Stress balance For a fluid to begin transporting sediment that is currently at rest on a surface, the boundary (or bed)
shear stress \tau_b exerted by the fluid must exceed the critical shear stress \tau_c for the initiation of motion of grains at the bed. This basic criterion for the initiation of motion can be written as: :\tau_b=\tau_c. This is typically represented by a comparison between a
dimensionless shear stress \tau_b* and a dimensionless critical shear stress \tau_c*. The nondimensionalization is in order to compare the driving forces of particle motion (shear stress) to the resisting forces that would make it stationary (particle density and size). This dimensionless shear stress, \tau*, is called the
Shields parameter and is defined as: :\tau*=\frac{\tau}{(\rho_s-\rho_f)(g)(D)}. And the new equation to solve becomes: :\tau_b*=\tau_c*. The equations included here describe sediment transport for
clastic, or
granular sediment. They do not work for
clays and
muds because these types of
floccular sediments do not fit the geometric simplifications in these equations, and also interact thorough
electrostatic forces. The equations were also designed for
fluvial sediment transport of particles carried along in a liquid flow, such as that in a river, canal, or other open channel. Only one size of particle is considered in this equation. However, river beds are often formed by a mixture of sediment of various sizes. In case of partial motion where only a part of the sediment mixture moves, the river bed becomes enriched in large gravel as the smaller sediments are washed away. The smaller sediments present under this layer of large gravel have a lower possibility of movement and total sediment transport decreases. This is called armouring effect. Other forms of armouring of sediment or decreasing rates of sediment erosion can be caused by carpets of microbial mats, under conditions of high organic loading.
Critical shear stress The
Shields diagram empirically shows how the dimensionless critical shear stress (i.e. the dimensionless shear stress required for the initiation of motion) is a function of a particular form of the particle
Reynolds number, \mathrm{Re}_p or Reynolds number related to the particle. This allows the criterion for the initiation of motion to be rewritten in terms of a solution for a specific version of the particle Reynolds number, called \mathrm{Re}_p*. :\tau_b*=f\left(\mathrm{Re}_p*\right) This can then be solved by using the empirically derived Shields curve to find \tau_c* as a function of a specific form of the particle Reynolds number called the boundary Reynolds number. The mathematical solution of the equation was given by
Dey.
Particle Reynolds number In general, a particle Reynolds number has the form: :\mathrm{Re}_p=\frac{U_p D}{\nu} Where U_p is a characteristic particle velocity, D is the grain diameter (a characteristic particle size), and \nu is the kinematic viscosity, which is given by the dynamic viscosity, \mu, divided by the fluid density, {\rho_f}. :\nu=\frac{\mu}{\rho_f} The specific particle Reynolds number of interest is called the boundary Reynolds number, and it is formed by replacing the velocity term in the particle Reynolds number by the
shear velocity, u_*, which is a way of rewriting shear stress in terms of velocity. :u_*=\sqrt{\frac{\tau_b}{\rho_f}}=\kappa z \frac{\partial u}{\partial z} where \tau_b is the bed shear stress (described below), and \kappa is the
von Kármán constant, where : \kappa = {0.407}. The particle Reynolds number is therefore given by: :\mathrm{Re}_p*=\frac{u_* D}{\nu}
Bed shear stress The boundary Reynolds number can be used with the Shields diagram to empirically solve the equation :\tau_c*=f\left(\mathrm{Re}_p*\right), which solves the right-hand side of the equation :\tau_b*=\tau_c*. In order to solve the left-hand side, expanded as :\tau_b*=\frac{\tau_b}{(\rho_s-\rho_f)(g)(D)}, the bed shear stress needs to be found, {\tau_b}. There are several ways to solve for the bed shear stress. The simplest approach is to assume the flow is steady and uniform, using the reach-averaged depth and slope. because it is difficult to measure shear stress
in situ, this method is also one of the most-commonly used. The method is known as the
depth-slope product.
Depth-slope product For a river undergoing approximately steady, uniform equilibrium flow, of approximately constant depth
h and slope angle θ over the reach of interest, and whose width is much greater than its depth, the bed shear stress is given by some momentum considerations stating that the gravity force component in the flow direction equals exactly the friction force. For a wide channel, it yields: :\tau_b=\rho g h \sin(\theta) For shallow slope angles, which are found in almost all natural lowland streams, the
small-angle formula shows that \sin(\theta) is approximately equal to \tan(\theta), which is given by S, the slope. Rewritten with this: :\tau_b=\rho g h S
Shear velocity, velocity, and friction factor For the steady case, by extrapolating the depth-slope product and the equation for shear velocity: :\tau_b=\rho g h S :u_*=\sqrt{\left(\frac{\tau_b}{\rho}\right)}, The depth-slope product can be rewritten as: :\tau_b=\rho u_*^2. u* is related to the mean flow velocity, \bar{u}, through the generalized
Darcy–Weisbach friction factor, C_f, which is equal to the Darcy-Weisbach friction factor divided by 8 (for mathematical convenience). Inserting this friction factor, :\tau_b=\rho C_f \left(\bar{u} \right)^2.
Unsteady flow For all flows that cannot be simplified as a single-slope infinite channel (as in the
depth-slope product, above), the bed shear stress can be locally found by applying the
Saint-Venant equations for
continuity, which consider accelerations within the flow.
Example Set-up The criterion for the initiation of motion, established earlier, states that :\tau_b*=\tau_c*. In this equation, :\tau*=\frac{\tau}{(\rho_s-\rho)(g)(D)}, and therefore :\frac{\tau_b}{(\rho_s-\rho)(g)(D)}=\frac{\tau_{c}}{(\rho_s-\rho)(g)(D)}. :\tau_c* is a function of boundary Reynolds number, a specific type of particle Reynolds number. :\tau_c*=f \left(\mathrm{Re}_p* \right). For a particular particle Reynolds number, \tau_c* will be an empirical constant given by the Shields Curve or by another set of empirical data (depending on whether or not the grain size is uniform). Therefore, the final equation to solve is: :\frac{\tau_b}{(\rho_s-\rho)(g)(D)}=f \left(\mathrm{Re}_p* \right).
Solution Some assumptions allow the solution of the above equation. The first assumption is that a good approximation of reach-averaged shear stress is given by the depth-slope product. The equation then can be rewritten as: :{\rho g h S}=f\left(\mathrm{Re}_p* \right){(\rho_s-\rho)(g)(D)}. Moving and re-combining the terms produces: :{h S}={\frac{(\rho_s-\rho)}{\rho}(D)}\left(f \left(\mathrm{Re}_p* \right) \right)=R D \left(f \left(\mathrm{Re}_p* \right) \right) where R is the
submerged specific gravity of the sediment. The second assumption is that the particle Reynolds number is high. This typically applies to particles of gravel-size or larger in a stream, and means the critical shear stress is constant. The Shields curve shows that for a bed with a uniform grain size, :\tau_c*=0.06. Later researchers have shown this value is closer to :\tau_c*=0.03 for more uniformly sorted beds. Therefore the replacement :\tau_c*=f \left(\mathrm{Re}_p* \right) is used to insert both values at the end. The equation now reads: :{h S}=R D \tau_c* This final expression shows the product of the channel depth and slope is equal to the Shield's criterion times the submerged specific gravity of the particles times the particle diameter. For a typical situation, such as quartz-rich sediment \left(\rho_s=2650 \frac{kg}{m^3} \right) in water \left(\rho=1000 \frac{kg}{m^3} \right), the submerged specific gravity is equal to 1.65. :R=\frac{(\rho_s-\rho)}{\rho}=1.65 Plugging this into the equation above, :{h S}=1.65(D)\tau_c*. For the Shield's criterion of \tau_c*=0.06. 0.06 * 1.65 = 0.099, which is well within standard margins of error of 0.1. Therefore, for a uniform bed, :{h S}={0.1(D)}. For these situations, the product of the depth and slope of the flow should be 10% of the diameter of the median grain diameter. The mixed-grain-size bed value is \tau_c*=0.03, which is supported by more recent research as being more broadly applicable because most natural streams have mixed grain sizes. If this value is used, and D is changed to D_50 ("50" for the 50th percentile, or the median grain size, as an appropriate value for a mixed-grain-size bed), the equation becomes: :{h S}={0.05(D_{50})} Which means that the depth times the slope should be about 5% of the median grain diameter in the case of a mixed-grain-size bed. ==Modes of entrainment==