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Summary Artificial lift systems are widely used in oil production, of which sucker rod pumps are conceptually among the simpler ones. The reciprocating movement of the plunger triggers the opening and closing of two ball valves, allowing fluid to be pumped to the surface. Their built-in ball valves are subject to long-time erosion and fail as a consequence of this damage mechanism. Understanding the principal damage mechanisms requires a thorough examination of the fluid dynamics during the opening and closing action of these valves. In this article, we present a fluid-structure interaction model that simultaneously computes the fluid flow in the traveling valve (TV), the standing valve (SV), and the chamber of sucker rod pumps during a full pump cycle. The simulations shed light on the causes of valve damage for standard and nonideal operating conditions of the pump. In particular, our simulations based on real pump operating envelopes reveal that the so-called “midcycle valve closure” is likely to occur. Such additional closing and opening events of the valves multiply situations in which the flow conditions are harmful to the individual pump components, leading to efficiency reduction and pump failure. This mechanism, hitherto unreported in the literature, is believed to constitute the primary cause of long-term valve damage. Our finite element method-based computational-fluid-dynamics model can accurately describe the opening and closing cycles of the two valves. For the first time, this approach allows an analysis of real TV speed versus position plots, usually called pump cards. The effects of stroke length, plunger speed, and fluid parameters on the velocity and pressure at any point and time inside the pump can thus be investigated. Identifying the damage-critical flow parameters can help suggest measures to avoid unfavorable operating envelopes in future pump designs. Our flow model may support field operations throughout the entire well life, ranging from improved downhole pump design to optimized pump operation or material selections. It can aid the creation of an ideal interaction between the valves, thus avoiding midcycle valve closure to drastically extend the mean time between failures of sucker rod pumps. Finally, our simulation approach will speed up new pump component development while greatly reducing the necessity for costly laboratory testing.

Recently, the authors considered a thin steady developed viscous free-surface flow passing the sharp trailing edge of a horizontally aligned flat plate under surface tension and the weak action of gravity, acting vertically, in the asymptotic slender-layer limit (J. Fluid Mech., vol. 850, 2018, pp. 924–953). We revisit the capillarity-driven short-scale viscous–inviscid interaction, on account of the inherent upstream influence, immediately downstream of the edge and scrutinise flow detachment on all smaller scales. We adhere to the assumption of a Froude number so large that choking at the plate edge is insignificant but envisage the variation of the relevant Weber number of$O(1)$. The main focus, tackled essentially analytically, is the continuation of the structure of the flow towards scales much smaller than the interactive ones and where it no longer can be treated as slender. As a remarkable phenomenon, this analysis predicts harmonic capillary ripples of Rayleigh type, prevalent on the free surface upstream of the trailing edge. They exhibit an increase of both the wavelength and amplitude as the characteristic Weber number decreases. Finally, the theory clarifies the actual detachment process, within a rational description of flow separation. At this stage, the wetting properties of the fluid and the microscopically wedge-shaped edge, viewed as infinitely thin on the larger scales, come into play. As this geometry typically models the exit of a spout, the predicted wetting of the wedge is related to what in the literature is referred to as the teapot effect.

Artificial lift pumps are widely used in oil production, and among them, sucker rod pumps are conceptually the simplest ones. The reciprocating movement of the plunger triggers the opening and closing of two ball valves, allowing fluid to be pumped to the surface. These valves are subject to long-time erosion and fail as a consequence of this damage mechanism. We demonstrate that understanding the principal damage mechanisms in the necessary depth and breadth requires a thorough examination of the fluid dynamics during the opening and closing action of the ball valves. This paper describes the basic ingredients and results of fluid–structure interaction model that simultaneously computes the fluid flow in the traveling valve, the standing valve, and the chamber of sucker rod pumps during a full pump cycle in an efficient and accurate way. The simulations provide necessary insight into the causes of valve damage for realistic standard as well as non-ideal operating conditions of the downhole pump. In particular, simulations based on real pump operating envelopes reveal that the phenomenon of so-called ‘‘mid-cycle valve closure’’ is likely to occur. Such additional closing and opening events of the ball valves multiply situations where the flow conditions are harmful to the individual pump components, leading to efficiency reduction and pump failure. The computational-fluid-dynamics model based on the finite-element method serves to accurately describe the opening and closing cycles of the two valves. Most importantly, this approach for the first time allows an analysis of real operating envelopes, derived from actual dynamometer cards. The combination of stroke length, plunger speed, fluid parameters, and velocity at any point inside the pump can thus be investigated at any time during the pump cycle. The flow parameters identified as critical in terms of damaging pump valves or other pump components can set the basis for taking measures to avoid unfavorable operating envelopes in future pump designs. Our comprehensive flow model may support field operations throughout the entire well life, ranging from improved downhole pump design to optimized pump operating modes and envelopes as well as in material selections. It is suggested to aid in adapting pump operating conditions to create an ideal interaction between the valves and avoiding the "mid-cycle valve closure". Specifically, a so-optimized pump design is expected to drastically extend the operation time before failure of sucker rod pumps. Finally, this type of simulation will speed up new pump or pump component development, and can eliminate or at least reduce the necessity of extensive and costly laboratory testing.

The steady laminar annular spread of a thin liquid film generated by a circular jet which impinges perpendicularly in direction of gravity on the centre of a rotating disc is examined both analytically and numerically. Matched asymptotic expansions of the flow quantities provide the proper means for studying the individual flow regimes arising due to the largeness of the Reynolds number formed with the radius of the jet, its slenderness and the relative magnitude of the centrifugal body force. This is measured by a suitably defined Rossby number,$Ro$. The careful analysis of jet impingement predicts a marked influence of gravity and surface tension on the film flow, considered in the spirit of a shallow-water approach, only through the vorticity imposed by the jet flow. Accordingly, associated downstream conditions are disregarded as the local Froude and Weber numbers are taken to be sufficiently large. Hence, the parabolic problem shaped from the governing equations in a rigorous manner describes the strongly supercritical spread of a developed viscous film past an infinite disc, essentially controlled by$Ro$. Its numerical solutions are discussed for a wide range of values of$Ro$. The different flow regimes reflecting varying effects of viscous shear and centrifugal force are elucidated systematically to clarify the surprising richness of flow phenomena. Special attention is paid to the cases$Ro\gg 1$and$Ro\ll 1$. The latter, referring to relatively high disc spin, implies a delicate breakdown of the asymptotic flow structure, thus requiring a specific analytical and numerical treatment. Finally, the impact of gravity and capillarity and thus of the disc edge on the film flow is envisaged in brief.

We consider the asymptotic structure of a steady developed viscous thin film passing the sharp trailing edge of a horizontally aligned flat plate under the weak action of gravity acting vertically and surface tension. The surprisingly rich details of the flow in the immediate vicinity of the trailing edge are elucidated both analytically and numerically. As a central innovation, we demonstrate how streamline curvature serves to regularise the edge singularity apparent on larger scales via generic viscous–inviscid interaction. This is shown to be provoked by weak disturbances of accordingly strong exponential downstream growth, which we trace from the virtual origin of the flow towards the trailing edge. They represent a prototype of the precursor to free interaction in the most general sense, which, interestingly, has not attracted due attention previously. Moreover, we delineate how an increased effect of gravity involves marginally choked flow at the edge.

Highlights of the asymptotic and numerical analysis of the steady axisymmetric swirl flow of a Newtonian liquid over a spinning disc and generated by a jet, impacting perpendicularly onto the latter in the direction of gravity, are presented. Ubiquitous in engineering applications and involving a myriad of disparate velocity and length scales, thus extreme aspect ratios, this flow configuration is an archetypical one for the application of dimensional reasoning and matched asymptotic expansions in fluid dynamics. Particular interest lies on the recent advances in the rigorous description of the thin developed layer relatively far from the jet, which is essentially parametrised by a suitably defined Rossby number, the influence of gravity on the thin film and its interplay with a finitely remote disc edge. The latter upstream influence explains the phenomenon of the hydraulic jump in developed flow. The clarification of long-standing and more recent controversies around this concludes the analysis.

The transport of energy in a moving fluid with a simply connected free surface is analyzed, taking into account the contribution of surface tension. This is done by following a "control volume" with arbitrary, specified velocity, independent of the flow velocity, and determining the rates of energy passing through the boundaries, as well as the energy dissipation in the bulk. In particular, a simple conservation equation for the surface area is written down, which clearly shows the contribution of the Laplace pressure at the free surface and the tangential surface tension forces at its boundary. It emerges as the mechanical conservation law for the surface energy in its general form. For a static control volume, all contributions from surface tension disappear, except that the pressure has to be modified by the Laplace contribution.

The viscous hydraulic jump still represents research in progress rather than a finalised edifice. The existing rigorous approaches show how this phenomenon is tied in with a bifurcation of the upstream flow adjacent to the guiding rigid plate of finite length, aligned perpendicularly to the direction of gravity. Here, this together with the upstream influence by the detached flow triggers transition from super‐ to subcritical flow (sensing its susceptibility to the upstream propagation of small disturbances). We present recent advances in the self‐consistent theory of single‐layer jumps continued as free shear layers.

The containers used for steel slag transportation to the recycling depot undergo high-temperature gradients and often deform plastically. Also, parts of the thereby solidified slag adhere to the pot walls, causing demolding problems and wear. A thorough finite-element analysis of the heat transfer, initiated by filling the pots and essentially driven by radiation, and the thermal stresses is performed. Due to periodic fill-in and discharge, these are assumed to admit a quasi-stationary state referring to the pot temperature before their emptying. The so obtained results aid optimizing the pot shape and the transport process in terms of minimizing maintenance and anti-adhesive cladding. A layer of solidified slag is found to exist throughout the transport process. Although the phase change occurs almost instantaneously at a liquid–solid interface, the “mushy zone” is considered in the accompanying analytical study of the associated Stefan-type problem. Good correlations of the predicted temperatures to the measured ones and the resultant surface damaging are obtained.

Wall shear rates are critical for a broad variety of fluidic phenomena and are taken into account in nearly every experimental or simulation study. Generally, shear rates are not observable directly but rather derived from other parameters such as pressure and flow, often assuming somehow idealized systems. However, there is a biological system which is able to constantly measure the wall shear as a part of a regulatory circuit: The blood circulation system takes advantage of shear rate sensor (protein)molecules (multimeric forms of von Willebrand Factor, VWF), which are dissolved in the blood plasma and dramatically change their conformation under shear conditions. The conformational changes are accompanied by several functional variations and therefore interplay with the regulation of the coagulation system. In this study, we use a recombinantly produced and therefore well-defined multimeric form of VWF as a sensor which directly responds to shear rates. Shear rates, up to 32.000 s-1, were obtained using a kind of micro-plate-to-plate rheometer capable of adsorbing shear-stretched VWF oligomeric molecules on a surface to conserve their differently stretched conformation and so allow detection of their elongation by atomic force microscopy. The laminar flow in this geometrically simple device has been characterized by adopting classical fluid dynamical models, in order to ensure well-known, stable shear rates which could be correlated quantitatively with an observed stretching of sensor molecules.

In this sequel to a rather recent paper on the classical problem of Helmholtz–Kirchhoff flows by Vic. V. Sychev (TsAGI Sci J 41(5):531–533, 2010), the representation of the flow far from the body and its specific implications discussed in that study are revisited. Here the concise derivation of these findings resorts to well-known Levi-Civita’s method and, alternatively, only fundamental properties of analytic functions and thin-airfoil theory. As particularly of interest when the well-known Kirchhoff parabola degenerates to an infinitely long cusp, integration constants debated controversially so far and important for the understanding and computation of those flows are specified by the integral conservation of momentum. Also, the parametric modification towards flows encompassing stagnant-fluid regions of finite extent and the previously unnoticed impact of higher-order terms on the associated high-Reynolds-number flows are addressed.

In this work we present a combined finite element model of a standing valve and a travelling valve in a down-hole pump used in oil extraction. The model allows the self-consistent simulation of several pump cycles without the need to assume artificial coupling conditions between two separate valves, at the cost of large variations in length of the system during a pump cycle. We discuss how to overcome the difficulties associated with such a model, such as avoiding solid–solid contact during valve closure and how to correctly implement the plunger kinematics. We then compare the behaviour of a down-hole pump operating according to ideal plunger kinematics to one with realistic plunger kinematics that are dominated by the characteristic oscillations of the sucker rod. In the latter case, we observed that transient reductions in the plunger velocity can lead to mid-cycle valve closure. This poses a threat in terms of additional damage to the components due to ball impacts and fluid pressure drops, which could in turn lead to undesirable accumulation of gas.

In the paper production, doctor (scraping) blades are placed in contact with press rolls during wet pressing so as to purge the surface of the rolls from processing water, contamination and stickies. The contact is achieved by mounting the blade on a holder, which is tilted around a rotation axis until the blade tip contacts with the roll. The contact force is determined by the supply pressure of the air forced through the tube that is placed at the bottom of the holder. Owing to contact, the blade wears off and needs to be replaced periodically. Our aim is to optimise the cleaning performance of the system by modelling the tribological contact between the doctor blade and press roll in order to achieve an optimum cleaning performance, thus increasing the blade lifetime and reducing energy consumption. The model is susceptible to an inextensive numerical evaluation as compared to that of a more advanced modelling approach, e.g. in terms of a full finite element analysis of the beam deflection. A first comparison with experimental findings is encouraging.

Doctor blades are commonly used in paper machines to keep the surface of rollers clean. Due to higher demanding conditions, the requirements for doctor blades have steadily increased. The wear rates must remain low, while simultaneously their cleaning function has to be ensured. For this reason, the paper industry has developed a high degree of empirical knowledge concerning the cleaning of roller surfaces. However, up to now, no systematic approach has been successfully applied to optimize the cleaning performance of the doctor blade-roller tribosystem. This study presents an attempt to model the system based on the force equilibrium conditions at the blade tip between hydrodynamic and contact forces. The change of the blade geometry due to wear is also taken into account. By these means, a non-dimensional group involving the key parameters is obtained. This allows for a systematic improvement of the cleaning efficiency, by targeted changes of the process parameters.

This study deals with a comparison between new experiments on the frictional behavior of porous journal bearings and its prediction by previous numerical simulations. The tests were carried out on bearings lubricated with polyalphaolefin (PAO)-based oils of distinct viscosities. The theoretical model underlying the simulations includes the effects of cavitation by vaporization and accounts for the sinter flow by virtue of Darcy's law. The effective eccentricity ratio corresponding to the experimentally imposed load is estimated by an accurate numerical interpolation scheme. The comparison focuses on the hydrodynamic branches of the Stribeck curve by dimensional analysis (DA), where the variations of the lubricant viscosity with temperature are of main interest. The numerically calculated values of the coefficient of friction are found to reproduce the experimentally obtained ones satisfactorily well in terms of overall trends; yet, the former lie predominantly below the measured ones, which results in a low-positive correlation between the two.

The classical analysis of turbulent boundary layers in the limit of large Reynolds number Re is characterised by an asymptotically small velocity defect with respect to the external irrotational flow. As an extension of the classical theory, it is shown in the present work that the defect may become moderately large and, in the most general case, independent of Re but still remain small compared to the external streamwise velocity for non-zero pressure gradient boundary layers. That wake-type flow turns out to be characterised by large values of the Rotta–Clauser parameter, serving as an appropriate measure for the defect and hence as a second perturbation parameter besides Re. Most important, it is demonstrated that also this case can be addressed by rigorous asymptotic analysis, which is essentially independent of the choice of a specific Reynolds stress closure. As a salient result of this procedure, transition from the classical small defect to a pronounced wake flow is found to be accompanied by quasi-equilibrium flow, described by a distinguished limit that involves the wall shear stress. This situation is associated with double-valued solutions of the boundary layer equations and an unconventional weak Re-dependence of the external bulk flow—a phenomenon seen to agree well with previous semi-empirical studies and early experimental observations. Numerical computations of the boundary layer flow for various values of Re reproduce these analytical findings with satisfactory agreement.

This contribution puts forward some recent advances in the rigorous (asymptotic) theory of gravity‐ (and capillarity‐)driven shallow flow of a viscous liquid past a horizontal plate, originating in jet impingement oblique to it. Hence, our concern is twofold: with steady developed flow over the distance from the jet centre to the trailing edge of the plate, referred to as a pronounced hydraulic jump blurred by viscous diffusion; with the predominantly inviscid transcritical limit arising near the edge due to scale reduction given an intrinsic expansive singularity taking place there. In the latter situation envisaged briefly, condensing nonlinear inertial effects, weak time dependence, and (very) weak streamline curvature as the essential ingredients into a distinguished limit demonstrates the generation of a weak (transcritical) hydraulic jump by a plate‐mounted obstacle.

The present theoretical article, dedicated to the memory of James Lighthill and his research contributions, is directed towards the central features of turbulent separation. The focus is on the time-mean equations, where the ensemble-averaged motion for an incompressible fluid is modelled as planar and steady. Specific major recent developments are discussed which are closely concerned in one way or another with turbulent separation at increased Reynolds numbers. These developments include in particular the behaviour of relatively thick boundary layers, on the one hand, and the intricate behaviour of breakaway separation on the other. The article brings the two behaviours together in a discussion of large-scale separation structure and accompanying interactions. The work presented here applies to quite general turbulence models and for specific cases enables direct comparisons to be made with experiments. The work also leads to new suggestions involving a combination of low- and high-intensity turbulence for the fluid motion around a bluff body and in its wake.

The paper deals with separation of a nominally steady and two-dimensional incompressible boundary layer from the smooth surface of a rigid blunt body in the presence of a front stagnation point, here denoted by PF. In agreement with earlier studies on the flow past a bluff body, it is argued that in the limit of vanishing viscosity, i.e. in the limit where the globally defined Reynolds number Re takes on arbitrarily large values, the solution of this problem is to be sought in the class of steady potential flows with free streamlines. Hence, it is first assumed that for sufficiently large values of Re the boundary layer upstream of separation is a fully developed turbulent one. Accordingly, it is demonstrated numerically and analytically, by adopting asymptotic techniques, how the well-known laminar flow taking place close to PF is gradually transformed into a fully developed turbulent boundary layer further downstream, which exhibits the well-established typical asymptotic two-layer splitting. However, as has been shown in a preceding study, this type of flow does not allow for a self-consistent rational description of separation, based on the nominally steady form of the equations of motion. From this result, supported by numerical and experimental evidence, the tentative but rather remarkable conclusion is drawn that the boundary layer along the smooth surface of a bluff body never attains a fully developed turbulent state, even in the limit Re→∞. Most important, these findings are seen to be independent of the choice of a specific turbulence closure.

A new rational theory of incompressible turbulent boundary-layer flows having a large velocity defect is presented on basis of the Reynolds-averaged Navier-Stokes equations in the limit of infinite Reynolds number. This wake-type formulation allows for, among others, the prediction of singular solutions of the boundary-layer equations under the action of a suitably controlled adverse pressure gradient, which are associated with the onset of marginally separated flows. Increasing the pressure gradient locally then transforms the marginal-separation singularity into a weak Goldstein-type singularity occurring in the slip velocity at the base of the outer wake layer. Interestingly, this behavior is seen to be closely related to (but differing in detail from) the counterpart of laminar marginal separation, in which the skin friction replaces the surface slip velocity. Most important, adopting the concept of locally interacting boundary layers results in a closure-free and uniformly valid asymptotic description of boundary layers that exhibit small, closed reverse-flow regimes. Numerical solutions of the underlying triple-deck problem are discussed.

We complement the recently achieved status quo of a self-consistent asymptotic theory: incompressible-flow separation from the perfectly smooth surface of a bluff rigid obstacle that perturbs an otherwise uniform flow in an unbounded domain. Here the globally formed Reynolds number, Re, takes on arbitrarily large values, and we are concerned with a long-standing challenge in boundary layer theory. Specifically, the external flow is sought in the class of potential flows with free streamlines, and the level of turbulence intensity, concentrated in the boundary layer undergoing separation, is measured in terms of distinguished limits. Their particular choices categorise the type of the viscous-inviscid interaction mechanism governing local separation and the strength of its downstream delay when compared with laminar-flow separation. In the case of extreme retardation, this implies the selection of a fully attached potential flow around a closed body, the singular member of the family of free-streamline flows. In turn, the asymptotic theory predicts the distance of the separation from the thus emerging rear stagnation point or trailing edge of the body to vanish at a rate much weaker than that given by \(1/\ln Re\), which plays a crucial role in the scaling of firmly attached turbulent boundary layers. Notably, the overall theory only resorts to specific turbulence closures when it comes to numerical v investigations.

A novel self-consistent description of time-mean two-dimensional turbulent-boundary-layer flow separating from a bluff body at arbitrarily large globally formed Reynolds numbers is presented. Contrasting with previous approaches, the theory deals with a sufficient delay of flow detachment or, correspondingly, increase of the turbulence intensity so as to both settle the question of the actual position of separation and trigger a turbulent boundary layer exhibiting a large relative streamwise velocity deficit. At separation, a generic variation of the velocity profile close to the body surface with the third power of the distance from it is detected. The Euler stage resulting from the breakdown of the incident boundary layer and governed by its vorticity is envisaged in detail. Specifically, an analytical solution to the central linear vortex-flow problem could be established. This represents the essential ingredient for the understanding of the multi-layered substructure of the flow more close to the surface, which completes the picture of gross separation at the Euler scale. Most important, the analysis does not resort to any specific turbulence closure. Concerning the canonical situation of circular-cylinder flow, a first comparison between the predicted and publicly available experimentally obtained values of the separation angle is encouraging.

The contribution deals with recent theoretical results concerning separation of a turbulent boundary layer from a blunt solid object in a uniform stream, accompanied by a numerical study. The investigation is restricted to incompressible nominally steady two-dimensional flow past an impervious obstacle surface. Then the global Reynolds number represents the only parameter entering the description of the Reynolds-averaged flow. It shall be large enough to ensure that laminar–turbulent transition takes place in a correspondingly small region encompassing the stagnation point. Consequently, the concomitant asymptotic hierarchy starts with the external Helmholtz–Kirchhoff potential flow, which detaches at an initially unknown point Open image in new window from the body, driving the turbulent boundary layer. It is found that the separation mechanism is inherently reminiscent of the transition process. The local analysis of separation not only fixes the actual scaling of the entire boundary layer but is also expected to eventually predict the position of Open image in new window in a rational way.

Massive flow separation from the surface of a plane bluff obstacle in an incompressible uniform stream is addressed theoretically for large values of the global Reynolds numberRe. The analysis is motivated by a conclusion drawn from recent theoretical results which is corroborated by experimental findings but apparently contrasts with common reasoning: the attached boundary layer extending from the front stagnation point to the position of separation never attains a fully developed turbulent state, even for arbitrarily largeRe. Consequently, the boundary layer exhibits a certain level of turbulence intensity that is linked with the separation process, governed by local viscous–inviscid interaction. Eventually, the latter mechanism is expected to be associated with rapid change of the separating shear layer towards a fully developed turbulent one. A self-consistent flow description in the vicinity of separation is derived, where the present study includes the predominantly turbulent region. We establish a criterion that acts to select the position of separation. The basic analysis here, which appears physically feasible and rational, is carried out without needing to resort to a specific turbulence closure.

This contribution reports on recent efforts with the ultimate goal to obtain a fully self-consistent picture of turbulent boundary layer separation. To this end, it is shown first how the classical theory of turbulent boundary layers having an asymptotically small streamwise velocity deficit can be generalised rigorously to boundary layers with a slightly larger, i.e. moderately large, velocity defect and, finally, to situations where the velocity defect is of O(1). In the latter case, the formation of short recirculation zones describing marginally separated flows is found possible, as described in a rational manner.

We present the most relevant recent findings that allow for a rational timeaveraged description of laminar-turbulent transition of an incompressible nominally two-dimensional and steady boundary layer along the impermeable surface of a rigid blunt body. Rigorous application of matched asymptotic expansions for sufficiently high values of both the Reynolds number and a turbulence-level gauge parameter shows that the presence of a leading-edge stagnation point is associated with the generation of a turbulent shear layer that exhibits an asymptotically small streamwise velocity deficit. Remarkably, however, the turbulence intensity never reaches its theoretically possible maximum that conforms to fully developed turbulent flow.

This paper reports on recent efforts with the ultimate goal to obtain a fully self-consistent picture of turbulent boundary layer separation. To this end, it is shown first how the classical theory of turbulent small-defect boundary layers can be generalised rigorously to boundary layers with a slightly larger, i.e. moderately large, velocity defect and, finally, to situations where the velocity defect is of O(1). In the latter case, the formation of short recirculation zones describing marginally separated flows is found possible, as described in a rational manner.

A most reliable prediction of the position of time-mean gross separation of incompressible turbulent boundary layer (BL) flow from the smooth impervious surface of a rigid and more-or-less blunt body not only still defies theoreticians but, needless to say, is also of great interest from an engineering point of view. The undeniable vital progress in computational techniques made in the recent past does not master this challenge presently in the form of sufficiently accurate solutions of the full (unsteady) Navier–Stokes equations. This is largely due to the fact that for practical applications, e.g. in aerodynamics, the relevant Reynolds numbers are still too high to be dealt with adequately.

A rational treatment of time-mean separation of a nominally steady turbulent boundary layer from a smooth surface in the limit Re → ∞, where Re denotes the globally defined Reynolds number, is presented. As a starting point, it is outlined why the ‘classical’ concept of a small streamwise velocity deficit in the main portion of the oncoming boundary layer does not provide an appropriate basis for constructing an asymptotic theory of separation. Amongst others, the suggestion that the separation points on a two-dimensional blunt body is shifted to the rear stagnation point of the impressed potential bulk flow as Re → ∞ — expressed in a previous related study — is found to be incompatible with a self-consistent flow description. In order to achieve such a description, a novel scaling of the flow is introduced, which satisfies the necessary requirements for formulating a self-consistent theory of the separation process that distinctly contrasts former investigations of this problem. As a rather fundamental finding, it is demonstrated how the underlying asymptotic splitting of the time-mean flow can be traced back to a minimum of physical assumptions and, to a remarkably large extent, be derived rigorously from the unsteady equations of motion. Furthermore, first analytical and numerical results displaying some essential properties of the local rotational/irrotational interaction process of the separating shear layer with the external inviscid bulk flow are presented.

A new rational theory of incompressible turbulent boundary layer flows having a large velocity defect is presented on basis of the Reynolds-averaged Navier–Stokes equations in the limit of infinite Reynolds number, here denoted by Re. The approach is essentially based on the assumption, strongly supported by dimensional reasoning and an asymptotic investigation of all commonly employed shear stress closures, that the Reynolds equations admit a further limit apart from the pure Eulerian one. This then implies the existence of a non-dimensional small parameter, denoted by �, which measures the slenderness the turbulent boundary layer in the formal limit 1/Re = 0. The resulting wake-type formulation of wall-bounded shear flows for � ! 0 allows for, among others, the prediction of singular solutions of the boundary layer equations under the action of a suitably controlled adverse pressure gradient which are associated with the onset of marginally separated flows. Increasing the pressure gradient locally then transforms the marginal-separation singularity into a weak Goldstein-type singularity occurring in the slip velocity at the base of the outer wake layer. Interestingly, this behaviour is seen to be closely related to (but differing in detail from) the counterpart of laminar marginal separation where the skin friction replaces the surface slip velocity. Most important, adopting the concept of locally interacting boundary layers gives rise to a closure-free and uniformly valid asymptotic description of turbulent boundary layers which exhibit small closed reverse-flow regimes. The according non-trivial eigensolutions of the underlying triple-deck problem which govern the mildly separating flow to leading order have been found numerically, see figure 1. For a more extensive outline of this theory the reader is referred to [1]. The main emphasis of the present investigation is on the effect of finite values of Re. As a highlight of the theory, it is demonstrated how the logarithmic law of the wall is gradually transformed into the well-known square-root variation with distance from the surface of the streamwise velocity on top of the viscous wall layer close to the locations of both separation and reattachment.

A novel rational description of incompressible two-dimensional time-mean turbulent boundary layer (BL) flow separating from a bluff body at an arbitrarily large globally formed Reynolds number, Re , is devised. Partly in contrast to and partly complementing previous approaches, it predicts a pronounced delay of massive separation as the turbulence intensity level increases. This is bounded from above by a weakly decaying Re -dependent gauge function (hence, the BL approximation stays intact locally), and thus the finite intensity level characterizing fully developed turbulence. However, it by far exceeds the moderate level found in a preceding study which copes with the associated moderate delay of separation. Thus, the present analysis bridges this self-consistent and another forerunner theory, proposing extremely retarded separation by anticipating a fully attached external potential flow. Specifically, it is shown upon formulation of a respective distinguished limit at which rate the separation point and the attached-flow trailing edge collapse as and how on a short streamwise scale the typical small velocity deficit in the core region of the incident BL evolves to a large one. Hence, at its base, the separating velocity profile varies generically with the one-third power of the wall distance, and the classical triple-deck problem describing local viscous–inviscid interaction crucial for moderately retarded separation is superseded by a Rayleigh problem, governing separation of that core layer. Its targeted solution proves vital for understanding the separation process more close to the wall. Most importantly, the analysis does not resort to any specific turbulence closure. A first comparison with the available experimentally found positions of separation for the canonical flow past a circular cylinder is encouraging.

By considering the time-mean motion, an asymptotic description of turbulent shear flows in the limit of high Reynolds numbers, conveniently carried out by adopting the method of matched asymptotic expansions, is inherently based on (a few) assumptions regarding the essential physical properties of the flow in the respective layer considered. Although this procedure has substantiated some classical results, as, for instance, the logarithmic law of the wall, it is, nevertheless, not completely satisfactory from a rational point of view: for this reason, we first demonstrate that a rigorous application of the matching principle alone is not capable of deducing the aforementioned wall law in the case of boundary layer flows by taking into account the widely accepted presumption of the so-called ‘inviscid’ nature of the fully turbulent main portion of the shear layer. Therefore, to some extent, the usual approaches turn out to be incomplete, for (i) the motivation for the assumptions commonly employed seems not stringent and, unfortunately, (ii) are, strictly speaking, insufficient for developing a complete asymptotic theory. To be more precise, assumptions that are usually expressed by dimensional considerations can not adequately enter the matching conditions derived from the non-dimensional form of the governing equations, which, however, is required for an asymptotic analysis. Consequently, the time-mean scaling of the asymptotic structure of the flow is seen to result from an asymptotic analysis of the Navier–Stokes equations by, as the only fundamental assumption, considering an asymptotically slender turbulent shear layer. Amongst others, it is demonstrated briefly how a combination of multiple-scales techniques and matched asymptotic expansions leads to the internal boundary layer splitting. Also, it is outlined how the latter also reflects the time-mean characteristics of the flow, which have already been adopted successfully in former studies, e.g. by using rather conventional mixing length arguments. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The contribution deals with a self-consistent description of time-mean turbulent lubricant flow, i.e. flow through a wedge-shaped gap confined by counter-sliding solid surfaces. The methods adopted are matched asymptotic expansions, where the slenderness or aspect ratio of the gap and the accordingly defined Reynolds number represent the perturbation parameters. The limit considered is conveniently approached from the typical aerodynamical problem of wing-ground interference. This then ties in with appropriate inflow and outflow conditions, which here appear quite naturally rather than form a common uncertainty in lubrication theory. As a remarkable finding, a lifting force as a consequence of the resultant pressure distribution can only be maintained for fully developed turbulent flow provided its asymptotic structure flow differs distinctly to that known from other turbulent internal flows as pipe or channel flows or classical turbulent boundary layers. The basic analysis is carried out without resorting to a specific Reynolds shear stress closure. However, the resulting requirements for asymptotically correct turbulence models are discussed. The theoretical study is accompanied by a numerical study of the boundary layer equations governing the fully turbulent core flow. Finally, the impact of cavitation, a phenomenon highly relevant in lubrication theory, on the novel flow structure is addressed.