Artificially thickened boundary layer turbulence due to trip wires of varying diameter

In this paper, we study the artificially thickened boundary layer flows downstream of the tripping configuration that consists of a set of trip wires of varying diameters. Single-point hot-wire measurements are executed at a fixed streamwise location in the adaptation region, where the boundary layer is in conditions ranging from understimulation to overstimulation. Comparisons of the mean flow, Reynolds stress, energy spectra, and higher-order turbulence statistics demonstrate that the tripping effects are significant in the outer region, by introducing the enhanced energetic large scales with increasing the trip-wire diameter. The emergence of large scales manifests the scale separation in the overstimulated conditions, implying that the boundary layer has the potential to simulate high-Reynolds-number flows in the current tripping configurations. Then we attempt to identify the effect of the emergent large scales by examining the scale interactions. Under the overstimulated conditions, the generated large-scale structures penetrate down to the wall and superimpose energy in the near-wall region. The cross-term of the scale-decomposed skewness factor reveals that the amplitude modulation (AM) of the large scales on small scales is enhanced in the near-wall region. On the other hand, the frequency modulation (FM) is discussed by the zero-crossings-based evidence. Both the AM and FM effects become more significant with increasing the trip-wire diameter and the free-stream velocity. Far away from the wall, a reversal mechanism occurs and becomes more noticeable due to the tripping influence on the external intermittency. Moreover, it is found that the intermittent geometry of the turbulent/nonturbulent interface exhibits a fractallike self-similar convolution behavior in the current artificially thickened wall turbulence.

Figure 1

Sketch of the hot-wire experimental setup and trip wires of varying diameters.

Figure 2

(a) Boundary-layer thicknesses $\delta$, (b) shape factor $H_{12}$, (c) Coles’ wake parameter $\mathrm{\Pi }$, and (d) friction coefficient $C_f$, as a function of ${\mathrm{Re}}_{\theta }$ for various $D_C$ and free-stream velocities. In subplot (b), the solid line represents the composite profile by Monkewitz et al. [26] as $H_{12}=1+\frac{\kappa I_{WW}}{ln\left({\mathrm{Re}}_{\theta }\right)}+\frac{{\kappa }^2{I}_{WW}\left(I_{WW}-C\right)}{\mathrm{l}{\mathrm{n}}^2\left({\mathrm{Re}}_{\theta }\right)}+\frac{{\kappa }^2{I}_{WW}\left(\kappa {I_{WW}}^2-I_{WW}-2\kappa I_{WW}C+\kappa C^2\right)}{\mathrm{l}{\mathrm{n}}^3\left({\mathrm{Re}}_{\theta }\right)}$, with $\kappa =0.41, I_{WW}=7.11$, and $C=3.3$. In subplot (c), solid line: ${\mathrm{\Pi }}_{\mathrm{num}}$ for the composite profile proposed in Chauhan et al. [28]; dashed line: ${\mathrm{\Pi }}_{\mathrm{num}}\pm 0.05$. In subplot (d), solid line: Coles-Fernholz skin-friction law $C_f=2{\left({\kappa }^{-1}\mathrm{lnR}{\mathrm{e}}_{\theta }+C\right)}^{-2}$, with coefficient $\kappa =0.384$ and $C=4.127$ [29]. (e) The linear relation between $H_{12}$ and $\mathrm{\Pi }$. The solid line represents the fit to all data points, viz. $H_{12}=0.27\mathrm{\Pi }+1.31$. (f) The relation between Reynolds numbers ${\mathrm{Re}}_{\theta }$ and ${\mathrm{Re}}_{\tau }$. The solid line indicates the relation of ${\mathrm{Re}}_{\tau }=1.13\times {\mathrm{Re}}_{\theta }^{0.843}$ by Schlatter and Örlü [13]. The data of the different tripping conditions in Rodríguez-López et al. [15] are involved for comparison: the two-rows-cylinder (2row20) case $▽$, the staggered-cylinder (2stag20) case $▾$, and the sawtooth (Saw) case . The empty and filled symbols denote the data at the streamwise locations of $x=1.3$ and $1.6$ m, respectively.

Figure 3

Inner-normalized mean velocity profiles (left column) and streamwise velocity-fluctuation variance profiles (right column) for the different $D_c$ at different free-stream speeds: (a) case L, (b) case M, and (c) case H. Left column: The solid black lines show the Musker profile [30] with constants $\kappa =0.41$ and $B=4.86$. Right column: Dashed lines indicate the inner peak at $y^{+}\approx 15$.

Figure 4

Premultiplied velocity spectra of the fluctuation signal for various tripping diameters at the high free-stream velocity. Case $D_c=1–20$ is labeled in each panel of the figure, correspondingly, in their right-hand panels, changes in spectrograms of all the tripping cases relative to case $D_c=2$ are shown, labeled as $\mathrm{\Delta }\left(D_c=1–20\right)$ in each panel. The horizontal dashed line represents the wavelength of ${\lambda }_x/{\delta }_{D_{c2}}=3$. The boundary layer thickness of case H2 ${\delta }_{D_{c2}}$ is chosen for normalization. The + symbol denotes the proposed outer peak of $y^{+}\approx 3.9{\mathrm{Re}}_{\tau }^{1/2}$ and ${\lambda }_x/\delta \approx 6$ [8].

Figure 5

Distribution of the discrepancy energy peak [(a) ${\mathrm{\Delta }}_P$], and the corresponding wall-normal height [(b) $y_L^{+}$] and wavelength [(c) ${\lambda }_L/{\delta }_{D_{c2}}$] against ${\mathrm{Re}}_{\tau }$ for all the tripping cases. The solid line in subplot (b) represents the relation of $y^{+}\approx 3.9{\mathrm{Re}}_{\tau }^{1/2}$ proposed by Mathis et al. [8].

Figure 6

(a) Power spectral density ${\varphi }_{uu}$ in the free-stream at $y/\delta =2$ for cases H2 and H20. (b) Distributions of power spectral density peaks normalized with that in the reference case ${\varphi }_m/{\varphi }_{{\delta }_2,m}$ against ${\mathrm{Re}}_{\tau }$.

Figure 7

Wall-normal profiles of the streamwise fluctuation skewness $S_u$ (left column) and flatness $F_u$ (right column) at different free-stream velocities: (a) case L, (b) case M, and (c) case H. Dash lines respectively represent $S_u=0$ and $F_u=3$.

Figure 8

Distribution of the negative peak value of the skewness $S_{u,\mathrm{NP}}$ against the normalized tripping diameter $D_c/{\delta }^{*}$. Square, circle, and diamond symbols represent the different free-stream velocities L, M, and H.

Figure 9

Effect of applying a high-pass filter, with cutoff frequency $f_C^{+}$, to the streamwise velocity statistics (a) $u_{\mathrm{rms}}^{+}$, (b) $S_u$ of the cases H2 (${\mathrm{Re}}_{\tau }=898$) and H20 (${\mathrm{Re}}_{\tau }=3546$) at $y^{+}=20$. Solid horizontal lines represent the unfiltered result of case L2 (${\mathrm{Re}}_{\tau }=482$) at the corresponding wall normal location. Solid vertical lines denote the location at which the filtered data of H2 and H20, respectively, reach the unfiltered values of $u_{\mathrm{rms}}^{+}$ in case L2 and get their minimum values of $S_u$.

Figure 10

Wall-normal profiles of $3\overline{\overline{u_L^{+}u{}_S^{+}{}^2}}$ at the different tripping conditions with different free-stream velocities: (a) case L, (b) case M, and (c) case H.

Figure 11

Wall-normal profiles of $3\overline{\overline{u_L^{+}u{}_S^{+}{}^2}}$ at the three cases. L2: ${\mathrm{Re}}_{\tau }=482$, H2: ${\mathrm{Re}}_{\tau }=898$, and H20: ${\mathrm{Re}}_{\tau }=3546$.

Figure 12

Comparisons of $3\overline{\overline{u_L^{+}u{}_S^{+}{}^2}}$ at two nominally matched Reynolds numbers: (a) ${\mathrm{Re}}_{\tau }\approx 1200$ (L20: ${\mathrm{Re}}_{\tau }=1237$, M8: ${\mathrm{Re}}_{\tau }=1180$, and H6: ${\mathrm{Re}}_{\tau }=1322$) and (b) ${\mathrm{Re}}_{\tau }\approx 2300$ (M20: ${\mathrm{Re}}_{\tau }=2253$ and H10: ${\mathrm{Re}}_{\tau }=2342$).

Figure 13

(a) An interval of large- and small-scale components ($u_L$ and $u_S$, blue and black lines), extracted from the fluctuation signal at $y^{+}\approx 15$ for case H2. (b) The TA $s\left(t_i\right)$ is shown in a black line, and the instants for upcrossing of zero are marked by red squares. (c) The frequency indicator function $k_P$ is shown in the green pulse, and $I_P$ is also plotted for reference.

Figure 14

The ratio of large-scale conditional averaging zero-crossing frequency $K_R$, as a function of the wall-normal coordinate $y^{+}$, in boundary layers for various tripping diameters for case H.

Figure 15

Comparisons of $K_R$ at two nominally matched Reynolds numbers (the same as in Fig. 12).

Figure 16

Wall-normal intermittency profile $\gamma \left(y\right)$ in inner and outer scaling (left and right column) for various $D_C$ for case H. Color lines represent the result by fitting $\gamma \left(y\right)$ to the error function.

Figure 17

Distribution of $\text{PDF}\left(l\right)$ at the wall-normal heights in the range of $0.1\le \gamma \left(y\right)\le 0.9$, under all the tripping effects for case H. The solid black line indicates an exponential distribution with a power-law scaling exponent of $\zeta \approx -\frac{4}{3}$. The square symbols represent the intermediate range between ${\lambda }_T$ and $l_{\mathrm{mean}}+l_{\mathrm{rms}}/2$.

Figure 18

The distribution of the number of boxes $N\left(r\right)$ in the size of $r$ covering the line intersection points of the turbulent/nonturbulent interface (TNTI) at the wall-normal heights in the range of $0.1\le \gamma \left(y\right)\le 0.9$, under all the tripping conditions for case H. The solid yellow line indicates an exponential distribution with a one-dimensional fractal dimension $D_1\approx 0.35$. The square symbols represent the intermediate range between ${\lambda }_T$ and $l_{\mathrm{mean}}+l_{\mathrm{rms}}/2$.

Figure 19

Premultiplied spectra of the streamwise fluctuation signal for the different tripping conditions at low free-stream speed (case L). The details are consistent with that shown in Fig. 4.

Figure 20

Premultiplied spectra of the streamwise fluctuation signal for the different tripping conditions at medium free-stream speed (case M). The details are consistent with that shown in Fig. 4.