Limin Wang, Ruifeng Hu

Center for Particle-Laden Turbulence, Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730 0 0 0, China

Keywords:Turbulent boundary layer Wall shear stress Inner-outer decomposition

ABSTRACT Fluctuating wall shear stress in turbulent channel flows is decomposed into small-scale and large-scale components.The large-scale fluctuating wall shear stress is computed as the footprints of the outer tur- bulent motions, and the small-scale one is obtained by subtracting the large-scale one from the total, which fully remove the outer influences.We show that the statistics of the small-scale fluctuating wall shear stress is Reynolds number independent at the friction Reynolds number larger than 10 0 0, while which is Reynolds number dependent or the low-Reynolds-number effect exists at the friction Reynolds number smaller than 10 0 0.Therefore, a critical Reynolds number that defines the emergence of universal small-scale fluctuating wall shear stress is proposed to be 10 0 0.The total and large-scale fluctuating wall shear stress intensities approximately follow logarithmic-linear relationships with Reynolds number, and empirical fitting expressions are given in this work.

Wall shear stress or wall friction is one of the most impor- tant parameters in wall-bounded turbulent flows, since it con- tributes significantly to the drag force of aircrafts or ships.On the other hand, in the fundamental research perspective, the mean wall shear stress 〈τw〉 determines the inner or viscous scales of near-wall turbulent flow, i.e., the friction velocityand the length scalelν=ν/uτ.Hereρis fluid density andνis fluid kinematic viscosity, respectively.

Although the Reynolds number scaling of mean wall shear stress has been well documented in literature [1] , no consen- sus on the statistics of the fluctuating wall shear stress has been reached.Alfredsson et al.[2] measured the fluctuating wall-shear stress with various types of hot-wire and hot-film sensors in tur- bulent boundary layers and turbulent channels, and reported the root-mean-squared (rms) value of the fluctuating wall shear stress is about 40% of the mean wall shear stress, i.e.,This finding was supported by early measurements in canonical wall-bounded turbulent flows [3-5] .In recent experimental stud- ies, the fluctuating wall shear stress was reported to be scattered as=(0.2 ?0.5)〈τw〉 in low Reynolds number range [6-8] .In a recent experimental investigation, Gubian et al.[9] suggested that the wall shear stress fluctuation may evolve to an asymptotic state atReτ= 600 .Here the friction Reynolds numberReτ=uτδ/νwithδthe outer scale of wall turbulence.

In contrast, high resolution direct numerical simulations (DNS) showed thatincreases with the Reynolds num- ber.Abe et al.[10] found the very-large-scale-motions in the outer layer of turbulent channel flows are responsible to the increase ofin the range ofReτ= 180 ?640 .The DNS of turbulent channel flows up toReτ= 1440 from Hu et al.[11] also reported an increase of wall shear stress fluctuations with Reynolds num- ber.In a DNS study of turbulent boundary layer, Schlatter and ?rlü[12] proposed that the fluctuating streamwise wall shear stress fol- lows a logarithmic-linear law ofwithReτ, which is,= 0.018 lnReτ+ 0.298 .It should be noted that a recent experimen- tal measurement of turbulent boundary layer [13] confirmed the log-linear law?lnReτat low-to-moderate Reynolds num- bers.Yang and Lozano-Durán [14] suggested?lnReτthrough a multifractal model of momentum cascade, whereis the in- tensity of fluctuating wall shear stress.

Although the debate on the Reynolds number dependence ofonReτstill exists [15,16] , it has been well accepted now that the outer large-scale and very-large-scale motions contribute to the near-wall velocity and wall shear stress fluctuations.The large-scale and very-large-scale motions are long streaky energy-containing structures prevailed in the logarithmic region [17,18] .The intensity of these structures increases with Reynolds number, and can impose strong superimposition and amplitude modulation effects on the near-wall turbulence [19,20] .The algebraic predic- tion model of Marusic and coworkers [21,22] provides a mathemat- ical formulation to quantitatively evaluate the effects by the outer structures.The model has also been extended and applied to wall shear stress prediction [23] .

The prediction model can be naturally regarded as a decompo- sition of near-wall small-scale motions and outer large-scale foot- prints.Wang et al.[24] decomposed the near-wall turbulent ve- locities into small-scale and large-scale components, with the for- mer one independent of Reynolds number atReτ≥10 0 0 .We start with this work here and apply it in the viscous sub-layer, where a linear relationship between the streamwise velocity component and the wall shear stress is known, so the mechanism of near- wall streamwise velocity can be easily extended to the wall shear stress.In this work, we decompose the fluctuating wall shear stress in a similar way with the same dataset, i.e., DNS data of turbu- lent channel flow atReτ= 180 ?5200 , as shown in Table 1 .TheReτ= 180 ～600 data are from our own DNS [24] and indicated by WHZ21, theReτ= 10 0 0 andReτ= 520 0 data are from Lee & Moser [25] indicated by LM15, and theReτ= 20 0 0 data are from Hoyas & Jiménez [26] indicated by HJ06.LxandLzare computation domain sizes in streamwise and spanwise, respectively.The outer length scale, i.e., boundary-layer thickness, channel half height or pipe radius, is denoted byδ.Δx+andΔz+are the streamwise and spanwise viscous-scaled grid size.ΔandΔare the viscous- scaled wall-normal grid spacings at the wall and the channel cen- tre, respectively.FD denotes finite difference scheme, and SP de- notes spectral method.More details about the dataset can be found in Wang et al.[24] and references therein.

Mathis et al.[23] extended the prediction model for velocity to wall shear stress, as

in which,Fxanddenote fast Fourier transform (FFT) and in- verse FFT along thexdirection, respectively,＜＞denotes the aver- age in time and in the spanwise direction, and overline indicates conjugate operation.HLτis a scale-dependent complex-valued ker- nel function for calculating footprints, representing the spectral linear stochastic estimation of outer velocity components in the near-wall region.Following Baars et al.[27] , we also use a band- width moving filter of 25% to smooth the original spectral ker- nel functions.The amplitude modulation coefficientsΓuτis deter- mined through an iterative procedure, that stops when the ampli- tude modulation factor AM is zero [21,22,27] , which can be written as

whereEL()denotes the envelope of, which is obtained by Hilbert transform.More details about the procedure can be found in Mathis et al.[22] and Baars et al.[27] .

On the other hand, we compare the premultiplied spectra ofatReτ= 10 0 0, 20 0 0 and 5200 with= 100 in Fig.2 c and d.It is seen that although there are slight differences among the premul- tiplied spectra at long wavelength, the results have been much im- proved with respect to Fig.2 a and b.As shown in Fig.2 c, the peak of the streamwise premultiplied spectra is located at≈10 0 0 .Figure 2 d shows the spanwise premultiplied spectra ofand it can be seen that all the spectra basically collapse well with each other, and the peak is located at≈100 .

Fig. 1. Variation of the magnitude of the scale-dependent superimposition coefficient |H Lτ| with streamwise wavelength at Re τ= 10 0 0,20 0 0 and 520 0:; (b)= 100 .

Fig. 2. The streamwise and spanwise premultiplied spectra ofatRe τ= 10 0 0, 20 0 0 and 520 0: = 100 .

Therefore, the above results demonstrate that similar to the fluctuating velocities, we should also use= 100 as the location of outer input signal to extract the universal small-scale wall shear stress.

Here we analyze the characteristics of small-scale fluctuating wall shear stress with= 100 .Wang et al.[24] reported the low- Reynolds-number effect of small-scale velocities, and here we will show this effect also exists in wall shear stress.

Figure 3 a shows the the magnitude of scale-dependent su- perimposition coefficientHLτatReτ= 180 ?600 with= 100 .Different with the results ofReτ= 10 0 0 ?5200 , Fig.3 demon- strates that at low Reynolds numbers the superimposition coef- ficient is Reynolds number dependent.Even we decreasesto 60, as shown in Fig.3 b, it is found thatHLis still Reynolds num- ber dependent.Figure 4 displays the streamwise and spanwise pre- multiplied spectra of the small-scale fluctuating wall shear stress, and it can be seen that the spectral energy distribution ofat these Reynolds numbers has Reynolds number effect.Therefore, the same with the velocity, the low-Reynolds-number effect also exists in the decomposition of wall shear stress.

Fig. 3. Variation of the magnitude of the scale-dependent superimposition coefficient |H Lτ| with streamwise wavelengthatRe τ= 180,310 and 600: (a)= 100 ; (b)= 60 .

Fig. 4. The streamwise and spanwise premultiplied spectra ofatRe τ= 180,310 and 600 with= 100 :

Fig. 5. Variations of total fluctuating wall shear stress intensity , large-scale fluctuating wall shear stress intensity , small-scale fluctuating wall shear stress intensity and demodulated small-scale fluctuating wall shear stress intensity with Reynolds numberRe τ.Dashed line: = 0.0143 lnRe τ+ 0.0763 .Dashed-dotted line: = 0.0143 lnRe τ?0.0820 .Dotted line: = 0.1553 .

Next, we analyze the Reynolds number dependence of the large-scale fluctuating wall shear stress, the small-scale fluc- tuating wall shear stressby subtracting the superimpositioneffect, and the small-scale fluctuating wall shear stressby demodulatingin the Reynolds number range of 180 ≤Reτ≤5200.

In this paper, we applied the improved inner-outer decomposi- tion model of Wang et al.[24] for near-wall turbulent velocities to the wall shear stress fluctuations, separating fluctuating wall shear stress into small-scale and large-scale components.The method- ology is principally based on the scaling-improved inner-outer de- composition method of Wang et al.[24] for near-wall turbulent ve- locities, in which the reference height of outer signal is set to be= 100 .We show that the decomposed small-scale fluctuating wall shear stress is Reynolds number independent atReτ≥10 0 0 , but Reynolds number dependent atReτ ＜10 0 0 (termed as the low-Reynolds-number effect).A critical Reynolds number responsi- ble for this transition is thus defined atReτ= 10 0 0 .The total and large-scale fluctuating wall shear stress intensities approximately follow logarithmic-linear relationships with Reynolds number, and empirical fitting expressions are given in this work.

Various empirical/semi-theoretical formulations have been pro- posed for howvaries with Reynolds number, however far less is known about the decomposition of the fluctuating compo- nent of wall-shear stress, i.e.,, and how its statistics change with Reynolds number.The present work could be use- ful for expanding our understanding of the Reynolds number ef- fect on the fluctuating wall shear stress at various scales, and pro- vide the critical Reynolds number for the emergence of universal small-scale fluctuating wall shear stress.In addition, a decompo- sition of the fluctuating wall shear stress has allowed us to refine the empirical fitting expression of the total and large-scale fluctu- ating wall shear stress.

Declaration of Competing Interest

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

Acknowledgement

The authors acknowledge the supports by grants from the National Natural Science Foundation of China (92052202 and 11972175).