Resistance to Flow

In the Common Sense Hydraulics blog on this page, I have pointed out that the first order solution to a hydrodynamic problem is governed by the actives forces of excess water pressure and gravitational pull resisted by the reactive friction force. It only makes sense that I devote this piece as a follow up on the reactive friction force. This force is caused by the resistance of a solid boundary to the fluid flow of mass and energy on it. Reactive forces are notoriously nonlinear, not only because of the fluid behavior itself, but also because of the mobility of the bed if it happens to be alluvial (further in Seabed Roughness of Coastal Waters).
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1. Flow Resistance Intro
Why understanding resistance to flow is important? One reason is that the first order solution, either in desktop analysis or in computational modeling cannot be achieved unless the reactive resistance is accurately understood and parameterized. In addition, without an accurate first order solution that dominates processes, advanced order solutions are unachievable. The second reason is that the stability and functions of a shoreline, and waterfront and maritime structures can be jeopardized if the understanding of erosion-sedimentation processes remains questionable.    
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2. History of Coastal Engineering
Before entering into the discussion of the topic, I am tempted to add a few lines on the history of coastal engineering. The official recognition and definition of it was launched at the First Conference on Coastal Engineering held in Long Beach, California in 1950. The conference proceedings contributed by some invited speakers gave birth to this new discipline of civil hydraulic engineering. In time, the discipline metamorphosed into several sub disciplines – not so much in an orderly fashion but rather in a confusing manner.
Apart from journal papers and conference proceedings, the initiative was followed by some remarkable books. Among them were three outstanding ones. The first was a volume {Estuary and Coastal Hydrodynamics. McGraw-Hill, 1966} edited by Arthur T. Ippen (1907 – 1974). The second was an initiative taken by the American Society of Civil Engineers (ASCE) that resulted in a volume {Sedimentation Engineering. ASCE, 1974} edited by Vito A. Vanoni (1904 – 1999). Contributed by outstanding scholars from around world, these two publications set the scientific background on which many future works were built upon.
Another publication {Shore Protection Manual (SPM), Vol. I and II} dealing with the guidelines on practical applications was initiated by U. S. Army Coastal Engineering Research Center (CERC). First published in 1973, these two SPM volumes soon became very popular with practicing engineers, and CERC continued issuing new editions, the last of which was in 1984. In later years, SPM reincarnated into the diverse and multiple Coastal Engineering Manuals (CEM).
During and following the publications of these three volumes, universities and research institutions of many different countries made significant contributions: notable among them were Delft Hydraulics, HR Wallingford, and Japan. An edited book, {History and Heritage of Coastal Engineering} by Nick Kraus published in 1996 chronicled the development of the discipline.
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3. Boundary Layer
Well, enough on the historical context for now. Let us try to focus on the topic of this piece. But before doing so, a very important concept requires clarification - and this concept is regarding the boundary layer. This layer of reduced flow velocity develops closed to the bed – from zero near the bed to the asymptote of free stream velocity up in the water column. The reduced velocity is mainly caused by the shearing resistance of the bed, and the loss of fluid flow energy in eroding and transporting alluvial sediments. The layer divided primarily into two is known as the boundary layer. A very thin viscous sub-layer occurs near the bed with a turbulent upper layer above. Interactions between the fluid shearing force and the bottom reactive force within this layer account for viscous and turbulent transfer of momentums within the water column. Described by an asymptotic logarithmic or a power function, the height of this layer changes in response to the change in flow velocities and roughness of the bed.
Among other investigators, my own works {Some Aspects of Turbulent Flow Structure in Large Alluvial Rivers. Journal of Hydraulic Research, Taylor & Francis, 1998} for the Flood Action Plan – River Survey Project, provide some insights into the flow structure of the boundary layer. The investigation showed that the bed-generated turbulence in the presence of high dune-scale bedforms reached the maximum above the bed at a height of 5 to 10% of water depth, with decaying of the strength from above that level to up in the water column. A bedform is a wavy undulation, mostly noticeable in a sandy bed, which is primarily asymmetric in unidirectional flows and symmetric in oscillatory short-wave environments. The image shown in this piece gives a snapshot of some small-scale bedforms.  
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4. Bed Resistance in Unidirectional Flow
How to characterize the bed resistance? The question can best be answered using the theory proposed by Daniel Bernoulli (1700 – 1782). His formulation shows that dynamic water pressure or kinetic energy is defined in terms of V^2, V being the mean flow velocity of the current. It is this dynamic pressure that is responsible for exerting drag on the bed. We will find out in later discussions that it is also this dynamic pressure that causes drag force on structures in water. Many, including one of the pioneering investigators Ralph Alger Bagnold (1896 – 1990) used this drag to formulate the bed shear force or the equivalent bottom reactive force and sediment transport. The formulation known as quadratic friction law defines this force as the product of a non-dimensional drag coefficient, water density and square of the current velocity.
The quadratic friction law is universally applicable in unidirectional flow such as in river as well as in oscillatory long-wave motions such as tide, tsunami or storm surge. The drag coefficient is related to its counterparts of other known resistance coefficients such as Chezy (Antoine Chezy, 1718 – 1798) coefficient C, Manning’s (Robert Manning, 1816 – 1897) n, and Darcy-Weisbach friction factor f. To give an idea, it can be shown that in a water depth of 10 m, and a representative bed-material size of 3 mm, the drag coefficient is 0.0014, C = 84.13 m^(1/2)/s, n = 0.0175 (s/m^(1/3)) and f = 0.0111. See more in Seabed Roughness of Coastal Waters.
When deformation of the alluvial bed occurs in high flow stages, additional resistance to flow is imposed. The migrating bedforms varying from ripples to large dunes are mostly asymmetric with a flatter stoss slope and steeper avalanche slope. In most instances, the larger the bedform, the larger is the resistance to flow. In one investigation {Bedform Dynamics and Sediment Transport – Report of an Investigation in the Jamuna River. Institution of Engineers Bangladesh, Paper #41-4-06, 1996} I worked on; it was found that bedforms accounted for 70% of the total bed resistance. 
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5. Bed Resistance in Oscillatory Flow
The bed resistance discussed so far is applicable for unidirectional flow, and for tide, tsunami and storm surge. How about the bed resistance to short-wave oscillatory motions? The quadratic friction law is similarly applicable in short-wave motions with the applicable velocity taken as the amplitude of bottom orbital velocity; and the drag coefficient is renamed as a wave friction factor. The wave orbital velocity is a bidirectional vector with a peak on either direction. This peak is the amplitude. In addition, a relation proposed by Swart {Offshore sediment transport and equilibrium beach profiles. Delft Hydraulics Publication No. 131, 1974} shows that the amplitude of the bottom orbital excursion affects the wave friction factor. As an example, at a water depth of 10 m, and a representative bed-material size of 3 mm, the friction factor is 0.0175 for a 2-meter 10-second wave. Similar to unidirectional flow, presence of bedforms which are mostly symmetric in short-wave environments, the frictional resistance is enhanced.
In a hydraulic environment where both long and short waves are active, the wave-current friction factor accounts for both. The resistances triggered by both the sources are added in some fashion that depends on the relative magnitudes of current and the amplitude of bottom wave orbital velocity.
Well, I am not sure whether I have managed to explain the topic in plain English as promised. I have tried to keep things as simple as possible with only some limited but unavoidable use of scientific jargon.
One more thing. It is important to mention that the materials covered in this piece are highly empirical, which means that there is no single value of coefficients applicable for all different cases, and for all hydraulic environments. Empirical coefficients are usually valid in orders of magnitudes, and they require verification to examine their applicability for specific cases. If that is not possible their limitations and uncertainties should be highlighted.                  
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Here is an anecdote to ponder:
The disciple asked the master, “Sir, how would you see the resistance to flow in a wider social context?”
The master smiled, “Well, the cause-effect, force-response or action-reaction duo is universally present everywhere. Things are even more complicated in a society than the complexities of fluid flow. What we are talking about here are positive actions or endeavors, and reactions of them. We used to hear from elders in our childhood that life was full of thorns. So resistances are there no matter how we don’t want them.”
“Could you elaborate please?”
“In a society, resistance could come from individuals or from a collective bunch. But while a reactive resistance in natural fluid flows is instantaneous, the same in social interactions could be instantaneous, delayed or even absent. If delayed, the reactive resistance can metamorphose into something different and subdued. Therefore it is often helpful to delay high emotional reactions. But in a society not all resistances are reactive. Some are active resistances which could either mean something good, or could mean something malicious. Some malicious resistances could even reach the scale of an obstacle.”
“Thank you Sir. When do you think active social resistances can become damaging.” 
 “Well, the effects could become very frustrating, ugly and damaging when a society promotes, or is founded upon division, mistrust and conflict.”

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- by Dr. Dilip K. Barua, 11 August 2016

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