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NASA NTRS1994

A hydrodynamic approach to cosmology: The mixed dark matter cosmological scenario

We compute the evolution of spatially flat, mixed cold and hot dark matter models containing both baryonic matter and two kinds of dark matter. Hydrodynamics is treated with a highly developed Eulerian hydrodynamic code (see Cen 1992). A standard particle-mesh (PM) code is also used in parallel to calculate the motion of the dark matter components. We adopt the following parameters: h equivalent to (sub 0)/100 km/s Mpc(exp -1) = 0.5, OMEGA(sub C) = 0.3, and OMEGA(sub B) = 0.06, with amplitude of the perturbation spectrum fixed by the Cosmic Background Explorer Satellite (COBE) Dark Matter Radiation (DMR) measurements (Smoot et al. 1992) being sigma (sub 8) = 0.67. Four different boxes are simulated with box sizes of L = (64, 16, 4, 1) h(exp -1) Mpc, respectively, the two small boxes providing good resolution but little valid information due to absence of large-scale power. We use 128(exp 3) approximate 10(exp 6.3) baryonic cells, 128(exp .3) cold dark matter particles, and 2 x 128(exp 3) hot dark matter particles. In addition to the dark matter we follow separately six baryonic species (H, H(+), He, He(+), He(++), e(-)) with allowance for both (nonequilibrium) collisional and radiative ionization in every cell. The background radiation field is also followed in detail with allowance made for both continuum and line processes, to allow nonequilibrium heating and cooling processes to be followed in detail. The mean final Zeldovich-Sunyaev y parameter is estimated to be y Bar = (5.4 + or - 2.7) x 10(exp -7) below currently attainable observations, with a rms fluctuation of approximately delta bar y = (0.6 + or - 3.0) x 10(exp -7) on arcminute scales. The rate of galaxy formation peaks at an even later epoch (z approximate 0.3) than in the standard (OMEGA = 1, sigma sub 8 = 0.67) cold dark matter (CDM) model (z approximate 0.5) and, at a redshift of z = 4, is nearly a factor of 100 lower than for the CDM model with the same value of sigma sub 8. With regard to mass function, the smallest objects are stabilized against collapse by thermal energy: the mass-weighted mass spectrum has a broad peak in the vicinity of M(sub B) = 10(exp 9.5) solar mass with a reasonable fit to the Schechter luminosity function if the ratio of baryon mass to blue light is approximately 4. In addition, one very large PM simulation was made in a box with size (320 h(exp - 1) Mpc) containing 3 x 200(exp 3) = 10(exp 7.4) particles. Utilizing this simulation we find that the model yields a cluster mass function which is about a factor of 4 higher than observed, but a cluster-cluster correlation length marginally lower than observed, but that both are closer to observations than in the (COBE) normalized CDM model. The one-dimensional pairwise velocity dispersion is 605 + or - 8 km/s at 1/h separation, lower than that of the DCM model normalized to COBE, but still significant higher than observations (Davis & Peebles 1983). A plausible velocity bias b(sub v) = 0.8 + or - 0.1 on this scale will reduce but not remove the discrepancy. The velocity auto-correlat ion function has a coherence length of 40/h Mpc, which is somewhat lower than the observed counterpart. In all these respects the model would be improved by decreasing the cold fraction of the dark OMEGA(sub CDM)/ (OMEGA(sub CDM) + OMEGA(sub HDB). But formation of galaxies and clusters of galaxies is much later in this model than in COBE-normalized CDM, perhaps too late. To improve on these constraints a larger ratio of OMEGA(sub CDM)/ (OMEGA(sub CDM) + OMEGA(sub HDM)) is required than the value of 0.67 adopted here. It does not seem possible to find a value for this ratio which would satisfy all tests. Overall, the model is similar both on large and intermediate scales to the standard CDM model normalized to the same value of sigma(sub B), but the problem with regard to late formation of galaxies is more severe in this model than in that CDM model. Adding hot dark matter, significantly improves the ability of the COBE-normalized CDM scenario to fit existing observations, but the model is in fact not as good as the CDM model with the same sigma(sub 8) and is still probably unsatisfactory with regard to several critical tests.

Cen, Renyue↗
NASA NTRS2008

Calculations of High-Temperature Jet Flow Using Hybrid Reynolds-Average Navier-Stokes Formulations

Two multiscale-type turbulence models are implemented in the PAB3D solver. The models are based on modifying the Reynolds-averaged Navier Stokes equations. The first scheme is a hybrid Reynolds-averaged- Navier Stokes/large-eddy-simulation model using the two-equation k(epsilon) model with a Reynolds-averaged-Navier Stokes/large-eddy-simulation transition function dependent on grid spacing and the computed turbulence length scale. The second scheme is a modified version of the partially averaged Navier Stokes model in which the unresolved kinetic energy parameter f(sub k) is allowed to vary as a function of grid spacing and the turbulence length scale. This parameter is estimated based on a novel two-stage procedure to efficiently estimate the level of scale resolution possible for a given flow on a given grid for partially averaged Navier Stokes. It has been found that the prescribed scale resolution can play a major role in obtaining accurate flow solutions. The parameter f(sub k) varies between zero and one and is equal to one in the viscous sublayer and when the Reynolds-averaged Navier Stokes turbulent viscosity becomes smaller than the large-eddy-simulation viscosity. The formulation, usage methodology, and validation examples are presented to demonstrate the enhancement of PAB3D's time-accurate turbulence modeling capabilities. The accurate simulations of flow and turbulent quantities will provide a valuable tool for accurate jet noise predictions. Solutions from these models are compared with Reynolds-averaged Navier Stokes results and experimental data for high-temperature jet flows. The current results show promise for the capability of hybrid Reynolds-averaged Navier Stokes and large eddy simulation and partially averaged Navier Stokes in simulating such flow phenomena.

Abdol-Hamid, Khaled S.↗
NASA NTRSDate not supplied

Representing the Sub-Grid Heterogeneity of Surface Precipitation in A General Circulation Model

Precipitation variability on spatial scales smaller than a typical general circulation model (GCM) grid box is often neglected, with the grid-mean precipitation rate being applied uniformly to underlying surface tiles. This reduces the extrema seen by the surface, with corresponding reductions in surface runoff and altered land-atmosphere fluxes. Here we present a novel approach to stochastically distribute precipitation across sub-grid surface tiles within a GCM. Based on 4 km Stage IV precipitation data, the scheme parameterizes the dry area fraction as a function of grid mean precipitation rate, and defines the relative distribution of intensities across non-dry surface tiles. To incorporate memory and mimic the persistence of precipitating storms, the relative intensity assigned to each sub-grid tile is determined by an autoregressive process. Using single column experiments, the scheme is shown to reproduce observed precipitation statistics at the scale of model surface tiles. We also document impacts on surface hydrology and energy partitioning, with notable increases in precipitation runoff, surface temperature variance, and the Bowen ratio.

GCM↗
NASA NTRS2001

Production, Outflow Velocity, and Radial Distribution of H2O and OH in the Coma of Comet C/1995 O1 [Hale-Bopp] from Wide Field Imaging of OH

Observations of OH are a useful proxy of the water production rate (Q(sub H2O)) and outflow velocity (V(sub out)) in comets. We use wide field images taken on 03/28/1997 and 04/08/1997 that capture the entire scale length of the OH coma of comet C/1995O1 (Hale-Bopp) to obtain Q(sub H2O) from the model-independent method of aperture summation. We also extract the radial brightness profile of OH 3080 angstroms out to cometocentric distances of up to 10(exp 6) km using an adaptive ring summation algorithm. Radial profiles are obtained as azimuthal averages and in quadrants covering different position angles relative to the comet-Sun line. These profiles are fit using both fixed and variable velocity two-component spherical expansion models to determine VOH with increasing distance from the nucleus. The OH coma of Hale-Bopp was more spatially extended than in previous comets, and this extension is best matched by a variable acceleration of H2O and OH that acted across the entire coma, but was strongest within 1-2 x 10(exp 4) km from the nucleus. This acceleration led to VOH at 10(exp 6) km that was 2-3 times greater than that obtained from a 1P/Halleytype comet at 1 AU, a result that is consistent with gas-kinetic models, extrapolation from previous observations of OH in comets with Q(sub H2O) > 10(exp 29)/s, and radio measurements of the outer coma Hale-Bopp OH velocity profile. When the coma is broken down by quadrant, we find an azimuthal asymmetry in the radial distribution that is characterized by an increase in the spatial extent of OH in the region between the orbit-trailing and anti-sunward directions. Model fits to this area and comparison with radio OH measurements suggest greater acceleration in this region, with VOH UP to 1.5 times greater at 10(exp 6) km radial distance than elsewhere in the coma.

Harris, Walter M.↗
NASA NTRS1994

The length-scaling properties of topography

The scaling properties of synthetic topographic surfaces and digital elevation models (DEMs) of topography are examined by analyzing their 'structure functions,' i.e., the qth order powers of the absolute elevation differences: delta h(sub q) (l) = E((absolute value of h(x + l) - h(x))(exp q)). We find that the relation delta h(sub 1 l) approximately equal cl(exp H) describes well the scaling behavior of natural topographic surfaces, as represented by DEMs gridded at 3 arc sec. Average values of the scaling exponent H between approximately 0.5 and 0.7 characterize DEMs from Ethiopia, Saudi Arabia, and Somalia over 3 orders of magnitude range in length scale l (approximately 0.1-150 km). Differences in appparent topographic roughness among the three areas most likely reflect differences in the amplitude factor c. Separate determination of scaling properties in the x and y coordinate directions allows us to assess whether scaling exponents are azimuthally dependent (anisotropic) or whether they are isotropic while the surface itself is anisotropic over a restricted range of length scale. We explore ways to determine whether topographic surfaces are characterized by simple or multiscaling properties.

Weissel, Jeffrey K.↗
NASA NTRS2003

Cool Flames in Propane-Oxygen Premixtures at Low and Intermediate Temperatures at Reduced-Gravity

The Cool Flame Experiment aims to address the role of diffusive transport on the structure and the stability of gas-phase, non-isothermal, hydrocarbon oxidation reactions, cool flames and auto-ignition fronts in an unstirred, static reactor. These reactions cannot be studied on Earth where natural convection due to self-heating during the course of slow reaction dominates diffusive transport and produces spatio-temporal variations in the thermal and thus species concentration profiles. On Earth, reactions with associated Rayleigh numbers (Ra) less than the critical Ra for onset of convection (Ra(sub cr) approx. 600) cannot be achieved in laboratory-scale vessels for conditions representative of nearly all low-temperature reactions. In fact, the Ra at 1g ranges from 10(exp 4) - 10(exp 5) (or larger), while at reduced-gravity, these values can be reduced two to six orders of magnitude (below Ra(sub cr)), depending on the reduced-gravity test facility. Currently, laboratory (1g) and NASA s KC-135 reduced-gravity (g) aircraft studies are being conducted in parallel with the development of a detailed chemical kinetic model that includes thermal and species diffusion. Select experiments have also been conducted at partial gravity (Martian, 0.3gearth) aboard the KC-135 aircraft. This paper discusses these preliminary results for propane-oxygen premixtures in the low to intermediate temperature range (310- 350 C) at reduced-gravity.

Pearlman, Howard↗
NASA NTRS2003

Production, Outflow Velocity, and Radial Distribution of H2O and OH in the Coma of Comet C/1995 O1 [Hale-Bopp] from Wide Field Imaging of OH

Observations of OH are a useful proxy of the water production rate (Q(sub H2O)) and outflow velocity (V(sub out)) in comets. From wide field images taken on 03/28/1997 and 04/08/1997 that capture the entire scale length of the OH coma of comet C/1995 O1 (Hale-Bopp), we obtain Q(sub H2O) from the model-independent method of aperture summation. With an adaptive ring summation algorithm, we extract the radial brightness distribution of OH 0-0 band emission out to cometocentric distances of up to 10(exp 6) km, both as azimuthal averages and in quadrants covering different position angles relative to the comet-Sun line. These profiles are fit using both fixed and variable velocity 2-component spherical expansion models to estimate V(sub OH) with increasing distance from the nucleus. The OH coma of Hale-Bopp was more spatially extended than previous comets, and this extension is best matched by a variable acceleration of H2O and OH that acted across the entire coma, but was strongest within 1-2 x 10(exp 4) km from the nucleus. Our models indicate that V(sub OH) at the edge of our detectable field of view (10(exp 6) km) was approx. 2-3 times greater in Hale-Bopp than for a 1P/Halley-class comet at 1 AU, which is consistent with the results of more sophisticated gas-kinetic models, extrapolation from previous observations of OH in comets with Q(sub H2O) greater than 10(exp 29)/s , and direct radio measurements of the outer coma Hale-Bopp OH velocity. The most probable source of this acceleration is thermalization of the excess energy of dissociation of H2O and OH over an extended collisional coma. When the coma is broken down by quadrants in position angle, we find an azimuthal asymmetry in the radial distribution that is characterized by an increase in the spatial extent of OH in the region between the orbit-trailing and anti-sunward directions. Model fits specific to this area and comparison with radio OH measurements suggest greater acceleration here, with V(sub OH) approx. 1.5 times greater at a 10(exp 6) km cometocentric distance than elsewhere in the coma. We discuss several mechanisms that may have acted within the coma to produce the observed effect.

Harris, Walter M.↗
NASA NTRS1992

Role of Si2N2O in the passive oxidation of chemically-vapor-deposited Si3N4

The results of two-step oxidation experiments on chemically-vapor-deposited Si3N4 and SiC at 1350 C show that a correlation exists between the presence of a Si2N2O interphase and the strong oxidation resistance of Si3N4. During normal oxidation, k sub p for SiC was 15 times higher than that for Si3N4, and the oxide scale on Si3N4 was found by SEM and TEM to contain a prominent Si2N2O inner layer. However, when oxidized samples are annealed in Ar for 1.5 h at 1500 C and reoxidized at 1350 C as before, three things happen: the oxidation k sub p increases over 55-fold for Si3N4, and 3.5-fold for SiC; the Si3N4 and SiC oxidize with nearly equal k sub p's; and, most significantly, the oxide scale on Si3N4 is found to be lacking an inner Si2N2O layer. The implications of this correlation for the competing models of Si3N4 oxidation are discussed.

Ogbuji, Linus U. J. T.↗
NASA NTRS2005

Numerical Study of High-Temperature Jet Flow Using RANS/LES and PANS Formulations

Two multi-scale-type turbulence models are implemented in the PAB3D solver. The models are based on modifying the Reynolds Averaged Navier-Stokes (RANS) equations. The first scheme is a hybrid RANS/LES model utilizing the two-equation (k(epsilon)) model with a RANS/LES transition function dependent on grid spacing and the computed turbulence length scale. The second scheme is a modified version of the Partially Averaged Navier-Stokes (PANS) model, where the unresolved kinetic energy parameter (f(sub k)) is allowed to vary as a function of grid spacing and the turbulence length scale. This parameter is estimated based on a novel two-stage procedure to efficiently estimate the level of scale resolution possible for a given flow on a given grid for Partial Averaged Navier-Stokes (PANS). It has been found that the prescribed scale resolution can play a major role in obtaining accurate flow solutions. The parameter f(sub k) varies between zero and one and equal to one in the viscous sub layer, and when the RANS turbulent viscosity becomes smaller than the LES viscosity. The formulation, usage methodology, and validation examples are presented to demonstrate the enhancement of PAB3D's time-accurate and turbulence modeling capabilities. The accurate simulations of flow and turbulent quantities will provide valuable tool for accurate jet noise predictions. Solutions from these models are compared to RANS results and experimental data for high-temperature jet flows. The current results show promise for the capability of hybrid RANS/LES and PANS in simulating such flow phenomena.

Abdol-Hamid, Khaled S.↗
NASA NTRS1990

The large-scale structures in turbulent plane Couette flow

Turbulent plane Couette flow has been numerically simulated at a Reynolds number, Re = U(sub w)(h)/nu = 6000. Preliminary examinations of the instantaneous velocity and vorticity fields revealed the existence of large scale eddies which grow in the flow direction with time and have the spanwise scale as large as the channel height. The persistence of the longitudinal eddies was not observed when the small scales in the spanwise direction were not well resolved, indicating that the growth of the large scale longitudinal eddies requires the contribution from the small scale motions in the spanwise direction. The statistical correlation in the flow agree well with the experimental results.

Lee, M. J.↗
NASA NTRS2006

Two Point Space-Time Correlation of Density Fluctuations Measured in High Velocity Free Jets

Two-point space-time correlations of air density fluctuations in unheated, fully-expanded free jets at Mach numbers M(sub j) = 0.95, 1.4, and 1.8 were measured using a Rayleigh scattering based diagnostic technique. The molecular scattered light from two small probe volumes of 1.03 mm length was measured for a completely non-intrusive means of determining the turbulent density fluctuations. The time series of density fluctuations were analyzed to estimate the integral length scale L in a moving frame of reference and the convective Mach number M(sub c) at different narrow Strouhal frequency (St) bands. It was observed that M(sub c) and the normalized moving frame length scale L*St/D, where D is the jet diameter, increased with Strouhal frequency before leveling off at the highest resolved frequency. Significant differences were observed between data obtained from the lip shear layer and the centerline of the jet. The wave number frequency transform of the correlation data demonstrated progressive increase in the radiative part of turbulence fluctuations with increasing jet Mach number.

Panda, Jayanta↗
NASA NTRS1995

The COBE normalization for standard cold dark matter

The Cosmic Background Explorer Satellite (COBE) detection of microwave anisotropies provides the best way of fixing the amplitude of cosmological fluctuations on the largest scales. This normalization is usually given for an n = 1 spectrum, including only the anisotropy caused by the Sachs-Wolfe effect. This is certainly not a good approximation for a model containing any reasonable amount of baryonic matter. In fact, even tilted Sachs-Wolfe spectra are not a good fit to models like cold dark matter (CDM). Here, we normalize standard CDM (sCDM) to the two-year COBE data and quote the best amplitude in terms of the conventionally used measures of power. We also give normalizations for some specific variants of this standard model, and we indicate how the normalization depends on the assumed values on n, Omega(sub B) and H(sub 0). For sCDM we find the mean value of Q = 19.9 +/- 1.5 micro-K, corresponding to sigma(sub 8) = 1.34 +/- 0.10, with the normalization at large scales being B = (8.16 +/- 1.04) x 10(exp 5)(Mpc/h)(exp 4), and other numbers given in the table. The measured rms temperature fluctuation smoothed on 10 deg is a little low relative to this normalization. This is mainly due to the low quadrupole in the data: when the quadrupole is removed, the measured value of sigma(10 deg) is quite consistent with the best-fitting the mean value of Q. The use of the mean value of Q should be preferred over sigma(10 deg), when its value can be determined for a particular theory, since it makes full use of the data.

Bunn, Emory F.↗
NASA NTRS1985

Mega-geomorphology: Mars vis a vis Earth

The areas of chaotic terrain, the giant chasma of the Valles Marineris region, the complex linear and circular depressions of Labyrinthus Noctis on Mars all suggest the possibility of large scale collapse of parts of the martian crust within equatorial and sub equatorial latitudes. It seems generally accepted that the above features are fossil, being perhaps, more than a billion years old. It is possible that parts of Earth's crust experienced similar episodes of large scale collapse sometime early in the evolution of the planet.

Sharp, R. P.↗
NASA NTRS1981

Computing aerodynamic sound using advanced statistical turbulence theories

It is noted that the calculation of turbulence-generated aerodynamic sound requires knowledge of the spatial and temporal variation of Q sub ij (xi sub k, tau), the two-point, two-time turbulent velocity correlations. A technique is presented to obtain an approximate form of these correlations based on closure of the Reynolds stress equations by modeling of higher order terms. The governing equations for Q sub ij are first developed for a general flow. The case of homogeneous, stationary turbulence in a unidirectional constant shear mean flow is then assumed. The required closure form for Q sub ij is selected which is capable of qualitatively reproducing experimentally observed behavior. This form contains separation time dependent scale factors as parameters and depends explicitly on spatial separation. The approximate forms of Q sub ij are used in the differential equations and integral moments are taken over the spatial domain. The velocity correlations are used in the Lighthill theory of aerodynamic sound by assuming normal joint probability.

Hecht, A. M.↗
NASA NTRS1994

Turbulent Diffusion and Concentration of "Chondrules" in the Protoplanetary Nebula

Turbulence is known to possess structure on many scales. The largest or integral scale L is usually taken to be the largest relevant dimension of the system; for the nebula as a whole, it would be on the order of a vertical gas scale height H approximately 0.1R where R is the distance from the sun. Turbulent kinetic energy cascades towards smaller scales; for fully developed, homogeneous turbulence, the turbulent kinetic energy peaks at the integral scale and is distributed within its "inertial range" according to the Kolmogorov energy spectrum dE(k) = E(sub 0)(kL)(sup -5/3)dk where k is the wavenumber or inverse wavelength. The smallest scale in a turbulent regime is the Kolmogorov scale (n)(sometimes referred to as the inner scale). On this scale, molecular viscosity can dissipate turbulent kinetic energy . Turbulent flows are characterized by their Reynolds number Re = UL/v, where U is a characteristic velocity scale and v is molecular viscosity. More energetic (higher Re) flows drive turbulence to smaller Kolmogorov scales: n = LRe(sup -3/4). Two different turbulent regimes are likely to be of importance; early stage convective turbulence in a hot nebula probably extends over the entire nebula scale height, and probably has typical eddy frequency comparable to the orbit frequency. At a later stage, midplane shear layer turbulence can be driven by the presence of a differentially rotating, densely settled particle layer, with different length and time scales.

Cuzzi, Jeffrey N.↗
NASA NTRS1998

Evidence for Langmuir Envelope Solitons in Solar Type III Burst Source Regions

We present observational evidence for the generation of Langmuir envelope solitons in the source regions of solar type III radio bursts. The solitons appear to be formed by electron beams which excite either the modulational instability or oscillating two-stream instability (OTSI). Millisecond data from the Ulysses Unified Radio and Plasma Wave Experiment (URAP) show that Langmuir waves associated with type III bursts occur as broad intense peaks with time scales ranging from 15 to 90 milliseconds (6 - 27 km). These broad field structures have the properties expected of Langmuir envelope solitons, viz.: the normalized peak energy densities, W(sub L)/n(sub e)T(sub e) approximately 10(exp -5), are well above the modulational instability threshold; the spatial scales, L, which range from 1 - 5 Langmuir wavelengths, show a high degree of inverse correlation with (W(sub L)/n(sub e)T(sub e))(sup 1/2); and the observed widths of these broad peaks agree well with the predicted widths of envelope solitons. We show that the orientation of the Langmuir field structures is random with respect to the ambient magnetic field, indicating that they are probably isotropic structures that have evolved from initially pancake-like solitons. These observations suggest that strong turbulence processes, such as the modulational instability or the OTSI, stabilize the electron beams that produce type III bursts.

Thejappa, G.↗
NASA NTRS1985

Interstellar turbulence, random density variations, and scintillation measurements

The presence of random electron variations suggests that the ionized interstellar medium is turbulent. In the interstellar plasma the presence of power spectra of such variations extending to spatial scales much less than a Coulomb mean free path, Lambda sub c, is required by analyses of measurements of scintillation and angular broadening of pulsar radio signals. The existence of corresponding variations in magnetic field strength could efficiently scatter cosmic rays and thus constrain cosmic-ray propagation. Unfortunately both the origin of the electron density variations and mechanisms by which these variations couple to fluctuations in magnetic field strength are unknown. It is conjectured that the small-scale density variations are generated by the convective distortion of initially large-scale isobaric entropy structures in the turbulent interstellar plasma. An investigation of the spectra of turbulent entropy structures, velocity, and magnetic fields at small spatial scales is made. The modifier small is employed to characterize length scales much less than the dimension, L, containing the bulk of the turbulent energy.

Higdon, J. C.↗
NASA NTRS1993

Probability density distribution of velocity differences at high Reynolds numbers

Recent understanding of fine-scale turbulence structure in high Reynolds number flows is mostly based on Kolmogorov's original and revised models. The main finding of these models is that intrinsic characteristics of fine-scale fluctuations are universal ones at high Reynolds numbers, i.e., the functional behavior of any small-scale parameter is the same in all flows if the Reynolds number is high enough. The only large-scale quantity that directly affects small-scale fluctuations is the energy flux through a cascade. In dynamical equilibrium between large- and small-scale motions, this flux is equal to the mean rate of energy dissipation epsilon. The pdd of velocity difference is a very important characteristic for both the basic understanding of fully developed turbulence and engineering problems. Hence, it is important to test the findings: (1) the functional behavior of the tails of the probability density distribution (pdd) represented by P(delta(u)) is proportional to exp(-b(r) absolute value of delta(u)/sigma(sub delta(u))) and (2) the logarithmic decrement b(r) scales as b(r) is proportional to r(sup 0.15) when separation r lies in the inertial subrange in high Reynolds number laboratory shear flows.

Praskovsky, Alexander A.↗