Starting with the general equation of force-momentum balance (see Figure 1)
Figure 1: Parameters of hydraulic jump on a rectangular prismatic channel
Where, M =momentum and F =force as defined in figure1. Analytical solution of Eq (1) requires evaluation of the shear force function Fs, which may be either shear or form drag. This is usually not possible unless empirical values of relevant coefficients and velocity at that location are incorporated into the equation. In the most simplified case of hydraulic jump in a horizontal rectangular channel with smooth bed and sides, the frictional resistance offered by the bed and sides is taken equal to zero, resulting into the classical equation (2)
According to Eq (2), a conjugate depth Y2 corresponding to the pre-jump depth Y1 and Froude number F1 must be provided to form a satisfactory jump. Also, a basin length corresponding to the length of the hydraulic jump, of approximately 4-6 Y2 would be required. Thus, designing stilling basins on the basis of Eq (2) resulted in uneconomical structures entailing deep excavations and longer paved lengths. Therefore, emphasis was laid on evolving designs that required lesser tail water depths as compared to Y2 and shorter lengths of paved aprons. With a re-look at Eq (1), the term Fs then gained significance.
ACCOUNTING FOR Fs
Analytical solution of Eq (1) with inclusion of the term Fs was, however, not possible because the relevant coefficients Cf and CD in expressions
Fs= Cf ρ ( V²1/2) Ar and Fs = CD ρ (V²1 /2) Ap (3)
were not known for the various types of blocks and roughness. Here, Cf and CD are the coefficients of shear and form drag respectively, ρ is density of water, V1 is the approach velocity and Ar and Ap are the areas of roughness and of projection of blocks respectively. The research therefore relied heavily on experiments.
As the shear drag resulting from roughness of the bed and sides was not expected to make significant reduction in the conjugate depth Y2 and jump length, a force generated as a form drag, caused by placing obstacles like cubes and blocks in the flow, was relied upon. Thus, the earlier designs included blocks of various shapes and sizes, just downstream of the initial depth Y1. These designs included several rows of such blocks, but as further development took place in the form of stilling basins designated as USBR basins, low Froude number stilling basins and SAF basin, only one row of such blocks was introduced. Also, special designs were evolved for site-specific applications such as Lower Bhavani and Pit 6 dam stilling basin to minimize tail water requirement and apron length. Addition of chute blocks at the beginning of the jump (to corrugate the incoming jet and aid some dissipation) and end sill at the end of the jump (to lift the flow leaving the basin and protect the unpaved portion from direct action of current) also increased efficiency of performance.
These designs resulted in smaller size structures as compared to the stilling basins without any appurtenances. The USBR type III basin, with chute blocks, one row of triangular shape baffle piers and end sill, required an apron length of about 2.75 Y2 and a tail water depth of about 0.83 Y2, against 6Y2 and Y2 respectively for a basin without any appurtenances. The SAF basin would permit even a length shorter than Y2 and a tail water depth of about 0.85Y2. The Bhavani stilling basin employing T-shape blocks and Pit 6 basin with special shape baffle piers also resulted in economical structures.
The above designs, however, suffered from one serious drawback that these could not be used for entrance velocities in excess of about 15 m/sec. The appurtenances, especially the baffle piers are likely to be subjected to excessive drag and even cavitation. Many instances of damage to the structures employing these designs have been reported in literature, although some structures with entrance velocity up to 20 m/sec have performed without any problem.
Limited application for entrance velocities higher than 15-20 m/sec together with risk of damage prompted designers to explore the alternative of shear drag to minimize the jump length and tail water requirements. Several ways to produce hydraulic jump on artificially roughened bed have been explored so far with varying degrees of success. Studies were performed with specific objectives starting from late seventies.
The roughened bed alternatives were: cubes of different shapes, strips, gravel laden beds and corrugations of various shapes as shown in Figure 2.
Figure 2: Roughened bed alternatives
For the alternatives of cubes and strips, the length of coverage, longitudinal and lateral spacing of cubes, height of cubes and the intensity of coverage were the variables. For the gravel bed alternative, the size distribution of gravels was the important parameter. For the corrugated beds, different shapes such as sine wave, trapezoidal and triangular forms with various heights and wave lengths of corrugations have been studied.
Of the various roughness elements, the cubes would offer a reduction in the range 10-20% in Y2/Y1 and 20-40% in jump length; strips would similarly offer a reduction of 10-15 % and 5-30% respectively. Gravel beds contribute towards reductions of 15-30 and 15-35% respectively. Corrugated beds resulted in 40% reduction in tail water depth requirement and 60% reduction in jump length. The above values correspond to the range of Froude numbers 5-9.
A comparison reveals that while both the forms of Fs contribute nearly the same amount of reduction in the tail water depth, the shear drag offers less reduction in the jump length, accept corrugated beds, which offer significant reduction. This is largely due to the increased shear stress resulting from the interaction of the supercritical flow above the corrugations with eddies trapped in the cavities of corrugations.
Although, significant research has been conducted to evaluate the contributions of various roughness elements towards reduction in Y2/Y1 and jump length, there appears to be very little headway towards its practical implementation. There is no information about any stilling basin that has been constructed specifically with any of the roughness elements mentioned above. The main reason for this is that it is impractical to reproduce most of the roughness elements in nature, except small size cubes or blocks. Strips are difficult to construct and maintain. Gravel beds require binding with cementing material to hold the material intact against high velocity flow. Although, rubble concrete may be a better alternative, the upper surface resisting the flow may become smooth during course of time and loose effectiveness. Corrugations in the shape of sine wave, trapezoids or triangles are similarly difficult to construct in concrete and maintain. The apron floors of large stilling basins are cast in panels of suitable size with contraction joints on all the four sides of panels. Besides, the corrugations of each panel must match with those on all the adjoining panels. The concrete of the panel is also provided with appropriate reinforcement. This type of construction is difficult with corrugations at the top. An alternative of fixing steel plates with the desired corrugations on the base concrete of panels in place of casting the corrugations in concrete may be thought of, but then there will be a problem of seepage of flow beneath the plates and resulting dynamic uplift peeling of the plate itself. The only practical way of construction would be with small size cubes or blocks that can be cast along with the base concrete of the panel. Another alternative may be to make rectangular corrugations as shown in Figure 3, instead of those involving curved boundaries like sine waves. This is basically large rectangular strips spaced at close interval, which would be easier to construct as on the base concrete of panels.
Figure 3: Details of a typical floor panel with rectangular corrugations.
Evolving practicable configurations
Although considerable research has been conducted on various types of roughness elements to produce hydraulic jump requiring lesser tail water depth and reduced length of jump, its direct application in practice has not made a head way as yet. Of all the alternatives, corrugated beds in the form of sine wave offer attractive results. However, there are difficulties in constructing stilling basin apron floors with corrugations as discussed above. A shape of corrugations made up of rectangular strips as shown in Figure 3 may be hydraulically efficient and easier to construct. Further research is therefore required to:
Optimize configurations of alternative design as shown schematically in Figure 3. This should include height, width and spacing of strips and the length of the apron required to be covered. If a part covered length results only in a marginal loss of efficiency over that of the entire length of the apron covered, this may result in an economical design.
- The hydraulic efficiency of such a design should be consistent with the structural safety in that this should be free from cavitation and excessive drag under the range of flow velocities anticipated.
Evolving appropriate technique of constructing apron floor in regard to determination of size of panels, stages of placement of concrete, construction of strips, contraction joints between the panels and water tight sealing between the panels.
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