April 30, 2013

 Predicting scour below ski jump spillways- Damle Equation  

Filed under: Uncategorized — Rajnikant Khatsuria @ 2:07 pm




Formerly Addl.Director

Central Water and Power Research Station

Pune, India


A detailed and reliable estimation of scour downstream of a ski jump spillway requires study on a comprehensive model. Yet, there are reservations regarding the acceptability of results in view of difficulty of simulating river bed, which might be composed of heterogeneous rock of complex mass. The scour levels thus obtained on a fully erodible bed are assumed to be of ultimate nature. In the preliminary assessment of a design, estimation of scour is often made with the use of a formula, although it gives an indication of single value of deepest possible scour unlike the areal pattern in the downstream vicinity.

 A question always arises as to which of the formulae should be used? There are some 50 formulae evolved by researchers in various countries.

 Scour prediction formulae are generally classified according to method of approach as:

  •  Theoretical analysis coupled with laboratory experiments like Yuditsky, Zvorykin, Mirtskhulava etc.
  • Theoretical analysis coupled with prototype observations like Taraimovich, Spurr etc.
  • Empirical relationships based on prototype observations, like Damle, Martins, Incyth etc.

 Another classification based on use of parameters may be as follows:

  • Formulae involving simplest parameters like discharge intensity, head and tail water depth, as for example Damle, Martins, Taraimovish etc.
  • Formulae involving additional parameter like bed material size, as for example Mason, Chee and Kung etc.
  • More complex formulae involving several parameters representing bed characteristics, rock properties and structure geometry etc. Most of these formulae have been developed by Russian researchers like Akhmedow, Mirtskhulava, Zvorykin etc.
  • Recently, Bollaert has proposed a comprehensive scour model based on experimental and numerical investigations of dynamic water pressure in rock joints.

However, according to author’s experience, use of even a comprehensive formula does not offer better result than obtained with a simpler formula. In this context, results given by Damle equation have been found to be quite reliable. A review of evolving of Damle equation is quite interesting and is the subject matter of this discussion.

Damle equation was evolved by a group of researchers at the Central Water and Power Research Station, Pune, India, ( a Government of India organization devoted to research in hydraulic and allied engineering), namely, P.M.Damle, C.P.Venkatraman and S.C.Desai in 1966. Data comprising results of scour observations from 6 prototype and 15 model studies was available then. The only hydraulic parameters responsible for causing scour were considered to be the stream power q x h (product of discharge intensity and head) and the tail water depth ds No account of bed material size was taken as it was assumed that the process of hydro-fracturing would ultimately lead to breakage of jointed rock mass into size that could be transported out of scour hole, to result into scour depth, considered to be ultimate. Study of prototype scour revealed that development of scour was in stages as the discharges of various magnitudes passed for different durations. Accordingly, the following set of equations was derived. Figure 1 shows the above mentioned relationships.




 Figure 1: Damle Equation with original constants

 It may be mentioned that several other formulae derived from the results of prototype observations or model studies in the form of ds=K qa x hb, where values of a and b were based on regression analysis. Only in Damle equation, the exponents of the product (qh) were predetermined to conform to the concept of stream power.

 Some 20 years later, from a study of scour development at Kariba dam, Mason and Arumugm (1985) found that the depth of scour was indeed a function of stream power viz. ds=F (Q0.5 H0.5) if it was assumed that the depth was proportional to the cube root of volume of scour. This substantiated the assumptions underlying Damle equation.

 It must be noted that equations developed from laboratory experiments under controlled conditions give scour depths that are ultimate and stabilized. This is not so for the field where scouring may still be in the process. The equations developed from field studies must be updated and refined as and when additional data becomes available. In the years that followed, data on scour in respect of several projects became available. Khatsuria (1992 and 2005) analyzed these data and plotted superimposed on the original data, which revealed that the coefficient for the ultimate scour K was 0.90 instead of 0.65. The revised plot is shown in Fig 2.



   Figure 2:  Damle Equation- Modifications based on additional Data

 Interestingly, a further refinement taking into effect latest data from Azamathulla (2008) also validates this equation (Figure 2). Only those data sets that represented deepest scour for a given (qh) value were plotted.

 It may thus be concluded that Damle equation is a reliable tool for prediction of scour downstream of ski jump spillways.


  1. Damle,P.M., Venkataraman,C.P. and Desai,S.C.- Evaluation of scour below ski-jump buckets of spillways, CWPRS Golden Jubilee Symposia, Vol 1, 1966
  2. Mason,P.J. and Arumugm,K- A review of 20 years of scour development at Kariba Dam, 2nd International Conference on The Hydraulics of Floods and Flood Control,Cambridge, England, September 1985
  3. Khatsuria, R.M- State of Art on Computation, Prediction and Analysis of Scour in Rocky Beds Downstream of Ski Jump Spillways, CWPRS Platinum Jubilee, 1992
  4. Khatsuria, R.M.- Hydraulics of Spillways and Energy Dissipators, Publishers- Marcel Dekkers, New York, 2005
  5. Azamathullah,H., Ghani,A., Zakaria,N.A., Lai,S.H.,Chang,C.K. and Leow,C.S.-Genetic programming to predict ski jump bucket spillway scour, Science Direct, 20(4), 2008 pp-477-484



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