Compressible Turbulent Channel Flow Research Articles

Direct numerical simulation of compressible turbulent channel flows over porous boundaries

The impact of porous boundaries on skin friction and heat transfer in compressible turbulent channel flows is explored through direct numerical simulations. Utilizing the volume-averaged Navier–Stokes equations for the porous media, we examine flows at Mach numbers Mb=0.3 and Mb=1.5, with friction Reynolds numbers ranging from 176 to 220, across three permeability levels: 0 (impermeable), 1.6×10−5, and 1.0×10−4. Our findings reveal a dual impact of porous boundaries on skin friction: a direct reduction due to averaged slip velocity at the interface and an indirect increase from amplified Reynolds shear stress due to enhanced wall-normal permeability. This permeability weakens the wall-blocking effect, leading to intensified wall-normal velocity fluctuations and consequently higher Reynolds stress. Additionally, the Mach number significantly influences flow structures, particularly at higher permeabilities, where it disrupts near-wall streaks at Mb=0.3 but has a diminished effect at Mb=1.5. Thermal analysis confirms the validity of the generalized Reynolds analogy for the mean temperature in these flows, with the turbulent Prandtl number approximating 1, in line with the strong Reynolds analogy. Moreover, a strong spatial correlation between heat flux and wall shear stress fluctuations underscores the intertwined dynamics of momentum and thermal transport at the porous–fluid interface.

Investigation of the inclination angles of wall-attached eddies for streamwise velocity and temperature fields in compressible turbulent channel flows

Investigation of the inclination angles of wall-attached eddies for streamwise velocity and temperature fields in compressible turbulent channel flows

On the streamwise velocity, temperature and passive scalar fields in compressible turbulent channel flows: a viewpoint from multiphysics couplings

It is generally believed that the velocity and passive scalar fields share many similarities and differences in wall-bounded turbulence. In the present study, we conduct a series of direct numerical simulations of compressible channel flows with passive scalars and employ the two-dimensional spectral linear stochastic estimation and the correlation function as diagnostic tools to shed light on these aspects. Particular attention is paid to the relevant multiphysics couplings in the spectral domain, i.e. the velocity–temperature ( $u-T$ ), scalar–temperature ( $g-T$ ) and velocity–scalar ( $u-g$ ) couplings. These couplings are found to be utterly different at a given wall-normal position in the logarithmic and outer regions. Specifically, in the logarithmic region, the $u-T$ and $u-g$ couplings are tight at the scales that correspond to the attached eddies and the very large-scale motions (VLSMs), whereas the $g-T$ coupling is robust in the whole spectral domain. In the outer region, $u-T$ and $u-g$ couplings are only active at the scales corresponding to the VLSMs, whereas the $g-T$ coupling is diminished but still strong at all scales. Further analysis indicates that although the temperature field in the vast majority of zones in a channel can be roughly treated as a passive scalar, its physical properties gradually deviate from those of a pure passive scalar as the wall-normal height increases due to the enhancement of the acoustic mode. Furthermore, the deep involvement of the pressure field in the self-sustaining process of energy-containing motions also drives the streamwise velocity fluctuation away from a passive scalar. The current work is an extension of our previous study (Cheng & Fu, J. Fluid Mech., vol. 964, 2023, A15), and further uncovers the details of the multiphysics couplings in compressible wall turbulence.

Turbophoresis and preferential accumulation of inertial particles in compressible turbulent channel flow: Effect of Mach number

Turbophoresis and preferential accumulation of inertial particles in compressible turbulent channel flow: Effect of Mach number

A wall model for large-eddy simulation of highly compressible flows based on a new scaling of the law of the wall

Wall modelling in large-eddy simulation (LES) is of high importance to allow scale resolving simulations of industrial applications. Numerous models were developed and validated for incompressible flows, including a simple quasi-analytical model based on Reichardt's formula that approximates the law of the wall. In this paper, a scaling is proposed to generalize this wall model to highly compressible flows. First, the results of wall-resolved LES (wrLES) of adiabatic compressible turbulent channel flows at $Re_\tau = 1000$ and at centreline Mach numbers of $M_c= 0.76$ and $1.5$ are presented. Then, three potential scalings of the incompressible wall model are proposed, and their a priori performance is evaluated : (i) the Howarth–Stewartson scaling, (ii) an improved Van Driest scaling and (iii) a new scaling obtained from a blending of those two. The results of wall-modelled LES (wmLES) of compressible channel flows using these three models are compared with the reference wrLES data, showing the superior accuracy of the hybrid scaling. The consistency of the new wall model at low Mach numbers is also verified by comparing the results of a wmLES at $M_c= 0.25$ with those of reference incompressible DNS data at $Re_\tau = 1000$ and $5200$ . Finally, the proposed wall model is also applied to a turbulent channel flow at $M_c=1.5$ and $Re_\tau =5200$ .

Total and static temperature statistics in compressible turbulent plane channel flow

The paper studies the statistics of total and static temperature (total $h_t$ and static $h$ enthalpy) in compressible turbulent plane channel flow using direct numerical simulations (DNS) data covering the range of centreline Mach numbers $0.3\lessapprox \bar {M}_{{{CL}}_x} \lessapprox 2.5$ and Huang–Coleman–Bradshaw friction Reynolds numbers $100\lessapprox Re_{\tau ^\star }\lessapprox 1000$ . For this class of very-cold-wall flows, the DNS data for correlation coefficients and joint probability density functions (p.d.f.s) show that $h_t'$ is invariably very strongly correlated with the streamwise velocity fluctuation $u'$ , in contrast to static temperature (static enthalpy $h'$ ) whose correlation with $u'$ weakens rapidly with increasing wall distance. We study various correlations and joint p.d.f.s of $h_t'$ and $h'$ in relation to the fluctuating velocity field, including the turbulent Prandtl number $Pr_{{T}}$ , and discuss the predictions of Reynolds analogy. The scaling of the mean enthalpy and the fluctuating enthalpy variance and fluxes with respect to inner and outer velocity scales is investigated. The complex behaviour and scaling of different terms in the transport equations for the enthalpy variance and fluxes are discussed.

Study of the linear models in estimating coherent velocity and temperature structures for compressible turbulent channel flows

Linear models, based on stochastically forced linearized equations, are deployed for spectral linear stochastic estimation (SLSE) of the velocity and temperature fluctuations in compressible turbulent channel flows with a bulk Mach number of 1.5. Through comparison with the direct numerical simulation (DNS) data, an eddy-viscosity-enhanced model (eLNS) outperforms the one not enhanced (LNS) in computing the coherence and amplitude ratio of streamwise velocity at different wall-normal heights, but they both largely deviate from DNS regarding the temperature prediction. For further investigation, the eigenspectra and pseudospectra of the linear operators are scrutinized. The eddy viscosity is shown to stabilize the eigenmodes and decrease the non-normality of the vortical modes. Consequently, the relative importance of acoustic and entropy modes increases, and they can contribute 20 % to 55 % of the response growth, which is not supported by DNS. Hence, it is an intrinsic defect of the eLNS model introduced by turbulence modelling. After a procedure of cospectrum decomposition, the contributions of acoustic and entropy components are filtered out. The resulting SLSE quantities for velocity, temperature and their coupling are basically agreeable with DNS, demonstrating that the coherent temperature fluctuation is dominated by advection and other vortical motions, instead of the compressibility effects. Moreover, a parameter study of Reynolds and Mach numbers (from 0.3 to 4) is conducted. The semi-local units are shown to well collapse the velocity SLSE quantities to the incompressible case for streamwise-elongated structures of high coherence.

Wall-pressure fluctuations in weakly compressible turbulent channel flow

Wall-pressure fluctuations in turbulent wall-bounded flows are detrimental in many applications because they can cause structural vibrations and acoustic radiation. Their spectral behavior at subconvective wavenumbers are to date poorly understood and predicted, particularly in low-Mach-number flows. In this study, compressible direct numerical simulation is employed to elucidate the low-wavenumber behavior of wall-pressure fluctuations in turbulent channel flow and the effect of flow Mach number in the nearly incompressible regime. Simulations are conducted at bulk Mach numbers of 0.4, 0.2, and 0.1, and friction Reynolds number of 180. In addition to the convective ridge that is virtually Mach-number independent, acoustic ridges representing longitudinal and oblique waves are clearly identified in the two-dimensional wavenumber-frequency spectrum. The acoustic peaks are orders of magnitude weaker than the convective peak and decay with flow Mach number, but remain distinctly identifiable even at Mach 0.1. The acoustic energy in the supersonic wavenumber range is significantly enhanced by the onset of the first oblique mode but not much affected by the higher modes. The effect of a small, two-dimensional surface hump is also considered, which is shown to elevate the spectral level of the fluctuating wall pressure in the subconvective wavenumber range by several decades due to acoustic diffraction by the hump. [Work supported by the U.S. Office of Naval Research.]

Momentum and heat flux events in compressible turbulent channel flows

Momentum and heat flux events in compressible turbulent channel flows

Particle dynamics in compressible turbulent vertical channel flows

In this work, we carry out direct numerical simulations of particle suspensions in the compressible turbulent vertical channel (TVC) flows with Mach number Ma = 1.5 and particle Stokes number St = 1–100. The compressibility effect is considered in the particle dynamic model for the first time in the study of compressible particle-laden wall turbulence. We find that in both incompressible and compressible flow, gravity weakens the wall-normal and spanwise fluctuations of particle velocities as the Stokes number increases. However, compared to the incompressible flow case, the compressible effect amplifies the mean velocity, fluctuations of velocity, and slip velocity of particle in the streamwise direction. The wall-normal and spanwise fluctuations of particle velocities are augmented by the compressible effect in the channel core region. Moreover, in the core region, the effect of fluid compressibility on the wall-normal and spanwise fluctuations of particle velocities attenuates as the Stokes number increases, indicating a competition between the compressible effect and the particle inertia effect. We, furthermore, conduct the quadrant analysis of the local fluctuation velocities of fluid at particle positions and observe preferential distributions in the second and the fourth quadrants at y+ = 12.5–13.5. For compressible TVC flows, the pattern of probability distributions is more elongated, and the percentage is slightly higher in the second and fourth quadrants than that of incompressible flows. This observation implies that more particles locate in the ejection and sweep events in compressible flows than that in incompressible flows, which is anticipated to influence the particle wall-normal transport.

Linear-model-based study of the coupling between velocity and temperature fields in compressible turbulent channel flows

It is generally believed that the temperature and the velocity fields are highly coupled in compressible wall-bounded turbulence. In the present study, we employ a linear model, i.e. the two-dimensional spectral linear stochastic estimation (SLSE), to study this coupling from the perspective of the multi-scale energy-containing eddies. Particular attention is paid to the relevant statistical characteristics of the temperature field. The connections of the two fields are found to be varied with the wall-normal position in the boundary layer. In a nutshell, their entanglement is strongest in the near-wall region, and only the extreme thermal events cannot be captured by SLSE. In the logarithmic region, only the scales that correspond to the attached eddies and the very large-scale motions (VLSMs) are firmly coupled. The near-wall footprints of the former are organized in an additive manner and fulfil the predictions of the celebrated attached-eddy model. In the outer region, the two fields are linearly coupled only at the scales corresponding to VLSMs. These findings are demonstrated to be insensitive to the Mach number effects and ascribed to the similarity between the momentum and the heat transfer in compressible wall turbulence. It is also shown that it is the Reynolds number rather than Mach number that acts as a key similarity parameter in constructing their coupling. The framework built in the present study may pave a way for investigating the multi-physics coupling in turbulence, and reinforcing our analysing and modelling capability to the interrelated problems.

Predicting near-wall turbulence with minimal flow units in compressible turbulent channel flows

Mach number effects on the near-wall turbulence in the absence of outer motions remain unclear so far. The present study extends the Minimal Flow Units (MFUs), a widely applied method to investigate near-wall turbulence free from the impact of large-scale motions in the outer region in incompressible channel flows, to compressible wall-bounded turbulence. The compressible near-wall turbulence in MFU proves accurate in replicating near-wall statistics, independent of Mach number and statistically equivalent to the universal signals extracted from the full-sized channel. It is further utilized as universal signals in the predictive models of compressible near-wall turbulence, which is capable of accurately predicting variances and joint probability density functions of velocity and temperature fluctuations.

Effects of mean shear on the vortex identification and the orientation statistics

This work compares the threshold applied to the swirling strength as well as the vortex orientation statistics in the total and fluctuating velocity fields using direct numerical simulations of compressible and incompressible turbulent channel flows. It is concluded that the difference in the swirling strength for vortex identification is minimal in the logarithmic region such that these two situations share the same threshold. Regarding the vortex orientation, the inclination angle remains similar. However, as the wall-normal distance increases, a more and more obvious distinction is noticed for its orientation with respect to the spanwise (z) direction. It is mainly due to their intrinsic differences and attendant contrasting preference for the vortex identification, i.e., vortices rotating in the −z direction for the total velocity field and in the z direction for the fluctuating one. These observations function as a reasonable explanation for various remarks in previous studies.

Helical model based on artificial neural network for large eddy simulation of compressible wall-bounded turbulent flows

Similar to the kinetic energy cascade, a helicity cascade is also a basic and key process in the generation and evolution of the turbulent flows. Furthermore, the helicity flux (HF) plays a crucial role between two scales in the helicity cascade. In this study, we will supply a new helical model constrained by the helicity flux for the large eddy simulation of the compressible turbulent flows. Then, in order to obtain a more precise HF, the local coefficient of the modeled HF is determined by the artificial neural network (ANN) method. The new model combines merits of the high robustness and the correlation with the real turbulence. In the test case of the compressible turbulent channel flow, the new model can supply a more accurate mean velocity profile, turbulence intensities, Reynolds stress, etc. Then, for the test in the compressible flat-plate boundary layer, the new model can also precisely predict the onset and peak of the transition process, the skin-friction coefficient, the mean velocity in the turbulent region, etc. Moreover, the ANN here is a semi-implicit method, and the new model would be easier to be generalized to simulate other types of the compressible wall-bounded turbulent flows.

Artificial-neural-network-based nonlinear algebraic models for large-eddy simulation of compressible wall-bounded turbulence

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

Predicting mean profiles in compressible turbulent channel and pipe flows

A robust modular iterative method is proposed to predict the mean profiles in compressible channel and pipe turbulence by solving a nonlinear system using the perfect gas assumption, Sutherland's viscosity law, the variable-property scaling models for compressible wall turbulence, the incompressible velocity law of the wall, and the temperature-velocity relation.

Compressibility effect in compressible turbulent channel flows

Compressibility effect in compressible turbulent channel flows

Scaling of pressure fluctuations in compressible turbulent plane channel flow

The purpose of the paper is to identify Mach-number effects on pressure fluctuations$p'$in compressible turbulent plane channel flow. We use data from a specifically constructed$(Re_{\tau ^\star },\bar {M}_{{CL}_x})$-matrix direct numerical simulation (DNS) database, with systematic variation of the centreline streamwise Mach number$0.32\leqslant \bar {M}_{{CL}_x}\leqslant 2.49$and of the HCB (Huanget al.,J. Fluid Mech., vol. 305, 1995, pp. 185–218) friction Reynolds number$66\leqslant Re_{\tau ^\star }\lessapprox 1000$. Strong$\bar {M}_{{CL}_x}$effects (enhanced by the increasingly cold-wall condition) appear for$\bar {M}_{{CL}_x}\gtrapprox 2$, for all$Re_{\tau ^\star }$, very close to the wall ($y^\star \lessapprox 15$). Compared with incompressible flow at the same$Re_{\tau ^\star }$, the wall root-mean-square$[p'_{rms}]^+_w$(in wall-units, i.e. scaled by the average wall shear stress$\bar {\tau }_w$) strongly increases with$\bar {M}_{{CL}_x}$. In contrast, the peak level across the channel,$[p'_{rms}]^+_{PEAK}$, slightly decreases with increasing$\bar {M}_{{CL}_x}$. In order to study the near-wall coherent structures we introduce a new wall-distance-independent non-local system of units, based for all$y$on wall friction and the extreme values of density and dynamic viscosity, namely, for cold walls$\{\bar {\tau }_w,\min _y\bar {\rho },\max _y\bar {\mu }\}$. The average spanwise distance between streaks, scaled by this length-unit, is nearly independent of$\bar {M}_{{CL}_x}$at constant$Re_{\tau ^\star }$. Using the in-plane (parallel to the wall) Laplacian$\nabla ^2_{xz}p'$we find that the$(+/-)\text {-}p'$wave-packet-like structures appearing inside the low-speed streaks ($y^\star \lessapprox 15$) with increasing$\bar {M}_{{CL}_x}\gtrapprox 2$are part of a more complex wave system with spanwise extent over several streaks, whose spatial density decreases rapidly with decreasing$\bar {M}_{{CL}_x}$or increasing$y^\star$. These$p'$wave packets appear to be collocated with strong$(+/-)$-$v'$events and could be responsible for compensating towards 0 the negative incompressible-flow correlation coefficient$c_{p'v'}$, with increasing$\bar {M}_{{CL}_x}$very near the wall.

Decomposition of the skin-friction coefficient of compressible boundary layers

We derive an integral formula for the skin-friction coefficient of compressible boundary layers by extending the formula of Elnahhas and Johnson [“On the enhancement of boundary layer skin friction by turbulence: An angular momentum approach,” J. Fluid Mech. 940, A36 (2022)] for incompressible boundary layers. The skin-friction coefficient is decomposed into the sum of the contributions of the laminar coefficient, the change of the dynamic viscosity with the temperature, the Favre–Reynolds stresses, and the mean flow. This decomposition is applied to numerical data for laminar and turbulent boundary layers, and the role of each term on the wall-shear stress is quantified. We also show that the threefold integration identity of Gomez et al. [“Contribution of Reynolds stress distribution to the skin friction in compressible turbulent channel flows,” Phys. Rev. E 79(3), 035301 (2009)] and the twofold integration identities of Wenzel et al. [“About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers,” J. Fluid Mech. 930, A1 (2022)] and Xu et al. [“Skin-friction and heat-transfer decompositions in hypersonic transitional and turbulent boundary layers,” J. Fluid Mech. 941, A4 (2022)] for turbulent boundary layers all simplify to the compressible von Kármán momentum integral equation when the upper limit of integration is asymptotically large. The dependence of these identities on the upper integration bound is studied. By using asymptotic methods, we prove that the multiple-integration identity of Wenzel et al. [“About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers,” J. Fluid Mech. 930, A1 (2022)] degenerates to the definition of the skin-friction coefficient when the number of integrations is asymptotically large.

Large-scale motions and self-similar structures in compressible turbulent channel flows

The present study investigates the scale characteristics of the log- and outer-region motions and structures in subsonic and supersonic wall turbulence. The energy distribution among the multiscale structures in the outer region is found to be dominated by the semilocal friction-Reynolds-number effects rather than the Mach-number effects. The geometrical characteristics of the self-similar structures populating the logarithmic region are also revealed by adopting a linear coherence spectrum.