Duo Wang,Heng Li,Bo-chao Cao,Hongyi Xu

Department of Aeronauticsand Astronautics, Fudan University,Shanghai 200433,China

Abstract:In-depth analysesof existing direct numerical simulations(DNS) data led to a logical and important classification of generic turbulent boundary layers(TBLs),namely Type-A,-Band -CTBL, based on the distribution patterns of the time-averaged wall-shear stress.Among these types,Type-A TBL and itsrelated law formulations were investigated in termsof the analytical velocity profiles independent on Reynolds number ( )Re .These formulations were benchmarked by the DNS data of turbulence on a zero-pressure-gradient flat-plate(ZPGFP).With reference to the analysis from von Karman in developing the traditional law-of-the-wall,the current study first physically distinguished the time-averaged local scale used by von Karman from the time-space-averaged scale defined in the current paper,and then derived the governing equations with the Re-independency under the time-space-averaged scales.Based on the indicator function (IDF)and TBL thickness,the sublayer partitions were quantitatively defined.Theanalytical formulationsfor entire ZPGFP TBL were derived,including the formula in the inner, buffer,semi-logarithmic(semi-log)and wake layers.The research profoundly understood the damping phenomenon and its controlling mechanism in the TBL with its associated mathematical expressions,namely the damping function under both linear and logarithmic coordinates.Based on these understandings and the quantified TBL partitions,the analytical formulations for the entire ZPGFP TBL were established and were further proved being uniform and consistent under both the time-averaged local and the time-space-averaged scales. Comparing to the traditional law,these formulations were validated by the existing DNS data with more accuracy and wider applicability.The findingsadvance thecurrent understandingsof the conventional TBL theory and its well-known foundation of law-of-the-wall.

Key words:Direct numerical simulation (DNS),wall-bounded turbulence,turbulent boundary layer (TBL),law-of-the-wall *Biography:Duo Wang (1993-),Male,Ph.D.Candidate,E-mail:16110290008@fudan.edu.cn Corresponding author:Hongyi Xu,E-mail:hongyi_xu@fudan.edu.cn

Introduction

Inspired by von Karman[1]and Cao and Xu’s work,Fu et al.[14]first defined the time-spaceaveraged scales for the Type-A TBL and applied these scales to the ZPGFPTBL.The study proved that the Type-A TBL governing equations were Re-independent under the time-space-averaged scales and therefore,were suitable applied to explore the law formulation for Type-A TBL.Consequently,the inner-layer law formulations were preliminarily developed for Type-A TBL based on the dual-control parameter formulations in Ref.[14]and were validated by the DNS data of the ZPGFP TBL in Schlatter[8].However,the study further found that the dual-control parameter formulations had a drawback of scaling inconsistency between the time-averaged local and the time-space-averaged scales.Consequently,the dual-control parameter formulations ceased being accurate near the neutral curve as explained in the following analysis.Within the context,Wang et al.[15]successfully developed the triple-control-parameter law formulations for the Type-B TBL,as represented by the Pressure-gradient driven square annular duct (PGDSAD)turbulence.Moreover,these formulations were,for the first time,extended to thermal TBL and were validated by the relevant DNS data,which opened the door for exploring the flow and thermal unified law for generic TBL.

1.Classification of generic TBLs

Obviously,different TBL configurations have their own TBL generation mechanisms,resulting in the distinctive types of TBL governed by different model equations.Historically,the conventional TBL theory was established based on the studies of ZPGFP TBL featured by the time-averaged local wall-shearstressτwsolely being the function of streamwise direction,i.e.,τw= τw( x).The feature can easily be understood by the Reynolds-averaged Navier-Stokes(RANS)governing equation (1)for the ZPGFP TBL

where x is the streamwise,y is the wall-normal direction,respectively and ( , , )u v w are timeaveraged velocity components.

2.On the scaling issuesfor Type-A TBL

Therefore,in order to better understand the traditional law from the perspective of time-spaceaveraged scale and to develop more accurate law for Type-A TBL,the current research revisited the ZPGFPTBL data[8]to more thoroughly investigatethe multi-control-parameter mechanisms in the law expressions for the Type-A TBL.The research pinpointed the drawbacks of the dual-controlparameter mechanisms in Refs.[13-14],and then first developed the more accurate and scaling-consistent formulations in the inner layer.Moreover, based on the thorough understandings of the inner-layer law formulae,the research extended the mathematics and their controlling mechanisms to the sublayers beyond the inner layer,i.e.,the buffer,semi-log and wake sublayers and thereby,successfully derived the scaling-consistent law formulations for the entire Type-A TBL.

Although the Type-A TBL of ZPGFPturbulence is geometrically simple,the investigations of the traditional base-line configuration,indeed,revealed an abundance of instantaneous turbulent characteristics such as the hairpin vortices,coherent structures and more recently the vortex forest in the TBL.Figure 1 schematically displays the patterns of ZPGFP TBL along the streamwise x direction with a variety of sublayers,including the inner,buffer,semi-log and wake sublayers,which was formed in the course of the turbulence evolution.The momentum exchange interactions are found being significantly enhanced by the ejection and sweeping processes which eventually evolved into the multi-scale eddy structures,namely from the -Λ shaped vortices to hairpin vortices and further to vortex-forest structures,as demonstrated by the third-generation vortex identification quantity of Liutex[16-17]using both the iso-surfaces and vortex core lines in Figs.1(b)and 1(c).The higher momentum fluids are continuously brought into the viscous inner and buffer layers by the sweeping process,resulting in the growing TBL thickness along the x direction and the local time-averaged wallshear-stresswτ asthe function of x.

Fig.1(a)ZPGFPTBL schematic drawing and the TBL natural transition

Fig.1 (b)(Color online)Liutex iso-surface[16]

Fig.1 (c)(Color online)LXC-Liutex core lines[18]

According to studies[1],the TBL velocity and length scales were first identified by nondimensionalizing the governing equation in viscous inner layer,which includes the two steps in Eq.(3):(1)making the left-hand side of the definition equal to unity and (2)reorganizing the right-hand side by nondimensionalizing the nominator and denominator in the partial derivative,giving rise to the appropriate velocity and length scales

where ρis thedensity,μisthe dynamic viscosity and u isthe time-averaged streamwise velocity.

Historically,the Re-independent model equations for Type-A TBL were derived by von Karman[1]only in the sublayers of viscous inner and semi-log layers based on the time-averaged local frictional (or TBL)scales.However,the systematical development of the law formulations for generic TBLs requires the Re-independent model be appropriately established for the entire TBL,which,from the author’s point of view,can only be achieved by introducing the time-space-averaged frictional scales introduced in the following section.

3.Re-independent mathematical model for Type- A TBL

Fig.2 Time-averaged local wall shear stressτw ( x)along the flat-plate surface[8]with the length scale L being the total length of flat plate and the stress scale τ w being the time-space-averaged wall shear stress taken over the L

With these definitions and relations,the traditional viscous linear law can be interpreted as a single-control-parameter form with+Δ being the only control-parameter,whereas the current work targets at developing the law formulation for each sublayer in a multi-control-parameter form with an improved accuracy and wider applicable scope comparing to the traditional law.For clarity in the following discussions,the superscripts of “*” and “+”denote the quantities under the time-averaged local and the time-space-averaged frictional scales,respectively.

4.Quantified partitions of TBL sublayers

As well known,a typical TBL contains multisublayers with their own physical and mathematical formation mechanisms. Following the traditional TBL descriptions in Coles[21],the valid range of each sublayer has to be first precisely defined before the law formulation can be accurately developed.However,the precise determination for each sublayer’s boundary has yet being achieved in the traditional law and the issue has haunted the fluid mechanics community for many years,particularly in the near-wall regions.The TBL outer boundary was commonly accepted as the location of 99%the incoming velocity proposed by Prantdl[2]and denoted by99δ .To the author’s point of view,the validity of thedefinition is justified by thefact that in the vicinity of TBL outer layer,the velocity gradient is trivial and therefore the velocity profile is reasonably chosen to describe the TBL boundary. However,both velocity profile and gradient are equally important in the near-wall region,suggesting that the two quantities,namely the velocity profileand itsgradient,need to be equally taken into account to determine the boundary of TBL sublayers near the wall.Obviously,the indicator function (IDF)in Refs.[8,12]can meet the requirementsup to the semi-log sublayer.

Therefore,the current study proposes to utilize the indicator function (IDF),taking the form of the product of viscous linear velocity profile and velocity gradient,i.e.

Fig.3 (Color online)IDFs and relevant characteristic partition points under semi-log coordinate with =Reθ 1 410,2 000,4 060 from Ref.[8](red, black, blue)

4.1 Viscous linear layer upper boundary of

4.2 Inner-layer upper boundary of

Fig.4 (Color online)Distributions of (a)Reynolds shear stressand from and (b)Turbulent fluctuations and for Type-A TBL at different Reynolds numbers[8]where the symbols of delta,gradient and square denote or d s*1 ,or and or

4.3 Buffer layer upper boundary of d s*3

the total shear stress,it is more appropriate to use the gradient-related IDF instead of the TBL profiles to define the sublayer boundaries,particularly for the inner layer where both velocity and its gradient are significant.Consequently,the physical features of the inner and buffer sublayers can be fully reflected in the law formulations.

4.4 Wake-layer lower and upper boundaries of (,

Table 1 TBL sublayer locations of ,,,andwith #= *, +for the TBLs at Reθ =1 410(TBL1),2 000(TBL2),4 060(TBL3)

Table 1 TBL sublayer locations of ,,,andwith #= *, +for the TBLs at Reθ =1 410(TBL1),2 000(TBL2),4 060(TBL3)

#d #s1 d #s2 d #s3 d #s4 d s5 TBL1*4.3234 9.4848 54.416 73.831 492.21 TBL2*4.3320 9.5319 56.923 100.67 671.11 TBL3*4.3730 9.6107 63.4107 190.79 1272.0 TBL1+ 4.0595 8.9059 51.095 69.325 462.17 TBL2+ 4.2929 9.4459 56.410 99.759 665.06 TBL3+ 4.7285 10.392 68.566 206.30 1375.4

5.Law-of-the-wall formulationsfor Type-A TBL

5.1 On TBL scaling issue in the law exploration

5.2 Law’s formulations for Type-ATBL

5.3 Scaling-consistency for the lawsbetween “*” and “+” scales

5.4 Determining the control parametersin law formulations

5.5 Validations of the triple-parameter law formulations

Fig.5 (Color online) Typical velocity profiles[8] of =Reθ 1410(TBL1),2 000(TBL2),4 060(TBL3)under (a)( ),(b)(u τ , lτ )scales and (c)Thetriple-control-parameter formulations comparing to theexperiment (exp.)data[12]under ( scales for the entire Type-A TBL

Table 2 Control parametersΔ #,,,i =1,2,3,# =* , +in theinner, buffer, wake layersand κ #,C #,#=*,+in the semi-log layer for the three typical TBLs

Table 2 Control parametersΔ #,,,i =1,2,3,# =* , +in theinner, buffer, wake layersand κ #,C #,#=*,+in the semi-log layer for the three typical TBLs

Δ ###1 1( , )Dε ##2 2( , )Dε ##( , )Cκ 3 3 TBL1*0(0.046,6.66)(4.80,6.80)(2.52,2.52)(0.42,5.40)TBL2*0(0.048,6.82)(3.39,5.00)(2.52,2.11)(0.42,5.41)TBL3*0(0.049,6.90)(1.72,3.05)(2.70,1.83)(0.41,5.28)TBL1+0.134(0.052,6.25)(5.45,6.80)(2.68,2.51)(0.40,5.92)TBL2+0.018(0.048,6.76)(3.50,5.07)(2.54,2.11)(0.42,5.49)TBL3+-0.145(0.042,7.54)(1.47,3.06)(2.49,1.81)(0.45,4.74)

Table 3(a)Quantitative error analyses of Eq.(15)and their derivatives in the viscouslinear layer 0≤y +

Table 3(a)Quantitative error analyses of Eq.(15)and their derivatives in the viscouslinear layer 0≤y +

E f f f-,=/v k k 1,1 0 0 v k E f f f ′ ′ ′ ′-( =max,avg)=/1, 1 0 0 k k Reθ+Δ E′ 1,max vE 1,max vE′ 1 410 0.1342 0.0091 0.0358 0.0294 0.1159 E 1,avg v 1,avg v 2 000 0.0183 0.0078 0.0308 0.0253 0.1000 4 060-0.1453 0.0078 0.0305 0.0257 0.0999

Table 3 (b)Quantitative error analyses of Eq.(16)and their derivatives in the viscous linear layer 0≤y +

Table 3 (b)Quantitative error analyses of Eq.(16)and their derivatives in the viscous linear layer 0≤y +

E f f f-,=/v k k 2, 2 0 0 E f f f ′ ′ ′ ′-( =max,avg)=/v k k 2,2 0 0 k Reθ+Δ E′ 2,max vE 2,max vE′1 410 0.1342 0.0103 0.0210 0.0219 0.0702 2 000 0.0183 0.0023 0.0275 0.0096 0.0956 4 060-0.1453 0.0077 0.0119 0.0212 0.0345 E v2,avg v2,avg

Table 3 (c)Quantitative error analyses of Eq.(10)and their derivatives in the viscous linear layer 0≤y +

Table 3 (c)Quantitative error analyses of Eq.(10)and their derivatives in the viscous linear layer 0≤y +

Reθ+Δ E v3,avg E f f f-,=/v k k 3, 3 0 0 E f f f ′ ′ ′ ′-( =max,avg)=/v k k 3, 3 0 0 k E′ 3,max vE v3,avg vE′3,max 1 410 0.1342 0.0139 0.0187 0.0192 0.0270 2 000 0.0183 0.0140 0.0202 0.0197 0.0278 4 060-0.1453 0.0140 0.0202 0.0199 0.0284

Table 4(a)Quantitative error analyses of Eq.(15)and their derivatives in the inner layer 0≤y #

Table 4(a)Quantitative error analyses of Eq.(15)and their derivatives in the inner layer 0≤y #

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Table 4 (b)Quantitative error analyses of Eq.(16)and their derivatives in the inner layer 0≤y #

Table 4 (b)Quantitative error analyses of Eq.(16)and their derivatives in the inner layer 0≤y #

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Table 4 (c)Quantitative error analyses of Eq.(10)and their derivatives in the inner layer 0≤y #

Table 4 (c)Quantitative error analyses of Eq.(10)and their derivatives in the inner layer 0≤y #

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Fig.6 (Color online)The inner-layer law formulation comparisons under ()scales at (a)Reθ=1410, (b)Reθ=2 000 and (c)Reθ=4 060 in Ref.[8],with the black delta and blue gradient denoting for thelocation of y +=and y +=, respectively,(d)Under ( , )u l τ τ scales with the black delta and blue gradient denoting for the location of y *=and y *=,respectively

6.Physical-mathematical connotations of the law formulations for Type-A TBL

7.Conclusions

In summary,the current research established the triple-control-parameter law formulations for Type-A TBL with the following conclusionsand comments:

(1)The damping mechanism as represented by the damping functions in Refs.[13-14,28]was intriguing and enlightening,which played an important role in establishing the law’sformulations.

(2)The IDF was found a crucially important function in quantitatively defining the sublayers in Type-A TBL,with which the TBL was precisely partitioned as the inner (linear and transitional),the buffer, the semi-log and the wake sublayers.

(4)Eqs.(10)-(13)were found well-fitting and consistently agreeing with the DNS data,which provided a strong predictive capability for the Type-A TBL turbulence and significantly enriched the contentsof theconventional TBL theory.

(5)The law formulations of Eqs.(10)-(13)derived in the current research is not a deny of the traditional law.Instead,these formulae are compatible with the traditional law and moreover,significantly expand the law’s applicability in terms of its applicable range (all the sublayers in Type-A TBL)and the predicative accuracy as well as its physical and mathematical connotations.

Acknowledgements

The work wassupported by the Talent Recruiting Program at Fudan University (Grant No.EZH2126503).The DNS data were provided by Schlatter from the KTH website.The experiment data were supplied by ?sterlund from KTH website.