Viscous Wall Layer Research Articles

Nonlinear neutral modes in zero-mass-flux Plane poiseuille-couette flow

SummaryPlane Poiseuille–Couette flow with zero mean advection velocity, a flow well known to be linearly stable, is analysed at asymptotically large Reynolds number R. A nonlinear neutral mode structure is uncovered which is self-sustained due to the interaction of the viscous wall layers with a predominantly inviscid nonlinear internal critical layer. The O(1) phase speed and wavenumber of the disturbance are computed as functions of the O(R−1/3) wave amplitude, with the formula for the amplitude determined analytically. Two distinct coherent structures are found: one involving a single critical layer and a second with two critical layers. The single critical layer case has a long wavelength/small amplitude limit for which an analytical solution exists, while the dual critical layer structure possesses two solution branches and exists above a threshold amplitude. Both states distort the mean flow by an O(R−1/6) amount, and in the long wavelength limit the distortion becomes comparable in size with the original basic flow. The relevance of this work to the experimental observation of self-sustained states in this flow is discussed.

Principle of fundamental resonance in hypersonic boundary layers: an asymptotic viewpoint

The fundamental resonance (FR) in the nonlinear phase of the boundary-layer transition to turbulence appears when a dominant planar instability mode reaches a finite amplitude and the low-amplitude oblique travelling modes with the same frequency as the dominant mode, together with the stationary streak modes, undergo the strongest amplification among all the Fourier components. This regime may be the most efficient means to trigger the natural transition in hypersonic boundary layers. In this paper, we aim to reveal the intrinsic mechanism of the FR in the weakly nonlinear framework based on the large-Reynolds-number asymptotic technique. It is found that the FR is, in principle, a triad resonance among a dominant planar fundamental mode, a streak mode and an oblique mode. In the major part of the boundary layer, the nonlinear interaction of the fundamental mode and the streak mode seeds the growth of the oblique mode, whereas the interaction of the oblique mode and the fundamental mode drives the roll components (transverse and lateral velocity) of the streak mode, which leads to a stronger amplification of the streamwise component of the streak mode due to the lift-up mechanism. This asymptotic analysis clearly shows that the dimensionless growth rates of the streak and oblique modes are the same order of magnitude as the dimensionless amplitude of the fundamental mode $(\bar {\epsilon }_{10})$ , and the amplitude of the streak mode is $O(\bar {\epsilon }_{10}^{-1})$ greater than that of the oblique mode. The main-layer solution of the streamwise velocity, spanwise velocity and temperature of both the streak and the oblique modes become singular as the wall is approached, and so a viscous wall layer appears underneath. The wall layer produces an outflux velocity to the main-layer solution, inclusion of which leads to an improved asymptotic theory whose accuracy is confirmed by comparing with the calculations of the nonlinear parabolised stability equations (NPSEs) at moderate Reynolds numbers and the secondary instability analysis (SIA) at sufficiently high Reynolds numbers.

On the near-wall instability of oblique flow over a wavy wall

The instability of a shear flow passing over a wavy wall with wave crests not perpendicular to the flow direction is investigated. The friction Reynolds number for the flow is large and the wave amplitude scaled on the wavelength is small compared with the viscous wall layer. The instability takes the form of a streamwise vortex of wavelength comparable to the viscous wall layer in which the basic flow adjusts to the presence of the wall. For a given wall amplitude, the instability considered is the first one to arise as the Reynolds number is increased and modes of wavelength comparable to the viscous layer grow much faster than modes of wavelength comparable to the wall wavelength. The instability is not driven by centrifugal or viscous effects but is a novel kind of cross-flow vortex instability associated with a spatially periodic flow; the existence of the instability is associated with the orientation of the wave crests. The instability is investigated for wavelengths comparable to the depth of the viscous wall layer; the limiting cases of large and small wavelengths are investigated asymptotically. At small wavenumbers and roughness heights the instability connects with disturbances of wavelength comparable to the wall wavelength. At high wavenumbers and roughness heights a new structure emerges and the disturbance moves away from the wall, that structure takes on a self-similar form with progressively faster variations in the streamwise and spanwise directions as the roughness height increases.

A new boundary layer integral method based on the universal velocity profile

A recently developed mixing length model of the turbulent shearing stress in wall bounded flows has been used to formulate a universal velocity profile (UVP) that provides an effective replacement for the widely used Coles wall-wake formulation. Comparisons with both direct numerical simulation and experimental data demonstrate the ability of the profile to approximate a wide variety of wall-bounded flows. The UVP is uniformly valid from the wall to the boundary layer edge and for all Reynolds numbers from zero to infinity. There is no presumption of logarithmic dependence of the velocity profile outside the viscous wall layer so the profile can accurately approximate low Reynolds number turbulent boundary layers. The effect of a pressure gradient is included in the UVP through the introduction of a modified Clauser parameter that correlates well with the parameters that determine the wake portion of the velocity profile. The inherent dependence of the UVP on Reynolds number, extended to include the effect of pressure gradient, enables it to be used as the basis of a new method for integrating the von Kármán boundary layer integral equation for a wide variety of attached wall bounded flows. To illustrate its application, the UVP is used to determine the zero-lift drag coefficient of the Joukowsky 0012 and NACA (National Advisory Committee for Aeronautics) 0012 airfoils over a wide range of chord Reynolds numbers.

A body in nonlinear near-wall shear flow: impacts, analysis and comparisons

Abstract

A universal velocity profile for smooth wall pipe flow

The most important unanswered questions in turbulence regard the nature of turbulent flow in the limit of infinite Reynolds number. The Princeton superpipe (PSP) data comprise 26 velocity profiles that cover three orders of magnitude in the Reynolds number from $Re=19\,639$, to $Re=20\,088\,000$ based on pipe radius and pipe centreline velocity. In this paper classical mixing length theory is combined with a new mixing length model of the turbulent shear stress to solve the streamwise momentum equation and the solution is used to approximate the PSP velocity profiles. The model velocity profile is uniformly valid from the wall to the pipe centreline and comprises five free parameters that are selected through a minimization process to provide an accurate approximation to each of the 26 profiles. The model profile is grounded in the momentum equation and allows the velocity derivative, Reynolds shear stress and turbulent kinetic energy production to be studied. The results support the conclusion that logarithmic velocity behaviour near the wall is not present in the data below a pipe Reynolds number somewhere between $Re=59\,872$, and $Re=87\,150$. Above $Re=87\,150$, the data show a very clear, nearly logarithmic, region. But even at the highest Reynolds numbers there is still a weak algebraic dependence of the intermediate portion of the velocity profile on both the near-wall and outer flow length scales. One of the five parameters in the model profile is equivalent to the well-known Kármán constant, $k$. The parameter $k$ increases almost monotonically from $k=0.4034$ at $Re=87\,150$ to $k=0.4190$ at $Re=20\,088\,000$, with an average value, $k=0.4092$. The variation of the remaining four model parameters is relatively small and, with all five parameters fixed at average values, the model profile reproduces the entire velocity data set and the wall friction reasonably well. With optimal values of the parameters used for each model profile, the fit to the PSP survey data is very good. Transforming the model velocity profile using the group, $u/u_{0}\rightarrow ku/u_{0}$, $y^{+}\rightarrow ky^{+}$ and $R_{\unicode[STIX]{x1D70F}}\rightarrow kR_{\unicode[STIX]{x1D70F}}$ where $R_{\unicode[STIX]{x1D70F}}$ is the friction Reynolds number, leads to a reduced expression for the velocity profile. When the reduced profile is cast in outer variables, the physical velocity profile is expressed in terms of $\ln (y/\unicode[STIX]{x1D6FF})$ and a new shape function $\unicode[STIX]{x1D719}(y/\unicode[STIX]{x1D6FF})$. In the limit of infinite Reynolds number, the velocity profile asymptotes to plug flow with a vanishingly thin viscous wall layer and a continuous derivative everywhere. The shape function evaluated at the pipe centreline is used to produce a new friction law with an additive constant that depends on the Kármán constant and a wall damping length scale.

Channel Flow Past A Near-Wall Body

Summary Near-wall behaviour arising when a finite sized body moves in a channel flow is investigated for high flow rates. This is over the interactive-flow length scale that admits considerable upstream influence. The focus is first on quasi-steady two-dimensional flow past a thin body in the outer reaches of one of the viscous wall layers. The jump conditions near the front of the body play an important part in the whole solution which involves an unusual multi-structured flow due to the presence of the body: flows above, below, ahead of and behind the body interact fully. Analytical solutions are presented and the repercussions for shorter and longer bodies are then examined. Second, implications are followed through for the movement of a free body in a dynamic fluid–body interaction. Particular key findings are that instability persists for all body lengths, the growth rate decreases like the $1/4$ power of distance as the body approaches the wall, and lift production on the body is dominated by high pressures from an unexpected flow region emerging on the front of the body.

Quality Measures of Mixing in Turbulent Flow and Effects of Molecular Diffusivity

Results from numerical simulations of the mixing of two puffs of scalars released in a turbulent flow channel are used to introduce a measure of mixing quality, and to investigate the effectiveness of turbulent mixing as a function of the location of the puff release and the molecular diffusivity of the puffs. The puffs are released from instantaneous line sources in the flow field with Schmidt numbers that range from 0.7 to 2400. The line sources are located at different distances from the channel wall, starting from the wall itself, the viscous wall layer, the logarithmic layer, and the channel center. The mixing effectiveness is quantified by following the trajectories of individual particles with a Lagrangian approach and carefully counting the number of particles from both puffs that arrive at different locations in the flow field as a function of time. A new measure, the mixing quality index Ø, is defined as the product of the normalized fraction of particles from the two puffs at a flow location. The mixing quality index can take values from 0, corresponding to no mixing, to 0.25, corresponding to full mixing. The mixing quality in the flow is found to depend on the Schmidt number of the puffs when the two puffs are released in the viscous wall region, while the Schmidt number is not important for the mixing of puffs released outside the logarithmic region.

Molecular dispersion in pre-turbulent and sustained turbulent flow of carbon dioxide

The average dispersion coefficients, Da¯, of two small molecules (acetonitrile and coronene) were measured under laminar, transient, and sustained turbulent flow regimes along fused silica open tubular capillary (OTC) columns (180 μm inner diameter by 20 m length). Carbon dioxide was used as the mobile phase at room temperature (296 K) and at average pressures in the range from 1500 to 2700 psi. The Reynolds number (Re) was increased from 600 to 5000. The measurement of Da¯ is based on the observed plate height of the non-retained analytes as a function of the applied Reynolds number. Da¯ values are directly estimated from the best fit of the general Golay HETP equation to the experimental plate height curves.The experimental data revealed that under a pre-turbulent flow regime (Re < 2000), Da¯ is 2–6 times larger (3.5 × 10−4 cm2/s) than the bulk diffusion coefficients Dm of the analyte (1.6 × 10−4 and 5.8 × 10−5 cm2/s for acetonitrile and coronene, respectively). This result was explained by the random formation of decaying or vanishing turbulent puffs under pre-turbulent flow regime. Yet, the peak width remains controlled exclusively by the slow mass transfer in the mobile phase across the inner diameter (i.d.) of the OTC. Under sustained turbulent flow regime (Re > 2500), Da¯ is about four to five orders of magnitude larger than Dm. The experimental data slightly overestimated the turbulent dispersion coefficients predicted by Flint-Eisenklam model (Da¯=4 cm2/s). The discrepancy is explained by the approximate nature of the general Golay equation, which assumes that Da¯ is strictly uniform across the entire i.d. of the OTC. In fact, both the viscous and buffer wall layers, in which viscous effects dominate inertial effects, cannot be considered as fully developed turbulent regions. Remarkably, the mass transfer mechanism in OTC under sustained turbulent flow regime is not only controlled by longitudinal dispersion but also by a slow mass transfer in the mobile phase across the thick buffer layer and the thin viscous layer. Altogether, these layers occupy as much as 35% of the OTC volume at Re = 4000. From a theoretical viewpoint, the general Golay HETP equation is only an approximate model which should be refined based on the actual profile of the analyte dispersion coefficient across the OTC i.d. In practice, the measured plate height of non-retained analytes under sustained turbulent flow of carbon dioxide are two orders of magnitude smaller than those expected under hypothetical laminar flow regime.

Scalar mixing in anisotropic turbulent flow

While turbulent mixing has been studied extensively in homogeneous turbulence, chemical engineering processes where mixing is important are anisotropic. In anisotropic turbulence, the interplay between convection and diffusion is critical. Flow in an infinite channel is utilized here with clouds of scalars released instantaneously at different distances from the wall and at Schmidt numbers between 0.7 and 2400. Qualitative and quantitative measures of mixing efficiency and intensity are defined and the dynamics of mixing are explored. It is found that molecular diffusivity can even hinder mixing in some instances, because it affects the development of the cloud of the released scalars from regions within the viscous wall layer. Another finding is that while one would expect mixing to occur mostly in the space between two points of release, considerable amount of mixing could take place outside of this region and closer to the wall. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2803–2815, 2018

On the effect of surface roughness in transonic high Reynolds number flow through slender channels

AbstractIn the present study transonic weakly three‐dimensional flows at high Reynolds numbers in slender channels, as they are found e.g. in micro nozzles and turbo‐machines of micro‐electromechanical‐ systems (MEMS), are considered. The channel height is taken to be so small that the viscous layers forming at the channel walls start to interact strongly rather than weakly with the predominantly inviscid core flow and, therefore, the classical boundary layer approach fails. The resulting viscous‐inviscid interaction problem is formulated using matched asymptotic expansions and found to be governed by a triple deck structure. As a consequence, the properties of the inviscid core region and the viscous wall layers have to be calculated simultaneously in the interaction region. Weakly three‐dimensional effects caused by surface roughness, upstream propagating flow perturbations and boundary layer separation are discussed. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Triple-deck analysis of transonic high Reynolds number flow through slender channels.

In this work, laminar transonic weakly three-dimensional flows at high Reynolds numbers in slender channels, as found in microsupersonic nozzles and turbomachines of micro-electro-mechanical systems, are considered. The channel height is taken so small that the viscous wall layers forming at the channel walls start to interact strongly rather than weakly with the inviscid core flow and, therefore, the classical boundary layer approach fails. The resulting viscous-inviscid interaction problem is formulated using matched asymptotic expansions and found to be governed by a triple-deck structure. As a consequence, the properties of the predominantly inviscid core region and the viscous wall layers have to be calculated simultaneously in the interaction region. Weakly three-dimensional effects caused by surface roughness, upstream propagating flow perturbations, boundary layer separation as well as bifurcating solutions are discussed. Representative results for subsonic as well as supersonic conditions are presented, and the importance of these flow phenomena in technical applications as, for example, a means to reduce shock losses through the use of deformed geometry is addressed.

Linear stability of miscible two-fluid flow down an incline

We show that miscible two-layer free-surface flows of varying viscosity down an inclined substrate are different in their stability characteristics from both immiscible two-layer flows, and flows with viscosity gradients spanning the entire flow. New instability modes arise when the critical layer of the viscosity transport equation overlaps the viscosity gradient. A lubricating configuration with a less viscous wall layer is identified to be the most stabilizing at moderate miscibility (moderate Peclet numbers). This also is in contrast with the immiscible case, where the lubrication configuration is always destabilizing. The co-existence that we find under certain circumstances, of several growing overlap modes, the usual surface mode, and a Tollmien-Schlichting mode, presents interesting new possibilities for nonlinear breakdown.

On internal fluid dynamics

Motivated by a range of applications (biomedical, industrial, engineering, environmental) this contribution is focussed on a mathematical study of (a) constriction/distortion and (b) branching in a vessel or network of vessels containing fluid flow. The central interest addressed is in medium-to-high Reynolds numbers where asymptotic approaches and matching yield much insight. The main reasoning, order arguments and scaling factors within various parts of the vessels are presented. Theory and corresponding analysis are described for aspect (a) in symmetric and nonsymmetric cases and aspect (b) over short or long length scales with or without viscous–inviscid interactions, where attention is given to side-branching, large networks, viscous wall layers, flow reversal, eddies and upstream influence. Three-dimensional effects for (a) and (b) are also investigated. A final discussion includes suggestions of future project topics.

The Walker Function

The quantitative description of turbulent flows is known to be severely hampered by the extremely rapid variations in the mean and higher-order statistics in the near-wall region. Some very early studies [1, 2, 3] showed that the basic structure of an attached turbulent boundary layer consists of a viscous wall layer, in which the turbulent and laminar stresses are of comparable magnitude, and a defect layer, in which the velocity profile may be expressed in terms of a small perturbation to the external flow solution [4]. Also, [1, 2, 3] showed that this structure naturally leads to a universal velocity solution that has logarithmic behavior and depends on the velocity and length scales based on the friction velocity.

Wall shape effects on multiphase flow in channels

Two-fluid flow is examined analytically and numerically for increased flow rates through a channel with surface roughness or branching or both. The viscosity and density ratios of the fluids are of order unity. There is much concern in terms of applications as well as fluid dynamical phenomena in configurations where one fluid is present only as a thin layer near an outer wall, leaving the other fluid occupying the channel core and part of a viscous wall layer. The interactive dynamics in both regions is studied and numerical and asymptotic analyses are performed. The major situations examined are: the flow to two symmetrically bifurcating daughters and the flow in a single channel over a rough wall, as well as a combination of the two situations. The principal phenomena of interest are conditions for flow reversal, the presence of upstream influence and the trajectories of the injected fluid as the density or viscosity ratio varies. Special relatively thin or thinning wall layers are produced when the core fluid viscosity increases or when the fluid travels downstream into a daughter vessel.

Weakly 3D Transonic Flows in Narrow Channels

AbstractSteady transonic flows through channels so slender that the classical boundary layer approach fails are considered. The resulting viscous inviscid interaction problem for weakly three‐dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes two‐dimensional in the leading order approximation. The behavior of the flow upstream of a surface mounted three‐dimensional obstacle will be demonstrated. (© 2010 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Local Interaction Theory for Laminar Transonic Flows in Slender Channels

AbstractTransonic flows through channels so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting viscous inviscid interaction problem for weakly three dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes planar in the leading order approximation. Representative solutions for subsonic as well as for supersonic flows disturbed by three dimensional surface mounted obstacles will be presented. (© 2009 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Sensitivity of incipient particle motion to fluid flow penetration depth within a packed bed

A discrete element method is applied to a three-dimensional analysis related to sediment entrainment on a micro-scale. Sediment entrainment is the process by which a fluid medium accelerates particles from rest and advects them upward until they are either transported as bedload or suspended by the flow. Modelling of the entrainment process is a critically important aspect for studies of erosion, pollutant resuspension and transport, and formation of bedforms in environmental flows. Previous discrete element method studies of sediment entrainment have assumed the flow within the particle bed to be negligible and have only allowed for the motion of the topmost particles. At the same time, micro-scale experimental studies indicate that there is a small slip of the fluid flow at the top of the bed, indicating the presence of non-vanishing fluid velocity within the topmost bed layers. The current study demonstrates that the onset of particle incipient motion, which immediately precedes particle entrainment, is highly sensitive to this small fluid flow within the topmost bed layers. Using an exponential decay profile for the inner-bed fluid flow, the discrete element method calculations are repeated with different fluid penetration depths within the bed for several small particle Reynolds numbers. For cases with slip velocity corresponding to that observed in previous experiments with natural sediment, the predicted particle velocity is found to be a few percent of the fluid velocity at the top of the viscous wall layer, which is a reasonable range of velocities for observation of incipient particle motion. This method for prescribing the fluid flow within the particle bed allows for the current discrete element method to be extended in future studies to the analysis of sediment entrainment under the influence of events such as turbulent bursting. Additionally, predictions for the slip velocities and fluid flow profile within the bed suggest the need for further experimental studies to provide the data necessary for additional improvement of the discrete element method models.

Transonic viscous inviscid interactions in narrow channels

AbstractUnsteady as well as steady transonic flows through channels which are so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows is formulated for both perfect and dense gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes one‐dimensional in the leading order approximation. The latter allows an interpretation of the flow in the core region by means of the theory of one‐dimensional transonic inviscid flow through a Laval nozzle while preserving the essential features of the interaction problem associated with the internal structure of pseudoshocks. The sensitivity of a separation bubble caused by a pseudoshock of sufficient strength to perturbations under the condition of choked flow will be demonstrated. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)