As computational fluid dynamics (CFD) advances, entropy generation minimization based on CFD becomes attractive for optimizing complex heat-transfer systems. This optimization depends on the accuracy of CFD results, such that accurate turbulence models, such as elliptic relaxation or elliptic blending turbulence models, become important. The performance of a previously developed elliptic blending turbulence model (the $\mathrm{SST} k\text{–}\omega \text{–}\phi \text{–}\alpha$ model) to predict the rate of entropy generation in the fully developed turbulent circular tube flow with constant heat flux was studied to provide some guidelines for using this class of turbulence model to calculate entropy generation in complex systems. The flow and temperature fields were simulated by using a CFD package, and then the rate of entropy generation was calculated in post-processing. The analytical correlations and results of two popular turbulence models (the realizable k–ε and the shear stress transport (SST) k–ω models) were used as references to demonstrate the accuracy of the $\mathrm{SST} k\text{–}\omega \text{–}\phi \text{–}\alpha$ model. The findings indicate that the turbulent Prandtl number (Pr_t) influences the entropy generation rate due to heat-transfer irreversibility. Pr_t = 0.85 produces the best results for the $\mathrm{SST} k\text{–}\omega \text{–}\phi \text{–}\alpha$ model. For the realizable k–ε and SST k–ω models, Pr_t = 0.85 and Pr_t = 0.92 produce the best results, respectively. For the realizable k–ε and the SST k–ω models, the two methods used to predict the rate of entropy generation due to friction irreversibility produce the same results. However, for the $\mathrm{SST} k\text{–}\omega \text{–}\phi \text{–}\alpha$ model, the rates of entropy generation due to friction irreversibility predicted by the two methods are different. The difference at a Reynolds number of 100,000 is about 14%. The method that incorporates the effective turbulent viscosity should be used to predict the rate of entropy generation due to friction irreversibility for the $\mathrm{SST} k\text{–}\omega \text{–}\phi \text{–}\alpha$ model. Furthermore, when the temperature in the flow field changes dramatically, the temperature-dependent fluid properties must be considered.

On the basis of a previously developed elliptic blending turbulence model (SST–k–ω–φ–α model), a scale-adaptive simulation (SAS) model is developed by following Menter and Egorov’s SAS concept. An SAS source term, which is related to the ratio of the modeled turbulence scale to the von Kármán length scale, is introduced into the corresponding length-scale determining equation. The major motivation of this study is that the conventional unsteady Reynolds-averaged Navier–Stokes (URANS) models provide only large-scale unsteadiness. The introduction of the SAS term allows the proposed SAS model to dynamically adjust to resolved structures in a URANS framework because this term is sensitive to resolved fluctuations. The predictive capabilities of the proposed SAS model are demonstrated by computing the complex flow configurations in three cases with flow separation from curved surfaces, namely, three-dimensional (3D) diffuser flow, two-dimensional (2D) periodic hills flow, and 2D U-turn duct flow. For comparison, the results predicted by the SST–k–ω–φ–α model and the Menter and Egorov’s SAS model (SST–SAS) are provided. The results are also compared with the relevant experimental, direct numerical simulation, and large eddy simulation data. The results show that the SST–k–ω–φ–α model cannot capture the critical features for all three flows, and that the SST–SAS model is able to predict the results reasonably well. The proposed SAS model is capable of resolving more portions of the turbulence structures, and it yields the best results in all the cases.

Numerical simulations were conducted to investigate high-pressure water jets in air. The Eulerian multiphase model was employed as the computational framework. Through simulating a high-pressure water jet impinging on a flat plate, two turbulence treatment methodologies were initially examined, demonstrating that the mixture turbulence modeling approach exhibits superior predictive capability compared to the per-phase turbulence modeling approach. Subsequent analysis focused on evaluating turbulence model effects on the impact pressure distribution on the flat plate. The results obtained from the elliptic–blending turbulence model (the SST k-ω-φ-α model) and the other two industry-standard two-equation turbulence models (the realizable k-ε model and the SST k-ω model) were comparatively analyzed against experimental data. The analysis revealed that the SST k-ω-φ-α model demonstrates superior accuracy near the stagnation region. The effects of bubble diameter and surface tension were further examined. Quantitative analysis indicated that the impact pressure exhibits a decrease with decreasing bubble diameter until reaching a critical threshold, below which diameter variations exert negligible influence. Furthermore, surface tension effects were found to be insignificant for impact pressure predictions when the nozzle-to-plate distance was maintained below 100 nozzle diameters (100D). Simulations of free high-pressure water jets were performed to evaluate the model’s capability to predict long-distance jet dynamics. While the axial velocity profile showed satisfactory agreement with experimental measurements within 200D, discrepancies in water volume fraction prediction along the jet axis suggested limitations in phase interface modeling at extended propagation distances.

Using two different methods, a previously developed elliptic blending model (the original $\mathrm{STT} k\text{-}\omega \text{-}\phi \text{-}\alpha$ model) is modified for sensitization to streamline curvature. One method involves modifying the dissipation term in the turbulent dissipation equation, while the other constructs a new formulation for the turbulent kinetic energy production term based on an explicit algebraic stress model. The capabilities of the proposed models are evaluated by applying them to three flows with curved surfaces; namely, the two-dimensional (2D) infinite serpentine passage flow, the 2D U-turn duct flow, and the 2D periodic hill flow. The $\mathrm{STT} k\text{-}\omega$ model with rotation and curvature correction (the $\mathrm{STT} k\text{-}\omega \text{-}CC$ model) is also used for comparison. The computed results are compared with the relevant direct numerical simulation, experimental, and large eddy simulation data from the literature. It is found that the two proposed models significantly improve upon the original $\mathrm{STT} k\text{-}\omega \text{-}\phi \text{-}\alpha$ model. Compared with the $\mathrm{STT} k\text{-}\omega \text{-}CC$ model, the two proposed models produce better results in the 2D infinite serpentine passage flow and the 2D periodic hill flow. The proposed models are similarly competitive with the $\mathrm{STT} k\text{-}\omega \text{-}CC$ model in the 2D U-turn duct flow.

Accurate prediction of entropy generation in nanofluid turbulent convection is essential for optimizing thermal system efficiency, yet remains challenging due to complex near-wall phenomena and thermal property variations with temperature. This study applied an elliptic blending turbulence model (SST k-ω-φ-α) to numerically analyze entropy generation in alumina-water nanofluid flow through a uniformly heated circular tube. The model’s performance was validated using both experimental data and established heat transfer and fluid flow correlations at small wall-bulk temperature difference condition, and its superiority was rigorously evaluated against two widely adopted turbulence models (SST k-ω and realizable k-ε). Results show that the SST k-ω-φ-α model demonstrates superior accuracy, with maximum deviations from classical Bejan’s correlations of only 3.75% in heat transfer irreversibility and 1.86% in friction irreversibility, both of which are lower than the deviations of the comparative turbulence models. The SST k-ω-φ-α model was further used to analyze the entropy generation under large wall-bulk temperature difference condition. It was found that the failure of classical Bejan’s correlation is due to the dominant influence of wall temperature on property evaluations and entropy mechanisms. Under large wall-bulk temperature difference condition, total entropy generation varies non-monotonically with Reynolds number (Re), revealing an optimal flow rate that minimizes irreversibility. On the whole, the effect of nanoparticle concentration depends on flow regime, being beneficial at low Re but detrimental at high Re. This work underscores the necessity of employing advanced numerical tools which are capable of resolving detailed near-wall physics using high-fidelity turbulence modeling, especially under significant temperature gradients, for reliable performance prediction in advanced thermal systems using nanofluids.

Comprehensive investigations were conducted on numerical methods for simulating high–pressure water jet in air impacting on a flat plate. The primary focus was on the influence of numerical methods on the impact pressure exerted on the flat plate. Taking into account the principal characteristics of high–pressure water jets in air, the Eulerian multiphase model was selected as the foundational approach for simulating multiphase flows. Firstly, two turbulence treatment methods were investigated, revealing that the mixture turbulence model method yields superior results compared to the per–phase turbulence model method. Subsequently, the effects of turbulence models on the impact pressure were examined. Two two–equation turbulence models (the realizable k–ε model and the SST k–ω model), widely employed in engineering, and a four–equation turbulence model (the SST k–ω–φ–α model) were comparatively analyzed. It was observed that the SST k–ω–φ–α model yields more accurate results in the vicinity of the stagnation point. The influences of the inlet boundary condition were also explored. It was determined that the inlet turbulence condition derived from a hydraulic diameter–based empirical formula, as provided in the commercial CFD software ANSYS Fluent, is unsuitable for numerical simulations of high–pressure water jets in air. Subsequently, the influence of bubble diameters in the secondary phase (air) was scrutinized. Results indicated that as the bubble diameter diminishes, the impact pressure likewise decreases. However, once the bubble diameter reaches a certain size, it exerts almost no effect on the impact pressure. Finally, it was also verified that surface tension has no effect on the impact pressure of high–pressure water jets in air when the nozzle exit is not so far from the flat plate.