Plane Poiseuille Flow Research Articles

THE EFFECT OF VISCOUS DISSIPATION ON THE ONSET OF INSTABILITY OF MAGNETO HYDRODYNAMIC FLOW IN A POROUS MEDIUM

The study of flow of fluid with its motion was study in the nineteenth century by Maurice Couette which was as a result of two parallel plates in which the movement are relative, such that the surfaces is moving laterally. The plates with varying radii, could be flat, parallel or two concentric which is generally referred to as plane Couette. Characteristics of fluid at rest or in motion may undergo changes and become unstable. The critical value or range of the parameters of flow which will give rise to instabilities is one of the problems of stability analysis. The stability of fluid motion may be tested by perturbing the fluid with a small sinusoidal disturbances on the parameter. In plane Couette where moving plates drive the flow and plane Poiseuille flow where a pressure gradient drives the flow, in the situation of weak flow and strong distortional elasticity, an asymptotic analysis will yields closed - form steady solutions with identical wall conditions which will focus on simulations will expose the effects due to wall anchoring conflicts and illustrate the induced morphology of the orientation distribution, stored viscoelastic stresses and non - Newtonian flow.

The Onset of Instability in A Magnetohydrodynamic Channel Flow through Porous Media of Casson Fluid

A detailed study is made on the stability of linear two-dimensional disturbances of Plane Poiseuille Flow (PPF) of Casson fluid through porous media in the presence of a vertical uniform magnetic field, B0 which is extremely useful in metals, mines, and fuels industries. Using the method of normal modes, the disturbance equations are derived. The resulting eigenvalue problem is then solved by the spectral collocation method using Chebyshev-based polynomials. The critical values of the triplets ( Rec, αc, cc ) are obtained for various values of the Casson parameter, η , Hartmann number, Ha , and porous parameter, σp. The stability of the system is discussed using the neutral stability curves for each value of the parameters present in the problem. It is found that the stability regions are enlarged for small values of η and large values of the porous parameter, σp and Hartmann number, Ha. It is also observed that the stability characteristics of plane Poiseuille flow in a porous medium are remarkably different from non-porous cases. The results obtained here contribute to the contemporary efforts to better understand the stability characteristics of PPF of Casson fluid flow through porous media in the presence of a uniform transverse magnetic field.

Two-dimensional strain rate imaging study using a polarization camera and birefringent aqueous cellulose nanocrystal suspensions

The birefringence response of aqueous cellulose nanocrystal (CNC) suspensions in a two-dimensional laminar flow is measured and studied. The suspensions have CNC concentrations of 1.0 wt% (weight percentage) and 1.2 wt%. Cellulose nanocrystals are optically anisotropic rod-like particles that align when subjected to local velocity gradients, whereas at rest, they remain randomly orientated by Brownian motion. The alignment causes birefringence, a phenomenon also known as flow-induced birefringence. We study the flow through an additively manufactured flow channel and measure the amount of birefringence as well as the position of the refractive index axes by using polarizers and a polarization camera. With the help of reference data published in a previous study (Lane, Rode, et al., 2022a), strain rates are derived from the birefringence measurements and compared with numerical simulations. Two flow situations are studied, a plane Poiseuille flow and the flow around a cosine-shaped constriction. The experimentally derived shear rates for the plane Poiseuille flow are consistent with theoretical and computational results. The derived strain rates for the flow around the cosine-shaped constriction show an unexpected asymmetric profile, with the strain rates in the contraction zone being larger than in the expansion zone. The averaged orientation of the CNCs in the flow is linked to the position of the refractive index axes. In the contraction zone, the CNCs tend to align parallel to the flow, whereas in the expansion zone, the CNCs tend to align perpendicular to the flow. The results of this study are discussed in the context of previous, similar studies. The asymmetric strain rate profile around the cosine-shaped constriction is thought to originate from history effects, and the alignment of the CNCs is influenced by extensional rates.

Rheodynamics of viscoelastic subdiffusive channel flows: Low Weissenberg number regime

Direct numerical simulations of viscoelastic, subdiffusive, and plane Poiseuille flow, representing thick polymer solutions including polymer melts, flows of liquid crystals, as well as biological flow such as mucus, are reported. The stress strain equation is fundamentally derived by relating the timescale exponent at the microscale (tα, with 0<α≤1) with the fractional order of the time derivative, α, of the corresponding non-linear equation in the continuum. The resultant stress constitutive equation is the fractional variant of the well-known upper convected Maxwell equation. In order to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia), the space of symmetric positive definite polymer conformation tensors is visualized using a Riemannian metric along with its three scalar invariants. Numerical simulations of the channel flow, in the regime of low to moderate Reynolds number and low Weissenberg number, effectively capture these flow-structures by providing (i) a better resolution of the instantaneous regions of elastic shocks (which are the alternating regions of expanded and compressed polymer volume, in comparison with the volume of the mean conformation tensor), and (ii) a better resolution to detect neighborhoods where the mean conformation tensor tends to be significantly different in comparison to the instantaneous conformation tensor, thereby corroborating the experimentally observed flow-instability transition of subdiffusive flows. Finally, the strength of the subdiffusive flow model and the invariant theory is highlighted through an application of an electrohydrodynamic micropump.

Numerical simulations of Phan-Thien-Tanner viscoelastic fluid flows based on the SPH method

Numerical simulations of viscoelastic fluid flows are always a hot issue in the field of computational fluid dynamics and the study of their complex rheological properties has important academic and engineering application value. In the present work, we develop a smoothed particle hydrodynamics (SPH) method for simulating transient viscoelastic fluid flows governed by the Phan-Thien-Tanner (PTT) constitutive equation. To improve the computational accuracy of the SPH method, the mixed symmetric correction algorithm of kernel gradient is implemented. To remove the particle clustering and unphysical fracture in fluid stretching which is named as the tensile instability, the artificial stress model is added into the momentum equation. We firstly apply the proposed SPH method to solve the plane Poiseuille flow of a PTT viscoelastic fluid, in which the effectiveness and advantage of the method are verified by comparing the SPH solution with those obtained by the finite volume method (FVM) and analyzing the l2 norm error of different SPH solutions to the FVM solution. Then, the method is employed to simulate the impact behavior of a PTT viscoelastic droplet with a rigid plate. In particular, we not only investigate the spreading behavior of PTT viscoelastic droplet after impacting the rigid plate, but also for the first time capture and analyze the bouncing behavior of droplet by decreasing the Reynolds number. The influences of the Reynolds number, the Weissenberg number, the solvent viscosity ratio, and the PTT elongational parameter on the droplet dynamics behavior are further deeply studied. Numerical results demonstrate that the SPH method proposed in this paper is a powerful computation tool for simulating PTT viscoelastic fluid flows and is capable of effectively describing their complex rheological properties and free surface variation characteristics.

Algebraic growth of 2D optimal perturbation of a plane Poiseuille flow in a Brinkman porous medium

The impact of various factors on the stability of a plane porous Poiseuille flow is investigated in detail. Both modal and non-modal linear stability analyses are used to study the effects of the porosity of the porous media and the ratio of effective viscosity to the fluid viscosity. The present analysis includes solving the linearized Navier-Stokes equation in the form of the Orr-Sommerfeld (O-S) type equation by applying the 2D perturbation to the basic mean flow. The Chebyshev collocation method is used to solve the Orr-Sommerfeld equation numerically. Through modal analysis, the accurate values of the critical triplets (αc,Rc,cc), the eigen-spectrum, the growth rate curves, and the marginal stability curves are studied. Then, by using non-modal analysis, the transient energy growth G(t) of two-dimensional optimal perturbations, the ε-pseudospectrum of the non-normal O-S operator (L), and the regions of stability, instability, and potential instability of the fluid flow system are investigated in detail. The collective results of both modal and non-modal analysis show that the porous parameter and the ratio of effective viscosity to the fluid viscosity have stabilizing effects on the fluid system due to the postponement of the onset of stability.

Analytical solution for channel flow of a Giesekus fluid with non-zero solvent viscosity

A semi analytical solution is obtained here for the fully developed plane Poiseuille flow of a Giesekus fluid with a Newtonian solvent. The fluid behaviour is described using the Deborah number, the mobility factor and an appropriate ratio of fluid viscosity to total viscosity as parameters. The given solution shows that the velocity increases significantly with rising the polymer concentration, confirming that dilution of the solution produces the same effect as an increase in resistance. The analysis demonstrates that there are limiting values of Deborah number related to the mobility parameter.

Inertial migration of a neutrally buoyant spheroid in plane Poiseuille flow

We study the cross-stream inertial migration of a torque-free neutrally buoyant spheroid, of an arbitrary aspect ratio $\kappa$ , in wall-bounded plane Poiseuille flow for small particle Reynolds numbers ( $Re_p\ll 1$ ) and confinement ratios ( $\lambda \ll 1$ ), with the channel Reynolds number, $Re_c = Re_p/\lambda ^2$ , assumed to be arbitrary; here $\lambda =L/H$ , where $L$ is the semi-major axis of the spheroid and $H$ denotes the separation between the channel walls. In the Stokes limit ( $Re_p =0)$ , and for $\lambda \ll 1$ , a spheroid rotates along any of an infinite number of Jeffery orbits parameterized by an orbit constant $C$ , while translating with a time-dependent speed along a given ambient streamline. Weak inertial effects stabilize either the spinning ( $C=0$ ) or tumbling orbit ( $C=\infty$ ), or both, depending on $\kappa$ . The asymptotic separation of the Jeffery rotation and orbital drift time scales, from that associated with cross-stream migration, implies that migration occurs due to a Jeffery-averaged lift velocity. Although the magnitude of this averaged lift velocity depends on $\kappa$ and $C$ , the shape of the lift profiles are identical to those for a sphere, regardless of $Re_c$ . In particular, the equilibrium positions for a spheroid remain identical to the classical Segre–Silberberg ones for a sphere, starting off at a distance of about $0.6(H/2)$ from the channel centreline for small $Re_c$ , and migrating wallward with increasing $Re_c$ . For spheroids with $\kappa \sim O(1)$ , the Jeffery-averaged analysis is valid for $Re_p\ll 1$ ; for extreme aspect ratio spheroids, the regime of validity becomes more restrictive being given by $Re_p \kappa /\ln \kappa \ll 1$ and $Re_p/\kappa \ll 1$ for $\kappa \rightarrow \infty$ (slender fibres) and $\kappa \rightarrow 0$ (flat disks), respectively.

Vorticity cascade and turbulent drag in wall-bounded flows: plane Poiseuille flow

Drag for wall-bounded flows is directly related to the spatial flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution to this wall-normal flux arises from nonlinear advection and stretching of vorticity, interpretable as a cascade. We study this process using numerical simulation data of turbulent channel flow at friction Reynolds number $Re_\tau =1000$ . The net transfer from the wall of spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is ‘down-gradient’ or outward from the wall, where most vorticity concentrates, and the other which is ‘up-gradient’ or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient/down-gradient transport occurs by a mechanism of correlated inflow/outflow and spanwise vortex stretching/contraction that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism, we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra identifying eddies involved in nonlinear vorticity transport in the two opposing directions and (iii) visualizations of coherent vortex structures which contribute to the transport. The ‘D-type’ vortices contributing to down-gradient transport in the log layer are found to be attached, hairpin-type vortices. However, the ‘U-type’ vortices contributing to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.

Transit Time Theory for a Droplet Passing through a Slit in Pressure-Driven Low Reynolds Number Flows.

Soft objects squeezing through small apertures are crucial for many in vivo and in vitro processes. Red blood cell transit time through splenic inter-endothelial slits (IESs) plays a crucial role in blood filtration and disease progression, while droplet velocity through constrictions in microfluidic devices is important for effective manipulation and separation processes. As these transit phenomena are not well understood, we sought to establish analytical and numerical solutions of viscous droplet transit through a rectangular slit. This study extends from our former theory of a circular pore because a rectangular slit is more realistic in many physiological and engineering applications. Here, we derived the ordinary differential equations (ODEs) of a droplet passing through a slit by combining planar Poiseuille flow, the Young-Laplace equations, and modifying them to consider the lubrication layer between the droplet and the slit wall. Compared to the pore case, we used the Roscoe solution instead of the Sampson one to account for the flow entering and exiting a rectangular slit. When the surface tension and lubrication layer were negligible, we derived the closed-form solutions of transit time. When the surface tension and lubrication layer were finite, the ODEs were solved numerically to study the impact of various parameters on the transit time. With our solutions, we identified the impact of prescribed pressure drop, slit dimensions, and droplet parameters such as surface tension, viscosity, and volume on transit time. In addition, we also considered the effect of pressure drop and surface tension near critical values. For this study, critical surface tension for a given pressure drop describes the threshold droplet surface tension that prevents transit, and critical pressure for a given surface tension describes the threshold pressure drop that prevents transit. Our solutions demonstrate that there is a linear relationship between pressure and the reciprocal of the transit time (referred to as inverse transit time), as well as a linear relationship between viscosity and transit time. Additionally, when the droplet size increases with respect to the slit dimensions, there is a corresponding increase in transit time. Most notably, we emphasize the initial antagonistic effect of surface tension which resists droplet passage but at the same time decreases the lubrication layer, thus facilitating passage. Our results provide quantitative calculations for understanding cells passing through slit-like constrictions and designing droplet microfluidic experiments.

Dynamics and proliferation of turbulent stripes in plane-Poiseuille and plane-Couette flows

The first long-lived turbulent structures observable in planar shear flows take the form of localized stripes, inclined with respect to the mean flow direction. The dynamics of these stripes is central to transition, and recent studies proposed an analogy to directed percolation where the stripes’ proliferation is ultimately responsible for the turbulence becoming sustained. In the present study we focus on the internal stripe dynamics as well as on the eventual stripe expansion, and we compare the underlying mechanisms in pressure- and shear-driven planar flows, respectively, plane-Poiseuille and plane-Couette flow. Despite the similarities of the overall laminar–turbulence patterns, the stripe proliferation processes in the two cases are fundamentally different. Starting from the growth and sustenance of individual stripes, we find that in plane-Couette flow new streaks are created stochastically throughout the stripe whereas in plane-Poiseuille flow streak creation is deterministic and occurs locally at the downstream tip. Because of the up/downstream symmetry, Couette stripes, in contrast to Poiseuille stripes, have two weak and two strong laminar turbulent interfaces. These differences in symmetry as well as in internal growth give rise to two fundamentally different stripe splitting mechanisms. In plane-Poiseuille flow splitting is connected to the elongational growth of the original stripe, and it results from a break-off/shedding of the stripe's tail. In plane-Couette flow splitting follows from a broadening of the original stripe and a division along the stripe into two slimmer stripes.

Predicting the energy stability limit of shear flows using weighted velocity components

Predicting the subcritical transition in fluid dynamic systems remains a challenging task, but recent advancements utilizing edge tracking methods, polynomial Lyapunov functions, and various energy norms have shown promise. In this study, we propose a novel approach by defining the general kinetic energy through weighted velocity components. The minimal Reynolds number is determined, where the derivative of this generalized energy with respect to time is zero. The procedure is similar to that of the well-known Reynolds–Orr equation. Unlike traditional methods, our approach does not necessitate the monotonic decay of the classic perturbation kinetic energy, resulting in a larger critical Reynolds number and reduced conservativeness of the Reynolds–Orr equation. However, the energy production of the pressure is not negligible, in contrast to the classical Reynolds–Orr equation. The pressure's implicit dependence on the velocity field complicates the variation process. To address this, a method is presented to handle the problem effectively. Our approach is then applied to analyze parallel flows, specifically the plane Couette and plane Poiseuille flows, wherein the problem can be further simplified using the complex Fourier transformation. The weights of velocity components are optimized to maximize the critical Reynolds number, resulting in a significant increase.

Flow and heat transfer analysis of Couette and Poiseuille flow of a hybrid nanofluid with temperature-dependent viscosity and thermal conductivity

The purpose of this research is to quantify the influence of a number of independent parameters (−0.5≤α1,β1,β2≤0.5) on the heat transmission rate and the temperature of a nanofluid hybrid flowing between parallel plates. The linked nonlinear momentum and energy equations are solved using the regular perturbation method for two different cases resembling the plane Couette flow and Poiseuille flow. By combining engine oil with multiwalled carbon nanotubes (MWCNT) and aluminum oxide nanoparticles, a hybrid nanofluid is produced with wide-ranging lubricating and industrial applications. It has been found that the hybrid nanofluid can function as a coolant provided the relevant parameters are adjusted appropriately. It is demonstrated that these additives have a tendency to stabilise the thermal and the volume flow rate, hence decreasing randomization that is necessary for a good lubricant to function. The findings of the study indicate that β1 has a greater influence on temperature, while β2 has a stronger effect on velocity. In addition to the variable model, the results for the constant model have also been deduced, and the novel study demonstrates that the considered variable viscosity model tends to produce a more uniform temperature distribution.

Calculation of linear stability of fluid flow in a flat channel with walls wavy across the flow

Using the full Navier-Stokes equations, the linear stability of a plane Poiseuille flow in a channel with a corrugated bottom wall is considered. The wall is corrugated across the flow, and the main flow has one velocity component. The perturbations of the velocity and pressure fields are three-dimensional with two wave numbers. The generalized eigenvalue problem is solved numerically. It is found that the critical Reynolds number, above which time-increasing disturbances appear, depends in a complex way on the dimensionless amplitude and the corrugation period. The magnitude of the ratio of the amplitude and the corrugation period divides the area of the dimensionless amplitude of the corrugation into two, where the dependences of the critical Reynolds number on the parameters of the corrugation are qualitatively different.

Structured input–output analysis of oblique laminar–turbulent patterns in plane Couette–Poiseuille flow

In this work, we employ structured input–output analysis to study the flow patterns in transitional plane Couette–Poiseuille flow (CPF). First, we focus on the well-studied intermediate laminar profile, which balances the shear and pressure effects. We show that the highest structured gain corresponds to perturbations with wavelengths associated with the oblique turbulent bands observed in experiments. In addition, the inclination angles of these structures show a Reynolds number dependence consistent with experimentally observed trends. We then examine the Reynolds number scaling of the maximal structured frequency response as the velocity profile is varied from plane Couette to Poiseuille flow. Our results demonstrate that, as expected, the scaling exponent increases over this range, but this increase is not monotonic. We attribute the variation in the shape of the scaling exponent curve as a function of the velocity profile shape to changes in flow patterns that dominate each associated flow regime. We then focus on the particular case of plane Couette flow and compute the structured response modes. Their behavior and structural features are consistent with results obtained through direct numerical simulation (DNS) studies. Finally, we employ the structured analysis framework to examine the temporal evolution of the dominant structures. For the well-studied cases of plane Couette and plane Poiseuille flows, the computed advection speeds of the oblique structures are consistent with those predicted through DNS.

Stability of diffusion flames under shear flow: Taylor dispersion and the formation of flame streets

Diffusion flame streets, observed in non-premixed micro-combustion devices, align parallel to a shear flow. They are observed to occur in mixtures with high Lewis number (Le) fuels, provided that the flow Reynolds number, or the Peclet number Pe, exceeds a critical value. The underlying mechanisms behind these observations have not yet been fully understood. In the present paper, we identify the coupling between diffusive-thermal instabilities and Taylor dispersion as a mechanism which is able to explain the experimental observations above. The explanation is largely based on the fact that Taylor dispersion enhances all diffusion processes in the flow direction, leading effectively to anisotropic diffusion with an effective (flow-dependent) Lewis number in the flow direction which is proportional to 1/Le for Pe≫1. Validation of the identified mechanism is demonstrated within a simple model by investigating the stability of a planar diffusion flame established parallel to a plane Poiseuille flow in a narrow channel. A linear stability analysis, leading to an eigenvalue problem solved numerically, shows that cellular (or finite wavelength) instabilities emerge for high Lewis number fuels when the Peclet number exceeds a critical value. Furthermore, for Peclet numbers below this critical value, longwave instabilities with or without time oscillations are obtained. Stability regime diagrams are presented for illustrative cases in a Le−Pe plane where various instability domains are identified. Finally, the linear analysis is supported and complemented by time dependent numerical simulations, describing the evolution of unstable diffusion flames. The simulations demonstrate the existence of stable cellular structures and show that the longwave instabilities are conducive to flame extinction.

A new elastic instability in gravity-driven viscoelastic film flow

We examine the linear stability of the gravity-driven flow of a viscoelastic fluid film down an inclined plane. The viscoelastic fluid is modeled using the Oldroyd-B constitutive equation and, therefore, exhibits a constant shear viscosity and a positive first normal stress difference in viscometric shearing flows; the latter class of flows includes the aforesaid film-flow configuration. We show that the film-flow configuration is susceptible to two distinct purely elastic instabilities in the inertialess limit. The first instability owes its origin entirely to the existence of a free surface and has been examined earlier [Shaqfeh et al., “The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number,” J. Non-Newtonian Fluid Mech. 31, 87–113 (1989)]. The second one is the analog of the centermode instability recently discovered in plane Poiseuille flow [Khalid et al., “Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow,” Phys. Rev. Lett. 127, 134502 (2021)] and owes its origin to the base-state shear; it is an example of a purely elastic instability of shearing flows with rectilinear streamlines. One may draw an analogy of the aforesaid pair of unstable elastic modes with the inertial free-surface and shear-driven instabilities known for the analogous flow configuration of a Newtonian fluid. While surface tension has the expected stabilizing effect on the Newtonian and elastic free-surface modes, its effect on the corresponding shear modes is, surprisingly, more complicated. For both the Newtonian shear mode and the elastic centermode, surface tension plays a dual role, with there being parameter regimes where it acts as a stabilizing and destabilizing influence. While the Newtonian shear mode remains unstable in the limit of vanishing surface tension, the elastic centermode becomes unstable only when the appropriate non-dimensional surface tension parameter exceeds a threshold. In the limit of surface tension being infinitely dominant, the free-surface boundary conditions for the film-flow configuration reduce to the centerline symmetry conditions satisfied by the elastic centermode in plane Poiseuille flow. As a result, the regime of instability of the film-flow centermode becomes identical to that of the original channel-flow centermode. At intermediate values of the surface tension parameter, however, there exist regimes where the film-flow centermode is unstable even when its channel-flow counterpart is stable. We end with a discussion of the added role of inertia on the aforementioned elastic instabilities.

Stability of plane Poiseuille and Couette flows of Navier–Stokes–Voigt fluid

Stability of plane Poiseuille and Couette flows of Navier–Stokes–Voigt fluid

Migration and heat transfer modeling of a neutrally buoyant melting particle in Poiseuille flow

A computational model is developed to simulate the hydrodynamic and heat transfer behavior of a melting cylindrical solid particle in plane Poiseuille flow between horizontal parallel plates. The two-dimensional transient conservation equations for mass, momentum, and energy are solved using a finite-volume scheme implemented on a deforming mesh, accounting for the rotation and non-uniform melting of the particle. An arbitrary Lagrangian–Eulerian (ALE) method is employed to directly track the moving and deforming solid–liquid interface during the particle migration and phase change. The developed model was validated by comparison with the previously reported numerical results for migration of a non-melting neutrally buoyant cylindrical particle in plane Poiseuille flow with heat transfer. The effects of flow Reynolds (Re), Grashof (Gr), and Stefan (Ste) numbers, as well as the initial position of the melting particle across the channel on the particle trajectory, melting rate, and average Nusselt (Nu) number were investigated. It was found that the melting rate increased by increasing Gr and Ste and decreased by increasing Re. The Nu did not change more than 20% with increasing Re number from 100 to 1000, but increased significantly by increasing Gr. It was also observed that by increasing Re, the particle migrates toward the channel center because of the stronger Magnus effect. Increasing Gr, on the other hand, pushes the particle to the bottom wall due to the stronger downward flow adjacent to the particle induced by buoyancy-driven convection (natural convection).

Study of compressible turbulent plane Couette flows via direct numerical simulation

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.