Outer Length Scale Research Articles

Reconstructing surface wave profiles from reflected acoustic pulses

Surface wave shapes are determined by analyzing underwater reflected acoustic signals. The acoustic signals (of nominal frequency 200 kHz) are forward scattered from the underside of surface waves that are generated in a wave tank and scaled to model smooth ocean swell. An inverse processing algorithm is designed and implemented to reconstruct the surface displacement profiles of the waves over one complete period. The inverse processing uses the surface scattered pulses collected at the receiver, an initial wave profile (two are considered), and a broadband forward scattering model based on Kirchhoff's diffraction formula to iteratively adjust the surface until it is considered optimized or reconstructed. Two physical length scales over which information can be known about the surface are confirmed. An outer length scale, the Fresnel zone surrounding each specular reflection point, is the only region where optimized surfaces resulting from each initial profile converge within a resolution set by the inner length scale, a quarter-wavelength of the acoustic pulse. The statistical confidence of each optimized surface is also highest within a Fresnel zone. Future design considerations are suggested such as an array of receivers that increases the region of surface reconstruction by a factor of 2 to 3.

Stokes flow singularity at the junction between impermeable and porous walls

AbstractFor two-dimensional, creeping flow in a half-plane, we consider the singularity that arises at an abrupt transition in permeability from zero to a finite value along the wall, where the pressure is coupled to the seepage flux by Darcy’s law. This problem represents the junction between the impermeable wall of the inflow section and the porous membrane further downstream in a spiral-wound desalination module. On a macroscopic, outer length scale the singularity appears like a jump discontinuity in normal velocity, characterized by a non-integrable$1/ r$divergence of the pressure. This far-field solution is imposed as the boundary condition along a semicircular arc of dimensionless radius 30 (referred to the microscopic, inner length scale). A preliminary numerical solution (using a least-squares variant of the method of fundamental solutions) indicates a continuous normal velocity along the wall coupled with a weaker$1/ \sqrt{r} $singularity in the pressure. However, inconsistencies in the numerically imposed outer boundary condition indicate a very slow radial decay. We undertake asymptotic analysis to: (i) understand the radial decay behaviour; and (ii) find a more accurate far-field solution to impose as the outer boundary condition. Similarity solutions (involving a stream function that varies like some power of$r$) are insufficient to satisfy all boundary conditions along the wall, so we generalize these by introducing linear and quadratic terms in$\log r$. By iterating on the wall boundary conditions (analogous to the method of reflections), the outer asymptotic series is developed through second order. We then use a hybrid computational scheme in which the numerics are iteratively patched to the outer asymptotics, thereby determining two free coefficients in the latter. We also derive an inner asymptotic series and fit its free coefficient to the numerics at$r= 0. 01$. This enables evaluation of the singular flow field in the limit as$r\ensuremath{\rightarrow} 0$. Finally, a uniformly valid fit is obtained with analytical formulas. The singular flow field for a solid–porous abutment and the general Stokes flow solutions obtained in the asymptotic analysis are programmed in Fortran for future use as local basis functions in computational schemes. Numerics are required for the intermediate-$r$regime because the inner and outer asymptotic expansions do not extend far enough toward each other to enable rigorous asymptotic matching. The logarithmic correction terms explain why the leading far-field solution (used in the preliminary numerics) was insufficient even at very large distances.

Reconstructing surface wave profiles from reflected acoustic pulses

Surface wave shapes are determined by analyzing underwater reflected acoustic signals. The acoustic signals (of nominal frequency 200 kHz) are forward scattered from the underside of surface waves that are generated in a wave tank and scaled to model smooth ocean swell. An inverse processing algorithm is designed and implemented to reconstruct the surface displacement profiles of the waves over one complete period. The inverse processing uses the surface scattered pulses collected at the receiver, an initial wave profile (two are considered), and a broadband forward scattering model based on Kirchhoff’s diffraction formula to iteratively adjust the surface until it is considered optimized or reconstructed. Two physical length scales over which information can be known about the surface are confirmed. An outer length scale, the Fresnel zone surrounding each specular reflection point, is the only region where optimized surfaces resulting from each initial profile converge within a resolution set by the inner length scale, a quarter-wavelength of the acoustic pulse. The statistical confidence of each optimized surface is also highest within a Fresnel zone. Future design considerations are suggested such as an array of receivers that increases the region of surface reconstruction by a factor of 2 to 3.

Multi-scale needle-network model of complex dendritic microstructure formation

We present a novel multi-scale Dendritic Needle Network (DNN) approach in order to model well-developed highly-ramified dendritic microstructures on the coarser scale of several crystal grains while retaining a faithful quantitative description of the transient dynamics of individual dendritic branches. This approach is intended to bridge the scale gap between phase-field and cellular automaton methods. The dynamics of each needle-like branch, characterized by its tip velocity V and radius ρ, is fixed by two conditions: (i) on the inner tip scale, a standard microscopic solvability condition relates ρ2V to the strength of surface tension anisotropy, and (ii) on the outer diffusion length scale, a flux balance condition relates the product ρV2 to a flux intensity factor extracted from a contour integral analogous to the J-integral of fracture mechanics. The method is tested for low supersaturation and reproduces the analytical solutions for both early stage and steady state growth dynamics. The results are directly compared with a quantitative phase-field simulation for an experimentally relevant supersaturation. We present as well an illustrative simulation for highly branched polycrystalline growth. This model should permit to investigate the macroscale grain evolution through the dynamics of individual primary dendrites and higher-order branches, controlled by both the intragrain history-dependent selection and the intergrain dendrite interactions.

Antitruncated stellar light profiles in the outer regions of STAGES spiral galaxies: bulge or disc related?

We present a comparison of azimuthally averaged radial surface brightness mu(r) profiles and analytical bulge-disc decompositions (de Vaucouleurs, r^(1/4) bulge plus exponential disc) for spiral galaxies using Hubble Space Telescope/Advanced Camera for Surveys V-band imaging from the Space Telescope A901/2 Galaxy Evolution Survey (STAGES). In the established classification scheme, antitruncated mu(r) profiles (Type III) have a broken exponential disc with a shallower region beyond the break radius r_brk. The excess light at large radii (r > r_brk) can either be caused by an outer exponential disc (Type III-d) or an extended spheroidal component (Type III-s). Using our comparisons, we determine the contribution of bulge light at r > r_brk for a large sample of 78 (barred/unbarred, Sa-Sd) spiral galaxies with outer disc antitruncations (mu_brk > 24 mag arcsec^-2). In the majority of cases (~85 per cent), evidence indicates that excess light at r > r_brk is related to an outer shallow disc (Type III-d). Here, the contribution of bulge light at r > r_brk is either negligible (~70 per cent) or too little to explain the antitruncation (~15 per cent). However in the latter cases, bulge light can affect the measured disc properties (e.g. mu_brk, outer scalelength). In the remaining cases (~15 per cent), light at r > r_brk is dominated by the bulge (Type III-s). Here, for most cases the bulge profile dominates at all radii and only occasionally (3 galaxies, ~5 per cent) extends beyond that of a dominant disc and explains the excess light at r > r_brk. We thus conclude that in the vast majority of cases antitruncated outer discs cannot be explained by bulge light and thus remain a pure disc phenomenon.

Near-wall turbulent characteristics along very long thin circular cylinders

Understanding the salient physics within the turbulent boundary layer of towed thin cylinders is paramount to the Navy sonar array communities. However, the required long array length to achieve wide acoustic aperture creates unique and consistent flow characteristics that suggest simplified tangential forcing expressions suitable for design purposes. One well-known fact is that the majority of the array surface experiences very thick turbulent boundary layers (TBL) and large Reynolds numbers. The resultant statistics are most commonly dependent on the inner and outer length scales. Herein, we resolve the near-wall TBL structure under those flow conditions by large-eddy simulation. The turbulent mean-flow statistics showed near-wall consistency using only inner scaling. But both inner and outer variables were found necessary to properly scale the turbulent fluctuations. An expression for the tangential wall-friction coefficient ( C t ) indicates two distinct flow regimes as characterized by the near-wall turbulent flow structure. The respective parameters appear independent of the outer length scale. Thickening (or thinning) the cylinder near their common threshold (defined by a radius-based Reynolds number) transitioned the turbulent character between the two regimes.

Estimates of Cn2 from Numerical Weather Prediction Model Output and Comparison with Thermosonde Data

Abstract Area-averaged estimates of Cn2 from high-resolution numerical weather prediction (NWP) model output are produced from local estimates of the spatial structure functions of refractive index with corrections for the inherent smoothing and filtering effects of the underlying NWP model. The key assumptions are the existence of a universal statistical description of small-scale turbulence and a locally universal spatial filter for the NWP model variables. Under these assumptions, spatial structure functions of the NWP model variables can be related to the structure functions of the atmospheric variables and extended to the smaller underresolved scales. The shape of the universal spatial filter is determined by comparisons of model structure functions with the climatological spatial structure function determined from an archive of aircraft data collected in the upper troposphere and lower stratosphere. This method of computing Cn2 has an important advantage over more traditional methods that are based on vertical differences because the structure function–based estimates avoid reference to the turbulence outer length scale. To evaluate the technique, NWP model–derived structure-function estimates of Cn2 are compared with nighttime profiles of Cn2 derived from temperature structure-function sensors attached to a rawinsonde (thermosonde) near Holloman Air Force Base in the United States.

Multi-wavelength characterization of the outskirts of spiral galaxies

AbstractWe have explored late-type spiral galaxies with truncated radial surface brightness profiles. Based on the study of optical color profiles, Bakos et al. (2008) suggested that truncated (TYPE II) galaxies have a generally older stellar population in their outer regions: one observable consequence of this would be a strong dependence of the structural parameters of the outer disk on the observing wavelength. To corroborate this result, we obtained surface brightness profiles using data from GALEX(UV), SDSS(optical), UKIDSS(NIR) and SPITZER-IRAC(IR). We have characterized the behaviour of the outer disk by obtaining the ratio of the inner and outer scale-lengths. Here we show the example of NGC0450. Our results suggest that there is an existing general trend of the scale-length ratio: from bluer to redder bands the scale-length ratio decreases, which is in accordance with the idea of the old smooth stellar disk in the outer disk.

Turbulent wall pressure fluctuation measurements on a towed model at high Reynolds numbers

Turbulent wall pressure fluctuation measurements were made in water on a towed model of length 129.8 (m) and diameter 3.8 (cm) for steady speeds from 6.2 (m/s) to 15.5 (m/s). The drag on the model was measured with a strut mounted load cell which provided estimates of the momentum thickness and friction velocity. Momentum thickness Reynolds numbers Reθ varied from 4.8 × 105 to 1.1 × 106. The ratio of momentum thickness to viscous length scale is significantly greater than for flat plate cases at comparable Reθ. The effectiveness of inner and outer velocity and length scales for collapsing the pressure spectra are discussed. The wavenumber–frequency spectra show a convective ridge at higher frequencies similar to flat plate boundary layers. At low frequencies, energy broad in wavenumber extends outside the convective ridge and acoustic cone, with no characteristic wave speed. Wall pressure cross-spectral levels scaled with similarity variables are shown to increase with increasing tow speed, and to follow decay constants consistent with flat plate cases. The convection velocities also display features similar to flat plate cases.

Study of the Log-Layer Structure in Wall Turbulence Over a Very Large Range of Reynolds Number

Studies of the logarithmic layer structure in turbulent boundary layers are presented that span three orders of magnitude change in Reynolds number. The experiments considered used two separate laboratory scale facilities, as well as the atmospheric surface layer at the SLTEST facility in Utah. Several experimental techniques were used in order to probe the three-dimensional nature of the flow structures. The main focus is on two-point correlation statistics at a given z/δ, which are found to agree well over all Reynolds numbers when scaled with an outer length-scale. Large-scale coherence recently noted in the logarithmic region of laboratory-scale boundary layers is also found to be present in the atmospheric surface layer flow. Recent findings regarding the influence of these large scale motions on the near-wall region are also presented.

A multiple-scales approach to crack-front waves

Perturbation of a steadily propagating crack with a straight edge is solved using the method of matched asymptotic expansions (MAE). This provides a simplified analysis in which the inner and outer solutions are governed by distinct mechanics. The inner solution contains the explicit perturbation and is governed by a quasi-static equation. The outer solution determines the radiation of energy away from the tip, and requires solving dynamic equations in the unperturbed configuration. The outer and inner expansions are matched via the small parameter ϵ = L/l defined by the disparate length scales: the crack perturbation length L and the outer length scale l associated with the loading. The method is illustrated for a scalar crack model and then applied to the elastodynamic mode I problem. The crack-front wave-dispersion relation is found by requiring that the energy release rate is unaltered under perturbation and dispersive properties of the crack-front wave speed are described for the first time. The example problems considered demonstrate the potential of MAE for moving-boundary-value problems with multiple scales.

Observations of upper ocean boundary layer dynamics in the marginal ice zone

Vertical profiles of turbulent kinetic energy dissipation and small‐scale hydrography were collected in the upper ocean boundary of both ice‐covered and ice‐free stations in the marginal ice zone of the Barents Sea in spring 2005. Together with shipboard wind measurements and current profiles, the mixed and mixing layer dynamics are studied. During the survey, shear production by the stress at the surface or under the ice dominated. Large upward turbulent heat fluxes ∼300–500 W m−2 were calculated for the mixing layers overlying the warm Atlantic Water which compared well with those obtained from an independent parameterization. At the top of the pycnocline, corresponding heat fluxes were 10–20 W m−2. Significant stabilizing buoyancy fluxes were estimated for ice‐drift stations owing to the melting of ice in response to the large heat fluxes. The mean dissipation profile in the ice‐free reference station agreed with the constant‐stress wall layer scaling within 30%. On the contrary, the observed dissipation profiles were enhanced in the upper half of under‐ice mixing layer or within 2.5 times the assessed pressure‐ridge keel depth. Deeper in the mixing layer the profiles relaxed toward the wall scaling. The variability of dissipation in the mixed layer was better captured by the shear production profile when local friction speed was used together with a mixing length profile modified by buoyancy fluxes. Significant correlations were found between dissipation integrated over the mixing layer and work done by stress under the ice and the wind work at 10 m height. The low correlation between the mixed layer depth and length scale for neutral conditions significantly increased when the reduction in the mixing length due to buoyancy fluxes was accounted for. The mixing depth is observed to be strongly correlated with the outer neutral planetary length scale.

Reynolds-Stress Enhancement Associated with a Short Fetch of Roughness in Wall Turbulence

DOI: 10.2514/1.22357 Particle-image velocimetry experiments are performed to study the response of smooth-wall turbulent channel flow to a short fetch of roughness (10 outer length scales long). The roughness studied herein is replicated from a surface scan of a damaged turbine blade and contains both large- and small-scale surface defects attributable to pitting, deposition, and spallation. Quadrant analysis is used to investigate the characteristics of Reynolds-stressproducing events within the internal layer that develops over the roughness. The mean Reynolds stress is dramatically increased in the presence of the roughness as compared to the smooth-wall baseline owing to an increasednumberofextremelyintenseejectionsandsweeps.Incontrast,inwardandoutwardinteractions,aswellas relativelyweakejectionandsweepevents,arefoundtobeinsensitivetothesurfaceconditions.Whilestressandspace fractions are found to be insensitive to the surface topology when all Reynolds-stress-producing events are included, stressandspacefractionsforonlythemostintenseejectionandsweepeventsvarysignificantlywiththelocalsurface topology. Nomenclature H = hyperbolic hole size h = channel half-height, m I = indicator function k = roughness height, m M = total number of streamwise gridpoints in an ensemble N = space fraction Re� = friction Reynolds number S = stress fraction u = fluctuating streamwise velocity, m=s

Correlation functions of temperature and velocity fluctuations in a turbulent atmosphere

von Kármán spectra of temperature and velocity fluctuations are widely used in theories of turbulence and wave propagation in random media, including sound propagation through a turbulent atmosphere. They are probably the simplest generalization of the Kolmogorov spectrum of homogeneous and isotropic turbulence which accounts for the outer scale of turbulence. However, the correlation functions of temperature and velocity fluctuations corresponding to the von Kármán spectra are proportional to the modified Bessel functions and are rather involved. In this paper, we propose to use much simpler correlation functions of temperature and longitudinal velocity fluctuations whose dependence on the distance R between observation points is given by exp(−R2/3/L2/3), where L is the outer length scale. It is shown that the spectra of the proposed correlation functions coincide with the Kolmogorov spectrum in the inertial subrange and are bounded in the energy subrange. For some problems, the proposed correlation functions can simplify analysis of sound propagation through a turbulent atmosphere. Examples of the use of these correlation functions in acoustic tomography of the atmosphere and calculations of the coherence function of a sound field are presented. [Work supported by ARO, Grant DAAD19-03-1-0104.]

Phenomenology treatment of magnetohydrodynamic turbulence with nonequipartition and anisotropy

Magnetohydrodynamics (MHD) turbulence theory, often employed satisfactorily in astrophysical applications, has often focused on parameter ranges that imply nearly equal values of kinetic and magnetic energies and length scales. However, MHD flow may have disparity magnetic Prandtl number, dissimilar kinetic and magnetic Reynolds number, different kinetic and magnetic outer length scales, and strong anisotropy. Here a phenomenology for such “nonequipartitioned” MHD flow is discussed. Two conditions are proposed for a MHD flow to transition to strong turbulent flow, which are extensions of (i) Taylor’s constant flux in an inertial range and (ii) Kolmogorov’s scale separation between the large and small scale boundaries of an inertial range. For this analysis, the detailed information on turbulence structure is not needed. These two conditions for MHD transition are expected to provide consistent predictions and should be applicable to anisotropic MHD flows, after the length scales are replaced by their corresponding perpendicular components. Second, it is stressed that the dynamics and anisotropy of MHD fluctuations are controlled by the relative strength between the straining effects between eddies of similar size and the sweeping action by the large eddies, or propagation effect of the large-scale magnetic fields, on the small scales, and analysis of this balance, in principle, also requires consideration of nonequipartition effects.

Refractivity structure constant and length scales from mm-wave propagation in the stably stratified troposphere

The refractivity structure constant C n 2 and length scales of turbulence are measured at mm-wavelengths with line-of-sight propagation experiments at an average height of 200 m above ground at Bern (Switzerland) and El Leoncito (Argentina) in the stably stratified troposphere. C n 2 at mm-wavelengths is of raised importance for scientific and commercial applications, as this frequency band has become easier accessible in recent years. At Bern, C n 2 varied predominantly between 10 - 14 and 10 - 13 m - 2 / 3 , whereas at the astronomical site of El Leoncito, with its desert-like conditions, it was approximately ten times less. Simultaneously measured potential refractivity gradients M and outer scale lengths of turbulence (proportional to integral scale) L 0 v and L 0 h , in vertical and horizontal directions respectively, has offered opportunities for the determination of the Ozmidov scale L R (proportional to Tatarskii's L 0 ) as well as the overturning turbulent length scale L t . This has been hardly done in the past, and probably never so far with spatially averaged length scales using remote measurement techniques. For weak stratification we found that L R is approximately one order of magnitude larger than L t and the latter is again one order of magnitude larger than L 0 v . During strongly stratified situations L R ≈ L t ≳ L 0 v which is in good agreement with other results from in situ and laboratory measurements. Finally, we empirically found that the turbulence anisotropy L 0 h / L 0 v is roughly proportional to the Richardson number with a proportionality factor of 30.

On the Generation, Propagation, and Reflection of Alfven Waves from the Solar Photosphere to the Distant Heliosphere

We present a comprehensive model of the global properties of Alfven waves in the solar atmosphere and fast solar wind. Linear non-WKB wave transport equations are solved from the photosphere to 4 AU, and for wave periods ranging from 3 seconds to 3 days. We derive a radially varying power spectrum of kinetic and magnetic energy fluctuations for waves propagating in both directions along a superradially expanding magnetic flux tube. This work differs from previous models in 3 major ways. (1) In the chromosphere and low corona, the successive merging of flux tubes on granular and supergranular scales is described using a 2D magnetostatic model of a network element. Below a critical merging height the waves are modeled as thin-tube kink modes, and we assume that all of the kink-mode wave energy is transformed into volume-filling Alfven waves above the merging height. (2) The frequency spectrum of horizontal motions is specified only at the photosphere based on prior analyses of G-band bright point kinematics. Everywhere else the amplitudes of outward and inward propagating waves are computed with no free parameters. We find that the wave amplitudes in the corona agree well with off-limb nonthermal line widths. (3) Nonlinear turbulent damping is applied to the results of the linear model using a phenomenological loss term. A single choice for the normalization of the turbulent outer-scale length produces both the right amount of damping at large distances (to agree with in situ measurements) and the right amount of heating in the extended corona (to agree with empirical wind acceleration models). In the corona, the modeled heating rate differs by more than an order of magnitude from a rate based on isotropic Kolmogorov turbulence.

Vortex motion in a weak background shear flow

A point vortex is introduced into a weak background vorticity gradient at finite Reynolds number. As the vortex spreads viscously so the background vorticity becomes wrapped around it, leading to enhanced diffusion of vorticity, but also giving a feedback on the vortex and causing it to move. This is investigated in the linear approximation, using a similarity solution for the advection of weak vorticity around the vortex, at finite and infinite Reynolds number. A logarithmic divergence in the far field requires the introduction of an outer length scale L and asymptotic matching. In this way results are obtained for the motion of a vortex in a weak vorticity field modulated on the large scale L and these are confirmed by means of numerical simulations.

Measurements and scaling of wall shear stress fluctuations

Measurements of velocity and wall shear stress fluctuations were made in an external turbulent boundary layer developed over a towed surface-piercing flat plate. An array of eight flush-mounted wall shear stress sensors was used to compute the space-time correlation function. A methodology for in situ calibration of the sensors for ship hydrodynamic applications is presented. The intensity of the wall shear stress fluctuations, τ rms/τ avg was measured as 0.25 and 0.36 for R θ =3,150 and 2,160 respectively. The probability density is shown to exhibit positive skewness, and lack of flow reversals at the wall. Correlations between velocity and wall shear stress fluctuations are shown to collapse with outer boundary layer length and velocity scales, verifying the existence of large-scale coherent structures which convect and decay along the wall at an angle of inclination varying from 10 to 13° over the range of Reynolds numbers investigated. The wall shear stress convection velocity determined from narrow band correlation measurements is shown to scale with outer variables. The space-time correlation of the wall shear is shown to exhibit a well-defined convective ridge, and to decay 80% over approximately 3δ for R θ =3,150.

Non-Kolmogorov features of differential image motion restored from the Multichannel Astrometric Photometer data

In this paper we present a new restoration of atmospheric turbulent parameters using published data on image motion observations with the Multichannel Astrometric Photometer. These data have been previously analyzed by Han (1989) whose estimate of the power of the phase structure function α = 1.64 was close to the 5/3 value expected for the Kolmogorov atmospheric turbulence. Investigating the temporal image motion spectrum, we show, however, that the experimental data do not follow predictions of the Kolmogorov model. The best fitting of the observed spectrum is achieved with α = 1.9−2.0, which is typical of non-classic phase distortions. The outer scale of turbulence was found to be 1800m or longer. Expressions for the non-Kolmogorov temporal spectrum of differential image motion allowing for a finite outer scale length are given.