Scaling Laws and Intermittency in Highly Compressible Turbulence
Alexei G. Kritsuk, Paolo Padoan, Rick Wagner, Michael L. Norman
aa r X i v : . [ a s t r o - ph ] J un Scaling Laws and Intermittency in HighlyCompressible Turbulence
Alexei G. Kritsuk † , Paolo Padoan, Rick Wagner,and Michael L. Norman Physics Department and Center for Astrophysics and Space Sciences,University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0424, USA † also Sobolev Astronomical Institute, St. Petersburg State University, St. Petersburg, Russia Abstract.
We use large-scale three-dimensional simulations of supersonic Euler turbulence to studythe physics of a highly compressible cascade. Our numerical experiments describe non-magnetizeddriven turbulent flows with an isothermal equation of state and an rms Mach number of 6. Wefind that the inertial range velocity scaling deviates strongly from the incompressible Kolmogorovlaws. We propose an extension of Kolmogorov’s K41 phenomenology that takes into accountcompressibility by mixing the velocity and density statistics and preserves the K41 scaling of thedensity-weighted velocity v ≡ r / u . We show that low-order statistics of v are invariant with respectto changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of v is − .
69 and the third-order structure function of v scales linearly with separation. We directlymeasure the mass dimension of the “fractal” density distribution in the inertial subrange, D m ≈ . Keywords:
ISM: structure — hydrodynamics — turbulence — fractals — methods: numerical
PACS:
INTRODUCTION
In the late 1930’s, Kolmogorov clearly realized that chances to develop a closed purelymathematical theory of turbulence are extremely low [1]. Therefore, the basic approachin [2, 3] (usually referred to as the K41 theory) was to rely on physical intuition andformulate two general statistical hypotheses which describe the universal equilibriumregime of small-scale fluctuations in arbitrary turbulent flow at high Reynolds number.Following the Landau (1944) remark on the lack of universality in turbulent flows [4],and with information extracted from new experimental data, the original similarity hy-potheses of K41 were then revisited and refined to account for intermittency effects[5, 6, 7]. While the K41 phenomenology became the cornerstone for all subsequent de-velopments in incompressible turbulence research [e.g., 8], there was no similar resultestablished for compressible flows yet [9, 10]. Historically, compressible turbulence re-search, preoccupied with a variety of specific engineering applications, was generally “An understanding of solutions to the [incompressible] Navier-Stokes equations” yet remains one of thesix unsolved grand challenge problems nominated by the Clay Mathematics Institute in 2000 for a $1M Millennium Prize l og k / < P ( k ) > log k/k min -1.95(2) E (k)-1.69(2) S (k)-1.53(2)E(k) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 1.4 1.6 1.8 2 2.2 2.4 l og S ( ℓ ) log ℓ/∆ u v w FIGURE 1.
Time average compensated power spectra ( left ) and third-order transverse structure func-tions ( right ) for velocity u and mass-weighted velocities v ≡ r / u and w ≡ r / u . The statistics of v clearly demonstrate a K41-like scaling. Notice strong bottleneck contamination in the spectra at highwavenumbers. lagging behind the incompressible developments. The two major reasons for this timelag were an additional complexity of analytical treatment of compressible flows and ashortage in experimental data for super- and hypersonic turbulence. In this respect, al-though limited to relatively low Reynolds numbers, direct numerical simulations (DNS)of turbulence (pioneered by Orszag and Patterson [14]) have occupied the niche of ex-periments at least for the most simple flows. One particularly important advantage ofDNS is an easy access to variables that are otherwise difficult to measure in the labora-tory or treat analytically.A traditionally straightforward approach to data analysis from DNS of compressibleturbulence includes computation of the “standard” statistics of velocity fluctuations. Inaddition, the diagnostics for density fluctuations are also computed and discussed as thedirect measures of compressibility. Quite naturally, both density and velocity statisticsdemonstrate strong dependence on the Mach number M in supersonic ( M ∈ [ , ] ) andhypersonic ( M >
3) regimes, while the variations in turbulent diagnostics at sub- ortransonic Mach numbers are rather small. For instance, at M ≈ ∼ k − / [16], at M ≈ ∼ k − . [20], and at M ≈ − .
07 [20].Based on the data from numerical experiments, it is well established that: (i) thevelocity power spectra tend to get steeper as the Mach number increases, reaching theBurgers slope of − − A reasonable measure of the delay is 60+ years passed between the appearance of incompressibleReynolds averaging [11] and mass-weighted Favre averaging for fluid flows with variable density [12],although see [13] for references to a few earlier papers that dealt with density-weighted averaging. ingular velocity structures increases from D s , u ∼ D s , u ∼ D m = D m ∼ . lossy compressibleturbulent cascade that would asymptotically match the incompressible Kolmogorov-Richardson energy cascade [2, 21] in the limit of very low Mach numbers. Sinceincompressible turbulence represents a degenerate case where the density is uncorrelatedwith the velocity, the phenomenology of the compressible cascade must include thiscorrelation. This essentially means that instead of velocity u , which is a single keyingredient of the K41 laws, one needs to consider a set of mixed variables, r / h u , where r is the density and h can take values 1, 2, or 3, depending on the statistical measure ofinterest [20]. For instance, if one is studying the scale-by-scale kinetic energy budget ina compressible turbulent flow, a mixed variable power spectrum with h = h = grid points. In this paper we present the highlights of the compressible cascade phe-nomenology verified in [20]. SCALING, STRUCTURES, AND INTERMITTENCY
Nonlinear interactions transfer kinetic energy supplied to the system at large scalesthrough the inertial range with little dissipation. Let us assume that the mean volume energy transfer rate in a compressible fluid, r u u /ℓ , is constant in a statistical steadystate [e.g., 23]. If this is true, then v p ≡ ( r / u ) p ∼ ℓ p / (1)for an arbitrary power p and, with the standard assumption of self-similarity of thecascade, the structure functions (SFs) of mixed variable v for compressible flows shouldscale in the inertial range as S p ( ℓ ) ≡ h| v ( r + ℓ ) − v ( r ) | p i ∼ ℓ p / . (2)In the limit of weak compressibility, the scaling laws (2) will reduce to the K41results for the velocity structure functions. The scaling laws S p ( ℓ ) ∼ ℓ z p , where z p = p / z p = p / + t p / [5]. The only exception is, perhaps, thethird order relation for the longitudinal velocity SFs, which is exact in the incompressible l og M ( ℓ ) / ℓ log ℓ/∆ z p / z pK41SL94BurgB02HS1HS2 v FIGURE 2.
Gas mass M ( ℓ ) as a function of the box size ℓ ( left ). The mass dimension D m is defined asthe log-log slope of M ( ℓ ) , see eq. (4). Relative exponents for structure functions of the transverse modifiedvelocities v versus order p and two hierarchical structure models with different parameters [HS1 & HS2,6] that fit the data for p ∈ [ , ] ( right ). Also shown are model predictions for the Kolmogorov-Richardsoncascade [K41, 2, 3], for intermittent incompressible turbulence [SL94, 6], for “burgulence” [Burg, 24],and for the velocity fluctuations in supersonic turbulence [B02, 25]. case and is known as the four-fifth law [3]. Our focus here is mostly on the low orderstatistics ( p ≤
3) for which the corrections are small. Since the power spectrum slope isrelated to the exponent of the second order structure function, the K41 slope of 5 / v ≡ r / u in the compressible case.Figure 1 shows the power spectra of u , v , and w ≡ r / u and the corresponding third-order transverse structure functions based on the simulations at Mach 6 [26, 20]. Thepower spectrum S ( k ) and the structure function of v clearly follow the K41 scaling: S ∼ k − . and S ∼ ℓ . [20], while the velocity power spectrum E ( k ) and struc-ture function have substantially steeper-then-K41 slopes: − .
95 and 1.29 [26]. At thesame time, the kinetic energy spectrum E ∼ k − . is shallow and both solenoidal anddilatational components of w have the same slope implying a single compressible en-ergy cascade with strong interaction between the two components [20]. These resultsbased on the high dynamic range simulations lend strong support to the scaling relationsdescribed by eq. (2) and to the conjecture from which they were inferred. Previous sim-ulations at lower resolution did not allow to measure the absolute exponents reliably dueto insufficient dynamic range and due to the bottleneck contamination [27].In 1951, von Weiszäker [28] introduced a phenomenological model for scale-invarianthierarchy of density fluctuations in compressible turbulence described by a simpleequation that relates the mass density at two successive levels to the corresponding scalesthrough a universal measure of the degree of compression, a , r n / r n − = ( ℓ n /ℓ n − ) − a . (3)The geometric factor a takes the value of 1 in a special case of isotropic compressionin three dimensions, 1 / IGURE 3.
Coherent structures in Mach 6 turbulence at resolution of 1024 . Projections along theminor axis of a subvolume of 700 × ×
250 zones for the density ( upper left ), the enstrophy ( upperright ), the dissipation rate ( lower left ), and the dilatation ( lower right ). The logarithmic grey-scale rampshows the lower values as dark in all cases except for the density. The inertial subrange structurescorrespond to scales between 40 and 250 zones and represent a fractal with D m ≈ .
4. The dominantstructures in the dissipation range ( ℓ < D ) are shocks with D m =
2. [Reprinted from [20].] and mass: u ∼ ℓ / + a , E ( k ) ∼ k − / − a , r ∼ ℓ − a , M ( ℓ ) ∼ ℓ D m ∼ ℓ − a , (4)where all the exponents depend on the compression measure a which is in turn afunction of the rms Mach number of the turbulent flow. We can now use the data fromnumerical experiments to verify the scaling relations (4). Since the first-order velocitystructure function scales as ℓ . [20], we can estimate a for the Mach 6 flow, a ≈ . D m ≈ .
38. It is indeed consistent with our direct measurement of the massdimension for the same range of scales, D m ≈ .
39, see Fig. 2.In strongly compressible turbulence at Mach 6, the density contrast between superson-ically moving blobs and their more diffuse environment can be as high as 10 . The mostcommon structural elements in such highly fragmented flows are nested bow-shocks[17]. Figure 3 shows an extreme example of structures formed by a collision of counter-propagating supersonic flows. On small scales within the dissipation range, these struc-tures are characterized by D m =
2, while within the inertial range D m ≈ . left ).he hierarchical structure (HS) model z p / z = g p + C ( − b p ) (5)[6] provides good fits to the data for the mass-weighted velocity v (see Fig. 2, right ).Here the codimension of the support of the most singular dissipative structures C ≡ − D s , v = ( − g ) / ( − b ) . (6)If the fit is limited to p ∈ [ , ] , two sets of model parameters b and g are formallyacceptable (models HS1 and HS2 in Fig. 2). The best-fit parameters of the HS1 model: b = / g = C = . b B = / g B = /
9) [25] and the Burgers’ model ( b Burg = g Burg =
0) [24]. TheHS2 model ( b = / g = /
9, and C = .
8) provides a fit of roughly the samequality for p ∈ [ , ] , but overestimates the scaling exponents z p at p >
4. Since thelevel of uncertainty in the high order statistics remains high even at a resolution of 1024 grid points, larger dynamic range simulations are needed to distinguish between the twooptions.If the HS1 option is confirmed, then Mach 6 turbulence is more intermittent thanincompressible turbulence ( b < b SL = /
3) and has the same degree of singularityof structures as burgulence. The singular dissipative structures with fractal dimension D s , v = . g = g SL ). In this case the fractal dimension of the most singular structures, D s , v = . CONCLUSION
Using large-scale Euler simulations of supersonic turbulence at Mach 6 we have demon-strated that there exists an analogue of the K41 scaling laws valid for both weakly andhighly compressible flows. The mass-weighted velocity v ≡ r / u – the primary vari-able governing the energy transfer through the cascade – should replace the velocity u in intermittency models for compressible flows at high Mach numbers. ACKNOWLEDGMENTS
This research was partially supported by a NASA ATP grant NNG056601G, by NSFgrants AST-0507768 and AST-0607675, and by NRAC allocations MCA098020S andMCA07S014. We utilized computing resources provided by the San Diego Supercom-puter Center and by the National Center for Supercomputer Applications.
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