Common Sense Hydraulics

It is always fun to go back to the basics. Revisiting basics helps us refine our understanding, and interestingly there is always something new to learn each time we do so. I remember seeing a line somewhere that said: Learning is like rowing upstream not to advance is to drop back. I have heard a somewhat similar line from professors: We are constantly challenged by our graduate students. But don’t lose sleep thinking about this – some of us are required to do it because of our profession, others may be driven by interests in exploring the horizons of mind and intellect. For now, let us try to explore hydraulics, an interesting field of water science and civil engineering. 
Hydraulics is the science of water in motion – in this blog we will focus on open-air hydraulics (a metaphorical image, credit: anon) in contrast to pressure flows as in pipes and conduits. Also, at a later time I intend to cover the hydraulics of sediment transport, which is vast in its own right. I have started the caption by suggesting that I will be talking about hydraulics as part of our common knowledge. But is there anything called common sense hydraulics? Well, it is all relative. We will try to see how human perception of water motion has changed from the common to the educated one.  
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Perhaps some knowledge is commonest of the commons. For example, apart from water that we use everyday either for drinking or for doing everything else, water is a wonder substance – a mystery – an uncertainty – a life blood – its fluidity intrigue us. Like all fluids, its behavior is determined by three fundamental properties known as viscosity, density and compressibility. The magnitudes of these properties are dependent on temperature, pressure and dissolved substances; and for most purposes water can be assumed to behave as incompressible. We have been seeing water motion from our childhood since we remember – flowing down the slope always finding the steepest one of least resistance. When obstructed the flowing water raises its head – or when we throw something solid in water, ripples radiate from the source of impact – or when dipped into water we feel lighter. These are some of our intuitive qualitative knowledge – we know their existence but most of us do not know how to explain them.
Water flow has been utilized in ancient times since urbanized civilizations started to take shape across different cultures. There are many examples of wonderful water works in history. The Grand Canal of China (5th century BCE) was initiated to facilitate water transfer, navigation and irrigation; the Indus Valley Civilization at Mohenjo-Daro and Harappa (4th century BCE) developed extensive water supply and drainage networks; and the Roman Aqueduct (3rd century BCE) was built to transfer water for human consumption and irrigation. These ancient marvels of water engineering must have required advanced knowledge of hydraulics. But so far we know there was no documented knowledge to establish how sophisticated they had been. The earliest known piece of workable relation on hydraulics, hydrostatics to be specific, came from Archimedes (287 – 212 BCE). The accidental discovery of the principle of buoyancy by Archimedes explained why weight was lighter in water than in air.     
Humanity had to wait for many centuries before major breakthroughs in hydraulics started to emerge. It was the European Renaissance (14th to 17th century CE) that paved the way for the rejuvenated thinking in everything including hydraulics.     
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Well, where to start in describing the vast of amount of knowledge in hydraulics generated since the 17th century. Perhaps we should begin from the principle of dynamic equilibrium – a signature characteristic of natural processes I have briefly introduced in the NATURE section.
Scientists try to understand fluid motion by considering a closed box known as a control volume. For simplicity, let us assume that there are no sources or sinks within the box. In one interpretation of the equilibrium principle, for parameters like mass, energy or force-acceleration duo, the rate of change of the parameter within the volume is the result of actions and reactions occurring at and across its surfaces.
In term of force-acceleration along a certain direction, the key tool is the Newton’s (1642 – 1727) 2nd Law of Motion. The active forces acting on the volume are those due to gravitational pull, excess water pressure, wind shear at the water surface and mean wave pressure. The reactive forces are the viscous friction at the bottom, and the momentum exchange by turbulence and eddy. These reactive forces dissipate some of the flow energy. In response to these forces, the change that occurs within the volume is the rate of change of flow velocity in time and space. In addition, when the actual application of the principle covers a large area, the rotation of the Earth adds another force. The translation of these descriptive accelerations and forces into mathematical terms is known as the Navier-Stokes equation. French engineer Claude-Louis Navier (1785 – 1836) and British mathematician George Gabriel Stokes (1819 – 1903) are credited to have developed the mathematical elaboration.          
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How can we make sense of all the terms in the formidable Navier-Stokes equation, which is only solvable on the platform of computational modeling? How significant and sensitive is each term relative to the other? In my 2017 Encyclopedia of Coastal Science Chapter Seabed Roughness of Coastal Waters - I have presented the equation in simple and easily understandable terms. Further, a technique known as Scale Analysis is very useful to answer these questions. The technique involves approximating the differential calculus terms in the equation by representative scale values. It essentially translates the complex differential equation into an algebraic one. The beauty of such an analysis is that it helps us readily understand the behavior of water motion.
In my presentations at the University of Central Florida in Orlando and at the Coastal Zone Canada Conference in Vancouver in 2008, I tried to show the usefulness of the technique. The scale analysis shows that in most natural flows a first-order approximation of the Navier-Stokes equation is governed by the actives forces of excess water pressure and gravitational pull resisted by the reactive friction force. The finding is not new however. This simple steady non-accelerating flow approximation has been worked out by Chezy (Antoine Chezy, 1718 – 1798), not by scale analysis but from observations. The relation got further refinement from Manning (Robert Manning, 1816 – 1897) nearly a century later that included pipe flow as well.
Another significant approximation results from balancing the rate of change of flow velocity in space – the so called convective acceleration, against the active forces of excess water pressure and gravitational pull. Again it turns out that this approximation has been worked out by Daniel Bernoulli (1700 – 1782) in the 18th century. Bernoulli formulation of irrotational water motion in a steady frictionless environment is very insightful. It says that for a given condition, water level and speed are reciprocal to each other – in other words when motion speeds up water level falls, and vice versa. It says that a balance occurs between standing water pressure and dynamic water pressure – or in other words, between potential energy, pressure energy and kinetic energy.
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Scale analysis also shows that the behavior of water motion is determined by the relative magnitudes of some basic forces. Among them, the gravitational pull, the force due to water motion or inertia, and the viscous friction are most important. William Froude (1810 – 1879) proposed a dimensionless number – a ratio of inertial force and gravitation pull, to show that water motion can be tranquil or torrential depending on the value of this number. Similarly Osborne Reynolds (1842 – 1912) showed that water particle motion can be parallel or erratically turbulent depending on the relative magnitudes of the inertial force and viscous friction. Apart from characterizing flows as such, these two definitions of dimensionless numbers paved the way for engineers to conduct easily manageable experiments on water motion by implementing miniature scaled replica of actual prototype conditions.  
Other notable developments include Newton’s gravitational theory explaining the reasons behind the tidal rise and fall of ocean water; and George Biddell Airy’s (1801 – 1892) simplified small amplitude wave theory based on solving what is known as unsteady Bernoulli equation. Also notable among the investigators was Blaise Pascal (1623 – 1662) who showed that a force imposed on water was equally transmitted in every direction – a mechanical behavior of water due largely to its incompressible property.
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Do these information sound common? Well, perhaps not so common – but educated common. The described relations are part of what are known as the process-based models.
Many hydraulic engineering problems can be reasonably solved or understood by simple process-based models like Chezy equation and Bernoulli equation. Someone might say: Come on, this is no-brainer. It turns out that an accomplishment appears no-brainer only after it is accomplished. Let me illustrate two simple examples from my own experience.
I had the opportunity to scientifically explain the reasons for drainage congestion faced by seaward building up of deltas in Bangladesh. The process of delta building lengthens a drainage channel and thus reduces its seaward slope – the slope being the ratio of the height difference between head and tail waters, and the channel length. The Chezy relation tells us that, no matter what one does either excavating or dredging locally, the drainage rate is not going to improve if the overall slope remains flatter. Therefore spending money on channel excavation or dredging is likely to prove wasteful unless the real reason is addressed. This and other findings were presented in the 1993 University of Southern California Sea Grant Symposium on Coastal Ocean Space Utilization.
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The other example is on characterizing morphological changes of erodible beaches by applying some simple ratios of wave height, wave period and sand particle settling velocity. These ratios were proposed by Robert G. Dean (1931 – 2015), one of the greatest talents in coastal engineering. Applied on Florida coast, the findings have shown that cross-shore morphology in some areas of Florida beaches is likely to remain erosive for 70% of the time in a year. This indicates that for ensuring a stable beach, either wave forcing needs to be reduced by engineered measures, or it should be replenished by borrowed sands on a recurring basis. No wonder, why Florida needs to make so much investments in beach replenishment each year. This and other findings including computational modeling were presented at the 2009 22nd Annual National Conference on Beach Preservation Technology.
The reason for illustrating these simple examples is that most often we do not need to conduct elaborate computational modeling to understand hydraulic processes. We should not forget to look into our backyard to find some educated common knowledge to generate solutions, or to evaluate and interpret the performance of a computational model.   
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Here is an anecdote to ponder:
The disciple asked the master, “Sir, one day I would like to walk on water?”
The master smiled, “You know, someone has told me once that common sense is something not so common. I have paid no attention to that until now.”

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- by Dr. Dilip K. Barua, 2 June 2016
 

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