In December 1997, Ali S. Fayad publicly announced the discovery of the explicit alternate flow depth in rectangular open channels while teaching at California State University, Sacramento.
HYDRAULIC APPLICATIONS WITH MNEMOSYNE
(c) Copyright 2000 Ali S. Fayad
In the summer of 1994 , the author was asked to teach the laboratory course in hydraulics at California State University Sacramento (CSUS). The author has been teaching as a lecturer at CSUS since 1984 but in the field of structures. Thence, the author began to prepare by reviewing the literature, solving, and deriving many of the formulas involved in the course. In the process of preparing notes, the author began a derivation of the alternate flow depth in rectangular open channels. The assumption at the time was that if the conjugate (sequent) depth of the hydraulic jump existed in an explicit form and was derivable in closed form solution, then the alternate depth must also exist in a similar manner. Half an hour into the derivation, which was neither particularly laborious nor exactly straightforward, a closed form solution emerged for the alternate flow depth. This occurred in August of 1994, and the closed form solution is presented later in this work. At the time, not realizing the significance of the discovery, the author proceeded to work on other formulas related to the hydraulics course whose contents and objectives are served by this work. Consequently, and due to numerous reasons beyond the scope of this narrative, little attention was paid to the fact that a closed form solution for the alternate flow depth has been derived. Over the next two years the author taught several sections of said laboratory course without anything noteworthy in this regards save for an encounter with an emeritus professor.
During the Fall of 1994 an emeritus professor, who had taught the course on numerous occasions and for several years, visited the author in the hydraulics lab and asked to check on the author's status and progress in teaching the course. The discussion that ensued encompassed the software developed by the author for the hydraulics lab. The alternate depth (for flow beneath a sluice gate) for rectangular open channels was part of the software. The emeritus professor wanted to know how the author was calculating the alternate depth for an idealized sluice gate, and the author simply said: "it was in explicit form". The emeritus professor looked a bit surprised (to say the least) and the author was not sure the exact reason for his response and facial expression. Three years later, it became evidently clear that the author inadvertently derived a closed form solution for the alternate depth which is as elegant (and closed form) as that for the sequent (i.e. conjugate) depth for the hydraulic jump. The significance of it was left in the hands of time. The author had to make sure, by literature search and by talking to various faculty, that this indeed was unavailable in current publication. Thence, the author was motivated to produce this set of practice problems along with the accompanying derivations.
INTRODUCTION TO HYDRAULIC APPLICATIONS
Three types of hydraulic applications that utilize the laws of fluid mechanics are identified in this work for purposes of organization. They are permeable flow, pressurized flow, and open channel flow. These are the typical realm of the civil engineer.
Permeable flow consists of the propagation of a liquid, such as water, within a solid or quasi-solid medium such as soil. This is feasible due to the voids within the matrix of the permeated substance. Thence, the permeable flow depends not only on the viscosity of the liquid but it also depends on the variation in the sizes of the constituents of the solids. An average flux intensity for a cross-sectional area of permeation and a corresponding flow direction are assumed because the process occurs on a very minute scale. A property known as the permeability of the medium is established that is typically direction and location dependent. The surface tension between the fluid and the medium increases as the sizes of the quasi-solid constituents become very minute. In the case of clay, which is considered impermeable in comparison to sandy or gravely soils, colloidal forces impede the permeation process.
The significance of permeable flow manifests in: dewatering projects during foundations construction; development of wells for irrigation and drinking; prevention of contamination in water retaining structures; prevention of moisture damage in basements or subterranean structures; painting applications. The flow of the fluid within the confines of the matrix of the solid medium depends on the following: (1)Fluid viscosity, (2) Average size of the voids along the flow, (3)Flow velocity ,(4)Hydraulic head differential in the direction of flow. These four criteria must be accounted for in all permeable flow problems. This hints to the fact that the solution of underground permeable flow is fraught with assumptions and simplifications. The availability of numerous charts and tables such as well-logs facilitates the real life investigative process.
Pressurized flow is typical of closed pipes, as in water supply application, where the flow completely fills the area of the pipe. The pressurized flow depends on the viscosity of the liquid, the diameter of the pipe, and quantity of flow. The Reynolds number is a dimensionless parameter which is a ratio of the inertia forces to those of viscosity. It predicts by its value the type of flow. Laminar flow indicates that the flow is proceeding in layers while turbulent flow indicates the presence of internal vortices with a net forward flow. A property known as the absolute roughness of the pipe establishes the friction loss that occurs in the pipe and the drop in the apparent energy of the water. The formation of a boundary (skin) layer masks the effects of the roughness as the turbulence increases. The flow equation in the turbulent region has been empirically developed and the Darcy-Weisbach equation is the one used in scientific circles.
Open Channel Flow
Free surface flow occurs in a canal that is open to the atmosphere. This constrains the pressure distribution in the water which increases with depth from zero relative pressure at the surface to a maximum value at the bottom of the channel. The open channel flow predominantly depends on the Froude number which is a ratio of the inertia forces to those of gravity. Absolute roughness of the lining of the channel is a factor in the flow regime leading to the loss of energy. The loss of energy is also dependent on the wet surface of the channel.
The water air interface in open channels is at zero relative (gage) pressure. There are two possible flow depths, subcritical and supercritical, for each energy level. These have been known in the hydraulics literature as the alternate flow depths. In theory, a sluice gate attains an alternate flow depth, to its upstream depth, downstream at the vena contracta. In practice the depth and location of the vena contracta are particularly difficult to ascertain. Factors such as the waviness of the water and the magnitude of the depth make the measurement process less certain than other locations in the flow. Approximate derivations of the alternate flow depth have been available for a long time. However, in December of 1997, Ali S. Fayad (Lecturer in Structures and Hydraulics at CSUS since 1984) announced the discovery of the explicit-closed-form derivation of the alternate flow depth.