Numerical study of the motions within a slowly precessing sphere at low Ekman number

2001 ◽  
Vol 437 ◽  
pp. 283-299 ◽  
Author(s):  
JÉRÔME NOIR ◽  
D. JAULT ◽  
P. CARDIN
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Numerical Study ◽  
Ekman Layer ◽  
Shear Layers ◽  
Nonlinear Interactions ◽  
Geostrophic Velocity ◽  
Ekman Number ◽  
Geostrophic Shear ◽  
Good Agreement

A geostrophic circulation and a pair of oblique oscillating shear layers arise in a spherical uid cavity contained in a slowly precessing rigid body. Both are caused by the breakdown of the Ekman boundary layer at two critical circles. We rely on numerical modelling to characterize these motions for very small Ekman numbers. Both the O(E1/5) amplitude of the velocity in the oscillating shear layer and the width (also O(E1/5)) of these oblique layers are the result of in ux into the interior from the regions where the Ekman layer breaks down. The oscillating motions are confined to narrow shear layers and their amplitude decays exponentially away from the characteristic surfaces. Nonlinear interactions inside the boundary layer drive the geostrophic shear layer attached to the critical circles. This steady layer, again of O(E1/5) thickness, contains O(E−3/10) velocities. Our results are in good agreement with the experimental measurement by Malkus of the geostrophic velocity arising in a slowly precessing spheroid.

On Integral Methods for Predicting Shear Layer Behavior

1969 ◽  
Vol 36 (4) ◽  
pp. 673-681 ◽  
Author(s):  
S. J. Shamroth
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Integral Equations ◽  
Boundary Layers ◽  
Specific Problem ◽  
Shear Layers ◽  
Near Wake ◽  
Integral Methods ◽  
Layer Behavior ◽  
Momentum Integral

The origin and consequences of a nonphysical constraint which may arise when boundary-layer momentum integral equations are used to predict the behavior of shear layers are examined. It is pointed out that should the constraint occur within the domain of integration of the momentum integral equations, the effect may either be catastrophic or significantly constrain the solution. Several methods of solution having the usual advantages associated with boundary-layer momentum integral equations, but free from this constraint, are proposed for the specific problem of the plane turbulent near wake. One method developed to avoid this constraint in the case of a plane turbulent near wake appears to be perfectly general, and therefore, it may be possible to apply this method to both boundary layers and wakes.

Linear and nonlinear barotropic instability of geostrophic shear layers

1991 ◽  
Vol 224 ◽  
pp. 49-76 ◽  
Author(s):  
L. J. Pratt ◽  
J. Pedlosky
Keyword(s):  
Shear Layer ◽  
Nonlinear Evolution ◽  
Shear Layers ◽  
Linear Instability ◽  
Barotropic Instability ◽  
Reasonable Estimate ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Geostrophic Shear

The linear, weakly nonlinear and strongly nonlinear evolution of unstable waves in a geostrophic shear layer is examined. In all cases, the growth of initially small-amplitude waves in the periodic domain causes the shear layer to break up into a series of eddies or pools. Pooling tends to be associated with waves having a significant varicose structure. Although the linear instability sets the scale for the pooling, the wave growth and evolution at moderate and large amplitudes is due entirely to nonlinear dynamics. Weakly nonlinear theory provides a catastrophic time ts at which the wave amplitude is predicted to become infinite. This time gives a reasonable estimate of the time observed for pools to detach in numerical experiments with marginally unstable and rapidly growing waves.

Scaling of square-prism shear layers

2018 ◽  
Vol 849 ◽  
pp. 1096-1119 ◽  
Author(s):  
D. C. Lander ◽  
D. M. Moore ◽  
C. W. Letchford ◽  
M. Amitay
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Power Law ◽  
Frequency Ratio ◽  
Shear Layers ◽  
Nonlinear Dependence ◽  
Front Face ◽  
Length Scale ◽  
Square Prism ◽  
Ratio Scaling

Scaling characteristics, essential to the mechanisms of transition in square-prism shear layers, were explored experimentally. In particular, the evolution of the dominant instability modes as a function of Reynolds number were reported in the range $1.5\times 10^{4}\lesssim Re_{D}\lesssim 7.5\times 10^{4}$. It was found that the ratio between the shear layer frequency and the shedding frequency obeys a power-law scaling relation. Adherence to the power-law relationship, which was derived from hot-wire measurements, has been supported by two additional and independent scaling considerations, namely, by particle image velocimetry measurements to observe the evolution of length and velocity scales in the shear layer during transition, and by comparison to direct numerical simulations to illuminate the properties of the front-face boundary layer. The nonlinear dependence of the shear layer instability frequency is sustained by the influence of $Re_{D}$ on the thickness of the laminar front-face boundary layer. In corroboration with the original scaling argument for the circular cylinder, the length scale of the shear layer was the only source of nonlinearity in the frequency ratio scaling, within the range of Reynolds numbers reported. The frequency ratio scaling may therefore be understood by the influence of $Re_{D}$ on the appropriate length scale of the shear layer. This length scale was observed to be the momentum thickness evaluated at a transition point, defined where the Kelvin–Helmholtz instability saturates.

Theoretical and Numerical Study of Vortex-Wake Flow Generated From Stack of Rectangular Bodies

2007 ◽  
Author(s):  
Khaled Alhussan
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Vortex Shedding ◽  
Numerical Study ◽  
The Body ◽  
Fluid Particle ◽  
Wake Flow ◽  
Free Shear Layer ◽  
Complex Flow ◽  
Discrete Vortex

Flow over external bodies has been studied extensively because of their many practical applications. For example, flow past a rectangular bodies, usually experiences strong flow oscillations and boundary layer separation in the wake region behind the body. As a fluid particle flows toward the leading edge of a rectangular body, the pressure of the fluid particle increases from the free stream pressure to the stagnation pressure. The boundary layer separates from the surface forms a free shear layer and is highly unstable. This shear layer will eventually roll into a discrete vortex and detach from the surface. A periodic flow motion will develop in the wake as a result of boundary layer vortices being shed alternatively from either side of the rectangular shapes. The periodic nature of the vortex shedding phenomenon can sometimes lead to unwanted structural vibrations, especially when the shedding frequency matches one of the resonant frequencies of the structure. The work to be presented herein is a theoretical and numerical analysis of the complex fluid mechanism that occurs over stack of rectangular bodies for different number of rectangular bodies, specifically with regard to the vortex shedding and generation of wake. A number of important conclusions follow from the current research. First, study of the actual flow configuration over rectangular bodies offers some insight into the complex flow phenomena. Second, the characteristics of the vortex and wakes change considerably with the number of bodies.

A new type of boundary layer in a rapidly rotating gas

1983 ◽  
Vol 126 ◽  
pp. 431-442 ◽  
Author(s):  
Takuya Matsuda ◽  
Keizo Nakagawa
Keyword(s):  
Boundary Layer ◽  
Temperature Jump ◽  
Ekman Layer ◽  
Infinite Length ◽  
Gaseous Flow ◽  
Ekman Number ◽  
New Type ◽  
Vertical Walls ◽  
Source Sink

Gaseous flow in a pie-shaped cylinder of infinite length rotating about the apex is considered. The horizontal flow is induced either by the temperature distribution or by the source/sink distribution on the walls θ = constant. It is found that along the vertical walls θ = constant the E½ boundary layer is formed, where E is the Ekman number. Although the equation governing the above boundary layer is very similar to that of the Ekman layer, it is a new type of boundary layer which may be called the buoyancy layer. Along the wall on which r is constant thermal boundary layers very similar to the Stewartson layers are found to be formed. The role of these layers is to mediate the temperature jump. These layers disappear in the incompressible limit.

Blunt Trailing Edge Shear Wake Development With Turbulent In-Flow Boundary Layers

2005 ◽  
Author(s):  
Dhanalakshmi Challa ◽  
Joe Klewicki
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Boundary Layers ◽  
High Speed ◽  
Velocity Ratio ◽  
Free Shear Layer ◽  
Low Speed ◽  
Axial Pressure Gradient ◽  
Layer Flow ◽  
Good Agreement

Experiments are conducted to explore the structural mechanisms involved in the post-separation evolution of a wall-bounded to a free-shear turbulent flow. At the upstream, both the boundary layers are turbulent. Experiments were conducted in a two-stream shear-layer tunnel, under a zero axial pressure gradient shear-wake configuration. A velocity ratio near 2 was explored. Profile data were collected with a single wire probe at various locations downstream of the blunt separation lip. With this set of measurements, mean profile, axial intensity and measures of profile evolution indicate that the predominant shift from turbulent boundary layer to free shear-layer like behavior occurs between the downstream locations x/θ = 13.7 & 27.4, where θ is the upstream momentum deficit thickness on the low-speed stream. The shear wake width is observed to be nominally constant with the downstream position. Axial velocity spectra show that the transition from boundary layer flow to shear flow occurs earlier in high-speed stream when compared to low speed stream. Strouhal number, Sto, of initial vortex rollup based on initial momentum thickness was found to be 0.034, which is in very good agreement with the existing literature. Other measures are in good agreement with linear stability considerations found in the literature.

Turbulent structure of high-amplitude pressure peaks within the turbulent boundary layer

2013 ◽  
Vol 735 ◽  
pp. 381-426 ◽  
Author(s):  
S. Ghaemi ◽  
F. Scarano
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Shear Layer ◽  
Pressure Field ◽  
Three Dimensional ◽  
High Amplitude ◽  
Shear Layers ◽  
Turbulent Structure ◽  
Streamwise Vortices ◽  
Ejection Event

AbstractThe positive and negative high-amplitude pressure peaks (HAPP) are investigated in a turbulent boundary layer at $R{e}_{\theta } = $ 1900 in order to identify their turbulent structure. The three-dimensional velocity field is measured within the inner layer of the turbulent boundary layer using tomographic particle image velocimetry (tomo-PIV). The measurements are performed at an acquisition frequency of 10 000 Hz and over a volume of $418\times 149\times 621$ wall units in the streamwise, wall-normal and spanwise directions, respectively. The time-resolved velocity fields are applied to obtain the material derivative using the Lagrangian method followed by integration of the Poisson pressure equation to obtain the three-dimensional unsteady pressure field. The simultaneous volumetric velocity, acceleration, and pressure data are conditionally sampled based on local maxima and minima of wall pressure to analyse the three-dimensional turbulent structure of the HAPPs. Analysis has associated the positive HAPPs to the shear layer structures formed by an upstream sweep of high-speed flow opposing a downstream ejection event. The sweep event is initiated in the outer layer while the ejection of near-wall fluid is formed by the hairpin category of vortices. The shear layers were observed to be asymmetric in the instantaneous visualizations of the velocity and acceleration fields. The asymmetric pattern originates from the spanwise component of temporal acceleration of the ejection event downstream of the shear layer. The analysis also demonstrated a significant contribution of the pressure transport term to the budget of the turbulent kinetic energy in the shear layers. Investigation of the conditional averages and the orientation of the vortices showed that the negative HAPPs are linked to both the spanwise and quasi-streamwise vortices of the turbulent boundary layer. The quasi-streamwise vortices can be associated with the hairpin category of vortices or the isolated quasi-streamwise vortices of the inner layer. A bi-directional analysis of the link between the HAPPs and the hairpin paradigm is also conducted by conditionally averaging the pressure field based on the detection of hairpin vortices using strong ejection events. The results demonstrated positive pressure in the shear layer region of the hairpin model and negative pressure overlapping with the vortex core.

An experimental study of turbulent separating and reattaching flows at a high Mach number

1972 ◽  
Vol 52 (3) ◽  
pp. 425-435 ◽  
Author(s):  
J. P. Batham
Keyword(s):  
Boundary Layer ◽  
Mach Number ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Shear Layers ◽  
Two Dimensional ◽  
Pressure Coefficients ◽  
High Mach ◽  
Incipient Separation ◽  
Good Agreement

Separating and reattaching flows in a two-dimensional compression corner were investigated experimentally at a Mach number of 7·0 and Reynolds numbers (based on the distance from the leading edge to the corner) of 4·75 × 106, 9·51 × 106 and 1·55 × 107. Heat-transfer measurements and Pitot traverses in the upstream boundary layer showed that the boundary layer had become fully turbulent at the start of the interactions. Increases in the Reynolds number gave increases in the length of separated shear layers and decreases in the corner angle required for incipient, separation. The reattachment pressure coefficients gave good agreement with the criterion of Batham (1969).

scholarly journals Internal shear layers from librating objects

2017 ◽  
Vol 826 ◽  
pp. 653-675 ◽  
Author(s):  
Stéphane Le Dizès ◽  
Michael Le Bars
Keyword(s):  
Asymptotic Solution ◽  
Shear Layer ◽  
Numerical Solutions ◽  
Shear Layers ◽  
Similarity Solutions ◽  
Curvature Radius ◽  
Ekman Number ◽  
Layer Structures ◽  
Two Parameters ◽  
Critical Latitude

In this work, we analyse the internal shear layer structures generated by the libration of an axisymmetric object in an unbounded fluid rotating at a rotation rate $\unicode[STIX]{x1D6FA}^{\ast }$ using direct numerical simulation and small Ekman number asymptotic analysis. We consider weak libration amplitude and libration frequency $\unicode[STIX]{x1D714}^{\ast }$ within the inertial wave interval $(0,2\unicode[STIX]{x1D6FA}^{\ast })$ such that the fluid dynamics is mainly described by a linear axisymmetric harmonic solution. The internal shear layer structures appear along the characteristic cones of angle $\unicode[STIX]{x1D703}_{c}=\text{acos}(\unicode[STIX]{x1D714}^{\ast }/(2\unicode[STIX]{x1D6FA}^{\ast }))$ which are tangent to the librating object at so-called critical latitudes. These layers correspond to thin viscous regions where the singularities of the inviscid solution are smoothed. We assume that the velocity field in these layers is described by the class of similarity solutions introduced by Moore & Saffman (Phil. Trans. R. Soc. Lond. A, vol. 264, 1969, pp. 597–634). These solutions are characterized by two parameters only: a real parameter $m$, which measures the strength of the underlying singularity, and a complex amplitude coefficient $C_{0}$. We first analyse the case of a disk for which a general asymptotic solution for small Ekman numbers is known when the disk is in a solid plane. We demonstrate that the numerical solutions obtained for a free disk and for a disk in a solid plane are both well described by the asymptotic solution and by its similarity form within the internal shear layers. For the disk, we obtain a parameter $m=1$ corresponding to a Dirac source at the edge of the disk and a coefficient $C_{0}\propto E^{1/6}$ where $E$ is the Ekman number. The case of a smoothed librating object such as a spheroid is found to be different. By asymptotically matching the boundary layer solution to similarity solutions close to a critical latitude on the surface, we show that the adequate parameter $m$ for the similarity solution is $m=5/4$, leading to a coefficient $C_{0}\propto E^{1/12}$, that is larger than for the case of a disk for small Ekman numbers. A simple general expression for $C_{0}$ valid for any axisymmetric object is obtained as a function of the local curvature radius at the critical latitude in agreement with this change of scaling. This result is tested and validated against direct numerical simulations.

Experimental Study on the Effect of Freestream Turbulence on the Development of an Inflectional Boundary Layer From the Semi-Circular Leading Edge of a Flat Plate

2013 ◽  
Author(s):  
A. Samson ◽  
S. Sarkar
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Flat Plate ◽  
Leading Edge ◽  
Separated Shear Layer ◽  
Bubble Length ◽  
Shedding Frequency ◽  
Velocity Spectra ◽  
Low Speed Wind Tunnel ◽  
Good Agreement

The characteristics of a boundary layer from the semi-circular leading edge of a flat plate has been investigated for two levels of stream turbulence (Tu = 0.5% and 7.7%) in a low-speed wind tunnel. Measurements of velocity and surface pressure were made along with a planar PIV to visualize flow structures for varying turbulence levels at a Reynolds number of 25000 (based on the leading edge diameter). At low stream turbulence the measurements reveal flow undergoes separation in the vicinity of leading-edge with reattachment in the downstream. Velocity spectra illustrates that the separated shear layer is laminar up to 20% of separation length and then the perturbations are amplified in the second half attributing to breakdown and reattachment. It is also evident that the shear layer is inviscidly unstable and the predominant shedding frequency when normalised with respect to the momentum thickness at separation shows a good agreement with previous studies. The bubble length is highly susceptible to change in Tu depicting an attached layer which grows into a fully turbulent profile at high Tu. Here, the spectra for an attached layer depicts a turbulent-like flow with band of frequencies from the beginning.

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