This article is a review and extension of several papers which presented a model for inertial range intermittency and anomalous scaling of velocity difference structure functions. The method of matched asymptotic expansions is used with the Navier–Stokes equation to derive a basic law for the instantaneous velocity difference between neighboring points in a turbulent flow. This is a power law with an undetermined exponent which must be specified statistically by making a physical hypothesis. (The simplest hypothesis, mean dissipation independent of Reynolds number makes the exponent be 1/3, the Kolmogorov 1941 result.) The resulting general theory is used to calculate structure functions, anomalous exponents and the probability density function of velocity differences and is extended to include the Reynolds number dependence of the instantaneous dissipation at a point and a theory for passive scalars. (A parameter in the instantaneous dissipation is adjusted to make the mean dissipation approximately independent of Reynolds number.)

The Navier-Stokes equations are expanded in asymptotic power series in a small parameter ϵ(=uτ∕Ue) which is determined as a function of Reynolds number by an asymptotic matching procedure. The present matched asymptotic expansion analysis differs from the more traditional approach by employing the unsteady Navier-Stokes equations instead of the unclosed Reynolds averaged equations. It is therefore not necessary to expand the Reynolds stress separately in the small parameter, and more importantly, the problem is mathematically well posed. The analysis is simpler, requires fewer assumptions, and gives information about all the velocity components. The main result of this analysis is an instantaneous log-law in the overlap region, of the form u+=κ−1ln(y+)+B, where the additive “constant” B is independent of y but depends on the outer scaled x,z,t variables. It is found that all the Reynolds stresses are constant in the overlap region. Estimates are made of the extent of the overlap region and the rate at which it is approached as Reynolds number tends to infinity.

The breakup of a liquid capillary filament is analyzed as a viscous potential flow near a stagnation point on the centerline of the filament towards which the surface collapses under the action of surface tension forces. The analysis given her is restricted to cases in which the neckdown is symmetric around the stagnation point. We find that the neck is of parabolic shape and its radius collapses to zero in a finite time; the curvature at the throat tends to zero much faster than the radius, leading ultimately to a microthread of nearly uniform radius. During the collapse the tensile stress due to viscosity increases in value until at a certain finite radius, which is about 1.5 microns for water in air, the stress in the throat passes into tension, presumably inducing cavitation there.

A model for inertial range intermittency and anomalous scaling of velocity structure functions is proposed. The Navier–Stokes equation is used to derive a basic Reynolds number dependent law for the instantaneous velocity difference between two points. This gives incomplete information about the dependence on scale and requires a statistical hypothesis in order to compute structure functions and other quantities. The specific assumptions made here relate the singular scaling exponent to the velocity amplitude. Anomalous exponents and Reynolds number dependent structure functions and velocity difference probability density functions are calculated which agree with the experiments.

: Stationary isotropic turbulence is often studied numerically by adding a forcing term to the Navier-Stokes equation. This is usually done for the purpose of achieving higher Reynolds number and longer statistics than is possible for isotropic decaying turbulence. It is generally accepted that forcing the Navier-Stokes equation at low wave number does not influence the small scale statistics of the flow provided that there is wide separation between the largest and smallest scales. It will be shown, however, that the spectral width of the forcing has a noticeable effect on inertial range statistics. A case will be made here for using a broader form of forcing in order to compare computed isotropic stationary turbulence with (decaying) grid turbulence. It is shown that using a forcing function which is directly proportional to the velocity has physical meaning and gives results which are closer to both homogeneous and non-homogeneous turbulence. Section 1 presents a four part series of motivations for linear forcing. Section 2 puts linear forcing to a numerical test with a pseudospectral computation.

Abstract : Figure 1 shows a snapshot of liquid fuel spray coming out of an injector nozzle in a realistic gas-turbine combustor. Here the spray atomization was simulated using a stochastic secondary breakup model (Apte et al. 2003a) with point-particle approximation for the droplets. Very close to the injector, it is observed that the spray density is large and the droplets cannot be treated as point-particles. The volume displaced by the liquid in this region is significant and can alter the gas-phase flow and spray evolution. In order to address this issue, one can compute the dense spray regime by an Eulerian- Eulerian technique using advanced interface tracking/level-set methods (Sussman et al. 1994; Tryggvason et al. 2001; Herrmann 2003). This, however, is computationally intensive and may not be viable in realistic complex configurations. We therefore plan to develop a methodology based on Eulerian-Lagrangian technique which will allow us to capture the essential features of primary atomization using models to capture interactions between the fluid and droplets and which can be directly applied to the standard atomization models used in practice. The numerical scheme for unstructured grids developed by Mahesh et al. (2003) for incompressible flows is modified to take into account the droplet volume fraction. The numerical framework is directly applicable to realistic combustor geometries. Our main objectives in this work are: * Develop a numerical formulation based on Eulerian-Lagrangian techniques with models for interaction terms between the fluid and particles to capture the Kelvin- Helmholtz type instabilities observed during primary atomization. * Validate this technique for various two-phase and particulate flows. * Assess its applicability to capture primary atomization of liquid jets in conjunction with secondary atomization models.

The Kolmogorov [Dokl. Akad. Nauk. SSSR 30, 299 (1941), hereafter K41] inertial range theory is derived from first principles by analysis of the Navier–Stokes equation using the method of matched asymptotic expansions without assuming isotropy or homogeneity and the Kolmogorov (K62) [J. Fluid Mech. 13, 82 (1962)] refined theory is analyzed. This paper is an extension of Lundgren [Phys. Fluids 14, 638 (2002)], in which the second- and third-order structure functions were determined from the isotropic Karman–Howarth [Proc. R. Soc. London, Ser. A 164, 192 (1938)] equation. The starting point for the present analysis is an equation for the difference in velocity between two points, one of which is a Lagrangian fluid point and the second, slaved to the first by a fixed separation r, is not Lagrangian. The velocity difference, so defined, satisfies the Navier–Stokes equation with spatial variable r. The analysis is carried out in two parts. In the first part the physical hypothesis is made that the mean dissipation is independent of viscosity as viscosity tends to zero, as assumed in K41. This means that the mean dissipation is finite as Reynolds number tends to infinity and leads to the K41 inertial range results. In the second part this dissipation assumption is relaxed in an attempt to duplicate the K62 theory. While the K62 structure is obtained, there are restrictions, resulting from the analysis which shows that there can be no inertial range intermittency as Reynolds number tends to infinity, and therefore the mean dissipation has to be finite as Reynolds number tends to infinity, as assumed in part one. Reynolds number-dependent corrections to the K41 results are obtained in the form of compensating functions of r/λ, which tend to zero slowly like Rλ−2/3 as Rλ→∞.

We consider the dynamics of axial velocity and of scalar transport in the stretched-spiral vortex model of turbulent fine scales. A large-time asymptotic solution to the scalar advection-diffusion equation, with an azimuthal swirling velocity field provided by the stretched spiral vortex, is used together with appropriate stretching transformations to determine the evolution of both the axial velocity and a passive scalar. This allows calculation of the shell-integrated three-dimensional spectra of these quantities for the spiral-vortex flow. The dominant term in the velocity (energy) spectrum contributed by the axial velocity is found to be produced by the stirring of the initial distribution of axial velocity by the axisymmetric component of the azimuthal velocity. This gives a k−7/3 spectrum at large wave numbers, compared to the k−5/3 component for the azimuthal velocity itself. The spectrum of a passive scalar being mixed by the vortex velocity field is the sum of two power laws. The first is a k−1 Batchelor spectrum for wave numbers up to the inverse Batchelor scale. This is produced by the axisymmetric component of the axial vorticity but is independent of the detailed radial velocity profile. The second is a k−5/3 Obukov–Corrsin spectrum for wave numbers less than the inverse Kolmogorov scale. This is generated by the nonaxisymmetric axial vorticity and depends on initial correlations between this vorticity and the initial scalar field. The one-dimensional scalar spectrum for the composite model is in satisfactory agreement with experimental measurement.

The Kolmogorov two-thirds law is derived for large Reynolds number isotropic turbulence by the method of matched asymptotic expansions. Inner and outer variables are derived from the Karman–Howarth equation by using the von Karman self-preservation hypothesis. Matching the resulting large Reynolds number asymptotic expansions yields the Kolmogorov law. The Kolmogorov similarity hypotheses are not assumed; only the Navier–Stokes equation is employed and the assumption that dissipation is finite. This indicates that the Kolmogorov results are a direct consequence of the Navier–Stokes equations.

This paper examines the aerodynamic characteristics of a freight pipeline system in which freight capsules are individually propelled by electrical motors. The fundamental difference between this system and the more extensively studied pneumatic pipeline is the different role played by aerodynamic forces. In a driven system the propelled capsules are resisted by aerodynamic forces and, in reaction, pump air through the tube. In contrast, in a pneumatic system external blowers pump air through the tubes and this provides thrust for the capsules. An incompressible transient analysis is developed to study the aerodynamics of multiple capsules in a cross linked two-bore pipeline. An aerodynamic friction coefficient is used as a cost parameter to compare the effects of capsule blockage and headway and to assess the merits of adits and vents. We conclude that optimum efficiency for off-design operation is obtained with long platoons of capsules in vented or adit connected tubes.

Stability and transition to turbulence are studied in a simple incompressible two-dimensional bounded swirling flow with a rectangular planform – a vortex in a box. This flow is unstable to three-dimensional disturbances. The instability takes the form of counter-rotating swirls perpendicular to the axis which bend the vortex into a periodic wave. As these swirls grow in amplitude the primary vorticity is compressed into thin vortex layers. These develop secondary instabilities which roll up into vortex tubes. In this way the flow attains a turbulent state which is populated by intense elongated vortex tubes and weaker vortex layers which spiral around them. The flow was computed at two Reynolds numbers by spectral methods with up to 2563 resolution. At the higher Reynolds number broad three-dimensional shell-averaged energy spectra are found with nearly a decade of Kolmogorov k−5/3 law and small-scale isotropy.

An isolated dynamic microburst is modelled in the laboratory by releasing an elevated volume of salt water solution which is slightly heavier than ambient fresh water, allowing it to fall onto a horizontal plate and develop into a turbulent microburst vortex ring. This experimental model exhibits many of the features of naturally occurring microbursts which are known to be a hazard to aviation. Flow visualization methods are used to observe the event and a high resolution video camera is used to record it. In the current investigation hot film velocity measurements are made at a number of radial positions near the ground. Realistic estimates of the wind field in real microbursts may be obtained from these measurements by means of a previously developed scaling law. Flow visualization and velocity measurements support a description of the microburst structure in its transition from a vortex dominated flow to a radial gravity current in its late stages.