## Abstract

A theoretical model of the laminar ‘calmed region’ following a three-dimensional turbulent spot within a transitioning two-dimensional boundary layer is formulated and discussed. The flow is taken to be inviscid, and the perturbation mean flow surface streamlines calculated represent disturbances to the basic slip velocity. Available experimental evidence shows a fuller, more stable, streamwise profile in a considerable region trailing the spot, with cross-flow ‘inwash’ towards the line of symmetry. Present results are in qualitative agreement with this evidence.

## 1. Introduction

The beneficial effects of the ‘calmed region’ at the rear of a turbulent spot in a laminar or transitional boundary layer are generally accepted to be considerable. In this region, first described by Schubauer & Klebanoff (1956), the flow is essentially laminar and the streamwise velocity profile relaxes from the high shear layer turbulent-like form to that of the original laminar layer. The relaxation takes place through a sequence of profiles that are fuller and more stable than that of the natural layer. The practical importance of this phenomenon includes the following: the boundary layer is thinned, laminar separation is delayed, an incipiently separating layer may reattach, and transition lengths are increased, all of which are advantageous in aerodynamics and acoustics, while compressors and turbines may be run at optimum speeds to minimize drag losses if interwake turbulent onset positions can be predicted and stabilized.

The present paper is an attempt to extend the mean flow calculation of Smith & Timoshin (2001), but for a zero pressure gradient, to the three-dimensional triggered spot of Doorly & Smith (1992), since a turbulent spot is certainly three‐dimensional. The aim is to involve the nonlinearity to the extent that the perturbation to the mean flow may be calculated and the calmed region identified. Experimentalists, particularly those who were concerned with the heat transfer, have noted that there was a near-wall concentration of activity in the calmed region. Therefore, we restrict ourselves to an examination of the wall streamlines of the perturbed mean Euler flow right across the width of the spot. We calculate both components of velocity, which in some aspects are found to be in agreement with the measured profiles of Seifert *et al*. (1994). It emerges that there are two cases depending on whether the initial perturbation to the slip velocity (negative displacement) is positive or negative. However, in either case, there is a considerable region of induced mean flow in the streamwise direction trailing the spot (see also the profiles of Wygnanski *et al*. 1976, 1982; Gostelow *et al*. 1997) and, as reported by Seifert *et al*. (1994), the transverse velocity is largely an inflow towards the centre-line.

The plan of the paper is as follows. In §2, we summarize the basic equations and leading-order solutions. Essentially, we extend the results of Doorly & Smith (1992) to include the velocity components. In §3, we calculate the mean flow perturbations on the boundary, and in §4, we illustrate these graphically. Section 5 compares the results with evidence available from previous experimental investigations.

## 2. The basic equations and the leading-order solutions

In this section, we describe the physical situation and present the leading-order linearized solution for long-scale disturbances to an oncoming two-dimensional boundary layer. The scenario is exactly that of Doorly & Smith (1992), hereafter referred to as DS. The velocity components and pressure are O(1) but the spatial and time-scales are of the same order as the boundary thickness: namely, O(*R*^{−1/2}), where *R* is the Reynolds number, taken to be large. The governing equations are the unsteady Euler equations which, for long-scale disturbances reduce near the wall *Y*=0, to the unsteady thin layer equations for the velocity **U**=(*U*, *V*, *W*):(2.1a)(2.1b)(2.1c)subject to(2.1d)An upper tier, in which the flow is potential, gives the relation(2.2)between the Fourier transforms of the pressure *P*(*X*, *Z*, *T*) and the (negative) displacement *A*(*X*, *Z*, *T*).

The inviscid triple-deck problem in (2.1*a*–*d*) and (2.2) is further discussed in DS and the references therein. It seems particularly applicable to spot disturbances of which the structure, identified by Gad-el-Hak *et al*. (1981), involves a main body overhanging the viscous wall layer.

The next step is to seek a solution of (2.1*a*–*d*) and (2.2) in the form(2.3)where . The leading-order approximation is available in DS, from which it may be shown that for scaled time , and on the boundary *Y*=0,(2.4a)(2.4b)Here, *σ*_{+}, *σ*_{−} are the roots of , *σ*_{+}>*σ*_{−}, as in DS eqn (3.8*c*,*d*). In addition, , , *c*=cos *θ* and , . An initial condition, , which gives a Gaussian in Fourier space, is employed as an example to obtain equations (2.4*a*,*b*). As in DS, these oscillatory terms are non-zero only inside the caustic *X*^{2}=8*Z*^{2}.

In addition to these oscillatory contributions, which are derived from the saddle points of the inverse Fourier transform integrals, we require the contributions from the origin *α*=*β*=0. It may be shown that this is a mean flow contribution, *U*_{0m}, *W*_{0m} where, with and (2.5a)(2.5b)These mean flow contributions are smaller than the oscillatory components in equations (2.4*a*,*b*) but turn out to be of importance. They are present both inside and outside the caustic.

The spot planform is illustrated in fig. 2 of DS. There *θ*_{s}=tan^{−1}(8^{−1/2}) is the bounding angle of the wedge, and we anticipate that the turbulent part of the spot is confined to the region *X*, *Z*=O(*T*) inside the wedge. The calmed laminar region behind the spot is expected to be confined to . In the following section we examine the O(*h*^{2}) terms in (2.4*a*,*b*). They satisfy(2.6a)(2.6b)(2.6c)with *V*_{1}=0 at *Y*=0, *U*_{1}→*A*_{1} and *W*_{1}→0 as *Y*→∞. We aim to find the mean flow contributions to *U*_{1} and *W*_{1} on *Y*=0 and to examine the induced streamline pattern formed by a combination of these O(*h*^{2}) terms and the O(*h*) contributions of (2.5*a*,*b*).

## 3. The mean flow contribution to the surface streamlines

To extract the mean flow contribution, *U*_{1m}, *W*_{1m} to *U*_{1} and *W*_{1} in (2.3), we argue that on the scale in which we are interested, the mean part of *P*_{1} is smaller than that of *A*_{1} and may be neglected when . In addition, the products on the right-hand sides of equations (2.6*a–c*) make a full solution in terms of double Fourier transforms intractable. Therefore, as noted in §2, we confine ourselves to a study of the streamlines on *Y*=0. Since *P*_{1m}=0 to the order considered, equations (2.6*a*–*c*) reduce to(3.1a)(3.1b)where, to calculate the forcing, we shall utilize the values of *U*_{0} and *W*_{0} on *Y*=0 as given in (2.4*a*,*b*) for values of . After some calculation, we find that(3.2a) (3.2b)where *s*=sin *θ* and the arbitrary functions of *X*, *Z* arising in the integration with respect to *T* have been set to zero. This is justified based on the assumption that they decay at worst algebraically and monotonically (note that the other terms decay exponentially), and are such that *U*_{1m} and *W*_{1m} are integrable over all space.

Diagrams to illustrate the mean flow contributions calculated in this and the preceding section follow.

## 4. Numerical results illustrating the calmed region

Experimental evidence (e.g. Seifert *et al*. 1994; Gostelow *et al*. 1997) shows that the calmed region lies behind the moving spot, and is characterized by a laminar profile with a distinctive property. The profile is fuller and more stable than that of the unperturbed boundary layer, with the overshoot in skin friction decaying to zero as the distance from the trailing edge of the spot increases. If the mean flow calculated in §§2 and 3 is to describe the excess over the undisturbed flow in the neighbourhood of the plate, then the streamwise component should be positive over a non-negligible region to the rear of the spot. To determine whether the present analysis matches such a profile, we first consider the sum of the mean flow contributions (2.5*a*) and (3.2*a*) on the centre-line . In terms of the original variable , the sum is(4.1)where , and . Since (3.2*a*) holds only for *X*>0 and *X*^{2}>8*Z*^{2}, outside this wedge-like region the term O(*h*^{2}) is replaced by a smaller contribution O(*h*^{2}*X*^{−3}) analogous to the complementary function term appearing at O(*h*). However, such a term is of smaller order than those to be retained and may be ignored.

We see from (4.1) that the two terms therein are of the same size if so that, because , we have as required. As suggested towards the end of §2, we take the region inside the wedge illustrated in fig. 2 of DS to describe the calmed region with , in which case (4.1) reduces to(4.2)when . In (4.2), the new variable defined by is taken to be O(1) in the region of current interest. The contribution to (4.2) from the term O(*h*^{3}) in (2.3) is, in the new variable , o(*T*^{−1}) and thus smaller than those displayed.

The expression for *U*_{mean} on the axis of symmetry given in (4.2) needs to be positive for a meaningful description of the flow perturbation on the centre-line of the calmed region. This is satisfied for when *h*>0, and for all if *h*<0. The sign of *h* is determined by the initial value *hA*_{0}(*X*, *Z*, 0) of the leading approximation to the negative displacement *A*(*X*, *Z*, *T*). Thus, the initial perturbation slip velocity has the sign of *h*, and the large time streamwise mean flow has in both cases a considerable expanse in which it is positive, a distinctive feature of the calmed region. Figure 1*a* shows on *Z*=0 from (4.2) as a function of for *h*>0 and *h*<0. In both cases, there is an algebraic decay to zero as increases into what is anticipated to be the main body of the spot. In addition, the *h*>0 situation exhibits a monotonic increase between its zero at and its maximum at (2/3)^{1/2}. This decrease in slip velocity as decreases away from the rear of the spot is also evident in the centre-line velocity profiles of Gostelow *et al*. (1997) as the skin friction overshoot returns to zero after the passage of the spot.

The expression for *W*_{mean} on the axis *Z*=0 corresponding to (4.2) for *U*_{mean} is derived in a similar way from (2.6*c*) and (3.2*b*). The result is that(4.3)where *s*=sin *θ* as before. Thus, the mean transverse velocity is towards the axis if *h*>0 but if *h*<0 is away from the axis for and towards it thereafter. The quantities are illustrated in figure 1*b*.

In the neighbourhood of the caustic both *U*_{1m} and *W*_{1m} in (3.2*a*,*b*) become singular. At the caustic and , and the term *O*(*h*^{2}) dominates the term *O*(*h*) given by (2.5*a*,*b*).

To illustrate the perturbation to the mean flow on the surface for all , we must return to (2.5*a*,*b*) and (3.2*a*,*b*) without setting *Z*=0. First, we define in terms of *Z* and by the same transformation as for , and ignore the term in *q* in (2.5*a*,*b*) as we did to obtain the first term of (4.2) after choosing . Thus, as in equations (4.2) and (4.3), only the terms in *q*^{−1} in (3.2*a*,*b*) need to be retained and the exponentials may be replaced by unity. The results are that, when and (i.e. inside the wedge),(4.4a)(4.4b)Here, and m(*θ*) are obtained from (3.2*a*,*b*) as(4.5a)(4.5b)Outside the boundary of the wedge, or caustic, on which they are both large and negative, *U*_{mean} and *W*_{mean} are obtained from (4.4*a*,*b*) by setting .

Three-dimensional plots for _{,} derived from (4.4*a*,*b*) by omitting the factor , are given in figures 2 and 3 for *h*>0 and for *h*<0. In all figures, the prolonged wedge and caustic are clearly seen. In figure 2, for the downstream wedge containing positive mean flow velocities, stands out clearly in both cases. Outside the caustic, the solutions are simply of opposite sign, and inside, where the term in in (4.5*a*) soon dominates as increases, the property of *U*_{mean} on the axis of symmetry seen in figure 1*a* is evident. That is, if *h*>0, *U*_{mean} on the axis has a positive maximum and thereafter decreases, but if *h*<0, then *U*_{mean} decreases monotonically into the main body of the spot. In figure 2, the main body is expected to lie within the wedge at values of that are O(1) or larger. Figures 3*a*,*b* show for *h*>0 and *h*<0, respectively. Again, the change of sign of *h* makes a significant difference outside the wedge only; inside, for both cases, ‘inwash’ is clearly the dominant feature. In §5 below, these results are discussed and compared qualitatively with experimental investigations.

## 5. Discussion and further comments

In the preceding section, we examined the mean flow perturbation caused by long-scale, three‐dimensional unsteady disturbances in a thin layer Euler flow. The governing equations are nonlinear, but triggering a small amplitude initial disturbance first enables a leading-order O(*h*) linear solution to be obtained explicitly. This is the solution of DS, the form of which enables two waves to be identified inside a caustic for large values of the time. This confined wedge-like region, although here laminar, displays similarities to a turbulent spot in a transitioning boundary layer. Secondly, the present work examines the mean flow that arises at O(*h*^{2}) from forcing by-products of the wave-like contributions based on the expectation that, as this is indeed a laminar property, it may describe the laminar ‘calmed’ region observed behind such a turbulent spot.

Comparison with experimental results is encouraging in two important respects. In the work of Gostelow *et al*. (1997) and Seifert *et al*. (1994), it is clear that there is experimentally an extensive region of positive mean flow in the positive *X*-direction, and also considerable inwash (i.e. flow towards the spot symmetry line). Both of these phenomena may be seen in figure 1 for and . However, an important feature of the experiments is not described here; namely, the decrease in wall shear stresses from the rear of the turbulent part of the spot where the shear stress is high, through the laminar calmed region with increasingly less full profiles (see Gostelow *et al*. 1997), to the mainly unperturbed fluid behind. Despite the qualitative agreement of the present theory with experiments concerned specifically with the calmed region, quantitative comparison remains to be attained.

## Footnotes

One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.

- © 2005 The Royal Society