Establecimiento del sistema de gestión de almacenes

The four non-dimensional boundary-layer parameters most commonly employed

to describe favourable pressure gradients are the acceleration parameter (K), the

boundary-layer shape factor (H), the momentum thickness Reynolds number (Reθ),

and the coefficient of friction (Cf). In an equilibrium sink-flow turbulent boundary

layer, these four parameters are constants with streamwise distance. The accel- eration parameter is also constant in the idealised laterally converging duct flow

geometry described in§4.1.1 and recreated experimentally by Murphy et al. (1983)

and Chambers et al. (1983); though it is unknown whether this type of flow has an equilibrium solution similar to the sink flow. It may be noted that the acceleration parameter is also a type of Reynolds number and is independent of the boundary layer behaviour itself. That this study involves a pressure gradient perturbation on a developed boundary layer, combined with the short contraction length relative to boundary-layer thickness, make it highly unlikely that any equilibrium behaviour will result.

The behaviour of the acceleration parameter was shown above in Figure 4.9 and the streamwise evolution of the remaining parameters is shown in Figure 4.12.

For reference, the wall shear stress (τw) has been included. The values of these

and other pertinent quantities are tabulated for the first measurement station in Table 4.1, along with the results from the zero-pressure-gradient experiments in the previous chapter. The techniques applied to calculate the integral parameters and the wall shear stress in these measurements are the same as those in the previous chapter. Here again, the measurements at the last station were conducted outside the calibration range and so will likely contain errors.

A striking feature of the results is that their streamwise variation remains so small throughout the test region despite the relatively large acceleration. The total contraction length is greater than 100δ∗(

0), where δ∗(

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 s(m) Reθ/Reθ(0) H/H(0) Cf/Cf(0) τw/τw(0)

Figure 4.12: The streamwise evolution of the Case 1 mean-flow parameters. Mea- surements are normalised with the initial values.

Quantity ∂P/∂s=0 ∂P/∂s<0 Qe (m⋅s−1) 0.122 0.124 Reθ 1080 923 Reτ 530 484 H 1.4 1.37 δθ (m) 0.0089 0.0074 δ∗ _{(m) 0.0125 0.0102} τw (kg⋅m−1s−2) 3.2×10−2 3.37×10−2 Cf 0.0043 0.0049

Table 4.1: The initial flow parameters of the turbulent boundary layer in the favourable pressure gradient compared with the zero-pressure-gradient values.

(2000) appears to be most comparable to the results here in terms of the streamwise behaviour and magnitude of the parameters. Their acceleration parameter took a

greater streamwise distance to reach its peak magnitude of K≈2.9×10−6 _however,

and their inlet Reynolds number is slightly lower than that used here (see Table A.1 in Appendix A).

Further noteworthy behaviour is the continued rise in the Reynolds number accompanied by a reduction in the coefficient of skin friction. The common response of a turbulent boundary layer to an accelerating perturbation is that, though the

Reθ continues to rise, Cf rises as well (see §2.3.2 also). Later downstream, as the

stabilising effects of the acceleration start to dampen the turbulence, Reθ starts

to decline as does Cf. Here however, the Reynolds number continues to increase

throughout the geometry even though the acceleration is large and the friction is falling. Some experiments such as the Case 1 test by Fernholz and Warnack (1998) show similar behaviour, but this only lasts while the acceleration is still relatively low, and it is usually followed later downstream by a more typical behaviour pattern.

Boundary-layer relaminarisation is also typified by large reductions in Cf.

Relaminarisation is a gradual process whose onset is difficult to define (Bourassa and Thomas, 2009). Further, relaminarisation is distinct from a ‘laminarescent’ flow as the process during which the turbulent boundary layer is undergoing re-transition back to the two-dimensional state, rather than one whose ability to produce tur- bulence is being strongly attenuated but increasing nonetheless. Aside from the reduction in friction, the remaining parameters measured here do not strongly sup- port the re-transition hypothesis as again, relaminarisation induced by acceleration is almost universally accompanied by a declining Reynolds number and by much

smaller values of the shape factor H. During relaminarisation, H declines down-

stream and reaches a minimum around the same streamwise location thatCf reaches

its peak. Downstream of this location,Cf starts to decline andH increases, as the

imum inH is seen here and the smallest value isHmin=1.3. Warnack and Fernholz (1998), Piomelli et al. (2000), Blackwelder and Kovasznay (1972), and Bourassa and Thomas (2009) all report values ofH<1.3 before the onset of a decline inCf. The thinning of the boundary layer here is shown in Appendix F, where the streamwise reduction in the boundary layer thickness measures is shown.

The favourable pressure gradient is generated by a lateral convergence of the duct walls, which in channel coordinates appears as(dU/dx) = (dW/dz). This extra strain perturbation may explain some of the behaviour seen here. Though few in number, the results of investigations into the impact of lateral strain per- turbations on a turbulent boundary layer show similar patterns to those measured here. McEligot and Eckelmann (2006) generate their favourable pressure gradient using a lateral convergence. Their experiment, however, is a better approximation of the idealised case of a converging duct, because a symmetric contraction should effectively be produced. The development length of their upstream boundary layer before contraction is also very short. A consideration of the lateral strain compo- nent here only follows from the idea that the boundary layer is perturbed by the convergence as clearly in the idealised case, the flow is entirely radial towards the sink and there is no spanwise straining.

Streamline convergence was investigated experimentally by Pompeo et al. (1993) and Panchapakesan et al. (1997). They isolated their experiments from streamwise strains by expanding the wall-normal flow area at the same time. Com- pared to a two-dimensional flow, the convergence was found to increase the rate

by which Cf declines. H either increased slightly or ‘flattened out’. These two

observations correspond somewhat with the behaviour in Figure 4.12 where a Cf

reduction was seen, in contradiction to the rising Reynolds number and decliningH.

The convergence was also found by Pompeo et al. and Panchapakesan et al. to trig- ger an increase in the growth rate of all three boundary-layer thicknesses. Though Figure F.1 showed that the boundary layer here is contracting, the reduction might

have been more severe otherwise. Neither Pompeo et al. nor Panchapakesan et al. see changes to the logarithmic behaviour of the inner layer, but convergence did appear to augment the wake component of the boundary.

The effects from the convergence appear to work in an opposite manner to a favourable pressure gradient. This is noted by Panchapakesan et al. (1997) who observed that, given their mean velocity results, it would be difficult to tell whether the flow was responding to a lateral convergence or to an adverse pressure gradient. This observation is also made by Coleman et al. (2009) in their numerical investigation of laterally-strained turbulent boundary layers.