Effects of gradient pressure on resistance law in roll waves

Sandro Longo

Department of Civil Engineering, University of Parma, Parco Area delle Scienze, 181/A 43100 Parma, Italy fax +39 0521 905924 e-mail: sandro.longo@unipr.it


Free surface instabilities of flows down inclined channels have been widely observed in Newtonian and non-Newtonian fluids. The observations of roll waves in a torrent by Maw in the late 19th century were followed by the description reported in Cornish (1934). Several Authors report observations of roll waves in mudflows in many areas. In addition Ishihara et al. (1954), Mayer (1959), Brock (1967), Julien & Hartley (1985, 1986) conducted several laboratory experiments with water streams. Roll waves are essentially controlled by Froude number, and can develop in laminar and in turbulent streams. The usual approach is linear stability analysis of the basic equations written in the long wave approximation: the basic flow field (the uniform flow) is perturbed assuming water depth and mean velocity small variations. The set of equations in perturbed variables is linearized and solved. If perturbations grow, the basic motion is linear asymptotic unstable, otherwise it is stable. Froude number correspondent to marginal stability (perturbation do not grow nor decay) depends on velocity profile and resistance law. In laminar flow, Yih (1954, 1963) and Benjamin (1957a, b), perturbed Navier-Stokes equations assuming free surface perturbation of sinus shape and recovered Orr-Sommerfeld equation finding a critical Froude number equal to 0.527. Chen, 1992, obtained similar result perturbing the shallow water 1-D equation including derivative of Coriolis coefficient of momentum along the stream motion.

In turbulent flow in rectangular channels, assuming a Chézy resistance law with a constant coefficient, Jeffreys (1925), Stoker (1957), Liggett (1975) found a critical Froude number equal to 2. Several researchers as Iwasa, 1954; Koloseus & Davidian, 1966; Berlamont & Vanderstappen, 1981 put in evidence the strong sensitivity of critical Froude number on velocity profile, Reynolds number, friction law. In particular according to Rouse, 1965; Rosso et al., 1990, the Darcy-Weisbach friction factor increases for increasing Froude number in supercritical streams. According to Brock, 1966, 1967 no firm conclusion on such a dependence could be drawn because experimental data were not enough accurate, especially the water depth measurements. Moreover an apparent increment of friction factor could better be explained as energy transfer from mean flow to waves. Montuori (1961) set up a diagram wherein the sufficient conditions for roll waves development can be obtained.

A finite amplitude wave theory in roll waves is due to Dressler (1949), who used several ideas by Thomas (1939). Dressler's theory, originally developed for fully turbulent flows, was extended to laminar flows for Newtonian fluids by Ishihara et al. (1954) and to power-law fluids by Ng & Mei (1994), which essentially focussed on pseudoplastic fluids (mud). Recently Prasad et al. (2000) applied Dressler's theory to flowing grains.

Most of the available results refer to the limit condition for roll waves existence, but no one can infer the determination of roll wave parameters (wavelength, wave height, and celerity) for a given system. Dressler (1949) separated the energy dissipation in the shock and the energy dissipation along the stream. For the former he verified that its mean contribution over a wavelength is function of the wavelength and has a maximum. He also inferred that if one were to compute the sum of all the energies over a wave length, one might expect the resulting curves to have a minimum for some value (or values) of the wave length, and such points might represent the stable solutions. He did not do work along this line because of the laboriousness of the calculations.

There are some experimental indications due to Ponce & Maisner (1993), who using Brock's data (1967) found that the observed periodic roll waves are those corresponding to the maximum growth rate (in linear stability analysis). Ng & Mei (1994), following Kapitza (1948), infer that the observed roll wave has the lowest amplitude corresponding to no energy loss across the shock. Longo (2003) found that the criterion proposed by Kapitza (1948) and Ng & Mei (1994), i.e. that the observed wavelength corresponds to zero dissipation in the shock, is not applicable for dilatant fluids in laminar condition.

It is evident that the existence of roll waves depends fundamentally upon resistance effects. It has been demonstrated by Dressler (1949) that finite amplitude periodic roll waves solutions cannot be obtained assuming a uniform resistance independent of the hydraulic radius, and that resistance must decrease as the water flows into deeper region and must decrease with velocity. The reduction of resistance in some part of the roll wave is essential in order to reduce energy dissipation by friction respect to the equivalent uniform stream. Lamberti & Longo (2000) carried out the calculation of energy dissipation suggested by Dressler. They found that neglecting dissipation due to friction in the shock (assumed of finite length) it is possible to find a zero energy budget corresponding to a situation where the dissipation by friction in the permanent wave profile plus the dissipation in the jump equals the energy gained by gravity. The balance is very sensitive to small variations in the parameters.

In the present paper we shall apply Dressler's model of periodic finite amplitude roll waves with a modified resistance law. The resistance law is derived considering the pressure gradient effect on boundary layer. In the roll waves the pressure gradient counteracts the boundary layer and reduces wall friction.


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