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ABSTRACT
This review aims to improve the reliability of Froude modelling in fluid flows where both the Froude number and Reynolds number are a priori relevant. Two concepts may help to exclude significant Reynolds number scale effects under these conditions: (i) self-similarity and (ii) Reynolds number invariance. Both concepts relate herein to turbulent flows, thereby excluding self-similarity observed in laminar flows and in non-fluid phenomena. These two concepts are illustrated with a wide range of examples: (i) irrotational vortices, wakes, jets and plumes, shear-driven entrainment, high-velocity open channel flows, sediment transport and homogeneous isotropic turbulence; and (ii) tidal energy converters, complete mixing in contact tanks and gravity currents. The limitations of self-similarity and Reynolds number invariance are also highlighted. Many fluid phenomena with the limitations under which self-similarity and Reynolds number invariance are observed are summarized in tables, aimed at excluding significant Reynolds number scale effects in physical Froude-based models.
Acknowledgements
The author would like to thank Dr Maarten van Reeuwijk, for helpful discussions and critical comments on an earlier version of this article, and Matthew Kesseler for linguistic suggestions.
Notation
| b | = | buoyancy b = −g(ρ−ρ0)/ρ0 (m s−2) |
| B | = | constant depending on the jet exit conditions (–) |
| cP | = | mean power coefficient (–) |
| C | = | void fraction (volume of air per unit volume of air and water) (–) |
| CD | = | drag coefficient (–) |
| CK | = | universal constant CK = 1.5 (–) |
| d | = | cylinder, jet source, pipe, plume source, rotor, sphere and turbine diameter (m) |
| dg | = | grain diameter (m) |
| dh | = | hydraulic diameter (equivalent pipe diameter) (m) |
| Do | = | dimensionless constant for air–water skimming flow (–) |
| e | = | entrainment rate (m s−1) |
| E(κ) | = | energy spectrum (m2 s−2) |
| f | = | frequency (s−1) |
| Fi, f | = | (vector-)function with i = 1, 2 or absent (–) |
| F | = | Froude number (–) |
| g | = | gravitational acceleration (m s−2) |
| g′ | = | reduced gravity g′ = g((ρ−ρ0)/ρ0) (m s−2) |
| Gxx(f) | = | power spectra (m2 s−3) |
| h | = | water depth, mixed boundary layer depth (m) |
| hc | = | critical flow depth (m) |
| hf | = | height of current front (m) |
| h1 | = | clear water depth (m) |
| h5 | = | current height behind the front (m) |
| h90 | = | depth where C = 90% (m) |
| k | = | roughness height (m) |
| K | = | dimensionless integration constant (–) |
| l | = | size of region of an eddy (m) |
| L | = | characteristic length (m) |
| m | = | scaling exponent (–) |
| M0 | = | momentum flux at the jet source (m4 s−2) |
| N | = | buoyancy frequency N = (∂b/∂z)1/2 (s−1) |
| q | = | specific discharge (m2 s−1) |
| Q | = | discharge (m3 s−1) |
| r | = | radial coordinate (m) |
| r0 | = | reference position (m) |
| r | = | position vector r = (x, y, z) (m) |
| R | = | Reynolds number (–) |
| Ri | = | Richardson number Ri = hΔb/u*2 (–) |
| ℜ | = | real number (–) |
| t | = | time (s) |
| u | = | velocity (m s−1) |
| u | = | vector of a scaled parameter (various) |
| = | mean of the velocity fluctuations (m s−1) | |
| = | Reynolds stresses (m2 s−2) | |
| u* | = | shear velocity u* = −(τw/ρ0)1/2 (m s−1) |
| u0 | = | reference value (various) |
| u1 | = | velocity of current front (conveyor belt velocity) (m s−1) |
| u∞ | = | free stream velocity (m s−1) |
| v | = | velocity vector (m s−1) |
| = | azimuthal velocity (m s−1) | |
| V | = | characteristic velocity (m s−1) |
| w | = | channel width (m) |
| x | = | streamwise coordinate (m) |
| x0 | = | virtual origin (m) |
| y | = | cross-flow coordinate (m) |
| z | = | vertical coordinate (m) |
| Z90 | = | dimensionless distance normal to chute Z90 = z/h90 (–) |
| αe | = | entrainment coefficient (–) |
| χ | = | dimensionless parameter (–) |
| Δb | = | buoyancy jump at the top of the mixed layer (m s−2) |
| ΔQ | = | volume-flux defect (m3 s−1) |
| Δu | = | longitudinal velocity difference (m s−1) |
| Δuc | = | velocity defect on the centreline (m s−2) |
| Δy | = | thickness of wake generator in cross-flow direction, width of slot (m) |
| = | dissipation rate (m2 s−3) | |
| = | initial circulation (m2 s−1) | |
| η | = | self-similarity variable (–) |
| ηK | = | Kolmogorov microscale (m) |
| κ | = | wave number κ = 2π/l (m−1) |
| λ | = | scale factor (–) |
| λg | = | Taylor microscale (m) |
| ν | = | kinematic viscosity (m2 s−1) |
| ω | = | vorticity (s−1) |
| π | = | numerical constant (–) |
| ρ | = | fluid density (kg m−3) |
| τw | = | shear stress (kg m−1 s−2) |
| θ | = | momentum thickness (m), azimuth (°) |
Subscripts
| c | = | centreline |
| dim | = | dimensionless |
| EI | = | energy-inertial |
| ID | = | inertial-dissipation |
| K | = | Kolmogorov |
| M | = | model |
| P | = | prototype, power |
| θ | = | azimuthal |
| 0 | = | initial, reference |
ORCID
Valentin Heller http://orcid.org/0000-0001-7086-0098