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Abstract
This work presents a numerical study of a skimming flow regime in a stepped spillway structure, which is a complex two-phase flow with self-aeration phenomenon. The numerical models employed consist in a hybrid approach capable of switching between interface resolving and interface modelling closures on a local level, and a state-of-the-art volume-of-fluid (VOF) method. A new switch criterion for the hybrid approach is proposed, allowing calculations on irregular grid cells. A detached eddy simulation (DES) turbulence model is selected in order to directly resolve the turbulent flow structures triggering the onset of self-aeration. The main flow properties computed in the non-aerated and aerated region are confronted with experimental measurements. Compared to the VOF method, the hybrid approach provides an overall better prediction of time-averaged air concentration, highlighting the positive outcome of modelling sub-grid interface structures.
Notation
| a1 | = | constant from k-ω SST model (–) |
| C | = | local time-averaged air concentration relative to water, also called void fraction (–) |
| Cα | = | coefficient of interface compression (–) |
| CL | = | cell edge length vector (m) |
| Cmean | = | depth-average air concentration defined in terms of Y90 (–) |
| d | = | water height normal to pseudo-bottom in the non-aerated region (m) |
| dc | = | critical depth (m) |
| dx | = | streamwise distance between probes (m) |
| dw | = | equivalent water height in the aerated region (m) |
| F* | = | roughness Froude number (–) |
| F1 | = | blending function from the k-ω SST turbulence model (–) |
| F2 | = | blending function from the k-ω SST turbulence model (–) |
| FDES | = | limiting function from the k-ω SST DES turbulence model (–) |
| Fs | = | source term for surface tension effort model (Pa m−1) |
| g | = | gravity acceleration (m s−2) |
| h | = | step height (m) |
| k | = | modelled turbulent kinetic energy (m² s−2) |
| L | = | step length (m) |
| Lt | = | turbulent length scale (m) |
| Lx | = | streamwise distance from the beginning of the chute (m) |
| M | = | source terms for interfacial momentum exchange models (Pa m−1) |
| n | = | power-law exponent (–) |
| ncellfaces | = | number of faces in each finite volume (–) |
| p | = | pressure (Pa) |
| = | production of k from the k-ω SST turbulence model (Pa s−1) | |
| qw | = | water discharge per unit width (m² s−1) |
| Rxx | = | interfacial auto-correlation function (–) |
| S | = | the invariant measure of the strain rate (s−1) |
| t | = | time (s) |
| Tt | = | lag time from cross-correlation between two signals (s) |
| Txx | = | auto-correlation time scale (s) |
| Tix | = | streamwise turbulence intensity (–) |
| Tiy | = | turbulence intensity normal to pseudo-bottom (–) |
| u | = | velocity field (m s−1) |
| U | = | streamwise velocity (m s−1) |
| u’ | = | streamwise velocity fluctuation (m s−1) |
| uc | = | compressive velocity (m s−1) |
| UI | = | interfacial streamwise velocity (m s−1) |
| Umax | = | maximum streamwise velocity (m s−1) |
| v’ | = | velocity fluctuation normal to pseudo-bottom (m s−1) |
| y | = | normal distance from the pseudo-bottom (m) |
| y+ | = | non-dimensional distance from the wall (–) |
| Y90 | = | distance from pseudo-bottom where air concentration equals 90% (m) |
| α | = | phase fraction function; air = 0; water = 1 (–) |
| β* | = | constant from k-ω SST model (–) |
| δ | = | boundary layer thickness (m) |
| ΔDES | = | filter width for the k-ω SST DES turbulence model (m) |
| ζ | = | time lag (s) |
| θ | = | slope of the stepped spillway structure (°) |
| μ | = | dynamic viscosity of the fluid (Pa s) |
| μt | = | turbulent viscosity (Pa s) |
| μM,eff | = | equivalent mixture viscosity (Pa s) |
| ρ | = | density of the fluid (kg m−3) |
| = | constant from k-ω SST model (–) | |
| = | constant from k-ω SST model (–) | |
| = | constant from k-ω SST model (–) | |
| = | normalized turbulent shear stress (–) | |
| ω | = | specific dissipation of turbulent kinetic energy (s−1) |
Subscripts
| 90 | = | value of variable where C = 90% |
| Φ | = | phase indicator |
| air | = | relative to phase air |
| M | = | mixture property |
| water | = | relative to phase water |