Sediment Transport / Shields Curve

In trying to maintain my Monday's show and tell, I began my morning by reading another paper from a file folder I have set aside for treasures I come across while browsing the internet. This particular paper, Sedimentation Investigations of Rivers and Reservoirs, produced by the USACE, details everything from formulation and planning sediment studies and sediment yield, to river and reservoir sedimentation. While it is quite expansive and detailed, the citations/references were a bit dated. As I read on I began wondering as to the relevance of the information currently, so I did a quick search on google to see what methods are still popularly in use. I knew both Shields parameter and grain Reynolds number were still applicable because those were pretty much the only method I was familiar with, and it hasn't been too terribly long since my sed/strat course. The majority of my search landed me towards closed subscription-required engineering pages, but I did manage to find a few papers that were informative. A third find during the writing of this post was a google book sample, but I will pontificate upon that a bit later. For now I will just say that despite its date, (originally published 18 Dec 1989), the USACE paper still held quite a bit of its relevance. One interesting excerpt caught my attention though, as it credits Rouse (ASCE 1975) for proposing the Shields curve, and Shields having utilized it in his analysis:

Although the experimental work and analysis were performed
by Shields, the curve termed the Shields Curve,
which is shown in Figure 9-1, was actually proposed by
Rouse (ASCE 1975). Shields curve may be expressed as
an equation, which is useful for computer programming.

~ Engineer Manual 1110-2-4000

I'm sure there was probably some obscure notation of Rouse in one of my textbooks, but I must have overlooked it or this would not have been a bit of a surprise to me. I looked up Rouse and discovered that he, in fact, was given little to no credit for his work. When Rouse introduced the Shields diagram, he did so with auxillary parameters. You can read more on this here.

The images to the left are that of a Shields diagram and correlating equation. (culled from EM 1110-2-4000). It's a widely used method of computation or anything other than very small Reynolds numbers, otherwise other empirical expressions are utilized. For general purposes though, the Shields diagram is a good starting point.

Further into the manual there is a section on Bank or Wall Shear Stress. Brownlie's approach appears to be the favored method, and the section is fairly well written describing the resistance equations and range of conditions. Duboy's concept where the significant assumption being that sediment
transport could be calculated using average cross-section
[hydraulic] parameters and that the main result of excess shear stress was transport of said sediment. (EM 1110-2-4000). There are a few more equations in this section, some of while were derived from Einstein, which I found interesting. Mostly because when I think of Einstein, I associate him with theoretical physics. 

I was curious as to what other methods were implemented in calculating shear stresses in banks or walls, so I did yet another internet search. This one yielded quite a bit of interesting reading material. One of which was a paper pretty much dedicated to hydraulic shear stresses, with several different environments/situations outlined and the correlating equations:  Shear Stress in Bends

Flow around bends creates secondary currents that exert higher shear forces on the channel bed
and banks than those found in straight sections. Several techniques are available for estimating
shear stress in bends. A relatively simple and widely used method, presented by U. S. Department of Transportation,2 estimates maximum shear stress on channel banks and bed occurring within bends. This equation, however, does not differentiate between bank and bed shear stress. The maximum bed/bank shear stress is primarily focused on the bank and bed on the outside portion of the bend .

Lastly was a link to a book on google books. This book, Introduction to bed, bank, and shore protection- by Gerrit J. Schiereck is by far the best [mathematically-heavy] book I have ever read. While I did not read the entire book, what I did read was so well written I forgot for a moment I was reading about math. In general, I like math, but it can be a love/hate relationship for me. I don't like to have to figure out what a writer is blundering through in addition to understanding the formulas. With this book you don't have to do that. The man is an artist, truly. When you can become so absorbed in what you are reading because they have grabbed your attention AND know how to write eloquently- well that is a book you just have to buy. So I did! You can catch a B&N link to it here, but it appears I bought the last copy. (At least I hope that is the case- my order went through, but you never know. I'll have to check my email when I finish up with this). The figure (Fig. 3.1, forces on a grain flow) at the beginning of this post comes from a section of his book. 

On page 52 of the google book Schiereck describes Shield's formula for uniform flow, and how it isn't always the best choice. He explains why using shear stress as the active force this isn't always the best choice. On p. 65 he goes in to describe another environment (a dam or a groyne ) where you can use Shields eq. in conjunction with a slope correction. If you have time, read p. 72-73, as the part about geotextiles particularly is interesting.

Sources:
Introduction to bed, bank, and shore protection- by Gerrit J. Schiereck
Shear Stress in Bends
USACE EM 1110-2-4000
Rouse and Shield



.