Description of turbulence scales and energy spectra

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -114-

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -115- frequency region with the increment in mean flow velocity, as shown in figure (4.8b). For downstream position of 49.8d an increment of Reynolds number from Re=5801 to Re=18436 have produced a displacement of peak from 3594 Hz to 7031 Hz.

(a) (b)

Figure (4. 8): (a) Energy spectrum at y/d=0.085 and y/d=3.14; along with flow at Re=5801, (b) comparison of energy spectrum for different Reynolds numbers; Re=5801 and 18436 at y/d=0.085.

However, this periodicity doesn’t exist in the boundary layer region where the turbulence is higher due to the wall shearing. The measurement of both velocity and temperature field at a maximum acquisition frequency being as high as 10 times of the maximum jet excitation frequency will remain unaffected by these frequency peaks.

The energy spectra and the corresponding inertial laws were determined for the points lying at y/d=3.14 and x/d=-10.2d, 12.9d, 36d and 49.8d for Re=5801 and 18436, see figure (4.9a-4.9h). A survey of the above results shows that the size of the inertial zone increases as the flow Reynolds number is increased. This phenomenon is usually reported by a one-dimensional spectra E11(K1)/(εν⁵)^1/4 plotted as a function of Kolmogorov scale K1.η, and the variation of spectrum for different Reynolds numbers based on the Taylor scale is shown, see Pope (2000).

1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03

1.00E+00 1.00E+02 1.00E+04

K- (m^-1) E(K)- (m3 /s2 )

-5/3 Measured

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01

1.00E+00 1.00E+02 1.00E+04

K- (m^-1) E(K)- (m3 /s2 )

-5/3 Meas ured

(a) (b)

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -116-

1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03

1.00E+00 1.00E+02 1.00E+04

K- (m^-1) E(K)- (m3 /s2 )

-5/3 Measured

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01

1.00E+00 1.00E+02 1.00E+04

K- (m^-1) E(K)- (m3 /s2 )

-5/3 Measured

(c) (d)

1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01

1.00E+00 1.00E+02 1.00E+04

K- (m^-1) E(K)- (m3 /s2 )

-5/3 Measured

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00

1.00E+00 1.00E+02 1.00E+04

K- (m-1) E(K)- (m3 /s2 )

-5/3 Measured

(e) (f)

1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01

1.00E+00 1.00E+02 1.00E+04

K- (m^-1) E(K)- (m3 /s2 )

-5/3 Measured

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00

1.00E+00 1.00E+02 1.00E+04

K- (m^-1) E(K)- (m3 /s2 )

-5/3 Measured

(g) (h)

Figure (4. 9): Energy spectra for the case o f Re=5801 obtained from y/d=3.14 and x/d=; (a) -10.2d, (c) 12.9d, (e) 36d and (g) 49.8d, and for the case of Re=18436 obtained from y/d=3.14 and x/d=; (b) -

10.2d, (d) 12.9d, (f) 36d and (h) 49.8d.

The elongation of inertial zone is also attributed to the shift of the measuring station along the longitudinal direction. Thus, at each successive plane a small increment in the inertial zone is observed.

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -117-

4.2.2 Turbulence length scale

Integral length scale

A review of basic turbulent parameters reveals that the flow field is characterized by low turbulence level in the central region. Because of this, the calculation of integral length scales outside the boundary layer is not feasible from the outlined definition, since their implementation carried significant errors or at times provided meaningless outcomes.

Results of integral length scale are therefore presented up to a distance of y/d=2.38 normal to the wall for all profiles. The profiles of integral length scale are shown in figure (4.10), which illustrates an increment in the integral length scale with respect to the flow Reynolds number. However, these profiles do not show any clear distinction in scale size with respect to the streamwise distance of the tunnel.

(a) (b)

Figure (4. 10): Profiles of integral length scale measured at x/d= -10.2d, 12.9d, 36d and 49.8d for the Reynolds number cases of (a) Re∞=5801 and (b) Re∞=18436.

Taylor micro scale

The initial assumption adopted for sampling frequency in hot wire measurement is found to be little under estimated as it is found that in the low turbulence central region the autocorrelation curves of longitudinal velocity fluctuation do not provide a sufficient number of points within the defined limits of its threshold value, for constructing the osculating parabolas. So, it becomes difficult to determine the inertial length scale in the central region of the test section from the graphical method. However, in the boundary layer region a sufficient number of points are obtained and the results are estimated with respect to the y- coordinate at each measuring station lying on the mid-span. It is observed that the velocity fluctuations increase with the increase in the flow Reynolds number, as was noticed by the results of root mean velocity (not presented here). Therefore, at higher flow rates the correlation between fluctuating velocity component is found to be weak within the limit of r→0, and so the calculaHon of the Taylor scales at high Reynolds numbers is halted in these

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -118- conditions as well. The only results for the Taylor micro scales, which are accessible, are at lowest Reynolds number exercised in the experiment as shown in figure (4.11).

Results depict small gradual increment in the size of the scale towards the edge of boundary layer for all profiles. It is also noticed that the diversity in scale size at different measuring stations is not easy to distinguish.

10 100 1000

10 100 1000

Y⁺

λ+

49.8d 36.0d 12.9d -10.2d

Figure (4. 11): Profiles of Taylor micro scale measured at x/d= -10.2d, 12.9d, 36d and 49.8d for Re∞=5801.

Kolmogorov scale

The determination of Kolmogorov scale η is quite a straight forward process, when results of energy dissipation rate are available. The limits for the distribution of profile appear to be similar as mentioned above. A positive shift in the profiles of the Kolmogorov scale are observed for each successively higher Reynolds number, when plotted on the scale of non- dimensionalized wall unit Y⁺. This positive shift is attributed to the increase of friction velocity (velocity associated to the viscous scale), which increases as well with the augmentation of mean flow rate. Also, a slight reduction in scale size is observed with the increment of the distance in x-direction for a region away from the wall. Figure (4.12) shows the variation of smaller scale within the boundary layer.

0 4 8 12 16 20

10 100 1000

Y⁺

η+

-10.2d 12.9d 36d 49.8d

0 4 8 12 16 20

10 100 1000 10000

Y⁺

η+

-10.2d 12.9d 36d 49.8d

(a) (b)

Figure (4. 12): Profiles of Kolmogorov scale measured at x/d= -10.2d, 12.9d, 36d and 49.8d for the Reynolds number cases of (a) Re=5801 and (b) Re=18436.

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -119-