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Two-dimensional turbulence models are compared with experimental measurements made using an array of instrumented towers. The spatial correlation coefficient, the two-point spectrum or cross spectrum, and the coherence function are discussed. The prediction techniques in general agree reasonably well with the experimental results. Measurements of the integral length scale however, do not correlate well with the prediction model.
An asymptotic form is derived for the longitudinal correlation function for isotropic homogeneous turbulence in an incompressible fluid governed by the Navier-Stokes equation. The result is obtained from an analysis of the algebraic-differential structure of the two-point correlation tensor contained in the complex-valued Fourier transform of the probability measure over the turbulence ensemble, which reveals that the Hopf characteristic functional (the complex-valued Fourier transform of the probability measure) satisfies the Fourier interference inequality. Consequences of the expression obtained, in which the correlation function is a positive definite function of the inverse cube of the spatial coordinate as it approaches infinity, are shown to include the nonexistence of the Loitsianskii invariant, and the solution is shown to be consistent with empirical formulas.
Consideration is given to an empirical formula for the longitudinal correlation function for grid-generated incompressible fluid turbulence at Reynolds numbers above 12,800. The formula, which relates the longitudinal correlation function to the inverse cube of a dimensionless geometrical ratio, is shown to minimize the global correlation integrals into which the two-point velocity correlation tensor has been substituted subject to a global constraint on the Sobolev concomitent of the longitudinal correlation function. Furthermore, the energy spectrum function associated with the empirical formula is shown to satisfy a tertiary Helmholtz-type linear condition throughout the initial period of decay.
Equations for the fluctuation correlation in an incompressible shear flow are derived on the basis of kinetic theory, utilizing the two-point distribution function which obeys the BBGKY hierarchy equation truncated with the hypothesis of 'ternary' molecular chaos. The step from the molecular to the hydrodynamic description is accomplished by a moment expansion which is a two-point version of the thirteen-moment method, and which leads to a series of correlation equations, viz., the two-point counterparts of the continuity equation, the Navier-Stokes equation, etc. For almost parallel shearing flows the two-point equation is separable and reduces to two Orr-Sommerfeld equations with different physical implications.
A model for the prediction of wake-related sound generation by a single airfoil is presented which assumes that the net force fluctuation on an airfoil can be expressed in terms of the net momentum fluctuation in the near wake of the airfoil. The model predicts a forcing function for sound generation which is related to the spectra of the two-point velocity correlations in the turbulent region near the airfoil trailing edge. Cross spectra were obtained from correlations of the longitudinal and transverse components of turbulence in the wake of a 36-in. chord airfoil in a wind tunnel using x-probe hot-wire sensors. Additional data were obtained from a 10-in. chord airfoil in a free-jet facility. Based on results from these tests, a scaling model was developed using the turbulent boundary-layer thickness as the primary length measure. The model was used to predict the sound radiated from a 2-in. chord airfoil for which acoustical data was available; good agreement was obtained both in level and spectral shape. The single-airfoil result is extended to a rotor geometry, and comparisons are studied for various aerodynamic parameters.
Void statistics of the galaxy distribution in the Center for Astrophysics Redshift Survey provide strong constraints on galaxy clustering in the nonlinear regime, i.e., on scales R equal to or less than 10/h Mpc. Computation of high-order moments of the galaxy distribution requires a sample that (1) densely traces the large-scale structure and (2) covers sufficient volume to obtain good statistics. The CfA redshift survey densely samples structure on scales equal to or less than 10/h Mpc and has sufficient depth and angular coverage to approach a fair sample on these scales. In the nonlinear regime, the void probability function (VPF) for CfA samples exhibits apparent agreement with hierarchical scaling (such scaling implies that the N-point correlation functions for N greater than 2 depend only on pairwise products of the two-point function xi(r)) However, simulations of cosmological models show that this scaling in redshift space does not necessarily imply such scaling in real space, even in the nonlinear regime; peculiar velocities cause distortions which can yield erroneous agreement with hierarchical scaling. The underdensity probability measures the frequency of 'voids' with density rho less than 0.2 -/rho. This statistic reveals a paucity of very bright galaxies (L greater than L asterisk) in the 'voids.' Underdensities are equal to or greater than 2 sigma more frequent in bright galaxy samples than in samples that include fainter galaxies. Comparison of void statistics of CfA samples with simulations of a range of cosmological models favors models with Gaussian primordial fluctuations and Cold Dark Matter (CDM)-like initial power spectra. Biased models tend to produce voids that are too empty. We also compare these data with three specific models of the Cold Dark Matter cosmogony: an unbiased, open universe CDM model (omega = 0.4, h = 0.5) provides a good match to the VPF of the CfA samples. Biasing of the galaxy distribution in the 'standard' CDM model (omega = 1, b = 1.5; see below for definitions) and nonzero cosmological constant CDM model (omega = 0.4, h = 0.6 lambda(sub 0) = 0.6, b = 1.3) produce voids that are too empty. All three simulations match the observed VPF and underdensity probability for samples of very bright (M less than M asterisk = -19.2) galaxies, but produce voids that are too empty when compared with samples that include fainter galaxies.
The evolution of structure in a postrecombination Friedmann-Robertson-Walker universe containing both gaseous baryons and cold dark matter (CDM) is studied by means of an Eulerian code coupled with a standard particle-mesh code. Ionization state and radiative opacity are calculated in detail, and the hydrodynamic simulations make it possible to compute properties of gas distribution on scales larger than three cell sizes. The model yields a soft X-ray background consistent with the latest cosmic nucleosynthesis values, and can accurately reproduce the galaxy-galaxy two-point correlation. The rate of galaxy formation peaks at a relatively late epoch. With regard to mass function, the smallest objects are stabilized against collapse by thermal energy: the mass-weighted mass spectrum peaks in the vicinity of m(b) = 10 exp 9.2 solar masses with a reasonable fit to the Schecter luminosity function if the baryon mass to blue light ratio is approximately 4. Overall, the simulations provide strong support for the CMD scenario. Of particular interest is that, while the baryons are not biased on scales greater than 1/h Mpc, the galaxies are, and that the 'galaxies' have a correlation function of the required slope and the correct amplitude.
There are many studies on the correlation functions of galaxies, of clusters of galaxies, even of superclusters (e.g., Groth and Peebles 1977; Davies and Peebles 1983; Kalinkov and Kuneva 1985, 1986; Bahcall 1988 and references therein) but not so many on pairs and groups of galaxies. Results of the calculations of two-point correlation fuctions for some catalogs of pairs and groups of galaxies are given. It is assumed that the distances to pairs and groups of galaxies are given by their mean redshifts according to R = sigma (sup n, sub i-1) V sub i/nH (sub 0), where n is the number of galaxies in the system and H sub 0 = 100 km s(exp -1) Mpc(exp -1).
The feasibility of determining two-point, two-time turbulent velocity correlations based on extending closure models for one-point, one-time turbulence correlations is demonstrated. The procedure used is based on a spatial moment integral formulation of the governing equations using approximate, parameterized trial functions for the two-point, two-time velocity correlations. Solution of the equations is shown to give a set of anisotropic length scales and the separation-time-dependent decorrelation of the ensemble averaged turbulent velocities. The analysis is simplified using the assumption of homogeneous stationary turbulence and a constant shear, unidirectional mean flow. It is shown that the anisotropic behavior of measured turbulence correlations can be characterized through this technique.
Pressure fluctuations in a turbulent channel flow are investigated by analyzing a database obtained from a direct numerical simulation. Detailed statistics associated with the pressure fluctuations are presented. Characteristics associated with the rapid (linear) and slow (nonlinear) pressure are discussed. It is found that the slow pressure fluctuations are larger than the rapid pressure fluctuations throughout the channel except very near the wall, where they are about the same magnitude. This is contrary to the common belief that the nonlinear source terms are negligible compared to the linear source terms. Probability density distributions, power spectra, and two-point correlations are examined to reveal the characteristics of the pressure fluctuations. The global dependence of the pressure fluctuations and pressure-strain correlations are also examined by evaluating the integral associated with Green's function representations of them. In the wall region where the pressure-strain terms are large, most contributions to the pressure-strain terms are from the wall region (i.e., local), whereas away from the wall where the pressure-strain terms are small, contributions are global. Structures of instantaneous pressure and pressure gradients at the wall and the corresponding vorticity field are examined.
Pressure fluctuations in a turbulent channel flow are investigated by analyzing a database obtained from a direct numerical simulation. Detailed statistics associated with the pressure fluctuations are presented. Characteristics associated with the rapid (linear) and slow (nonlinear) pressure are discussed. It is found that the slow pressure fluctuations are larger than the rapid pressure fluctuations throughout the channel except very near the wall, where they are about the same magnitude. This is contrary to the common belief that the nonlinear source terms are negligible compared to the linear source terms. Probability density distributions, power spectra, and two-point correlations are examined to reveal the characteristics of the pressure fluctuations. The global dependence of the pressure fluctuations and pressure-strain correlations are also examined by evaluating the integral associated with Green's function representations of them. In the wall region where the pressure-strain terms are large, most contributions to the pressure-strain terms are from the wall region (i.e., local), whereas away from the wall where the pressure-strain terms are small, contributions are global. Structures of instantaneous pressure and pressure gradients at the wall and the corresponding vorticity field are examined.
Single point and two-point fluctuating static pressure measurements have been made in the initial mixing and transition regions in a co-annular circular jet. Space scales, correlation and time scales in the convected frame, and convection velocities have been determined for both broad band and narrow band frequency components. Analytical functionals describing the structural behavior of the turbulent pressure field have been developed from the data which are suitable for direct application in predicting the sound power level radiated from this turbulent field. Effects of both area and velocity ratios are incorporated in the model.
To examine how peculiar velocities can affect the two-, three-, and four-point redshift correlation functions, we evaluate volume-average correlations for configurations that emphasize and minimize redshift distortions for four different volume-limited samples from each of the CfA, SSRS, and IRAS redshift catalogs. We present the results as the correlation length r(sub 0) and power index gamma of the two-point correlations, bar-xi(sub 0) = (r(sub 0)/r)(exp gamma), and as the hierarchical amplitudes of the three- and four-point functions, S(sub 3) = bar-xi(sub 3)/bar-xi(exp 2)(sub 2) and S(sub 4) = bar-xi(sub 4)/bar-xi(exp 3)(sub 2). We find a characteristic distortion for bar-xi(sub 2), the slope gamma is flatter and the correlation length is larger in redshift space than in real space; that is, redshift distortions 'move' correlations from small to large scales. At the largest scales (up to 12 Mpc), the extra power in the redshift distribution is compatible with Omega(exp 4/7)/b approximately equal to 1. We estimate Omega(exp 4/7)/b to be 0.53 +/- 0.15, 1.10 +/- 0.16, and 0.84 +/- 0.45 for the CfA, SSRS, and IRAS catalogs. Higher order correlations bar-xi(sub 3) and bar-xi(sub 4) suffer similar redshift distortions but in such a way that, within the accuracy of our ananlysis, the normalized amplitudes S(sub 3) and S(sub 4) are insensitive to this effect. The hierarchical amplitudes S(sub 3) and S(sub 4) are constant as a function of scale between 1 and 12 Mpc and have similar values in all samples and catalogs, S(sub 3) approximately equal to 2 and S(sub 4) approximately equal to 6, despite the fact that bar-xi(sub 2), bar-xi(sub 3), and bar-xi(sub 4) differ from one sample to another by large factors (up to a factor of 4 in bar-xi(sub 2), 8 for bar-xi(sub 3), and 12 for bar-xi(sub 4)). The agreement between the independent estimations of S(sub 3) and S(sub 4) is remarkable given the different criteria in the selection of galaxies and also the difference in the resulting range of densities, luminosities, and locations between samples.
The problem of closure in turbulence in the case of two-point correlations resides in the existence of two unknowns E and W, the energy spectrum function and the transfer function, respectively, in the spectrum equation. In the case of weak turbulence, W is negligible. In case of higher correlations, closure can be effective by neglecting the inertia term in the highest order term used. Specifying a certain number of spectra at an initial time is also a way of getting around the closure problem. A simple case of turbulent shear flow is then considered, where two-point correlation equations are used and the velocity is broken into mean and fluctuating components. This yields a differential equation for the energy spectrum, the three terms of which are the energy spectrum, production term and dissipation term. They are plotted for a particular time. Similar analyses and comparisons with experiment are made for pipe and boundary layer flows.
The proper orthogonal decomposition is utilized to provide one mode estimates of various turbulence quantities in the near-wall region of the turbulent boundary layer. The experimental data of Herzog (1986) and the direct numerical simulation (DNS) results of Zang (1991) proved the two-point correlation tensor. The correlation tensor is then used as the kernel in the integral eigenvalue problem from which the proper orthogonal modes are obtained. The one mode estimates of the turbulent kinetic energy, dissipation, and triple moments all exhibit the proper asymtotic behavior near the wall. As a validation test, a model for the triple moments is developed and its near-wall behavior is studied. This information can then be used to obtain an estimate of the damping function near the wall, which is then compared to known existing behavior. Since this is an initial investigation, the streamwise and spanwise dependence is surpressed and only the wall normal direction (y+) behavior of various turbulence statistics near the wall is studied. Before examining one mode estimates of the triple moments and the dissipation (on which the model is based), various second order moments are examined. If these do not exhibit the proper behavior near the wall (as discussed by Speziale et al 1990) it does not seem reasonable to expect the one mode estimate of the triple moment and dissipation to have the right behavior. Only the results obtained using the experimental data of Herzog are discussed since the results from the simulations of Zang exhibit similar behavior as discussed above.
Measurements of the total energy, cross helicity, and magnetic helicity of the solar wind at 1, 2.8, and 5 AU are presented. These quantities are the three rugged invariants of three-dimensional ideal incompressible MHD turbulence theory. The theoretical technique for measuring the magnetic helicity from the matrix of two-point correlations is shown. The length scales characterizing the magnetic helicity are found to be equal to or greater than those which characterize the magnetic energy. The magnetic helicity typically lies at scales larger than the magnetic correlation length, consistent with the expectations of the inverse cascade and selective decay hypotheses of three-dimensional MHD turbulence. At smaller scales, the magnetic helicity oscillates in sign. Measurements of the cross helicity are not fully consistent with the usual interpretation in terms of outward propagating Alfvenic functions. Especially during the interval at 5 AU the cross helicity is found to oscillate in sign indicating fluctuations propagating both outward and inward.
The structure of pressure-pressure correlations at the interface of an incompressible steady-state turbulent flow with a rigid boundary was investigated. For the sake of completeness, the absolute value of the correlation between two random varying functions is herein defined as a number greater than or equal to zero and less than or equal to unity which is a measure of that fraction of one of the functions that 'follows' the second function (or vice versa). It was found that the soughtafter correlations can be determined by consideration of the high Re Navier-Stokes equation, but that the complexity of boundary layer turbulence, in particular the inhomogeneity perpendicular to the boundary and the anisotropy due to convective flow gradients, makes the structure of said correlations extremely difficult to assess. One of the earlier researchers in this field described the quantity under present consideration as 'a quantity which is beyond assessment.' Nonetheless, it was found that under some rather simplifying assumptions the determination of the required structure necessitates the formulation of the related structure of second order two-point correlations of turbulent velocity gradients, as well as third order two-point correlations of velocity gradients. The presence of these latter gradients is due to the nonlinearity in the turbulence ('turbulence self-interaction'). Both of these correlations are scaled, although not similarly, by factors dependent upon the magnitude of the convective flow, which can be modeled using a log law approximation. Fourth order correlations, although present, can be ignored, since they constitute 'higher order terms.' In a slightly more complex situation, it was found that convective flow gradients also have to be incorporated. At the moment, no definitive algebraic information peculiar to pressure-pressure correlations is available in the most highly idealized cases.
Upper limits on the angular correlation function, w(theta), of gamma-ray bursts are obtained which constrain the sources to be very nearby or very distant. The angular clustering properties of the sources in two recent gamma-ray burst catalogs (Golenetskii et al., 1986 and Atteia et al., 1987) have been derived by determining their two-point w(theta) values, and the correlations of both data sets are consistent with w(theta) = 0. It is shown that if the spatial correlation of gamma-ray bursts resembles that of galaxies or galactic clusters, present gamma-ray burst catalogs must be complete to distances in excess of 100 Mpc, while if gamma-ray bursts are associated with a Galactic disk population, then the upper limits on w(theta) constrain the sampling depth to less than two disk scale heights.