Viscous Wall Layer Research Articles

Mechanisms for particle transfer and segregation in a turbulent boundary layer

Particle transfer in the wall region of turbulent boundary layers is dominated by the coherent structures which control the turbulence regeneration cycle. Coherent structures bring particles toward and away from the wall and favour particle segregation in the viscous region, giving rise to non-uniform particle distribution profiles which peak close to the wall. The object of this work is to understand the reasons for higher particle concentration in the wall region by examining turbulent transfer of heavy particles to and away from the wall in connection with the coherent structures of the boundary layer. We will examine the behaviour of a dilute dispersion of heavy particles – flyashes in air – in a vertical channel flow, using pseudo-spectral direct numerical simulation to calculate the turbulent flow field at a shear Reynolds number Reτ = 150, and Lagrangian tracking to describe the dynamics of particles. Drag force, gravity and Saffman lift are used in the equation of motion for the particles, which are assumed to have no influence on the flow field. Particle interaction with the wall is fully elastic. As reported in several previous investigations, we found that particles are transferred by sweeps – Q2 type events – in the wall region, where they preferentially accumulate in the low-speed streak environments, whereas ejections – Q4 type events – transfer particles from the wall region to the outer flow. We quantify the efficiency of the instantaneous realizations of the Reynolds stresses events in transferring different size particles to the wall and away from the wall, respectively. Our findings confirm that sweeps and ejections are efficient transfer mechanisms for particles. In particular, we find that only those sweep and ejection events with substantial spatial coherence are effective in transferring particles. However, the efficiency of the transfer mechanisms is conditioned by the presence of particles to be transferred. In the case of ejections, particles are more rarely available since, when in the viscous wall layer, they are concentrated under the low-speed streaks. Even though the low-speed streaks are ejection-like environments, particles remain trapped for a long time. This phenomenon, which causes accumulation of particles in the near-wall region, can be interpreted in terms of overall fluxes toward and away from the wall by the theory of turbophoresis. This theory, proposed initially by Caporaloni et al. (1975) and re-examined later by Reeks (1983), can help to explain the existence of net particle fluxes toward the wall as a manifestation of the skewness in the velocity distribution of the particles (Reeks 1983). To understand the local and instantaneous mechanisms which give rise to the phenomenon of turbophoresis, we focus on the near-wall region of the turbulent boundary layer. We examine the role of the rear-end of a quasistreamwise vortex very near to the wall in preventing particles in the proximity of the wall from being re-entrained by the pumping action of the large, farther from the wall, forward-end of a following quasi-streamwise vortex. We examine several mechanisms for turbulence structures near the wall and we find that the mechanism based on the archetypal quasi-streamwise structures identified by Schoppa & Hussain (1997), the parent–offspring regeneration cycle for near-wall quasi-streamwise vortices discussed by Brooke & Hanratty (1993), and the mechanism based on coherent packets of hairpin vortices, the fundamental super-structure characterized by Adrian, Meinhart & Tomkins (2000), all depict the same characteristic pattern which is responsible for particle trapping very near to the wall.

Transonic laminar interacting boundary layers in narrow channels

AbstractThe investigation of transonic viscous‐inviscid interactions is hampered by the fact that the nonlinear small disturbance equation which governs the external flow has to be solved simultaneously with the nonlinear boundary layer equations. This poses an extremely difficult numerical problem which has been treated so far with limited success only. However, if the medium is confined in a sufficiently narrow channel, the flow outside the viscous wall layers is one‐dimensional in the leading order approximation which in turn allows the derivation of a solution in closed form. This significantly simplifies the construction of numerical solutions which nevertheless display essential features of transonic flows associated with the transition from subsonic to supersonic conditions or/and vice versa.

Observations of the formation of streamwise vortices by rotation of arch vortices

Vortices, oriented in the streamwise direction, play an important role in generating Reynolds shear stresses (and turbulence) for flow over a smooth wall. Many of these have been observed to originate from tiny streamwise vortices, located in the immediate vicinity of the wall. In this paper we identify a different process that forms about 30 percent of the streamwise vortices at locations, away from the wall, in the middle of the viscous wall layer. This is accomplished by examining the changes, with time, of the turbulent field obtained from a direct numerical simulation of turbulent flow in a channel. Streamwise vortices create a shear layer by pumping low momentum fluid from the wall. One or more small spanwise vortices are formed at the top of this layer. These grow in size and rotate in the direction of flow. Previous investigators have suggested that spanwise vortices could have a direct role in the formation of streamwise vortices. This paper describes, in detail, a process by which this is accomplished. Of particular interest is the need to recognize that the shear layer is asymmetric since it is formed by a single streamwise vortex rather than a pair of counter-rotating vortices.

Steady flow in a dividing pipe

The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the bifurcation, but wall shear on the curved wall is found to increase from its undisturbed value downstream.

Boundary wave-vortex interaction in channel flow with a wavy wall at high Reynolds numbers

Asymptotic and scaling analyses at high Reynolds numbers are used to investigate channel flow over a wavy wall. Flow is divided into a core away from the boundaries and two viscous wall layers adjacent to the boundaries. The spanwise dependent amplitudes of the Tollmien–Schlichting (TS) and boundary waves contribute to a source term in the system for the longitudinal vortices. The presence of this source term requires streamwise variation of either the TS wave or the boundary wave. For the case where the initial TS wave does not exist, the boundary wave, due to the wavy wall, controls the existence, scales, variations and shape of the longitudinal vortices. For the case where the initial TS wave is present, the wavy wall affects the TS wave and the vortex motion through the boundary wave. The TS wave is subjected to secondary three-dimensional instability for favorable boundary waves due to particular forms of the wavy wall, while the secondary instability can be suppressed by special boundary waves due to some other particular wavy walls.

Characterizing boundary-layer instability at finite Reynolds numbers

When the Reynolds number is treated as an asymptotically large number in a boundary-layer stability analysis, it is possible to identify Reynolds number scalings at which different terms in the governing equations balance. These balances determine which physical mechanisms are operating under which circumstances and enable a systematic treatment of the various parameter regimes to be carried out. Linear waves can grow by viscous mechanisms if the velocity profile is noninflexional and both viscous and inviscid instabilities are present for inflexional profiles. When disturbances become nonlinear the resulting dynamics depend upon the type of instability and, in particular, on whether the critical layer lies within the viscous wall layer or is separate from it. Therefore, the classification of a wave as viscous or inviscid is important to the theory of transition. However, at finite Reynolds numbers the boundaries separating different types of instability become blurred. Moreover, certain asymptotic theories are known to give poor quantitative agreement with experiment while others remain untested by detailed experimental comparison. This paper is concerned with identifying the domains where the different asymptotic theories are most relevant so as to facilitate their comparison with experiment. Numerical solutions of the Orr-Sommerfeld equation are presented that indicate that in wind-tunnel experiments the instability driving transition is esssentially viscous even for adverse pressure gradient boundary layers, and that inviscid instability waves would be difficult to observe. For an inviscid wave near the upper branch there could be a lower frequency viscous wave at the same point in the flow with amplitude 50 times larger in a typical practical situation.

The effect of buoyancy on upper-branch Tollmien-Schlichting wave

We investigate the effect of buoyancy on the upper-branch linear stability characteristics of an accelerating boundary-layer flow. The presence of a large thermal buoyancy force significantly alters the stability structure. As the factor G (which is related to the Grashof number of the flow, and defined in Section 2) becomes large and positive, the flow structure becomes two layered and disturbances are governed by the Taylor-Goldstein equation. The resulting inviscid modes are unstable for a large component of the wavenumber spectrum, with the result that buoyancy is strongly destabilizing. Restabilization is encountered at sufficiently large wavenumbers. For G large and negative the flow structure is again two layered. Disturbances to the basic flow are now governed by the steady Taylor-Goldstein equation in the majority of the boundary layer, coupled with a viscous wall layer. The resulting eigenvalue problem is identical to that found for the corresponding case of lower-branch Tollmien-Schlichting waves, thus suggesting that the neutral curve eventually becomes closed in this limit.

On the neutral curve of the flat-plate boundary layer: comparison between experiment, Orr–Sommerfeld theory and asymptotic theory

The neutral stability curve for the flat-plate boundary layer has been calculated using the Orr–Sommerfeld equation and compared to those obtained using upper- and lower-branch scalings. The Orr–Sommerfeld results agree well with the lower-branch scaling at Reynolds numbers relevant to experiment, but agree well with the upper-branch scaling only for Rδ > 105. It is shown that the critical layer only emerges from the viscous wall layer when Rδ > 105. This suggests that for Rδ < 105, when the critical layer lies within the viscous wall layer, the disturbance has a triple-deck structure, even for the upper branch of the neutral curve (which can be reached if the phase jump across the critical layer is retained).The transition from a triple-deck to a five-deck structure with increasing Reynolds number on the upper branch occurs relatively abruptly and can be associated with a square-root branch point in the Tietjens function. Essentially, the lower- and upper-branch scalings pertain to two different modes, the first possessing a triple-deck structure, the second a five-deck structure. The modes are connected at the branch point, and the neutral curves of each mode join to give a single curve close to this branch point. The asymptotic expansions for the upper- and lower-branch neutral curves depend upon the analyticity of the dispersion relationship, and so the proximity of the branch point indicates where these expansions will be liable to inaccuracies. This explains the poor neutral-curve predictions made by five-deck analyses at the Reynolds numbers where transition occurs.

The Nonlinear Critical‐Wall Layer in a Parallel Shear Flow

The nonlinear critical layer theory is developed for the case where the critical point is close enough to a solid boundary so that the critical layer and viscous wall layers merge. It is found that the flow structure differs considerably from the symmetric “eat's eye” pattern obtained by Benney and Bergeron [1] and Haberman [2]. One of the new features is that higher harmonics generated by the critical layer are in some cases induced in the outer flow at the same order as the basic disturbance. As a consequence, the lowest‐order critical layer problem must be solved numerically. In the inviscid limit, on the other hand, a closed‐form solution is obtained. It has continuous vorticity and is compared with the solutions found by Bergeron [3], which contain discontinuities in vorticity across closed streamlines.

On the nonlinear development of the most unstable Görtler vortex mode

The nonlinear development of the most unstable Görtler vortex mode in boundary-layer flows over curved walls is investigated. The most unstable Görtler mode is confined to a viscous wall layer of thickness O(G−1/5) and has spanwise wavelength O(G−1/5); it is, of course, most relevant to flow situations where the Görtler number G [Gt ] 1. The nonlinear equations governing the evolution of this mode over an O(G−3/5) streamwise lengthscale are derived and are found to be of a fully non-parallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Görtler equations valid for O(1) Görtler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite-difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flow field as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.

Linear, hydrodynamic flow in a rotating saturated porous medium

Linear, steady, axisymmetric flow of a homogeneous fluid in a rigid, bounded, rotating, saturated porous medium is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The dynamics of the interior region and vertical boundary layers are investigated as functions of the Ekman number E(=v/ΩL 2) and rotational Darcy 3 numberN(=kΩ/v) which measures the ratio between the Coriolis force and the Darcy frictional term. IfN≫E −1/2, the permeability is sufficiently high and the flow dynamics are the same as those of the conventional free flow problem with Stewartson'sE 1/3 andE 1/4 double layer structure. For values ofN≤E −1/2 the effect of porous medium is felt by the flow; the Taylor-Proudman constraint is no longer valid. ForN≪E −1/3 the porous medium strongly affects the flow; viscous side wall layer is absent to the lowest order and the fluid pumped by the Ekman layer, returns through a region of thicknessO(N −1). The intermediate rangeE −1/3≪N≪E −1/2 is characterized by double side wall layer structure: (1)E 1/3 layer to return the mass flux and (ii) (NE)1/2 layer to adjust the interior azimuthal velocity to that of the side wall. Spin-up problem is also discussed and it is shown that the steady state is reached quickly in a time scaleO(N).

Prediction of liquid circulation in viscous bubble columns

AbstractA new theory to predict liquid circulation was derived and compared with applicable literature data. A single dimensionless group, ε(gRd2/v2c), is shown to control mixing intensity under low‐viscosity conditions, but two separate groups (ε and gRd2/v2c) are necessary for viscous systems. Closure was accomplished using the Prandtl model for Reynolds stress. Locally varying mixing length was taken to be proportional to bubble size and bubble concentration. The model was constructed based on the premise that two distinct zones exist: a mainly turbulent core joined to a thin viscous wall layer.

A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls

An asymptotic theory is developed for two- and three-dimensional disturbances growing in a two-dimensional boundary layer over a compliant wall. The theory exploits the multideck structure of the boundary layer to derive asymptotic approximations at a high Reynolds number for the perturbation wall pressure and viscous stresses. These quantities can be regarded as driving the wall and, accordingly, the equation(s) of motion for the wall is (are) used as the characteristic equation(s) for finding the eigenvalue(s). The main assumptions are that the amplitude of the disturbance is sufficiently small for linear theory to hold, the Reynolds number is large, the disturbance wavelength is long compared with the boundary-layer thickness, and the critical and viscous wall layers are well separated. The theory was developed to study the travelling-wave flutter instability discussed by Carpenter and Garrad, i.e., the Class B instability of Benjamin and Landahl. Under certain limiting processes both the upper-branch and conventional triple-deck scalings for the Tollmien-Schlichting instability can be obtained with the present approach. Accordingly, the theory also gives a reliable qualitative guide to the effect of anisotropic wall compliance on the Tollmien-Schlichting instability.

On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien‐Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien‐Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien‐Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien‐Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien‐Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex‐wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien‐Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien‐Schlichting wave amplitude.

Ejection mechanisms in the sublayer of a turbulent channel

The structure of the vorticity field in the viscous wall layer of a turbulent channel is studied by examining the results of a fully resolved direct numerical simulation. It is shown that this region is dominated by intense three-dimensional shear layers in which the dominant vorticity component is spanwise. The advection and reproduction processes of these structures are examined and shown to be consistent with the classical generation mechanism for two-dimensional Tollmien–Schlichting waves. This process is fundamentally different from the usually accepted mechanism involving hairpin vortices.

Effect of favorable pressure gradients on turbulent boundary layers

AbstractLarge favorable pressure gradients can cause the drag of a turbulent boundary layer on a solid surface to decrease; if large enough, such gradients can cause a return to laminar flow. Scaling relations and a computational model for the viscous wall layer, recently developed in this laboratory are used to interpret this phenomenon.

Balance of Turbulent Energy in the Linear Wall Region of Channel Flow

A description of the energy balance in the wall region of turbulent flow is of importance both in understanding the physical process of turbulent motion and in developing useful closure models. In the present note it is shown that the presence of a very small viscous wall layer is in contradiction with the energy balance calculated by Townsend if it assumed that channel and pipe flows have a similar energy balance in this region.

Amplitude-dependent neutral modes in the Hagen-Poiseuille flow through a circular pipe

An investigation is described for the nonlinear stability, at large Reynolds numbers R , of the Hagen-Poiseuille flow through a pipe of circular cross section when subjected to three-dimensional disturbances of typical relative size δ large enough to yield only a vanishingly small phase shift across the critical layer. A crucial size is δ = O ( R -⅓ ) since then this small phase shift is in tune with the small phase shift produced by the viscous wall layers. The critical layer, which is fully nonlinear and three-dimensional, and the wall layers, where the disturbance is greater than the basic flow and flow reversal occurs, are discussed in detail. Neutral solutions are then found to exist in the range c 01 < c 0 < 1 with N = 1, where c 0 is the non-dimensional wavespeed, c 01 = 0.284 and N is the azimuthal wavenumber; there is also evidence to suggest that no similar neutral solutions exist outside that range. The amplitude-dependence of the neutral modes follows and it is such that the cut-off value c 0 = c 01 + is approached as the amplitude shrinks, whereas centre modes with c 0 → 1 - are produced as the amplitude becomes relatively large.

Unsteady viscous supersonic flow over a short flap

The method of matched asymptotic expansions is used to investigate the unsteady flow over a short flap deflected through an angle of order R−1/4 in a supersonic stream in a characteristic time ∼ R−1/4. The length of the flap and the pressure difference at the trailing edge are assumed to be such that an interaction region is formed near the flap for which the asymptotic description in the limit R → ∞ is analogous to the description of the flow in the neighborhood of the trailing edge of an oscillating plate [7]. A numerical method is proposed for calculating the unsteady flow and the heat fluxes in the viscous wall layer. The results of calculations make it possible to estimate the influence of the viscous interaction in unsteady flows on the aerodynamic coefficients of control elements of aircraft and the distribution of the heat fluxes over the surface of the body.

Removal of Goldstein's singularity at separation, in flow past obstacles in wall layers

It is shown that, in the flow of a viscous wall layer past a relatively steep obstacle at the wall, the Goldstein (1948) singularity generated in the classical boundary-layer approach to separation is removable in a physically sensible fashion. The removal is effected by means of a sequence of local double structures, the last of which arises just beyond separation owing to the occurrence of a further singularity which is also removable and describes the necessary complete breakaway of the viscous layer from the wall. The novel forms of the local pressure–displacement relations are the key elements allowing the solution to retain physical reality throughout. Beyond the breakaway the reattachment process takes place only at a relatively large distance downstream, before the motion returns to its original uniform shear form. The present flow configuration, the first we know of where Goldstein's singularity proves to be removable, has important applications in both internal and external flows at high Reynolds numbers and these are also discussed.