Production Terms Research Articles

Research on data assimilation approach of wind turbine airfoils in stall conditions

This study aims to analyze the applicability of the turbulence model constants obtained through data assimilation on various airfoils in stall conditions, thereby offering the potential for computational resource savings. Through the wind tunnel experiments and published test data at Reynolds numbers ranging from the order of 105 to 106, the ensemble Kalman filter method was proposed to optimize the constants of the SA (Spalart-Allmaras) model, and its efficacy was validated. The assimilated constants obtained from YA-21 and S809 airfoils were applied separately to other airfoils with similar separation degrees for comparative analysis of their assimilation effects. Based on this, the influence of each model constant on numerical simulation was explored, and the pressure distributions were compared before and after assimilation as well as on other airfoils. Additionally, the impact of variations in airfoil thickness and Reynolds number on assimilated results was investigated. The results suggest that the constants that impact assimilation outcomes appreciably under stall conditions pertain to production and diffusion terms. When the Reynolds numbers are on the order of 105, assimilated constants derived from the 21% thickness airfoil exhibited optimization effects on the NACA4415 and S809 airfoil, providing a more accurate depiction of separated flow over an airfoil than the original constants conditions. The optimization effect persisted when the Reynolds number reached the order of 106. As the primary factor in the production term, Cb1 became sensitive to changes in Reynolds number exceeding other constants. However, the applicability of thick airfoils is slightly degenerated. Thicker airfoils are more susceptible to changes in the constants of the production and diffusion terms, which makes the assimilated constants need to be applied with caution. These findings demonstrate the feasibility of the mentioned approach, suggesting that assimilated constants from a medium-thickness airfoil can be partially used to replace the self-assimilated constants of other airfoils.

Turbulence modulation in liquid–liquid two-phase Taylor–Couette turbulence

We investigate the coupling effects of the two-phase interface, viscosity ratio and density ratio of the dispersed phase to the continuous phase on the flow statistics in two-phase Taylor–Couette turbulence at a system Reynolds number of $6\times 10^3$ and a system Weber number of 10 using interface-resolved three-dimensional direct numerical simulations with the volume-of-fluid method. Our study focuses on four different scenarios: neutral droplets, low-viscosity droplets, light droplets and low-viscosity light droplets. We find that neutral droplets and low-viscosity droplets primarily contribute to drag enhancement through the two-phase interface, whereas light droplets reduce the system's drag by explicitly reducing Reynolds stress due to the density dependence of Reynolds stress. In addition, low-viscosity light droplets contribute to greater drag reduction by further reducing momentum transport near the inner cylinder and implicitly reducing Reynolds stress. While interfacial tension enhances turbulent kinetic energy (TKE) transport, drag enhancement is not strongly correlated with TKE transport for both neutral droplets and low-viscosity droplets. Light droplets primarily reduce the production term by diminishing Reynolds stress, whereas the density contrast between the phases boosts TKE transport near the inner wall. Therefore, the reduction in the dissipation rate is predominantly attributed to decreased turbulence production, causing drag reduction. For low-viscosity light droplets, the production term diminishes further, primarily due to their greater reduction in Reynolds stress, while reduced viscosity weakens the density difference's contribution to TKE transport near the inner cylinder, resulting in a more pronounced reduction in the dissipation rate and consequently stronger drag reduction. Our findings provide new insights into the physics of turbulence modulation by the dispersed phase in two-phase turbulence systems.

Comparative study on predicting turbulent kinetic energy budget using high-order upwind scheme and non-dissipative central scheme

The accurate computation of different turbulent statistics poses different requirements on numerical methods. In this paper, we investigate the capabilities of two representative numerical schemes in predicting mean velocities, Reynolds stress and budget of turbulent kinetic energy (TKE) in low Mach number flows. With concerns on numerical order of accuracy, dissipation and dispersion properties, a high-order upwind scheme with relatively good dispersion and dissipation and a second-order non-dissipative central scheme with perfect dissipation but poor dispersion are adopted for this comparative study. By carrying out a series of numerical simulations including Taylor-Green vortex, turbulent channel flow at Reτ=180\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{\ au }=180$$\\end{document} and turbulent flow over periodic hill at Reb=10595\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{b}=10595$$\\end{document}, it can be obtained that although the high-order upwind scheme lacks perfection of dissipation in high wave number range, it still demonstrates superior predictive capability compared with the second-order non-dissipative central scheme, especially with relatively coarse grids. Finally, by taking the high-order upwind scheme as a suitable selection for turbulence simulation, the turbulent flow over a 30P30N multi-element airfoil is investigated as an application study. After briefly comparing the simulated profiles and spectrum with reference experimental results as validation, the budget of TKE is analyzed to locate the dominant flow structures and regions. It is found that the production and dissipation terms behave in a “monopole” pattern in the locations with strong shears and wakes. Whereas the advection and diffusion terms show an “inward” pattern and an “outward” pattern, which indicate the spatial transport of TKE between the center of the shear layer and nearby locations.

Improving turbulence modeling for gas turbine blades: A novel approach to address flow transition and stagnation point anomalies

Accurate prediction of temperature and Heat Transfer Coefficient (HTC) distributions over gas turbine blades is crucial for the design process and life assessment of these components. Numerical studies of flow over gas turbine blades face significant challenges in accurately simulating two complex phenomena: (1) the transition of flow from laminar to turbulent, and (2) stagnation point flow at the leading edge. Many turbulence models tend to overpredict the temperature on turbine blades, leading to incorrect identification of hot-spot regions and, consequently, erroneous estimations of blade life. This paper investigates the performance of various turbulence models in simulating flow and heat transfer over gas turbine vanes. The study includes three full turbulence models, i.e., Spalart-Allmaras (SA), Shear Stress Transport k−ω (SST-kw), and v2−f (V2F), as well as two transitional models, i.e., Transition SST (Trans-SST) and k−kL−ω (k-kl-w). Simulation results indicate that the v2−f, Trans-SST, and k−kL−ω models can detect flow transition. However, the transition length and onset location predicted by the Trans-SST and k−kL−ω models do not align with experimental data. Conversely, the v2−f model suffers from over-predictions at the leading edge due to stagnation point anomaly. To address these issues and due to capacities of the V2F model, this study proposes two modifications to enhance the performance of the V2F model. First, the production term of turbulent kinetic energy is redefined to mitigate the stagnation point anomaly. Second, the model is recalibrated to improve the prediction of flow transition. The new model, named the Production Modified V2F (PMV2F) model, shows promising results in predicting temperature and heat transfer coefficients.

Numerical studies of hydrodynamic relativistic astrophysical jet: The modification of vorticity transport equation

In this Letter, we have carried out the two-dimensional numerical simulation of axisymmetric relativistic jet in cylindrical coordinates by employing higher order finite volume method in PLUTO [Mignone et al., “PLUTO: A numerical code for computational astrophysics,” Astrophys. J. Suppl. Ser. 170, 228 (2007)] solver. The modified vorticity transport equation has been proposed for relativistic flow by taking the curl of the momentum equation, which shows significant change in the baroclinic vorticity production term due to relativistic effect. Both mathematical analysis and numerical results show that the vorticity production term due to baroclinic torque is heavily influenced due to the presence of specific enthalpy gradient and square of Lorentz factor gradient in a relativistic fluid flow.

Floating particles transport through the free surface vortex technique: A novel numerical study to assess the interaction among different scales of vortex structures

Fat, oil, and grease (FOG) floating particles in the sump of sewage pumping stations will accumulate together to form rigid layers, resulting in failure for pump device. To overcome this, the free surface vortex (FSV) technique has been considered and applied to transport floating particles toward the submerged suction pump inlet. This paper investigates the potential of vortices as a means of downward motion of FOG. The entrainment capacity of FSV is investigated by numerical simulations using a coupled level-set and volume-of-fluid method. Two coherent structures are decomposed by proper orthogonal decomposition: FSV represented by the first two orders with high energy content and spiral vortex bands represented by low energy and high order models. The extracted ridges of the finite-time Lyapunov exponent (FTLE) delineate different regions of the flow field and effectively capture the evolution of Lagrangian coherent structures. The floating particles in the sump are first caught by the dividing line formed by the FTLE ridges, mixed in the entrainment zone, and then merged into the vortex. The enstrophy production term dominates the development of vorticity. Subject to the influence of flow velocity gradients, both radial and tangential vortices undergo a transition into axial vortices. This transformation enhances the vortex's capacity to entrain particles within the vortex core area, leading to their rapid inward spiraling toward the vortex center and eventual expulsion due to the vortex's entrainment effect.

Direct numerical simulation of fully-developed supersonic turbulent channel flows with dense vapors

This work aims to investigate the impact of the molecule-complexity effect and the non-ideal effect on wall-bounded turbulent flows by applying direct numerical simulation (DNS) to fully-developed channel flows of two typical organic vapors: R1233zd(E) and octamethyltrisiloxane (MDM). For each vapor, three thermodynamic states are analyzed: one in the dilute-gas region, one near the saturation line, and one in the supercritical region. For mean flow fields, it is found that, due to smaller Prandtl and Eckert numbers, both the molecule-complexity effect and the non-ideal effect reduce the mean temperature rise from the cold wall to the channel center. Meanwhile, the molecule-complexity effect weakens the mean density drop, while the non-ideal effect strengthens the drop. Furthermore, once the density and viscosity variations are considered, the mean streamwise velocity profiles of dense vapors are practically the same as the ideal gas. For turbulent fluctuations, it is found that the correlations between T′, p′, and ρ′ in dense vapors are more complicated than the ideal gas: for the ideal gas, fluctuations are dominated by “vorticity mode”; hence, ρ′ and T′ are strongly related to u′ but independent of p′; however, for dense vapors, “acoustic mode” can also play an important role. A newly derived equation illustrates that, through the “acoustic mode,” the molecule-complexity effect obviously enhances the positive correlation between ρ′ and p′, while the non-ideal effect can enhance the positive correlation between T′ and p′. Further analysis of instantaneous flow fields shows that p′ is isotropic. The isotropic character affects fluctuation magnitudes but has limited effect on the specified wall-direction turbulent transport. Consequently, Walz's equation and Reynolds analogy in terms of enthalpy are still valid. Finally, a comparison between the DNS energy budget and k equation of Reynolds-averaged Navier–Stokes (RANS) model has been carried out. Results show that obvious deviation happens on the production term in spite of the careful selection of eddy viscosity model.

Large-eddy simulation of shock train in convergent-divergent nozzles with isothermal walls

ABSTRACT High-order accurate Large-eddy simulations (LES) have been carried out for compressible, turbulent flow through two bell-shaped convergent-divergent nozzles- one with a short divergent portion and the other with a longer one. An explicit filtering approach based on approximate deconvolution method is used for the LES. The nozzles have isothermal walls and circular cross-sections. The incoming flow is a turbulent, subsonic fully developed pipe flow at centreline Mach number of around 0.4 and friction Reynolds number 216. An adaptive filter is used to capture the shocks appearing in the divergent portion of the nozzles. The complicated behaviour of the mean flow, Reynolds stresses, production and pressure-strain terms of the axial Reynolds stress budget in the shock train region is shown. The broadband nature of the shock train oscillations having multiple dominant frequencies in these nozzles is observed in the present LES.

On the comparison between phenomenological and kinetic theories of gas mixtures with applications to flocking

We study the comparison between the phenomenological and kinetic models for a mixture of gases from the viewpoint of collective dynamics. In the case in which constituents are Eulerian gases, balance equations for mass, momentum, and energy are the same in the main differential part, but production terms due to the interchanges between constituents are different. They coincide only when the thermal and mechanical diffusion are sufficiently small. In this paper, we first verify that both models satisfy the universal requirements of conservation laws of total mass, momentum, and energy, Galilean invariance and entropy principle. Following the work of Ha and Ruggeri (ARMA 2017), we consider spatially homogeneous models which correspond to the generalizations of the Cucker Smale model with thermal effect. In these circumstances, we provide analytical results for the comparison between two resulting models and also present several numerical simulations to complement analytical results.

A modified algebraic turbulent heat flux model for non-equilibrium and rotating flow and its application in film cooling simulation of a rotating turbine blade

Thermal fields and cooling flows over rotating turbine blades are usually a non-equilibrium anisotropic flow, and most turbulence models in RANS simulations do not provide acceptable accuracy. A modified model is proposed and assessed here. This is based on an explicit algebraic heat flux model that is a low-Reynolds-number version of the higher order generalized gradient diffusion hypothesis turbulent heat flux model. The presented model, called “Non-equilibrium and Sensible to Rotation Algebraic Heat Flux Model” or “NSR-AHFM” includes additional production terms added to the second moment closure of turbulent heat flux transport equation that considers system rotation and non-equilibrium effects in velocity field, while maintaining the frame invariance during transformation to a rotating coordinate system. The model was initially tested through a-priori analysis, comparing the model to corresponding quantities obtained from Direct Numerical Simulations, allowing for the identification of suitable model parameters. Subsequently, the proposed modified model was implemented in a CFD code to a representative configuration (i.e., rotating turbine blade) and results were compared against available experimental data from literature. The present model was undergone a comparative analysis with other similar models in both stationary and rotating channel flow scenarios. Also, the present model was applied to a film cooling simulation of a rotating turbine blade in different rotational speeds. The findings indicate that the present model exhibits superior performance when compared to other models. Moreover, its behavior aligns consistently with the underlying physics governing the flow dynamics of the rotating channel.

A novel nitrogenous coumarin: The promising antiviral therapy against white spot syndrome virus in the culture of shrimp seedlings

White spot disease (WSD), attributed to the white spot syndrome virus (WSSV), represents a substantial danger to the international shrimp sector, resulting in considerable losses in both production and economic terms. Nevertheless, there are currently no approved therapeutic measures to control the outbreak. This study involved the synthesis of 15 new compounds for the assessment of their anti-WSSV efficacy in Litopenaeus vannamei post-larvae. Among these compounds, N-(4-methyl-2-oxo-2H-chromen-7-yl)acetamide (P13) exhibited a maximum antiviral response of > 90 %, with a median effective concentration (EC50) < 12 mg/L, making P13 the most promising antiviral compound. In the cases where the larvae were simultaneously incubated with WSSV and P13, as well as where the larvae were first infected with WSSV and then treated with P13, the replication of WSSV was significantly inhibited. However, pre-treatment with P13 had minimal impact on WSSV multiplication, suggesting a potential therapeutic effect. Simultaneously, the cumulative mortality rate of WSSV-infected post-larvae decreased in the presence of P13. After the shrimp larvae were infected with WSSV, continuously changing the aquacultural water containing DMSO resulted in a mortality rate of 80 % at 120 hours post-infection (hpi); whereas continuously changing the aquaculture water containing P13 resulted in a mortality rate of 45 %. Given that P13 remains stable in farmed water for up to two days, it is recommended to regularly replace the medication to eliminate the virus threat. In conclusion, P13 demonstrates a positive anti-WSSV activity and holds potential for further development as an antiviral therapy against WSSV infection.

Long term relationships of electron density with solar activity

Greenhouse gases such as carbon dioxide and methane that are causing climate change may cause long term trends in the thermosphere and ionosphere. The paper aims to contribute to explore long term effects in the ionosphere focusing on the impact of solar activity changes. Peak electron density data derived from vertical sounding measurements covering 65 years at the ionosonde stations Juliusruh (JR055), Boulder (BC840) and Kokubunji (TO536), have been utilized to estimate the long-term behavior of daytime ionospheric F2 layer ionization in relation to the solar 10.7 cm radio flux index F10.7. In parallel, Global Navigation Satellite System (GNSS) based vertical total electron content (TEC) data over the ionosonde stations in combination with the peak electron density data have been used to derive the equivalent slab thickness τ for estimating long-term behavior in the time period 1996-2022. A new approach has been developed for deriving production and loss term proxies for studying long-term ionization effects from F2 layer peak electron density and TEC data. The derived coefficients allow estimating the long-term variation of atomic oxygen and molecular nitrogen concentrations including their ratio during winter months. The noon-time slab thickness values over Juliusruh correlate well with the decrease of F10.7 and the F2 layer peak height and enable estimating the neutral gas temperature. The equivalent slab thickness decreases by about 20 km per decade in the period 1996-2022, indicating a thermospheric cooling by about 100 K per decade for Juliusruh. Whereas the oxygen concentration decreases, the loss term, considered as a proxy for molecular components of the neutral gas, in particular N2, increases with the long-term solar activity variation. Considering 11 years averages of the production and loss terms under wintertime conditions, the long-term study reveals for the O/N2 ratio a percentage decrease of 5% per decade and for F10.7 about 3.1% per decade in a linear approach referred to the year 1970. Linear models of 11 years averaged NmF2 and foF2 from corresponding F10.7 show a very close correlation with the temporal variation of F10.7 until about 1990. The root mean square errors are in the order of 1.0 -1.3 ‧1010m-3 for NmF2 and 0.03-0.05 MHz for foF2. After 1990 the linear models clearly deviate from F10.7 at all selected ionosonde stations indicating a non-local effect.

Numerical analyses of water swirls in aquaculture tanks

Water swirls were observed in both model and prototype aquaculture tanks with bottom drains. This phenomenon was seldom replicated in studies of flow fields of water in aquaculture tanks with computational fluid dynamics. A commonality of these studies is that the linear eddy-viscosity turbulence models were employed. This study delves into the water swirls from the aspect of the generation of the turbulence kinetic energy; both the linear and nonlinear eddy-viscosity turbulence models are used, as well as a Reynolds stress turbulence model. A novel identification method is utilized to pinpoint vortex structures. Numerical results affirm the outstanding performance of the Reynolds stress model to estimate the flow fields of water in aquaculture tanks. Both the linear and nonlinear eddy-viscosity models overestimate the kinetic turbulence energy in the stagnation zone of swirls. The overestimation can be restrained by including the effect of the streamline curvature correction (SCC) to the turbulence production term. Even so, the SCC does not help to improve the estimating accuracy between the Reynolds stress tensor and the mean rate-of-strain tensor. While the nonlinear model offers some improvements, the Reynolds stress model provides a more effectvie solution. After that, we calculate the flow fields of water in aquaculture tanks with different inlet–outlet configurations with the Reynolds stress turbulence model. Both the vertical and horizontal scales of water swirls increase with inlet velocities and Froude numbers, where the Froude number is based on the bottom draining velocity and the water depth above the bottom drain. The vertical distributions of the water residence time in aquaculture tanks are shown to improve the design of tanks.

The effect of variations in experimental and computational fidelity on data assimilation approaches

AbstractWe conduct a comprehensive analysis of two data assimilation methods: the first utilizes the discrete adjoint approach with a correction applied to the production term of the turbulence transport equation, preserving the Boussinesq approximation. The second is a state observer method that implements a correction in the momentum equations alongside a turbulence model, both applied to fluid dynamics simulations. We investigate the impact of varying computational mesh resolutions and experimental data resolutions on the performance of these methods within the context of a periodic hill test case. Our findings reveal the distinct strengths and limitations of both methods, which successfully assimilate data to improve the accuracy of a RANS simulation. The performance of the variational model correction method is independent of input data and computational mesh resolutions. The state observer method, on the other hand, is sensitive to the resolution of the input data and CFD mesh.

An innovative modification to the Menter shear-stress transport turbulence model employing the symbolic regression approach

Drawing from the non-equilibrium link between the production Pk and dissipation ε of turbulent kinetic energy (TKE), we advocate for the introduction of a limiter to modulate the TKE production term within the Menter shear-stress transport (SST) model. The original SST model is made more sensitive to the adverse pressure gradient (APG) by Bradshaw's assumption. Bradshaw's assumption introduces the equilibrium condition Pk/ε = 1 in most regions of the turbulent boundary layer. In the APG flows with Pk≫ε, the equilibrium condition suppresses the magnitude of TKE (k) within the boundary layer, resulting in an early separation problem. To address this issue, we employ the symbolic regression (SR) to scrutinize the physical correlation between Pk/ε and local turbulence parameters, obtaining an approximate function FSR that encapsulates the relationship between Pk/ε, Sk/ε, and y+ in the APG flow. Following its incorporation into the original SST model in the form of a limiter, the FSR evolves into the SST-Symbolic Regression Evolution model. The SST-SRE is then applied to four cases with APGs. The modification leads to an increase in the skin-friction coefficient Cf in the APGs region and causes a downstream shift in the separation location, improving the consistency with high-accuracy data and experimental results. It is demonstrated that this correction can improve the early separation problem in the Menter SST turbulence model.

Moderately dense granular gas of inelastic rough spheres

A kinetic theory for moderately dense gases of inelastic and rough spherical molecules is developed from the Enskog equation where a macroscopic state is characterised by 29 scalar fields which correspond to the moments of the distribution function: mass density, hydrodynamic velocity, pressure tensor, absolute temperature, translational and rotational heat fluxes, hydrodynamic angular velocity and angular velocity flux. The balance equations for the 29 scalar fields are obtained from a transfer equation derived from the Enskog equation where the kinetic and potential parts of the new moments of the distribution function and production terms are calculated from Grad’s distribution function for the basic fields. The transition from the 29 field theory to an eight field theory—with mass density, hydrodynamic velocity, absolute temperature and hydrodynamic angular velocity—leads to the determination of the transport coefficients of the Navier–Stokes and Fourier laws. The transport coefficients are functions of the normal and tangential restitution coefficients and of the local equilibrium radial distribution function. The transport coefficients in the limiting case of elastic rough spheres is also determined.

Field inversion machine learning augmented turbulence modeling for time-accurate unsteady flow

Field inversion machine learning (FIML) has the advantages of model consistency and low data dependency and has been used to augment imperfect turbulence models. However, the solver-intrusive field inversion has a high entry bar, and existing FIML studies focused on improving only steady-state or time-averaged periodic flow predictions. To break this limit, this paper develops an open-source FIML framework for time-accurate unsteady flow, where both spatial and temporal variations of flow are of interest. We augment a Reynolds-Averaged Navier–Stokes (RANS) turbulence model's production term with a scalar field. We then integrate a neural network (NN) model into the flow solver to compute the above augmentation scalar field based on local flow features at each time step. Finally, we optimize the weights and biases of the built-in NN model to minimize the regulated spatial-temporal prediction error between the augmented flow solver and reference data. We consider the spatial-temporal evolution of unsteady flow over a 45° ramp and use only the surface pressure as the training data. The unsteady-FIML-trained model accurately predicts the spatial-temporal variations of unsteady flow fields. In addition, the trained model exhibits reasonably good prediction accuracy for various ramp angles, Reynolds numbers, and flow variables (e.g., velocity fields) that are not used in training, highlighting its generalizability. The FIML capability has been integrated into our open-source framework DAFoam. It has the potential to train more accurate RANS turbulence models for other unsteady flow phenomena, such as wind gust response, bubbly flow, and particle dispersion in the atmosphere.

A modification of production term for the shear stress transport turbulence model based on a new wall-law for adverse pressure gradient

This paper presents an attempt to improve the predictive accuracy of the Menter shear stress transport (SST) turbulence model for adverse pressure gradient (APG) flows. The foundation of this work lies in the newly modified law of the wall and turbulent kinetic energy (TKE) transport equation within the APG turbulent boundary layer. The modification of the law of the wall is achieved through a newly proposed variable λ, which is derived theoretically and calibrated using direct numerical simulation data. Based on the new law of the wall, we analyze the TKE transport equation of the SST model at APG and demonstrate that the analytical solution satisfying the law of the wall cannot achieve equilibrium in the current SST model. Further theoretical analysis reveals that this imbalance is caused by the TKE production term Pk, and the decomposition of the wall friction coefficient Cf indicates that Pk directly affects the prediction of flow separation. Based on the analysis and the new law of the wall, an additional term for Pk is suggested to correct Pk and improve the model's predictive accuracy for APG flows. Numerical validation results show that the correction leads to an increase in Pk within the APG turbulent boundary layer, resulting in more accurate predictions of Cf and the mean velocity profile in the inner layer of the turbulent boundary layer.

Numerical modeling of turbulent air-water mixture flows in spilling breaking waves

This paper presents a numerical study of turbulent air-water mixture flows under spilling breaking waves by using a mixture-flow model. The model is extended from the earlier 2D numerical framework NEWFLUME, which solves the Reynolds-averaged Navier-Stokes (RANS) equations for mean flow motions. The turbulence closure is accomplished by a modified k−ε two-equations model, in which the effect of entrained bubbles on turbulence production and dissipation is considered by incorporating additional buoyancy terms. A transient equation based on two-fluid formulation is solved for air bubble transport. The present model is validated by comparing the simulated free surface displacement, void fraction, mean velocity and turbulence kinetic energy against existing experimental data and the previous numerical results. The effects of the additional convective term in transient equation of void fraction and the bubble-induced turbulence term in turbulence transport equation are further explored. The results reveal that the additional convective term in the transient equation of void fraction plays an essential role of suppressing unphysical turbulence growth and bringing correct amount of air entrainment into water during wave breaking processes. The additional bubble-induced turbulence production and dissipation terms become effective after wave breaks when the void fraction has been appropriately modeled, and they can further reduce the simulated turbulence intensities in spilling breaking wave, rendering a better agreement with the experimental measurements.

Integral methods for friction decomposition and their extensions to rough-wall flows

A primary objective of integral methods, such as the momentum integral method, is to discern the physical processes contributing to skin friction. These methods encompass the momentum, kinetic energy and angular momentum integrals. This paper reformulates existing integrals based on the double-averaged Navier–Stokes equations, and extends their application to flows over rough walls. Our derivation yields distinct decompositions for the bottom-wall viscous friction coefficient, denoted as $C_S$ , and the roughness element drag coefficient $C_R$ . The decompositions comprise three terms: a viscous term, a turbulent term and a roughness (dispersive) term – regardless of the flow configuration, be it channel or boundary layer. Notably, when these integrals are evaluated for laminar flow scenarios, only the viscous term remains significant. In addition, we elucidate the spatial distributions of the terms within these decompositions. To demonstrate the practicality of our formulations, we apply them to analyse data from direct numerical simulations of turbulent half-channel flows. These flows feature aligned and staggered cubical roughness at various packing densities. Our analyses, based on kinetic-energy-oriented decompositions, reveal that when the surface coverage density $\lambda _p$ is small, the dominant terms within the decompositions are the viscous and turbulent terms. With increasing $\lambda _p$ , the viscous dissipation term decreases, while the turbulent production term increases and then decreases. These variations arise from a subdued near-wall cycle and the development of a shear layer at the height of the cubes.