We study the statistically steady states of the forced dissipative three-dimensional homogeneous isotropic turbulence at scales larger than the forcing scale in real separation space. The probability density functions (p.d.f.s) of longitudinal velocity difference at large separations are close to, but deviate from, Gaussian, measured by their non-zero odd parts. The analytical expressions of the third-order longitudinal structure functions derived from the Kármán–Howarth–Monin equation prove that the odd-part p.d.f.s of velocity differences at large separations are small but non-zero. Specifically, when the forcing effect in the displacement space decays exponentially as the displacement tends to infinity, the odd-order longitudinal structure functions have a power-law decay with an exponent of $-$2, implying a significant coupling between large and small scales. Under the assumption that forcing controls the large-scale dynamics, we propose a conjugate regime to Kolmogorov's inertial range, independent of the forcing scale, to capture the odd parts of p.d.f.s. Thus, dynamics of large scales departs from the absolute equilibrium, and we can partially recover small-scale information without explicitly resolving small-scale dynamics. The departure from the statistical equilibrium is quantified and found to be viscosity-independent. Even though this departure is small, it is significant and should be considered when studying the large scales of the forced three-dimensional homogeneous isotropic turbulence.

Direct numerical simulations of temporally developing mixing layers have been performed to investigate the effects of compressibility on statistics and structures near the interfaces of high- and low-speed large-scale structures (LSSs), covering a range of convective Mach numbers from $M_c=0.2$ to $1.8$ and Taylor Reynolds numbers up to 290. The interfaces of LSSs are directly defined by the isosurface of zero fluctuating streamwise velocity. The characteristic velocity jump at the interfaces grows rapidly in the transition stage and then decreases until reaching a gradual self-similar stage. The evolution of velocity jump is evidently delayed as the convective Mach number increases. The interface layer is formed by the shearing motion of neighbouring LSSs. A small vortical motion characterized by the Kolmogorov scale is induced in the interface layer by shear-dominated outer regions. The strengths of outer shearing motion and central vortical motion are greater at the leading edge, but smaller at the trailing edge, which is also reflected in a larger velocity jump at the leading edge. As the convective Mach number increases, the small-scale vortical structure is obviously suppressed by compressibility. At high convective Mach number $M_c=1.8$, the compressive shear-dominated outer regions are linked with a sheet-like structure passing through the centre of the expansion region corresponding to a small-scale vortical structure. The compressibility and shearing strength near the interface are highly dependent on the interface orientation.

The wall cooling effect on the spectra and structures of thermodynamic variables are investigated in hypersonic turbulent boundary layers. The density and temperature can be divided into the acoustic and entropic modes based on the Kovasznay decomposition. The intensities of the pressure and the acoustic modes of density and temperature attain the maximum values near the wall, while those of the entropy and the entropic modes of density and temperature achieve their primary peaks near the edge of the boundary layer. In the near-wall region, the pressure and the acoustic modes of density and temperature are significantly enhanced when the wall is strongly cooled, which can be attributed to the appearance of the travelling-wave-like alternating positive and negative structures. Moreover, the intensities of the entropy and the entropic modes of density and temperature become stronger near the wall as the wall temperature decreases, due to the appearance of the streaky entropic structures (SES). The SES are mainly caused by the advection effect of the strong positive wall-normal gradient of the mean temperature associated with ejection and sweep events. It is also found that the profiles of the intensities of the entropy, density and temperature are similar to each other far from the wall, which is mainly due to the reason that the entropic modes are dominant in the fluctuating density and temperature in the far-wall region. The acoustic modes of density and temperature only have significant contributions in the near-wall region.

The Reynolds number and wall cooling effects on correlations between the thermodynamic variables are systematically investigated in hypersonic turbulent boundary layers by direct numerical simulations. The correlations between the thermodynamic variables and the streamwise velocity are also analysed. The Kovasznay decomposition is introduced to decompose the fluctuating density and temperature into the acoustic and entropic modes. It is found that in the strongly cooled wall cases, the travelling-wave-like alternating positive and negative structures (TAPNSs) are found in the fluctuating pressure and the acoustic modes of density and temperature, and the streaky entropic structures (SESs) are identified in the fluctuating entropy and the entropic modes of density and temperature near the wall. Furthermore, both the acoustic and the entropic modes of density and temperature give significant contributions to the correlations involving density and temperature in the near-wall region, while these correlations are almost totally contributed by the entropic modes in the far-wall region. The entropic modes of the density and temperature are almost linearly correlated with the fluctuating entropy. Therefore, the fact that the fluctuating entropy is strongly correlated with the fluctuating density and temperature far from the wall is mainly due to the dominance of the entropic modes in the fluctuating density and temperature. Moreover, the fluctuating temperature is strongly positively correlated with the fluctuating streamwise velocity near the wall in strongly cooled wall cases, which can be ascribed to the appearance of the TAPNSs and SESs.

In this paper, we propose artificial-neural-network-based (ANN-based) nonlinear algebraic models for the large-eddy simulation (LES) of compressible wall-bounded turbulence. An innovative modification is applied to the invariants and the tensor bases of the nonlinear algebraic models through using the local grid widths along each direction to normalise the corresponding gradients of the flow variables. Furthermore, the dimensionless model coefficients are determined by the ANN method. The modified ANN-based nonlinear algebraic model (MANA model) has much higher correlation coefficients and much lower relative errors than the dynamic Smagorinsky model (DSM), Vreman model and wall-adapting local eddy-viscosity model in the a priori test. The significantly more accurate estimations of the mean subgrid-scale (SGS) fluxes of the kinetic energy and temperature variance are also obtained by the MANA models in the a priori test. Furthermore, in the a posteriori test, the MANA model can give much more accurate predictions of the flow statistics and the mean SGS fluxes of the kinetic energy and the temperature variance than other traditional eddy-viscosity models in compressible turbulent channel flows with untrained Reynolds numbers, Mach numbers and grid resolutions. The MANA model has a better performance in predicting the flow statistics in supersonic turbulent boundary layer. The MANA model can well predict both direct and inverse transfer of the kinetic energy and temperature variance, which overcomes the inherent shortcoming that the traditional eddy-viscosity models cannot predict the inverse energy transfer. Moreover, the MANA model is computationally more efficient than the DSM.

Enstrophy production and flow topology are numerically investigated for statistically stationary compressible isotropic turbulence in vibrational non-equilibrium with a large-scale thermal forcing. The net enstrophy production term is decomposed into solenoidal, dilatational and isotropic dilatational terms based on the Helmholtz decomposition. From the full flow field perspective, the net enstrophy production mainly stems from the solenoidal term. For the dilatational and isotropic dilatational terms, although their local magnitudes can be considerable, the positive values in the compression region and the negative values in the expansion region cancel out on average. For the solenoidal component of the deviatoric strain-rate tensor, the statistical properties of its eigenvalues and alignments between vorticity and its eigenvectors are nearly independent of the local dilatation and vibrational relaxation. The solenoidal components of enstrophy production along three eigendirections are thus mainly affected by the vorticity. For the dilatational component of deviatoric strain-rate tensor, the statistical properties of its eigenvalues and alignments between vorticity and its eigenvectors closely relate to the local dilatation and vibrational relaxation. The dilatational components of enstrophy production along three eigendirections are therefore affected by the vorticity, eigenvalues and alignments between the vorticity and eigenvectors. The topological classification proposed by Chong et al. (Phys. Fluids, vol. 2, issue 5, 1990, pp. 765–777) is employed to decompose the flow field into various flow topologies. In the strong compression and strong expansion regions, the relaxation effects on the volume fractions of flow topologies and their relative contributions to the local enstrophy production are significant.

The effects of compressibility on the statistics and coherent structures of a temporally developing mixing layer are studied using numerical simulations at convective Mach numbers ranging from $M_c=0.2$ to $1.8$ and at Taylor Reynolds numbers up to 290. As the convective Mach number increases, the streamwise dissipation becomes more effective to suppress the turbulent kinetic energy. At $M_c=1.8$, the streamwise dissipation increases much faster than the other two components in the transition region, even larger than pressure–strain redistribution, correlating with the streamwise elongated vortical structures at a higher level of compressibility. We confirm the existence of the large-scale high- and low-speed structures in the mixing layers, which accompany the spanwise Kelvin–Helmholtz rollers at low convective Mach number and dominate the mixing layer at higher convective Mach number. Conditional statistics demonstrate that the large-scale low-speed structures are lifted upwards by a pair of counter-rotating quasi-streamwise rollers flanking the structures. The small-scale vortical structures have an apparent preference for clustering into the top of the low-speed regions, which is directly associated with high-shearing motions on top of the low-speed structures. The high-speed structures statistically exhibit central symmetry with the low-speed structures. The statistics and dynamics of large-scale high- and low-speed structures in the compressible mixing layers resemble those in the outer region of the turbulent boundary layers, which reveals the universality of the large-scale structures in free shear and wall-bounded turbulence. A conceptual model is introduced for the large-scale high- and low-speed structures in turbulent mixing layers.

The decompositions of the skin-friction and heat-transfer coefficients based on the twofold repeated integration in hypersonic transitional and turbulent boundary layers are analysed to give some major reasons of the overshoot phenomena of the wall skin friction and heat transfer. It is shown that the overshoot of the skin-friction coefficient is mainly caused by the drastic change of the mean velocity profiles, especially the strong negative streamwise gradient of the mean streamwise velocity far from the wall; and the overshoot of the heat-transfer coefficient is primarily due to the viscous dissipation, especially the strong positive vertical gradient of the mean streamwise velocity near the wall. These observations are different from the previous observations that the Reynolds shear stress and Reynolds heat flux are the reasons, respectively. Further investigations show that the above observations are independent of the set-up of the wall blowing and suction parameters, which indicates the universality of the major reasons of the overshoot phenomena in our numerical simulations. In the hypersonic turbulent boundary layers, it is observed that the strongly cooled wall temperature and the high Mach number can slightly enhance the contribution of the Reynolds shear stress, and weaken the contribution of the mean convection, mainly due to the strong compressibility effect. Moreover, the magnitudes of the relative contributions of the mean convection, pressure dilatation, viscous dissipation and the Reynolds heat flux increase as the wall temperature increases.

Kinetic energy flux (KEF) is an important physical quantity that characterizes cascades of kinetic energy in turbulent flows. In large-eddy simulation (LES), it is crucial for the subgrid-scale (SGS) model to accurately predict the KEF in turbulence. In this paper, we propose a new eddy-viscosity SGS model constrained by the properly modelled KEF for LES of compressible wall-bounded turbulence. The new methodology has the advantages of both accurate prediction of the KEF and strong numerical stability in LES. We can obtain an approximate KEF by the tensor-diffusivity model, which has a high correlation with the real value. Then, using the artificial neural network method, the local ratios between the real KEF and the approximate KEF are accurately modelled. Consequently, the SGS model can be improved by the product of that ratio and the approximate KEF. In LES of compressible turbulent channel flow, the new model can accurately predict mean velocity profile, turbulence intensities, Reynolds stress, temperature–velocity correlation, etc. Additionally, for the case of a compressible flat-plate boundary layer, the new model can accurately predict some key quantities, including the onset of transitions and transition peaks, the skin-friction coefficient, the mean velocity in the turbulence region, etc., and it can also predict the energy backscatters in turbulence. Furthermore, the proposed model also shows more advantages for coarser grids.

The effect of wall temperature on the transfer of kinetic energy in a hypersonic turbulent boundary layer for different Mach numbers and wall temperature ratios is studied by direct numerical simulation. A cold wall temperature can enhance the compressibility effect in the near-wall region through increasing the temperature gradient and wall heat flux. It is shown that the cold wall temperature enhances the local reverse transfer of kinetic energy from small scales to large scales, and suppresses the local direct transfer of kinetic energy from large scales to small scales. The average filtered spatial convection and average filtered viscous dissipation are dominant in the near-wall region, while the average subgrid-scale flux of kinetic energy achieves its peak value in the buffer layer. It is found that the wall can suppress the inter-scale transfer of kinetic energy, especially for the situation of a cold wall. A strong local reverse transfer of fluctuating kinetic energy is identified in the buffer layer in the inertial range. Helmholtz decomposition is applied to analyse the compressibility effect on the subgrid-scale flux of kinetic energy. A strong transfer of the solenoidal component of fluctuating kinetic energy is identified in the buffer layer, while a significant transfer of the dilatational component of fluctuating kinetic energy is observed in the near-wall region. It is also shown that compression motions have a major contribution to the direct transfer of fluctuating kinetic energy, while expansion motions play a marked role in the reverse transfer of fluctuating kinetic energy.

The transfer of internal energy fluctuation is numerically investigated for the stationary compressible isotropic turbulence in vibrational non-equilibrium with large-scale thermal forcing. We observe the spectra of velocity, solenoidal pressure component, density and temperatures all exhibiting the $k^{-5/3}$ scaling in the inertial range. The Helmholtz decomposition results reveal that the solenoidal velocity component predominates over the dilatational component. Fluctuations of the solenoidal velocity and pressure components are nearly insensitive to the turbulent Mach number and vibrational relaxation, while those of the dilatational velocity and pressure components are closely related to them. In addition, the weak and strong acoustic equilibrium hypotheses are verified. On global average, the dissipation of translational–rotational energy fluctuation stems mainly from the thermal conduction and vibrational relaxation, while effects of the dilatation and viscosity are negligible. For the vibrational energy fluctuation, the dilatation effect is insignificant, while the dissipation due to the thermal conduction is roughly equivalent to the production owing to the vibrational relaxation. The cascades of translational–rotational and vibrational energy fluctuations are mainly dominated by the solenoidal component of filtered velocity. The direct subgrid-scale (SGS) fluxes of translational–rotational and vibrational energy fluctuations due to the dilatational component of filtered velocity in the compression region are balanced by the reverse SGS fluxes in the expansion region. On the other hand, the dependencies of the SGS fluxes due to the solenoidal component of filtered velocity on the local compressibility are closely related to the relaxation effect. The sensibility gradually disappears as the relaxation effect weakens.

Interscale kinetic energy transfer in chemically reacting compressible isotropic turbulence is studied using numerical simulations at turbulent Mach numbers 0.2 and 0.8 for isothermal and exothermic reactions. At low turbulent Mach number $M_{t}=0.2$ for exothermic reaction, heat release greatly enhances expansion and compression motions, and induces the formation of shocklets which are not observed for isothermal reaction at the same turbulent Mach number. It is found that heat release through exothermic reactions enhances both the positive and the negative components of pressure–dilatation, as well as positive and negative components of subgrid-scale (SGS) kinetic energy flux, indicating an increase of both forward-scatter and backscatter of kinetic energy. The SGS flux of kinetic energy has a tendency to be from large scales to small scales. The solenoidal components of pressure–dilatation and SGS kinetic energy flux are minorly influenced by heat release of reaction, while the dilatational components of pressure–dilatation and SGS kinetic energy flux are significantly intensified by heat release at all length scales. Heat release enhances the kinetic energy of the dilatational mode through the pressure work, and leads to the increase of dilatational kinetic energy transfer from large scales to small scales through dilatational SGS flux, as well as the viscous dissipation of dilatational kinetic energy at small scales. Taylor Reynolds number has a minor influence on the qualitative statistical properties of interscale kinetic energy transfer terms.

Helicity, as one of only two inviscid invariants in three-dimensional turbulence, plays an important role in the generation and evolution of turbulent flows. Through theoretical analyses, we find that there are two channels in the helicity cascade process, which differs dramatically from the traditional viewpoint. In this paper, we have conducted important research on the newly proposed dual-channel helicity cascade theory, including vortex dynamic processes, intermittent discrepancies, tensor geometries, etc. The first channel mainly originates from the vortex twisting process, and the second channel mainly originates from the vortex stretching process. Antisymmetric tensors are introduced to the derivations of dual-channel helicity cascade theory, and a complex rotation frame leads to a higher helicity transfer efficiency. By analysing data from direct numerical simulations of typical turbulent flows, we find that these two channels behave differently. The ensemble averages of helicity flux in different channels are equal in homogeneous and isotropic turbulence, while they are different in other types of turbulent flows. The intermittency of the second channel is stronger than that of the first channel. In addition, we find a novel mechanism of hindered or even inverse energy cascades, which could be attributed to the second-channel helicity flux.

The effects of flow topology on the subgrid-scale (SGS) kinetic energy flux in compressible isotropic turbulence is studied. The eight flow topological types based on the three invariants of the filtered velocity gradient tensor are analysed at different scales, along with their roles in the magnitude and direction of kinetic energy transfer. The unstable focus/compressing (UFC), unstable node/saddle/saddle (UN/S/S) and stable focus/stretching (SFS), are the three predominant topological types at all scales; they account for at least 75 % of the flow domain. The UN/S/S and SFS types make major contributions to the average SGS flux of the kinetic energy from large scales to small scales in the inertial range. The unstable focus/stretching (UFS) topology makes a contribution to the reverse SGS flux of kinetic energy from small scales to large scales. In strong compression regions, the average contribution of the stable node/saddle/saddle (SN/S/S) topology to the SGS kinetic energy flux is positive and is predominant over those of other flow topologies. In strong expansion regions, the UFS topology makes a major contribution to the reverse SGS flux of the kinetic energy. As the turbulent Mach number increases, the increase of volume fraction of the UFS topological regions leads to the increase of the SGS backscatter of kinetic energy. The SN/S/S topology makes a dominant contribution to the direct SGS flux of the compressible component of the kinetic energy, while the UFS topology makes a dominant contribution to the reverse SGS flux of the compressible component of the kinetic energy.

Cascades of temperature and entropy fluctuations are studied by numerical simulations of stationary three-dimensional compressible turbulence with a heat source. The fluctuation spectra of velocity, compressible velocity component, density and pressure exhibit the $-5/3$ scaling in an inertial range. The strong acoustic equilibrium relation between spectra of the compressible velocity component and pressure is observed. The $-5/3$ scaling behaviour is also identified for the fluctuation spectra of temperature and entropy, with the Obukhov–Corrsin constants close to that of a passive scalar spectrum. It is shown by Kovasznay decomposition that the dynamics of the temperature field is dominated by the entropic mode. The average subgrid-scale (SGS) fluxes of temperature and entropy normalized by the total dissipation rates are close to 1 in the inertial range. The cascade of temperature is dominated by the compressible mode of the velocity field, indicating that the theory of a passive scalar in incompressible turbulence is not suitable to describe the inter-scale transfer of temperature in compressible turbulence. In contrast, the cascade of entropy is dominated by the solenoidal mode of the velocity field. The different behaviours of cascades of temperature and entropy are partly explained by the geometrical properties of SGS fluxes. Moreover, the different effects of local compressibility on the SGS fluxes of temperature and entropy are investigated by conditional averaging with respect to the filtered dilatation, demonstrating that the effect of compressibility on the cascade of temperature is much stronger than on the cascade of entropy.

Kinetic energy transfer in compressible isotropic turbulence is studied using numerical simulations with solenoidal forcing at turbulent Mach numbers ranging from 0.4 to 1.0 and at a Taylor Reynolds number of approximately 250. The pressure dilatation plays an important role in the local conversion between kinetic energy and internal energy, but its net contribution to the average kinetic energy transfer is negligibly small, due to the cancellation between compression and expansion work. The right tail of probability density function (PDF) of the subgrid-scale (SGS) flux of kinetic energy is found to be longer at higher turbulent Mach numbers. With an increase of the turbulent Mach number, compression motions enhance the positive SGS flux, and expansion motions enhance the negative SGS flux. Average of SGS flux conditioned on the filtered velocity divergence is studied by numerical analysis and a heuristic model. The conditional average of SGS flux is shown to be proportional to the square of filtered velocity divergence in strong compression regions for turbulent Mach numbers from 0.6 to 1.0. Moreover, the antiparallel alignment between the large-scale strain and the SGS stress is observed in strong compression regions. The inter-scale transfer of solenoidal and compressible components of kinetic energy is investigated by Helmholtz decomposition. The SGS flux of solenoidal kinetic energy is insensitive to the change of turbulent Mach number, while the SGS flux of compressible kinetic energy increases drastically as the turbulent Mach number becomes larger. The compressible mode persistently absorbs energy from the solenoidal mode through nonlinear advection. The kinetic energy of the compressible mode is transferred from large scales to small scales through the compressible SGS flux, and is dissipated by viscosity at small scales.

We investigate how compressibility affects the turbulent statistics from a Lagrangian point of view, particularly in the parameter range where the flow transits from the incompressible type to a state dominated by shocklets. A series of three-dimensional simulations were conducted for different types of driving and several Mach numbers. For purely solenoidal driving, as the Mach number increases a new self-similar region first emerges in the Lagrangian structure functions at sub-Kolmogorov time scale and gradually extends to larger time scale. In this region the relative scaling exponent saturates and the saturated value decreases as the compressibility becomes stronger, which can be attributed to the shocklets. The scaling exponent for the inertial range is still very close to that of incompressible turbulence for small Mach number, and discrepancy becomes visible when the Mach number is high enough. When the driving force is dominated by the compressive component the shocklet-induced self-similar region occupies a much wider range of time scales than that in the purely solenoidal driving case. Regardless of the type of driving force, the probability density functions of the velocity increment collapse onto one another for the time scales in the new self-similar region after proper normalization.

A mass-conserving lattice Boltzmann model based on the Bhatnagar-Gross-Krook (BGK) model is proposed for non-Newtonian fluid flows. The equilibrium distribution function includes the local shear rate related with the viscosity and a variable parameter changing with the shear rate. With the additional parameter, the relaxation time in the collision can be fixed invariable to the viscosity. Through the Chapman-Enskog analysis, the macroscopic equations can be recovered from the present mass-conserving model. Two flow problems are simulated to validate the present model with a local computing scheme for the shear rate, and good agreement with analytical solutions and/or other published results are obtained. The results also indicate that the present modified model is more applicable to practical non-Newtonian fluid flows owing to its better accuracy and more robustness than previous methods.

Using a simulated highly compressible isotropic turbulence field with turbulent Mach number around 1.0, we studied the effects of local compressibility on the statistical properties and structures of velocity gradients in order to assess salient small-scale features pertaining to highly compressible turbulence against existing theories for incompressible turbulence. A variety of statistics and local flow structures conditioned on the local dilatation – a measure of local flow compressibility – are studied. The overall enstrophy production is found to be enhanced by compression motions and suppressed by expansion motions. It is further revealed that most of the enstrophy production is generated along the directions tangential to the local density isosurface in both compression and expansion regions. The dilatational contribution to enstrophy production is isotropic and dominant in highly compressible regions. The emphasis is then directed to the complicated properties of the enstrophy production by the deviatoric strain rate at various dilatation levels. In the overall flow field, the most probable eigenvalue ratio for the strain rate tensor is found to be −3:1:2.5, quantitatively different from the preferred eigenvalue ratio of −4:1:3 reported in incompressible turbulence. Furthermore, the strain rate eigenvalue ratio tends to be −1:0:0 in high compression regions, implying the dominance of sheet-like structures. The joint probability distribution function of the invariants for the deviatoric velocity gradient tensor is used to characterize local flow structures conditioned on the local dilatation as well as the distribution of enstrophy production within these flow structures. We demonstrate that strong local compression motions enhance the enstrophy production by vortex stretching, while strong local expansion motions suppress enstrophy production by vortex stretching. Despite these complications, most statistical properties associated with the solenoidal component of the velocity field are found to be very similar to those in incompressible turbulence, and are insensitive to the change of local dilatation. Therefore, a good understanding of dynamics of the compressive component of the velocity field is key to an overall accurate description of highly compressible turbulence.

In the traditional hybrid RANS/LES approaches for the simulation of wall-bounded fluid turbulence, such as detached-eddy simulation (DES), the whole flow domain is divided into an inner layer and an outer layer. Typically the Reynolds-averaged Navier–Stokes (RANS) equations are used for the inner layer, while large-eddy simulation (LES) is used for the outer layer. The transition from the inner-layer solution to the outer-layer solution is often problematic due to the lack of small-scale dynamics in the RANS region. In this paper, we propose to simulate the whole flow domain by large-eddy simulation while enforcing a Reynolds-stress constraint on the subgrid-scale (SGS) stress model in the inner layer. Both the algebraic eddy-viscosity model and the one-equation Spalart–Allmaras (SA) model have been used to constrain the Reynolds stress in the inner layer. In this way, we improve the LES methodology by allowing the mean flow of the inner layer to satisfy the RANS solution while small-scale dynamics is included. We validate the Reynolds-stress-constrained large-eddy simulation (RSC-LES) model by simulating three-dimensional turbulent channel flow and flow past a circular cylinder. Our model is able to predict mean velocity, turbulent stress and skin-friction coefficients more accurately in turbulent channel flow and to estimate the pressure coefficient after separation more precisely in flow past a circular cylinder compared with the pure dynamic Smagorinsky model (DSM) and DES using the same grid resolution. Furthermore, the computational cost of the RSC-LES is almost the same as that of DES.