Depth–slope product

Depth–slope product

The depth–slope product is used to calculate the shear stress at the bed of an open channel containing fluid that is undergoing steady, uniform flow. It is widely used in river engineering, stream restoration, sedimentology, and fluvial geomorphology. It is the product of the water depth and the mean bed slope, along with the acceleration due to gravity and density of the fluid.

Contents

Formulation

The use of the depth–slope product — in computing the bed shear-stress — specifically refers to two assumptions that are widely applicable to natural river channels: that the angle of the channel from horizontal is small enough that it can be approximated as the slope by the small-angle formula, and that the channel is much wider than it is deep, and sidewall effects can be ignored. Although it is a simplistic approach to find the shear stress in what can often be a locally unsteady fluvial system, when averaged over distances of kilometers, these local variations average and the depth–slope product becomes a useful tool to understand shear stress in open channels such as rivers.

Depth and hydraulic radius

The first assumption is that the channel is much wider than it is deep, and the equations can be solved as if the channel were infinitely wide. This means that side-wall effects can be ignored, and that the hydraulic radius, Rh, can be assumed to be equal to the channel depth, h.

R_h = \frac{A}{P}

where A is the cross sectional area of flow and P is wetted perimeter. For a semicircular channel, the hydraulic radius would simply be the true radius.

For an approximately rectangular channel (for simplicity in the mathematics of the explanation of the assumption),

A = bh,

where b is the width (breadth) of the channel, and

P = b + 2h.

For b>>h,

P \rightarrow b,

and therefore

R_h \rightarrow \frac{b h}{b} = h.

Formally, this assumption can generally be held to hold when the width is greater than about 20 times the height; the exact amount of error accrued can be found by comparing the height to the hydraulic radius. For channels with a lower width-to-depth ratio, a better solution can be found by using the hydraulic radius instead of the above simplification.

Pressure

The total stress on the bed of an open channel of infinite width is given by the hydrostatic pressure acting on the bed. For a fluid of density ρ, an acceleration due to gravity g, and a flow depth h, the pressure exerted on the bed is simply the weight of an element of fluid, ρg, times the depth of the flow, h. From this, we get the expression for the total pressure, Pb, acting on the bed.

P_b=\rho g h \,

Shear stress

In order to convert the pressure into a shear stress, it is necessary to determine the component of the pressure that provides shear on the bed. For a channel that is at an angle α from horizontal, the shear component of the stress acting on the bed /\left(\tau_b\right), which is the component acting tangentially to the bed, equals the total pressure times the sine of the angle α.

\tau_b=\rho g h \sin{\left(\alpha\right)} \,

In natural rivers, the angle α is typically very small. As a result, the small-angle formula states that:

\sin{\left(\alpha\right)}\approx\tan{\left(\alpha\right)}

The tangent of the angle α is, by definition, equal to the slope of the channel, S.

\tan{\left(\alpha\right)}\equiv S

From this, we can arrive at the final form of the relation between bed shear stress and depth–slope product:

\tau_b=\rho g h S \,

Scaling

Assuming a single, well-mixed, homogeneous fluid and a single acceleration due to gravity (both are good assumptions in natural rivers, and the second is a good assumption for processes on Earth, or any planetary body with a dominant influence on the local graviational field), the only two variables that determine the boundary shear stress are the depth and the slope. This is the significance of the name of the formula.

\tau_b\propto h S

For natural streams, in the mks or SI system (units of pascals for shear stress), a typical useful relationship to remember is that:

\tau_b=10000 \left[\frac{\mathrm{kg}}{\mathrm{m}^2 \ \mathrm{s}^2}\right] h S

for water with a density of 1000 kg/m3 and approximating the acceleration due to gravity as 10 m/s2 (the error in this assumption is typically much smaller than the error from measurements).

Uses

Bed shear stress can be used to find:

See also

References

  • Leopold, Wolman, and Miller (1964), Fluvial Processes in Geomorphology, Dover Publications, Mineola, NY, USA, 535 pp.

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Sediment transport — is the movement of solid particles ( sediment ) due to the movement of the fluid in which they are immersed. This is typically studied in natural systems, where the particles are clastic rocks (sand, gravel, boulders, etc.) or clay, and the fluid …   Wikipedia

  • Shear stress — SI symbol: τ SI unit: pascal Derivations from other quantities: τ = F / A …   Wikipedia

  • river — river1 riverless, adj. riverlike, adj. /riv euhr/, n. 1. a natural stream of water of fairly large size flowing in a definite course or channel or series of diverging and converging channels. 2. a similar stream of something other than water: a… …   Universalium

  • climate — /kluy mit/, n. 1. the composite or generally prevailing weather conditions of a region, as temperature, air pressure, humidity, precipitation, sunshine, cloudiness, and winds, throughout the year, averaged over a series of years. 2. a region or… …   Universalium

  • coal mining — Coal was very important in the economic development of Britain. It was used as fuel in the factories built during the Industrial Revolution and continued to be important until the 1980s. The main coalfields are in north east England, the north… …   Universalium

  • lake — lake1 /layk/, n. 1. a body of fresh or salt water of considerable size, surrounded by land. 2. any similar body or pool of other liquid, as oil. 3. (go) jump in the lake, (used as an exclamation of dismissal or impatience.) [bef. 1000; ME lak(e) …   Universalium

  • Lake — /layk/, n. Simon, 1866 1945, U.S. engineer and naval architect. * * * I Relatively large body of slow moving or standing water that occupies an inland basin. Lakes are most abundant in high northern latitudes and in mountain regions, particularly …   Universalium

  • GEOGRAPHICAL SURVEY — Names The name Ereẓ Israel (the Land of Israel) designates the land which, according to the Bible was promised as an inheritance to the Israelite tribes. In the course of time it came to be regarded first by the Jews and then also by the… …   Encyclopedia of Judaism

  • Life Sciences — ▪ 2009 Introduction Zoology       In 2008 several zoological studies provided new insights into how species life history traits (such as the timing of reproduction or the length of life of adult individuals) are derived in part as responses to… …   Universalium

  • mining — /muy ning/, n. 1. the act, process, or industry of extracting ores, coal, etc., from mines. 2. the laying of explosive mines. [1250 1300; ME: undermining (walls in an attack); see MINE2, ING1] * * * I Excavation of materials from the Earth s… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”