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The '''depth–slope product''' is used to calculate the [[shear stress]] at the bed of an [[open channel flow|open channel]] containing [[fluid]] that is undergoing [[steady flow|steady]], [[uniform flow|uniform]] flow. It is widely used in [[river engineering]], [[stream restoration]], [[sedimentology]], and [[Geomorphology#Fluvial|fluvial geomorphology]]. It is the [[product (mathematics)|product]] of the water [[wikt:depth|depth]] and the mean bed [[slope]], along with the [[gravitational acceleration|acceleration due to gravity]] and [[density]] of the fluid.
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==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]], <math>R_h</math>, can be assumed to be equal to the channel depth, <math>h</math>.
 
:<math>R_h = \frac{A}{P}</math>
 
where <math>A</math> is the [[cross sectional area]] of flow and <math>P</math> 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),
 
:<math>A = b h </math>,
 
where <math>b</math> is the width (breadth) of the channel, and
 
:<math>P = b+2h</math>.
 
For b>>h,
 
:<math>P \rightarrow b</math>,
 
and therefore
 
:<math>R_h \rightarrow \frac{b h}{b} = h</math>.
 
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 (mechanics)|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]] <math>\rho</math>, an [[gravitational acceleration|acceleration due to gravity]] <math>g</math>, and a flow depth <math>h</math>, the pressure exerted on the bed is simply the weight of an element of fluid, <math>\rho g</math>, times the depth of the flow, <math>h</math>. From this, we get the expression for the total pressure, <math>P_b</math>, acting on the bed.
 
:<math>P_b=\rho g h \,</math>
 
===Shear stress===
 
In order to convert the pressure into a shear stress, it is necessary to determine the [[Euclidean vector|component]] of the pressure that provides shear on the bed. For a channel that is at an angle <math>\alpha</math> from horizontal, the shear component of the stress acting on the bed <math>/\left(\tau_b\right)</math>, which is the component acting tangentially to the bed, equals the total pressure times the sine of the angle <math>\alpha</math>.
 
:<math>\tau_b=\rho g h \sin{\left(\alpha\right)} \,</math>
 
In natural rivers, the angle <math>\alpha</math> is typically very small. As a result, the small-angle formula states that:
 
:<math>\sin{\left(\alpha\right)}\approx\tan{\left(\alpha\right)}</math>
 
The tangent of the angle <math>\alpha</math> is, by definition, equal to the [[slope]] of the channel, <math>S</math>.
 
:<math>\tan{\left(\alpha\right)}\equiv S</math>
 
From this, we can arrive at the final form of the relation between bed shear stress and depth–slope product:
 
:<math>\tau_b=\rho g h S \,</math>
 
===Scaling===
 
Assuming a single, well-mixed, [[wikt:homogeneous|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.
 
:<math>\tau_b\propto h S</math>
 
For natural streams, in the [[Mks system of units|mks]] or [[International System of Units|SI]] system (units of [[Pascal (pressure)|pascals]] for shear stress), a typical useful relationship to remember is that:
 
:<math>\tau_b=10000 \left[\frac{\mathrm{kg}}{\mathrm{m}^2 \ \mathrm{s}^2}\right] h S</math>
 
for water with a density of 1000&nbsp;kg/m<sup>3</sup> and approximating the acceleration due to gravity as 10&nbsp;m/s<sup>2</sup> (the error in this assumption is typically much smaller than the error from measurements).
 
==Uses==
 
Bed shear stress can be used to find:
* The vertical [[velocity]] profile within the fluid flow
* The ability of the [[fluid]] to [[Sediment transport|carry sediment]]
* The rate of shear [[Dispersive mass transfer|dispersion]] of contaminants and tracers.
 
==See also==
* [[Sediment]]
* [[Fluid mechanics]]
 
==References==
*Leopold, Wolman, and Miller (1964), ''Fluvial Processes in Geomorphology,'' Dover Publications, Mineola, NY, USA, 535 pp.
 
{{DEFAULTSORT:Depth-slope product}}
[[Category:Fluid dynamics]]
[[Category:Geology]]
[[Category:Civil engineering]]
[[Category:Environmental engineering]]

Latest revision as of 21:00, 3 September 2014

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