scales. Energy tends tobe exchanged between harmonics of the base frequencyof 10 Hz , which is well in line with the fact that the non-linear term in the Navier-Stokes equation introducesfrequency doubling, sum- and difference-frequencies as xperimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes 5 Fig. 5
Schematic (not to scale) of the setup of the oscillatingairfoil in a turbulent jet. The airfoil is depicted in blue andthe measurement points in red. shown analytically in [21] . From the figure it can alsobe inferred that the center of gravity of the inducedspectral peaks moves towards higher frequencies as theflow is traced downstream. The alternation of the domi-nance of the first (10 Hz ) and second (20 Hz ) harmonic,the ‘wave behavior’, is in agreement with Josserand etal. [7], who observed that energy transfers primarilyamong wavenumbers which are strongly correlated tothe fundamental frequency. Moreover, it shows that theenergy can flow in both directions, but the global en-ergy transfer remains towards higher frequencies.3.2 Oscillations in turbulent jet – large airfoil (HWA)The LDA measurements and flow visualization fromthe previous section showed that the 50 × mm air-foil required a significant amplitude to create the de-sired modulation to ensure injection and isolation ofthe development of a single mode into the jet. Thus,new measurements were conducted with a larger (spanand chord) airfoil, which was oscillating with a smalleramplitude to introduce less energy. The dimension ofboth the chord and the span dimension of the airfoilwas increased to 50 and 210 mm , respectively. A hot-wire anemometer could, to sufficient accuracy, be usedin this less turbulent flow because the power spectrumcould be processed online, and the system could bemore easily traversed to cover more downstream mea-surement points.A Mini CTA 54 T
30 system for measurements in airfrom Dantec Dynamics was used for acquisition. The55 P
11 straight single wire probe (Tungsten, d = 5 µm and length 1 mm ) was connected through the straight55 H
20 support through a 4 . m cable. Calibration was This is easily observed e.g. by substituting a velocity wavetraveling in the x-direction u ( x ) = U cos( k x x ) into ( u · ∇ ) u performed across 10 velocity points, from 1 ms − to30 ms − and measuring the pressure difference with an F CO
560 Furness Anemometer. The record length ofeach measurement was set to 120 s and each signal wassampled with a rate of 30 kHz . The data was trans-ferred to a computer passing through an anti-aliasingfilter and processed with the Dantec MiniCTA v . Re D = 4 . · , keeping a distance of 10 D betweenthe jet exit and the leading edge of the airfoil. The spec-tra were measured at increasing axial distances between1 D and 50 D from the airfoil trailing edge. These spectraare displayed in one graph with an offset of 10 − dB between each for the sake of comparison of the peaks.Several features of this plot are noteworthy. It canbe observed that the low order harmonics are createdat an early stage, and several frequencies are observedalready close to the trailing edge of the flap; note thatthe closest practical position of measurement was 1 D from the airfoil trailing edge. Note that the third har-monic remains weaker than higher harmonics acrossthe full downstream development. This phenomenonis explained in a companion paper [20] as a result ofthe finite region for the interaction of the participat-ing Fourier components. The spectral window, causedby the finite interaction region, modifies the shape ofthe spectrum in comparison to a spectrum created inan infinite region. Furthermore, the generation of thethird harmonic in particular requires the prior creationof the 2 nd harmonic in order to take place [20].Figure 7 shows that the energy associated with theinjected frequency of 10 Hz depletes downstream, whilethe 20 Hz second harmonic peak initially increases andsubsequently decreases after reaching its maximum aroundapproximately 8 D . A similar behavior can be seen forthe higher harmonics. The energy of the generated peakseventually becomes absorbed in the developed turbu-lence further downstream. This behavior is analogousthat depicted in Figure 6: the transfer of energy can beboth direct and indirect, but the net effect shifts theconcentration of energy towards higher frequencies [20,21].The striking stability and sharpness of the higherharmonics in Figure 7 is noteworthy. Even at the latestages just before being absorbed, the positions of the Margherita Dotti et al.
Fig. 6
Dynamic evolution of a ‘wave-like behavior’ of the spectral harmonic peaks for distances of 1 D , 2 D , 4 D , 6 D , 8 D and10 D behind the airfoil trailing edge, respectively.xperimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes 7 Fig. 7
Downstream spectrum development for Re D = 4 . · and 10 D between the jet exit and the airfoil leading edge. (a)Full view and (b) zoom-in on the peak development. Table 1
Reynolds number dependence of the time for ab-sorption of the spectral peaks. Re D Absorption time [s]2 . · . . · . . · . . · . . · . peaks remain at their precise integer values of the airfoilexcitation frequency and are apparently not smearedout by the surrounding, presumed Stochastic, turbulentvelocity fluctuations. This behavior may partly arisedue to the fact that the injected mode is characterizedby a much larger energy than the ones relative to thegeneral turbulence. Consequently, the interaction be-tween harmonics wavenumbers remains more efficient,leading to the persistence of the initial structure fardownstream in the jet (historically referred to as ‘per-manence of large eddies’). This result lends strong sup-port to the ideas of turbulence dependency upon initialconditions (in contrast to assuming universality), in linewith our recent results [19,20,21].The time for absorption of the peaks in Figure 7and corresponding parametric experimental investiga-tions can be found from the integration of the down-stream decaying velocity over the downstream distance.The results are summarized in Table 1. The absorptiontime of the peaks, which should be of significance toturbulence modelers, depends in the current case onthe Reynolds number. In this regard, it has been no-ticed that the peaks are, independently of Reynoldsnumber, consistently completely absorbed around ap-proximately the same downstream spatial position, i.e. ∼ D after the trailing edge of the airfoil. This appearsto be independent of the fact that the peaks from thenon-linear interactions are more pronounced for higherReynolds numbers. 3.3 Vortex shedding in laminar jet core (HWA)Vortex shedding measurements were also carried out inthe laminar core of a round turbulent jet with exit di-ameter D = 100 mm and contraction ratio 2 . × mm . As Figure 8 shows, the rod spanextends across the entire nozzle diameter.The objective was to isolate the workings of the non-linear term by isolating the downstream development ofa sharply defined Fourier mode injection using a shed-ding generator. This allowed for measurement of thevery initial generation as well as the downstream devel-opment of the triadic interactions. This was not possi-ble with the oscillating airfoil generated shedding, sincemeasurements were only practical just behind the flaptrailing edge where the energy distribution had alreadyhad ample time to develop across the airfoil chord. Thissetup, on the other hand, allows measurements fromeven the initial generation of the base frequency.In Figure 9 a flow visualization of the flow behindthe vertical rod is shown. The laminar jet core aroundthe sharp-edged rod is clearly visible in Figure 9 alongwith the Kelvin-Helmholtz vortices developing fartherdownstream. Downstream of the rod, the flow vortexshedding formed by the sharp edge imposes a singlefundamental frequency.The same hot-wire anemometer system from Dan-tec Dynamics, as described previously, was employed.For the current experiment, the signal was captured us-ing a PicoScope 5444 B by Pico Technology. This scopeallowed the change in both the waveform signal andthe flow spectrum to be visualized in nearly real time, Margherita Dotti et al.
Fig. 8
Jet with an extended laminar core, flat rod and hot-wire anemometer probe. allowing therefore to quickly find identify the desiredmeasurement points. The measurements were acquiredwith 0 . mm intervals from 0 . mm to 10 mm down-stream from the vertical rod at Re D = 1 . · . Fig-ure 10 depicts a schematic of the linear region of the ac-quisition positions (in blue) from a top and a side view,respectively. The spanwise position was kept fixed to adistance of 1 mm from the rod edge.A total of 100 time-series of 0 . s each were ac-quired in each measurement point with a sampling rateof 20 kHz , a hardware resolution of 12 bits and sam-pling interval 50 µs . A spectrum was produced for eachspatial point, cutting out the DC part of the signaland employing a Hamming window function, with arange of 50 kHz . A computer simulation applying a one-dimensional projection of Navier-Stokes equation ontothe instantaneous flow direction is reported in [20]. Theprogram assumes a time record of the velocity as in-put and computes the time development of the velocitytrace employing multiple incremental passes through afluid control volume exposed to the effect of the termsin the Navier-Stokes equation.One result from this calculation is quoted and com-pared to one of the measurements. The measured time trace from the spatial point closest to the rod has beenused as input to the computer program, so that theinitial condition for the development of the velocity inthe flow is identical to the initial velocity trace usedin the program. The program uses a rectangular low-pass filter, which cuts out the DC part of the signaland some high frequency peaks due to external noiseon the time traces. The spectra are then computed em-ploying a Hamming window function, just like the Pi-coScope does. Note, the program uses only the termsin the Navier-Stokes equation without further approxi-mations.Figure 11 reports the downstream evolution of theexperimental (blue) and the computational (red) down-stream dynamic power spectra. The links in the figurecaptions provide moving images displaying the down-stream evolution of the repeated workings of the non-linear term on the initial modal injection. Note that theacquisition instrument introduces a filter, manifestingitself as a slow variation across frequency, which shouldbe disregarded when interpreting the results.The plots in Figure 11(a) and 11(b) display respec-tively the first spectrum measured close to the vortexshedding rod and the spectrum measured at a distanceof 10 mm downstream. Figures 11(c) and 11(d) displaythe corresponding results from the simulation. Fromthese results, the delays in the cascade wavenumberinteractions are particularly evident. Indeed, in con-trast to Figure 7, where the low order harmonics arealready present even in the most upstream measure-ment, in Figure 11 only the initial peak is present atthe initial measurement point.The time scale for vortex shedding to develop andreach an ‘asymptotic’ configuration (like the one shownin Figure 11b) has been computed for the consideredexperiment and was found to be 6 ms . The time wasestimated from the integration of the downstream ve-locity over the downstream distance, as for the time forabsorption of the peaks of Table 1. The energy transfer between different scales of veloc-ity fluctuations is a key process in the development ofturbulence, and knowledge of the efficiency and timescales for these energy exchanges is crucial for the un-derstanding of turbulence theory and for further devel-opment of engineering models. The exchange betweenharmonics of the originally injected frequency is well inline with the fact that frequency doubling and sum- anddifference-frequencies are to be expected as a result ofthe actions of the nonlinear term of the Navier-Stokesequation [21]. xperimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes 9
Fig. 9
Vortex shedding flow visualization.
Fig. 10
Schematic (not to scale) of the vortex shedding ex-periment in a laminar jet core, including the sharp-edged rod(in grey) to inject a sharp frequency and the region of themeasurement points (indicated by a blue line).
Three experiments were performed where a singleFourier mode was injected into a well-defined turbu-lent flow and the development of the velocity powerspectrum was traced along the downstream direction.The downstream convection has been considered as asuccessive exposure of the initial velocity trace to theNavier-Stokes equation in a small fluid control volume.This point of view allowed us to compare the devel-opment of the measured spectra to the spectra com-puted by a one-dimensional computer simulation withthe measured initial time trace as input and to computethe power spectra as this time trace was transformed by incremental repeated exposures to the Navier-Stokesequation.Although the particular form of the measured andcomputed power spectra depends on the initial positionof the measurement point and the way the Fourier modewas injected, a number of common properties were re-vealed: – Higher order frequency components were formed suc-cessively as a fluid element passing near the pointof injection evolved downstream in time. – The higher frequencies were exact multiples of theinjected frequency and they retained their well-definedsharp spectral frequency – even when submergedinto high-intensity turbulence. – Far downstream the spectrum tends to an asymp-totic form whose energy is slowly being absorbedinto the surrounding turbulence. – The absorption distance does not change signifi-cantly with Reynolds number in the investigatedrange Re = 22 . − . – The transfer of energy between modes is seen tobe both direct and indirect, but it occurs predomi-nantly from low wavenumbers towards higher wavenum-bers. – A delay in the interactions between the injectionharmonics is evident in all experiments. The initialdevelopment is particularly carefully isolated in themeasurement of the vortex shedding from a sharp-edged rod in a laminar jet core. – Influence of the initial spectral components on thefar downstream spectrum is clearly evident, in par-ticular in the oscillating airfoil measurements in aturbulent jet.All in all, the measurements and computer simula-tions illustrate the dynamic nature of the triadic inter-
Fig. 11
Comparison of velocity power spectra at 1 mm (left) and 10 mm (right) from the rod edge. Blue: Mea-surements, video available at https://doi.org/10.11583/DTU.12016827.v1. Red: Computer simulations, video available athttps://doi.org/10.11583/DTU.12016869.v1. actions between injected energetic Fourier componentsthrough the downstream development of the velocityand power spectrum, as predicted from [20,21].The authors hope that these results can be usedby turbulence modelers to improve their methods, inparticular the results on the generation and absorptionof the harmonics of the initial injected frequency. Acknowledgements
Benny Edelsten is acknowledged forhis assistance with the hot-wire experiments.
Conflict of interest
The authors declare that they have no conflict of inter-est.
References
1. Kolmogorov AN., “The local structure of turbulencein incompressible viscous fluid for very large Reynoldsnumbers”, Proc. R. Soc. Lond. A, (1991), 434, 9–13.https://doi.org/10.1098/rspa.1991.0075 (1991)2. Lvov V., Procaccia I., “Hydrodynamic turbulence: a19th century problem with a challenge for the 21stcentury”. In: Boratav O., Eden A., Erzan A. (eds)Turbulence Modeling and Vortex Dynamics. LectureNotes in Physics, vol 491. Springer, Berlin, Heidelberg.https://doi.org/10.1007/BFb0105026 (1997)3. Richardson LF., Lynch P., “Weather Prediction byNumerical Process”. 2nd edn. Cambridge: CambridgeUniversity Press (Cambridge Mathematical Library).https://doi.org/10.1017/CBO9780511618291 (2007)4. Kraichnan RH., “The structure of isotropic turbulence atvery high Reynolds numbers”, Journal of Fluid Mechan-ics. Cambridge University Press, 5(4), pp. 497–543, doi:10.1017/S0022112059000362 (1959)5. From https : collectedon 15.03.2020xperimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes 116. George WK., “A 50-Year Retrospective and the Fu-ture”, in: Whither Turbulence and Big Data in the21st Century?, Springer International Publishing, https : //doi.org/ . / − − − − − / − / //