## Abstract

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triple-deck problem describing local viscous–inviscid interaction crucial for moderately retarded separation is superseded by a Rayleigh problem, governing separation of that core layer.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.

## 1. Motivation and introduction

Let us consider incompressible, isoviscous cross-flow around a (closed) rigid, blunt obstacle at rest in the frame of reference with a smooth impervious cylindrical surface aligned with the spanwise flow direction in an otherwise unbounded domain. Let the Reynolds number *Re* formed with the speed of the undisturbed uniform flow infinitely far upstream of the body, a characteristic value of the radius of its surface curvature and the (constant) kinematic viscosity, take on arbitrarily large values. We are then concerned with the formation of a nominally steady and two-dimensional turbulent boundary layer (BL) along the body surface, which finally undergoes gross (i.e. massive, developed or break-away) separation. We furthermore disregard any effects of free-stream turbulence, surface tension at play along the two dividing streamlines separating the free-stream region and that of rather weakly recirculating flow, and body forces such as gravity. By ignoring the complex picture of the flow in the region representative for the obstacle dimensions and the wake emerging further downstream, our concern is with a (most) rigorous description of the local separation process, in particular the prediction of its location, by exploiting the time- or Reynolds-averaged Navier–Stokes or Reynolds equations for . As in the preceding studies surveyed next, an *ab initio* theory is the ultimate goal of the efforts: turbulence modelling is invoked only when it comes to quantitative predictions as the asymptotic properties of the flow are universal.

Notwithstanding the undeniable progress made in the past decades by rigorous application of perturbation methods, its completion still poses one of the most challenging problems in theoretical hydrodynamics. As achieving seminal progress towards its solution, the outstanding study [1] must be viewed as a milestone: it was not until its advent that the regularization of the inevitable Brillouin–Villat (B–V) singularity forming at detachment of the external potential flow, which then encloses a dead-water cavity further downstream, was addressed correctly within the framework of triple-deck theory. Here, the well-established picture of strictly laminar (steady) flow served as a starting point, where we refer to the textbook [2] and the corresponding references therein for a seminal overview. However, the first attempt to deal with turbulent gross separation must be attributed to the asymptotic treatment [3] of the turbulent flow past an inclined flat plate as the external potential admits a singularity of B–V type at the trailing edge. Most important, in contrast to the laminar-flow situation, cf. [2], in this typical setting (forming the starting point for advanced studies of the practically important class of flows around thin-aerofoil of more general shape), the angle of attack may be even of *O*(1). This finding merely relies on the assumption of a fully developed turbulent BL, but the actual mechanism of separation, i.e. the flow in the viscous wall layer and the near-wake, could be largely ignored. (Unfortunately, the analysis is then also not capable of assessing the violation of the Kutta condition by viscous effects in turbulent trailing-edge flow.) The series of papers [4–6] represent a definite leap forward in the description of separation from a bluff body insofar as the regularization of the pronounced B–V singularity at play is taken into account in a self-consistent manner. One has to concede, however, that the proposed theory is a purely local one, and no attempt was made to embed the local structure of the flow into a global setting at length scales comparable to the characteristic body dimensions. In addition, and as pointed out first in [7], the proposed three-tiered structure of the BL implies a mismatch in the overlap of its main and the viscous wall layer. This inherently occurs, because the relative streamwise velocity deficit in the former flow region is of *O*(1) rather than measured by the principal perturbation parameter , with *κ* denoting the von Kármán constant, as is the case in the well-established classical two-tiered picture of a turbulent BL. There the slenderness of the BL also scales with *ϵ*. The large velocity defect is nevertheless deemed required in the light of the mechanisms governing laminar-flow separation. This idea was taken up in the later study [7], but there the description of the BL flow is accomplished through the introduction of a slenderness parameter independent of *Re*. Here, realistic values for this quantity are determined on an empirical basis or are extracted from the small constants appearing in algebraic but also more sophisticated turbulence models when these are examined within the framework of the associated formal two-parameter asymptotic analysis. The wall slip arising at the base of the resulting wake-type velocity profile in the outer main part of the again three-tiered BL was considered to resolve the deficiencies of the previous approaches. However, matching the velocity profiles in the intermediate and the innermost viscous wall layer once more proves inconsistent with that prior stipulated *Re*-independence.

A yet seriously weaker reasoning in the theory put forward in [7]—as apparently unrecoverable in an attempt to amend and complete it—is given by the absence of the B–V singularity in the external flow as its regularity intrinsically inhibits massive separation of the BL. The latter discrepancy bears resemblance to that encountered in the fundamental study [8] on the extension of laminar BL flow through a point of vanishing skin friction indicating the onset of gross separation. This finally supports the necessity to seek the external potential flow in the class of free-streamline or Kirchhoff flows.

Eventually, the theory presented in [1] and refined in [9] masters all the shortcomings arising in the forerunner studies. At first, it adopts the classical two-tiered structure of the incident turbulent BL to the extent that it exhibits the well-known asymptotically small velocity defect in its main tier. Simultaneously, however, the BL thickness is reduced drastically in its order of magnitude when compared with that of a fully developed turbulent BL, owing to the process of laminar–turbulent transition shortly downstream of the front stagnation point or leading edge of the obstacle as demonstrated in [10]. This embedding of the BL into a global flow picture implies that the actual BL represents a member of the family of *slightly underdeveloped* turbulent BLs parametrized by the novel (here redefined) gauge factor which measures the relative level of the turbulence intensity or, in other words, the ratio between the slenderness and the relative velocity deficit of *O*(*ϵ*). The limit *T*=1 refers to the classical flow picture and the situation of a perfectly laminar BL. Next, the effective dependence of *T* on *Re* and hence the scaling of the BL follows from the interaction mechanism governing the separation process. As shown in [9], this mechanism is twofold: an internal triple-deck structure with the viscous wall layer serving as the main deck is embedded in a Euler stage, evolving from the local breakdown of the oncoming small-deficit main layer and accounting for weak perturbations about the local state of the flow therein. A linear vortex-flow problem is the central ingredient of the Euler region, as in [3]. However, the local flow structures differ distinctly in detail as a consequence of the reduced turbulence intensity level in [9]. By this modification, the neglect of explicit viscous effects in [3] renders this analysis of trailing-edge flows not only incomplete but questionable. Furthermore, in [9], the strength of the B–V singularity is measured by the so-called B–V parameter *k* (>0), which increases monotonically for increasing distances between leading-edge stagnation and separation (relative to the typical body dimension ) [9–12]. We recall that gross separation of purely laminar flow applies if *k*=*O*(*Re*^{−1/16}) [2], but a considerable level of turbulence intensity in the BL requires *k*=*O*(1). Formally, *k* takes the place of the inclination angle in [3], but in the present setting of self-induced separation, its effective value remains arbitrary as long as it refers to semi-open cavities embedded in the Kirchhoff flows [9].

In other words, the existing theory of turbulent separation is indeed self-consistent both locally and globally, but has proved incomplete in the latter sense. To be more specific, the required match of the solution of the upper-deck equations, describing the internally induced potential flow, and that of the vortex-flow problem, at play on the BL scale, does not yield a constraint fixing the value of *k*, in contrast to original expectations.

Specifically, the viscous–inviscid interaction described by the triple deck and regularizing the B–V singularity in the near-wall regions of the BL results in the appealingly concise relationship
1.1
Herein, *u*_{d} denotes the flow speed owing to the Kirchhoff flow at detachment, made non-dimensional with , and the omitted proportionality constant of *O*(1) is part of the triple-deck solution; see [1,9] and the study [13] on the non-interactive flow. Thorough parametric (numerical) studies of the class of Kirchhoff flows [12] indicate that for a given circulation these depend continuously and uniquely on the parameter *k*, where the semi-open stagnant-flow wake confined by the free streamlines closes when *k* exceeds a certain threshold. Its then finite streamwise extent shrinks monotonically when *k* increases further. Finally, when *k* increases above all bounds it reduces to a single point which then forms the trailing edge of the body. Hence, this singular limit points to a perfectly attached potential flow, obtained formally for . Guided by this view, we condense the two central shortcomings of the available theory into the following statements:

(i) neither the correct value of

*k*could be determined so far under the premise that*k*=*O*(1),(ii) nor a self-consistent flow picture could be completed in the specific limit ,

*T*=1 as anticipated in [1], i.e. with fully developed turbulence found to be tied in with a fully attached potential flow.

More precisely, considering the inviscid local self-interaction of the BL governed by a vortex-flow problem, neglected in [1] in the case addressed in (ii), in the follow-up analysis in [11] led to an inconsistency. This renders the theory in [1] not full-fledged. Consequently, settling the question
*How do k and T vary with Re in a distinguished limit* , *yet to be suitably defined?*

is crucial for completing the theory of gross separation as described by these three asymptotic key parameters. Specifically, the formulation of the extreme flow case mentioned in (ii) demands a critical revision and refinement. We hence commence the analysis by scrutinizing the external flow for large values of *k* before we establish the distinguished limit and focus on the consequences.

## 2. Kirchhoff flow with infinitesimally small cavity

In the following, we tacitly refer to the configuration sketched in figure 1*a*.

For the sake of simplicity, the body and, consequently, the time-mean flow are assumed to have an axis of symmetry aligned with the unperturbed flow. Hence, the flow is free of circulation. The value of the B–V parameter *k* shall be so large that the cavity emerging in the inviscid-flow limit and exhibiting fluid at rest is closed. Let the natural coordinates *s*, *n* pointing along and normal from the contour of the body, respectively, and having their origin in its leading edge (front stagnation or branching point of the flow) be non-dimensional with , the velocity components *u*, *v* in the streamwise and normal directions, respectively, with , and the local pressure difference *p* with respect to flow stagnation as well as the components *r*_{ij} (*i*, *j* stand for *s*, *n*) of the Reynolds stress tensor with the dynamic head formed with . We furthermore introduce a streamfunction *ψ* such that *u*=∂*ψ*/∂*n*. The Kirchhoff flow is obtained in the inviscid limit [*u*,*v*,*p*]∼[*u*^{0},*v*^{0},*p*^{0}](*s*,*n*;*k*) as , such that *u*_{s}(*s*;*k*):=*u*^{0}(*s*,0;*k*) denotes the speed it exerts on the body surface. The B–V singularity occurs immediately upstream of the position of inviscid-flow detachment, *s*=*s*_{d}(*k*), with d*s*_{d}/d*k*>0. We provide the classical result [2,9,14] for
2.1
Here, the remainder terms are affected by the local surface curvature, but the subsequent analysis is not. We now have *u*_{d}(*k*)=*u*_{s}(*s*_{d},*k*), equal to the flow speed along the free streamlines downstream of detachment.

The aforementioned modification of the Kirchhoff flow towards a fully attached flow is completed in the limit where the two detachment points, the one considered given by (*s*,*n*)=(*s*_{d},0), have merged at the thus arising trailing edge (rear stagnation point), (*s*,*n*)=(*s*_{t},0) with . Hence, this adjustment for deserves to be elucidated from an asymptotic viewpoint. One conveniently considers this limiting behaviour by letting the distance *d*(*k*):=*s*_{t}−*s*_{d}(*k*) become arbitrarily small. At the body scale, we then write . In turn, we have as *σ*:=*s*_{t}−*s*→0_{+}, with the constant λ (>0) depending on the overall body shape. In these expansions, the remainder terms stem from matching with the flow in the emerging subregion of streamwise and lateral extent *d* that encompasses the detachment and stagnation points and the accordingly shrunken cuspidal cavity. We note the well-known results
2.2
referring to the important canonical case of attached flow about a circular cylinder (unit disc).

The stretched (locally Cartesian) variables (*x*,*y*):=(−*σ*,*n*)/*d* are appropriate for that subregion, locally symmetric with respect to *x*=0 and *y*=−*x*. In that region, we have the inner expansions , and conveniently describe the leading-order potential flow in terms of the complex potential , *z*:=*x*+i*y*, with . It can be expressed analytically by virtue of a conformal mapping of the flow region in the physical complex *z*-plane for *x*<0 onto the interior of the upper quarter-unit circle in an auxiliary *χ*-plane (figure 1*b*): the origin *χ*=0, the boundary of that quarter-circle, and the branching points *χ*=−1 and *χ*=i of are the images of , the free streamline (separatrix) confining the stagnant–flow cusp, its point *z*=i of merging with its counterpart stemming from the flow along the surface for *x*>0 or *s*>*s*_{t}, and the location of flow detachment, *z*=−1, respectively. On the separatrix, the kinematic and dynamic boundary conditions, respectively, and equal to the initially unknown flow speed at detachment hold. One then readily uses the representations , . Elimination of *χ* confirms the match with the outer stagnating flow as for . Finally, subsequent integration provides for the entire flow region in the upper half of the *z*-plane
2.3
The above mapping onto the two branching points provides a constraint expressed by the last finding, hereafter tacitly invoked when is used. We note the branch cuts in the -plane along the images and of the merged streamlines and the separatrix, respectively. The latter is represented by a quarter of the tetracuspid (asteroid) *x*^{2/3}+*y*^{2/3}=1 (−1<*x*<0, 0<*y*<1).

The B–V singularity is revealed in the form as *z*→−1. As the gauge pressure is given by Bernoulli's law or , we find for
2.4
Specifically, with , *x*≤−1, being the surface slip exerted by the leading-order inner solution, one extracts from the implicit representation (2.3) for an inverse one for ,
2.5
This gives rise to the expansions for
2.6a,b
with *g* taken from (2.1). In figure 2*a,* is plotted versus *x* together with the asymptotes given explicitly by (2.6). The behaviours (2.6*a*) and agree by matching. Most importantly, (2.6*b*) recovers the universal part of the B–V singularity in the form (2.1) provided the following important leading-order estimates hold for
2.7
This dependence of *d* on *k* completes our analysis of the limiting state of Kirchhoff flows.

Two side issues are noteworthy: at first, the inner leading-order solution (2.3) can also be extracted from well-known Kolscher's solution [16] describing the family of flows past a splitter plate. Second, it can be generalized in a physically viable manner (from the viewpoint of the high-*Re* limit). To this end, we consider only the region *x*≥0 and substitute *z* by [*z*^{2α}+*H*(1/2−*α*)]^{1/2} with some *α*>0 and *H* denoting the Heaviside step function. Then, (2.3) recovers the patterns of potential flows around an internal stagnation zone adjacent to a kinked wall formed by the ray and the positive *x*-axis: for 1/4<*α*<1/2, it is convex towards the flow and the cavity located along the ray immediately downstream of *z*=0; for it is concave and the flow symmetric with respect to the line ; for *α*=1/2, the flow represents a special member of the Batchelor–Sadovskii potential-vortex flows. Flows parametrized by *α* were calculated first in [17] by a more elaborate conformal-mapping technique.

## 3. Incident boundary layer

We next consider the slightly underdeveloped oncoming two-tiered turbulent BL when approaching the inner Chaplygin-flow square region of extent *d*. We focus only on the behaviour of the core layer. This will first extend the existing theory as available by [9] towards the limiting case .

### (a) Splitting of small-defect flow revisited

According to [9], in a sublayer of wall-normal extent *δ*_{l}, say, the Reynolds stresses *r*_{ij} scale with (*δ*_{l}∂_{u}/∂_{n})^{2}. Hereafter, partial derivatives are indicated by subscripts unless this may cause confusion. For 0<*s*<*s*_{d}, thus the inner expansion
3.1
describes the core layer exhibiting a velocity defect of *O*(*ϵ*). Here, *δ* denotes the thickness of the BL having a rather abrupt outer edge (the disregard of a thin overlayer on top of the core layer does not affect the present analysis), and the streamfunction *F* as well as the shape functions *R*_{ij}, Δ of the Reynolds stresses and the BL edge are quantities of *O*(1). Then *F* and the shear stress function *R*:=*R*_{sn} are governed by the leading-order BL problem
3.2
subjected to initial conditions reflecting the match with the transitional flow as *s*→0 and to be supplemented with a suitable closure for *R*. The match with the viscous wall layer for *η*→0 is provided via that of the Reynolds shear and the wall shear stress, associated with the celebrated logarithmic law of the wall: *R*∼1, ; the intercept *C* is part of the solution of (3.2). The outer expansion of the potential flow, i.e. of *u*_{s}, as triggers the regular corresponding one , again to be considered for *σ*→0_{+}.

As shown in [1,11], the least-degenerate behaviour of , , in this limit means a rapid increase of the velocity deficit and the BL thickness. Specifically, examination of (3.2) yields for
3.3
Thus, the flow near the trailing edge is represented by the shape functions *F*_{t} and *R*_{t} of *O*(1), and the factor *f* accounts for its upstream history. Inspection of the streamwise component of the Reynolds equations shows that the flow in the core layer becomes predominantly inviscid immediately upstream of stagnation. Measures *σϵF*_{η} the velocity deficit as *u*_{s}=*O*(*σ*), we have . This result then follows from casting the assertion *u*∂*u*/∂*s*≫∂*r*_{sn}/∂*y* into the order-of-magnitude estimate *σ*^{2}*ϵF*_{η}/*σ*≫*T*(*σϵF*_{η})^{2}/*δ*, which provides , confirmed by (3.3). Accordingly, the left-hand side of the BL equation in (3.2) is satisfied identically to leading order. On account of the logarithmic variations of *σ*, (3.2) assumes the universal form
3.4
Because the behaviour (3.3) accompanies a splitting of the core layer, the leading-order stresses at the base of its main part and, in turn, the inhomogeneities in (3.2) vanish. Therefore, the logarithmic law of the wall still holds in the overlap between the new intermediate and the viscous wall layer, whereas a reduced wall slip on its top owing to a positive value of *F*_{t}′(0) and a square-root behaviour now provides the match of the new core and the intermediate layer for
3.5
This terminal state of the BL was not taken into account in [1], but recognized first in [11].

Remarkably, such a tree-tiered flow structure associated with a moderately large velocity deficit (here given by the growth rate of in (3.3) taking place at a streamwise extent *σ*≪1) is known to apply to turbulent BLs prone to mild separation. On condition of leading-order equilibrium, these are also governed by (3.4) (see [15]). Hence, we chose the pre-factors in (3.3) so as to obtain (3.2) in this specific form and adopt the numerical solution to this problem put forward [11]. It employs an asymptotically correct algebraic mixing-length model for *R* and is presented in figure 2*b*. The near-wall behaviour and the associated linear variation of *R*_{t} according to (3.4) are clearly discerned. Our experience with other algebraic models is that they provide numerically close results.

### (b) Large-defect flow

The mechanism of breakdown of the BL limit, initiated by its above splitting, is twofold: it is due to

(I) an increased relative velocity defect of

*O*(1) in the core layer, so that the linearization of the convective terms in the BL equation, see (3.2), ceases to be valid or/and(II) the emergence of a square region where

*σ*/*δ*=*O*(1), giving rise to an Euler stage.

According to (3.3), a velocity deficit of *O*(1) as considered in case (I) applies if *σ* is shortened and *δ* has increased to *O*(*τ*) and *O*(*μ*), respectively, with
3.6
Using these quantities, we next consider case (II) given the small-defect assumption can be maintained: *σ*/*τ*≫1. On condition that *k* is so large that *d*≪*σ* and the presence of the B–V singularity is not perceptible on the considered length scale *σ*, this situation refers to case (ii), §1, and is therefore here discarded. For values of *k* just so large that *σ*/*d*=*O*(1), i.e. *d*/*τ*≫1, we arrive at the terminal extension of the theory put forward in [1,9]. Then, (1.1) must be replaced by *T*∝*ϕ*(*d*)*d*^{−8/9}(λ*Re*)^{−4/9}*ϵ*^{−2} upon substitution of (2.7) and with the additional factor introduced to account for the increase of *δ*/*ϵ*^{2} according to (3.3) (so that the *O*(1)-constant of proportionality now depends on the upstream history of the flow). However, this estimate still relies upon the concept of viscous–inviscid interaction, described by an internal triple-deck problem, and the linear response of the square core region, now exhibiting a moderately large velocity defect, on the B–V singularity. Hence, as seen from the above estimates and the definition of *τ* in (3.6), the ratio of *δ* and the thickness of the viscous wall layer and, in turn, the gauge factor *T* essentially still depend algebraically on *Re*. A desired drastic increase of the turbulence intensity is therefore not accomplished by this limit.

We are thus left with case (I). Reverting to the estimate given in §3*a*, we now find that the inertia terms are larger than the wall-normal gradient of the Reynolds shear stress by a factor of or . The formation of a fully nonlinear Euler stage (as indicated by ‘and’ initiating case (I) above) is prevented in favour of its BL approximation, which accommodates a pronounced velocity defect. Hence, further analytical progress is enabled when we require that
3.7
This finally represents a considerably weaker constraint for the turbulence intensity level than that given by (1.1). The interplay with the splitting of the Kirchhoff flow as *d*→0 is captured in terms of a least-degenerate distinguished limit governed by a similarity parameter, *D*:=*d*/*τ*, taken as of *O*(1). The potential flow in the inner region considered in §2 then drives that large-defect BL.

Hence, *x* and *Y* :=*n*/*μ* represent appropriate (locally Cartesian) BL coordinates of *O*(1). We accordingly expand , where we note that *r*_{sn}=*O*(*Tu*^{2}). Then, the leading-order streamfunction satisfies Bernoulli's law,
3.8
expressed in von-Mises variables. Here, the specification of the Bernoulli function follows from the match with the splitting small-defect flow upstream, where the factor 3/4 (proving computationally advantageous subsequently) has its source in the choice of the proportionality constants in (3.6). Hence, represents the vorticity carried along the streamlines. The first relationship in (3.8) is valid for or and as long as (see (2.5))
3.9
For *x*≤*x**, the core layer exerts the slip on the flow regions more close to the surface, and and the rescaled BL thickness are determined by
3.10

The BL structure is essentially characterized by far upstream. Without resorting to a specific form of , one detects the generic small-deficit structure such a BL admits there: and , with *η*, from here on redefined as , taken as of *O*(1). Specifically, we have as , and matching with the oncoming small-deficit BL is confirmed by virtue of (2.6*a*).

Let us first consider the two limiting cases
3.11
3.12
In the first, one assumes that *s*=*O*(*τ*) but the B–V singularity is not taken into account, which points to the situation addressed in issue (ii), §1; the second provides the match with the existing theory when evaluated for sufficiently low values of *d*, unsatisfactory from the viewpoint (i), §1. Here, both these shortcomings are surmounted by taking *D* as of *O*(1) and fixing its value by having the canonical representation (3.8), (3.10) of the BL flow terminating at the most downstream position, given by that of the B–V singularity: *x**=−1, . This maximum retardation of the breakdown is due to the regularization of both singularities on the basis of the full Reynolds equations that is required to take place at a single streamwise scale much shorter than *d*. As a result, the following separation criterion relevant for *σ*/*d*=*O*(1) gains awareness:
3.13
by (3.6). In the currently proposed theory, this criterion replaces that given in [1].

The numerical solution of (3.4) in [15] provides (sensitive to the values of the model input parameters, in particular that of the von Kármán constant, *κ*≈0.384). Hence, and are sorted out by (3.13). In order to distinguish between the limits considered in the forerunner studies and here, it is instructive to plot the flow speed imposed on the BL in dependence of *D* and *σ* related to the scale *τ*, representative of the occurrence of a velocity defect of *O*(1) (figure 3*a*). Most importantly, the canonical flow around the unit cylinder yields *f*=1/3 by (3.3) and (2.2). From these numbers and (3.13), the separation angle *s*_{d} as a function of *Re* is obtained (figure 3*b*). We recall that the separation point collapses with the trailing edge, *s*_{d}=*π*, at . A comparison of the theoretical predictions with the not so comprehensive set of measured data available, summarized in [21], emphasizes the still notable distance of the separation point from the trailing edge for not too large (but experimentally and practically feasible) values of the Reynolds number *Re*_{2}:=2*Re* (as usual in experiments, formed with the diameter rather than with the radius of the cylinder). This is mostly due to the quite weak decay of *d* as , but a probably insufficient suppression of three-dimensionality of the nominal flow in experiments is also worth mentioning. For the compilation in figure 3*b*, the data points were extracted from the early pioneering wind-tunnel studies [18–20]. We considered only values of *Re*_{2} so large that the attached BLs were confidently viewed as developed turbulent ones in these studies (*transcritical* regime; [18], 1.5 [19], 1.31 [20]). The data scatter fairly when individual studies are compared, and owing to their scarcity in that regime the trend ∂*s*_{d}/∂*Re*>0 is not reproduced so far. Apparently, the highest value of *Re*_{2} of about 1.782×10^{7}, achieved in the study [20], is still hardly surmountable in experiments.

The transformation of the initially moderately-large- to a large-defect BL with a wake-like velocity profile and finally a state of incipient separation at *x*=−1 is visualized in figure 4. Smooth cubic splines are used to interpolate the data points obtained by numerical evaluation of (2.5), (3.10), (3.8) for *D*=*D** and , with *i*=0,1,2,… and *ω*:=0.425. In figure 4, the growth of the BL is conveniently attenuated via the use of the rescaled BL coordinate

The criterion (3.13) crucially affects the analysis of in the limit *x*→0_{−} as it induces the formation of a sublayer. We infer from (3.8), (3.10) and (2.6*b*), (3.5) the following expansions for
3.14
3.15
Here, is found. For (3.14), (3.15) and the subsequent results, cf. figure 4. A novel one-third-power law given by (3.15) replaces the half-power behaviour holding for *Y* →0 and −*S*=*O*(1), according to (3.8) and (3.5). It characterizes the velocity profile at separation and the base of the core tier. Its advent is accompanied by a splitting of this layer owing to the apparent non-uniformity of the expansion (3.14). It is seen first from (3.8) subjected to (3.13) that the B–V singularity is superseded by an even stronger one at the base of the main tier as . Inspection of (2.6*b*) and (3.5) reveals that the expression for in (3.8) is retained in full if *ζ*:=*bY*/*S*^{3/4} with *b*:=4*a*^{2}/(6^{1/4}*D*^{*3}), *a* given in (3.5), is kept fixed. We then have with *c*:=3*D*^{*4}/(8*a*^{2}), which, by the specific choice of the constants *b* and *c*, yields the problem *f*′(*ζ*)=(1+*f*^{1/2})^{1/2}, *f*(0)=0. Integration provides
3.16
The inversion of the first representation in (3.16) follows by Cardano's method. One finds that *f*′′(*ζ*)>0 and *f*′′′(*ζ*)<0, cf. figure 4, and *f*∼*q*^{4/3}+2*q*^{2/3}+*O*(1) as . This allows for a direct match of the expression for in the sublayer with the two-term expansion (3.14) and confirms the anticipated suppression of terms having the form in (3.14) with a gauge function *γ* much larger than . These describe eigensolutions of the convective operator in the streamwise momentum equation. Here, the inviscid nature of the flow prevents their occurrence. In addition, (3.16) reveals the transition from the square-root law, yet holding for *q*→0, towards the newly found one-third-power law further away from the wall.

The behaviour of the core layer immediately upstream of the B–V singularity differs strikingly from that in [9] (where the singularity is correspondingly weaker), as a consequence of the *O*(1)-velocity deficit ruling here. The emerging inviscid–vortex flow or Rayleigh problem of self-interacting flow, envisaged next, represents a cornerstone in the current theory of separation. Owing to the BL limit and the associated nonlinearity and arising surface slip at the base of the core layer in the oncoming flow, it is not only set apart from that derived in [3] and recovered in [9] but also the central one in [11], which does not allow for a solution confirming consistency of the limit proposed in [1]: cf. item (ii) at the end of §1. However, the mechanism elucidated below proves promising as the vorticity originating in the BL upstream enters directly the perturbations considered rather than it forming an inhomogeneity.

## 4. Local Euler stage

The sudden decrease of the surface slip causes the core layer to split into two tiers. Hence, a shortened streamwise length scale comes into play. In the resulting near-wall sublayer, and become of comparable magnitude. For a sufficiently severe shortening, the classical hierarchical BL concept is in general superseded locally by a strong interaction of the BL with the inviscid flow generated on its top, governed by the so induced pressure gradient. Owing to the absence of the above eigensolutions, here this typical mechanism is replaced by a radically different one.

### (a) Self-interacting inviscid flow

Let us first recall a classical interactive regime, leading to a triple-deck problem. In the main part of the (large-deficit) BL, the velocity profile is given by the separating one, here λ*τU*(*Y*), slightly perturbed by the aforementioned eigensolutions, but with *γ* now superseded by initially unknown displacement functions that vary in streamwise direction, and contributions accounting for a balance of the convective terms with the pressure gradient in the linearized streamwise momentum equation, here about [*u*,*v*]∼[λ*τU*(*Y*),0]. Then, both the induced pressure and the dominant displacement function are determined by the interaction mechanism, which finally implies a regularization of a singularity encountered if the classical hierarchical flow description is continued. Interestingly, here, this process is stymied as we are left with dominant disturbances of the second type (where the shortened streamwise scale promotes the predominance of the convective over the stress terms). However, the regularization of the B–V singularity then has to be accomplished by a virtually inviscid flow in the core region itself where the pressure gradient is neither imposed nor, as a consequence, its wall-normal component negligibly small (as is classically the case). From this, the strongest shortening of the scale in *s*-direction possible ensues, measured by the (local) thickness of the BL, and makes way for a linearized Euler, i.e. a Rayleigh, stage superseding the shear-layer structure. We so arrive at the leading-order approximations
4.1
by virtue of (2.7) and (3.6), and for *X*=*O*(1). Here, the ‘frozen’ state of the BL condensed in is given by its upstream history, whereas results from solving the accordingly linearized Reynolds equations. This follows by inspection analysis, formally even for *T*=*O*(1). The relationships (4.1) and the underlying smallness of the principal perturbation parameter *μ*/*τ*, however, again reflect the assumption (3.7): it just delays the emergence of a (then linearized) Euler stage.

### (b) Bernoulli sublayer

The sublayer more close to the surface represents the continuation of the corner region where *Y* =*O*(*S*^{3/4}), addressed at the end of §3*b*. Hence, we stipulate that *P*(*X*,*Y*) is bounded for *Y* →0 and find that *p*=*O*(*u*^{2}), *r*_{ij}=*O*(*Tu*^{2}) in this region, where with some constant *A*>0. Then, the ratio of ∂*r*_{sn}/∂*n* and ∂*p*/∂*s* is of *O*(*T*(*τ*/*μ*)^{3/4}) and vanishingly small by (3.7). Consequently, the subregion represents an inviscid shear layer to leading order where Bernoulli's law holds in full. We find that as the stretched streamfunction satisfies
4.2
The Bernoulli function is found by matching with the corner region far upstream. To this end, we infer from (4.1) that *X*=−*SD***τ*/*μ* and note that . Furthermore, from (2.4), we deduce the dominant varying contribution to immediately upstream of the B–V singularity, according to (2.6*b*). From matching for *S*→0_{+} and the expression for *p* in (4.1) for then follows in this limit. When we conveniently choose *A*:=3^{3/4}*D*^{*7/4}/(2^{7/4}*c*), matching of *ψ* results in . Independently, matching of *u* yields far upstream for . We hence identify with .

The rescaled surface pressure is part of the solution of the Rayleigh problem scrutinized below. It follows that the solution of (4.2), expressed by , , and (3.16), exists as long as . We fix a translational invariance of , and hence *Ψ*(*X*,*Y*), *P*(*X*,*Y*) with respect to a shift in *X* that reflects the accuracy of the leading-order approximations without any loss of generality. Hence, let *X*=*Y* =0 designate the separation point within the current asymptotic accuracy. We then complete the analysis of the sublayer by noting two results of particular interest. At first, represents the surface slip exerted on the near-wall regions around separation, which are strongly affected by the Reynolds and the viscous shear stresses. Secondly, matching *ψ* with its two-term expansion in the core region, given by (4.1) as free of eigensolutions proportional to *U*(*Y*), verifies (3.15) and yields for
4.3
This indicates non-uniformity of (4.1) and a new singularity immediately upstream of separation.

### (c) The Rayleigh stage—weakening the pressure singularity

In leading order, the momentum equations for the streamwise and wall-normal directions governing the Rayleigh stage in the (locally Cartesian) coordinates *X*, *Y* read
4.4a,b

By eliminating either *Ψ* or *P*, one obtains
4.5a,b

In (4.5*b*), a potentially *Y* -dependent additive function arising from integration with respect to *X* is set to zero as a consequence of the conditions of matching with the flow upstream. The inviscid perturbed flow described by (4.4), (4.5) exhibits a vorticity given by the negative right-hand side of (4.5*b*). By integration of (4.4*a*), one recovers the linearized Bernoulli's law in the form *P*=−*U*^{2}(*Ψ*/*U*)_{Y}. Again, here any additive *Y* -dependent function is seen to be constant upon differentiation with respect to *Y* , substitution of *P* in (4.4*b*), and by (4.5*b*). Finally, it turns out to be zero upon insertion of in the limit *Y* →0 into (4.3) and by inspection of (3.15).

The strategy to solve (4.4) does not impose any strong restrictions on *U*(*Y*) apart from its generic near-wall behaviour (3.15). We stress that the detailed form *U*(*Y*) depends on the turbulence closure used to solve (3.4), which confirms the required continuity of *P* and *P*_{Y} at the local BL edge, . We find that *U*_{e}−*U*(*Y*)=*O*((*Y* _{e}−*Y*)^{2}) with as *Y* →*Y* _{e}, cf. [15], and have to assume that *U*(*Y*)=*U*_{e} for *Y* >*Y* _{e}.

Because the ultimate goal of our effort is the determination of the surface slip , i.e. , we advantageously exploit (4.5*a*) rather than (4.5*b*) in the further analysis. As will become evident below, the appearance of *P*_{Y} rather than of *P* on the right-hand side of (4.5*a*) supports this preference. The boundary conditions are given by the required match with the external flow in the subregion considered in §2 and the Bernoulli sublayer. The first owing to (2.4) expressed through *Z*:=*X*+i*Y* yields for
4.6
The second ensues from (4.3) and by inspection of (4.4*b*) in the form of a regularity condition for
4.7
Here and in the following, the restriction to negative values of *X* allows for a possible drastic change of the flow structure downstream of separation, accompanied by a breakdown of its current description. More precisely, by the homogeneity of the equations (4.5), their solutions are, in general, composed of a regular and an irregular contribution. Specifically, the irregular ones to *Ψ* and *P* are then associated with the emergence of eigensolutions in (4.1) and, with the aid of the one-third-power law (3.15), found to be of, respectively, *O*(*Y* ^{1/3}) and *O*(*Y* ^{5/3}) as *Y* →0; their suppression gives (4.7). Because (4.6) must provide a match with the behaviour of *P* as for *Y* =*O*(1) (via that of *P* near *Y* =*Y* _{e}), (4.5*a*) with (4.7) yields for
4.8
The gauge function *g* with 1≫*g*(*X*)≫*X*^{−2} as is introduced to account for higher-order perturbations in (4.6) and remains undetermined at this stage. The upstream behaviour (4.8) includes the aforementioned one of . For , (4.8) indeed complies with (4.6) for any rate of decay of *U*′/*U* as *Y* _{e}−*Y* →0_{+}. On condition that (4.7) holds even for (sufficiently large) positive values of *X*, the downstream counterpart to (4.8) starts with some small *X*-dependent gauge function, predicting the expected decay of *P* as . However, this obviously raises a mismatch with the corresponding behaviour derived from (4.6). This can be absorbed only if the aforementioned change of the flow structure is indeed anticipated by relaxing (4.7) sharply, i.e. disrupting it at a singular point which must coincide with the separation point. We then complement (4.7) to Wiener–Hopf-type boundary conditions, typical for problems describing flow separation in an inviscid limit. Hence, the irregular contribution to *P* supersedes the regular one at *X*=0 as we have for
4.9
We then arrive at for *X*>0, and the new separation singularity is associated with a singular behaviour of *P* for *Z*→0. Finally, we find for
4.10
It is noted that (4.5*a*) prevents *P*(*X*,*Y*) and its derivatives from assuming local extrema. Thus, the relationships , and (4.6), the last giving *P*_{X}>0 as , imply *P*≤0 and *P*_{X}≥0 for *Y* ≥0.

Let us state the findings (4.7) and (4.9) more precisely as we obtain for
4.11a,b
On the other hand, *P* admits a Maclaurin expansion as *X*→0 for *Y* >0 owing to the dominance of the *X*-derivatives in (4.5*a*). However, the rather abrupt change from a regular to an irregular behaviour of *P* as *Y* →0 is accomplished in a smooth manner in a subregion formed by the half-circle 0≤*ϑ*≤*π* with in the limit |*Z*|→0. There that regular expansion breaks down as all terms in (4.5*a*) are retained in leading order. One then finds by matching that *P* admits a purely algebraic behaviour as |*Z*|→0. Specifically, we have *P*∼|*Z*|^{β}*G*_{β}(*ϑ*) with some constant *β*>0 and *G*_{β} satisfying . Setting , yields the general Legendre equation for the index 5/6:
4.12
Its general solution consists of the linear combination of the associated Legendre functions on the cut (Ferrer's functions) , [22]. Then, the expansion (4.11*a*), which also ceases to be valid in the circular subregion, and (4.11*b*) expressed in |*Z*| and *t* determine the constants *c*_{P}, *c*_{Q} and the degree *l*. Therefore, we consider the expansions *M*_{l}∼*a*_{∓}+*b*_{∓}(1±*t*)^{5/6}+*O*(1±*t*) as *t*→∓1_{±}; for the four (*l*-dependent) coefficients that constitute *a*_{∓}, *b*_{∓} by respective linear combinations we refer to [22]. Matching the representation of *P* in the circular region and (4.11*a*) (*t*→−1_{+}), (4.11*b*) (*t*→1_{−}) indicates that
4.13
must not vanish (*M*_{l}(−1)≠0), whereas
4.14
must both vanish. The last requirement is equivalent to solving the eigenvalue problem for the eigenvalue *l* formed by (4.12) when subjected to *M*_{l}′(−1)=*M*_{l}(+1)=0. By (4.14), a non-trivial dupel *c*_{P}, *c*_{Q} is only possible if . This yields the discrete set *l*=0,1,2,…, where the non-degenerate case is given by *l*=0, i.e. *β*=5/6. Then (4.12) can be integrated in closed form to give the linearly independent solutions , [22]. This provides the eigensolution and . From (4.13), we detect a pressure singularity near separation considerably weaker than the original B–V singularity as we have for
4.15
It is noted that in the above Maclaurin expansion the two leading *Y* -dependent coefficients are unknown, reflecting the ellipticity of (4.5). However, the above local analysis ascribed the two arising coefficients *c*_{P}, *c*_{Q} to the remaining unknown one *c*_{Q}.

In figure 5*a,* the azimuthal variation of *P* in the circular subregion is plotted. The resultant final splitting of the flow considered in this study is illustrated in figure 5*b*.

### (d) On solving the Rayleigh problem

We close the investigations by considering the semi-analytical/numerical strategy we adopt (in a subsequent study) to solve the Rayleigh problem posed by (4.5*a*), (4.6), (4.7), (4.9), *P*(0,0)=0, and continuity of *P* and *P*_{Y} near *Y* =*Y* _{e}. Here, the main difficulty arises from the largely unspecified form of *U*(*Y*). Denoting a wavenumber and the generalized Fourier transform of —the singularity given by (4.15) is sufficiently weak—with respect to *X*, the corresponding transform of *P*_{X} is expressed as ; we remark that as by (4.8). We then obtain from (4.5*a*) for
4.16
Here, the last condition suppresses the exponential growth of for *Y* >*Y* _{e} and agrees with (4.6). We obtain for *Y* ≥*Y* _{e} and find for
4.17
with *U*_{r} introduced in (4.10). Hence, is smooth for and for and 0<*Y* <*Y* _{e}. Owing to the inhomogeneous boundary condition, the two-point boundary value problem (4.16) then has a solution symmetric in and real and unique for all except a set of discrete values of where the corresponding (self-adjoint) homogeneous problem has a non-trivial solution. This is solely given by its fundamental solution forming the irregular contribution to *P*. In addition, must exhibit a basically exponential tail in the wavenumber space. Specifically, (4.16) suggests the expansion for both and large and a yet unknown gauge function of subexponential variation. A near-wall region of non-uniformity arises for where we have with satisfying , owing to (3.15) and subjected to and exponential decay rather than growth of for . The solution with *K*_{5/6} specifying a modified Bessel function of the second kind provides as , matching the above representation for . This yields , leaving the decay of dominated by which makes (4.16) amenable to its efficient numerical solution via the discrete cosine transform.

The apparent impossibility to represent solely by the regular fundamental solution of the transformed equation (4.5*a*) confirms the splitting of the near-wall conditions into (4.7) and (4.9). The crucial step then is the determination of . By (4.7) and (4.9), its real and imaginary parts form a Hilbert pair. The same holds for the regular (and the irregular) fundamental solution, which results in two further equations for these parts. This procedure yields the (unique) distribution *P*(*X*) exhibiting the intrinsic singularity (4.15) and, finally, proves the self-consistency of the theory within the order of approximation considered.

A further promising aspect supporting existence of the solution is provided by the well-established theory of Beltrami equations. It is sensible to straightly consider the equation *P*_{XX}+*P*_{YY}=(2/*C*)(*C*_{X}*P*_{X}+*C*_{Y}*P*_{Y}), which reduces to (4.5*a*) when the function *C*(*X*,*Y*), taken as given, is specified by *U*(*Y*). It can be rewritten as (*P*_{X}/*C*^{2})_{X}+(*P*_{Y}/*C*^{2})_{Y}=0. (As an appealing interpretation, this expresses the continuity of a potential flow with Cartesian velocity components *P*_{X}, *P*_{Y} and a density distribution *C*(*Y*)^{−2}.) Hence, it is equivalent to a system of Beltrami equations: [*P*_{X},*P*_{Y}]=*C*^{2}[*V* _{Y},−*V* _{X}]; specifically, here (4.4) yields *V* (*X*,*Y*)=−*Ψ*_{X}/*U*. These are known to be invariant against a conformal mapping, *X*+i*Y* ↦*ξ*+*iυ*, say, such that [*Π*,*Υ*](*ξ*,*υ*):=[*P*,*V* ](*X*,*Y*) and *Γ*(*ξ*,*υ*):=*C*(*X*,*Y*) satisfy [*Π*_{ξ},*Π*_{υ}]=*Γ*^{2}[*Υ*_{υ},−*Υ*_{ξ}]. Accordingly, the original equation is transformed into *Π*_{ξξ}+*Π*_{υυ}=(2/*Γ*)(*Γ*_{ξ}*Π*_{ξ}+*Γ*_{υ}*Π*_{υ}). The idea here is to construct the above conformal mapping such that *Γ* is given in the simplest form conceivable.

## 5. Conclusions and further outlook

A rigorous theory on turbulent bluff-body separation under the premise that the turbulence intensity attains the highest level possible in the limit of large Reynolds numbers *Re* was presented. One intriguing finding is the necessity of a large velocity deficit in the BL flow on a small streamwise length scale that measures the distance *d* of the position of the B–V singularity from the trailing edge which emerges in the limit of a fully attached potential flow, seen as a limiting state of Kirchhoff flows. It was shown, for the first time, how this BL evolves (in a universal manner) from an initially firmly attached turbulent BL having a small velocity deficit towards one on the brink of separation. There the deficit has become large, and the velocity profile admits a novel one-third power variation with distance from the wall. Insofar as a generic separation criterion was formulated by virtue of the distinguished limit *d*→0 as considered, the present theory represents the ‘missing link’ bridging the gap between the cases addressed in (i) and (ii) at the end of §1. The (preliminary) analysis of the Rayleigh problem finally ensuing from the breakdown of the shear layer limit raises hope that its solution allows for completing the theory to a fully self-consistent flow description.

Open points of massive interest currently under investigation include the numerical solution of the Rayleigh problem by a pseudo-spectral method, in particular the distributions of the surface pressure and the associated slip velocity that drives the multi-structured near-wall flow. Disregarded in the present approach, the latter is crucially affected by the shear stresses which bring that slip to rest. Therefore, it is first of vital interest how the new pressure singularity encountered in the Rayleigh stage close to separation and the associated transition of the surface pressure are modified by the Bernoulli sublayer. Subsequent efforts will aim at disclosing how the so altered singularity emerging at the base of the region where a Euler flow prevails and more close to the separation point affects that near-wall flow. As the regularization of this singularity can only be accomplished beyond the inviscid limit considered here, the near-wall flow must abandon its hitherto passive role locally. Eventually, we expect to detect a novel mechanism of viscous–inviscid flow interaction as the classical triple-deck structure is suppressed by the extreme retardation of separation. The question arises whether, unlike in the forerunner study [9], here the logarithmic law of the wall is eradicated when the wall shear stress changes sign at the effective position of separation. A most comprehensive understanding of the near-wall flow will then refine the separation criterion and hence complete the local picture of gross separation.

Further issues concern the full Euler stage replacing the Rayleigh stage once the present constraint on the turbulence intensity level is relaxed to allow for a fully developed turbulent flow. Regarding the flow on the body scale and beyond, the current theory predicts zero body drag in the limit , but this may be in doubt if a further transition of the separated shear layers towards massive fully developed turbulent ones has taken place. First ideas are reported in the survey paper [23]; for more recent developments, we refer to the consecutive papers [24,25]. Finally, the validation of the theory by comparison with measurements deserves intensified attention. For the time being, this can be seen as promising in the light of the surprising scarcity of publicly available data for a most two-dimensional averaged flow around a circular cylinder.

## Acknowledgements

This research was partially supported by the Austrian Research Promotion Agency (FFG) within the Austrian Competence Centers for Excellent Technologies (COMET) K2 Programme (FFG grant no. 824187, project acronym: *XTribology*).

## Footnotes

One contribution of 15 to a Theme Issue ‘Stability, separation and close body interactions’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.