Velocity Expansion Research Articles

Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers

Experiments were done on sink flow turbulent boundary layers over a wide range of streamwise pressure gradients in order to investigate the effects on the mean velocity profiles. Measurements revealed the existence of non-universal logarithmic laws, in both inner and defect coordinates, even when the mean velocity descriptions departed strongly from the universal logarithmic law (with universal values of the Kármán constant and the inner law intercept). Systematic dependences of slope and intercepts for inner and outer logarithmic laws on the strength of the pressure gradient were observed. A theory based on the method of matched asymptotic expansions was developed in order to explain the experimentally observed variations of log-law constants with the non-dimensional pressure gradient parameter (Δp=(ν/ρU3τ)dp/dx). Towards this end, the system of partial differential equations governing the mean flow was reduced to inner and outer ordinary differential equations in self-preserving form, valid for sink flow conditions. Asymptotic matching of the inner and outer mean velocity expansions, extended to higher orders, clearly revealed the dependence of slope and intercepts on pressure gradient in the logarithmic laws.

Wavelet approximation of earthquake strong ground motion-goodness of fit for a database in terms of predicting nonlinear structural response

Reduced dimensionality representation of strong ground motion records as a superposition of a relatively small number of pulses is studied. Such representation is obtained by the expansion of velocity in orthogonal wavelet series using the Fast Wavelet Transform, and approximation by only the largest energy terms in the series. The Coiflet 5 wavelet family is used, which is orthogonal, smooth and nearly symmetric. The goodness of the approximation is examined for the EQUINFOS for USA Part I database, as representative of a large variety of strong motion records (it consists of 494 three-component records from 106 earthquakes recorded in the western US between 1933 and 1984). The goodness of fit is measured in terms of closeness of predicting several input and output characteristics of a nonlinear oscillator representing a structure: (i) energy of the input ground motion (proportional to integral of velocity squared), (ii) the peak input power, and (iii) the time of collapse of a bi-linear oscillator (considering also collapse due to dynamic instability). The results show very high degree of correlation of these characteristics as estimated from the actual record and from its approximations, even for small number of pulses (relative to the number that would represent the ground velocity exactly). Such reduced representation of strong ground motion is useful for extracting such pulses from strong motion records to study their nature, and for development of new algorithms for the synthesis of artificial earthquake strong motion records.

Stress singularities and the formation of birefringent strands in stagnation flows of dilute polymer solutions

We consider stagnation point flow away from a wall for creeping flow of dilute polymer solutions. For a simplified flow geometry, we explicitly show that a narrow region of strong polymer extension (a birefringent strand) forms downstream of the stagnation point in the UCM model and extensions, like the FENE-P model. These strands are associated with the existence of an essential singularity in the stresses, which is induced by the fact that the stagnation point makes the convective term in the constitutive equation into a singular point. We argue that the mechanism is quite general, so that all flows that have a separatrix going away from the stagnation point exhibit some singular behaviour. These findings are the counterpart for wall stagnation points of the recently discovered singular behaviour in purely elongational flows: the underlying mechanism is the same while the different nature of the singular stress behaviour reflects the different form of the velocity expansion close to the stagnation point.

Galactic rotation curve and the effect of density waves from data on young objects

Based on currently available data on the three-dimensional field of space velocities of young (<50 Myr) open star clusters and the radial velocities of HI clouds and star-forming (HII) regions, we have found the Galactic rotation curve in the range of Galactocentric distances 3 kpc <R<12 kpc using the first six terms of the Taylor expansion of the angular velocity of Galactic rotation in Bottlinger's equations. The Oort constants found are A= 15.5+-0.3 km/s/kpc and B=-12.2+-0.7 km/s/kpc. We have established that the centroid of the sample moves relative to the local standard of rest along the Galactic Y axis with a velocity of -6.2+-0.8 km/s. A Fourier spectral analysis of the velocity residuals from the derived rotation curve attributable to density waves reveals three dominant peaks with wavelengths of 2.5, 1.4, and 0.9 kpc and amplitudes of 4.7, 2.6, and 3.6 km/s, respectively. These have allowed us to estimate the distances between the density wave peaks, 1.9, 2.4, and 3.2 kpc as R increases, in agreement with the description of the density weave as a logarithmic spiral. The amplitude of the density wave perturbations is largest in the inner part of the Galaxy, about 9 km/s, and decreases to approximately 1 km/s in its outer part. A spectral analysis of the radial velocities of young open star clusters has confirmed the presence of periodic perturbations with an amplitude of 5.9+-1.1 km/s and a wavelength 1.7+-0.5 kpc. It shows that the phase of the Sun in the density wave is close to -90 degrees and the Sun is located in the interarm space near the outer edge of the Carina-Sagittarius arm.

Evaluation of performance in some ultrasonic procedures for non-invasive thermal estimation into hyperthermia phantoms

To dispose of a precise and non-invasive temperature measurement over the treated area becomes very important during hyperthermia treatments, in order to make possible an optimization of their healing effects. A performance analysis of some previously reported ultrasonic techniques, which were proposed for non-invasive temperature estimation, is made in this work over phantoms mimicking human tissues. A first technique is based on discrete scattering modeling for tissue characterization and spectral analysis of frequency distributions related to average scatters spacing [R. Seip &amp; E.S. Ebbini]. Other technique uses certain relations between tissue temperature changes and time-shifts in echoes due to thermally induced sound velocity changes and expansions [R. Maass-Moreno &amp; C.A. Damianou]. And finally, the third procedure analyzed here, is a recently proposed alternative, based on phase demodulation processing, to estimate the indirect effects of echoes time-shifts in the phase domain [M. Vázquez, A. Ramos, et al.]. The three options are analyzed for the same simulated &amp; measured multi-pulse echo patterns, looking for detecting possible advantages and inconveniences in each case. Finally, some improvement paths over the basic estimation methods are also explored.

Analysis of time-resolved laser plasma ablation using an imaging spectra technique

Pulsed laser deposition (PLD) is extensively employed for the growth of thin films. The laser-material interaction involves complex processes of heating, melting, vaporization, ejection of atoms, ions and molecules, shock waves, plasma initiation, expansion and deposition onto a substrate. The understanding of the spatial and temporal distribution of a plasma parameters in a laser-produced plasma is important to the control of thin film growth process. In this work we have studied the dynamics of laser ablated graphitic carbon plasma expanding into vacuum using a spectroscopic imaging suitable as an in situ & automated diagnostic sampling technique for PLD. Time-resolved spectra, that were also spatially resolved in one dimension along the axis of plasma expansion, were obtained using a time-gated intensified charge-coupled device (ICCD) coupled to a stigmatic Czerny-Turner spectrograph. Plasma parameters such as electron density, temperature and plume velocity expansion were extracted directly from the analysis of the C II (2s 2 3d-2s 2 4f) transition.

Simple and accurate scheme for fluid velocity interpolation for Eulerian–Lagrangian computation of dispersed flows in 3D curvilinear grids

Particle tracking in turbulent flows in complex domains requires accurate interpolation of the fluid velocity field. If grids are non-orthogonal and curvilinear, the most accurate available interpolation methods fail. We propose an accurate interpolation scheme based on Taylor series expansion of the local fluid velocity about the grid point nearest to the desired location. The scheme is best suited for curvilinear grids with non-orthogonal computational cells. We present the scheme with second-order accuracy, yet the order of accuracy of the method can be adapted to that of the Navier–Stokes solver. An application to particle dispersion in a turbulent wavy channel is presented, for which the scheme is tested against standard linear interpolation. Results show that significant discrepancies can arise in the particle displacement produced by the two schemes, particularly in the near-wall region which is often discretized with highly-distorted computational cells.

Flow Downstream of a Cluster of Nine Jets

This work is an experimental investigation of the flow downstream of a low emission nozzle. The nozzle is a 3×3 square matrix of nine small swirling air jets, has a design swirl number of 0.8, and operates at a Reynolds number of 40,000. Particle image velocimetry (PIV) was used to map the velocity field under nonburning and atmospheric conditions for the first 18 jet diameters downstream of the nozzle exit plane. Seeding was liquid injected into the air stream and drops were sized to filter out those larger than 3×3pixels. The results showed that the cluster blends into a single jet-like flow 12 jet diameters downstream with the axial component of the velocity displaying self-similar properties. Lateral jet interaction slows the decay of the axial component of the velocity and jet expansion in the developed region while accelerating the decay of the radial component.

Electromagnetic quarkonium decays at orderv7

We compute S-wave and P-wave electromagnetic quarkonium decays at order v^7 in the heavy-quark velocity expansion. In the S-wave case, our calculation confirms and completes previous findings. In the P-wave case, our results are in disagreement with previous ones; in particular, we find that two matrix elements less are needed. The cancellation of infrared singularities in the matching procedure is discussed.

Trajectory analysis for non-Brownian inertial suspensions in simple shear flow

We analyse pair trajectories of equal-sized spherical particles in simple shear flow for small but finite Stokes numbers. The Stokes number, $\mbox{\textit{St}} \,{=}\, \dot{\gamma} \tau_p$, is a dimensionless measure of particle inertia; here, $\tau_p$ is the inertial relaxation time of an individual particle and $\dot{\gamma}$ is the shear rate. In the limit of weak particle inertia, a regular small-$\mbox{\textit{St}}$ expansion of the particle velocity is used in the equations of motion to obtain trajectory equations to the desired order in $\mbox{\textit{St}}$. The equations for relative trajectories are then solved, to $O(\mbox{\textit{St}})$, in the dilute limit, including only pairwise interactions. Particle inertia is found to destroy the fore–aft symmetry of the zero-Stokes trajectories, and finite-$\mbox{\textit{St}}$ open trajectories suffer net transverse displacements in the velocity gradient and vorticity directions. The vorticity displacement remains $O(\mbox{\textit{St}})$, while the scaling of the gradient displacement increases from $O(\mbox{\textit{St}})$ for far-field open trajectories, to $O(\mbox{\textit{St}}^{{1}/{2}})$ for open trajectories with $O(\mbox{\textit{St}}^{{1}/{2}})$ upstream gradient offsets. The gradient displacement also changes sign, being negative close to the plane of the reference sphere (the shearing plane) on account of dominant lubrication interactions, and then becoming positive at larger off-plane separations. The transverse displacements accompanying successive pair interactions lead to a diffusive behaviour for long times. The shear-induced diffusivity in the vorticity direction is $O(\mbox{\textit{St}}^2\phi \dot{\gamma} a^2)$, while that in the gradient direction scales as $O(\mbox{\textit{St}}^2 \ln \mbox{\textit{St}}\,\phi \dot{\gamma} a^2)$ and $O(\mbox{\textit{St}}^2 \phi \ln (1/\phi) \dot{\gamma} a^2)$ in the limits $\phi \,{\ll}\, \mbox{\textit{St}}^{{1}/{3}}$ and $\mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi \,{\ll}\, 1$, respectively. Further, the region of zero-Stokes closed trajectories is destroyed, and there exists a new attracting limit cycle whose location in the shearing plane is, at leading order, independent of $\mbox{\textit{St}}$. The extension of the present analysis to include a generic linear flow, and the implications of the finite-$\mbox{\textit{St}}$ trajectory modifications for coagulating systems are discussed.

Anomalies in sound velocity and thermal expansion related to charge order and ferromagnetic transitions in Pr 0.65Ca 0.35(Mn 1−zCo z)O 3

The charge order (CO), ferromagnetic (FM) and metal–insulator (M–I) transitions have been studied for Pr 0.65Ca 0.35(Mn 1− Z Co Z )O 3 ( Z=0–0.10) by measuring the sound velocity v s( T), magnetization M( T), electrical resistivity ρ( T) and thermal expansion L( T)/ L R. The v s( T) hardening has been found to precisely monitor the CO transition, while the L( T)/ L R shrink has been found to be sensitive to the onset of the FM-metallic phase. The competition of CO, FM-metallic and FM-insulating phases has been studied from the viewpoint of the enhanced electron–phonon interaction through the Jahn–Teller effect.

Fully Nonlinear Boussinesq-Type Equations for Waves and Currents over Porous Beds

The paper introduces a complete set of Boussinesq-type equations suitable for water waves and wave-induced nearshore circulation over an inhomogeneous, permeable bottom. The derivation starts with the conventional expansion of the fluid particle velocity as a polynomial of the vertical coordinate z followed by the depth integration of the vertical components of the Euler equations for the fluid layer and the volume-averaged equations for the porous layer to obtain the pressure field. Inserting the kinematics and pressure field into the Euler and volume-averaged equations on the horizontal plane results in a set of Boussinesq-type momentum equations with vertical vorticity and z -dependent terms. A new approach to eliminating the z dependency in the Boussinesq-type equations is introduced. It allows for the existence and advection of the vertical vorticity in the flow field with the accuracy consistent with the level of approximation in the Boussinesq-type equations for the pure wave motion. Examination of...

Analysis of the brittle fragmentation of an expanding ring

The current research extends our previous work on brittle fragmentation of a one-dimensional bar [F. Zhou, J.-F. Molinari, K.T. Ramesh, A cohesive-model based fragmentation analysis: effects of strain rate and initial defects distribution, International Journal of Solids and Structures 42 (2005) 5181–5207] to a circular ring that is dynamically expanded (e.g. by explosive loading). The expanding ring test is a convenient and effective technique to study the dynamic fracture and fragmentation properties of materials under high strain rate tensile loading. In this paper we describe the basic elastodynamic equations that govern the expansion of the ring, and develop a numerical integration scheme based on characteristic line differential relationships to solve these equations. A dynamic crack initiation criterion and a cohesive crack growth model are used to describe the failure and fracture process. With this methodology we investigate the fragmentation processes of rings under two typical loading conditions: forced constant velocity expansion and free expansion. The effects of ring size and the fracture properties of the material are investigated. The Rayleigh distribution function is used to describe the statistical characteristics of the fragment populations. Finally we discuss the energy conversions during the fragmentation process.

Dissipation and entropy generation in fully developed forced and mixed laminar convection

A power series expansion of the axial velocity has been used to solve the coupled partial differential equations for fully-developed mixed convection in a vertical tube with uniform heat flux at the wall, uniform internal volumetric heat source and viscous dissipation. Explicit analytical expressions have been obtained for the temperature and velocity profiles as well as for the axial pressure gradient. They were satisfactorily validated for the case of forced convection with both heat source and dissipation, and also for cases of mixed convection with heat source and with or without dissipation. The power series solution has been used to study the effect of viscous dissipation on mixed convection. A Brinkman number of 0.5 results in a significant decrease of both the pressure gradient and the Nusselt number with respect to their values for Br = 0 . The entropy generation rate has also been calculated for both forced and mixed convection; in both cases the contribution of internal friction is significantly higher than that of conduction.

On flame holes and local extinction in lifted-jet diffusion flames

This work examines the dynamics of confined bluff body flames in the process of lean blowoff (LBO) using simultaneous stereo-PIV (particle image velocimetry), OH PLIF (planar laser induced fluorescence) and formaldehyde (CH2O) PLIF. Flames at high density ratios blow off in at least two distinct stages: stage 1, where intermittent extinction occurs along the flame front, but the flame and flow remain qualitatively similar to stable conditions, and stage 2, where there is permanent downstream flame extinction and large-scale changes in dynamic flow characteristics. This paper particularly focuses on stage 2 processes, with the goal of understanding what ultimately leads to irrecoverable flame blowoff. A test facility was developed with the flexibility to achieve two goals: (1) approach LBO by keeping the parameters that influence its hydrodynamic stability approximately constant, particularly flow velocity (ubulk) and gas expansion ratio (σ), and (2) compare near-LBO dynamics under conditions where, well away from blowoff, the flame is globally stable (high σ case) and globally unstable (low σ case, where the Bénard-Von Karman, BVK, instability of the flow is present). The latter case was of particular interest as most prior detailed diagnostic studies of LBO have been performed at high σ, BVK-suppressed conditions. We find that the transient blowoff process remains largely unchanged in the high and low σ cases, apparently due to the fact that the BVK instability reappears in either case under conditions very close to LBO. In all cases, blowoff is preceded by permanent downstream extinction that moves progressively closer to the bluff body as LBO is approached. We also find that near-LBO dynamics are intrinsically three dimensional, due to both secondary instabilities of the shear layer and large out-of-plane motions believed to be due to confinement effects associated with bluff body-wall interactions. These three-dimensional structures often manifest themselves as burning reactant fingers which are caught in the backflow of the recirculation zone; under very near LBO conditions they impinge on the back of the bluff body and extinguish as well. At the very edge of blowoff, the recirculation zone is no longer composed of hot products and is unable to autoignite the oncoming reactant flow, leading to global extinction. The characteristic time associated with this feedback between downstream extinction and wake structure alteration that leads to blowoff is about two orders of magnitude larger than the characteristic flow time, D/ubulk. We also discuss several implications of these results on computations of LBO - in particular, LBO's intrinsically three-dimensionality and the need for many flow through times to capture it.

Cross-linked alginate–guar gum beads as fluidized bed affinity media for purification of jacalin

Copolymeric beads consisting of cross-linked alginate–guar gum were found to fluidize well. The terminal velocity and bed expansion index were found to be 1339.4 cm h −1 and 5.9, respectively. The calculated particle Reynold number of 400 indicated laminar flow. The residence time distribution curves with bovine serum albumin showed intermediate dispersion behaviour according to axial dispersion model. The presence of guar gum in the beads led to binding of jacalin, with maximum binding capacity of 35 × 10 4 U ml −1 beads. The dynamic binding capacity at 5% breakthrough reflected efficient loading of the lectin. The haemagglutinating activity could be recovered at the level of 88% and the preparation was found to be 50-fold purified. SDS-PAGE showed a single band for the purified preparation.

Brownian dynamics simulations of a flexible polymer chain which includes continuous resistance and multibody hydrodynamic interactions

Using methods adapted from the simulation of suspension dynamics, we have developed a Brownian dynamics algorithm with multibody hydrodynamic interactions for simulating the dynamics of polymer molecules. The polymer molecule is modeled as a chain composed of a series of inextensible, rigid rods with constraints at each joint to ensure continuity of the chain. The linear and rotational velocities of each segment of the polymer chain are described by the slender-body theory of Batchelor [J. Fluid Mech. 44, 419 (1970)]. To include hydrodynamic interactions between the segments of the chain, the line distribution of forces on each segment is approximated by making a Legendre polynomial expansion of the disturbance velocity on the segment, where the first two terms of the expansion are retained in the calculation. Thus, the resulting linear force distribution is specified by a center of mass force, couple, and stresslet on each segment. This method for calculating the hydrodynamic interactions has been successfully used to simulate the dynamics of noncolloidal suspensions of rigid fibers [O. G. Harlen, R. R. Sundararajakumar, and D. L. Koch, J. Fluid Mech. 388, 355 (1999); J. E. Butler and E. S. G. Shaqfeh, J. Fluid Mech. 468, 204 (2002)]. The longest relaxation time and center of mass diffusivity are among the quantities calculated with the simulation technique. Comparisons are made for different levels of approximation of the hydrodynamic interactions, including multibody interactions, two-body interactions, and the "freely draining" case with no interactions. For the short polymer chains studied in this paper, the results indicate a difference in the apparent scaling of diffusivity with polymer length for the multibody versus two-body level of approximation for the hydrodynamic interactions.

Comparison of kinetic calculation techniques for the analysis of electron swarm transport at low to moderate E/N values

We present a comparison between results for the electron velocity distribution function (evdf), and transport and rate coefficients of an electron swarm obtained under different assumptions for the space and angular dependence of the evdf. Several solution techniques for the Boltzmann equation as well as Monte Carlo simulations have been tested. The comparison is made in neon at a constant and homogeneous reduced electric field in the range 10 Td ≤ E/N ≤ 500 Td taking into account the production of electrons in ionizing collisions. The results show that to obtain an accurate description of the electron swarm we need to take into account the variation in space of the electron density in the representation of the evdf. In what regards the angular dependence on velocity we discuss criteria to estimate the importance of the anisotropy of the evdf for any gas. Depending on the solution technique and on the E/N value, we find good to excellent agreement between the Boltzmann results obtained with a half-range method, a multi-term Legendre expansion, an elliptic approximation and the Monte Carlo results. The accuracy of the transport and rate coefficients obtained with each approach is evaluated and it is found that although the two-term velocity expansion is not sufficiently accurate to be used for cross section fitting, the corresponding rate and transport coefficients can generally be used in discharge modelling.

State-space inflow modelling for lifting rotors with mass injection

AbstractIn the field of rotorcraft dynamics, it is significant that the induced inflow field is well understood and modeled. A large number of methodologies have been developed in the past years, among which the state-space model is recognised for its advantage in real-time simulation, preliminary design, and dynamic eigenvalue analysis. Recent studies have shown success in representing the induced flow field everywhere above the rotor plane even with mass source terms on the disk as long as they have zero net flux of mass injection when integrated over the disk. Nevertheless, non-zero net mass influx is expected in numerous situations, such as ground effect, tip drive rotors, etc; and the incapability of previous models limits the utilisation of the methodology in these cases. This work presents an extended potential-flow, state-space model derived from the potential-flow momentum equation by means of a Galerkin approach. The induced velocity and pressure perturbation are expanded in terms of closed-form, time-dependent coefficients and space-dependent associated Legendre functions and harmonics. Non-zero net mass flux terms are represented by the involvement of associated Legendre functions with equal degrees and orders. Validation, as well as discrepancies, of the inclusion of such terms is investigated. Numerical simulation of frequency response in axial and skew-angle flight is presented and compared with exact solutions obtained by the convolution integral. Also the study shows that, unlike other pressure distribution responses, non-zero mass influx exhibits a high sensitivity to the choice of the number of states in the velocity expansion. Error analyses are performed to show this sensitivity.

Minimum Fluidization Velocity and Bed Expansion Characteristics of Hydrotreating Catalysts in Ebullated-Bed Systems

Experiments were conducted to study the hydrodynamic characteristics of a gas−liquid−solid fluidized bed containing commercial hydrotreating catalyst extrudates with both water and oil distillates as the liquid phase. The minimum fluidization velocity and bed expansion data were contrasted with predictions from empirical correlations reported in the literature. Empirical correlations tend to yield relatively smaller errors in reproducing the systems with water than with the organic liquids. Bed expansion predictions for the petroleum distillates systems were substantially improved when the sphericity factor of the extrudates was incorporated to the Begovich-Watson equation.