Shields Diagram

The Shields diagram is a threshold graph for initiation of sediment grain movement that is widely used by sedimentologists and is well established by experimental work. ^[1] It has more rigour than the Hjulström diagram and a more general application - it can be used for wind as well as water, and for a variety of conditions in water. ^[1] Where the Hjulström diagram uses flow velocity directly and is valid only for 1 m water depth in freshwater, the Shields diagram uses two dimensionless parameters that absorb the relevant fluid and grain properties, making it applicable across a much wider range of environments.

The Two Dimensionless Parameters

The Shields diagram is more complex than the Hjulström diagram because it involves two dimensionless relationships. ^[1]

Dimensionless Shear Stress (θ_t)

Instead of flow velocity as a measure of critical shear, the Shields diagram uses the dimensionless bed shear stress, expressed as: ^[1]

θ_t = τ₀ / (ρ_s − ρ)gD

where τ₀ is boundary shear stress, ρ_s is density of the particles, ρ is density of the fluid, g is gravitational acceleration, and D is particle diameter. ^[1] The value of dimensionless shear stress increases with increasing bed shear stress and increasing velocity; it decreases with increasing density and size of the particles. ^[1]

By incorporating shear stress, grain size, and grain and fluid densities into a single term, the dimensionless shear stress is considerably more informative than flow velocity alone. An increase in the dimensionless shear stress can reflect either an increase in flow velocity and shear stress or a decrease in grain size or grain density. ^[1]

Grain Reynolds Number (R_eg)

The mean grain size parameter used in the Hjulström diagram is replaced in the Shields diagram by the grain Reynolds number R_eg, another dimensionless quantity. ^[1] The grain Reynolds number differs from the ordinary Reynolds number: the length or water depth value L is replaced by particle diameter d, and the flow velocity U is replaced by shear velocity U*. ^[1] It is a measure of turbulence at the grain-fluid boundary and increases with increasing grain size if shear velocity and kinematic viscosity remain constant. ^[1]

Reading the Shields Diagram

Points above the curve indicate that noncohesive grains on the bed are fully in motion; points below indicate no motion, as in the Hjulström diagram. ^[1] Beginning of motion is determined by the dimensionless shear stress, which increases with increasing bed shear stress under a given set of conditions for grain density, fluid density, and grain size. ^[1]

The critical dimensionless shear stress required to initiate grain movement depends upon the grain Reynolds number - which in turn is a function of grain size, kinematic viscosity, and turbulence. ^[1] The dimensionless bed shear stress increases slightly with increasing grain Reynolds number above about 5 to 10, although it remains mainly between 0.03 and 0.05. ^[1] At lower Reynolds numbers, the value increases steadily up to a value of 0.1 or higher. ^[1]

The Role of the Viscous Sublayer

The greater rate of increase of θ_t at lower grain Reynolds numbers is related to the presence of the viscous sublayer. ^[1] When the bed is composed of small particles on the order of fine sand or smaller, a smooth boundary and hydraulically smooth flow result - the particles lie entirely within the viscous sublayer, where flow is essentially nonturbulent and instantaneous velocity variations are less than in the lower part of the overlying turbulent boundary layer. ^[1] For coarser particles, the viscous sublayer is so thin that the grains project through the layer into the turbulent flow. ^[1]

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