**4. Energy spectra and structure functions in boundary layer flows**

Atmospheric data come from anemometers frequently located 10 m height. These values contain climatic system contributions and components of the boundary layer itself. That is, measured data include wind velocity variations corresponding to time scales from some hours to fractions of one second. Usually power spectra are employed to analyze these at‐ mospheric records. The Van der Hoven spectrum, obtained in Brookhaven, Long Island, NY, USA [24], represents the energy of the longitudinal velocity fluctuation on the complete frequency domain. Two peaks can be distinguished in this spectrum, one corresponding to the 4-day period or 0.01 cycles/hour (macro-meteorological peak), and another peak be‐ tween the periods of 10 minutes and 3 seconds associated to the boundary layer turbulence (micro-meteorological peak). A spectral valley, with fluctuations of low energy, is observed between the macro and micro-meteorological peak. This region is centered on the period of 30 minutes and allows dividing the mean flow and the velocity fluctuations. This spectral characteristic confirms that interaction between climate and boundary layer turbulence is negligible and permits considering both aspects independently.

Velocity fluctuations with periods lower than one hour define the micro-meteorological spectral region or the atmospheric turbulence spectrum. Interest of wind load and disper‐ sion problems is concentrated on this spectral turbulence region. In 1948 von Kármán sug‐ gested an expression for the turbulence spectrum with which his name is related, and 20 years later this spectral formula started to be used for wind engineering applications. Some deficiencies in fitting data measured in atmospheric boundary layer were pointed later and Harris [5] shown a modified formulae for the von Kármán spectrum.

According ESDU [3], the von Kármán formula for the dimensionless spectrum of the longi‐ tudinal component of atmospheric turbulence is:

$$\frac{fS\_u}{\left(\sigma\_u\right)^2} = \frac{4X\_u(z)}{\left[1 + 70, 78X\_u(z)^2\right]^{5/6}}\tag{3}$$

where *Su* is the spectral density function of the longitudinal component, *f* is the frequency in Hertz and *σ<sup>u</sup> <sup>2</sup>* is the variance of the longitudinal velocity fluctuations. The dimensionless fre‐ quency *Xu*(*z*) is *fL(z)/U(z)*, being L the integral scale. This spectrum formula satisfies the Wie‐ ner-Khintchine relations between power spectra and auto-correlations and provides a Kolmogorov equilibrium range in the spectrum. However, the von Kármán expression pro‐ vides no possibility to fit other measured spectral characteristics [5].

Two situations of spectral analysis of boundary layer flow are presented next from different wind tunnel studies and atmospheric data. These cases resume a typical spectral evaluation of a boundary layer simulation and a spectral comparison of different boundary layer flows. Finally, a discussion of the use of structure functions applied to the analysis of velocity fluc‐ tuations is presented.

#### **4.1. Spectral evaluation of a wind tunnel boundary layer simulation**

A first example of spectral analysis is that corresponding to the Counihan boundary layer simulation described on previous section. Longitudinal velocity fluctuations were measured by the hot wire anemometer system and the uncertainty associated with the measured data is the same as previously mentioned. In this case, spectral results from longitudinal velocity fluctuations were obtained by juxtaposing three different spectra from three different sam‐ pling series, obtained in the same location, each with a sampling frequency, as given in Ta‐ ble 1, as low, mean and high frequencies. The series were divided in blocks to which an FFT algorithm was applied [25]. In Fig. 6, four spectra obtained at height z=0.233, 0.384, 0.582 and 0.966 m are shown. Values of the spectral function decrease as the distance from the tunnel floor z is increased. An important characteristic of the spectra is the presence of a clear region with a -5/3 slope, characterizing Kolmogorov's inertial sub-range.

The comparison of the results obtained through the simulations with the atmospheric boun‐ dary layer is made by means of dimensionless variables of the auto-spectral density *fSu/σ<sup>u</sup> 2* and of the frequency *Xu*(*z*) using the von Kármán spectrum, given by the expression of Eq. (3). Kolmogorov's spectrum will have, therefore, a -2/3 exponent instead of -5/3. The com‐ parison was realized for spectra measured at different heights, but only is presented the spectrum obtained at *z* = 0.233 m (Fig. 7). The agreement is very good, except for the highest frequencies affected by the action of the low-pass filter.


**Table 2.** Data acquisition conditions for spectral analysis.

hours to fractions of one second. Usually power spectra are employed to analyze these at‐ mospheric records. The Van der Hoven spectrum, obtained in Brookhaven, Long Island, NY, USA [24], represents the energy of the longitudinal velocity fluctuation on the complete frequency domain. Two peaks can be distinguished in this spectrum, one corresponding to the 4-day period or 0.01 cycles/hour (macro-meteorological peak), and another peak be‐ tween the periods of 10 minutes and 3 seconds associated to the boundary layer turbulence (micro-meteorological peak). A spectral valley, with fluctuations of low energy, is observed between the macro and micro-meteorological peak. This region is centered on the period of 30 minutes and allows dividing the mean flow and the velocity fluctuations. This spectral characteristic confirms that interaction between climate and boundary layer turbulence is

Velocity fluctuations with periods lower than one hour define the micro-meteorological spectral region or the atmospheric turbulence spectrum. Interest of wind load and disper‐ sion problems is concentrated on this spectral turbulence region. In 1948 von Kármán sug‐ gested an expression for the turbulence spectrum with which his name is related, and 20 years later this spectral formula started to be used for wind engineering applications. Some deficiencies in fitting data measured in atmospheric boundary layer were pointed later and

According ESDU [3], the von Kármán formula for the dimensionless spectrum of the longi‐

*X z*

where *Su* is the spectral density function of the longitudinal component, *f* is the frequency in

quency *Xu*(*z*) is *fL(z)/U(z)*, being L the integral scale. This spectrum formula satisfies the Wie‐ ner-Khintchine relations between power spectra and auto-correlations and provides a Kolmogorov equilibrium range in the spectrum. However, the von Kármán expression pro‐

Two situations of spectral analysis of boundary layer flow are presented next from different wind tunnel studies and atmospheric data. These cases resume a typical spectral evaluation of a boundary layer simulation and a spectral comparison of different boundary layer flows. Finally, a discussion of the use of structure functions applied to the analysis of velocity fluc‐

A first example of spectral analysis is that corresponding to the Counihan boundary layer simulation described on previous section. Longitudinal velocity fluctuations were measured by the hot wire anemometer system and the uncertainty associated with the measured data is the same as previously mentioned. In this case, spectral results from longitudinal velocity

*<sup>2</sup>* is the variance of the longitudinal velocity fluctuations. The dimensionless fre‐

(3)

<sup>2</sup> 5/6 <sup>2</sup> 4 () 1 70,78 ( )

é ù <sup>+</sup> ë û

*u u <sup>u</sup> <sup>u</sup> fS X z*

negligible and permits considering both aspects independently.

Harris [5] shown a modified formulae for the von Kármán spectrum.

s<sup>=</sup>

vides no possibility to fit other measured spectral characteristics [5].

**4.1. Spectral evaluation of a wind tunnel boundary layer simulation**

tudinal component of atmospheric turbulence is:

204 Wind Tunnel Designs and Their Diverse Engineering Applications

Hertz and *σ<sup>u</sup>*

tuations is presented.

**Figure 6.** Power spectra of the longitudinal velocity fluctuation for a boundary layer simulation.

This evaluation was realized at high velocity (*Ug* ≈ 27 m/s) being the resulting Reynolds number value of Re ≈ 4×106 . The juxtaposing technique used to improve the spectral resolu‐ tion is today unnecessary because of the fact that is possible to utilize a large sample size. However, sample series were limited to 32000 values for this analysis and three spectra were juxtaposed.

A scale factor of 250 for this boundary layer simulation was obtained through the procedure proposed by Cook [4], by means of the roughness length *z0* and the integral scale *Lu* as pa‐ rameters. The values of the roughness length are obtained by fitting experimental values of velocity to the logarithmic law of the wall, while integral scale is given by fitting the values of the measured spectrum to the design spectrum.

**Figure 7.** Comparison of the dimensionless spectrum obtained at z = 0.233 m and the von Kármán spectrum.

#### **4.2. Spectral comparison of different boundary layer flows**

A second study based on results of different boundary layer flows was realized. Measure‐ ments of the longitudinal fluctuating velocity obtained in three different wind tunnels were selected for this analysis. All selected velocity samples correspond to neutral boundary layer flow simulations developed in appropriate wind tunnels. The analysis was complemented using measurements realized in a smooth tube flow and in the atmosphere.

Wind tunnel and smooth tube measurements were realized by a constant hot-wire anemom‐ eter previously described. Atmospheric data were obtained using a Campbell 3D sonic ane‐ mometer [26], for which the resolution is 0.01 m/s for velocity measurements. Table 3 indicates a list of sampling characteristics, being *z* the vertical position (height), *U* the mean velocity, *σ<sup>u</sup> 2* the variance of fluctuations velocity, *facq* the acquisition frequency, *Lu* the integral scale and *ReL* the Reynolds number associated to *Lu*.

One of the three wind tunnels used to obtain the wind data employed in this experimental analysis is the *"Jacek Gorecki"* wind tunnel described on a previous section. The second is the "*TV2"* wind tunnel of the Laboratorio de Aerodinámica, UNNE, too. The "*TV2"*, smaller, is also an open circuit tunnel with dimensions of 4.45×0.48×0.48m (length, height, width). The study was complemented by the analysis of measurements realized on atmospheric boun‐ dary layer simulations performed in the closed return wind tunnel "Joaquim Blessmann" of the Laboratório de Aerodinâmica das Construçoes, Universidade Federal de Rio Grande do Sul, UFRGS [12]. The simulations of natural wind on the atmospheric boundary layer were performed by means of the Counihan [16] and Standen [17] methods, with velocity distribu‐ tions corresponding to a forest, industrial or urban terrain. The tube measurement was ob‐ tained in the centre of a 60 mm diameter smooth tube. Atmospheric data were obtained in a micrometeorological station located at Paraiso do Sul, RS, Brasil [26, 27].


**Table 3.** Measurement characteristics for spectral analysis.

This evaluation was realized at high velocity (*Ug* ≈ 27 m/s) being the resulting Reynolds

tion is today unnecessary because of the fact that is possible to utilize a large sample size. However, sample series were limited to 32000 values for this analysis and three spectra were

A scale factor of 250 for this boundary layer simulation was obtained through the procedure proposed by Cook [4], by means of the roughness length *z0* and the integral scale *Lu* as pa‐ rameters. The values of the roughness length are obtained by fitting experimental values of velocity to the logarithmic law of the wall, while integral scale is given by fitting the values

0.1 1 10 100

Dimensionless frequency

A second study based on results of different boundary layer flows was realized. Measure‐ ments of the longitudinal fluctuating velocity obtained in three different wind tunnels were selected for this analysis. All selected velocity samples correspond to neutral boundary layer flow simulations developed in appropriate wind tunnels. The analysis was complemented

Wind tunnel and smooth tube measurements were realized by a constant hot-wire anemom‐ eter previously described. Atmospheric data were obtained using a Campbell 3D sonic ane‐ mometer [26], for which the resolution is 0.01 m/s for velocity measurements. Table 3 indicates a list of sampling characteristics, being *z* the vertical position (height), *U* the mean

One of the three wind tunnels used to obtain the wind data employed in this experimental analysis is the *"Jacek Gorecki"* wind tunnel described on a previous section. The second is the

the variance of fluctuations velocity, *facq* the acquisition frequency, *Lu* the integral

 measured (z=0.233 m) von Kármán spectrum

**Figure 7.** Comparison of the dimensionless spectrum obtained at z = 0.233 m and the von Kármán spectrum.

using measurements realized in a smooth tube flow and in the atmosphere.

. The juxtaposing technique used to improve the spectral resolu‐

number value of Re ≈ 4×106

of the measured spectrum to the design spectrum.

206 Wind Tunnel Designs and Their Diverse Engineering Applications

0.01

**4.2. Spectral comparison of different boundary layer flows**

scale and *ReL* the Reynolds number associated to *Lu*.

Dimensionless spectrum

0.1

1

juxtaposed.

velocity, *σ<sup>u</sup>*

*2*

The measurements realized in the J. Gorecki wind tunnel at high velocity were used to ana‐ lyze the sampling effects on the spectral characteristics. Five different samplings were realiz‐ ed for measurements Gorecki WT-HV(+) at z= 0.21 m. Sampling characteristics like frequency acquisition *facq*, low pass frequency *flp* and sampling time *ts* are indicated in Table 4. Resulting superposed spectra are shown in Fig. 8 where it is possible to see a good definition of the inertial sub-range (-5/3 slope) and the effect of the low pass filter.


**Table 4.** Measurement characteristics for analysis of sampling effects.

Fig. 9 shown spectral density functions *Su* corresponding to measurements indicated in Ta‐ ble 3. High frequencies in the atmosphere spectrum correspond to low frequencies in the smooth pipe. The same spectra in dimensionless form are presented in Figs. 10 and 11. The frequency is non dimensionalised by *fLu/U* in Fig. 10 and by *fz/U* in Fig. 11, according to pa‐ rameters usually employed in wind engineering. In general, preliminary results permit veri‐ fying the good behavior of the wind tunnel spectra and a good definition of the inertial range (slope -5/3). The inertial sub-region is narrower for low velocity measurements (LV).

Spectral special features in smooth tube and atmosphere appear in Fig. 9 and in the dimen‐ sionless comparison too (Figs. 10 and 11). This particular behavior is a product of the uni‐ form flow in the centre of the smooth tube, that is, not a boundary layer flow is being analyzed. In the atmospheric flow case, this type of behavior is possibly due to the existence of a convective turbulence component at low frequencies because of that atmospheric stabil‐ ity is not totally neutral. This behavior was verified in the case of measurements realized in near-neutral atmosphere. The existence of a low frequency convective component was de‐ tected in three dimensional measurements obtained at the atmosphere [28]. The aliasing ef‐ fect is perceived at high frequencies due to high pass filter is not used for sample acquisition of atmospheric data.

**Figure 8.** Spectral superposition for different sampling.

**Figure 9.** Power spectra for measurements indicated on Table 3.

frequency is non dimensionalised by *fLu/U* in Fig. 10 and by *fz/U* in Fig. 11, according to pa‐ rameters usually employed in wind engineering. In general, preliminary results permit veri‐ fying the good behavior of the wind tunnel spectra and a good definition of the inertial range (slope -5/3). The inertial sub-region is narrower for low velocity measurements (LV).

Spectral special features in smooth tube and atmosphere appear in Fig. 9 and in the dimen‐ sionless comparison too (Figs. 10 and 11). This particular behavior is a product of the uni‐ form flow in the centre of the smooth tube, that is, not a boundary layer flow is being analyzed. In the atmospheric flow case, this type of behavior is possibly due to the existence of a convective turbulence component at low frequencies because of that atmospheric stabil‐ ity is not totally neutral. This behavior was verified in the case of measurements realized in near-neutral atmosphere. The existence of a low frequency convective component was de‐ tected in three dimensional measurements obtained at the atmosphere [28]. The aliasing ef‐ fect is perceived at high frequencies due to high pass filter is not used for sample acquisition

1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04

Slope -5/3

**f [Hz]**

of atmospheric data.

208 Wind Tunnel Designs and Their Diverse Engineering Applications

1.00E-03

**Figure 8.** Spectral superposition for different sampling.

1.00E-02

1.00E-01

1.00E+00

sp1 sp2 sp3 sp4 sp5

1.00E+01

**Su [(m/s)2/Hz]** 

1.00E+02

1.00E+03

1.00E+04

1.00E+05

**Figure 10.** Comparison of dimensionless spectra using *fLu/U*.

**Figure 11.** Comparison of dimensionless spectra using *fz/U.*

The superposition technique allows defining precisely the sub-inertial range and extending the frequency analysis interval. Besides, it is possible defining adequately the sampling characteristics and optimizing the measuring time. In general, the spectral comparison real‐ ized using *fLu/U* (Fig. 10) indicates better coincidence [27, 28]. However, the analysis realized up to now is preliminary and it should be studied in depth. For example, the methods for the parameter *Lu* calculation should be analyzed, the application of other parameters to ob‐ tain the dimensionless frequency at smaller scales and other measurements must be ana‐ lyzed looking for the improvement of the scale modeling.

A different approach to analyze velocity fluctuations will be presented below. This is based on the high order moments of velocity increments. Small scales to characterize the boundary layer flows will be used and a new representation of energy spectra will be evaluated.

#### **4.3. Statistical moments of velocity fluctuations**

Previous type of spectral analysis is usually employed in Wind engineering. The following study is realized using velocity structure functions of turbulent boundary layer flow. These statistical moments are utilized by atmospheric physical researchers. The approach consid‐ ers scales smaller than the integral scale *Lu* and, therefore is presumably more suitable for applications to turbulent diffusion studies. Apart from integral scales, the mean dissipation rate, the Kolmogorov and Taylor micro-scales could be obtained. On other hand, results from this type of study can be employed to analyze the Kolmogorov constant and, indirect‐ ly, for application to pollution dispersion models [30, 31].

Kolmogorov`s laws for locally isotropic turbulence [32, 33] were originally derived for struc‐ ture functions from the von Kármán-Howarth-Kolmogorov equation [34],

$$S\_3(r) = -\frac{4}{5}\varepsilon r + 6\nu \frac{d}{dr} S\_2(r) \tag{4}$$

valid for *r<<Lu* in the limit of very large Reynolds number, where *Sp*(*r*)= *<sup>u</sup>*(*<sup>x</sup>* <sup>+</sup> *<sup>r</sup>*)−*u*(*x*) *<sup>p</sup>* is the structure function of order *p*, *ν* is the kinematic viscosity, *ε* is the mean dissipation rate, and ⋅ represents statistical expectation operator.

Kolmogorov deduced the following relations for second and third-order structure functions:

0.001

**Figure 11.** Comparison of dimensionless spectra using *fz/U.*

lyzed looking for the improvement of the scale modeling.

ly, for application to pollution dispersion models [30, 31].

**4.3. Statistical moments of velocity fluctuations**

0.001 0.01 0.1 1 10

smooth tube atmosphere Blessmann WT-LV Gorecki WT-LV Gorecki WT-HV TV2 WT-LV TV2 WT-HV

**fz/U**

The superposition technique allows defining precisely the sub-inertial range and extending the frequency analysis interval. Besides, it is possible defining adequately the sampling characteristics and optimizing the measuring time. In general, the spectral comparison real‐ ized using *fLu/U* (Fig. 10) indicates better coincidence [27, 28]. However, the analysis realized up to now is preliminary and it should be studied in depth. For example, the methods for the parameter *Lu* calculation should be analyzed, the application of other parameters to ob‐ tain the dimensionless frequency at smaller scales and other measurements must be ana‐

A different approach to analyze velocity fluctuations will be presented below. This is based on the high order moments of velocity increments. Small scales to characterize the boundary layer flows will be used and a new representation of energy spectra will be evaluated.

Previous type of spectral analysis is usually employed in Wind engineering. The following study is realized using velocity structure functions of turbulent boundary layer flow. These statistical moments are utilized by atmospheric physical researchers. The approach consid‐ ers scales smaller than the integral scale *Lu* and, therefore is presumably more suitable for applications to turbulent diffusion studies. Apart from integral scales, the mean dissipation rate, the Kolmogorov and Taylor micro-scales could be obtained. On other hand, results from this type of study can be employed to analyze the Kolmogorov constant and, indirect‐

0.01

**fSu/**s**u**

**2**

0.1

1

210 Wind Tunnel Designs and Their Diverse Engineering Applications

$$\mathcal{S}\_2(r) = \mathcal{C}(\varepsilon r)^{2/3} \tag{5}$$

$$S\_3(r) = -\frac{4}{5}\varepsilon r,\tag{6}$$

valid for *η<<r<<Lu*, where *η* =(*ν* <sup>3</sup> / *ε*) 1/4 is the Kolmogorov microscale, and C≈2 is the Kolmo‐ gorov constant [29, 34].

The third-order structure function Eq.(6), also known as the four-fifths law, is straightfor‐ wardly obtained from Eq.(4) since, for very large Reynolds number, the second term in the right hand side of Eq.(4) can be neglected. The four-fifths law is of special interest in the stat‐ istical theory of turbulence because, besides being an exact relation, it allows a direct identi‐ fication of the mean dissipation energy per unit mass with the mean energy transfer across scales [35].

The two-thirds law Eq. (3), on the other hand, is not an exact relation; it was obtained using dimensional arguments and introducing a nondimensional constant that should be empiri‐ cally determined. The second-order structure function provides information about the ener‐ gy content in all scales smaller than *r*. Moreover, the famous Kolmogorov energy spectrum *<sup>E</sup>*(*k*)=*Ck<sup>ε</sup>* 2/3 *k* <sup>−</sup>5/3 is derived from Eq. (5).

Table 5 shows the results of the analysis for four experiments selected from the analysis de‐ scribed in section 4.2. The distinct columns report the mean wind speed *U*, height *z*, inertial range (*ra, rb*), integral scale *Lu*, mean dissipation rate *ε*, Kolmogorov microscale *η*, Taylor's microscale based Reynolds number *Reλ*. The mean dissipation rate, *ε*, was determined by the best fit of S3(r), Eq. (2), in the inertial range. The Kolmogorov microscale was computed by *η* =(*ν* <sup>3</sup> / *ε*) 1/4 and Taylor's microscale based Reynolds number Re*<sup>λ</sup>* =*σuλ* / *ν* was computed from *λ* = *σ<sup>u</sup>* <sup>2</sup> / (∂*<sup>x</sup> u*)<sup>2</sup> 1/2 , where (∂*<sup>x</sup> u*)<sup>2</sup> was indirectly estimated with the aid of the isotrop‐ ic relation *ε* =15*ν* (∂*<sup>x</sup> u*)<sup>2</sup> .


**Table 5.** Main turbulence characteristics from laboratory and atmospheric turbulence data.

Experimental evaluations of second and third-order structure functions for the J. Gorecki wind tunnel are shown in Fig. 12. In the K41 picture, the estimation of the second-order structure function constant is reduced to an estimation of the skewness *S* =*S*3(*r*) / (*S*2(*r*))3/2; however, differently from *S*2(*r*), *S*3(*r*) displays some noise (Fig. 12). This behavior is ob‐ served in all datasets.

One immediate consequence of the similarity arguments assumed in K41 is that graphical representation of distinct turbulence spectra should collapse in a single-curve after a proper normalization with characteristic velocity and length scales. Another consequence, which follows from dimensional analysis, is the scaling *Sp*(*r*)≈*<sup>r</sup> <sup>p</sup>*/3 for a structure function of order *p*, with *η<<r<<Lu*. However, inertial range physics has been proved to be much more com‐ plex than previously assumed in the K41. A remarkable consequence of this complexity, which has close relation with the small scale intermittency phenomenon [33], is the existence of anomalous scaling concerning structure functions exponents, *Sp*(*r*)≈*<sup>r</sup> <sup>ζ</sup><sup>p</sup>* , where ζ*p* is non linear function of *p*. The multifractal formalism was then introduced by Parisi and Frisch in order to provide a robust framework, allowing the analysis and interpretation for a general class of complex phenomena presenting anomalous scaling.

One important difference between the multifractal interpretation of turbulence and the (monofractal) K41 theory is the assumption of a local similarity scaling for small scales. The global scaling similarity assumed in the K41 theory is still at the core of the most wind tun‐ nel and atmospheric turbulence modeling [5, 28, 34]. The local scaling similarity ideas of the multifractal formalism, on the other side, provide a new vocabulary, enabling interpretation and comparison of diverse multiscale phenomena. Although the multifractal formalism has been used in many areas of applied physics, does not share the same popularity in the fields of engineering.

According to the multifractal universality [36], a single-curve collapse of distinct experimen‐ tal turbulence spectra is obtained by plotting log*E*(*k*)/ logRe against log*k* / logRe, after hav‐ ing properly normalized *E*(*k*) and *k.* On the other hand, an alternative similarity plot has been proposed by Gagne et al. [7] based on an intermittency model, but still compatible with the multifractal formalism. These authors propose that a better merging of experimental spectra can be obtaining by plotting *β*log(*aE*(*k*)(*εν* <sup>5</sup> ) <sup>−</sup>1/4) against *β*log(*ck*(*ν* <sup>3</sup> / *ε*) 1/4), with *<sup>β</sup>* =1 / log(Re*<sup>λ</sup>* / *<sup>R</sup>* <sup>∗</sup>), where Re*λ* is the Taylor scale based Reynolds number and the empirical constants *R\** = 75, *a* = 0.154, and *c* = 5.42 were determined to provide the best possible super‐ position in their dataset.

*z [m] U [m/s] (ra, rb)[cm] Lu [m]* ε [m*2/s3]* η [mm] *Re*<sup>λ</sup>

*Smooth tube* 0.03 38.89 0.35-1.10 0.034 52.9 0.08 174

*Atmosphere* 10.00 4.51 30-600 36.30 0.045 0.51 13141

*Gorecki WT-HV* 0.21 16.77 2.0-9.0 0.39 33.0 0.10 1311

*TV2 WT-HV* 0.04 11.69 0.3-2.0 0.13 48.8 0.09 629

Experimental evaluations of second and third-order structure functions for the J. Gorecki wind tunnel are shown in Fig. 12. In the K41 picture, the estimation of the second-order structure function constant is reduced to an estimation of the skewness *S* =*S*3(*r*) / (*S*2(*r*))3/2; however, differently from *S*2(*r*), *S*3(*r*) displays some noise (Fig. 12). This behavior is ob‐

One immediate consequence of the similarity arguments assumed in K41 is that graphical representation of distinct turbulence spectra should collapse in a single-curve after a proper normalization with characteristic velocity and length scales. Another consequence, which

*p*, with *η<<r<<Lu*. However, inertial range physics has been proved to be much more com‐ plex than previously assumed in the K41. A remarkable consequence of this complexity, which has close relation with the small scale intermittency phenomenon [33], is the existence

linear function of *p*. The multifractal formalism was then introduced by Parisi and Frisch in order to provide a robust framework, allowing the analysis and interpretation for a general

One important difference between the multifractal interpretation of turbulence and the (monofractal) K41 theory is the assumption of a local similarity scaling for small scales. The global scaling similarity assumed in the K41 theory is still at the core of the most wind tun‐ nel and atmospheric turbulence modeling [5, 28, 34]. The local scaling similarity ideas of the multifractal formalism, on the other side, provide a new vocabulary, enabling interpretation and comparison of diverse multiscale phenomena. Although the multifractal formalism has been used in many areas of applied physics, does not share the same popularity in the fields

According to the multifractal universality [36], a single-curve collapse of distinct experimen‐ tal turbulence spectra is obtained by plotting log*E*(*k*)/ logRe against log*k* / logRe, after hav‐ ing properly normalized *E*(*k*) and *k.* On the other hand, an alternative similarity plot has been proposed by Gagne et al. [7] based on an intermittency model, but still compatible with the multifractal formalism. These authors propose that a better merging of experimental

for a structure function of order

, where ζ*p* is non

**Table 5.** Main turbulence characteristics from laboratory and atmospheric turbulence data.

212 Wind Tunnel Designs and Their Diverse Engineering Applications

follows from dimensional analysis, is the scaling *Sp*(*r*)≈*<sup>r</sup> <sup>p</sup>*/3

class of complex phenomena presenting anomalous scaling.

of anomalous scaling concerning structure functions exponents, *Sp*(*r*)≈*<sup>r</sup> <sup>ζ</sup><sup>p</sup>*

served in all datasets.

of engineering.

In Fig. 13 the plot proposed by Gagne et al. [7] is presented for laboratory and atmospheric turbulence data. Despite the fact that data comprise very different scales, *Lu* ≈ 102 m for at‐ mospheric data, and *Lu* ≈ 10-1 m for smooth pipe, the merging of spectra is reasonably good, also regarding the fact that the originally proposed empirical constants have been used in the present dataset.

In this representation the slopes remain unchanged, but the extent of inertial range presum‐ ably has the same length for all spectra. Although a solid ground for the physics behind the representation is lacking, it is clear that the properties provided by such a representation can be very useful for physical analysis and modeling of turbulence.

**Figure 12.** Second and third-order structure functions for the Gorecki WT measurement*.*

**Figure 13.** Single-curve spectral collapse from laboratory and atmospheric turbulence data (Table 5), as proposed by Gagne et al. [7].
