The present work investigates the causes of the circular hydraulic jump for both low- and high-viscosity liquids in an effort to address a recent dispute in the research community. We first validate our numerical model against existing experiments and then study the effects of different parameters involved in the problem. The influences of viscosity, gravity, and surface tension on the formation of the jump are comprehensively explored. We observe a significant difference in the mechanisms behind the hydraulic jump for low- and high-viscosity liquids, which have rarely been reported. Surface tension is found to be responsible for the low-viscosity jump, while gravity dominates the high-viscosity jump, which partially resolves the recent noise regarding the cause of the jump in a consistent manner.

The transient flow of a circular liquid jet impinging on a horizontal disk, and the hydraulic jump formation, are examined theoretically and numerically. The interplay between inertia and gravity is particularly emphasised. The flow is governed by the thin-film equations, which are solved along with a force balance across the jump. The unsteadiness of the flow is caused by a linearly accelerating jet from an initial to a final steady state. To validate the predicted boundary-layer flow evolution, an analytical development is conducted for small distance from impingement, and for small time. In addition, the predictions of the film profile and jump location are compared against numerical simulation for the transient flow, and are further validated against experiment for steady flow. The evolutions of the film thickness and the wall shear stress in the developing boundary-layer region are found to be similar to those reported for a fluid lying on a stretching surface. The flow responds to the jet acceleration quasi-steadily near impingement but exhibits a long-term transient behaviour near the jump. Analysis of the jump evolution is considered in the range 5 < Fr < 40 for the Froude number (based on the jet radius and velocity). For Fr < 10, the jump reaches the final state instantly when the jet acceleration ceases. At higher Froude number, the jump settles at a later time, exhibiting an overshoot in the thickness due to the dominance of inertia.

Density-unweighted methods in large-eddy simulations (LES) of turbulence have received little attention, and the modeling of unclosed terms using density-unweighted methods even less. We investigate the density-unweighted subgrid-scale (SGS) closure problem for LES of decaying compressible isotropic turbulence at initial turbulent Mach numbers 0.4 and 0.8. Compared to the LES with Favre (density-weighted) filtering, there are more unclosed SGS terms for density-unweighted LES, which can be reconstructed using different SGS models, including the gradient model (GM), approximate deconvolution model (ADM), dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), and the dynamic iterative approximate deconvolution (DIAD) models proposed by Yuan et al. “Dynamic iterative approximate deconvolution models for large-eddy simulation of turbulence,” Phys. Fluids 33, 085125 (2021). We derive GM models suitable for density-unweighted methods. We also, for the first time, apply the DIAD model to investigate compressible turbulence. In the a priori tests, the correlation coefficients of the GM, ADM, and DIAD models are larger than 0.9. Particularly, the correlation coefficients of DIAD models exceed 0.98 and the relative errors are below 0.2, which is superior to that in other SGS models. In the a posteriori tests of the density-unweighted LES, the DIAD model shows great advantages over other SGS models (including GM, ADM, DSM, and DMM models) in predicting the various statistics and structures of compressible turbulence, including the velocity spectrum, probability density functions (PDFs) of SGS fluxes and the instantaneous spatial structures of SGS heat flux, SGS kinetic energy flux, and vorticity.

Existing works have shown that the small-scale errors of turbulence can be completely eliminated through data assimilation (DA), provided that all the large-scale Fourier modes below a critical wavenumber $k_c \approx 0.2 \eta^{-1}$ are continuously enforced, where $\eta$ is the Kolmogorov length scale. Here, we further explore the DA-based small-scale reconstruction problem for which the large-scale data is insufficient. Under such conditions, an unexpected artificial jump in the energy spectrum is observed. To alleviate this issue and improve the reconstruction accuracy, several approaches have been attempted, including ensemble averaged assimilation, temporally sparse data assimilation (TSDA) and filtering the penalty term in the assimilation. It is shown that ensemble averaging can tangibly reduce the reconstruction error, but the resulted energy spectrum is invariably lower than the true spectrum; TSDA can effectively remove the jump in the energy spectrum, but the reduction of the reconstruction error is limited; Filtering the penalty term can also rectify the energy spectrum, but it makes the reconstruction error larger. Based on these observations, we re-scale the ensemble averaged solution according to the rectified energy spectrum. Both the energy spectrum and the small-scale reconstruction accuracy have been improved by the re-scaled ensemble average (RES-EA) method. Further, we also test the current approach in the spatial nudging-based reconstruction of turbulence. Again, enhanced predictions are obtained for both the energy spectrum and the instantaneous turbulent field, invariably demonstrating the effectiveness and robustness of the proposed method.