Flow PatternsThe deflection perturbs the supercritical approach flow and a shock wave occurs. The simulations indicated two main effects: •A pronounced shock wave propagates along the external bend wall, starting at the deflection start and including a first wave maximum within the bend. If the bend is sufficiently long, a second wave maximum occurs at the inner wall. The shock wave affects the downstream pipe flow and a cross-wave pattern is observed.•According to Gisonni and Hager (2002), the first shock wave maximum (at the outer wall) occurs at about θ=45°. If the bend manhole ends there, the wave often impinges at the manhole end (transition from U- to O-shaped), leading to spray and turbulence phenomena. At worst, the flow chokes, partially generating a hydraulic jump.We defined three conditions to interpret the flow behavior across the manhole (Fig. 4): 1.Ordinary condition (subscript ord). A free-surface supercritical flow regime across the bend manhole emerges. The maximum shock wave at the outer wall characterizes the flow pattern within the manhole [Fig. 4(a)].2.Limit condition (subscript lim). Hydraulic anomalies occur within the manhole and, possibly, in the downstream pipe. Pronounced spray is observed at the manhole end. The outlet cross section is close to submergence and behaves similar to a gate. The supercritical flow regime is maintained in the manhole, whereas the filling ratio yd=hd/D along the downstream pipe increases [Fig. 4(b)].3.Failure condition (subscript fail). The supercritical flow regime breaks down with choking in the manhole. A hydraulic jump moves up into the approach flow pipe, and the downstream pipe is pressurized, with yd≈1 [Fig. 4(c)].Hydraulic CapacityAs suggested by Del Giudice et al. (2000) and Gisonni and Hager (2002), the nondimensional parameter to express the discharge across a bend is (6) Combining Eqs. (5) and (6) obtains Q*=q/yo1.5.The hydraulic capacity (subscript C, limit condition) of a bend manhole with supercritical flow is reached for a specific discharge QC when failure occurs. Thus, for the limit condition equal to the manhole capacity, QC*=QC/(gyoC3D5)0.5 is defined.Fig. 5 shows tested Q* versus yo for the 45° bend, including BMs 1, 8, and 9 as examples (Table 3). For a certain BM and a numerically fixed yo, the increase of Q* led to the transition from the ordinary condition (circles) via the limit condition (triangles, similar to QC) to the failure condition (squares). Globally, Q* increased with yo, probably due to the reduced mechanical energy of the supercritical approach flow. Comparing the BMs [Figs. 5(a–c)] suggests that the increase of Ra/D and the addition of a straight extension L increases the capacity of the bend manhole. BM 1 could convey flows only with yoC≤0.4, whereas yoC≤0.6 was possible for BMs 8 and 11. Furthermore, for a certain yo, the capacity (triangles) of BMs 8 and 11 was larger than that of BM 1. In particular, the effect of Ra on the bend manhole capacity is evident in Figs. 5(a and b), with Ra/D increasing from 1 [Fig. 5(a)] to 3 [Fig. 5(b)]. An increase of Ra resulted in increased QC*.Figs. 5(b and c) show the effect of the straight extension on the capacity. As expected, a longer L/D increased the capacity. This was due to the position of the shock wave maximum located at about δ=45° (Gisonni and Hager 2002). Thus, the wave was distant from the manhole end if an extension was present. Equally, the straight extension did not increase yoC, which was equal to 0.6 for θ=45°.The values Q* versus yo of the tested 90° bends are shown in Fig. 6 with three BMs as examples. For a fixed value of yo, QC* (triangles) increased with increasing Ra/D and with the addition of a 2D-long straight extension. In all the BMs in Fig. 6, yoC was a constant 0.6. For a given bend curvature radius and extension length, the 90° bend had a larger capacity than the 45° bend. Again, this circumstance probably was related to the typical position of the shock wave maximum, located at about δ=45°, and, therefore, far from the manhole end for 90° bends.Using the present data, an empirical equation to estimate the hydraulic capacity of the bend manholes as a function of the geometrical and hydraulic parameters was derived. Thus, QC* was estimated for 0.30≤yo≤0.70 as (7) with k and α depending on Ra/D, L/D, and θ as follows: (8) (9) where θ is expressed in radians. Eqs. (7)–(9) are valid in the inspected range of the influential parameters, that is, for 1≤Ra/D≤3, 0≤L/D≤2, and 45°≤θ≤90°. According to these equations, the effects of the increase of the bend curvature radius Ra and of the straight extension L contribute equally to the increase of bend capacity. If θ increases, then QC* also increases, but the positive effect on the capacity tends to decrease as yoC increases. The accuracy of Eq. (7) is shown in Fig. 7, which compares the observed QC* in the numerical simulations (subscript s) with those computed (subscript comp) using Eqs. (7)–(9). Eq. (7) can predict QC* quite satisfactorily. Most points were within a confidence interval of ±5%, and the overall correlation coefficient R2=0.92. Fig. 7 also includes the data from the physical models recorded by Del Giudice et al. (2000) and Gisonni and Hager (2002), and indicates a good fit (R2=0.82).Gisonni and Hager (2002) proposed two distinct equations to derive QC* for 45° and 90° bend manholes with Ra=3D and with [Eq. (10)] or without [Eq. (11)] an L=2D-long straight extension (10) QC*=[3sinθ(1−yoC)]+yoC(11) Fig. 8 compares these two equations with our Eq. (7), and includes our simulated points for BM 1 and BM 7.

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