Reynolds Number

The fundamental differences in laminar and turbulent flow arise from the ratio of inertial forces to viscous forces within a moving fluid. ^[1] Inertial forces - related to the scale and velocity of fluids in motion - tend to cause fluid turbulence, while viscous forces - which increase with increasing viscosity of a fluid - resist deformation and thus tend to suppress turbulence. ^[1] The Reynolds number captures this competition between the two force types as a single dimensionless value, making it possible to predict whether a given flow will be laminar or turbulent without needing to observe the flow directly.

The Reynolds number R_e is expressed as: ^[1]

R_e = UL / ν = ULρ / μ

where U is the mean velocity of flow, L is some length (commonly water depth) that characterises the scale of flow, and ν is kinematic viscosity. ^[1] Being dimensionless, the Reynolds number can be used to compare scaled-down laboratory models of flow systems with full-scale natural flow systems, a major advantage in hydraulic modelling studies. ^[1]

Interpreting the Reynolds Number

When viscous forces dominate - as in highly concentrated mud flows with high effective viscosity - Reynolds numbers are small and flow is laminar. ^[1] Very low flow velocity or shallow depth also produces low Reynolds numbers and laminar flow. ^[1] This makes intuitive sense from the formula: if U is small, or L is small (shallow water), or ν is large (viscous fluid), Re will be small and the flow can remain organised as parallel layers.

When inertial forces dominate and flow velocity increases - as in the atmosphere and most flow in rivers - Reynolds numbers are large and flow is turbulent. ^[1] This confirms the earlier point that most flow under natural conditions is turbulent: rivers are deep, fast, and carry low-viscosity water, so Re is always large.

Note from the Reynolds number formula that an increase in viscosity can have the same effect as a decrease in flow velocity or flow depth. ^[1] This equivalence is powerful: it means that very cold, very viscous flows can behave like slower, shallower flows of warmer water. It also explains why glacier ice - despite moving at metres per year - flows in a laminar regime: ice has such enormous viscosity that Re remains extremely small.

The Critical Reynolds Number

The transition from laminar flow to turbulent flow takes place above a critical value of Reynolds number, which commonly lies between 500 and 2000 and which depends upon the boundary conditions such as channel depth and geometry. ^[1] Under a given set of boundary conditions, therefore, the Reynolds number can be used to predict whether flow will be laminar or turbulent and to derive some idea of the magnitude of turbulence. ^[1]

The range 500-2000 rather than a single fixed value reflects the reality that the transition depends on boundary geometry: smooth channels allow laminar flow to persist to higher Reynolds numbers before it breaks down into turbulence, while rough or irregular channels force an earlier transition. In laboratory pipes, transition typically occurs near Re = 2000; in natural channels with rough beds, the transition can begin at lower values.

Grain Reynolds Number

In sediment transport, a related parameter called the grain Reynolds number R_eg is used, which differs from the ordinary Reynolds number in important ways. ^[1] The length or water depth value L of the ordinary Reynolds number is replaced by particle diameter d, and the flow velocity U is replaced by shear velocity U*. ^[1] The grain Reynolds number is a measure of turbulence at the grain-fluid boundary. ^[1]

The grain Reynolds number increases with increasing grain size if shear velocity and kinematic viscosity remain constant - so an increase in the grain Reynolds number means an increase in grain size, an increase in shear velocity and turbulence, or a decrease in kinematic viscosity. ^[1] The grain Reynolds number is plotted on the horizontal axis of the Shields diagram and determines which flow regime a grain experiences at the bed - specifically, whether the grain lies within a viscous sublayer (low R_eg) or projects into turbulent flow (high R_eg).

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