## Abstract

The stability of developing entry flow in both two-dimensional channels and circular pipes is investigated for large Reynolds numbers. The basic flow is generated by uniform flow entering a channel/pipe, which then provokes the growth of boundary layers on the walls, until (far downstream) fully developed flow is attained; the length for this development is well known to be (Reynolds number)×the channel/pipe width/diameter. This enables the use of high-Reynolds-number theory, leading to boundary-layer-type equations which govern the flow; as such, there is no need to impose heuristic parallel-flow approximations. The resulting base flow is shown to be susceptible to significant, three-dimensional, transient (initially algebraic) growth in the streamwise direction, and, consequently, large amplifications to flow disturbances are possible (followed by ultimate decay far downstream). It is suggested that this initial amplification of disturbances is a possible and alternative mechanism for flow transition.

## 1. Introduction

The transition to turbulence found experimentally in pipe flows remains a significant paradox in fluid mechanics. The famous Reynolds (1883) experiment undertaken in Manchester is one of the landmark experiments in fluid dynamics. However (and ironically), although linear stability analysis on the (fully developed) problem can be performed in an entirely rational/rigorous fashion, without any need for the use of heuristic arguments, such as parallel-flow approximations, the overwhelming evidence from such theories indicates flow *stability* (Drazin & Reid 1981; Schmid & Henningson 2001). Reynolds (1883) suggested that the flow was turbulent for Reynolds numbers (based on the pipe diameter and mean flow velocity) greater than approximately 2000, while later experiments (Pfenninger 1961) indicated that the critical Reynolds number can be increased to approximately 100 000 if the experiment is conducted with great care (i.e. if the disturbances to the flow are kept to a minimum). The lower limit has also been further investigated, by (amongst others) Binnie & Fowler (1947), Leite (1959), Wygnanski & Champagne (1973) and Darbyshire & Mullin (1995); the conclusion from these studies is that Reynolds's (1883) value is close to the lower critical limit. These experimental observations then beg the question of how/if the critical Reynolds number may be related to the amplitude of the disturbance field (implying a nonlinear mechanism). Some very recent experimental work by Hof *et al*. (2003) suggests the scaling law that the amplitude of perturbation required to cause transition is inversely proportional to the Reynolds number.

Another suggested resolution of the dichotomy between experiment and theory (of the fully-developed problem) has been linked to temporal transient growth. The first significant analysis of transient growth in internal flows was presented by Gustavsson & Hultgren (1980), for the case of plane Couette flow (another very simple base flow for which experiments and linear stability theory disagree). However, it should be pointed out that the mechanism described by Gustavsson & Hultgren (1980) involving a resonance between Orr–Sommerfeld and Squire modes is quite different from that of the present paper (and indeed of more recent transient growth studies), which may be regarded as being linked to the non-orthogonality of the Orr–Sommerfeld operator. Subsequently, Gustavsson & Hultgren's (1980) ideas were developed by Gustavsson (1991) for the case of plane Poiseuille flow. Later, Butler & Farrell (1992) obtained more quantitative results for plane Couette flow and plane Poiseuille flow (and the Blasius boundary layer), and went on to determine the ‘optimal’ disturbance, i.e. the initial disturbance leading to maximum energy growth. Alternative but related arguments based on the spectrum of eigenvalues, in particular the non-orthogonality of eigenfunctions arising from the Orr–Sommerfeld equation (in the case of plane Couette and Poiseuille flow), were proposed by Trefethen *et al*. (1992) in their study of transient growth. The underlying argument for this approach is that although normal-mode analysis indicates stability, the non-orthogonality results in a significant transient response to disturbances, prior to eventual temporal decay; this opens up the possibility of nonlinearity then leading to breakdown/transition.

A further proposed line of investigation that has been followed in the past has been to study the effects of flow development. These are well known (Wilson 1969) to persist at relatively large distances downstream of the inlet, specifically distances of the order of the Reynolds number diameters downstream. The developing-flow approach (based on the Orr–Sommerfeld methodology) has been used by a number of authors including Tatsumi (1952), Huang & Chen (1974*a*,*b*), Garg & Gupta (1981), Gupta & Garg (1981), Abbot & Moss (1994) and da Silva & Moss (1994). However, the base flow in this (developing) regime is non-parallel, and these aforementioned studies all utilized the parallel-flow approximation, and as such, neglect some (possibly important) terms in the stability analysis. A comparison between the experimental results of Tatsumi (1952) and the theoretical work of Huang & Cheng (1974*a*,*b*) was presented by Sarpkaya (1975). A more recent investigation is due to Savenkov (1993), in which high-Reynolds-number asymptotic analysis (based on the triple-deck theory of Stewartson 1969) was employed on the developing flow profile. This study focused on (lower-branch Tollmien–Schlichting) modes at very high frequency. A further related study is that of Smith & Bodonyi (1980), who also investigated the role of upper-branch modes on the stability of developing channel and pipe flows.

In this paper, although the underlying basis for our study is on developing flow, we adopt a quite different approach to previous investigations of this type. We assume (as in previous studies) that the incoming flow into the channel or pipe is uniform. We may expect that in the immediate vicinity of the inlet, Blasius-type boundary layers form on the wall(s) of the pipe (channel). Quite recently (Luchini 1996), it has been shown that Blasius boundary layers are susceptible to three-dimensional disturbances which exhibit algebraic growth in the streamwise direction. Other still more recent work includes that of Andersson *et al*. (1999) and Luchini (2000), who showed that if the disturbance spanwise wavelength was comparable to the boundary-layer thickness (whereas the streamwise scale is much longer), then the disturbance fields exhibited a maximum response at a finite downstream location, before decaying downstream (i.e. the growth was bounded). A study of the effects of unsteadiness on modes of this general type was undertaken by Duck & Dry (2001), and it was concluded that unsteadiness has a strong tendency to reduce the response to imposed disturbances. The effects of nonlinearity were incorporated into the study of Duck (2003), in which it was found that breakdowns (i.e. singularities) were a quite common feature of these disturbed flows, suggesting a strong link to the boundary-layer transition process. Indeed, Kachanov (1994) has proposed a connection between algebraic growth and by-pass transition. The same general approach was used to study the effect of flow disturbances on two-dimensional similarity states by Duck & Owen (2004).

On account of the Blasius-like flow near the inlet, we may expect the imposition of *three-dimensional* disturbances on developing pipe and channel flows in the first instance to therefore trigger growth. Given the long lengthscales involved in the streamwise direction, there must certainly exist the potential for significant linear growth. One of the complicating features of flows of the two classes to be studied is that, unlike previous studies, the base flow itself evolves downstream (i.e. is of non-self-similar form), and this must be taken fully into account. However, one of the appealing features of the analysis in this paper is that it is based entirely on rigorous asymptotic analysis (in particular boundary-layer theory), with no need for recourse to *ad hoc* approximations, such as parallel-flow approximations, as have been necessary in previous theoretical studies of developing flows of this type.

The layout of this paper is as follows. In §2, developing channel flows are considered. It is very clear that great care must be taken in the numerical work for both the base state and the disturbance field. One particular feature is the use of appropriate coordinates and dependent variables, to ensure the computed quantities are all regular functions of the chosen coordinate system. In §3 the related problem of developing Hagen Poiseuille flow is studied. Some comparison with experimental results and our conclusions are presented in §4.

## 2. Developing plane Poiseuille flow

We consider first the development of entry flow within a two-dimensional channel. The base (developing) flow received a great deal of attention some years ago. Van Dyke (1970) and Wilson (1971) considered the flow resulting from a uniform flow entering a two-dimensional channel (the configuration of interest to this section), and showed how, in the immediate vicinity of the inlet, symmetrical Blasius-type boundary layers form on the two walls. Further downstream, the two boundary layers merge and the flow asymptotes towards fully developed flow through a sequence of exponential eigensolutions (Wilson 1969; see also Stocker & Duck 1995), with the rate of asymptote being controlled primarily by the eigenvalue of smallest magnitude.

We take coordinates *L*(*xRe*, *y*, *z*), origin at the leading edge of the lower wall, which lies along *y*=0 (the upper wall lying along *y*=2) for 0<*x*<∞, with the flow directed along the positive *x* direction; *z* is the spanwise direction. Here *L* is the semi-width of the channel, and the Reynolds number , with *U*_{∞} the incoming (uniform) flow velocity and *ν* the kinematic viscosity of the fluid (assumed to be incompressible). Throughout this paper, it is assumed that , enabling the use of the boundary-layer equations. Note, in particular, that the streamwise (*x*) scale is long, in line with classical entrance-flow work at large Reynolds numbers. The flow velocity is then written(2.1)while the pressure develops in the form(2.2)*ρ* being the density of the fluid. Substituting equations (2.1) and (2.2) into the Navier–Stokes equations, and taking the leading-order (in *Re*) terms leads to(2.3)(2.4)(2.5)(2.6)where (*L*/*U*_{∞})*t* denotes dimensional time. Notice that the leading-order pressure term is solely dependent on the downstream location, while the *y* and *z* components of the momentum equation involve the *third* term in the pressure expansion. These equations may be regarded, quite simply, as a quite standard form of the three-dimensional boundary-layer equations, being found in corner flows (Rubin 1966), and are inherent in more recent work on algebraic growth in boundary layers (Luchini 1996, 2000; Andersson *et al*. 1999).

We now decompose the flow into a base flow and a perturbation, as follows:(2.7)*δ* being a measure of the small amplitude perturbation. Here it has been assumed there is no crossflow component to the base flow.

### (a) The base flow

The streamwise development of the basic flow in the entry regime is now very well understood, but from a numerical point of view (of particular relevance to the present study) great care must be exercised in order to obtain high-quality, accurate solutions. This necessitates incorporating the singular behaviour of the base flow into the numerical scheme, close to the channel entrance, mimicking the asymptotic structures which describe the formation of the two boundary layers, as described by Van Dyke (1970) and Wilson (1971). To this end, we embed a double structure to the problem. It is also useful to introduce a streamfunction (*ψ*) for the base flow, together with the vorticity (*ζ*), defined as follows:(2.8)Close to *x*=0, we then develop (and match) ‘inner’ and ‘outer’ solutions as follows: for the inner solution (close to *y*=0)(2.9)where , . Substitution of the above into equations (2.4) and (2.8) leads to(2.10)(2.11)The following standard boundary conditions, representing impermeability and no slip are appropriate on the wall:(2.12)We shall defer discussion of the other required conditions until after a consideration of an outer region. In this other zone, given that we are assuming uniform inlet conditions (i.e. *U*_{0}=1 at *ξ*=0), and in consideration of the need to match with the inner layer, we write(2.13)which on substitution into equations (2.4) and (2.8) leads to(2.14)(2.15)It is necessary to impose symmetry about *y*=1, namely(2.16)

Close to the inlet, i.e. as *ξ*→0, conditions near the wall approach those of the Blasius boundary layer (and this is confirmed by setting *ξ*=0 in equation (2.11)). This provides the necessary initial condition to be imposed on *Ψ*_{0} at *ξ*=0, which implies(2.17)(*δ*^{*} representing displacement effects), and so matching between the inner and outer solutions at *ξ*=0 leads to the conclusion that(2.18)which then provides initial conditions to be imposed on and at the inlet. The numerical approach implemented is based on a second-order, finite-difference scheme, utilizing a standard Crank–Nicolson procedure to extend the solution downstream, together with Newton iteration in order to handle the inherent nonlinearity to the problem. The overall treatment of the problem necessitates the use of four first-order equations by means of introducing and in the inner region, and treating equations (2.10) and (2.11) as first-order equations for and , respectively, and and in the outer region, and treating equations (2.14) and (2.15) as first-order equations for and , respectively. Overall, the differencing is based on the Keller (1978) box scheme. A further key point is to ensure that the *η* and *y* grids mesh properly, which leads to some constraints on the numerical grid. If we assume that the *η* grid extends from *η*=0 to *η*=*η*_{∞} in steps of Δ*η*, while the *y* grid extends from *y*=*η*_{∞}*ξ* to *y*=1 in steps of Δ*y*, then this leads to the constraint that(2.19)where Δ*ξ* is the (uniform) step size in the *ξ*-wise direction. As the solution is marched downstream in this region, one *y* grid point is discarded at each *ξ* step, in order that the upper grid point always corresponds to the line of symmetry, *y*=1.

A downstream location must be chosen (denoted here by *ξ*_{c}), at which the double grid changes to a single grid. In previous studies (e.g. Smith 1974) the transition has been made to a uniform (*x*,*y*) grid, however, after a significant amount of numerical experimentation, the most effective (i.e. accurate) scheme appeared to be one in which the (*ξ*,*η*) grid was retained beyond *ξ*=*ξ*_{c}, with the solution on the (*ξ*,*y*) grid transferred to the (*ξ*,*η*) grid at *ξ*=*ξ*_{c}. By far the most efficient means to do this is if(2.20)which makes this transfer a trivial task. Taking equations (2.19) and (2.20) together leads to the condition that(2.21)and so if there are *J η*-points such that Δ*η*=*η*_{∞}/(*J*−1), and *K y* points at *ξ*=0 such that Δ*y*=1/(*K*−1), then we must have(2.22)In order to properly ‘patch’ the inner and outer grid solutions, the following conditions were applied at the interface (for *ξ*≤*ξ*_{c}):(2.23)(2.24)(2.25)(2.26)Note that the value of *η*_{∞} (typically 20) was chosen to ensure the results to be numerically insensitive to this value. Beyond *ξ*=*ξ*_{c}, in order that the uppermost grid point coincides with the line of symmetry as the solution is progressed downstream, one (*η*) point is discarded at each *ξ* station. Of necessity, this requires a non-uniform Δ*ξ* in this region, with the set of streamwise (*ξ*) locations being given by(2.27)where *M* denotes the local number of gridpoints across the channel, which decreases by 1 at each streamwise location. Consequently, the step size is given by(2.28)This causes no numerical difficulties, provided sufficient gridpoints are retained, i.e. if the value of *M* is sufficiently large. The uppermost boundary conditions are then(2.29)In this manner, there is a smooth grid development downstream—other grids the author experimented with, possessing less smooth qualities, were very susceptible to spurious solution oscillations. The numerical grid employed is shown schematically in figure 1.

### (b) The flow perturbation

We now turn to consideration of perturbations to the base flow described above. Given the linear nature of the disturbances, we seek flow perturbations of the form(2.30)disturbances which are generally three-dimensional (if *β*≠0) and unsteady (if *ω*≠0). Note that there can be no perturbation to the leading-order pressure term *P*_{0}, which has no *y* or *z* dependency.

Let us consider the ‘inner’ development of the disturbance field first. In keeping with the singular development of the baseflow, we introduce the following working (‘starred’) variables:(2.31)The scalings above are chosen to mimic those of the base flow (see equation (2.9)), together with the transverse (*η*) scaling introduced for the base flow. Here (and in the outer flow zone), it is found advantageous to eliminate the third pressure term, , and this is facilitated by introducing a modified vorticity function,(2.32)together with (scaled and regularized) baseflow variables(2.33)The continuity equation, written in terms of the ‘starred’ variables is(2.34)The streamwise disturbance momentum equation takes the form(2.35)while a combination of the crossflow (*z*) and transverse (*y*) momentum equations, which eliminates the pressure term from both equations, may be written as(2.36)In the ‘outer’ zone, we adopt an entirely analogous approach. We define scaled velocity components as(2.37)Here the *ξ* scalings are chosen to lead to consistent matching between the inner and outer solutions, in particular, the transverse velocity component which is the dominant velocity component that ‘drives’ the perturbation flow in the core, as *ξ*→0, as verified below. We again introduce a modified vorticity, this time by(2.38)while continuity is written as(2.39)and we define regularized base-flow velocity components(2.40)The streamwise momentum equation is then(2.41)Eliminating the pressure term between the crossflow and transverse momentum equations leads to(2.42)

In keeping with the base flow, the system was treated as a system of first-order equations, through the introduction of derivative variables, i.e. , and in the inner zone and , and in the outer zone.

For the disturbance field, the patching conditions at the interface between the two zones are then(2.43)

These conditions arise directly by connecting equation (2.31) with equation (2.37). The boundary conditions on *η*=0 are , while the condition of symmetry along *y*=1 leads to , while that for antisymmetric modes (which are also permissible) is .

The final set of conditions that must be addressed are those for the disturbance field at the inlet (*ξ*=0). This is a key step and builds in the expectation that three-dimensional flow disturbances will be susceptible to streamwise algebraic growth, on account of the Blasius-like inlet profiles, which are known (Luchini 1996) to be prone to these types of instabilities. Accordingly, as *ξ*→0 we expect(2.44)Substituting equation (2.44) into equations (2.32) and (2.34)–(2.36) and then setting *ξ*=0 results in the system(2.45)(2.46)(2.47)(2.48)This system replicates that found by Luchini (1996) (also investigated by Duck *et al*. 1999), and represents an eigenvalue problem for *λ*. The key point here is that this system *does* possess a single (positive; and real) eigenvalue *λ* (with a value of approximately 0.426) when *β*≠0 when correspond to the Blasius state. Note the mode has a spanwise wavelength comparable to the channel width, and so as *ξ*→0, this wavelength is much longer than the boundary-layer thickness. Further downstream, as the boundary layer fills the channel, the boundary-layer thickness becomes comparable to the spanwise wavelength; in this respect, there is some analogy with the transient growth in boundary layers studied by Andersson *et al*. (1999) and Luchini (2000), with one important difference—in the present study the base flow evolves downstream (in a fully non-similarity manner). It is the eigenfunction associated with the ‘Luchini’ eigenvalue that provides the initial conditions for the inner disturbance field at *ξ*=0, and so we set(2.49)at *ξ*=0.

Taking the limit as *ξ*→0 in the outer equations leads to the conclusion that for symmetric modes(2.50)while for antisymmetric modes(2.51)Clearly as *ξ* increases, additional terms will become important, modifying the algebraic growth downstream, as determined by our numerical scheme.

One final point of detail should now be mentioned, concerning the normalization procedure. The set of eigenfunctions may of course be normalized quite arbitrarily; however, in general, this leads to results that are dependent on the (streamwise) gridsize. This issue may be resolved if the input disturbance is normalized consistent with equation (2.44); such that , Δ*ξ* representing here the size of the first step size in *ξ*, and *λ* takes on the value as discussed above. Indeed, given the algebraic nature of the eigensolutions, this is a perfectly natural choice of normalization. In this manner, disturbance fields which are effectively grid dependent may be calculated, and this then enables us to undertake a meaningful comparison of results as the various parameters are varied.

Second-order differencing was then used to approximate the perturbation system, coupled with a routine Crank–Nicolson procedure in the streamwise direction. The bandedness of the resulting system (at each streamwise location) was then fully exploited in the inversion process, which then required no iteration (on account of its linear nature).

### (c) Results

The first set of results we present is for the case of steady perturbations (*ω*=0), for a variety of values of *β*; certainly, in the flat-plate case studied by Luchini (2000), it was found that it was the steady disturbances that exhibited the largest growth rates and Duck & Dry (2001) showed how, in the flat-plate case, there is a profound difference between steady and unsteady disturbances (in their case, linear steady disturbances exhibited unbounded algebraic growth downstream, whereas unsteady disturbances always decayed downstream, in an oscillatory manner). Results are shown in figure 2; it should be noted that all results shown were checked for numerical accuracy, which is at least as good as graphical accuracy (this remark is applicable to all the results presented in this paper). In these figures, results for a range of crossflow wavenumbers (*β*) are shown. It should also be noted that for steady perturbations, *U**, *V**, *u** and *v** are purely imaginary, whereas *W**, *Θ**, *w** and *θ** are purely real. Comparing figure 2*a* with *b* indicates that the antisymmetric modes generally exhibit a stronger response than corresponding symmetric modes (based on *Θ** at the wall). These figures also indicate that the maximum attained by this quantity is observed at some finite value of *β*. Additional computations indicated this to occur at *β*≈0.7 in the case of symmetric modes and *β*≈1.5 in the case of antisymmetric modes. However, it should be emphasized that this is just one measure of the response of the flow to perturbations. Figure 2*c*,*d* shows the variation of the streamwise component of the perturbation wall shear for selected values of spanwise wavenumbers. These figures indicate that as *β* increases, so does the maximum value of . In fact, this trend is readily confirmed by inspection of equation (2.44).

We now turn to consider a selection of unsteady results. Figure 3*a* shows results for the case *ω*=10, *β*=1. The unsteadiness causes disturbance quantities to become complex. These results indicate a more oscillatory behaviour for flow quantities—presumably linked to the oscillatory behaviour of perturbation flow quantities found generally in the unsteady flat plate case (Duck & Dry 2001).

Finally, it should be stressed that in all these calculations, the inlet perturbation (forcing) was very small, and the flows clearly exhibit a significant response.

## 3. Developing Hagen Poiseuille flow

Although the basic methodology here follows that used in the previous section on plane Poiseuille flow, there are a number of important yet subtle differences. The developmental length is again long compared with the pipe diameter, and fully developed flow is again approached exponentially downstream (Bramley 1986). In this case we take (*xLRe*, *Lr*, *ϕ*) in the axial, radial and azimuthal coordinates (this somewhat non-standard notation is adopted in order to follow the work of the previous section as closely as possible), where *L* denotes the pipe radius. The corresponding velocity vector is given by equation (2.1), and the pressure develops in the form of equation (2.2) (but with *z* replaced by *ϕ*). Taking the boundary-layer approximation to the Navier–Stokes equations, in a manner similar to that described in the previous section, leads to(3.1)(3.2)(3.3)(3.4)

These may be regarded as a non-axisymmetric version of the quasi-cylindrical equations (e.g. Hall 1972). Let us consider the appropriate system of ‘inner’ and ‘outer’ coordinates to be used in this case. Perhaps an obvious choice would be (for the inner) and 1−*r* (for the outer). However, this choice led to a variety of irregularities at the inlet, which in turn caused numerical difficulties downstream. After a substantial amount of experimentation, the final choice (which has excellent numerical properties) was found to be for the inner transverse coordinate, together with *y*=(1/2)(1−*r*^{2}) as the outer transverse coordinate (and so , which implies the pipe wall lies along *y*=0 and the axis along *y*=1/2). Again the notation has been deliberately chosen to follow that of the previous section as closely as possible, and indeed the resulting set of equations has much resemblance to the channel-flow set of equations. To this end, we again decompose the flow field into its components in a manner similar to equation (2.7), but with one minor alteration, namely(3.5)Here, the minus sign has been inserted into the definition of the radial velocity component (*V*) such that positive *V*_{0} and denotes flow directed in the positive *y* direction, i.e. radially *inwards*.

### (a) The base flow

In consideration of the base flow, it is then possible to define a (Stokes) streamfunction (*ψ*) and vorticity function (*ζ*) as follows:(3.6)

For the inner solution we again utilize equation (2.9), while equations (2.10) and (2.11) must be modified by the inclusion of curvature terms (these are omitted in the interests of brevity); equations (2.12) are again applicable. In the outer zone, equation (2.13) may be used, while equations (2.14) and (2.15) must be modified to include curvature terms. The symmetry conditions (2.16) are applicable here (but must be implemented along *y*=1/2), together with the matching conditions (2.43).

Regarding the inlet conditions (i.e. the conditions at *ξ*=0), these are the Blasius conditions for the inner region, which must be matched on to the outer region (through equation (2.17) again). As *ξ*→0, again it is found that , implying that(3.7)The formulation described above, coupled with the matching of the grids in an identical manner to that described in the channel case, and the switch to a single (*η*,*ξ*) grid at *ξ*=*ξ*_{c} yielded smooth and accurate baseflows, appropriate for the computation of the associated perturbation flowfields. One further minor difference with the channel-flow situation is that equation (2.27), namely the location of the streamwise gridpoints beyond *ξ*_{c}, is replaced by(3.8)and correspondingly, equation (2.29) is replaced by(3.9)(since 0≤*y*≤1/2). Note that results for the developing base flow have been presented by Mohanty & Asthana (1978).

### (b) The flow perturbation

We may again invoke a form similar to equation (2.30) for the disturbance field, namely(3.10)where here, clearly, *β* must be restricted to integer values. For the inner zone, we chose to work with the starred variables given by equation (2.31) and a natural variable to introduce was a modified vorticity function, described by(3.11)The channel continuity and momentum equations (2.34)–(2.36) are modified with the inclusion of curvature terms (but are not presented here).

Turning now to consider the nature of the flow perturbations in the zone, we again use equation (2.37), and introduce the modified vorticity function by(3.12)The net result is the system (2.39), (2.41) and (2.42) with the inclusion of curvature effects. The boundary conditions are zero velocity at the wall (*y*=0) together with boundedness/zero conditions along *y*=1/2.

The inlet conditions for are again those corresponding to the Luchini (1996) eigenvalues (for precisely the same reasons as in the planar case), and hence given by equation (2.44), while for the zone at the inlet leads us to the conclusion that as *ξ*→0, *u*^{*}→0, *θ*^{*}→0, while *v*^{*} is described by(3.13)subject to and *v*^{*}(*y*=1/2)=0. The appropriate solution is(3.14)implying the (important) restriction that *β*≥2 (and of course an integer). *β*=1 is inadmissible because of the need for zero radial velocity at *r*=0 (*y*=1/2). It follows that(3.15)Finally, for precisely the same reasons stated with regard to the channel-flow case, a normalization based on (Δ*ξ*)^{λ} was again implemented.

### (c) Results

Figure 4*a*,*b* shows the streamwise distributions of and for a range of values of *β* for the steady case (*ω*=0); the conditions of disturbance quantities being purely real/imaginary are again applicable in this case. The results for *Θ*^{*}(*η*=0) show a fair degree of qualitative resemblance to the corresponding channel results, presented in figure 2, with one important exception, namely that the perturbation response extends over a much shorter streamwise scale. The maximum response (in *Θ*^{*}(*η*=0)) occurs for *β*=2. The results shown in figure 4*b* also show a close similarity with those shown in figure 2*c*,*d*, although our earlier remarks with regard to the streamwise extent are again applicable, and indeed a concentration (and increase in magnitude) of response of the streamwise component of wall shear is again observed with increasing wavenumber *β*.

Figure 5 gives an indication of the response of the core flow with variations in wavenumber *β*, where(3.16)This is clearly an entirely arbitrary measure of the core response, but nonetheless will give some indication of the response of the core to the imposed flow perturbations. This figure suggests that the largest core response is provoked at the smaller wavenumbers, although, as noted already, maximum responses are very much dependent upon the particular flow quantity (and indeed radial location) under consideration.

We now consider some unsteady calculations. Figure 6 shows streamwise distributions of *Θ*^{*}(*η*=0) and for the case *ω*=100, *β*=2. As with the planar case, oscillatory behaviour is an inevitable consequence of the introduction of unsteadiness (cf. figure 3*a*).

In order to judge the more global effect of unsteadiness, in figure 7 we show the variation of the response function with frequency, *ω*, for *β*=2, 3, 4. This suggests that unsteadiness serves to reduce the value of the response function, i.e. the response of the core.

## 4. Comparison with experiment and conclusions

All preceding results in this paper have involved feeding an eigensolution (one corresponding to the Luchini (1996) mode) at the inlet. In order to mimic some of the experimental work that has been undertaken in the past on pipe flows, we now consider an alternative trigger to the disturbance field, namely one that models the injection/suction of fluid at a finite downstream location, through a very narrow slot (the approach followed in some previous experimental work). To this end, specifically, the perturbation to the flow field was forced as follows:(4.1)all other boundary conditions being homogeneous; this form was deliberately chosen in order to introduce smooth perturbations into the flow (discontinuous forcing had a strong tendency to trigger large, sustained and spurious numerical oscillations). In this manner, the effect is azimuthal forcing of the fluid over a short downstream extent (if , then this streamwise extent is , centred about *ξ*=*ξ*_{0}). It is important that the forcing comprises an azimuthal/swirl component, in order to trigger the cross-flow and hence the Luchini (1996) type modes. Since the results of the previous section indicate that the ‘most dangerous’ modes are those corresponding to *ω*=0 (the steady case) with *β*=2, these were the parameters chosen for these calculations. Note, however, that from the results of the previous section, moderate values of frequency input (*ω*) are unlikely to affect our results significantly. The main point of interest here is how the slot location (i.e. *ξ*_{0}) affects the flow response. Figure 8*a*,*b* shows the variation of for the selected values of *ξ*_{0} as indicated, for *Λ*=1000 and *Λ*=100 000, respectively. Note that as *ξ*_{0} decreases, the distributions shown in figure 8*b* (and, as expected, to a lesser extent figure 8*a*) closely approach the *β*=2 eigensolution-type results shown in figure 4*b* (suitably scaled linearly).

Let us now consider some similarities and differences between these results and theoretical and experimental data that have been presented in the past (including some recent results of Williams 2001), together with a number of related issues. Much of the work in the past has been concerned with axisymmetric disturbances, while the present model inherently assumes non-axisymmetry (the fundamental assumption in the leading-edge disturbance field is one of three-dimensionality, thus making these modes quite distinct from those to which Squire's theorem is applicable). Equally, since the present theory is based on the assumption of infinite Reynolds number, it cannot, by its very nature, predict critical Reynolds numbers. Further, since we do not employ a normal-mode analysis in the streamwise direction, the concept of a streamwise wavenumber is, in our context (and indeed the context of developing, non-parallel flows), meaningless; it is therefore not possible to compare with table 1 of Sarpkaya (1975) for example. Indeed, a number of comparisons between neutral stability curves (predicted using the normal-mode approach on developing flows) and experimental observations show quite large discrepancies (Sarpkara 1975; Williams 2001). Sarpkaya (1975) describes three modes of flow disturbance used in experiment. In the first, an electromagnetic device was used to generate disturbances through vibration; in the second, a sleeve was oscillated harmonically in the azimuthal direction; in the third, two sets of tripping wires were inserted and withdrawn from the pipe (using two sets of electromagnets). In the more recent work of Williams (2001), two disturbance mechanisms were employed. In the first, water was injected and withdrawn though a single cylindrical hole, perpendicular to the wall. In the second case, the injection/withdrawal was by means of six holes, each making an angle of 32° to the wall. In any case, equation (4.1) must be regarded as an idealization to any experimental configuration.

In the work of Williams (2001), there appears to be quite a marked difference (up to a factor of three) in results for critical amplitudes of disturbance necessary to provoke transition between the two experimental configurations, with the six-jet case requiring smaller amplitudes than the single-jet case. One (possible) explanation is that in the former, there is an explicit forcing of the azimuthal velocity component, while in the latter case this is not so, i.e. there might exist two (or more) competing mechanisms for transition.

It is worth noting that there does seem to be a marked disparity between the experimental results (discussed by Sarpkaya 1975) and the theoretical predictions of Huang & Cheng (1974*a*,*b*), with respect to critical Reynolds numbers (based on normal-mode analysis), and since the location of the flow regime corresponds very much with that described in the previous section in which flow perturbations grow/persist, the mechanism described in this paper is certainly a candidate to describe this disparity. Sarpkaya (1975) also references a thesis of Mackrodt (1971) in which the suggestion is made that the instability observed in fully-developed Poiseuille flow may be induced by swirl (originating from the fluid reservoir); effectively the present paper addresses the issue of disturbance development owing to infinitesimal amounts of swirl in the inlet region, and it is clear that significant flow disturbances can result.

Since the present results are linear in nature, and are non-axisymmetric (with azimuthal wavenumbers greater than or equal to two), it should be noted that all the disturbances reported in this paper for pipe flows must have zero centreline velocities; as a consequence, any experimental observations of these modes, taken along the pipe axis would inevitably measure only *nonlinear* effects (leading to axisymmetric modes), and therefore would not be an accurate assessment of linear disturbance development.

Figure 8*a*,*b* indicates that as *ξ*_{0} increases, i.e. as the forcing is applied at distances further downstream, although the initial reaction of the flow increases, the flow disturbances immediately decay. On the other hand, in the case of forcing applied close to the leading edge (i.e. smaller values of *ξ*_{0}), the initial impulse is proceeded by a period of quite sustained growth, followed by ultimate decay (similar to the eigen-response of figure 4*b*). Consequently, although the initial response increases significantly with an increase in *ξ*_{0}, the disturbance does not seem to persist especially far downstream; these two competing trends, therefore, make drawing firm conclusions difficult. To some degree, this increase in the magnitude in the initial response of is because of the multiplicative *ξ* in equation (4.1), however, it is worth bearing in mind that when transforming back from our working variables (in particular ) to more physical variables (in particular ) there is then a one-to-one correspondence in the ordering of these quantities with regard to *ξ*, i.e. , on account of equation (2.31). There are various other subtleties when comparing between experimental studies (involving developing flows) and theory. It is not clear if it is the magnitude of the injection/suction velocity that is key here or if it is the magnitude of the impulse that is important. Further, the same distributions of azimuthal velocity in the experiments will likely have more impact as the location of the forcing is moved downstream, as the boundary-layer similarity form becomes more susceptible to perturbations (in particular this is again evidenced by equation (2.31)). It is clear that the present results show a marked trend towards disturbance decay/stabilization as *ξ*_{0} increases (arguably mimicking the increase in critical Reynolds number beyond the minimum critical Reynolds number location); however, again, it is difficult to deduce critical Reynolds number results with infinite Reynolds number theories. Fig. 2 of Sarpkaya (1975) (taken from the work of Tatsumi 1952) indicates that the most dangerous mode (i.e. that exhibiting the lowest critical Reynolds number) is non-axisymmetric (although axisymmetric modes are comparatively only slightly less unstable) and occurs when the disturbance generators are located at a downstream location of *ξ*_{0}≈0.155. Somewhat intriguingly, this location does seem to coincide with the point of maximum flow response in the eigensolution triggered results (see figure 4*a*,*b*).

The analysis of this paper (implicitly) confirms the linear stability of fully developed flow. Some of the seemingly contradictory trends/results found experimentally in problems of this type could well be explained by the existence of different transition mechanisms. Certainly, the model proposed in this paper is quite different from the classical Tollmien–Schlichting route, which involves several transitional stages; the by-pass route, on the other hand, proposed by a number of authors (see Kachanov 1994) is a real alternative to the classical transition process. This alternative route involves disturbances of somewhat longer lengthscale than Tollmien–Schlichting waves, and thus the selection of the mechanism will likely be determined by the details of the disturbance field.

It is interesting, however, that the recent experimental results of Hof *et al*. (2003) suggest that the amplitude of perturbation necessary to cause transition scales is , which therefore, tantalizingly is the same as the fundamental scalings introduced at the start of this paper, i.e. equation (2.1), although any precise linkage has yet to be established. Other experimental results on the fully developed flow by Eliahou *et al*. (1998) describe the transition process (later stages of which are described by Han *et al*. 2000), in particular focusing on flow forcing which is equivalent to *β*=2 in the present paper. Interestingly, these authors observe the growth of higher harmonics, which, given a sufficiently large perturbation lead to transition. Certainly, this strongly suggests the importance of nonlinear effects, the inclusion of which is the next, natural step for the analysis presented here.

## Acknowledgements

Thanks to Professor Tom Mullin for arousing the author's interest in problems of this type, and to Professor Anatoly Ruban for suggesting the importance of antisymmetric modes.

## Footnotes

As this paper exceeds the maximum length normally permitted, the authors have agreed to contribute to production costs.

- Received March 11, 2004.
- Accepted October 6, 2004.

- © 2005 The Royal Society