Coherent structures over two distinct, organized wall perturbations – a transverse sinusoidal bump with and without small-scale longitudinal grooves – are studied using direct numerical simulations. Large-scale spanwise rollers (SRs) form via shear layer rollup past the bump peak, enveloping a large separation bubble (SB) for both a smooth wall (SW) and a grooved wall (GW). In a GW, small-scale alternatingly spinning jets emanating from the crests’ corners merge with the shear layer, altering the SRs compared with SRs in a SW. The underlying coherence of the highly turbulent SRs is educed via phase-locked ensemble averaging. Coherent vorticity contours of SRs are ellipses tilted downward, hence causing co-gradient Reynolds stress. The limited streamwise length of SB precludes SR tumbling, unlike in a free shear layer. The coherent field reveals minibubbles attached to the bump’s downstream wall with circulation opposite to that of the SB – they are larger, stronger and more numerous in GW than in SW – reducing skin friction. Compared with SW, the swirling jets in GW increase coherent production while decreasing incoherent production. Additionally, the jets push the SRs to travel faster and farther before reattachment. The SB experiences two different modes of oscillation due to high-frequency advection of the shear layer SR and low-frequency breathing of the SB, where the former dominates in GW and the latter in SW. Negative production is caused by counter-rotating vortex dipoles inducing flow ejections (for both SW and GW) and single vortices penetrating the grooves – both occurring in the region of flow acceleration.

A theory of incompressible turbulent plane jets (TPJs) is proposed by advancing an improved boundary layer approximation over the limiting classical – retaining more terms in the momentum balance equations. A pressure deficit inside the jet (with respect to the ambient) must exist due to transverse turbulence (Miller & Comings, J. Fluid Mech., vol. 3, 1957, pp. 1–16; Hussain & Clarke, Phys. Fluids, vol. 20, 1977, pp. 1416–1426). Contrary to the universally accepted invariance of the total momentum flux $J_T(x)$ (non-dimensionalized by its inlet value) as a function of the streamwise distance $x$, we prove that $J_T(x) >1$ – a condition that all TPJs must satisfy; surprisingly, prior theories and most experiments do not satisfy this condition. This motivated us to apply Lie symmetry analysis with translational and dilatational transformations of the modified equations (incorporating $J_T>1$), which yields scaling laws for key jet measures: the mean streamwise and transverse velocities $U(x,y)$ and $V(x,y)$, the turbulence intensities, the Reynolds shear stress $-\rho \,\overline {u'v'}(x,y)$, the mean pressure $P(x,y)$, etc. Experiments satisfying $J_T(x)>1$ validate our predictions for all jet measures, including, among others, the profiles of $U$, $V$ and $-\rho \,\overline {u'v'}$. We further predict $U \sim x^{-0.24}$, $V \sim x^{-0.45}$, $-\rho \,\overline {u'v'}\sim x^{-0.69}$, the mass flux $Q_m \sim x^{0.55}$, and $J_T$ increases to approximately 1.5. Contrary to the classical linear jet spread, we find sublinear spread, with the jet half-width growing like $b(x)\sim x^{0.79}$, indicating a narrower jet. Our predictions differ notably from most results reported in the literature. These contradictions demand revisiting jet studies involving carefully designed facilities and boundary conditions, and highly resolved simulations.

Turbulent boundary layers (TBLs) over surface perturbations like bumps with roughness – notably altering heat and mass transfer, drag, etc. – are prevalent in nature (mountains, dunes, etc.) and technology. We study a channel flow with a transverse bump on one wall superimposed with small-scale longitudinal grooves via direct numerical simulation (DNS) of incompressible flow. Turbulence statistics and dynamics are compared between grooved wall (GW) and smooth wall (SW) bumps. Streamwise spinning jets emanating from the crests’ corners alter the flow structure within the separation bubble (SB), extending the SB length by 30 % over that for SW, and have lingering effects far downstream. Grooves decrease skin friction but increase the bump's form drag by 25 %. In GW, the peaks of turbulence intensity and production decrease by 20 % and shift downstream, compared with SW. Three regions of negative production, found upstream as well as downstream of the bump, are explained in terms of two separate mechanisms: normal and shear productions. Separation upstream of the bump occurs always for GW, but intermittently for SW. Within the downstream SB, counter-rotating minibubbles form intermittently for SW but always for GW. Interestingly, a minibubble causes streamwise vorticity reversal of the upstream moving secondary flow around each crest corner. The wall pressure in GW is invariant in the spanwise direction and is explained in terms of its non-local nature and its connection with outer structures. The grooved bump unearths rich TBL flow physics – upstream separation, dynamics of the downstream minibubble, altered reattachment dynamics and negative production.

A transcritical domain with a sharp two-phase interface may exist during the early times of liquid hydrocarbon fuel injection at supercritical pressure. Thus, two-phase dynamics are sustained before substantial heating of the liquid and drive the early three-dimensional deformation and atomisation. A recent study of a transcritical liquid jet showed distinct deformation features caused by interface thermodynamics, low surface tension and intraphase diffusive mixing. In the present work, the compressible vortex identification method $\lambda _\rho$ is used to study the vortex dynamics in a cool liquid n-decane transcritical jet surrounded by a hotter oxygen gaseous stream at supercritical pressures. The relationship between vortical structures and the liquid surface evolution is detailed, along with the vorticity generation mechanisms, including variable-density effects. The roles of hairpin and roller vortices in the early deformation of lobes, the layering and tearing of liquid sheets and the formation of fuel-rich gaseous blobs are analysed. At these high pressures, enhanced intraphase mixing and ambient gas dissolution affect the local liquid structures (i.e. lobes). Thus, liquid breakup differs from classical sub-critical atomisation. Near the interface, liquid density and viscosity drop by up to 10 % and 70 %, respectively, and the liquid is more easily affected by the vortical motion (e.g. liquid sheets wrap around vortices). Despite the variable density, compressible vorticity generation terms are smaller than the vortex stretching and tilting. Layering traps and aligns the vortices along the streamwise direction while mitigating the generation of new rollers.

Helicity, an invariant under ideal-fluid (Euler) evolution, has a topological interpretation in terms of writhe and twist for a closed vortex tube, but accurately quantifying twist is challenging in viscous flows. With a novel helicity decomposition, we present a framework to construct the differential twist that establishes the theoretical relation between the total twisting number and the local twist rate of each vortex surface. This framework can characterize coiling vortex lines and internal structures within a vortex – important in laminar–turbulence transition, and in vortex instability, reconnection and breakdown. As a typical example, we explore the dynamics of vortex rings with differential twist via direct numerical simulation (DNS) of the Navier–Stokes equations. Two twist waves with opposite chiralities propagate towards each other along the ring and then collide whence the local twist rate rapidly surges. Local vortex surfaces are squeezed into a disk-like dipole structure containing coiled vortex lines, leading to vortex bursting. We derive a Burgers-equation-like model to quantify this process, which predicts a bursting time that agrees well with DNS.

Compressible turbulent plane Couette flows are studied via direct numerical simulation for wall Reynolds numbers up to $Re_w=10\ 000$ and wall Mach numbers up to $M_w=5$. Various turbulence statistics are compared with their incompressible counterparts at comparable semilocal Reynolds numbers $Re^*_{\tau,c}$. The skin friction coefficient $C_f$, which decreases with $Re_w$, only weakly depends on $M_w$. On the other hand, the thermodynamic properties (mean temperature, density and others) strongly vary with $M_w$. Under proper scaling transformations, the mean velocity profiles for the compressible and incompressible cases collapse well and show a logarithmic region with the Kárman constant $\kappa =0.41$. Compared with wall units, the semilocal units yield a better collapse for the profiles of the Reynolds stresses. While the wall-normal and spanwise Reynolds stress components slightly decrease in the near-wall region, the inner peak of the streamwise component notably increases with increasing $M_w$ – indicating that flow becomes more anisotropic when compressible. In addition, the near-wall turbulence production decreases as $M_w$ increases – due to rapid wall-normal changes of viscosity caused by viscous heating. The streamwise and spanwise energy spectra show that the length scale of near-wall coherent structures does not vary with $M_w$ in semilocal units. Consistent with those in incompressible flows, the superstructures (the large-scale streamwise rollers) with a typical spanwise scale of $\lambda _z/h\approx 1.5{\rm \pi}$ become stronger with increasing $Re_w$. For the highest $Re_w$ studied, they contribute about $40\,\%$ of the Reynolds shear stress at the channel centre. Interestingly, flow visualization and correlation analysis show that the streamwise coherence of these structures degrades with increasing $M_w$. In addition, at comparable $Re^*_{\tau,c}$, the amplitude modulation of these structures on the near-wall small scales is quite similar between incompressible and compressible cases – but much stronger than that in plane Poiseuille flows.

Well-resolved direct numerical simulations (DNS) have been performed of the flow in a smooth circular pipe of radius $R$ and axial length $10{\rm \pi} R$ at friction Reynolds numbers up to $Re_\tau =5200$ using the pseudo-spectral code OPENPIPEFLOW. Various turbulence statistics are documented and compared with other DNS and experimental data in pipes as well as channels. Small but distinct differences between various datasets are identified. The friction factor $\lambda$ overshoots by $2\,\%$ and undershoots by $0.6\,\%$ the Prandtl friction law at low and high $Re$ ranges, respectively. In addition, $\lambda$ in our results is slightly higher than in Pirozzoli et al. (J. Fluid Mech., vol. 926, 2021, A28), but matches well the experiments in Furuichi et al. (Phys. Fluids, vol. 27, issue 9, 2015, 095108). The log-law indicator function, which is nearly indistinguishable between pipe and channel up to $y^+=250$, has not yet developed a plateau farther away from the wall in the pipes even for the $Re_\tau =5200$ cases. The wall shear stress fluctuations and the inner peak of the axial turbulence intensity – which grow monotonically with $Re_\tau$ – are lower in the pipe than in the channel, but the difference decreases with increasing $Re_\tau$. While the wall value is slightly lower in the channel than in the pipe at the same $Re_\tau$, the inner peak of the pressure fluctuation shows negligible differences between them. The Reynolds number scaling of all these quantities agrees with both the logarithmic and defect-power laws if the coefficients are properly chosen. The one-dimensional spectrum of the axial velocity fluctuation exhibits a $k^{-1}$ dependence at an intermediate distance from the wall – also seen in the channel. In summary, these high-fidelity data enable us to provide better insights into the flow physics in the pipes as well as the similarity/difference among different types of wall turbulence.

Well-resolved direct numerical simulations of turbulent open channel flows (OCFs) are performed for friction Reynolds numbers up to $Re_\tau =2000$. Various turbulent statistics are documented and compared with the closed channel flows (CCFs). As expected, the mean velocity profiles of the OCFs match well with the CCFs in the near-wall region but diverge notably in the outer region. Interestingly, a logarithmic layer with Kárman constant $\kappa =0.363$ occurs for OCF at $Re_\tau =2000$, distinctly different from CCF. Except for a very thin layer near the free surface, most of the velocity and vorticity variances match between OCFs and CCFs. The one-dimensional energy spectra reveal that the very-large-scale motions (VLSMs) with streamwise wavelength $\lambda _x>3 h$ or spanwise wavelength $\lambda _z>0.5 h$ contribute the most to turbulence intensity and Reynolds shear stress in the overlap and outer layers (where h is the water depth). Furthermore, the VLSMs in OCFs are stronger than those in CCFs, resulting in a slightly higher streamwise velocity variance in the former. Due to the footprint effect, these structures also have significant contributions to the mean wall shear stress, and the difference between OCF and CCF enlarges with increasing $Re_\tau$. In summary, the free surface in OCFs plays an essential role in various flow phenomena, including the formation of stronger VLSMs and turbulent kinetic energy redistribution.

The dynamics of two slender Hopf-linked vortex rings at vortex Reynolds numbers ($Re \equiv \varGamma /\nu, \mathrm {circulation/viscosity}$) $2000$, $3000$ and $4000$ is studied using direct numerical simulations of the incompressible Navier–Stokes equations. Under self-induction, the initially perpendicularly placed vortex rings approach each other and reconnect to form two separate vortex rings. The leading ring is closely cuddled and further undergoes secondary reconnection to form two even smaller rings. At high $Re$, the leading ring and the subsequent smaller rings are unstable and break up into turbulent clouds consisting of numerous even smaller-scale structures. Although the global helicity $H$ remains constant before reconnection, it increases and then rapidly decays during reconnection – both the growth and decay rates increase with $Re$. In the two higher $Re$ (i.e. 3000 and 4000) cases, $H$ further rises after the first reconnection and reaches a quasi-plateau with the asymptotic value continuously increasing with $Re$ – suggesting that $H$ for viscous flows is not conserved at very high $Re$. Further flow analysis demonstrates that significant numbers of positive and negative helical structures are simultaneously generated before and during reconnection, and their different decay rates is the main reason for the complex evolution of $H$. By examining the topological aspects of the helicity dynamics, we find that, different from $H$, the sum of link and writhe ($L_k+W_r$) continuously drop during reconnection. Our results also clearly demonstrate that the twist, which increases with $Re$, plays a significant role in the helicity dynamics, particularly at high $Re$.

Topological transition and helicity conversion of vortex torus knots and links are studied using direct numerical simulations of the incompressible Navier–Stokes equations. We find three topological transitional routes (viz. merging, reconnection and transition to turbulence) in the evolution of vortex knots and links over a range of torus aspect ratios and winding numbers. The topological transition depends not only on the initial topology but also on the initial geometry of knots/links. For small torus aspect ratios, the initially knotted or linked vortex tube rapidly merges into a vortex ring with a complete helicity conversion from the writhe and link components to the twist. For large torus aspect ratios, the vortex knot or link is untied into upper and lower coiled loops via the first vortex reconnection, with a helicity fluctuation including loss of writhe and link, and generation of twist. Then, the relatively unstable lower loop can undergo a secondary reconnection to split into multiple small vortices with a similar helicity fluctuation. Surprisingly, for moderate torus aspect ratios, the incomplete reconnection of tangled vortex loops together with strong vortex interactions triggers transition to turbulence, in which the topological helicity decomposition fails due to the breakdown of vortex core lines.

We study the sound generation mechanism of initially subsonic viscous vortex reconnection at vortex Reynolds number $Re~(\equiv \text {circulation}/\text {kinematic viscosity})=1500$ through decomposition of Lighthill's acoustic source term. The Laplacian of the kinetic energy, flexion product, enstrophy and deviation from the isentropic condition provide the dominant contributions to the acoustic source term. The overall (all time) extrema of the total source term and its dominant hydrodynamic components scale linearly with the reference Mach number $M_o$; the deviation from the isentropic condition shows a quadratic scaling. The significant sound arising from the flexion product occurs due to the coiling and uncoiling of the twisted vortex filaments wrapping around the bridges, when a rapid strain is induced on the filaments by the repulsion of the bridges. The spatial distributions of the various acoustic source terms reveal the importance of mutual cancellations among most of the terms; this also highlights the importance of symmetry breaking in the sound generation during reconnection. Compressibility acts to delay the start of the sequence of reconnection events, as long as shocklets, if formed, are sufficiently weak to not affect the reconnection. The delayed onset has direct ramifications for the sound generation by enhancing the velocity of the entrained jet between the vortices and increasing the spatial gradients of the acoustic source terms. Consistent with the near-field pressure, the overall maximum instantaneous sound pressure level in the far field has a quadratic dependence on $M_o$. Thus, reconnection becomes an even more dominant sound-generating event at higher $M_o$.

A slender trefoil knotted vortex is studied using direct numerical simulation of the Navier–Stokes equations for vortex Reynolds numbers ($Re \equiv \varGamma /\nu$, circulation/viscosity) up to 12 000. For initially zero twist ($T_{w,0}=0$), neither the writhe $W_r$ nor the global helicity $H$ is conserved. Initially $W_r$ slowly decreases, then suddenly drops during reconnection and becomes almost constant thence; its evolution is almost $Re$ independent. Before reconnection, $H$ also gradually decreases but sharply increases during reconnection. The evolution of $H$ after reconnection strongly depends on $Re$. While steadily decreasing at low $Re$, $H$ significantly increases before eventually decaying at high $Re$. Flow visualization, helicity decomposition and helical wave decomposition reveal that significant amounts of positive and negative twist helicities are simultaneously generated before and during reconnection. Also, the small leading and large trailing rings resulting from asymmetric reconnection have respectively negative and positive twists, which then decay differently due to different initial values, geometries and mutual induction. In particular, at high $Re$, the twist in the small ring, under stretching by the large trailing ring, decays much faster and even switches sign to become positive by the writhe-to-twist conversion – the main reason for the ‘transient growth’ of $H$. Simulations with non-zero initial twists ($T_{w,0}=7.48$ and $22.48$) reveal that the overall dynamics is similar to the $T_{w,0}=0$ case. Hence, the evolution of the trefoil knotted vortex is mainly governed by $W_r$, not $T_w$, although the latter is found to play an essential role in enstrophy growth as well as energy cascade.

Polarized vortical structures (i.e. with axial flow, thus coiled vortex lines) are generic to turbulent flows – hence the importance of their dynamics, interactions and cascade. Direct numerical simulations of two anti-parallel polarized vortex tubes are performed for vortex Reynolds numbers $Re$ ($\equiv \varGamma /\nu$) up to $9000$ and initial polarization strength $q$ (ratio of peak axial to azimuthal velocities) between $0$ and $4/3$. For both counter- and co-polarized cases, although the reconnection is delayed as $q$ increases – mainly due to weakened self-induction – it is more rapid and more complete for small $q$. Enstrophy growth and energy cascade are suppressed for weak polarization ($q < 1/2$) due to depleted nonlinearity, but are enhanced for strong polarization ($q > 1/2$) due to instability and/or transient growth. When counter-polarized, numerous structures with both positive and negative helicity densities (i.e. $\pm h$) are generated. For large $q$, strong axial flows opposite to the initial flows occur – causing polarization reversals. For the co-polarized cases, although $+h$ predominates, $-h$ structures also form and interact with positive ones – leading to helicity cascade to small scales. As $Re$ increases, small scales are more numerous: for counter-polarized cases, the threads undergo successive reconnections in a cascade – akin to the unpolarized case; for co-polarized cases, the newly formed vortex ring breaks up with numerous hairpin vortices wrapping around it. Increasing $q$ alters the energy spectrum in the inertial range with a scaling varying from $k^{-5/3}$ for the unpolarized case to $k^{-7/3}$ for the strongly polarized case, which seems to be associated with the enhanced vortex spiralling. In addition, for the strongly co-polarized cases, a $k^{-4/3}$ helicity spectrum develops. Furthermore, most of the energy and helicity in the inertial range with scale $L$ transfer to scales between $0.3L$ and $0.4L$. Therefore, polarization can significantly alter the dynamics of vortex reconnection as well as turbulence cascade.

Reconnection plays a significant role in the dynamics of plasmas, polymers and macromolecules, as well as in numerous laminar and turbulent flow phenomena in both classical and quantum fluids. Extensive studies in quantum vortex reconnection show that the minimum separation distance $\delta$ between interacting vortices follows a $\delta(t) \sim t^{1/2}$ scaling. Due to the complex nature of the dynamics (e.g. the formation of bridges and threads as well as successive reconnections and avalanche), such scaling has never been reported for (classical) viscous vortex reconnection. Using direct numerical simulation of the Navier–Stokes equations, we study viscous reconnection of slender vortices, whose core size is much smaller than the radius of the vortex curvature. For separations that are large compared to the vortex core size, we discover that $\delta (t)$ between the two interacting viscous vortices surprisingly also follows the 1/2-power scaling for both pre- and post-reconnection events. The prefactors in this 1/2-power law are found to depend not only on the initial configuration but also on the vortex Reynolds number (or viscosity). Our finding in viscous reconnection, complementing numerous works on quantum vortex reconnection, suggests that there is indeed a universal route for reconnection – an essential result for understanding the various facets of the vortex reconnection phenomena and their potential modelling, as well as possibly explaining turbulence cascade physics.

Recognizing the fact that the finite-time singularity of the Navier–Stokes equations is widely accepted as a key issue in fundamental fluid mechanics, and motivated by the recent model of Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019a, pp. 930–967; J. Fluid Mech., vol. 870, 2019b, R1) on this issue, we have performed direct numerical simulation (DNS) for two colliding slender vortex rings of radius $R$. The separation between the two tipping points $2s_{0}$ and the scale of the core cross-section $\unicode[STIX]{x1D6FF}_{0}$ are chosen as $\unicode[STIX]{x1D6FF}_{0}=0.1s_{0}=0.01R$; the vortex Reynolds number ($Re=\text{circulation/viscosity}$) ranges from 1000 to 4000. In contrast to the claim that the core remains compact and circular, there is notable core flattening and stripping, which further increases with $Re$ – akin to our previous finding in the standard anti-parallel vortex reconnection. Furthermore, the induced motion of bridges arrests the curvature growth and vortex stretching at the tipping points; consequently, the maximum vorticity grows with $Re$ substantially slower than the exponential scaling predicted by the model – implying that, for this configuration, even physical singularity is unlikely. Our simulations not only shed light on the longstanding question of finite-time singularities, but also further delineate the detailed mechanisms of reconnection. In particular, we show for the first time that the separation distance $s(\unicode[STIX]{x1D70F})$ before reconnection follows 1/2 scaling exactly – a significant DNS result.

Viscous anti-parallel vortex reconnection is studied by means of direct numerical simulation for vortex Reynolds numbers $Re$ ($\equiv \unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$, circulation/viscosity) up to 40 000. To suppress the inherent symmetry breaking due to the Kelvin–Helmholtz (planar jet) instability, as prevalent in prior studies, and to better explore the progression of the mechanism details, the simulation is performed by imposing symmetry and using double-precision arithmetic. We show, for the first time, the evidence of vortex reconnection cascade scenario initially proposed by Melander and Hussain (CTR Report, 1988), who suggested that the remnant threads, following the first reconnection, undergo successive reconnections in a cascade. Secondary reconnection (the details distinctly captured and visualized at a lower $Re=9000$) leads to the successive generation of numerous small-scale structures, including vortex rings, hairpin-like vortex packets and vortex tangles. As $Re$ increases, the third and higher generations of reconnection form a turbulent cloud avalanche consisting of a tangle of fine vortices. The energy is rapidly transferred to finer scales during reconnection, and a distinct - 5/3 inertial range is observed for the kinetic energy spectrum, associated with numerous resulting fine-scale bridgelets and thread filaments. In addition, we also discover an inverse cascade at large scales through the accumulation of bridgelets. The separation distance $\unicode[STIX]{x1D6FF}(t)$ before the first reconnection is found to scale as $t^{3/4}$, which is different from the typical 1/2 scaling for classical and quantum vortex filament reconnections. Both peak enstrophy and its production rate grow with $Re$ faster than the power law suggested by Hussain and Duraisamy (Phys. Fluids, vol. 23, 2011, 021701). Our simulations not only reveal the detailed mechanisms of high-$Re$ reconnection, but also shed light on the physics of turbulence cascade and present the reconnection avalanche as a realistic physical model for turbulence cascade.

Spanwise wall oscillation has been extensively studied to explore possible drag control methods, mechanisms and efficacy – particularly for incompressible flows. We performed direct numerical simulation for fully developed turbulent channel flow to establish how effective spanwise wall oscillation is when the flow is compressible and also to document its drag reduction (${\mathcal{D}}{\mathcal{R}}$) trend with Mach number. Drag reduction ${\mathcal{D}}{\mathcal{R}}$ is first investigated for three different bulk Mach numbers $M_{b}=0.3$, $0.8$ and $1.5$ at a fixed bulk Reynolds number $Re_{b}=3000$. At a given velocity amplitude $A^{+}$ ($=12$), ${\mathcal{D}}{\mathcal{R}}$ at $M_{b}=0.3$ agrees with the strictly incompressible case; at $M_{b}=0.8$, ${\mathcal{D}}{\mathcal{R}}$ exhibits a similar trend to that at $M_{b}=0.3$: ${\mathcal{D}}{\mathcal{R}}$ increases with the oscillation period $T^{+}$ to a maximum and then decreases gradually. However, at $M_{b}=1.5$, ${\mathcal{D}}{\mathcal{R}}$ monotonically increases with $T^{+}$. In addition, the maximum ${\mathcal{D}}{\mathcal{R}}$ is found to increase with $M_{b}$. For $M_{b}=1.5$, similar to the incompressible case, ${\mathcal{D}}{\mathcal{R}}$ increases with $A^{+}$, but the rate of increase decreases at larger $A^{+}$. Unlike the flow behaviour when incompressible, the flow surprisingly relaminarizes when it is supersonic (at $A^{+}=18$ and $T^{+}=300$) – this enigmatic behaviour requires further detailed studies for different domain sizes, $Re_{b}$ and $M_{b}$. The Reynolds number effect on ${\mathcal{D}}{\mathcal{R}}$ is also investigated. Although ${\mathcal{D}}{\mathcal{R}}$ generally decreases with $Re_{b}$, it is less affected at small $T^{+}$, but drops rapidly at large $T^{+}$. We introduce a simple scaling for the oscillation period as $T^{\ast }=T_{C}^{+}l_{I}^{+}/l_{C}^{+}$, with $l_{I}^{+}$ and $l_{C}^{+}$ denoting the mean streak spacing for incompressible and compressible cases, respectively. At the same semi-local Reynolds number $Re_{\unicode[STIX]{x1D70F}c}^{\ast }\equiv Re_{\unicode[STIX]{x1D70F}}\sqrt{\overline{\unicode[STIX]{x1D70C}}_{c}/\overline{\unicode[STIX]{x1D70C}}_{w}}/(\overline{\unicode[STIX]{x1D707}}_{c}/\overline{\unicode[STIX]{x1D707}}_{w})$ (subscripts $c$ and $w$ denote quantities at the channel centre and wall, respectively), ${\mathcal{D}}{\mathcal{R}}$ as a function of $T^{\ast }$ exhibits good agreement between the supersonic and strictly incompressible cases, with the optimal oscillation period becoming $M_{b}$-invariant as $T_{opt}^{\ast }\approx 100$.

As a counterpart of energy cascade, turbulent momentum cascade (TMC) in the wall-normal direction is important for understanding wall turbulence. Here, we report an analytic prediction of non-universal Reynolds number ($Re_{\unicode[STIX]{x1D70F}}$) scaling transition of the maximum TMC located at $y_{p}$. We show that in viscous units, $y_{p}^{+}$ (and $1+\overline{u^{\prime }v^{\prime }}_{p}^{+}$) displays a scaling transition from $Re_{\unicode[STIX]{x1D70F}}^{3/7}$ ($Re_{\unicode[STIX]{x1D70F}}^{-6/7}$) to $Re_{\unicode[STIX]{x1D70F}}^{3/5}$ ($Re_{\unicode[STIX]{x1D70F}}^{-3/5}$) in turbulent boundary layer, in sharp contrast to that from $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ ($Re_{\unicode[STIX]{x1D70F}}^{-2/3}$) to $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ ($Re_{\unicode[STIX]{x1D70F}}^{-1/2}$) in a channel/pipe, countering the prevailing view of a single universal near-wall scaling. This scaling transition reflects different near-wall motions in the buffer layer for small $Re_{\unicode[STIX]{x1D70F}}$ and log layer for large $Re_{\unicode[STIX]{x1D70F}}$, with the non-universality being ascribed to the presence/absence of mean wall-normal velocity $V$. Our predictions are validated by a large set of data, and a probable flow state with a full coupling between momentum and energy cascades beyond a critical $Re_{\unicode[STIX]{x1D70F}}$ is envisaged.

A recently developed symmetry-based theory is extended to derive an algebraic model for compressible turbulent boundary layers (CTBL) – predicting mean profiles of velocity, temperature and density – valid from incompressible to hypersonic flow regimes, thus achieving a Mach number ($Ma$) invariant description. The theory leads to a multi-layer analytic form of a stress length function which yields a closure of the mean momentum equation. A generalized Reynolds analogy is then employed to predict the turbulent heat transfer. The mean profiles and the friction coefficient are compared with direct numerical simulations of CTBL for a range of $Ma$ from 0 (e.g. incompressible) to 6.0 (e.g. hypersonic), with an accuracy notably superior to popular current models such as Baldwin–Lomax and Spalart–Allmaras models. Further analysis shows that the modification is due to an improved eddy viscosity function compared to competing models. The results confirm the validity of our $Ma$-invariant stress length function and suggest the path for developing turbulent boundary layer models which incorporate the multi-layer structure.