This study investigates the influence of wind tunnel ground conditions (stationary/moving) on flow topology and passive scalar dispersion in the wake of the Ahmed body with rear slant angles, $\phi$ = 25$^\circ$ and 40$^\circ$. We implement field measurements of both velocity and scalar concentrations in the wake, for both the ground conditions, within the same experimental set-up, allowing for structural correlation between wake topology and scalar dispersion. Particle image velocimetry measurements reveal the existence of a third spanwise vortex (vortex G) near the stationary wind tunnel ground, due to the floor boundary layer, for both of the Ahmed bodies ($\phi$ = 25$^\circ$, 40$^\circ$). Concentration field measurements performed using quantitative smoke visualisation show higher scalar dispersion in the wake of both Ahmed bodies for the stationary ground condition. Comparing the velocity and concentration fields further identifies vortex G as the primary physical driver for the enhanced vertical dispersion of the scalar, observed in stationary ground conditions. To quantify the dispersion and characterise these effects, we introduce dispersion parameters, such as non-dimensional dispersion ($\mathscr{D}$) and dispersion length scales ($\mathscr{L}_y, \mathscr{L}_z$). These parameters confirm that, while lateral dispersion remains relatively insensitive to wind tunnel ground conditions, the presence of vortex G in stationary ground conditions leads to an overestimation of vertical dispersion by up to $\approx$29 % ($\phi$ = 25$^\circ$) and $\approx$49 % ($\phi$ = 40$^\circ$). This study quantifies the overestimated dispersion, identifies the vortical structures responsible for scalar redistribution, provides physical insight into the wake dispersion phenomenon and highlights the importance of correct wind tunnel ground conditions in the vehicle wake dispersion studies.

Recently, flow-reversal mechanisms in Rayleigh–Bénard (RB) convection and controlling strategies via modifying local temperature boundaries have received increasing attention due to the impact on heat-transfer efficiency and extreme eruption events. We consider an alternative possibility of altering fluid density: an added scalar field that induces double-diffusive convection, implemented by imposing local iso-concentration bands on the horizontal plates. In addition to the Rayleigh number (${\textit{Ra}}$) and Prandtl number (${\textit{Pr}}$), the system is governed by the Lewis number ($Le$), buoyancy ratio ($Br$), normalised bandwidth ($\delta$) and normalised band-centre-to-midline distance ($c$). We examine the influence of $\delta$ and $c$ on flow reversal at ${\textit{Ra}}=5\times 10^7$, ${\textit{Pr}}=2$, $Le =1$ and ${\textit{Br}}=1.5$. Paired bands effectively reduce reversal frequency, with stronger suppression for larger $\delta$; the optimal band position is $c=0.2$. Fourier mode analysis reveals a previously underappreciated role of the $3\times 3$ roll structure in reversal suppression, whose mean energy correlates positively with the single-roll structure. In standard RB convection, turbulence destabilises the symmetric $2\times 2$ roll configuration, causing frequent reversals owing to competition with asymmetric (1, 1) and (3, 3) modes. The concentration bands enhance the (1, 1) and (3, 3) modal energies, especially at the optimal band position, producing a steady mean flow structure comprising a large-scale circulation and four corner rolls. Despite the local boundary modification, scaling laws for the response parameters (Nusselt number (${\textit{Nu}}$) and Reynolds number (${\textit{Re}}$)) remain close to standard RB convection: ${\textit{Nu}}\sim Ra^{1/3}$ and ${\textit{Re}}\sim Ra^{4/9}Pr^{-2/3}$. These findings demonstrate an effective approach to suppress flow reversal and alter heat transfer efficiency.

Shock tube experiments are essential in understanding the environment encountered by hypersonic vehicles. Such experiments provide information used to determine rate constants of chemical, relaxation and radiative processes taking place in non-equilibrium plasmas. These constants are significant drivers of uncertainty in surface heat flux predictions. Recent work has shown that flow non-uniformities in real shock tube experiments can be misinterpreted as a need to alter these parameters; however, no comprehensive model exists to decouple the effects. We show that there is a rigorous method to achieve this by using experimental measurements as boundary conditions and including their effects via reverse time integration. This method improves over previous implementations by rigorously enforcing conservation laws, incorporating two-temperature, non-equilibrium thermochemistry and explicitly modelling both forward- and backward-running sound waves in the shock tube test slug through a method of characteristics formulation. This approach allowed the effect of shock speed variation in highly non-equilibrium tests, specifically those relevant to Titan entry, to be studied for the first time. A validation study showed that properties predicted by the method were found to agree with results from a viscous, two-dimensional axisymmetric Navier–Stokes solver within 1.5 %. When applied to shock tube test cases from the EAST and T6 facilities for simulation of lunar return and Titan entry representative conditions, the method offered improved agreement with experimentally measured oxygen 777 nm and 240–440 nm radiance, respectively, when compared with previous implementations, particularly towards the rear of the test slug where forward-running sound waves from the driver become influential.

This study explores partial synchronisation in turbulent channel flows using sequential variational data assimilation with sparse observations, emphasising the roles of model and observation uncertainties. Unlike previous work that focused on synchronisation using direct numerical simulation, this study considers synchronisation under imperfect models and noisy data. In the first part, a synchronisation map is constructed, revealing invariance with respect to variations in the predictive model, Reynolds number and mesh resolution. Full synchronisation emerges above a critical level of equivalent observation density. At lower observation densities, modal synchronisation is observed, where the energies of dominant modes evolve independently of initial conditions. As data become sparser, the system transitions to a non-synchronisation regime, with assimilated flows exhibiting minimal correlation with the observations. The second part of this study uses the master flow interpolated from down-sampled sparse observations. The delay-coordinate strategy is introduced to enhance the modal synchronisation. Results indicate that the optimal $\sigma$ lies near the threshold between modal synchronisation and non-synchronisation. This demonstrates that the modal synchronisation serves as a critical prerequisite for leveraging historical information in data assimilation to improve the accuracy of turbulence reconstruction. These findings extend the scope of synchronisation theory and provide valuable guidance for advancing data assimilation methodologies.

In compressible gas–particle flows the dispersion of particle clouds driven by a blast is widely observed in extreme natural and engineering scenarios. Whereas prior research has primarily focused on planar shock or blast-driven configurations, this study investigates a gas–particle system combining a finite-source blast with supersonic inflow. Accordingly, the compressible multiphase particle-in-cell method is employed to simulate the flow. The resulting waves including main shock, contact surface and secondary shock are parametrically investigated, where the main shock radius follows an approximate power law to time. Driven primarily by the drag force, a simplified two-stage scaling law for spanwise leading particle dispersion is derived: a time-squared dependence during the blast-dominated stage and growth behaviour ranging from linear to logarithmic in the subsequent flow-impingement stage. Furthermore, four dispersion morphologies are identified: compressed, uniform, eroded and jetting, each explained by specific wave–particle interaction mechanisms. Finally, a phase diagram correlating these morphologies with the inflow Mach number, Stokes number and pressure ratio is constructed. These findings reveal the coupled mechanisms in gas–particle systems driven by a blast and supersonic inflow, providing a predictive basis for impulse effects and particle dispersion.

Quantifying the contribution of vortex structures to wall forces is essential for identifying the primary sources of forces. The traditional force-element method focuses on the contribution of flow structures to the resultant force. However, the contribution of flow structures to the distributed force cannot be identified. This work proposes a distributed force element method to address this issue. Inspired by the framework of matched asymptotic expansions, the method resolves the surface pressure by matching the fundamental solutions of the outer wave and inner flow regions. The pressure is thus decomposed into contributions from a convective acceleration term, a boundary acceleration term and a boundary vorticity term. The method is implemented by solving the resulting linear system with singular value decomposition. The volume source is further decomposed into direct radiation and boundary scattering components. It is found that compared with the direct radiation component, the boundary scatter component decays fast in the wake. Consequently, the direct radiation component is dominant in the far wake. A finite-domain pressure correction is proposed based on the direct radiation component. The distributed force element method is validated using several benchmark cases: two-dimensional configurations including laminar flow around stationary and oscillating circular cylinders; three-dimensional cases comprising laminar flow past a sphere, subcritical flow past a sphere and laminar flow over an inclined spheroid. The results suggest that the proposed distributed force element method enables the precise quantification of how flow structures in the wake and around the bluff bodies contribute to the surface pressure.

The present study establishes a general theory for fluid-element rotation and intrinsic vorticity decompositions within the framework of vorticity kinematics. We propose two direction-dependent vorticity decompositions (DVDs) based on the analysis of rotation of directed material line and surface elements, with the rigid-rotation and spin modes of vorticity being explicitly defined. Intrinsic coupling relations are then derived for a pair of orthogonal line and surface elements, demonstrating their complementary roles in both kinematics and geometry. Notably, the surface-element-based spin mode is shown to coincide with the relative vorticity in the generalized Caswell formula, thereby providing a faithful representation of surface shear stress in Newtonian fluids. Correspondingly, another two DVDs are constructed based on the geometry of streamlines and streamsurfaces in the field description. Furthermore, within the characteristic algebraic description, in terms of the rotational invariants $(\psi ,\gamma )$ in the real Schur form of the velocity gradient tensor, two invariant vorticity decompositions (IVDs) are formulated. The first IVD with positive spin aligns with the Liutex-shear decomposition, which corresponds to the Klein–Kaden–Betz (KKB) mechanism by which wrapping shear layers form axial vortices. The second IVD is indispensable for understanding unidirectional swirling motion around a point on the rotation-axis-normal plane ${\mathcal{P}}$, corresponding to an anti-KKB mechanism/phenomenon characterized by the negative spin. Importantly, it is proved that the DVD vorticity modes are rigorously bounded by the IVD vorticity modes $(R_{N}^{\pm },s_{N}^{\pm })=(2\psi ^{\pm },\gamma ^{\pm })$ on ${\mathcal{P}}$. Finally, distinctive features and applicability of these kinematic tools are demonstrated with representative examples. The results indicate that a coupled IVD–DVD approach provides a powerful diagnostic tool for unravelling the subtle structures and fundamental physics inherent to complex flow fields.

The influence of hydrodynamic interactions on the schooling behaviour of fish is still poorly understood. This paper numerically investigates the collective motion of two parallel fish that move freely in both the longitudinal and lateral directions, focusing on the effects of wavelength and phase difference on their stable formations, swimming speed and energy efficiency. It is found that two parallel fish can achieve stable formations in both longitudinal and lateral directions, only via the hydrodynamic interactions. Three distinct modes are classified based on the cycle-averaged longitudinal speed, i.e. the steady slow mode, the steady fast mode and the fluctuating fast mode; which mode occurs depends on the wavelength and phase difference. Compared to a single fish, two fish in the steady slow mode swim slower, whereas they swim faster in both the steady fast and fluctuating fast modes, with a maximum speed increase of 12 % observed in the latter mode. Moreover, the fish school exhibits higher propulsive efficiency than a single fish in most cases. Furthermore, the power consumption and propulsive efficiency of each fish in different modes are discussed in detail. Finally, the mechanism behind the stable formations has been analysed. These results may shed some light on understanding the underlying mechanisms of fish schooling behaviour.

Planar particle image velocimetry (PIV) measurements were conducted to investigate turbulent flows through a square duct roughened by transverse rectangular ribs of four blockage ratios (${\textit{Br}}=0.1$, 0.15, 0.2 and 0.25) at a bulk Reynolds number of ${\textit{Re}}_b = 9400$. In contrast to the classical two-dimensional (2-D) rib-roughened boundary-layer flows, the turbulent flow studied here is intrinsically three-dimensional (3-D) and inhomogeneous, complicated by not only the internal shear layers (ISLs) triggered by the rib crests, but also the intense interaction of the four boundary layers developing over duct sidewalls. It is observed that turbulent motions near the rib crest are mainly dominated by the ejection and sweep events. As the blockage ratio increases, the magnitudes of Reynolds stresses near the rib crest increase significantly attributed to enhanced sweep events and large-scale flapping motions. The results of temporal auto-correlations and spatial two-point auto-correlations show that both temporal and spatial integral scales of turbulence structures are dominated by the streamwise velocity fluctuations, which increase as the rib height increases. Based on proper orthogonal decomposition (POD) analyses, it is interesting to observe that the ISL near the rib crest is dominated by both the low- and high-frequency flapping motions characteristic of the first POD mode.

The effectiveness of ultrasonic absorptive coatings (UACs) in achieving delay in turbulent transition on a hypersonic boundary layer over a 3$^\circ$ half-angle cone was investigated under flight-like free-stream disturbance conditions. Tests were conducted at the Boeing/AFOSR Mach 6 Quiet Tunnel at Purdue University for four free-stream Reynolds numbers ranging from $9.0\times 10^6\,$ to $14.3 \times 10^6\,\rm m^{-1}$. Silicon-carbide-coated carbon foams with pore densities of 60, 100 and 200 pores per inch (X0.6, X1, X2) were fabricated as three frustums to vary streamwise location and porous section length. Solid–porous configurations were constructed to analyse the effect of foam length and position. Axisymmetric direct numerical simulations (DNS) and linear-stability theory (LST) analysis were performed to support the experimental findings, modelling the porous foams as time-domain impedance boundary conditions. The UACs influence boundary-layer transition primarily by modifying wall impedance and providing acoustic absorption to weaken second-mode resonance. All porous foams exhibited this behaviour, with the X1 foam achieving the most effective transition delay, strongly dependent on placement. Downstream positioning (59.2–74.3 cm) produced a 13.6 % relative delay, whereas upstream extension (44.1–74.3 cm) led to initial stabilisation followed by a downstream overshoot in second-mode amplitude. The X0.6 and X2 foams showed similar trends. Both LST and DNS predict attenuation of the high-frequency second-mode band and delayed amplification of adjacent low-frequency modes, explaining the overshoot and placement sensitivity. A detailed comparison of $N$-factors shows excellent agreement among experiment, LST and DNS, reinforcing the validity of the combined methodology and the consistency of the identified instability mechanisms.

Natural convection within a heated, inclined slot with wavy walls is investigated. The coupling between the heating and topography patterns determines the properties of the flow and the effectiveness of this interaction can vary significantly as the slot inclination is adjusted. The analysis is two-dimensional and is likely to be a useful model for the flow in a slot of large cross-stream aspect ratio. Typically, the flow topology consists of a combination of rolls and stream tubes that carry the fluid along the conduit. It is shown that a judicious choice of inclination angle and careful positioning of the grooves relative to the temperature pattern can yield a flow rate greater than that achievable within a smooth slot. There is an optimal inclination and phase difference between the groove and heating patterns for which the flow rate is the greatest. The most effective inclination angle is a function of the wavelength and amplitude of the grooves, the heating intensity and the fluid Prandtl number.

While surfactants are known to affect fluid–fluid interfaces, their impact on solid–liquid interfaces is an open problem. Here, we show that surfactants carried by a spreading nonpolar droplet can dynamically alter the solid–liquid interfacial energy, leading to a new spreading regime beyond the classical Tanner’s law and known Marangoni regimes. We develop a theoretical framework that combines the new spreading mechanism, governed by the solid–liquid interfacial energy gradient, together with capillarity. Experiments across twelve distinct combinations of nonpolar solvents, surfactants, and substrates confirm our theoretical predictions for the transition from Tanner’s law to the newly uncovered spreading regime. Our findings provide predictive control for applications in coatings, printing, microfluidics and surface engineering.

The instability of a liquid film in a nanotube is significantly influenced by van der Waals forces. A theoretical framework based on the axisymmetric Stokes equations is developed to investigate their effects through linear stability analysis. The model reveals that van der Waals forces markedly enhance perturbation growth, reduce the dominant wavelength, and lower the critical film thickness that distinguishes collapse from non-collapse regimes. Direct numerical simulations of the Navier–Stokes equations both confirm these theoretical predictions and extend the analysis into the nonlinear regime. In this regime, van der Waals forces are found to alter the interfacial morphology and suppress the formation of satellite lobes. Both rupture and collapse follow a universal temporal scaling law with exponent $1/3$, and exhibit self-similar behaviour near the singularity.

The interaction of a pair of unequal strength counter-rotating vortices is examined using a variety of visualization methods, including volumetric particle image velocimetry. Developed vortex cavitation in the cores of the vortices is also used to characterize the interaction of the initially parallel vortices. A pair of hydrofoils was used to generate two nearly parallel vortices with varying attack angle combinations conditions over a modest range of Reynolds numbers. The vortex pairs that are produced undergo an instability that was first analysed by Crow (1970 AIAA J., vol. 8 (12), pp. 2172–2179), where the vortices interact through mutual induction, eventually leading to large deformations. Velocimetry is used to determine the characteristics for three regimes of the flow: the upstream region, effectively the initial condition of the parallel vortex pair; a midstream region where the vortices are interacting during the linear regime of the instability; a downstream region where the vortical flow is strongly three-dimensional resulting from the nonlinear vortex interactions. Properties of the vortices were measured in all three regions, including the local circulation, core size, eccentricity and velocity along the vortex axis. The rate of vortex stretching for the secondary (weaker vortex) was characterized as it undergoes strong deformation. The observed development of the instability was compared with the predictions of the theory by Crow.

This study examines how wavy orientation and undulation-induced geometric variations regulate vortex formation, wake transitions and aerodynamic performance in sinusoidally wavy cylinders. Using three-dimensional (3-D) simulations at a Reynolds number Re = 100, we analyse the transition from two-dimensional (2-D) to 3-D wakes across varying spanwise wavelengths and undulation configurations. A novel framework is introduced for classifying vortex structures, analyisng centreline trajectories and decomposing vortex structures, revealing how geometric variations induce distinct 3-D vortical structures. At short wavelengths, vortices originate from bluff regions and diminish in a continuous manner, stabilising the wake. At longer wavelengths, phase-dependent vortex onset leads to localised interactions, disrupting wake coherence and delaying stabilisation. A key discovery is the role of transverse recirculating flow in wake stabilisation, which induces reverse impingement, redirects fluid and weakens spanwise vortex coherence. Additionally, wavy orientation strongly influences vortex evolution and dislocation, altering vortex trajectories and wake stability. To further clarify these wake transitions, a classification framework is introduced, defining distinct phases such as vortex stretching, break-up and re-symmetrisation. The relationship between force characteristics and wake stabilisation is also established, with wavy orientation and undulation geometry regulating the transition from quasi-2-D spanwise vortical flow to 3-D spiral flow. A critical wavelength is identified where drag and lift fluctuations are minimised, with elliptical-section undulations achieving superior aerodynamic performance through enhanced vortex synchronisation. These findings provide new insights into vortex control strategies, with applications in bio-inspired propulsion, passive flow control and energy-efficient aerodynamic designs across engineering and industrial fields.

This study investigates the role of vibrational and chemical non-equilibrium mechanisms in the evolution of pressure-Hessian and velocity gradient tensors in high-temperature compressible turbulence. Specifically, it focuses on reacting air mixtures relevant to aerospace applications. Understanding these mechanisms is essential for accurately predicting turbulent flows encountered during atmospheric re-entry of spacecraft and cruise flights of hypersonic vehicles. We employ direct numerical simulation (DNS) of isotropic compressible decaying turbulence using the hy2Foam solver on the OpenFOAM platform, with detailed finite-rate chemistry and vibrational energy exchanges among five species (N$_{2}$, O$_{2}$, NO, N and O). Our findings reveal that vibrational and chemical non-equilibrium mechanisms do influence the statistics of turbulent flows in a reacting air mixture. Specifically, chemical non-equilibrium processes associated with species production dominate the evolution of the pressure-Hessian tensor in air mixtures. Vibrational non-equilibrium, significant in a nitrogen-only flow, becomes insignificant in a reacting air mixture. Additionally, air mixture interactions result in an increase in the vortical fluctuations and a decrease in the dilatational fluctuations, along with a reduction in the strength of the pressure-Hessian tensor relative to the velocity gradient tensor. These results highlight the importance of accurately modelling chemical and vibrational non-equilibrium mechanisms in high-temperature compressible turbulent flows.

The evaporation of multicomponent sessile droplets is key in many physicochemical applications such as inkjet printing, spray cooling and micro-fabrication. Past fundamental research has primarily concentrated on single drops, though in applications they are rarely isolated. Here, we experimentally explore the effect of neighbouring drops on the evaporation process, employing direct imaging, confocal microscopy and particle tracking velocimetry. Remarkably, the centres of the drops move away from each other rather than towards each other, as we would expect due to the shielding effect at the side of the neighbouring drop and the resulting reduced evaporation on that side. We hypothesise that pinning-induced motion mediated by suspended particles in the droplets (due to contamination or added on purpose) is the cause of this counter-intuitive behaviour. We also discuss an alternative interpretation, namely that the repulsion between the two droplets is caused by thermal Marangoni flow as is the case for a pair of pure droplets on an isothermal substrate (Malachtari and Karapetsas, J. Fluid Mech. vol. 978, 2024, p. A8), but give the arguments why that interpretation is not applicable in our case of binary droplets. To further support our interpretation, with the help of direct numerical simulations we explore the relative contributions of the replenishing flow and of the solutal and thermal Marangoni flows to the overall flow dynamics in one droplet. Finally, as further evidence, the azimuthal dependence of the radial velocity in the drop is compared with the evaporative flux and a perfect agreement is found.

The propulsion of a flapping wing or foil is emblematic of bird flight and fish swimming. Previous studies have identified hallmarks of the propulsive dynamics that have been attributed to unsteady effects such as the formation and shedding of edge vortices and wing–vortex interactions. Here, we show that several key features of heaving flight are captured by a quasi-steady aerodynamic model that aims to predict stroke-averaged forces from wing motions without explicitly solving for the flows. We address the forward dynamics induced by up-and-down heaving motions of a thin plate with a nonlinear model which involves lift and drag forces that vary with speed and attack angle. Simulations reproduce the well-known transition for increasing Reynolds number from a stationary state to a propulsive state, where the latter is characterised by a Strouhal number that is conserved across broad ranges of parameters. Parametric, sensitivity and stability analyses provide physical interpretations for these results and show the importance of accounting for the flow regimes which are demarcated by Reynolds number and angle of attack. These findings extend the phenomena of unsteady locomotion that can be explained by quasi-steady modelling, and they broaden the conditions and parameter ranges over which such models are applicable.

Direct numerical simulations of turbulence in a flexible pipe with imposed standing-wave vibration are performed to reveal the flow dynamics inside an oscillating pipe. We choose the parameters of standing-wave vibration with small amplitude as the most unstable mode in flow-induced free vibration. The flow is driven under the condition of constant mass flow rate, with the bulk Reynolds number, based on the bulk velocity and pipe diameter, being ${\textit{Re}}_b$ = 5300. In response to the imposed vibration, the evolution of the flow inside manifests obvious space–time-dependent characteristics. Specifically, the streamwise velocity fluctuation is enhanced downstream of the crest – the convex region on the internal pipe wall – an event often accompanied by localised flow separation. Meanwhile, the two other components of velocity fluctuation are augmented downstream of the trough – the concave region of the wall’s sinusoidal undulation. This is attributed to the wall deformation, which forces a redistribution of turbulent kinetic energy among the components. The latter process gives rise to a high-level fluctuation of wall shear stresses, leading to the intermittent variation of the drag force in that region. In addition, secondary flow emerges in the form of a typical counter-rotating vortex pair due to the bending of pipe, with the vortex cores located near the wall. The temporal variation of the magnitude of secondary flow lags slightly behind the pipe vibration and its maximum occurs closer to the node where the pipe displacement is consistently zero. Moreover, the secondary flow intensity increases with the increasing of steepness and a slight drag reduction can be achieved with relatively low-wavenumber vibration.

In this work, the classical Prandtl relation for the skin-friction law of incompressible turbulent channel and pipe flows is generalised to compressible cases. Specifically, based on the law of the wall and asymptotic analysis, a skin-friction transformation is proposed to map the skin-friction law of compressible turbulent channel and pipe flows to the classical Prandtl relation. It has been theoretically proven that the skin-friction coefficient $C_{\!f,i}$ and the bulk Reynolds number $\textit{Re}_{b,i}$ for compressible turbulent channel and pipe flows, where the subscript $i$ denotes the transformed quantity obtained from the proposed skin-friction transformation, adhere to the Prandtl relation, expressed as $\sqrt {2/C_{\!f,i}}\propto \ln (\textit{Re}_{b,i}\sqrt {C_{\!f,i}/2})$. Moreover, it is quantitatively verified that the transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$, obtained from direct numerical simulations (DNS) of compressible turbulent channel flows with bulk Mach numbers ranging from $0.2$ to $4$, and friction Reynolds numbers from $200$ to $2000$, elegantly collapse into the Prandtl relation for the incompressible skin-friction law. Additionally, the transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$ from DNS of compressible turbulent pipe flows, with bulk Mach numbers ranging from $1.5$ to $3$, and friction Reynolds numbers from $200$ to $1000$, are also unified with the Prandtl relation for the incompressible skin-friction law.