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A primer on turbulence in hydrology and hydraulics: The power of dimensional analysis

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The apparent random swirling motion of water is labeled “turbulence,” which is a pervasive state of the flow in many hydrological and hydraulic transport phenomena. Water flow in a turbulent state can be described by the momentum conservation equations known as the Navier–Stokes (NS) equations. Solving these equations numerically or in some approximated form remains a daunting task in applications involving natural systems thereby prompting interest in alternative approaches. The apparent randomness of swirling motion encodes order that may be profitably used to describe water movement in natural systems. The goal of this primer is to illustrate the use of a technique that links aspects of this ordered state to conveyance and transport laws. This technique is “dimensional analysis”, which can unpack much of the complications associated with turbulence into surprisingly simplified expressions. The use of this technique to describing water movement in streams as well as water vapor movement in the atmosphere is featured. Particular attention is paid to bulk expressions that have received support from a large body of experiments such as flow‐resistance formulae, the Prandtl‐von Karman log‐law describing the mean velocity shape, Monin‐Obukhov similarity theory that corrects the mean velocity shape for thermal stratification, and evaporation from rough surfaces. These applications illustrate how dimensional analysis offers a pragmatic approach to problem solving in sciences and engineering. This article is categorized under: Science of Water > Methods Science of Water > Hydrological Processes
The variables impacting the mean velocity gradient du/dz. These include the height from the ground z, and the kinematic surface stress . As z increases, du/dz visually decreases. Also, in the limit of τo = 0, du/dz = 0. Because τo is determined from a balance between frictional forces resisting the driving forces to ensure no acceleration, a τo = 0 either implies no forcing (i.e., the flow is not moving) or a “free slip” condition. In the free slip condition, the flow no longer senses the presence of the surface and the velocity is finite at the surface
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A uniform rectangular channel characterized by water depth h, width B, length Lc, slope So, and surface roughness characterized by a protrusion height r into the flow. The bed and one of the side stresses resisting the flow τo are shown along with the gravitational force Fg driving the flow along So
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Two solutions for the logistic map differing only by a small perturbation in initial conditions (left). The histogram or probability density function (PDF) for the two solutions are also compared (right). Despite the differences in the temporal dynamics of the two solutions, their statistics (i.e., PDF) are virtually the same. These are the reasons why turbulence, which abides by Navier–Stokes (NS), is treated in a statistical manner
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Measured longitudinal velocity time series u(t) at z = 1 cm above a smooth open channel showing irregularity in turbulence (left). The measurements were sampled at 100 Hz when the water depth attained a uniform value of h = 10 cm. Measured u(t) in the atmosphere at z = 5 m above a grass field (right) again illustrating similar irregularity to those reported in the smooth open channel flow. The measurements were sampled at 56 Hz during daytime conditions
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The shape of the energy spectrum E(k) ([L]3[T]−2) as a function of wavenumber k ([L]−1) or inverse of eddy sizes. K41 applies far from scales where energy is introduced (low k or large scales LI) into the spectrum or removed by the action of fluid viscosity at small scales (or high k) commensurate to and smaller than η. Also, K41 assumes that the rate of energy production P, energy transfer across scales T, and TKE dissipation rate ɛ ([L]2[T]−3) are all equal. This assumption is labeled as “conservative” cascade—where the energy content varies across scales but the energy transfer rate (P = T = ɛ) across k remains constant independent of k
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Evaporation rate or water vapor flux FE from a bare soil surface characterized by a roughness r. The flow above the surface is an outcome of a balance between the driving forces and the frictional forces acting at the solid–air interface. The water vapor concentration at the surface and some distance from the surface is characterized by ΔC. The water vapor exchange between the moist roughness elements and the dry air aloft at the air–solid interface is only permissible through molecular diffusion. For this reason, the molecular diffusion coefficient of water vapor in air, Dm, is included in the analysis
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