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

Abstract

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

Hydrology without dimensions

“…In fact, the regular behavior of ϕ(C I , m) reminds of the behavior of the Darcy friction factor plotted in the Moody diagram for pipe flow (Munson et al, 2013). For a detailed description of dimensional analysis in turbulence, see Barenblatt (1996) and Katul et al (2019).…”

“…It is also useful to note that in Bonetti et al (2020) the Pi theorem was applied to (31) considering l y , U , and δ as repeating variables, which instead leads to Π δ = ϕ δ (C I ), with Π δ = Π CR C I and ϕ δ (C I ) = ψ CR (C I )C −1 I . In the turbulence analogy, the present analysis corresponds to choosing viscosity instead of density (which is the typical choice; see Katul et al (2019)) as one of the repeating variables in the physical law for the wall-shear stress. Thus, for the wall-sear stress, τ = µΣ * , where µ is the dynamic viscosity and Σ * is the slope of the streamwise velocity profile at the wall, the physical law is τ = f τ (V, L, ρ, µ, ), where ρ is the density, V the mean velocity, L the characteristic lateral dimension, and the roughness height.…”

“…As observed by Hooshyar et al (2020), the presence of a mean logarithmic profile of elevation at an intermediate distance from the domain boundaries is similar to the classic results for the turbulent velocity profile (see also Barenblatt, 1996). In wallbounded turbulence (Katul et al, 2019), the logarithmic profile for the mean velocity profile is obtained from the Pi theorem applied to the physical law for the mean gradient, dū dy = f (y, u * , ρ, L, µ, ), where u * = τ ρ is the friction velocity (the other quantities have the same meaning as in the previous Subsection) and then assuming complete self-similarity with respect to both the bulk and local Reynolds numbers. From a phenomenological point of view, the analogy between the two phenomena is found in the resemblance between progressive penetration to smaller scales of ridges and valleys into the landscape and the intensification of vorticity producing smaller and smaller vortices (i.e., smaller Kolmogorov scales).…”

“…In his 1638 'Dialogues Concerning Two New Sciences' (Galilei, 1914), he deduced that geometrically similar objects are not equally strong under the their own weight: "A small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size". Since this discovery of scaling laws for complex biological materials, dimensional analysis has continued to fascinate many scientists, from Fourier to Maxwell, Reynolds, Rayleigh, Kolmogorov and Taylor, and contributed to numerous new results in several fields (e.g., Barenblatt, 1996;Szirtes, 2007;Bolster et al, 2011;Katul et al, 2019). These methods have been extremely useful in complex problems lacking closed form solutions due to nonlinear interactions and multi-physics as well as in design of experiments and numerical simulations and the rational interpretation of their results.…”

“…Looking at some of the most striking applications of the Pi theorem and self-similarity (e.g., the famous example of the atomic bomb of Taylor (Barenblatt, 1996) and the Kolmogorov spectrum of turbulence (Katul et al, 2019)), it is easy to be lured by the promise of a mathematical structure offering a powerful dimension reduction in the space of variables and parameters, even for vaguely formulated problems. In spite of the straightforward steps for its application, a brute force approach to dimensional analysis rarely leads to useful results.…”