Reynolds Normal Stresses Research Articles

Effect of an incoming Gaussian wave packet on underlying turbulence

We present a simulation-based study of the effect of a passing wave packet on underlying fully developed turbulence. We propose a novel wave-phase-resolved simulation method inspired by Helmholtz decomposition to directly couple the turbulence simulation with instantaneous wave orbital motions without wave-phase averaging. We also introduce a boundary condition treatment for the turbulence at the wave surface, which allows the turbulence simulation to be conducted in a rectangular domain while retaining the wave-phase effect. The results obtained from the proposed method reveal considerable variations in turbulence statistics, including the enstrophy and Reynolds normal stresses, during wave packet passage. Most changes occur rapidly when the narrow bandwidth around the wave packet core passes. Further analyses of the energy spectra indicate that the enhancement of turbulence occurs across a wide range of scales, with the near-surface small-scale motions experiencing the most significant intensification. Meanwhile, large-scale motions with scales comparable to the boundary layer depth are also enhanced. The mechanisms underlying the Reynolds normal stress variation at different length scales are related to the energy transfer from the wave orbital straining to turbulence through production, the pressure–strain effect, the pressure diffusion and the wave advection. By assessing the turbulence statistics and dynamics impacted by a wave packet in detail, this study provides an improved understanding of the response of a developed turbulent flow to a transient wave field. The proposed simulation method also proves to be a promising phase-resolved approach for efficiently modelling the wave effect on turbulence.

Separation and reattachment fixed Reynolds-stress model for iced airfoil and wing simulations

Differential Reynolds-stress models (RSMs) are the higher level of Reynolds-averaged Navier-Stokes (RANS) modeling and generally considered to have more potential than eddy viscosity models (EVMs) when applied to separated flows. Nevertheless, some peculiarities have been observed in simulating flows over iced wings. The first is the numerical stability problem caused by non-streamlined wall surfaces. To weaken this, a wall-boundary-natural scale λ=ω−1/8 is introduced into SSG/LRR RSM in this paper. The second is to remedy the problem of the unsatisfactory non-equilibrium characteristic in separating shear layer (SSL), a correction determined by anisotropy of Reynolds normal stresses is developed. The last focuses on the unphysical backbending of the streamlines near stagnation/reattachment (SR) points. Since the SSL correction will worsen this defect, a SR correction is carried out to blended with it. Four two-dimensional iced airfoils and a three-dimensional iced wing are simulated and then the above measures are confirmed to be the positive.

Scaling of coherent structures in compressible wall-bounded turbulence

Semi-local scales have been widely used in compressible wall-bounded turbulence, but it is still unclear whether they are applicable to the scaling of coherent structures, especially under conditions of high Mach number and cold wall temperature. By scrutinizing the direct numerical simulation dataset at different Mach numbers and wall temperatures, this paper demonstrates that the coherent structures normalized by semi-local scales are universal in size. In addition to this, we find that the ratios of Kolmogorov scales to semi-local scales are independent of Mach number and wall temperature. Thus, Kolmogorov scales can achieve the same scaling effect as the semi-local scales. The velocity spectra are also compared to verify the current scaling method quantitatively. A method to determine the threshold for the vortex identification criterion is proposed, allowing the same threshold for different cases to obtain vortices of similar size. The scaling of other statistics including turbulent kinetic energy, streamwise Reynolds normal stress, and root mean square of fluctuating vorticity is also investigated. A new velocity scale is proposed based on the total-stress-based transformation for mean streamwise velocity, which can collapse the profiles of these statistics more accurately than the semi-local velocity scale. The present paper demonstrates that through appropriate normalization, the structures and statistics of compressible turbulence become universal, reaffirming the validity of Morkovin's hypothesis even for the present high Mach number and cold wall cases.

An implicit large-eddy simulation perspective on the flow over periodic hills

The periodic hills simulation case is a well-established benchmark for computational fluid dynamics solvers due to its complex features derived from the separation of a turbulent flow from a curved surface. We study the case with the open-source implicit large-eddy simulation (ILES) software Lethe. Lethe solves the incompressible Navier–Stokes equations by applying a stabilized continuous finite element discretization. The results are validated by comparison to experimental and computational data available in the literature for Re = 5600. We study the effect of the time step, averaging time, and global mesh refinement. The ILES approach shows good accuracy for average velocities and Reynolds stresses using less degrees of freedom than the reference numerical solution. The time step has a greater effect on the accuracy when using coarser meshes, while for fine meshes the results are rapidly time-step independent when using an implicit time-stepping approach. A good prediction of the reattachment point is obtained with several meshes and this value approaches the experimental benchmark value as the mesh is refined. We also run simulations at Reynolds equal to 10600 and 37000 and observe promising results for the ILES approach.

The Role of Turbulence in an Intense Tropical Cyclone: Momentum Diffusion, Eddy Viscosities, and Mixing Lengths

Abstract Improved representation of turbulent processes in numerical models of tropical cyclones (TCs) is expected to improve intensity forecasts. To this end, the authors use a large-eddy simulation (with 31-m horizontal grid spacing) of an idealized category 5 TC to understand the role of turbulent processes in the inner core of TCs and their role on the mean intensity. Azimuthally and temporally averaged budgets of the momentum fields show that TC turbulence acts to weaken the maximum tangential velocity, diminish the strength of radial inflow into the eye, and suppress the magnitude of the mean eyewall updraft. Turbulent flux divergences in both the vertical and radial directions are shown to influence the TC mean wind field, with the vertical being dominant in most of the inflowing boundary layer and the eyewall (analogous to traditional atmospheric boundary layer flows), while the radial becomes important only in the eyewall. The validity of the downgradient eddy viscosity hypothesis is largely confirmed for mean velocity fields, except in narrow regions which generally correspond to weak gradients of the mean fields, as well as a narrow region in the eye. This study also provides guidance for values of effective eddy viscosities and vertical mixing length in the most turbulent regions of intense TCs, which have rarely been measured observationally. A generalized formulation of effective eddy viscosity (including the Reynolds normal stresses) is presented. Significance Statement This study uses a turbulence-resolving simulation of a category 5 tropical cyclone to understand the role of turbulence in intense storms. Results show that turbulence clearly modulates storm structure and intensity. This study provides guidance for the values of turbulent quantities (which are usually parameterized in comparatively coarse operational TC forecast models) in scarcely observed regions of intense storms. Furthermore, a complete formulation of the effective eddy viscosities is proposed, incorporating contributions from typically ignored Reynolds normal stress terms.

Hypersonic turbulent boundary layer over the windward side of a lifting body

In the present study, we performed direct numerical simulations for a hypersonic turbulent boundary layer over the windward side of a lifting body, the HyTRV model, at Mach number $6$ and attack angle 2 $^{\circ }$ to investigate the global and local turbulent features, and evaluate its difference from canonical turbulent boundary layers. By scrutinizing the instantaneous and averaged flow fields, we found that the transverse curvature on the windward side of the HyTRV model induces the transverse opposing pressure gradients that push the flow on both sides towards the windward symmetry plane, yielding significant effects of the azimuthal inhomogeneity and large-scale cross-stream circulations, moderate and azimuthal independent influences of adverse pressure gradient, and negligible impact of the mean flow three-dimensionality. Further inspecting the local turbulent statistics, we identified that the mean and fluctuating velocity become increasingly similar to the highly decelerated turbulent boundary layers over flat plates in that the mean velocity deficit is enhanced, and the outer layer Reynolds stresses are amplified as it approaches the windward symmetry plane, and prove to be self-similar under the scaling of Wei & Knopp (J. Fluid Mech., vol. 958, 2023, A9) for adverse-pressure-gradient turbulent boundary layers. Conditionally averaged Reynolds stresses based on strong sweeping and ejection events demonstrated that the Kelvin–Helmholtz instability of the strong embedded shear layer induced by the large-scale cross-stream circulations is responsible for the turbulence amplification in the outer layer. The strong Reynolds analogy that relates the mean velocity and temperature was refined to incorporate the non-canonical effects, showing considerable improvements in the accuracy of such a formula. On the other hand, the temperature fluctuations are still transported passively, as indicated by their resemblance to the velocity. The conclusions obtained in the present study provide potentially profitable information for turbulent modelling modification for the accurate predictions of skin friction and wall heat transfer.

On the spatio-temporal dynamics of cavitating turbulent shear flow over a microscale backward-facing step: A numerical study

The influence of cavitation on the mean characteristics and unsteady behavior of turbulent separated flows was comprehensively investigated in this study over a microscale backward-facing configuration at the Reynolds number (ReD) of 7440. The computational approach took both compressibility and finite mass transfer (Thermodynamic non-equilibrium) into account, to accurately capture the effects of shock waves, as well as to capture baroclinic phenomena on vortex dynamics within the turbulent separated flow. The compressibility effects were handled by using appropriate equation of states for each phase and for the mixture. Phase-change was considered through a transport equation for the vapor volume fraction, allowing for finite mass transfer contributions. Additionally, a wall adaptive large eddy simulation (LES) approach was utilized for simulating turbulent structures and their effects. The findings reveal that vapor development diminishes the mean growth rate of the shear layer and delays its reattachment to a longer distance from the step. Moreover, analysis of Reynolds normal and shear stresses, as well as the root mean square (RMS) of pressure fluctuations, demonstrates that the formation and collapse of vapor packets significantly influence turbulence decay and production in the second half of the shear layer and reattachment. It was also observed that both mean pressure and pressure fluctuations increased in vicinity of the reattachment region when cavitation was present, which was attributed to the condensation and collapse events. Spectral analysis further indicates the emergence of two dominant low frequency modes, linked to the displacement of the reattachment point. In the presence of cavitation, the frequencies associated with dominant Power Spectral Densities (PSDs) were smaller than those in the absence of cavitation. Additionally, each of these low frequencies corresponded to a specific vapor transport mechanism within the Turbulent Separation Bubble (TSB). Furthermore, it is shown that cavitation leads to a significantly higher spectral energy of high frequency fluctuations within the reattachment region in comparison to the condition where cavitation is absent. This can be attributed to the frequent collapse of bubbles in this region. At the end, we employed Spectral Proper Orthogonal Decomposition (SPOD) for modal analysis. This method offers valuable insights into the coherent structures and associated frequencies that arise in both the presence and absence of cavitation, which provides a deeper understanding of the effect of cavitation on the coherent structures and their dynamics.

Effect of Tip Gap Size on the Tip Flow Structure and Turbulence Generation in a Low Reynolds Number Compressor Cascade

Abstract Efficient and compact axial compressors are currently undergoing rapid development for use in microcooling systems and small-scale vehicles. Limited experimental work concentrates on the inner flow field of the compressors working at such low Reynolds numbers (Re∼104). This study examines the vortical structures and the resulting turbulence production in the transitional flow over a C4 compressor blade at a Reynolds number Re of 24,000, with a specific focus on the impact of tip clearance. The particle image velocimetry measurements reveal the tip flow structures in detail, including the tip leakage vortex (TLV) and its induced complex vortical structures. The tip secondary flow at the low Reynolds number can be divided as the tip leakage flow (TLF)/vortex and transitional boundary layer both at the end walls and the blade surfaces. The TLV propagates at the highest spanwise positions and farthest pitchwise positions at the middle tip gap size (τ/C = 3%) for the three tip gap sizes investigated. The tip flow fluctuations decrease from τ/C = 5% to τ/C = 3% and then increase from τ/C = 3% to τ/C = 1%. The spatial distribution, streamwise evolution, and individual Reynolds normal stress components contributing to the turbulent kinetic energy (TKE) are discussed. The primary contributors to the turbulence generation are examined to elucidate the flow mechanism leading to the distinct anisotropic turbulence structure in the tip region with various tip gap sizes.

Wave-phase dependence of Reynolds shear stress in the wake of fixed-bottom offshore wind turbine via quadrant analysis

There has been an increase in recognition of the important role that the boundary layer turbulent flow structure has on wake recovery and concomitant wind farm efficiency. Most research thus far has focused on onshore wind farms, in which the ground surface is static. With the expected growth of offshore wind farms, there is increased interest in turbulent flow structures above wavy, moving surfaces and their effects on offshore wind farms. In this study, experiments are performed to analyze the turbulent structure above the waves in the wake of a fixed-bottom model wind farm, with special emphasis on the conditional averaged Reynolds stresses, using a quadrant analysis. Phase-averaged profiles show a correlation between the Reynolds shear stresses and the curvature of the waves. Using a quadrant analysis, Reynolds stress dependence on the wave phase is observed in the phase-dependent vertical position of the turbulence events. This trend is primarily seen in quadrants 1 and 3 (correlated outward and inward interactions). Quantification of the correlation between the Reynolds shear stress events and the surface waves provides insight into the turbulent flow mechanisms that influence wake recovery throughout the wake region and should be taken into consideration in wind turbine operation and placement.

Computational study of thermal and fluid mixing behaviour of a dual jet flow using LES approach

The primary goal of any thermal system which demands efficient cooling is to improve heat transmission. For a turbulent jet, this condition becomes far more significant. The present work has employed large-eddy simulation (LES) for an offset ratio of 2.0 to investigate the transfer in heat and flow characteristics of a dual jet. An isothermal boundary wall condition is considered for a Reynolds number of 7500. The present findings are evaluated by comparing with the published experimental and numerical studies on wall jets. A comparative analysis here exhibits the profiles of Reynolds normal stresses, Reynolds shear stresses, turbulence kinetic energy, Nusselt number and turbulent heat fluxes. A comprehensive investigation of the mixing and flow properties of the dual jet is presented by using the Q-criterion of the iso-surface for mean velocity, pressure and vorticity. The present outcomes show that an efficient thermal transmission from the wall occurs in the recirculation region. The magnitudes of the profiles of Reynolds normal stresses, Reynolds shear stresses, r.m.s. temperature and turbulent heat fluxes are observed to be low in the combined region. Roller-like structure formation and near-wall eddies are noticed which spread in the combined region indicating large heat loss due to the turbulent flow.

Pipe to annular flow conversion: Numerical study

Large Eddy Simulation (LES) of turbulent flow at the entrance to an annular pipe section has been performed. The flow over a single circular rod, coaxially immersed in a circular pipe is considered. The Reynolds number, based on mean velocity and hydraulic diameter of the annular section, is ReDh=40530. The study aims to gain insights into the turbulence characteristics of the primary and secondary separation bubbles, developing on a side surface of the rod. Key findings include tiny toroidal vortices in the primary bubble's front, forming a vortex train with intermittently shed vortex structures. Large-scale helical stripes have been reproduced at the entrance to the annular section. Tilting of the axially-oriented streaks was observed close to the rod surface. It resulted in a slight anisotropy of the normal Reynolds stresses. The LES data could refine classic Reynolds-averaged Navier-Stokes-based turbulence models.

Consistent outer scaling and analysis of adverse pressure gradient turbulent boundary layers

Under adverse pressure gradient (APG) conditions, the outer regions of turbulent boundary layers (TBLs) are characterized by an increased velocity defect $U_{e}-U$ , an outwards shift of the peak value of the Reynolds shear stress $-\langle uv\rangle$ and an appearance of the outer peak value of the Reynolds normal stress $\langle uu\rangle$ . Here $U_{e}$ is the TBL edge velocity. Scaling APG TBLs is challenging due to the non-equilibrium effects caused by changes in the APG. To address this, the response distance of TBLs to non-equilibrium conditions is utilized to extend the Zagarola–Smits scaling $U_{zs} = U_{e}({\delta ^{*} }/{\delta })$ and ensure that the original properties of the Zagarola–Smits scaling are maintained as $Re \to \infty$ . Here $\delta ^{*}$ is the displacement thickness and $\delta$ is the boundary layer thickness. Based on the established correlation between $U_{e}-U$ and $-\langle uv\rangle$ , the scaling is extended to $-\langle uv\rangle$ . Furthermore, considering the coupling relationship between Reynolds stress components, the scaling is extended to encompass each Reynolds stress component. The proposed consistent scaling is verified using five non-equilibrium databases and five near-equilibrium databases, successfully collapsing the data of the TBL outer region. The pressure gradient parameter $\beta =({\delta ^{*} }/{\rho u_{\tau }^{2} }) ({\mathrm {d} P_{e} }/{\mathrm {d}\kern0.7pt x})$ of these databases spans two orders of magnitude. Here $P_{e}$ is the boundary layer edge pressure, $u_{\tau }$ is the friction velocity and $\rho$ is the density. Finally, the influence of the APG on the inner and outer regions of TBLs is analysed using the mean momentum balance equation. The analysis suggests that the shift of the $-\langle uv\rangle$ peak to the outer region under APG conditions is due to an insufficient inertia term near the inner region to balance the APG. It is observed that the APG promotes interaction between the inner and outer regions of TBLs, but the inner and outer regions still retain distinctive properties.

Numerical study of mixing vane spacer grid effect on subchannel mixing in a 5 × 5 rod bundle

Turbulent flow in rod bundle with spacer grid without mixing vanes and with mixing vanes of inclination angles from 10° to 30° was investigated by Reynolds Stress Transportation (RST) model under Reynolds numbers (Re) from 3.96 × 103 to 3.96 × 105. The predicted results were validated by experimental data of lateral velocity field in 5 × 5 rod bundle. The cross-flow velocity was relatively well predicted against experimental data, while lateral Reynolds normal stress was underpredicted near spacer grid and overpredicted far from spacer grid. The mixing vanes in one sub-channel generate high pressure region at the upstream side of mixing vane and low-pressure region at the downstream side of mixing vane, forming secondary flow structures. The lateral velocity increases with inclination angle and Re increasing. The lateral Reynolds stress increases with inclination angle increasing and decreases with Re increasing. The anisotropic turbulent states were studied by Lumley triangle method. The mixing vanes are the main factor dominating the turbulent states in rod bundle. The space grid resistance follows decreasing power function of Re and increases with inclination angle increasing. The secondary flow intensity follows an exponentially decaying function of distance from spacer grid and increases with inclination angle and Re increasing. The turbulent mixing coefficient model is an exponentially decreasing function of the distance from spacer grid, and it increases with inclination angle increasing and decreases with Re increasing. This paper studied the inclination angle and Re effect on the sub-channel mixing in a relatively large range of inclination angle and Re, improving the understandings of the sub-channel mixing phenomena.

A numerical investigation of momentum flux and kinetic energy transfers between turbulent wind and propagating waves

This paper focuses on simulating turbulent flow over propagating waves by solving the full Navier–Stokes equations in a moving frame. A careful comparison of flow statistics with previous experimental and numerical results demonstrates, to some extent, the rationality of simplifying wind waves as turbulent flow over moving wave boundaries. The phase-averaging method is then applied to investigate the momentum and energy transfers between turbulent wind and waves propagating at slow, intermediate and fast speeds. The results suggest that the dominant mechanism for producing Reynolds shear stress (RSS) and turbulent kinetic energy (TKE) is related to the wave age. Slow waves produce RSS and TKE similar to a two-dimensional shear turbulence. However, a fast wave enhances the streamwise Reynolds normal stress, the windward side's negative RSS and the gradient of both streamwise and vertical velocities, leading to additional RSS and TKE productions that can be ignored under the slow wave regimes. A strengthening wave–turbulence exchange is also found for fast waves. The intermediate wave can be regarded as a transitional condition determining this change.

Fluid Flow and Effect of Turbulence Model on Large-Sized Triple-Offset Butterfly Valve

The performance a valve has been frequently estimated with numerical methods owing to limitations such as cost and place. In this study, for the triple-offset butterfly valves, the different sizes in various disc-opening cases was numerically conducted using different turbulence models of the two-equation turbulence models of k–ε, k-ω, and Reynolds stress model. The numerical calculations were validated against experimentally obtained valve flow test results. The numerical effect with the different turbulence models were analyzed with respect to the disc-opening cases. From the numerical analysis, the Reynolds stress model exhibits the most pronounced turbulence effects among the various turbulence models showing higher value of Reynolds normal stress near the valve disc region. The sensitivity of the turbulence model constants was examined using the 300 mm valve to observe the sensitivity of the turbulence model parameters in the two-equation turbulence models.

Compressibility effects in supersonic and hypersonic turbulent boundary layers subject to wall disturbances

In the present study, we investigate the compressibility effects in supersonic and hypersonic turbulent boundary layers under the influence of wall disturbances by exploiting direct numerical simulation databases at Mach numbers up to 6. Such wall disturbances enforce extra Reynolds shear stress on the wall and induce mean streamline curvature in rough wall turbulence that leads to the intensification of turbulent motions in the outer region. The turbulent and fluctuating Mach numbers, the density and the velocity divergence fluctuation intensities suggest that the compressibility effects are enhanced by the increment of the free-stream Mach number and the implementation of the wall disturbances. The differences between the Reynolds and Favre average due to the density fluctuations constitute approximately $9\,\%$ of the mean velocity close to the wall and $30\,\%$ of the Reynolds stress near the edge of the boundary layer, indicating their non-negligibility in turbulent modelling strategies. The comparatively strong compressive events behaving as eddy shocklets are observed at the free-stream Mach number of $6$ only in the cases with wall disturbances. By further splitting the velocity into the solenoidal and dilatational components with the Helmholtz decomposition, we found that the dilatational motions are organized as travelling wave packets in the wall-parallel planes close to the wall and as forward inclined structures in the form of radiated waves in the vertical planes. Despite their increased magnitudes and higher portion in the Reynolds normal and shear stresses, the dilatational motions show no tendency of contributing significantly to the skin friction and the production of turbulent kinetic energy due to their mitigation by the cross-correlation between the solenoidal and dilatational velocity components.

Physical interpretation of neural network-based nonlinear eddy viscosity models

Neural network-based turbulence modeling has gained significant success in improving turbulence predictions by incorporating high–fidelity data. However, the interpretability of the learned model is often not fully analyzed, which has been one of the main criticisms of neural network-based turbulence modeling. Therefore, it is increasingly demanding to provide physical interpretation of the trained model, which is of significant interest for guiding the development of interpretable and unified turbulence models. The present work aims to interpret the predictive improvement of turbulence flows based on the behavior of the learned model, represented with tensor basis neural networks. The ensemble Kalman method is used for model learning from sparse observation data due to its ease of implementation and high training efficiency. Two cases, i.e., flow over the S809 airfoil and flow in a square duct, are used to demonstrate the physical interpretation of the ensemble-based turbulence modeling. For the flow over the S809 airfoil, our results show that the ensemble Kalman method learns an optimal linear eddy viscosity model, which improves the prediction of the aerodynamic lift by reducing the eddy viscosity in the upstream boundary layer and promoting the early onset of flow separation. For the square duct case, the method provides a nonlinear eddy viscosity model, which predicts well secondary flows by capturing the imbalance of the Reynolds normal stresses. The flexibility of the ensemble-based method is highlighted to capture characteristics of the flow separation and secondary flow by adjusting the nonlinearity of the turbulence model.

Direct numerical simulation of turbulent convection in a channel roughened with circular-arc ribs

Turbulent flows through a channel roughened with circular-arc ribs of different pitch-to-height ratios (P/H=3.0, 5.0 and 7.5) at a fixed Reynolds number are studied using direct numerical simulations (DNS). It is interesting to observe that in contrast to the classical channel flow over transverse square bars, the flow separation point in a channel roughened with circular-arc ribs is sensitive to the testing conditions and varies along the rib arc. The influences of the pitch-to-height ratio on the flow separation, statistical moments of the velocity field, and the anisotropy of turbulence stresses are investigated. The dynamics of coherent structures are studied by examining the skewness of the flow field, characteristics of the joint probability density functions (JPDF) of velocity fluctuations, quadrant analysis of Reynolds stresses, instantaneous velocity field, swirling strength, and temporal–spatial two-point auto-correlations. It is found that the flow anisotropy becomes suppressed as the value of P/H increases due to significant enhancement of the spanwise Reynolds normal stress 〈w′w′〉. It is also interesting to observe that the lifespan of turbulence structures becomes increasingly shortened at the midspan between two adjacent ribs as the pitch-to-height ratio increases.

Budgets of Reynolds stresses in film cooling with fan-shaped and cylindrical holes

The compressible budget terms in the transport equations of Reynolds stresses are examined from the large eddy simulation (LES) result of the film cooling. The capability of LES and the statistical post-processing procedure were first validated. The compressible Reynolds stress budget terms are then analyzed for both fan-shaped and cylindrical cooling films. The balance of all budget terms is shown. The effect of the blowing ratio on each budget term is examined. The mechanisms by which energy is extracted from the mean flow and distributed among the normal Reynolds stresses are highlighted. The sources of anisotropy in the Reynolds stress distributions are examined in detail, and their relation to the flow patterns of the mean and instantaneous flow is explored. The downstream development of the Reynolds stress budgets is studied, and it is shown that the jets of both fan-shaped and cylindrical films can be split into a near field and a far field with different properties. Far downstream of the cooling films, the Reynolds stress budgets near the wall present similarities with the Reynolds stress budgets in a boundary layer, while the Reynolds stress budgets further away from the wall resemble budgets in a free-shear flow. It is shown that the budgets of the Reynolds stress in the three-dimensional wall jets object of this study obey approximate similarity laws. These laws are based on easily obtained integral scales but need to be modified by suitable powers of the distance from the orifice producing the jet.

Entropy: An Inspiring Tool for Characterizing Turbulence-Combustion Interaction in Swirling Flames via Direct Numerical Simulations of Non-Premixed and Premixed Flames.

This article focuses on entropy generation in the combustion field, which serves as a useful indicator to quantify the interaction between turbulence and combustion. The study is performed on the direct numerical simulations (DNS) of high pressure non-premixed and premixed swirling flames. By analyzing the entropy generation in thermal transport, mass transport, and chemical reactions, it is found that the thermal transport, driven by the temperature gradient, plays a dominant role. The enstrophy transport analysis reveals that the responses of individual terms to combustion can be measured by the entropy: the vortex stretching and the dissipation terms increase monotonically with the increasing entropy. In high entropy regions, the turbulence behaves as the "cigar shaped" state in the non-premixed flame, while as the axisymmetric state in the premixed flame. A substantial increase in the normal Reynolds stress with the entropy is observed. This is due to the competition between two terms promoted by the entropy, i.e., the velocity-pressure gradient correlation term and the shear production term. As a result, the velocity-pressure gradient correlation tends to isotropize turbulence by transferring energy increasingly from the largest streamwise component to the other smaller normal components of Reynolds stress and is dominated by the fluctuating pressure gradient that increases along the entropy. The shear production term increases with the entropy due to the upgrading alignment of the eigenvectors of strain rate and Reynolds stress tensors.