We systematically study the dissipative anomaly in compressible magnetohydrodynamic (MHD) turbulence using direct numerical simulations, and show that the total dissipation remains finite as viscosity diminishes. The dimensionless dissipation rate $\mathcal {C}_{\varepsilon }$ fits well with the model $\mathcal {C}_{\varepsilon } = \mathcal {C}_{\varepsilon,\infty } + \mathcal {D}/R_L^-$ for all levels of flow compressibility considered here, where $R_L^-$ is the generalized large-scale Reynolds number. The asymptotic value $\mathcal {C}_{\varepsilon,\infty }$ describes the total energy transfer flux, and decreases with increase of the flow compressibility, indicating non-universality of the dimensionless dissipation rate in compressible MHD turbulence. After introducing an empirically modified dissipation rate, the data from compressible cases collapse to a form similar to the incompressible MHD case depending only on the modified Reynolds number.

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.

The third-order law links energy transfer rates in the inertial range of magneto- hydrodynamic (MHD) turbulence with third-order structure functions. Anisotropy, a typical property in the solar wind, challenges the applicability of the third-order law with the isotropic assumption. To shed light on the energy transfer process in the presence of anisotropy, we conducted direct numerical simulations of forced MHD turbulence with normal and hyper-viscosity under various strengths of the external magnetic field ($B_0$), and calculated three forms of third-order structure function with or without averaging of the azimuthal or polar angles with respect to $B_0$ direction. Correspondingly, three estimated energy transfer rates were obtained. The result shows that the peak of normalized third-order structure function occurs at larger scales closer to the $B_0$ direction, and the maximum of longitudinal transfer rates shifts away from the $B_0$ direction at larger $B_0$. Compared with normal viscous cases, hyper-viscous cases can attain better separated inertial range, thus facilitating the estimation of the energy cascade rates. We find that the widespread use of the isotropic form of the third-order law in estimating the energy transfer rates is questionable in some cases, especially when the anisotropy arising from the mean magnetic field is inevitable. In contrast, the direction-averaged third-order structure function properly accounts for the effect of anisotropy and predicts the energy transfer rates and inertial range accurately, even at very high $B_0$. With limited statistics, the third-order structure function shows a stronger dependence on averaging of azimuthal angles than the time, especially for high $B_0$ cases. These findings provide insights into the anisotropic effect on the estimation of energy transfer rates.

Data reconstruction of rotating turbulent snapshots is investigated utilizing data-driven tools. This problem is crucial for numerous geophysical applications and fundamental aspects, given the concurrent effects of direct and inverse energy cascades. Additionally, benchmarking of various reconstruction techniques is essential to assess the trade-off between quantitative supremacy, implementation complexity and explicability. In this study, we use linear and nonlinear tools based on the proper orthogonal decomposition (POD) and generative adversarial network (GAN) for reconstructing rotating turbulence snapshots with spatial damages (inpainting). We focus on accurately reproducing both statistical properties and instantaneous velocity fields. Different gap sizes and gap geometries are investigated in order to assess the importance of coherency and multi-scale properties of the missing information. Surprisingly enough, concerning point-wise reconstruction, the nonlinear GAN does not outperform one of the linear POD techniques. On the other hand, the supremacy of the GAN approach is shown when the statistical multi-scale properties are compared. Similarly, extreme events in the gap region are better predicted when using GAN. The balance between point-wise error and statistical properties is controlled by the adversarial ratio, which determines the relative importance of the generator and the discriminator in the GAN training.

The mean flow behaviour of a turbulent boundary layer over rough walls is expected to exhibit symmetries that govern the flow dynamics. In particular, when roughness elements are arranged in a spanwise symmetric manner, the mean flow above them should also exhibit spanwise symmetry. This symmetrical consideration has garnered substantial empirical support. We conduct direct numerical simulations (DNS) of flow over aligned cube arrays to test such symmetry considerations further. We vary the surface coverage density from 0.25 % to 6.25 %, and employ an averaging time of about 100 large-eddy turnover times, which is longer than the typical averaging time in prior DNS studies of rough-wall boundary layers. The results suggest the presence of spanwise asymmetry in the mean flow. Specifically, we observe the development of a prominent secondary vortex on one side of the cubical roughness, accompanied by a relatively smaller secondary vortex on the other side. This asymmetry becomes most pronounced when the surface coverage density is approximately 0.59 %, and diminishes as the coverage density approaches either a low or a high value. We also establish that this mean flow asymmetry is robust across variations in the domain size, the initial condition, and the placement of the cubes in the spanwise direction.

The statistical characteristics and the evolution of the backflow structures are investigated in wall-bounded flows at Reynolds numbers up to $Re_{\tau }=1000$. The backflow is caused by the joining of large-scale high- and low-speed structures in the vicinity of the wall and is formed at the tail tip of the low-speed structure. The distribution density of the backflow structures and the percentage area of the backflow region on the wall both increase with the Reynolds number. The backflow structures have an average lifespan of 8 wall units which is found to be slightly longer in the pipe than the channel, and they are convected downstream at the average velocities of the buffer region of approximately 10 wall units, similar to Cardesa et al. (J. Fluid Mech., vol. 880, 2019, R3). The backflow structures occasionally split and merge, and can form detached from the wall. Evidence shows that the split, merged and wall-detached backflow structures are caused by the near-wall structures. The split backflow structures are on average, larger and more spanwise-elongated which are split due to the spanwise shearing of the near-wall streaks. A backflow structure is formed detached from the wall when the trailing end of its carrier low-speed structure ‘sits’ on the near-wall high-speed streaks. The wall-detached backflow structures tend to become wall-attached by approaching the wall when undergoing a similar life cycle to the normal backflow of growth and decay with spanwise elongation because the backflow region at the tail of the low-speed structure is continuously pressed down to the wall by the high-speed structure driven by persistent vortical structures in the buffer region.

The mixing-layer analogy is due to Raupach, Finnigan & Brunet (Boundary-Layer Meteorol., vol. 25, 1996, pp. 351–382). In the analogy, the flow in the roughness sublayer of a homogeneous deep vegetation canopy boundary layer is analogous to a plane mixing layer rather than a surface layer. Evidence for the analogy includes the inflected velocity profile, which resembles the velocity profile in a plane mixing layer, and, most notably, the following estimate as a result of the Kelvin–Helmholtz instability: $\varLambda _x=8.1L_s$, where $\varLambda _x$ is the spacing of the large-scale eddies, and $L_s$ is the shear length. The mixing-layer analogy has been very successful in vegetation canopy flow research, but has received only limited support in urban-canopy flow research. This work revisits Raupach et al.'s mixing-layer analogy, and we present the evidence for the mixing-layer analogy in urban-canopy flows: the exponential velocity profile in the canopy layer, i.e. $(U-U_c)/(U_h-U_c)=\exp (z/L_m)$, and $L_m\sim [(U_h/U_c-1)(U_h/U_c+3)]^{-1}$. Here, $z$ is the vertical coordinate, $L_m$ is the attenuation length and is a measure of the largest eddy in the canopy layer, $U_h$ is the wind speed at the canopy crest and $U_c$ is the velocity in the inactive layer. We conduct direct numerical simulations of various deep homogeneous urban-canopy flows and test the above two scalings. We also discuss why Raupach et al.'s analogy has not seen as many successes in urban-canopy flows as in vegetation canopy flows.

We perform a theoretical investigation of the instability of a helical vortex filament beneath a free surface in a semi-infinite ideal fluid. The focus is on the leading-order free-surface boundary effect upon the equilibrium form and instability of the vortex. This effect is characterised by the Froude number $F_r = U(gh^*)^{-{1}/{2}}$ where $g$ is gravity, and $U = \varGamma /(2{\rm \pi} b^*)$ with $\varGamma$ being the strength, $2{\rm \pi} b^*$ the pitch and $h^*$ the centre submergence of the helical vortex. In the case of $F_r \rightarrow 0$ corresponding to the presence of a rigid boundary, a new approximate equilibrium form is found if the vortex possesses a non-zero rotational velocity. Compared with the infinite fluid case (Widnall, J. Fluid Mech., vol. 54, no. 4, 1972, pp. 641–663), the vortex is destabilised (or stabilised) to relatively short- (or long-)wavelength sub-harmonic perturbations, but remains stable to super-harmonic perturbations. The wall-boundary effect becomes stronger for smaller helix angle and could dominate over the self-induced flow effect depending on the submergence. In the case of $F_r > 0$, we obtain the surface wave solution induced by the vortex in the context of linearised potential-flow theory. The wave elevation is unbounded when the $m$th wave mode becomes resonant as $F_r$ approaches the critical Froude numbers ${\mathcal {F}} (m) = (C_0^*/U)^{-1} (mh^*/b^*)^{-{1}/{2}}$, $m=1, 2, \ldots,$ where $C_0^*$ is the induced wave speed. We find that the new approximate equilibrium of the vortex exists if and only if $F_r < {\mathcal {F}}(1)$. Compared with the infinite fluid and $F_r \rightarrow 0$ cases, the wave effect causes the vortex to be destabilised to super-harmonic and long-wavelength sub-harmonic perturbations with generally faster growth rate for greater $F_r$ and smaller helix angle.

The accelerations of tracer and light particles (bubbles) in compressible homogeneous isotropic turbulence are investigated by using data from direct numerical simulations up to turbulent Mach number $M_t =1$. For tracer particles, the flatness factor of acceleration components, $F_a$, increases gradually for $M_t \in [0.3, 1]$. On the contrary, $F_a$ for bubbles develops a maximum around $M_t \sim 0.6$. The probability density function of longitudinal acceleration of tracers is increasingly skewed towards the negative value as $M_t$ increases. By contrast, for light particles, the skewness factor of longitudinal acceleration, $S_a$, first becomes more negative with the increase of $M_t$, and then goes back to $0$ when $M_t$ is larger than $0.6$. Similarly, differences among tracers and bubbles appear also in the zero-crossing time of acceleration correlation. It is argued that all these phenomena are intimately linked to the flow structures in the compression regions close to shocklets.

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.

We use Navier–Stokes-based linear models for wall-bounded turbulent flows to estimate large-scale fluctuations at different wall-normal locations from their measurements at a single wall-normal location. In these models, we replace the nonlinear term by a combination of a stochastic forcing term and an eddy dissipation term. The stochastic forcing term plays a role in energy production by the large scales, and the eddy dissipation term plays a role in energy dissipation by the small scales. Based on the results in channel flow, we find that the models can estimate large-scale fluctuations with reasonable accuracy only when the stochastic forcing and eddy dissipation terms vary with wall distance and with the length scale of the fluctuations to be estimated. The dependence on the wall distance ensures that energy production and energy dissipation are not concentrated close to the wall but are evenly distributed across the near-wall and logarithmic regions. The dependence on the length scale of the fluctuations ensures that lower wavelength fluctuations are not excessively damped by the eddy dissipation term and hence that the dominant scales shift towards lower wavelengths towards the wall. This highlights that, on the one hand, energy extraction in wall turbulence is predominantly linear and thus physics-based linear models give reasonably accurate results. On the other hand, the absence of linearly unstable modes in wall turbulence means that the nonlinear term still plays an essential role in energy extraction and thus the modelled terms should include the observed wall distance and length scale dependencies of the nonlinear term.

We study the decay of compressible magnetohydrodynamic (MHD) turbulence, emphasizing exchanges of energy between compressive and incompressive kinetic energies, magnetic energy, and thermal energy. A recently developed high order finite difference code is employed for compressible runs with a Mach number up to 2. Varying the nature of the initial conditions (magnitudes of velocity and magnetic fluctuations), and initial Mach numbers permits examination of various dynamical regimes characterized here by the changes between different energy reservoirs. Acoustic waves are responsible for the oscillatory exchange between compressive kinetic and thermal energy through the pressure dilatation term. The results indicate that exchange between kinetic and magnetic energy is dominated by interactions involving the solenoidal velocity. Several systematic rapid adjustments are found to be reproducible with simple scalings derived from the empirical data.

The Ginzburg–Landau equation (GLE) can phenomenologically model several key features of non-equilibrium systems including those in fluid mechanics. Its validity in real flows, however, remains questionable. Here, we show that the linear GLE can be formulated such that it has the same Wentzel–Kramers–Brillouin (WKB) approximation as for the linear global stability problem in open shear flows. We use the GLE to model the linear global modes of three different wakes and find that it can accurately capture the linear growth rate and frequency to first order in the WKB approximation. Furthermore, we find that it can also provide the shapes of the direct and adjoint eigenvectors and the regions of maximal structural sensitivity. The proposed model requires only the basic flow as input, but gives robust predictions and is computationally inexpensive. As well as opening up new possibilities for GLE-based control strategies, the proposed model makes accurate stability calculations possible, even for some computationally intractable open shear flows.

The characteristics and dynamics of the uniform-momentum zones (UMZ) and UMZ interfaces in a fully developed turbulent pipe flow are studied using direct numerical simulation at $Re_{\unicode[STIX]{x1D70F}}=500$. The multiple UMZs detected from the probability density functions of the instantaneous streamwise velocity following de Silva et al. (J. Fluid Mech., vol. 786, 2016, pp. 309–331) showed similarities to both turbulent channel and boundary layer flows (TBL): the hierarchical structural distribution of thinner UMZs with thinner interfaces nearer the wall, accompanied with sharper and larger jumps in the streamwise velocity at the UMZ interface. The conditional average results indicate that channel and pipe are very similar quantitatively whereas pipe and TBL display significant discrepancies. The innermost UMZs in pipe flow exhibit different behaviours to the other UMZs in pipes. The contortion of the UMZ interface representing the meandering of coherent motions with high- and low-momentum streaks is examined three-dimensionally. The meandering of UMZ in both two and three dimensions intensifies away from the wall and is always wavier in the azimuthal direction than the streamwise direction. The UMZs in the near-wall region capture the small-scale velocity fluctuation of the near-wall cycle and show asymmetric modulation of $Q2$ ejections over $Q4$ sweeps. The asymmetric modulation of ejections over sweeps decreases from the wall towards the pipe centre and the opposite trend of elevated $Q4$ sweeps is observed for the innermost UMZs. Near the wall, the ejection regions are very spiky compared to the flat sweep regions whereas, in the pipe centre, the large-scale ejections are relatively flat and the sweep regions are spikier.

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.

In self-excited combustion systems, the application of open-loop forcing is known to be an effective strategy for controlling periodic thermoacoustic oscillations, but it is not known whether and under what conditions such a strategy would work on thermoacoustic oscillations that are not simply periodic. In this study, we experimentally examine the effect of periodic acoustic forcing on a prototypical thermoacoustic system consisting of a ducted laminar premixed flame oscillating quasiperiodically on an ergodic $\mathbb{T}^{2}$ torus at two incommensurate natural frequencies, $f_{1}$ and $f_{2}$. Compared with that of a classical period-1 system, complete synchronization of this $\mathbb{T}_{1,2}^{2}$ system is found to occur via a more intricate route involving three sequential steps: as the forcing amplitude, $\unicode[STIX]{x1D716}_{f}$, increases at a fixed forcing frequency, $f_{f}$, the system transitions first (i) to ergodic $\mathbb{T}_{1,2,f}^{3}$ quasiperiodicity; then (ii) to resonant $\mathbb{T}_{1,f}^{2}$ quasiperiodicity as the weaker of the two natural modes, $f_{2}$, synchronizes first, leading to partial synchronization; and finally (iii) to a $P1_{f}$ limit cycle as the remaining natural mode, $f_{1}$, also synchronizes, leading to complete synchronization. The minimum $\unicode[STIX]{x1D716}_{f}$ required for partial and complete synchronization decreases as $f_{f}$ approaches either $f_{1}$ or $f_{2}$, resulting in two primary Arnold tongues. However, when forced at an amplitude above that required for complete synchronization, the system can transition out of $P1_{f}$ and into $\mathbb{T}_{1,2,f}^{3}$ or $\mathbb{T}_{2,f}^{2}$. The optimal control strategy is to apply off-resonance forcing at a frequency around the weaker natural mode ($f_{2}$) and at an amplitude just sufficient to cause $P1_{f}$, because this produces the largest reduction in thermoacoustic amplitude via asynchronous quenching. Analysis of the Rayleigh index shows that this reduction is physically caused by a disruption of the positive coupling between the unsteady heat release rate of the flame and the $f_{1}$ and $f_{2}$ acoustic modes. If the forcing is applied near the stronger natural mode ($f_{1}$), however, resonant amplification can occur. We then phenomenologically model this $\mathbb{T}_{1,2}^{2}$ thermoacoustic system as two reactively coupled van der Pol oscillators subjected to external sinusoidal forcing, and find that many of its synchronization features – such as the three-step route to $P1_{f}$, the double Arnold tongues, asynchronous quenching and resonant amplification – can be qualitatively reproduced. This shows that these features are not limited to our particular system, but are universal features of forced self-excited oscillators. This study extends the applicability of open-loop control from classical period-1 systems with just a single time scale to ergodic $\mathbb{T}^{2}$ quasiperiodic systems with two incommensurate time scales.

In this study, the behaviours of subgrid-scale (SGS) turbulence are investigated with direct numerical simulations when an isotropic turbulence is brought to interact with imposed rapid waves. A partition of the velocity field is used to decompose the SGS stress into three parts, namely, the turbulent part $\unicode[STIX]{x1D749}^{T}$, the wave-induced part $\unicode[STIX]{x1D749}^{W}$ and the cross-interaction part $\unicode[STIX]{x1D749}^{C}$. Under strong wave straining, $\unicode[STIX]{x1D749}^{T}$ is found to follow the Kolmogorov scaling $\unicode[STIX]{x1D6E5}_{c}^{2/3}$, where $\unicode[STIX]{x1D6E5}_{c}$ is the filter width. Based on the linear Airy wave theory, $\unicode[STIX]{x1D749}^{W}$ and the filtered strain-rate tensor due to the wave motion, $\tilde{\unicode[STIX]{x1D64E}}^{W}$, are found to have different phases, posing a difficulty in applying the usual eddy-viscosity model. On the other hand, $\unicode[STIX]{x1D749}^{T}$ and the filtered strain-rate tensor due to the turbulent motion, $\tilde{\unicode[STIX]{x1D64E}}^{T}$, are only weakly wave-phase-dependent and could be well related by an eddy-viscosity model. The linear wave theory is also used to describe the vertical distributions of SGS statistics driven by the wave-induced motion. The predictions are in good agreement with the direct numerical simulation results. The budget equation for the turbulent SGS kinetic energy shows that the transport terms related to turbulence are important near the free surface and they compensate the imbalance between the energy flux and the SGS energy dissipation.

Based on recent numerical simulations and field experiments, the mechanism behind wake meandering is increasingly accepted to be through the amplification of upstream disturbances owing to the convectively unstable nature of the flow. In this paper, we deduce a low-order phenomenological model for the far-wake region, which is based on a modified form of the complex Ginzburg–Landau (CGL) equation for flows that are in the amplifier regime, i.e. are only convectively unstable. The model reproduces the main qualitative features of wake meandering: (i) its origin via amplification of upstream structures, (ii) dependence of oscillation frequency on the upstream disturbance amplitude (higher amplitudes lead to lower frequencies), (iii) shift towards lower frequencies as the wake flow evolves in the streamwise direction and, to an extent, (iv) the transfer of energy from very low frequencies towards relatively higher frequencies. Additionally, the model also predicts the increase in the meandering amplitude and an advancement in its onset with increasing thrust coefficient. To our knowledge, this is the first low-order dynamical system in the literature that models wake meandering. The model coefficients are obtained from the mean flow local stability results that we show correctly account for the changing operating conditions and thus pave way for the prediction of wake meandering features. Its low order makes it suitable to use inside an energy farm design model, where it can help to mitigate the adverse effects of wake meandering.

In an earlier paper (Wan et al., J. Fluid Mech., vol. 697, 2012, pp. 296–315), the authors showed that a similarity solution for anisotropic incompressible three-dimensional magnetohydrodynamic (MHD) turbulence, in the presence of a uniform mean magnetic field $\boldsymbol{B}_{0}$, exists if the ratio of parallel to perpendicular (with respect to $\boldsymbol{B}_{0}$) similarity length scales remains constant in time. This conjecture appears to be a rather stringent constraint on the dynamics of decay of the energy-containing eddies in MHD turbulence. However, we show here, using direct numerical simulations, that this hypothesis is indeed satisfied in incompressible MHD turbulence. After an initial transient period, the ratio of parallel to perpendicular length scales fluctuates around a steady value during the decay of the eddies. We show further that a Taylor–Kármán-like similarity decay holds for MHD turbulence in the presence of a mean magnetic field. The effect of different parameters, including Reynolds number, mean field strength, and cross-helicity, on the nature of similarity decay is discussed.

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.