July 5, 2009

Supercritical contraction on inclined plane: A new topic for research

Filed under: Uncategorized — Rajnikant Khatsuria @ 2:46 pm

           Contractions along spillway chutes are provided whenever reduction in the channel width becomes necessary. This may be due to geological, topographical, economical or hydraulic consideration. Since the flow along a spillway chute is supercritical, any deviation in the stream lines creates oblique standing waves that can propagate downstream unless a properly designed contraction is provided to minimize this effect.

             A supercritical contraction has three parameters viz. contraction ratio (b2/b1), upstream Froude number F1 (determined by discharge, channel width and flow depth prior to contraction) and angle of contraction θ or length of contraction Lc. The phenomenon of oblique standing waves has been studied extensively, both theoretically and experimentally, for the past half century. However, a direct solution to the governing equations is not possible and either a trial and error procedure or graphical solution is resorted to. The earlier approaches of Ippen and Dawson [1951] involved solution through four-quadrant graphs and that of Sturm [1985] of twin graphs, but as the analytical treatments improved, today  it is possible to design an optimal contraction by a single algebraic equation or a chart, refer Hsu et al.[2004], Subramanya et al.[1974]. In an optimal contraction, the positive and negative disturbances will cancel each other so that the flow is calm in the channel downstream of contraction.

       It is worth mentioning here that all the analytical and experimental studies related to supercritical contraction have involved only the three parameters listed above. The transition is assumed to be on a horizontal plane. The only change in the velocity is due to change in the width of the waterway. However, when a contraction is provided on a spillway chute having an inclination to horizontal, an additional parameter of chute slope α is introduced. Now the gravitational effect is also accompanied simultaneously along with the contraction in the width to affect the velocity. None of the analytical treatments so far has included the effect of channel slope on the performance of contraction. Although several contractions have been provided on various spillway chutes, their designs have not included the parameter of channel inclination.

        Recently, the first attempt towards inclusion of channel inclination has been made by a group of Algerian researchers; refer Berreksi et al. [2008]. They have applied St. Venant’s equations governing two-dimensional unsteady free surface flows, incorporating therein the parameter α, which in turn led to a system of non-linear hyperbolic equations. These are solved numerically by applying second order two step McCormack explicit finite difference scheme. Although there is experimental verification for the case of contraction on a horizontal plane, i.e. α=0, there is no direct verification for the case of inclined plane. The authors have applied the scheme to a rectangular channel with no contraction i.e (b2/b1) equal to unity, on an inclined plane, with good correlation between analytical and experimental results. It was assumed that the same scheme would apply equally well to a contraction on inclined plane with different values of (b2/b1) and α. Figure 1 shows a schematic of results for a contraction on an inclined plane.



                   Figure 1:  Supercritical contraction on inclined plane- A shematic


            It would be seen that, as anticipated, flow depths decrease and velocities increase with increasing inclination. Another noteworthy feature of this analysis is that there is only one occurrence of wave, that too, just downstream of the end of contraction, along the centre line of contraction. Along the side walls, the water surface falls considerably and that the effect of increasing inclination is not significant.

           The above results have opened up a new chapter of studies on supercritical contractions on inclined planes. Further studies are required specifically to address the following issues:

  1. Correlation of analytical and experimental studies with different configurations of contraction on inclined planes.
  2. It is apparent that there would be an optimal configuration of such a contraction that would give acceptable conditions in the downstream channel.
  3. With steeply inclined channels, air entrainment and resulting bulking of flow also come into play.
  4. The results of such studies should be generalized on the lines of contractions on horizontal plane, in the form of algebraic equations or design charts that would obviate the need for complex computational procedures.



  1.  Berreksi,A, Kettab,A, Remini, B, and Benmamar, S (2008)-“Computation of two-dimensional unsteady supercritical flows in open channel contraction of spillway chutes”- Dam Engineering, Vol.XIX  Issue 3  December 2008, pp 149-168
  2. Ippen, A.T. and Dawson, J.H (1951)-“Design of channel contractions”- Symposium on high velocity flow in open channels, Trans ASCE, 116, pp 326-346
  3. Hsu, Ming-His, Su, Tung-Hung and Chang, Tsang-Jung (2004)-“Optimal channel contraction for supercritical flows”- Jnl of Hydraulic Research, Vol 42 No.6 (2004) pp 639-644
  4. Sturm, Terry W, (1985)-“ Simplified design of Contractions in Supercritical Flow”- Jnl of Hyd Engg, Vol111 No. 5 May 1985.pp 871-875
  5. Subramanya,S and Dakshinamurty,S  (1974)-“Design of Supercritical Flow Contractions by Least Square Error Minimization Technique”- 5th Australasian Conference on Hydraulics and Fluid Mechanics, Uni.of Canturbury, New Zealand, 1974

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