An example of steady laminar flow at large Reynolds number

1960 ◽  
Vol 9 (4) ◽  
pp. 593-602 ◽  
Author(s):  
Iam Proudman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layers ◽  
Tangential Velocity ◽  
Fluid Flows ◽  
The Other ◽  
Large Reynolds Number ◽  
Rotational Flows ◽  
Flow Domain ◽  
Source Of Information

The purpose of this note is to describe a particular class of steady fluid flows, for which the techniques of classical hydrodynamics and boundary-layer theory determine uniquely the asymptotic flow for large Reynolds number for each of a continuously varied set of boundary conditions. The flows involve viscous layers in the interior of the flow domain, as well as boundary layers, and the investigation is unusual in that the position and structure of all the viscous layers are determined uniquely. The note is intended to be an illustration of the principles that lead to this determination, not a source of information of practical value.The flows take place in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted. When fluid is extracted through both walls there are boundary layers on both walls and the flow outside these layers is irrotational. When fluid is extracted through one wall and injected through the other, there is a boundary layer only on the former wall and the inviscid rotational flow outside this layer satisfies the no-slip condition on the other wall. When fluid is injected through both walls there are no boundary layers, but there is a viscous layer in the interior of the channel, across which the second derivative of the tangential velocity is discontinous, and the position of this layer is determined by the requirement that the inviscid rotational flows on either side of it must satisfy the no-slip conditions on the walls.

Annulus Wall Boundary-Layer Measurements in a Four-Stage Compressor

1986 ◽  
Vol 108 (1) ◽  
pp. 2-6 ◽  
Author(s):  
N. A. Cumpsty
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Layer Thickness ◽  
Boundary Layers ◽  
Boundary Layer Thickness ◽  
Tip Clearance ◽  
The Other ◽  
Wall Boundary Layer ◽  
Multistage Compressors ◽  
Full Design

There are few available measurements of the boundary layers in multistage compressors when the repeating-stage condition is reached. These tests were performed in a small four-stage compressor; the flow was essentially incompressible and the Reynolds number based on blade chord was about 5 • 104. Two series of tests were performed; in one series the full design number of blades were installed, in the other series half the blades were removed to reduce the solidity and double the staggered spacing. Initially it was wished to examine the hypothesis proposed by Smith [1] that staggered spacing is a particularly important scaling parameter for boundary layer thickness; the results of these tests and those of Hunter and Cumpsty [2] tend to suggest that it is tip clearance which is most potent in determining boundary-layer integral thicknesses. The integral thicknesses agree quite well with those published by Smith.

scholarly journals On the identification of well-behaved turbulent boundary layers

2017 ◽  
Vol 822 ◽  
pp. 109-138 ◽  
Author(s):  
C. Sanmiguel Vila ◽  
R. Vinuesa ◽  
S. Discetti ◽  
A. Ianiro ◽  
P. Schlatter ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Velocity Profile ◽  
Skin Friction ◽  
Boundary Layers ◽  
Shape Factor ◽  
Outer Region ◽  
The Other ◽  
Diagnostic Plot ◽  
Other Hand

This paper introduces a new method based on the diagnostic plot (Alfredsson et al., Phys. Fluids, vol. 23, 2011, 041702) to assess the convergence towards a well-behaved zero-pressure-gradient (ZPG) turbulent boundary layer (TBL). The most popular and well-understood methods to assess the convergence towards a well-behaved state rely on empirical skin-friction curves (requiring accurate skin-friction measurements), shape-factor curves (requiring full velocity profile measurements with an accurate wall position determination) or wake-parameter curves (requiring both of the previous quantities). On the other hand, the proposed diagnostic-plot method only needs measurements of mean and fluctuating velocities in the outer region of the boundary layer at arbitrary wall-normal positions. To test the method, six tripping configurations, including optimal set-ups as well as both under- and overtripped cases, are used to quantify the convergence of ZPG TBLs towards well-behaved conditions in the Reynolds-number range covered by recent high-fidelity direct numerical simulation data up to a Reynolds number based on the momentum thickness and free-stream velocity $Re_{\unicode[STIX]{x1D703}}$ of approximately 4000 (corresponding to 2.5 m from the leading edge) in a wind-tunnel experiment. Additionally, recent high-Reynolds-number data sets have been employed to validate the method. The results show that weak tripping configurations lead to deviations in the mean flow and the velocity fluctuations within the logarithmic region with respect to optimally tripped boundary layers. On the other hand, a strong trip leads to a more energized outer region, manifested in the emergence of an outer peak in the velocity-fluctuation profile and in a more prominent wake region. While established criteria based on skin-friction and shape-factor correlations yield generally equivalent results with the diagnostic-plot method in terms of convergence towards a well-behaved state, the proposed method has the advantage of being a practical surrogate that is a more efficient tool when designing the set-up for TBL experiments, since it diagnoses the state of the boundary layer without the need to perform extensive velocity profile measurements.

Modeling mass transfer and substrate utilization in the boundary layer of biofilm systems

1998 ◽  
Vol 37 (4-5) ◽  
pp. 139-147 ◽  
Author(s):  
Harald Horn ◽  
Dietmar C. Hempel
Keyword(s):  
Boundary Layer ◽  
Mass Transfer ◽  
Reynolds Number ◽  
Film Theory ◽  
Substrate Utilization ◽  
The Other ◽  
Concentration Profiles ◽  
Hydrodynamic Conditions ◽  
Time Axis ◽  
Transfer Coefficients

The use of microelectrodes in biofilm research allows a better understanding of intrinsic biofilm processes. Little is known about mass transfer and substrate utilization in the boundary layer of biofilm systems. One possible description of mass transfer can be obtained by mass transfer coefficients, both on the basis of the stagnant film theory or with the Sherwood number. This approach is rather formal and not quite correct when the heterogeneity of the biofilm surface structure is taken into account. It could be shown that substrate loading is a major factor in the description of the development of the density. On the other hand, the time axis is an important factor which has to be considered when concentration profiles in biofilm systems are discussed. Finally, hydrodynamic conditions become important for the development of the biofilm surface when the Reynolds number increases above the range of 3000-4000.

Aero-optics of subsonic turbulent boundary layers

2012 ◽  
Vol 696 ◽  
pp. 122-151 ◽  
Author(s):  
Kan Wang ◽  
Meng Wang
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layers ◽  
Dominant Role ◽  
Elevation Angle ◽  
Turbulent Boundary Layers ◽  
Density Field ◽  
Small Scale ◽  
Number Range ◽  
Scale Turbulence

AbstractCompressible large-eddy simulations are carried out to study the aero-optical distortions caused by Mach 0.5 flat-plate turbulent boundary layers at Reynolds numbers of ${\mathit{Re}}_{\theta } = 875$, 1770 and 3550, based on momentum thickness. The fluctuations of refractive index are calculated from the density field, and wavefront distortions of an optical beam traversing the boundary layer are computed based on geometric optics. The effects of aperture size, small-scale turbulence, different flow regions and beam elevation angle are examined and the underlying flow physics is analysed. It is found that the level of optical distortion decreases with increasing Reynolds number within the Reynolds-number range considered. The contributions from the viscous sublayer and buffer layer are small, while the wake region plays a dominant role, followed by the logarithmic layer. By low-pass filtering the fluctuating density field, it is shown that small-scale turbulence is optically inactive. Consistent with previous experimental findings, the distortion magnitude is dependent on the propagation direction due to anisotropy of the boundary-layer vortical structures. Density correlations and length scales are analysed to understand the elevation-angle dependence and its relation to turbulence structures. The applicability of Sutton’s linking equation to boundary-layer flows is examined, and excellent agreement between linking equation predictions and directly integrated distortions is obtained when the density length scale is appropriately defined.

Pressure gradient effects on the large-scale structure of turbulent boundary layers

2013 ◽  
Vol 715 ◽  
pp. 477-498 ◽  
Author(s):  
Zambri Harun ◽  
Jason P. Monty ◽  
Romain Mathis ◽  
Ivan Marusic
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Pressure Gradient ◽  
Boundary Layers ◽  
Amplitude Modulation ◽  
Large Scale ◽  
Outer Region ◽  
Adverse Pressure Gradient ◽  
Turbulent Boundary Layers ◽  
Small Scale

AbstractResearch into high-Reynolds-number turbulent boundary layers in recent years has brought about a renewed interest in the larger-scale structures. It is now known that these structures emerge more prominently in the outer region not only due to increased Reynolds number (Metzger & Klewicki, Phys. Fluids, vol. 13(3), 2001, pp. 692–701; Hutchins & Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28), but also when a boundary layer is exposed to an adverse pressure gradient (Bradshaw, J. Fluid Mech., vol. 29, 1967, pp. 625–645; Lee & Sung, J. Fluid Mech., vol. 639, 2009, pp. 101–131). The latter case has not received as much attention in the literature. As such, this work investigates the modification of the large-scale features of boundary layers subjected to zero, adverse and favourable pressure gradients. It is first shown that the mean velocities, turbulence intensities and turbulence production are significantly different in the outer region across the three cases. Spectral and scale decomposition analyses confirm that the large scales are more energized throughout the entire adverse pressure gradient boundary layer, especially in the outer region. Although more energetic, there is a similar spectral distribution of energy in the wake region, implying the geometrical structure of the outer layer remains universal in all cases. Comparisons are also made of the amplitude modulation of small scales by the large-scale motions for the three pressure gradient cases. The wall-normal location of the zero-crossing of small-scale amplitude modulation is found to increase with increasing pressure gradient, yet this location continues to coincide with the large-scale energetic peak wall-normal location (as has been observed in zero pressure gradient boundary layers). The amplitude modulation effect is found to increase as pressure gradient is increased from favourable to adverse.

The Numerical Simulation of Turbulent Flow in Semi-Closed Rotor-Stator Cavity

2016 ◽  
Author(s):  
Guohu Luo ◽  
Shengde Wang ◽  
Hong Shen ◽  
Zhenqiang Yao
Keyword(s):  
Reynolds Number ◽  
Turbulent Flow ◽  
Boundary Layers ◽  
Pressure Difference ◽  
Tangential Velocity ◽  
Reynolds Stress Model ◽  
Mean Flow ◽  
Closed Cavity ◽  
Turbulent Field ◽  
Effects Of Rotation

The present work numerically considered the turbulent flow in a semi-closed rotor-stator cavity with a superimposed throughflow based on Reynolds Stress Model (RSM). The mean flow structure and turbulent field in the semi-closed cavity (SC) were identified by comparison with the flow in open cavity (OC) and closed cavity (CC). Then the effects of rotation Reynolds number, ranging from 1 × 106 to 4 × 106, on the flow in SC were investigated. The superimposed flow noticeably decreases the tangential velocity, resulting that the pressure difference between central hub and periphery in SC is greater than the OC but less than the CC. The flow in SC belongs to Stewartson type in the region between inlet and outlet, but to Bachelor type between outlet and periphery. Around the outlets, the flow is greatly affected, especially for turbulent field, where the turbulence intensities maintain at higher levels outside the two boundary layers. With the increase of Reynolds number, the tangential velocity goes up, resulted the attenuation of jet impinging effects, the shrinking of affected zones by outlets and the enlargement of pressure difference. Moreover, with the Bödewadt layer moving toward the central hub, the turbulence intensities increase inside two boundary layers but decrease outside them. Consequently, the flow is transited to Stewartson and then Batchelor type.

The boundary layer in crossed-fields m.h.d.

1966 ◽  
Vol 25 (2) ◽  
pp. 229-240 ◽  
Author(s):  
W. R. Sears
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Flat Plate ◽  
Flow Problem ◽  
Inviscid Flow ◽  
Magnetic Reynolds Number ◽  
Large Reynolds Number ◽  
Zero Values ◽  
Plate Solution ◽  
A Current

This study of the boundary layer of steady, incompressible, plane, crossed-fields m.h.d. flow at large Reynolds numberReand magnetic Reynolds numberRmbegins with a review of Hartmann's case, where a boundary layer occurs whose thickness is proportional to (Re Rm)−½. Following this clue, it is shown that in general the boundary layer is a ‘local Hartmann boundary layer’. Its profiles are always exponential and it is determined completely by local quantities. The skin friction and the total electric current in the layer are proportional to the square root of the magnetic Prandtl number, i.e. to (Rm/Re)½. Thus the exterior-flow problem, the solution of which precedes a boundary-layer solution, generally involves a current sheet at the fluid-solid interface.This inviscid-flow problem becomes tractable if (Rm/Re)½is small enough to permit a linearized solution. The flow field about a flat plate at zero incidence is calculated in this approximation. It is pointed out that the thin-cylinder solutions of Sears & Resler (1959), which pertain toRm/Re= 0, can immediately be extended to small, non-zero values of this parameter by linear combination with this flat-plate solution.

Estimating amplitude ratios in boundary layer stability theory: a comparison between two approaches

2001 ◽  
Vol 439 ◽  
pp. 403-412 ◽  
Author(s):  
RAMA GOVINDARAJAN ◽  
R. NARASIMHA
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Small Difference ◽  
Detailed Comparison ◽  
The Other ◽  
Boundary Layer Stability ◽  
Mean Flow ◽  
Amplitude Ratios ◽  
The Stability ◽  
Local Reynolds Number

We first demonstrate that, if the contributions of higher-order mean flow are ignored, the parabolized stability equations (Bertolotti et al. 1992) and the ‘full’ non-parallel equation of Govindarajan & Narasimha (1995, hereafter GN95) are both equivalent to order R−1 in the local Reynolds number R to Gaster's (1974) equation for the stability of spatially developing boundary layers. It is therefore of some concern that a detailed comparison between Gaster (1974) and GN95 reveals a small difference in the computed amplitude ratios. Although this difference is not significant in practical terms in Blasius flow, it is traced here to the approximation, in Gaster's method, of neglecting the change in eigenfunction shape due to flow non-parallelism. This approximation is not justified in the critical and the wall layers, where the neglected term is respectively O(R−2/3) and O(R−1) compared to the largest term. The excellent agreement of GN95 with exact numerical simulations, on the other hand, suggests that the effect of change in eigenfunction is accurately taken into account in that paper.

scholarly journals Structure of high Reynolds number boundary layers over cube canopies

2019 ◽  
Vol 870 ◽  
pp. 460-491 ◽  
Author(s):  
Jérémy Basley ◽  
Laurent Perret ◽  
Romain Mathis
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layers ◽  
Coherent Structures ◽  
High Reynolds Number ◽  
Stereoscopic Particle Image Velocimetry ◽  
Roughness Sublayer ◽  
Large Field ◽  
Normal Velocity ◽  
High Reynolds

The influence of a cube-based canopy on coherent structures of the flow was investigated in a high Reynolds number boundary layer (thickness $\unicode[STIX]{x1D6FF}\sim 30\,000$ wall units). Wind tunnel experiments were conducted considering wall configurations that represent three idealised urban terrains. Stereoscopic particle image velocimetry was employed using a large field of view in a streamwise–spanwise plane ($0.55\unicode[STIX]{x1D6FF}\times 0.5\unicode[STIX]{x1D6FF}$) combined to two-point hot-wire measurements. The analysis of the flow within the inertial layer highlights the independence of its characteristics from the wall configuration. The population of coherent structures is in agreement with that of smooth-wall boundary layers, i.e. consisting of large- and very-large-scale motions, sweeps and ejections, as well as smaller-scale vortical structures. The characteristics of vortices appear to be independent of the roughness configuration while their spatial distribution is closely linked to large meandering motions of the boundary layer. The canopy geometry only significantly impacts the wall-normal exchanges within the roughness sublayer. Bi-dimensional spectral analysis demonstrates that wall-normal velocity fluctuations are constrained by the presence of the canopy for the densest investigated configurations. This threshold in plan area density above which large scales from the overlying boundary layer can penetrate the roughness sublayer is consistent with the change of the flow regime reported in the literature and constitutes a major difference with flows over vegetation canopies.

scholarly journals Assessment of inner–outer interactions in the urban boundary layer using a predictive model

2019 ◽  
Vol 875 ◽  
pp. 44-70 ◽  
Author(s):  
Karin Blackman ◽  
Laurent Perret ◽  
Romain Mathis
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Predictive Model ◽  
Boundary Layers ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Limited Range ◽  
Rough Wall ◽  
Wall Boundary ◽  
Layer Flow

Urban-type rough-wall boundary layers developing over staggered cube arrays with plan area packing density, $\unicode[STIX]{x1D706}_{p}$, of 6.25 %, 25 % or 44.4 % have been studied at two Reynolds numbers within a wind tunnel using hot-wire anemometry (HWA). A fixed HWA probe is used to capture the outer-layer flow while a second moving probe is used to capture the inner-layer flow at 13 wall-normal positions between $1.25h$ and $4h$ where $h$ is the height of the roughness elements. The synchronized two-point HWA measurements are used to extract the near-canopy large-scale signal using spectral linear stochastic estimation and a predictive model is calibrated in each of the six measurement configurations. Analysis of the predictive model coefficients demonstrates that the canopy geometry has a significant influence on both the superposition and amplitude modulation. The universal signal, the signal that exists in the absence of any large-scale influence, is also modified as a result of local canopy geometry suggesting that although the nonlinear interactions within urban-type rough-wall boundary layers can be modelled using the predictive model as proposed by Mathis et al. (J. Fluid Mech., vol. 681, 2011, pp. 537–566), the model must be however calibrated for each type of canopy flow regime. The Reynolds number does not significantly affect any of the model coefficients, at least over the limited range of Reynolds numbers studied here. Finally, the predictive model is validated using a prediction of the near-canopy signal at a higher Reynolds number and a prediction using reference signals measured in different canopy geometries to run the model. Statistics up to the fourth order and spectra are accurately reproduced demonstrating the capability of the predictive model in an urban-type rough-wall boundary layer.

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