Dimensionless Gap Research Articles

Experiments on the flow past a square cylinder placed near a wall

Experiments are reported on the flow past a square cylinder placed near a wall at Re = 22,000 (based on the cylinder diameter, D). Visualization studies were carried out for various widths G of the gap between cylinder and wall. Over the range of the dimensionless gap width G/ D = 0.35–0.5, the fraction of time with periodic shedding motion increases from zero to one. Below this range, the shedding motion is completely suppressed and, above it, regular shedding occurs at all times. Detailed two-component laser-Doppler velocimeter measurements were carried out for a gap width of G/ D = 0.75. Phase-averaged statistics were obtained with a special signal-processing procedure; the phase relation was determined from a pressure signal measured on the cylinder side wall. In addition to time-averaged mean-velocity and stress components, phase-averaged components are presented. A detailed comparison is made with the results obtained previously for the flow past a free-standing cylinder away from walls. The Strouhal number is roughly the same and so are many flow features, particularly in the base region near the cylinder, but further downstream the flow with a nearby wall develops a distinct asymmetry, with the vortex trajectories moving away from the wall and the wake developing an oblique character. Differences are noted between the inner and the outer vortices, and the greatest differences in the free-standing cylinder case are the considerably larger values of the u fluctuations in the near wake. The flow topology was found to be similar to that in the flow past two side-by-side cylinders.

Solution of the gap equation in neutron matter

Abstract The problem of solving the gap equation for S-wave pairing in pure neutron matter is considered for the case that the pairing matrix elements V ( p , p ′) are calculated directly from a realistic bare neutron-neutron potential containing a strong short-range repulsion. The original gap equation is replaced identically by a coupled set of equations: a non-singular quasilinear integral equation for the dimensionless gap function χ ( p ) defined by Δ ( p ) = Δ F χ ( p ) and a non-linear algebraic equation for the gap magnitude Δ F = Δ ( p F ) at the Fermi surface. This reformulation admits a robust and rapidly convergent iteration procedure for the determination of the gap function. The treatment may be extended to singlet or triplet pairing in non-zero angular momentum states. S-wave pairing is investigated numerically for the Reid-soft-core interaction. Although the pairing matrix elements of this potential are everywhere positive, non-trivial solutions of the gap equation are obtained on the range 0 p F p c = 1.7496… fm −1 of Fermi momenta, with the gap parameter Δ F reaching a maximum of some 3 MeV near p F = 0.85 fm −1 . Numerical results are also provided for the highly realistic Argonne υ 14 and υ 18 interactions. Within the context of the new computational scheme, a condition for closure of the gap is derived in terms of the first zero p 0 of the gap function Δ ( p ). It is shown that Δ F vanishes exponentially not only in the low-density limit p F → 0, but also as the Fermi momentum rises and approaches the upper critical value p c specified by p F = p 0 ( p F ), beyond which there exists no non-trivial solution of the gap equation. The numerical results for the function Δ ( p ) in neutron matter display a remarkable universality of structure, visible especially in the stability of p 0 under variation of density. Upon renormalizing the gap equation in terms of the vacuum S-wave scattering amplitude, this behavior is seen to be a manifestation of the resonant nature of the neutron-neutron interaction at low energy, which leads to a scattering amplitude of nearly separable form.

Viscoelastic flow between two opposing orifices

The flow of the Oldroyd-B fluid between two opposing orifices is simulated using a finite element method in which the non-Newtonian stress is solved by integrating the strain history along the streamlines. The energy analysis of Mackay et al. [J. Rheol., 39 (1994) 1–14] is investigated by dividing the rate of energy dissipated into three parts, due to the first normal stress difference N 1 , the shear stress τ rz and the second normal stress difference N 2 . The numerical results show that the flow between two opposing orifices, an unsteady elongational flow in a Lagrangian sense, could serve as a reasonable approximation to the steady-state uniaxial elongational flow when the dimensionless gap Z 0/R 0 is of the order one. At larger gaps, the error can be several hundred percent. Other flow configurations are also investigated using the same numerical method, demonstrating that pressure drop measurement does not always yield a good approximation to the tensile force generated in a nearly elongational flow.

Flow visualization of the elastic Taylor-Couette instability in Boger fluids

Flow visualization is performed on an elastically-dominated instability in several similar Boger fluids in Taylor-Couette flow. The onset and evolution of secondary flow are observed over a range of shear rates using reflective mica platelet seeding. Sequences of ambiently and sheet-illuminated images were digitally processed. Rotation of the inner cylinder was ramped from rest to its final value over a time on the order of a polymer relaxation time. Dilute solutions of high molecular weight polyisobutylene in oligomeric polybutene manifest a flow transition at a Deborah number, De s = λ s γ ≈ 1.5 with a Taylor number of 0.00022 in a cell with dimensionless gap ratio δ = 0.0963. At this transition, simple azimuthal shearing is replaced by steady, roughly square, axisymmetric counter-rotating vortices grossly similar to the well-known Taylor vortex flow that is observed at De s = 0, Ta = 3612. At De s = 3.75, Ta = 0.0014, an axisymmetric oscillatory secondary flow develops initially but is replaced by the steady vortices. At De s = 7.5, Ta = 0.0054, the oscillatory and vortex flow coexist and possess an irregular cellular cross-section. A wide span of growth rates is observed: the ratio of onset to polymer relaxation time ranges from 170000 at De s = 1.5 to O(10) at De s > 5. The role of inertia was explored through changing the solvent viscosity. A transition similar to the one that occurs at De s = 3.75, Ta = 0.0014, from the base azimuthal shearing flow to axisymmetric vortices, was also observed with a much lower viscosity fluid at De s = 3.3, Ta = 74.

Solution for the oscillation of a newtonian fluid in a co-axial viscometer

When one cup of a co-axial viscometer oscillates, the measured moment on the other (stationary) cup shows a phase lag, partly due to the inertial effect of the fluid within the gap between the cups. In this paper such an effect is illustrated by a new exact solution of the Navier-Stokes equation, which is derived herein by a scheme of reducing it to a two-point boundary value problem for ODEs. Our numerical results indicate that, as the Womersley number a or the dimensionless gap width δ increases. the fluid velocity profile within the gap gradually deviates from the linear one and transits to that of the boundary layer type, with the result that the moment decreases in the magnitude and lags behind in the phase. With the advantage of high accuracy and excellent stability, the scheme proposed herein can readily be extended to solve other linear periodic problems.

Single-phase turbulent mixing in rod bundles

In the subchannel analysis of rod bundles, the mass exchange between adjacent subchannels is quantified by mixing coefficients, or mixing factors. This process of turbulent cross-flow mixing was found to be related to the phenomenon of quasi-periodic “flow pulsations” across the gaps between adjacent subchannels. By means of hot-wire measurements of spectra of the components of the turbulent fluctuating velocities in the gap regions and a phenomenological model for the flow pulsations, mixing factors were determined for several four-rod bundles with different aspect ratios ( P/ D, W/ D). Comparison of the results with the literature shows good agreement. A correlation for the mixing factor as a function of the dimensionless gap width, obtained by linear regression of the new data, is proposed.

NATURAL CONVECTION FROM DISCRETE HEAT SOURCES IN A VERTICALLY VENTED RECTANGULAR ENCLOSURE

Natural convection in a discretely heated, vertically vented enclosure has been investigated experimentally. A vertically vented enclosure is one whose top and bottom boundaries are partially open, allowing ambient fluid to be drawn in by buoyancy. Mach-Zehnder interferometry was used to visualize the temperature field within the enclosure and to determine the local and average heat transfer characteristics of the discrete heat sources. A smoke-generation technique was used to visualize the flow structure. The experimental parameters investigated include Grashof number, vent gap width, and heater locations for a dual heater configuration and a single-enclosure aspect ratio. The vent gap width was varied between a fully open condition (discretely healed parallel plates) and nearly closed. The Mach-Zehnder interferograms and local heat transfer results suggest a complex buoyancy-driven flow field that depends intimately on the rent gap width. For some geometric configurations, the heat transfer coefficient was found to be nearly uniform across the length of the heater, while for others traditional boundary-layer-type heat transfer characteristics were found. In general, the heat transfer coefficient increased with increased vent gap spacing. However, a maximum in average Nusselt number was observed for a dimensionless gap spacing of G/W – 0.67 for one of the two dual-heater placement configurations studied. The heat transfer characteristics were compared to those of a single isolated plate and unobstructed parallel-plate channels.

SURFACE ROUGHNESS EFFECTS ON FLOW BETWEEN STATIONARY- AND ROTATING DISCS

The turbulent outward fluid flow between stationary and a rotating disc enclosed by a cylindrical housing has been investigated experimentally . This type of flow occurs in many applications of engineering interests . Most of the flows enco-untered in practice are turbulent specially in turbo-machinery. The effect of surface roughness on the friet.ional voment of the rotating disc as well as on the radial pressure distributi-ons has been investigated for different relative roughness ratios of the rotating disc CR/k),16005CR/k35I6401 . Here k is the mean particle diameter of the used grain size and R is the outer disc radius . The flow is investigated at a dimensionless gap height CID of 10.013 SCHmh/10.15 0.401, mass flow coefficient CCw er 11.0x103 5CCwapWeR35 4.2x103 and rotational Reynolds number (Re) of 13x1055 (Re•peR2/0)5. 1.21x 1061. Radial pressure and frictional moment for rough discs are compared with the experimental results obtained on smooth rotating discs . The results have showed that the surface roughness affects strongly upon the radial pressure distribution, the frictional moment and the axial thrust force .

A purely elastic transition in Taylor-Couette flow

Experimental evidence of a non-inertial, cellular instability in the Taylor-Couette flow of a viscoelastic fluid is presented. A linear stability analysis for an Oldroyd-B fluid, which is successful in describing many features of the experimental fluid, predicts the critical Deborah number,Dec, at which the instability is observed. The dependence ofDec on the value of the dimensionless gap between the cylinders is also determined.

Excitation energies of the fractional quantized Hall effect in high-mobility AlxGa1-xAs/GaAs heterostructures.

Disorder reduces the excitation gap of the fractional quantized Hall effect. We analyze recent experiments on ${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As/GaAs heterostructures for filling factors \ensuremath{\nu}=p/3 (p=1,2,4) and \ensuremath{\nu}=p/5 (p=2,3). The disorder is specified from a fit of experimental results on the mobility with theory. Our calculations on the disorder-renormalized excitation gap are in good agreement with experimental results on three different samples, if the dimensionless gap parameter \ensuremath{\Delta}^ (in case of vanishing disorder) is fixed to \ensuremath{\Delta}^(\ensuremath{\nu}=p/3)=0.035 and \ensuremath{\Delta}^(\ensuremath{\nu}=p/5)=0.015.

Forward roll coating with deformable rolls: A simple one-dimensional elastohydrodynamic model

A simple one-dimensional elastohydrodynamic model of forward roll coating with deformable rolls is formulated and solved. It predicts the flow rate, pressures, and roll surface deflections in the nip between rollers, all as functions of one parameter, which can be conveniently chosen as either a dimensionless roll gap setting or load between rolls. The model predicts that for the small-deflection limit the coated-film thickness varies as R(μ V) W −1, while for the large-deflection limit the coated-film thickness varies as R 2 5 (μ V) 1 2 E − 1 3 W − 1 6 (μ = viscosity, V = roll speed, E = Young's modulus, W = load, R = roll radius). The range of applicability of these limits is defined, as are the operating characteristics in the intermediate region. Predictions of minimum roll surface separation compare well with published data.

CREEPING FLOW AROUND AND THROUGH A PERMEABLE SPHERE MOVING WITH CONSTANT VELOCITY TOWARDS A SOLID WALL

The problem of creeping flow of a newtonian fluid around and through a permeable sphere that is moving towards an impermeable wall with constant velocity is solved in terms of the streamfunction and the pressure. The permeability of the sphere is assumed to be continuous, uniform and isotropic, The flow in the sphere is modelled with Darcy's law, and a Beavers-Joseph-Saffman slip-flow boundary condition is assumed at the boundary. Sample streamlines and isobars are calculated. The hydrodynamic correction factor to Stoke's law, ƒ is calculated as a function of the dimensionless permeability, κ, the dimensionless slip factor, β, and the dimensionless gap length, δ. As expected, for typical κ and β values the values of ƒ are substantially smaller than those for an impermeable sphere. An important result is that as δ decreases the value of ƒincreases much more slowly than it does for an impermeable sphere; furthermore, ƒ is finite as δ→0. For moderate and large κ values the ƒ vs δ curve has a maximum.

Rotational and translational motion of a sphere parallel to a wall

Rotational and translational velocities of a sphere moving parallel to a plane wall were determined experimentally as a function of wall-sphere separation. The measurements were carried out under creeping flow conditions for dimensionless gap widths ε in the range 0.006–1.8. Good agreement between the experimental data and the results calculated from exact solutions was found for the whole range of gap widths investigated. The ranges of applicability of solutions by the method of reflections and by lubrication theory are discussed.

電解加工間げき形状の理論解析

Equilibrium electrode gap thickness is theoretically analyzed along a rectilinear path for electrolyte flow. Parameters of little effects on machining accuracy are neglected on basis of assumptions which were commonly used in the conventional ECM gap theories, so that very simple basic equations are obtained without introducing momentum equations for electrolyte flow. The effect of hydrogen gas void fraction, β on electrolyte conductivity is given by a fractional expression of f(β)=(1-β)/(1+λβ) which is an accurate approximation for f(β)=(1-β)n in the range of n=1.43-1.58, in order to cancel (1-β) included in a dominator in a basic equation. Thus for h* of dimensionless gap thickness is induced a quadratic equation of x*, including only ρH* of dimensionless hydrogen gas density, as a x*-dependent parameter. An approximation for ρH* with a straight line leads to determination of gap profiles in the form of parts of hyperbola without numerical calculation by electronic computer. Gap profiles are roughly classified into the types of ever-decreasing, intermediate-peaked and ever-increasing, with an index of dimensionless parameter A1 depending on voltage Eohm· A result of theoretical analysis denies the existence of the conventional optimal machining condition to form a constant gap along the flow path. In order to minimize gap variation along the flow path is proposed an alternative way of optimization, in which Eohm is periodically changed so that conditions of increasing and decreasing effects on the gap thickness compensate each other most.

Slender channel theory in flow between two cylinders

We consider high-Reynolds-number flow between a rotating inner circular cylinder and a fixed non-circular outer cylinder. The dimensionless gap width δ and the Reynolds number R are assumed to be related by Rδ=λ where λ is a constant. Then the governing equations (to first approximation) are shown to be the classical boundary-layer equations, but with unusual boundary conditions. Numerical solutions and analytic approximations are found in various cases.

Wind tunnel performance data for two- and three-bucket Savonius rotors

Fifteen configurations of a Savonius rotor wind turbine were tested in the Vought Corporation Systems Division 4.9- x 6.1-m Low Speed Wind Tunnel to determine aerodynamic performance. The range of values of the varied parameters was as follows: number of buckets, 2 and 3; nominal freestream velocity, 7 and 14 m/s; Reynolds number per meter, 4.32 x 10/sup 5/ and 8.67 x 10/sup 5/; rotor height, 1 and 1.5 m; rotor diameter (nominal), 1 m; bucket overlap, 0.0 to 0.1 m. The measured test variables were torque, rotational speed, and tunnel conditions. It is concluded that increasing Reynolds number and/or aspect ratio improves performance. The recommended configuration consists of two sets of two-bucket rotors, rotated 90 deg apart, with each rotor having a dimensionless gap width of 0.1 to 0.15.

Functional Dependence of Torque Coefficient of Coaxial Cylinders on Gap Width and Reynolds Numbers

Dimensional considerations on the torque coefficient of coaxial cylinders, inner one rotating and outer one at rest, result in the expression CM = f(T/Ri, Re) where Re is the Couette Reynolds number, T is the clearance between the cylinders, and Ri is the inner radius. Experimental data published in the literature and additional data obtained by the authors are analyzed to determine the above functional relationship. It is shown that, depending on the Couette Reynolds number and dimensionless gap width, four different regimes exist and that the fractional characteristics of a coaxial cylinder can accordingly be optimized.