scholarly journals On the equatorial Ekman layer

2016 ◽  
Vol 803 ◽  
pp. 395-435 ◽  
Author(s):  
Florence Marcotte ◽  
Emmanuel Dormy ◽  
Andrew Soward
Keyword(s):  
Shear Layer ◽  
Layer Structure ◽  
Ekman Layer ◽  
Far Field ◽  
Integral Boundary ◽  
Electrically Conducting ◽  
Incompressible Viscous Flow ◽  
Inner Sphere ◽  
Non Local ◽  
Layer Problem

The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the small Ekman number limit $E\ll 1$, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the $E^{1/2}$ Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder $\mathscr{C}$, is geostrophic. Stewartson identified a complicated nested layer structure on $\mathscr{C}$, which comprises relatively thick quasigeostrophic $E^{2/7}$- (inside $\mathscr{C}$) and $E^{1/4}$- (outside $\mathscr{C}$) layers. They embed a thinner ageostrophic $E^{1/3}$ shear layer (on $\mathscr{C}$), which merges with the inner sphere Ekman layer to form the $E^{2/5}$-equatorial Ekman layer of axial length $E^{1/5}$. Under appropriate scaling, this $E^{2/5}$-layer problem may be formulated, correct to leading order, independent of $E$. Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate $z$) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using a non-local integral boundary condition at finite $z$ to account for the far-field behaviour. Adopting $z^{-1}$ as a small parameter we extend Stewartson’s similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.

Characteristics of Leeward Shear Layer Structure of Hollow-Cone Spray in Gaseous Crossflow

AIAA Journal ◽  
2021 ◽  
Vol 59 (1) ◽  
pp. 405-409
Author(s):  
Haibin Zhang ◽  
Shilin Gao ◽  
Bofeng Bai ◽  
Yechun Wang
Keyword(s):  
Shear Layer ◽  
Layer Structure ◽  
Hollow Cone ◽  
Hollow Cone Spray

The structure of a shear layer bounding a separation region. Part 2. Effects of free-stream turbulence

1988 ◽  
Vol 192 ◽  
pp. 577-595 ◽  
Author(s):  
I. P. Castro ◽  
A. Haque
Keyword(s):  
Shear Layer ◽  
Free Stream ◽  
Layer Structure ◽  
Low Frequency ◽  
Flow Region ◽  
Turbulence Structure ◽  
Mean Flow ◽  
Free Stream Turbulence ◽  
Turbulence Data ◽  
Upstream Flow

Detailed measurements throughout the separated region behind a flat plate placed normal to a turbulent stream are reported. A long, central, downstream splitter plate prevented vortex shedding and led to a relatively extensive reversed flow region. Mean flow and turbulence data are compared with results obtained in the (nominal) absence of free-stream turbulence, and attention is concentrated on the changes in the shear-layer structure resulting from the different nature of the upstream flow.Many aspects of the results confirm those obtained recently by other workers. Free-stream turbulence enhances shear-layer entrainment rates, reduces the distance to reattachment and modifies the relatively low-frequency ‘flapping’ motion of the shear layer. In addition, however, extensive use of pulsed wire anemometry has allowed detailed measurements of the turbulence structure throughout the flow and it is shown that this is also modified significantly by the stream turbulence.

scholarly journals On the boundary layer structure near a highly permeable porous interface

2016 ◽  
Vol 798 ◽  
pp. 88-139 ◽  
Author(s):  
Mohit P. Dalwadi ◽  
S. Jonathan Chapman ◽  
Sarah L. Waters ◽  
James M. Oliver
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Layer Structure ◽  
Three Dimensional ◽  
Interfacial Stress ◽  
Far Field ◽  
Dimensional Problem ◽  
Flow Through ◽  
High Reynolds ◽  
Porous Obstacle

The method of matched asymptotic expansions is used to study the canonical problem of steady laminar flow through a narrow two-dimensional channel blocked by a tight-fitting finite-length highly permeable porous obstacle. We investigate the behaviour of the local flow close to the interface between the single-phase and porous regions (governed by the incompressible Navier–Stokes and Darcy flow equations, respectively). We solve for the flow in these inner regions in the limits of low and high Reynolds number, facilitating an understanding of the nature of the transition from Poiseuille to plug to Poiseuille flow in each of these limits. Significant analytical progress is made in the high Reynolds number limit, and we explore in detail the rich boundary layer structure that occurs. We derive general results for the interfacial stress and for the conditions that couple the flow in the outer regions away from the interface. We consider the three-dimensional generalization to unsteady laminar flow through and around a tight-fitting highly permeable cylindrical porous obstacle within a Hele-Shaw cell. For the high Reynolds number limit, we give the coupling conditions and interfacial stress in terms of the outer flow variables, allowing information from a nonlinear three-dimensional problem to be obtained by solving a linear two-dimensional problem. Finally, we illustrate the utility of our analysis by considering the specific example of time-dependent forced far-field flow in a Hele-Shaw cell containing a porous cylinder with a circular cross-section. We determine the internal stress within the porous obstacle, which is key for tissue engineering applications, and the interfacial stress on the boundary of the porous obstacle, which has applications to biofilm erosion. In the high Reynolds number limit, we demonstrate that the fluid inertia can result in the cylinder experiencing a time-independent net force, even when the far-field forcing is periodic with zero mean.

The flow caused by the differential rotation of a right circular cylindrical depression in one of two rapidly rotating parallel planes

1972 ◽  
Vol 53 (4) ◽  
pp. 647-655 ◽  
Author(s):  
M. R. Foster
Keyword(s):  
Shear Layer ◽  
Differential Rotation ◽  
Layer Structure ◽  
Uniform Flow ◽  
General Shape ◽  
Geostrophic Flow ◽  
Vertical Boundary ◽  
Taylor Column ◽  
The Way

The flow induced by the differential rotation of a cylindrical depression of radius a in one of two parallel rigid planes rapidly rotating about their common normal at speed Q is studied. A Taylor column bounded by the usual Stewartson layers arises, but the shear-layer structure is rather different from any previously studied. The Ei-layers (E = v/ωa2) smooth the discontinuity in the geostrophic flow, but the way in which this is accomplished is related to the possible singu-larities of the E1/3-layer solutions. The fact that the 1/4-layer is partially free and partially attached to a vertical boundary accounts for the new joining conditions for the 1/4-layer. The drag on a right circular cylindrical bump in uniform flow is given in addition to some general comments on the applicability of these joining conditions to the motion of an axisymmetric object of quite general shape.

Smart Blending: Functional Fine-Scale Structures Formed by Intelligent Agitations in Multi-Component Melts

2003 ◽  
Author(s):  
D. A. Zumbrunnen
Keyword(s):  
Mechanical Properties ◽  
Fluid Mechanics ◽  
Layer Structure ◽  
Chaotic Advection ◽  
Fine Scale ◽  
Electrically Conducting ◽  
Molecular Scale ◽  
Small Feature ◽  
Scale Structures

In viscous melts, turbulence often does not arise. Consequently, opportunities exist for controllably organizing melt components into functional structures that can have very small feature sizes. In this paper, concepts and results of smart blending are described. Smart blending entails the controllable development in situ of a variety of fine-scale structures in the melt by intelligent agitations. Once formed, structures may be useful in the melt or may be captured in applicable products by extrusion and solidification. Chaotic advection is an enabling recent sub-field of fluid mechanics for smart blending since it provides a means to stretch and fold melt domains and evolve a multi-layer structure leading to derivative arrangements, or indirectly manipulate solid additives. Applications include the production of plastics with enhanced mechanical properties, electrically conducting plastics and glasses, low permeation films, membranes, and nano- and molecular-scale composites.

scholarly journals How the turbulent/non-turbulent interface is different from internal turbulence

2019 ◽  
Vol 866 ◽  
pp. 216-238 ◽  
Author(s):  
G. E. Elsinga ◽  
C. B. da Silva
Keyword(s):  
Shear Layer ◽  
Flow Pattern ◽  
Large Scale ◽  
Layer Structure ◽  
Compressive Strain ◽  
Turbulent Jets ◽  
Scalar Fields ◽  
Passive Scalar ◽  
Flow Topology ◽  
Definition Of

The average patterns of the velocity and scalar fields near turbulent/non-turbulent interfaces (TNTI), obtained from direct numerical simulations (DNS) of planar turbulent jets and shear free turbulence, are assessed in the strain eigenframe. These flow patterns help to clarify many aspects of the flow dynamics, including a passive scalar, near a TNTI layer, that are otherwise not easily and clearly assessed. The averaged flow field near the TNTI layer exhibits a saddle-node flow topology associated with a vortex in one half of the interface, while the other half of the interface consists of a shear layer. This observed flow pattern is thus very different from the shear-layer structure consisting of two aligned vortical motions bounded by two large-scale regions of uniform flow, that typically characterizes the average strain field in the fully developed turbulent regions. Moreover, strain dominates over vorticity near the TNTI layer, in contrast to internal turbulence. Consequently, the most compressive principal straining direction is perpendicular to the TNTI layer, and the characteristic 45-degree angle displayed in internal shear layers is not observed at the TNTI layer. The particular flow pattern observed near the TNTI layer has important consequences for the dynamics of a passive scalar field, and explains why regions of particularly high scalar gradient (magnitude) are typically found at TNTIs separating fluid with different levels of scalar concentration. Finally, it is demonstrated that, within the fully developed internal turbulent region, the scalar gradient exhibits an angle with the most compressive straining direction with a peak probability at around 20$^{\text{o}}$. The scalar gradient and the most compressive strain are not preferentially aligned, as has been considered for many years. The misconception originated from an ambiguous definition of the positive directions of the strain eigenvectors.

Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extention method

2018 ◽  
pp. 14-23 ◽  
Author(s):  
N. N. Vassiliev ◽  
I. N. Parasidis ◽  
E. Providas
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Chemical Engineering ◽  
Direct Method ◽  
Boundary Value ◽  
Medical Science ◽  
Integral Boundary ◽  
Local Boundary ◽  
Non Local

Introduction:Boundary value problems for differential and integro-differential equations with multipoint and non-local boundary conditions often arise in mechanics, physics, biology, biotechnology, chemical engineering, medical science, finances and other fields. Finding an exact solution of a boundary value problem with Fredholm integro-differential equations is a challenging problem. In most cases, solutions are obtained by numerical methods.Purpose:Search for necessary and sufficient solvability conditions for abstract operator equations and their exact solutions. Results: A direct method is proposed for the exact solution of a certain class of ordinary differential or Fredholm integro-differential equations with separable kernels and multlpolnt/lntegral boundary conditions. We study abstract equations of the formBu = Au -gF(Au) = fandB1u = A2u -qF(Au) -gF(A2u) = fwith non-local boundary conditionsΦ(u ) =NѰ(u )andΦ(u ) =NѰ(u ),Φ(Au) =DF(Au) +NѰ(Au), respectively, where A is a differential operator,qandgare vectors,DandNare matrices, andF,ΦandѰare functional vectors. This method is simple to use and can be easily incorporated into any Computer Algebra System (CAS). The upcoming Part 2 of this paper will be devoted to decomposition method for this problem where the operatorB1is quadratic factorable.

An experimental investigation of turbulent flow over a two-dimensional obstacle on a flat plate

2021 ◽  
Vol 263 (2) ◽  
pp. 4459-4470
Author(s):  
Shivam Sundeep ◽  
Xin Zhang ◽  
Siyang Zhong ◽  
Huanxian Bu
Keyword(s):  
Shear Layer ◽  
Large Scale ◽  
Pressure Fluctuations ◽  
Wall Pressure ◽  
Far Field ◽  
Two Dimensional ◽  
Vortical Structures ◽  
Wall Pressure Fluctuations ◽  
Field Noise ◽  
Velocity Spectra

Aeroacoustic and aerodynamic characteristics of the turbulent boundary layer encountering a large obstacle are experimentally investigated in this paper. Two-dimensional obstacles with a square and a semi-circular cross-section mounted on a flat plate are studied in wind tunnel tests, with particular interests in the shear layer characteristics, wall pressure fluctuations, and far-field noise induced by the obstacles. Synchronized measurements of the far-field noise and the wall pressure fluctuations were conducted using microphone arrays in the far-field and flush-mounted in the plate, respectively. Additionally, the streamwise and wall-normal velocity fluctuations behind the obstacle were measured using the X-wire probe. The measured velocity profiles, spectra, and wall pressure spectra are compared, showing that the rectangular obstacle has a significant impact on both the turbulent flow and far-field noise. The large-scale vortical structures shed from the obstacles can be identified in the wall pressure spectra, the streamwise velocity spectra, and the wall pressure coherence analysis. Within the shear layer, the pairing of vortices occurs and the frequency of the broadband peak in the velocity spectra decreases as the shear layer grows downstream. Further eddy convective velocities of large-scale vortical structures inside the shear layer were analyzed based on the wall pressure fluctuations.

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