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Experimental study on the local similarity scaling of the turbulence spectrum in the turbulent boundary layer XIA ZhenYan1,2†, JIANG Nan1,2, TIAN Yan1,2 & WANG YuChun1,2 1 2
Department of Mechanics, Tianjin University, Tianjin 300072, China; Tianjin Key Laboratory of Modern Engineering Mechanics, Tianjin 300072, China
The streamwise fluctuating velocity in the turbulent boundary layer is measured under approximately medium Reynolds Number by hot wire in order to investigate the scaling properties of the overlapped turbulent spectrum among energy-containing area, inertial subrange and dissipation range based on FFT analysis. The experiment indicates that the high Reynolds flow reported before is not indispensable to produce −1 scaling. So far as the measured position is provided with much higher spatial resolution and enough closing to the wall, −1 scaling is determinate to exist when approaching medium Reynolds. The scaling ranges are supposed to begin at inner scale and end in outer scale, which reveals the local similarity of the energy spectrum over the energy-containing eddies near the wall. In the logarithmic area (y+ > 130), −5/3 scaling occurs in the energy spectrum, while moving away from the wall with Reynolds numbers increasing, the inertial subrange extends to the lower wavenumbers. On the condition k1η 0.1, the curves of the turbulence spectrum in the logarithmic layer are superposed, which expresses the similarity of turbulence energy distributed in Komogorov scaling area and exhibits local isotropy characteristics by virtue of the viscous dissipation. turbulent boundary layer, −1 spectrum, −5/3 spectrum, energy-containing eddy, inertial subrange, local isotropy
The investigation on the turbulent boundary layer is at all times the focus and difficulty in fluid mechanics. It is still controversial about the scaling characteristics of energy spectrum in wall turbulence and especially in the similarity of large scale turbulent structures near the wall region[1,2]. At present the prevalent opinions and relative experimental reports make sure that only high Reynolds number can produce −1 spectral overlapped region, whereas how much such range of Reynolds number is not determinately known yet and synchronously the arising position or length of −1 spectral overlapped region cannot draw a definite conclusion. In the condition of the non-homogeneous and anisotropic turbulence, such as near the wall region of the turbulent boundary layer ( y+ < 30), the effect of viscous shearing strength is obvious, however when entering into the absolute tur-
bulent layer, the viscous effects can be ignored and the － Reynolds stress plays a dominant role[3 6]. Consequently the scaling characteristics of turbulent spectrum alter with the position of the boundary layer, which means all kinds of turbulent structures exhibit different traits. The scaling characteristic of k1−1 (k1 is streamwise wave number) in streamwise fluctuating velocity spectrum becomes one of the presumptions about the large scale structures in the turbulent flow[7]. The above essential point of view is that the overlapped similar wave Received November 27, 2007; accepted January 1, 2009 doi: 10.1007/s11433-009-0102-5 † Corresponding author (email:
[email protected]) Supported by the National Natural Science Foundation of China (Grant Nos.
10832001 and 10872145), the Program for New Century Excellent Talents in Universities of Education Ministry of China, and the Plan of Tianjin Science and Technology Development (Grant No. 06TXTJJC13800)
Sci China Ser G-Phys Mech Astron | Jun. 2009 | vol. 52 | no. 6 | 900-908
number region lies in the spectral curves denoted by non-dimensional inner scale and outer scale inside a certain wave number scope, where the viscous effect and the thickness of the boundary layer can be ignored, so it results in the existence of −1 slope in the turbulent spectrum curve, which assumes that the two scales are both in effect. According to the explanation of Tchen in 1953, the actions between the mean flow and turbulent structures induce the wave number region with −1 spectral scaling between the producing turbulence area and the inertial subrange. After this the scaling character of −1 spectrum has been probed by other methods such as similarity analysis, dimensional analysis and coherent structures method near the wall region and so on[7]. Because the control of turbulence mainly involves in how to restrain the movement of these structures carrying on the great mass of turbulent energy, and besides, how to dissipate the energy of large scale turbulent structures in large-eddy simulations (LES), that is to say, whether the sub-grid stress model is accurate or not and the conditions of in or out flow are reasonable, the above mentioned issues have relations with the flowing characteristics and the energy distributing of large turbulent structures[1]. Consequently experimental study on the scaling characteristics of energy spectrum in large scaling turbulent structures has an important theoretical and practical meaning. In homogeneous and isotropic turbulence, the occurring −5/3 power in inertial subrange is expressed by dimension analysis in the famous K41 theory and validated in high Reynolds number experiments[8]. Nevertheless in shear turbulent flow, such non-homogeneous and anisotropic turbulence as the turbulent boundary layer, how the scaling characters of the turbulent spectrum in inertial region to be transformed all the same needs to be further investigated.
1 Scaling analysis of the wall turbulent spectrum By reason of low wave number region of streamwise fluctuating velocity corresponding to the large eddies with the same scale as the thickness of the turbulent boundary layer[9,10], the outer scaling form should be introduced into this turbulent spectrum: φ11 (k1δ ) φ11 (k1 ) = = g1 (k1δ ), (1) δ uτ2 uτ2
where k1 is streamwise wave number, δ is the boundary
layer thickness, uτ is friction velocity, φ11(k1) denotes the energy spectrum intensity of streamwise fluctuating velocity per unit wave number k1, φ11(k1δ ) is the energy spectral form per unit nondimensional wave number k1δ. Because the medium scale eddies, which are of the same size as the distance y from the wall, contribute energy to the middling range of the wave number, the function described in inner scaling is adopted as follows: φ11 (k1 y ) φ11 (k1 ) = = g 2 (k1 y ), (2) uτ2 yuτ2 where φ11(k1y) is the PSD of the streamwise fluctuating velocity per unit nondimensional wave number k1y. The approximatively homogeneous and isotropic small scale eddies involved in viscous movement exist in the high wave number range of turbulent spectrum, where the wave number spectrum complies with the classical Komogorov scaling rule: φ11 (k1η ) φ11 (k1 ) = = g3 (k1η ), ηv2 v2 (3) 1/ 4 1/ 4 ⎛ ν uτ3 ⎞ ⎛ ν 3κ y ⎞ v = ⎜⎜ ⎟⎟ , η = ⎜⎜ 3 ⎟⎟ , ⎝ κy ⎠ ⎝ uτ ⎠ where v is Komogorov velocity and η is Komogorov scale, κ is a constant, φ11(k1η) is the PSD of the streamwise fluctuating velocity per unit nondimensional wave number k1η. The overlapped region I, where inner scaling and outer scaling both work effectively is present in the superposed range between low and high wave number. Then the formulas (1) and (2) are entirely valid, which predicates the following equation:
φ11 (k1 ) = δ uτ2 g1 (k1δ ) = yuτ2 g 2 (k1 y ).
(4)
This equation displays that the wave numbers of the overlapped region I varies along with k1−1. Although much theoretic analysis presumes that overlapped region is present, there is not enough experimental verification. Antonia could not find −1 scaling of the wave number spectrum[11]. Morrison et al.[1] did not observe the overlapped region I in high Reynolds number, and thereby the conclusion was provided that −1 scaling of the spectrum is not broadly applicable and but only incompletely similar according to the result of his experiment; Del Alamo et al.[12] had a opinion that the velocity scaling in these two scale regions should be different from each other which leads to inexistence of overlapped region, and then he suggested that −1 spectrum must be modi-
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fied by logarithm; Nickels[2] observed −1 scaling of spectrum in high Reynolds number and discussed the occurring conditions, nevertheless the height of the boundary layer dominated by entirely similar wave number region, length of overlapped region and the minimum occurring Reynolds number are still short of exact answer. In the superposed range between medium and high wave numbers of turbulent spectrum, there exists the overlapped region II under the conjunct effects by both inner scaling and Komogorov scaling. Since the formula (2) and (3) are simultaneously effective, the following equation comes into use:
φ11 (k1 ) = yuτ2 g 2 (k1 y ) = η v 2 g3 (k1η ).
(5)
The above equation implies that the wave number spectrum of overlapped region II, which is also called inertial subrange, will be evolved in k1−5/3. Leveque observed the development of turbulent spectrum when moving from the wall[13], and could not find the obvious transition of the slope in energy spectrum curves from large scale region to inertial subrange, so that he was not able to find the −1 scaling of spectrum and subsequently announced a fantastic phenomenon that a uniform scaling traverses through the above-mentioned two parts of the wave number ranges. A discussion about this will be offered in the later part of this article. This experiment was performed to measure the streamwise velocity nearby the wall in the turbulent boundary layer on two kinds of experimental conditions approaching to the medium Reynolds numbers. A power analyzing method was used to research the scaling characteristics of each wave number range of turbulent spectrum in the boundary layer, which verifies the existence of −1 scaling spectrum and the characters of −5/3 scaling spectrum, and canvasses the relative problems such as the forming conditions of −1 scaling, occurring positions, the length of the overlapped region in the turbulent spectrum.
2 The experimental equipment and technique The wind tunnel was operated in the straight suction mode with low turbulent intensity in this experiment, whose test segment shows the dimension, 4.5-m length, 350-mm width and 450-mm height of the rectangle section. The wind velocity range can be changed from 0.57 to 41 m/s. The experimental equipment is configured in 902
Figure 1. The test flat plate in 3-m length is made up of stainless steel and fixed on the horizontal center of the test section, whose leading edge is symmetry wedged in order to decrease the influence of the anterior plate to the flow. A piece of sandpaper is embedded in the leading edge of the plate to accelerate the transition of the boundary layer. The hot-wire probe fixed by the coordinates moving system at the location of 1.5 m downstream from the leading edge can measure the instantaneous velocity of the flow along the vertical different heights from the plate. TSI IFA300 Constant Temperature Anemometer with four channels is used to acquire the velocity signal of the flow, which can continue to tail the varieties of the velocity and there is no need of quadrate wave testing. A 12 bits A/D card up to 1 MHz sampling frequency is assembled with a high gain signal amplifier and a low-pass filter for the resisting mixture and eliminating electromagnetic noise, so the SNR arrives at 72 dB.
Figure 1 system.
The schematic diagram of the experiment and measurement
Two kinds of momentum thickness Reynolds number of experiment are respective Reθ = 5.6×103 and Reθ = 5.2×103, which are close to medium Reynolds number (Reθ ≤5×103). The mean velocity profile measured in Reθ = 5.6×103 is shown in Figure 2. According to the invariable linear log relationship emerging in logarithmic region of the mean velocity profile, the skin friction velocity uτ is achieved by nonlinear fits method. The energy spectrum expressed in frequency is calculated by Fourier transform from the temporal series of each
XIA ZhenYan et al. Sci China Ser G-Phys Mech Astron | Jun. 2009 | vol. 52 | no. 6 | 900-908
spatial point measured in this experiment. Being based upon the Tayler frozen hypothesis, the frequent spectrum is converted into wave number spectrum. The formula, k1=2πf /U, denotes the relationship between frequency and wave number, where U is the mean velocity of each measured point and f is frequency.
Figure 2
The mean velocity profile.
3 Experimental results analysis 3.1 The −1 scaling spectrum in the turbulent boundary layer
The streamwise nondimensional wave number spectrum by pre-multiplied factors in the logarithmic region with Reynolds number Reθ = 5.6×103 are shown in Figures 3(a)－(b). These graphs are respectively described by inner scaling (k1y) and outer scaling (k1δ ). Figure 3(a) lays out an obvious horizontal overlapped region in 0.2 ≤ k1 y ≤ 0.4, which belongs to the wave number scope of −1 scaling area in inner scaling. With the measured position moving out of the wall, the wave number spectrum in the low wave number range of the overlapped region (corresponding to the outer scaling
region in Figure 3(b)) gradually deviates from −1 scaling region; however, all curves of measured positions tend to be converged in the higher wave number range. Figure 3(b) shows that there exists a horizontal overlapped region in 12 ≤ k1δ ≤ 30 for the same position of the turbulent boundary layer in Figure 3(a), which falls into −1 scaling region of wave number spectrum expressed by outer scaling; however, in the high wave number section of the overlapped region (corresponding to the inner scaling region in Figure 3(a)), the wave number spectrum deviates little by little from −1 scaling of spectrum when moving away from the wall. By analyzing Figures 3(a) and (b)，the wave number scope of inner scaling extends to higher wave number and simultaneously the wave number part of outer scaling spreads to lower one. The above tendency leads to a horizontal common overlapped range in both scaling regions ( 0.2 ≤ k1 y ≤ 0.4 for the inner scaling, 12 ≤ k1δ ≤ 30 for the outer scaling), which confirms the Reynolds similarity hypothesis introduced by Townsend and existence of the −1 scaling even in non-high Reynolds number of this experiment. For the wave number scope of the horizontal overlapped region, Nickels put forward that −1 scaling region starts from k1δ = 21 of outer scaling wave number spectrum and ends in k1 y = 0.4 of inner scaling wave number spectrum. Because the outer scaling region corresponds to large scale energycontaining eddies in low wave number and further extends to the lower wave number section, nevertheless the medium scale eddies denoted by inner scaling expand to higher wave number section. Thereby it is much more reasonable to adopt the two kinds of scaling as abovementioned to validate the exact overlapped region, which respectively correspond to the inner scaling for
Figure 3 The pre-multiplied streamwise −1 spectrum (Reθ = 5.6×103). (a) Inner scaling; (b) outer scaling. XIA ZhenYan et al. Sci China Ser G-Phys Mech Astron | Jun. 2009 | vol. 52 | no. 6 | 900-908
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indicating the originated position and the outer scaling for identifying the terminative locality in turbulent spectrum. The results of the experiment make it clear that the overlapped region of −1 scaling should start from k1 y = 0.2 and end in k1δ = 30. The curves of streamwise wave number spectrum in Reθ = 5.2×103 are shown in Figures 4(a)－(b). Compared with Figures 3(a)－(b), whether using inner scaling or outer scaling, a segment of overlapped region with −1 slope always exists in logarithmic region ( 65 ≤ y + ≤ 130 ), which shows that −1 scaling of the wall turbulent spectrum is present in this Reynolds number and the scaling range of the wave number spectrum is coincident with the situations in Reθ = 5.6×103. It is not determinate and conclusive for the occurring height of −1 scaling in the turbulent spectrum up to now[1,2]. Nickels suggested that −1 scaling of spectrum appears from y+ ≥ 100. in experimental Reynolds Reθ = 37540. For this experiment the result shows that the range of −1 scaling begins from logarithmic region, which is equivalent to 0.0073 ≤ y/δ ≤ 0.0261 in outer scaling. In spite of the difference of experimental conditions between the above two flows, the originated position of −1 scaling described by outer scaling in this experiment is almost similar to the result of Nickels, namely 0.007 ≤ y/δ ≤ 0.0140 (corresponding to 100≤ y+≤200). Morrison could not observe −1 scaling in round pipeline experiment with two kinds of Reynolds number since his minimum nondimensional measured height denoted by outer scaling in the boundary layer is
Figure 4 904
y / δ = 0.03, which becomes greater than the upper limit
0.0261 in this experiment. Compared with above experimental results, it is more appropriate for the originated height of −1 scaling region from the wall to be expressed in outer scaling, whose value should be located at y/δ ≥ 0.007 nearby the wall and approximates the calculating or experimental results from Perry et al.[9,10]. 3.2 The −5/3 spectrum scaling in the turbulent boundary layer
The curves of the streamwise compensated spectrum in the logarithmic region are shown in Figures 5(a)－(b). From Figure 5(a), there exists a horizontal inner scaling region in 2 ≤ k1 y ≤ 7 belonging to the wave number range of −5/3 scaling spectrum, which corresponds to the overlapped section between the inner region and the Komogorov region. The curves in the low wave number area converge little by little, which indicate that the characteristics of inner scaling are obvious; however, in the higher wave number range the curves of the overlapped region in −5/3 scaling spectrum disperse with each other and ceaselessly decline when moving to the wall, which shows that the inner scaling characters gradually disappear; moreover, the wave number spectrum starts to enter into the absolute Komogorv scaling region. In the position far from the wall y+ = 1062 ( y / δ = 0.213), the nondimensional wave number scope in the horizontal scaling region, namely the inertial subrange, extends to the maximum. The measured position is still located in the logarithmic region, but has
The streamwise −1 spectrum (Reθ = 5.2×103). (a) Inner scaling; (b) outer scaling.
XIA ZhenYan et al. Sci China Ser G-Phys Mech Astron | Jun. 2009 | vol. 52 | no. 6 | 900-908
Figure 5
The streamwise compensated spectrum (Reθ = 5.6×103). (a) Inner scaling; (b) Komogorov scaling.
come into the intersection between wall region and out region. To the same position in the turbulent boundary layer as shown in Figure 5(b), the wave number spectrum of −5/3 scaling lies in the horizontal scaling range 0.02 ≤ k1η ≤ 0.07. The high wave number sections of
0.05 (shown in Figure 6(b)). From the above curves the two scaling regions do not overlap with each other, which indicates the inexistence of the common scaling trait in the −5/3 spectrum of the measured position. In 0.2 ≤ k1 y ≤ 0.4 of Figure 6(a) the curves of the wave
above curves are superposed so as to display the local homogeneous and isotropic Komogorv scaling traits, where the viscous effect causes the similarity in this wave number region of turbulent spectrum. Whereas the curves of the low wave number sections spread around and the Komogorv scaling characters disappear, which means the curves go into the inner scaling region (Figure 5(a)). Summing up Figures 5(a)－(b), the wave number of the inner scaling region becomes lower and the wave number of the Komogorv scaling region becomes higher, which produces the common horizontal overlapped region between 2 ≤ k1 y ≤ 7 (corresponding to 0.02≤
number spectrum are overlapped nearby the wall
k1η ≤ 0.07). When leaving out of the wall, the inertial
subrange sequentially expands to the left part of the turbulent spectrum curves, which reveals that the local homogeneous and isotropic wave number range continuously extends to the lower wave number range ( k1η ) with increased Reynolds number. The compensated spectra nearby the wall of the logarithmic region are shown in Figures 6(a)－(b). Obviously the horizontal overlapped region appears both in the inner scaling region and in the Komogorv scaling region. The overlapped region of inner scaling is located in 1 ≤ k1 y ≤ 2 (shown in Figure 6(a)), corresponding to 0.01 ≤ k1η ≤ 0.02 in Figure 6(b), but the overlapped region of Komogorv scaling emerges in 0.03 ≤ k1η ≤
( 87 ≤ y + ≤ 130 ), which displays the same −1 scaling region as Figure 3(a). Compared with Figures 5(b) and 6(b), for k1η 0.1 (corresponding to k1 y ≥ 10 ) in the dissipation range, the curves of the wave number spectrum are almost superposed, which indicates that the shear stress vanishes and viscous dissipation predominates in the Komogorv scaling region far away from the overlapped II, and makes the dissipation range close to the local homogeneous and isotropic turbulence. This kind of turbulence similarity spreads throughout the whole logarithmic region of the turbulent boundary layer. The curves of the streamwise nondimensional wave number spectrum are shown in Figures 7(a)－(b). The u ⋅v formula, Re y =U + ⋅ y + , defines the local Reynolds
ν
number. Nearby the wall, 65 ≤ y + ≤ 130 , namely 1034 ≤ Re y ≤ 2327 , produces −1 scaling, where the
wave number range corresponds to 0.2 ≤ k1 y ≤ 0.4 or 0.002 ≤ k1η ≤ 0.004 (point F and P in Figure 7). For k1 y 1, the large scale (low wave number) non-iso-
tropic attached eddies (for example hairpin eddy) nearby the wall are mainly devoted to the energy-containing movement. With the increased wave number, the turbulent energy received from the mean flow is transformed
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Figure 6 Overlap scaling regions of the streamwise compensated spectrum (Reθ = 5.6×103). (a) Overlap region I; (b) overlap region II.
Figure 7
The streamwise spectrum (Reθ = 5.6×103). (a) Inner scaling; (b) Komogorov scaling.
from the lower to the higher wave number region by virtue of the non-stability of the flow and shear effect. In the energy transfer process viscous dissipation can be ignored. Passing through the viscous diffuse or the reaction including break-up and combination of eddies, these large scale attached eddies from viscous sublayer produce the layered eddies with the characteristic scale proportional to the distance y from the wall. These energy-containing eddies are provided with the same characteristic velocity uτ and kinetic periodicity, which makes the mean velocity profile in the absolute turbulent layer near the wall take on the linear logarithmic form. Here Reynolds stress and shear stress are almost constant, so the energy cascade course brings forth some similarity. In the process of the energy transferring from low wave number (large scale) to high wave number (small scale) the self-similarity leads to produce −1 scaling region in streamwise velocity wave number spectrum, hence uτ becomes the characteristic velocity 906
scaling in overlapped region I. Nikora[7] put forward that the Komogorv eddy cascade overlapped model to explain the occurrence of −1 scaling of spectrum and presumed that the scaling area of −1 scaling in the turbulent spectrum lies in y / δ ≤ k1 y ≤ 1 . From this experimen-
tal result, the horizontal scaling range of −1 scaling is located in the above presumed range. The formula, k1 y = 1 , becomes the interface between overlapped region I and II, which belongs to the nondimensional wave number of inner scaling region and is intervenient between the outer scaling region and Komogorv scaling region, so it can be regarded as a disjunctive point to separate the shear effect of low wave number region from the viscous action of high wave number region. When the wave number continuously adds up to k1 y 1 , the viscous effect is enhanced little by little and the wave number starts entering into the overlapped region II, where the inner scaling region and Komogorv
XIA ZhenYan et al. Sci China Ser G-Phys Mech Astron | Jun. 2009 | vol. 52 | no. 6 | 900-908
scaling region come together into the overlapped area. From now on the transfer and dissipation of energy entirely reaches a balanced situation, thereby the wave number comes into the local isotropic inertial subrange. The −5/3 spectrum scaling (corresponding to point N and M in Figure 7) lies in the wave number range of 2 ≤ k1 y ≤ 7 or 0.02 ≤ k1η ≤ 0.07 throughout all over of the turbulent boundary layer. Leveque could not determine the starting and ending position of overlapped region II and was unable to find −5/3 scaling of spectrum. For the high wave number region when k1η 0.1, the wave number spectrum of the whole logarithmic region in Figure 7(b) turns to approximate superposition, which displays a similar distributing of wave number spectrum denoted by k1η in the effect of viscous dissipation. In other words, the characteristic scaling: viscous length scaling η and Komogorv velocity v exists in the whole Komogorv scaling region. Because the turbulent structures of this wave number region are regarded as small scale eddies closing to homogeneous and isotropic turbulence, they mainly viscously dissipate the energy of the large scale structures and are distributed in all around the large scale attached eddies. When the wave number is greater than point N in 130 < y+≤1062 (2327≤Rey≤24774), with moving out of the wall the gradually increasing scaling regresses to −5/3 power law in the homogeneous and isotropic turbulence throughout inner scaling and outer scaling region including the separated points between the overlapped region I and II. But for 65≤y+≤130, namely 1034≤Rey≤2327, in the wave number k1η ≤ 0.01 (corresponding to k1 y ≤ 1), the scaling of spectrum actually turns to transition and gradually evolves to −1 scaling of spectrum. However Leveque could not observe the obviously shifted slope in his wave number region of the turbulent spectrum because −1 scaling of spectrum is a short piece of segment in wave number region. If too long wave number region embodying all outer and inner scaling is used to calculate the slope of energy spectrum, a real scaling value will be sunk into the computing process.
4 Conclusion By measuring the streamwise fluctuating velocity of the turbulent boundary layer in approximately medium
Reynolds number, the result shows that −1 spectrum scaling exists in the logarithmic region (50 ≤ y + ≤ 130) near the wall, which corresponds to 0.0073 ≤ y δ ≤ 0.0261 in outer scaling. It is validated that the occurring position of the turbulent boundary layer for −1 scaling region can be appropriately expressed in outer scaling, which should be much near the wall in y / δ ≥ 0.007. The high Reynolds number is not the necessary condition for producing −1 scaling of spectrum; furthermore, the much closer and denser spatial resolving measurement is the key factor. In order to reasonably identify the overlapped range between low wave number energy-containing eddies expressed in outer scaling and medium scale eddies defined in inner scaling, the −1 scaling of spectrum should begin from inner scaling and end in outer scaling. In this experiment −1 scaling of spectrum starts from k1y = 0.2 and terminates at k1δ = 30. By virtue of the viscous diffusion or the activation including break-up and combination of eddies, the large scale attached eddies originating from viscous sublayer produce layered eddies with the characteristic scale proportional to the distance y from the wall, moreover, the process of large scale energy-containing eddies transferring energy step by step to the smaller ones keeps similarity, and the streamwise mean velocity profile takes on the linear logarithmic form. The above-mentioned factors lead to −1 spectrum scaling . For the logarithmic region, 130 < y+≤1062 (2327≤ Rey≤24774), there exists −5/3 spectrum scaling in 2≤ k1y≤7. With moving away from the wall, the gradually increasing scaling comes back to −5/3 power law of the homogeneous and isotropic turbulence throughout outer scaling and inner scaling region containing the separated points between the overlapped regions I and II, and besides, the range of inertial subrange extends to low wave number (k1η) along with Reynolds number increasing step by step. When k1η 0.1 (corresponding to k1y≥10), the curves of the wave number spectrum in each measured position of the turbulent boundary layer are closely superposed and the flow approximates to the homogeneous and isotropic turbulence. The above similarity dominates the whole logarithmic region in the turbulent boundary layer, which means that there exists the characteristic scaling, namely viscous length scaling η and Komogorv velocity v.
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