Self-Similar Intermediate Structures in Turbulent Boundary Layers At Large Reynolds Numbers
a r X i v :m a t h /9908044v 1 [m a t h .N A ] 10 A u g 1999Self-similar Intermediate Structures
in Turbulent Boundary Layers
at Large Reynolds Numbers
G.I.Barenblatt,A.J.Chorin Department of Mathematics and Lawrence Berkeley National Laboratory University of California Berkeley,California 94720,USA and V.M.Prostokishin P.P.Shirshov Institute of Oceanology Russian Academy of Sciences 36Nakhimov Prospect Moscow 117218,Russia Abstract.Processing the data from a large variety of zero-pressure-gradient boundary layer ?ows shows that the Reynolds-number-dependent scaling law,which the present authors obtained earlier for pipes,gives an accurate description of the velocity distribution in a self-similar intermediate region of distances from the wall adjacent to the viscous sublayer.The
appropriate length scale that enters the de?nition of the boundary layer Reynolds number is found for all the ?ows under investigation.
Another intermediate self-similar region between the free stream and the ?rst intermedi-ate region is found under conditions of weak free stream turbulence.The e?ects of turbulence in the free stream and of wall roughness are assessed,and conclusions are drawn.
1Introduction
Asymptotic laws for wall-bounded turbulent shear?ows at large Reynolds numbers are considered.Classical examples of such?ows are the?ows in pipes,channels,and boundary layers.This class of?ows is of major fundamental and practical importance.All these?ows share as dimensional governing parameters the shear stress at the wallτand the?uid’s properties,its densityρand dynamic viscosityμ.From these parameters two important quantities can be formed:the dynamic or friction velocity u?=(τ/ρ)1
u?=
1
ν
(1)
where y is the distance from the wall;the constantsκ(the von K′a rm′a n constant)and C should be identical for all turbulent wall-bounded shear?ows at high Reynolds numbers, and the law(1)should be valid in intermediate regions between,on one hand,the viscous sublayer and,on the other,the external parts of the?ows,e.g.vicinity of the axis in pipe ?ow,or vicinity of the external?ow in the boundary layer.In1932,L.Prandtl,the greatest mechanician of this century,came to the law(1)using a di?erent approach,but e?ectively with the same basic assumption.The law(1)is known as the von K′a rm′a n-Prandtl universal logarithmic law.More recent derivations which,however,follow the same ideas and the same basic assumption,often in an implicit form,can be found in monographs by Landau and Lifshits(1987),Monin and Yaglom(1971),Schlichting(1968)and in a recent textbook by Spurk(1997).
According to the von K′a rm′a n-Prandtl law(1),all experimental points corresponding to the intermediate region should collapse on a single universal straight line in the traditional coordinates lnη,φ.
Subsequent investigations showed,however,that this is not what happens.First,the experiments showed systematic deviations from the universal logarithmic law(1)even if one is willing to tolerate a variation in the constantsκand C(from less than0.4to0.45forκ,and from less than5.0to6.3for C).Furthermore,using analytic and experimental arguments, the present authors showed[Barenblatt(1991,1993);Barenblatt and Prostokishin(1993); Barenblatt,Chorin and Prostokishin(1997b);Chorin(1998)]that the fundamental von K′a rm′a n hypothesis on which the derivation of the universal law(1)was based,i.e.the assumption that the in?uence of viscosity disappears totally outside the viscous sublayer, is inadequate.In fact,this hypothesis should be replaced by the more complicated one of incomplete similarity,so that the in?uence of viscosity in the intermediate region remains, but the viscosity enters only in power combination with other factors.This means that the in?uence of the Reynolds number,i.e.both of the viscosity and the external length scale, e.g.the pipe diameter,remains and should be taken into account in the intermediate region.
For the readers’convenience we present here brie?y the concept of incomplete similarity;
a more detailed exposition can be found in Barenblatt,(1996).The mean velocity gradient ?y u in turbulent shear?ows can be represented in the general form suggested by dimensional analysis
u?
?y u=
=(C0ln Re+C1)ηc/ln Re.(2)
u?
where the constants C0,C1andαmust be universal.The scaling law(2)was compared with what seemed(and seems to us up to now)to be the best available data for turbulent
pipe?ows,obtained by Nikuradze(1932),under the guidance of Prandtl at his Institute in G¨o ttingen.The comparison has yielded the following values for the coe?cients
c=3√2(3) when the Reynolds number Re was taken in the form
ˉu d
Re=
√2)η3/2ln Re(5) or,equivalently
φ= √2α ηα,α=3
ln 2αφ3+5α =lnη,α=3
α
authors(Barenblatt,Chorin(1996,1997)),it was demonstrated that the scaling law(2)is compatible with the properly modi?ed IMM procedure.The method of vanishing viscosity (Chorin,(1988,1994))was used in this modi?cation.
Let us turn now to shear?ows other than?ows in pipes.By the same logic,the scaling law(5)should be also valid for an intermediate region adjacent to the viscous sublayer for all good quality experiments performed in turbulent shear?ows at large Re.
The?rst question is,what is the appropriate de?nition of the Reynolds number for these ?ows which will make the formula(5)applicable?This is a very important point—if the universal Reynolds-number-independent logarithmic law were valid,the de?nition of the Reynolds number would be irrelevant provided it were su?ciently large.For the scaling law (5)this is not the case.Indeed,if the scaling law(5)has general applicability it should be possible to?nd,for every turbulent shear?ow at large Reynolds number,an appropriate de?nition of the Reynolds number which will make the scaling law(5)valid.
There exists nowadays a large amount of data for an important class of wall-bounded turbulent shear?ows:turbulent zero-pressure-gradient boundary layers.These data were obtained over the last25years by various authors using various set-ups.For boundary layers the traditional de?nition of the Reynolds number is
Uθ
Reθ=
(9)
ν
is properly determined.Moreover,we show that for all the?ows where the turbulence in the external?ow is small,there exists a sharply distinguishable second intermediate region
between the?rst one where the scaling law(5)is valid and the external homogeneous?ow. The average velocity distribution in this second intermediate region is also self-similar of scaling type:
φ=Bηβ(10)
where B andβare constants.
However,a Reynolds number dependence of the powerβwas not observed.Within the accuracy of the experimental dataβis close to1/5.When the turbulence in the external homogeneous?ow becomes signi?cant,the second self-similar region deteriorates and the powerβdecreases with growing external turbulence until the second intermediate region disappears completely.
2The?rst group of zero-pressure-gradient boundary layer experiments
We will explain later why we divided the experimental data into three groups.Here it is su?cient to note that all available sets of experimental data were eventually taken into account.
The original data were always presented by their authors in the form of graphs in the tra-ditional(lnη,φ)plane,suggested by the universal logarithmic law(1).The shape of original graphs was always similar to the one presented qualitatively in Figure1a.Therefore,the ?rst rather trivial step was to replot the data in the doubly logarithmic coordinates(lgη,φ) appropriate for revealing the scaling laws.The result was instructive:for all experiments of the?rst group(in chronological order),speci?cally:Collins,Coles,Hike,1(1978);Erm and Joubert(1991);Smith2(1994);Naguib3(1992),and Nagib and Hites4(1995);Krogstad and Antonia(1998),the data outside the viscous sublayer(lgη>1.5)have the characteristic shape of a broken line,shown qualitatively in Figure1b and quantitatively in Figures2–6.
Thus,the two straight lines forming the broken line that were revealed in the lgη,lgφplane have as equations
(I)φ=Aηα;(II)φ=Bηβ.(11) The coe?cients A,α,B,βwere obtained by us through statistical processing.
We assume as before that the e?ective Reynolds number Re has the form(9):Re=UΛ/ν, where U is the free stream velocity andΛis a length scale.The basic question is,whether one can?nd in each case a length scaleΛwhich plays the same role for the intermediate region(I)of the boundary layer as the diameter does for pipe?ow?In other words,whether it is possible to?nd a length scaleΛ,perhaps in?uenced by individual features of the?ow,so that the scaling law(5)is valid for the?rst intermediate region(I)?To answer this question we have taken the values A andα,obtained by statistical processing of the experimental data in the?rst intermediate scaling region,and then calculated two values ln Re1,ln Re2, by solving the equations suggested by the scaling law(5):
1 3ln Re1+5
2ln Re2
=α.(12)
If these values ln Re1,ln Re2obtained by solving the two di?erent equations(12)are indeed close,i.e.,if they coincide within experimental accuracy,then the unique length scaleΛcan be determined and the experimental scaling law in the region(I)coincides with the basic scaling law(5).
Table1shows that these values are close,the di?erence slightly exceeds3%in only two cases;in all other cases it is less.Thus,we can introduce for all these?ows the mean Reynolds number
Re= 2(ln Re1+ln Re2)(13) and consider Re as an estimate of the e?ective Reynolds number of the boundary layer?ow. Naturally,the ratio Reθ/Re=θ/Λis di?erent for di?erent?ows.
3Zero-pressure-gradient boundary layer beneath a tur-bulent free stream:The experiments of Hancock and Bradshaw
The experiments of Hancock and Bradshaw(1989)revealed a new feature important for our analysis.Examination of these experimental data suggested that we separate the other experiments into two groups.In the Hancock and Bradshaw experiments the free stream was turbulized by a grid in all series,except one.Thus,processing the data from these experiments we were able not only to compare the scaling law(5)with experimental data once again but also to investigate the in?uence of the turbulence of the external?ow on the second self-similar intermediate region.The results of the processing are presented in Table 2and Figures7and8.In both Table2and Figures7and8the intensity of turbulence is shown by the value of u′/U,where u′is the mean square velocity?uctuation in the free stream.
Table2
Figure ReθαA ln Re1ln Re2ln Re u′/U Reθ/Reβ
Hancock,P.E.and Bradshaw,P.(1989)
Fig.8a4,6800.1408.6610.6710.7110.690.00030.110.20
Fig.8b2,9800.1388.7710.8610.9110.880.0240.060.18
Fig.8c5,7600.1378.8010.9110.9510.930.0260.10–
Fig.8d4,3200.1508.229.9110.009.950.0410.21–
Fig.8e3,7100.1229.4912.1112.3012.200.0400.02–
Fig.8f3,1000.1289.1311.4811.7011.590.0580.03–
Fig.8g3,8600.1299.0711.3811.6311.500.0580.04
First of all,our processing showed that the?rst self-similar intermediate layer is clearly seen in all these experiments,both in the absence of the external turbulence,and in its presence.The values of ln Re1and ln Re2are close.This means that the basic scaling law (5)is valid in the intermediate region adjacent to the viscous sublayer.At the same time, the second self-similar region is clearly observed and well-de?ned only when the external turbulence is weak(Figure8a and to a lesser extent,Figure8b)so that the external turbu-lence leads to a drastic reduction of the powerβ,and even to the deterioration of the second self-similar intermediate region so thatβbecomes indeterminate.We illustrate the in?uence of the free stream turbulence additionally by Figure7(b).
The experiments of Hancock and Bradshaw are instructive because they suggest at least one possible reason for the destruction of the intermediate self-similar region adjacent to the external?ow that is observed in the experiments of the next group.
4The remaining group of zero-pressure-gradient bound-ary layer experiments
In this section the results of the processing are presented for all the remaining series of experiments.For all of them we used the data presented in the form of graphs in the review of Fernholz and Finley(1996).The results of the processing are presented in Table3and in Figures9–15.
All the data reveal the self-similar structure in the?rst intermediate region adjacent to the viscous sublayer.The scaling laws obtained for this region give values of ln Re1and ln Re2,close to each other,although the di?erence between ln Re1and ln Re2is sometimes larger than in the experiments of the?rst group.The scaling law(5)is con?rmed by all these experiments.At the same time,for this group of experiments the second self-similar structure adjacent to the free stream turns out to be less clear-cut,if it is there at all. Therefore,for this group of experiments,we did not present the estimates for the values of β.Nevertheless we note that when it was possible to obtain estimates ofβthey always gave aβless than0.2.Note also that for all these experiments the number of experimental points belonging to the region adjacent to the free stream was less than for the experiments of the?rst group:this was an additional argument for our reluctance to show here the second self-similar layer.As explained in Section3,we suggest that the turbulence of the external ?ow in the experiments of this remaining group was more signi?cant.
5Checking universality The universal form of the scaling law
ψ=1
√
in these experiments the experimental points lie much below the bisectrix.Furthermore,in these experiments ln Re1and ln Re2di?ered signi?cantly,and we therefore picked the value ofαthat corresponds to ln Re1.The result is a pair of lines parallel to the bisectrix but far below it(Figure16e,(b)).
More generally,it is very likely that any outside cause that increases the level of turbu-lence should also increase the e?ective viscosity,and thus shift the points in the(lnη,ψ) plane downwards.A case in point is the set of experiments of Hancock and Bradshaw(1989) discussed above,where turbulence was created by a grid in the free stream.The parallel downward shift is indeed observed(Figure16f),and it is of the same order of magnitude as the shift in the experiments of Nagib et al.Note that in the experiments of Nagib et al.the second intermediate region is intact,and it is therefore likely that the shift in the universal description of the?rst intermediate region is due to the disturbance close to the wall,i.e., to roughness,just as in the experiment of Zagarola et al.(1996).
6Conclusion
The Reynolds-number-dependent scaling law
φ=u
√2)η3/2ln Re,η=u?y
References
Barenblatt,G.I.,1996.Scaling,Self-Similarity and Intermediate Asymptotics,Cambridge University Press.
Barenblatt,G.I.1991.On the scaling laws(incomplete self-similarity with respect to Reynolds number)in the developed turbulent?ow in pipes,C.R.Acad.Sci.Paris,se-ries II,313,309–312.
Barenblatt,G.I.,1993.Scaling laws for fully developed shear?ows.Part1:Basic hypotheses and analysis,J.Fluid Mech.248,513–520.
Barenblatt,G.I.and Chorin,A.J.,1996.Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulence,Proc.Nat.Acad. Sciences USA93,6749–6752.
Barenblatt,G.I.and Chorin,A.J.,1997.Scaling laws and vanishing viscosity limits for wall-bounded shear?ows and for local structure in developed turbulence,Comm.Pure Appl. Math.50,381–398.
Barenblatt,G.I.,Chorin,A.J.,Hald,O.H.,and Prostokishin,V.M.,1997.Structure of the zero-pressure-gradient turbulent boundary layer,1997.Proc.Nat.Acad.Sciences USA 94,7817–7819.
Barenblatt,G.I.,Chorin,A.J.,Prostokishin,V.M.,1997,a.Scaling laws in fully developed turbulent pipe?ow:discussion of experimental data,https://www.360docs.net/doc/fc12079882.html,A94a, 773–776.
Barenblatt,G.I.,Chorin,A.J.and Prostokishin,V.M.,1997,b.Scaling laws in fully developed turbulent pipe?ow,Applied Mechanics Reviews50,no.7,413–429.
Barenblatt,G.I.and Prostokishin,V.M.,1993.Scaling laws for fully developed shear?ows. Part2.Processing of experimental data,J.Fluid Mech.248,521–529.
Chorin,A.J.,1988,Scaling laws in the vortex lattice model of turbulence,Commun.Math. Phys.114,167-176.
Chorin,A.J.,1994.Vorticity and Turbulence,Springer,New York.
Chorin,A.J.,1998.New perspectives in turbulence.Quart.Appl.Math.56,767–785.
Erm,L.P.and Joubert,P.N.,1991.Low Reynolds-number turbulent boundary layers,J. Fluid Mech.230,1–44.
Fernholz,H.H.and Finley,P.J.,1996.The incompressible zero-pressure-gradient turbulent boundary layer:an assessment of the data.Progr.Aerospace Sci.32,245–311.
Hancock,P.E.and Bradshaw,P.,1989.Turbulence structure of a boundary layer beneath a turbulent free stream,J.Fluid Mech.205,45–76.
Krogstad,P.-?A.and Antonia,R.A.,1998.Surface roughness e?ects in turbulent boundary layers,Experiments in Fluids,in press.
Landau,L.D.and Lifshits,E.M.,1987.Fluid Mechanics,Pergamon Press,New York.
Monin,A.S.and Yaglom,A.M.,1971.Statistical Fluid Mechanics,vol.1,MIT Press, Boston.
Nagib,H.and Hites,M.,1995.High Reynolds number boundary layer measurements in the NDF.AIAA paper95–0786,Reno,Nevada.
Naguib,A.N.,1992.Inner-and Outer-layer e?ects on the dynamics of a turbulent boundary layer,Ph.D.Thesis.Illinois Institute of Technology.
Nikuradze,J.,1932.Gesetzmaessigkeiten der turbulenten Stroemung in glatten Rohren, VDI Forschungheft,No.356.
Prandtl,L.,1932.Zur turbulenten Stroemung in Rohren und laengs Platten,Ergeb.Aero-dyn.Versuch.,Series4,Goettingen,18–29.
Schlichting,H.,1968.Boundary Layer Theory,McGraw-Hill,New York.
Spurk,J.,1997.Fluid Mechanics,Springer,New York.
von K′a rm′a n,Th.,1930.Mechanische¨Ahnlichkeit und Turbulenz,Proc.3rd International Congress for Applied Mechanics,C.W.Oseen,W.Weibull,eds.AB Sveriges Litogra?ska Tryckenier,Stockholm,vol.1,pp.85–93.
Zagarola,M.V.,Smits,A.J.,Orszag,S.A.,and Yakhot,V.,1996.Experiments in high Reynolds number turbulent pipe?ow,AIAA paper96-0654,Reno,NV.
Zagarola,M.V.and Smith,A.J.,1998.Mean-?ow scaling of turbulent pipe?ow,J.Fluid Mech.373,33–79.
Table1
Figure ReθαA ln Re1lnRe2lnRe Reθ/ReβCollins,D.J.,Coles,D.E.,and Hiks,J.W.(1978)
Fig.2(a)5,9380.1299.1011.4311.6311.530.060.203 Fig.2(b)6,8000.1259.2311.6612.0011.830.050.195 Fig.2(c)7,8800.1239.4111.9712.2112.090.040.202
Erm,L.P.and Joubert,P.N.(1991)
Fig.3(a)6970.1637.839.239.209.220.070.202 Fig.3(b)1,0030.1597.969.469.439.450.080.192 Fig.3(c)1,5680.1567.979.479.629.540.110.202 Fig.3(d)2,2260.1488.269.9810.1410.060.100.214 Fig.3(e)2,7880.1408.6610.6710.7110.690.060.206 Naguib,A.M.(1992)and Hites,M.and Nagib,H.(1995)
Fig.4(a)4,5500.1567.879.309.629.460.360.22 Fig.4(b)6,2400.1488.249.9410.1410.040.270.20 Fig.4(c)9,5900.1438.3710.1710.4910.330.310.206 Fig.4(d)13,8000.1318.9411.1511.4511.300.170.193 Fig.4(e)21,3000.1388.6110.5810.8710.730.470.22 Fig.4(f)29,9000.1308.9911.2411.5411.390.340.204 Fig.4(g)41,8000.1249.3011.7812.1011.940.270.201 Fig.4(h)48,9000.1249.2811.7412.1011.920.330.192
Smith,R.W.(1994)
Fig.5(a)4,9960.1468.3610.1510.2710.210.180.20 Fig.5(b)12,9900.1299.1911.5911.6311.610.120.167 Krogstad,P.-A.and Antonia,R.A.(1998)
Fig.612,5700.1468.3810.1810.2710.230.450.201
Table3
Figure ReθαA ln Re1ln Re2ln Re Reθ/Re
Winter,K.G.and Gaudet,L.(1973)
Fig.9(a)32,1500.1338.8611.0211.3211.170.45 Fig.9(b)42,2300.1229.3711.9012.3012.100.24 Fig.9(c)77,0100.11510.3013.5113.0413.270.13 Fig.9(d)96,2800.10710.5613.9614.0213.990.08 Fig.9(e)136,6000.10310.8314.4314.5614.500.07 Fig.9(f)167,6000.10111.2015.0714.8514.960.05 Fig.9(g)210,6000.10011.1514.9815.0014.990.06
Purtell,L.P.,Klebanov,P.S.,and Buckley,F.T.(1981)
Fig.10(a)1,0020.1707.398.478.828.640.18 Fig.10(b)1,8370.1647.629.148.879.000.23 Fig.10(c)5,1220.1498.119.7210.079.890.26
Erm,L.P.(1988)
Fig.11(a)2,2440.1538.049.609.809.700.14 Fig.11(b)2,7770.1548.139.759.749.750.16
Petrie,H.L.,Fontaine,A.A.,Sommer,S.T.and Brungart,T.A.(1990) Fig.1235,5300.1199.7612.5712.6112.590.12
Bruns,J.,Dengel,P.,Fernholz,H.H.(1992)and Fernholz,H.H.,Krause,E.,Nockemann,M.and Schober,M.(1995) Fig.13(a)2,5730.1518.4610.329.9310.130.10 Fig.13(b)5,0230.1448.8511.0010.4210.700.11 Fig.13(c)7,1390.1488.4910.3710.1410.250.25 Fig.13(d)16,0800.1428.4510.3110.5610.430.47 Fig.13(e)20,9200.378.5110.4110.9510.680.48 Fig.13(f)41,2600.1328.6310.6211.3610.980.70 Fig.13(g)57,7200.1308.7110.7611.5411.140.84
Djenidi,L.and Antonia,R.A.(1993)
Fig.14(a)1,0330.1548.209.879.749.810.06 Fig.14(b)1,3200.1508.3710.1710.0010.080.06
Warnack,D.(1994)
Fig.15(a)2,5520.1528.2910.039.879.950.12 Fig.15(b)4,7360.1498.209.8710.079.970.22
Figure1.(a)Schematic representation of the experimental data in traditional coordinates lnη,φ.(b)Schematic representation of the experimental data in(lnη,lnφ)coordinates for experiments of the?rst group.
Figure2.(a)The experiments by Collins,Coles,Hike,(1978).Reθ=5,938.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure2.(b)The experiments by Collins,Coles,Hike,(1978).Reθ=6,800.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure2.(c)The experiments by Collins,Coles,Hike,(1978).Reθ=7,880.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure3.(a)The experiments by Erm and Joubert,(1991).Reθ=697.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure3.(b)The experiments by Erm and Joubert,(1991).Reθ=1,003.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure3.(c)The experiments by Erm and Joubert,(1991).Reθ=1,568.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure3.(d)The experiments by Erm and Joubert,(1991).Reθ=2,226.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure3.(e)The experiments by Erm and Joubert,(1991).Reθ=2,788.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure4.(a)The experiments by Naguib,(1992).Reθ=4,550.Both self-similar interme-diate regions(I)and(II)are clearly seen.
Figure4.(b)The experiments by Naguib,(1992).Reθ=6,240.Both self-similar interme-diate regions(I)and(II)are clearly seen.
Figure4.(c)The experiments by Nagib and Hites,(1995).Reθ=9,590.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure4.(d)The experiments by Nagib and Hites,(1995).Reθ=13,800.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure4.(e)The experiments by Nagib and Hites,(1995).Reθ=21,300.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure4.(f)The experiments by Nagib and Hites,(1995).Reθ=29,900.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure4.(g)The experiments by Nagib and Hites,(1995).Reθ=41,800.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure4.(h)The experiments by Nagib and Hites,(1995).Reθ=48,900.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure5.(a)The experiments of Smith,(1994).Reθ=4,996.The?rst self-similar inter-mediate region(I)is clearly seen,the second region(II)can be revealed.
Figure5.(b)The experiments of Smith,(1994).Reθ=12,990.The?rst self-similar intermediate region(I)is clearly seen,the second region(II)can be revealed.
Figure6.The experiments of Krogstad and Antonia,(1998).Reθ=12,570.Both self-similar intermediate regions(I)and(II)are clearly seen.
Figure7.(a)The experiments by Hancock and Bradshaw,(1989)–a general view.?,see Figure8(a);+,see Figure8(b);×,see Figure8(c);2,see Figure8(d);?,see Figure8(e);?,see Figure8(f);?,see Figure8(g).