Ming YU, Ylu FU, Zhigong TANG, Xinxu YUAN, Chunxio XU

a State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China

b Key Laboratory of Applied Mechanics of Ministry of Education, Institute of Fluid Mechanics, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

KEYWORDS Turbulent flow;Direct numerical simulation;Supersonic flow;Boundary layer flow;Mach number effects

Abstract 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.

1.Introduction

Wall-bounded turbulence is ubiquitously encountered in natural sciences and engineering applications.1–3Recent years have seen the ever-increasing computational resources that enable us to accurately predict their aerodynamic performances.4However, the inevitable high Reynolds number of turbulence requiring huge sums of mesh grids to capture the vortices at the smallest scales, especially in the near-wall region, impedes the application of Direct Numerical Simulation (DNS) that is capable of replicating turbulent motions to practical engineering problems.5,6

Near-wall turbulence in the absence of motions in the outer regions is self-sustained and composed of regeneration cycles of low-speed streaks, quasi-streamwise vortices and bursting events.7–10This has been widely proved by previous numerical studies in turbulent channel flows.11Limiting the sizes of the computational domain in the wall-parallel directions leads to the laminarization of turbulence above a certain location from the wall, where the turbulence is no longer ‘healthy’,12–14and the large-scale motions, if there is any, are weakened, or even eliminated.Such flows in the limited-sized computational domains are the commonly known ‘Minimal Flow Units’(MFUs).7With these favourable features of MFUs,the statistics and dynamics of small-scale motions in the near-wall region can be discussed free from the impacts of the motions wider than the computational domain.15–17

In a subsequent study, Yin et al.23applied the near-wall healthy turbulent fluctuations in MFUs to the predictive models as the universal signals.The predictive model was firstly proposed by Marusic et al.20(herein referred to as the‘MMH model’).It was suggested that the near-wall turbulent fluctuations can be expressed as the summation of the amplitude-modulated universal signals (small-scale motions)and the large-scale signals imprinted by the largescale motions in the outer region.This model was later refined by Baars et al.24, who incorporated the non-universal superposition effects at different scales with the Spectral Linear Stochastic Estimation (SLSE) and phase shift between modulation and superposition25.These models concern only the streamwise component of the velocity fluctuation.Agostini and Leschziner26further extended the predictive model to the other two velocity components and considered the unsymmetrical modulation effects.The predictive model proposed by Yin et al.23was based on the MMH model,encompassing the modifications above, except that the universal signals were substituted by the turbulence in MFUs.The accurately predicted near-wall turbulent fluctuation intensities, spectra and Joint Probability Density Function (JPDF) distributions confirm the validity of the predictive models, and more importantly,support that the near-wall turbulence in MFU is indeed equivalent to the universal signals a posteriori.Such predictive models are useful because they point the way to provide the precisely accurate near-wall turbulent fluctuations for Large Eddy Simulations (LES) as off-wall boundary conditions, as it has recently been achieved by Wang et al.27

The literature survey above merely concerns the incompressible wall-bounded turbulence, where we witnessed the successful applications of the predictive models in the LES,with the near-wall turbulence in the MFU serving as the universal signals.The related research in compressible turbulence,however, remains vacant, to the best of our knowledge.The predictive models for near-wall velocity and temperature fluctuations were proposed based on the MMH model by Helm and Martin.28,29The present authors recently contributed to this realm of research30by incorporating the refinements made by Baars et al.24and Agostini and Leschziner.26Moreover,the predictive model for temperature fluctuation was derived to be consistent with the generalized Reynolds analogy.The questions remain whether the near-wall turbulence in the MFU is Mach number independent and whether,like in incompressible flows, it is equivalent to the universal signals.This work attempts to confirm these two aspects, and utilizes the nearwall turbulence in MFU as the universal signals in the predictive models of compressible wall-bounded turbulence.

The remainder of this paper is organized as follows.The physical model and numerical methods used in the present study are briefly introduced in Section 2.The Mach number independence of the near-wall turbulence is demonstrated in Section 3.The consistency between the near-wall turbulence in MFU and the universal signals is discussed in Section 4.The refined predictive model and its performance are stated in Section 5.Conclusions are drawn in Section 6.

2.Physical model and numerical schemes

The physical model under scrutiny is the compressible channel flow with constant mass and heat flux, which replicates the fully developed turbulence in a long wind tunnel, as adopted in our previous studies.31–33The flow is governed by the Navier-Stokes equations for compressible Newtonian fluids.The uniformly distributed body force and heat sink are added in the equations to balance the momentum loss and the heat generated by viscosity.The streamwise (x), wall-normal (y)and spanwise (z) velocity components are represented by u, v and w respectively, and density, pressure and temperature by ρ, p and T respectively.The state equation of perfect gas p=ρRT is adopted, with R the gas constant.The variation of the dynamic viscosity μ is determined by the Sutherland’s law and the heat-conductivity as κ=μcp/Pr, with Pr=0.71 the Prandtl’s number.Periodic conditions are applied in the wall-parallel directions.The nonslip and impermeable conditions for velocity and the isothermal condition for temperature, set as the recovery temperature of the ‘upstream’nominal free-stream flow, are applied on the upper and lower walls.Under this wall temperature, the wall heat transfer is trivial enough that the velocity statistics are dimly affected,as reported by Yu et al.31.The conservative governing equations are solved numerically with the finite-difference method with minor adjustments on the body force and heat sinks according to the physical model considered herein.The convective terms are approximated by the seventh-order upwind scheme, the viscous terms by the eighth-order central scheme,and the time-advancement by the third-order TVD Runge-Kutta scheme.For detailed descriptions on the physical model and numerical implements, please refer to the work of Yu et al.31and Yu and Xu33.

Table 1 Numerical setup.

3.Mach number independence

Fig.1 Distributions of (a) mean velocity under van Driest transformation, (b) mean temperature T-, compared with GSRA (Eq.(1),solid diamonds), (c) variances of density-weighted velocity fluctuations and (d) temperature fluctuations defined in Eq.(2).

where the subscript h represents the value at the healthy turbulent heightfor MFU cases, and that at the channel center for full-sized channel cases.This modification is necessary,for the laminarization of the flow abovewould invalidate GRA.The mean temperature distributions agree with Eq.(1)below y+h.For Case M4, the mean temperature distributions also conform with the full-sized channel Case F1.These consistencies suggest that the MFU is capable of accurately predicting the mean velocity and temperature distributions.

The variances of density-weighted velocity and temperature fluctuations normalized by viscous scales below y+=100 are presented in Figs.1(c)-(d), defined as

Fig.2 JPDF distributions of (a) P（u ′′+,v′′+), (b) P（u ′′+,w′′+) and (c) P（u ′′+,T′′+) at y+ =30.

Fig.3 Pre-multiplied spanwise spectra normalized by viscous scales, (a) kzE*uu, (b) kzE*vv, (c) kzE*ww, (d) kzE+TT.

Fig.4 JPDF distributions at y+ =30, (a) P（u *,v*), (b) P（u *,w*), (c) P（u *,T+).

4.Universal signals

In this section,we discuss the equivalence of the near-wall turbulence in MFU with the universal signals20.The universal signals are extracted from Case F2, obtained by removing the superposition and modulation effects of the large-scale motions from the original fluctuations as follows:

where ?denotes the spectral coefficient, and the superscript c its complex conjugate.As shown in Fig.3,the spectra distributions at small scales are nearly collapsed below y+≈80, indicating that the turbulent fluctuation intensities at these scales are roughly the same.It can also be inferred that, like the incompressible channel flows,12,14the compressible turbulence in MFUs is accurate in predicting the fluctuation intensities at all resolved scales, and that the near-wall small-scale motions are not much affected by the different Reynolds number.

Fig.5 Instantaneous streamwise velocity distributions.Contours: -0.5Ub(blue) - 0 (green) -0.5Ub (red).

The JPDF distributions at y+=30 are displayed in Fig.4.Within the high-speed region u*>0, the results of the universal signals and MFUs are nearly identical.For P u*,w*（), the wide-spread spanwise velocity fluctuations w*induced by the strong dispersive motions are identical,so are the strong correlations between u*and T+.Within the region of the extreme low-speed events (u*≤-4), there are slight discrepancies between two groups of results.However, due to their lowlevel probability, the discrepancies do not constitute considerable disparities to low-order statistical moments.Based on preceding observations, we can conclude that the healthy nearwall turbulence in MFUs is statistically equivalent to the universal signals.

5.Refined predictive model

As an application of the previous conclusions, we apply the near-wall turbulent fluctuations in MFUs as the alternative to the universal signals in the predictive models proposed by Yu and Xu.30This alternation has already been proved successful in incompressible channel flows.23According to Yu and Xu,30the predictive models for near-wall velocity and temperature fluctuations are cast as

Fig.6 Near-wall fluctuation variances (a) R*uiui, (b) R+TT.

Fig.7 JPDF distributions at y+ =30, (a),(d),(g) P（u *,v*); (b),(e),(h) P（u *,w*); (c),(f),(i) P（u *,T+).(a)-(c) Ensemble-averaged results.Conditional-averaged results within large-scale, (d)-(f) high-speed regions u*L >+u*L,rms and (g)-(i) low-speed regions u*L <-u*L,rms.

6.Conclusions

In this paper,we studied near-wall turbulence in the compressible turbulent channel flows in the Minimal Flow Units(MFUs) in order to examine whether this widely applied method in incompressible turbulence can be adopted in compressible flows as well.Statistical results indicate that the near-wall turbulence within the healthy channel height y+his Mach number independent, and consistent with the universal signals extracted from the flows at a higher Reynolds number.The MFU data are further utilized as an alternative to the universal signals in the predictive models for the near-wall turbulent prediction.Excellent agreements between the predicted and DNS results suggest that these models are capable of predicting the near-wall turbulence.We are optimistic about the application of these models to the LES to provide accurate off-wall boundary conditions, which have been successfully applied to the incompressible turbulent channel flows.27,40

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Key R&D Program of China (No.2019YFA0405201), the National Numerical Windtunnel Project, Open Project of State Key Laboratory of Aerodynamics, China (No.SKLA-20200102)and the National Natural Science Foundation of China(Nos.92052301, 12202469).