3.1. Instantaneous streak patterns

The PVP induced by MVG (figure 3) leads to the formation of the high- and low-speed streaks by transporting the high- and low-momentum fluids through the lift-up mechanism, which has been reported in numerous studies performed in laminar boundary layers (e.g. Fransson & Talamelli Reference Fransson and Talamelli2012; Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013). The instantaneous streak pattern around the MVG in a turbulent boundary layer is illustrated in figure 4$(a)$. Upstream of the MVG ($x^\ast /h < 0$) are typically the naturally occurring large-scale structures associated with high- and low-momentum regions with self-similar spanwise length scales that vary with distance from the wall (Tomkins & Adrian Reference Tomkins and Adrian2003). Downstream of the MVG ($x^\ast /h > 0$) the flow patterns become more organised with high- and low-speed region spacings that are scaled with the spanwise distance ($\varLambda _z$) between the MVG pairs. A closer view is presented in figure 4$(b)$. The isosurfaces of vortical structures are computed using the $\lambda _2$-criterion (Jeong & Hussain Reference Jeong and Hussain1995) and coloured by the mean streamwise velocity. High-speed streaks are flanked by the low-speed streaks and vortical structures on both sides. The vortical structures are commonly observed slightly above and outwards of the low-speed streaks, reflecting ejection events associated with the lift-up mechanism. To quantitatively measure the streamwise streaks, the growth amplitude of the streamwise streaks is defined as

(3.1)\begin{equation} A_{ST}(x) = \frac{1}{U_\infty}\int_{-\varLambda_z/2}^{\varLambda_z/2} \int_{0}^{\delta} | \bar{u}(x,y,z^\ast)-U(x,y) |\,\textrm{d} {y}\,\textrm{d} {z^\ast},\end{equation}

which considers the spanwise periodicity of the streaks by integrating the MVG-induced fluctuation over one complete spanwise wavelength (Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013). The MVG-induced-vortex strength is based on the integral of the squared streamwise vorticity (enstrophy) over one complete spanwise wavelength as

(3.2)\begin{equation} {\gamma} = \frac{h^2}{U_\infty^2}\int_{-\varLambda_z/2}^{\varLambda_z/2} \int_{0}^{\delta} \bar{\omega}_x^2(x,y,z^\ast) \,\textrm{d} {y}\,\textrm{d} {z^\ast}.\end{equation}

Figure 4. $(a)$ Instantaneous visualisation of the high-speed (cyan) and low-speed (blue) streaks ($u'^+ = \pm 2.5$). $(b)$ Closer views of the instantaneous flow patterns of high-speed and low-speed streaks (same colour code as $(a)$). Vortical structures are detected by the $\lambda _2-{criterion}$ and coloured by the mean streamwise velocity.

Figure 5$(a)$ shows the streak amplitude and induced-vortex strength based on (3.1) and (3.2) at $0 < x^\ast /h < 1000$, normalised by initial values at $x=x_M$. Based on these trends, the flow is split into three distinct regions. In the first region $(\text {I})$, where $5 < x^\ast /h < 50$, the vortex decreasing strength is coupled to the streak formation. The vortex strength decay follows a power-law relation of $\gamma /\gamma _{{M}} \sim (x^\ast /h)^{-1.1}$ for $5< x^\ast /h<50$ (blue line). The decay rate maybe different due to the definition used. Lögdberg et al. (Reference Lögdberg, Fransson and Alfredsson2009) reported the vortex circulation decays exponentially at a rate of $-0.0164$ with respect to the downstream distance, but the calculation considered was the vortex area enclosed by five per cent of the maximum value of the second invariant of the velocity gradient $(0.05{Q}_{max})$, which did not take into account the secondary (induced) effects. Figure 5(b,c) shows the streamwise evolution of the PVP in the first region $(\text {I})$, which is defined as the position at the local maximum of the absolute streamwise vorticity. The results qualitatively agree with those obtained by Lögdberg et al. (Reference Lögdberg, Fransson and Alfredsson2009). Overall, region I reflects the fact that energy is being redistributed from PVP to the streamwise velocity component (Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013). At region $(\text {II})$, where $50< x^\ast /h<500$, the streak amplitude settles to a somewhat steady value of $A_{ST}/A_{ST,M} \simeq 7.5$. The stabilisation effect presumably results from the interactions of streaks generated between the MVG pairs, forming long and steady high- and low-speed regions. On the other hand, the vortex strength decreases steadily at a rate $\gamma /\gamma _{{M}} \sim (x^\ast /h)^{-0.1}$. Region (III) denotes the range $x^\ast /h > 500$, which is the onset of equilibrium. The streak amplitude decreases due to viscous dissipation (Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013), and the mean flow and turbulent statistics are less dependent on the effect of MVGs and collapse to the smooth wall case, as shown in the following section.

Figure 5. $(a)$ Streak amplitude $A_{ST}$ (black $\circ$) and vorticity strength $\gamma$ (red $\diamond$), normalised by the values at $x=x_{M}$. Blue solid line in region (I) approximates a power-law relation $\gamma /\gamma _{M} \sim (x^\ast /h)^{-1.1}$ for $5< x^\ast /h<50$, and solid green line within region (II) indicates $\gamma /\gamma _{M} \sim (x^\ast /h)^{-0.1}$. Streamwise evolution of vortex centre in $(b)$ side view and $(c)$ top view. Red for clockwise and blue for counter-clockwise rotation vortices. In $(b)$, the solid line represents the Reynolds number based on the displacement thickness and the dash-dotted line represents the Reynolds number based on the momentum thickness.

3.2. Mean flow characteristics

The mean streamwise and secondary flow topology are illustrated in figure 6 where the time-averaged streamwise velocity (contours) and cross-flow velocity (streamlines) are plotted at $x^\ast /h =5, 25, 50$ and $500$. In the near-wake region $(x^\ast /h = 5)$, opposite secondary motion is observed close to the wall (indicated by the red arrow on figure 6$a$), which is induced by rotation of the PVP, and the PVP lift away from the wall and move apart along the downstream (figure 6$b\text {--}d$) due to self-induction (Lögdberg et al. Reference Lögdberg, Fransson and Alfredsson2009). The contour colours illustrate the mean streamwise velocity downstream in the $z\text {--}y$ cross-section views. The downwash motion induced by the PVP causes the high-momentum region to form between the MVG, while the low-momentum region is formed along the sides and interacts with the low-momentum region originating from the adjacent MVG pairs (Fransson & Talamelli Reference Fransson and Talamelli2012). The black contour lines at $\bar {u}=0.6U_\infty$ illustrate the development of regions along the spanwise direction and show the respective changes in the wall-normal direction. The mean flow exhibits an initial momentum deficit and a fast momentum recovery after the streaks have settled. The mean velocity deficit is estimated based on the mean velocity profile, as shown in figure 7. The flow exhibits an apparent velocity defect at $x^\ast /h =5$. The velocity defect in the mean flow profile can be estimated by comparing the downward shift of the log-law constants, as shown by the dash-dotted lines in figure 7. This yields ${\rm \Delta} U^+ = 2$ at $y/h \simeq 0.5$, corresponding to the half-blade height.

Figure 6. Mean streamwise velocity (colour contours) and secondary flow topology (streamlines) at the downstream of the MVG ($a\text {--}d$): $x^\ast /h = 5,25,50$ and $500$.

Figure 7. Mean streamwise velocity. Downstream locations are at $x^\ast /h \simeq 5$ (black $\circ$), $25$ (red $\circ$), $50$ (blue $\circ$) and $1000$ (purple $\circ$). Light blue dashed, dotted and solid lines represent smooth wall DNS TBL at $Re_\tau \simeq 500,1000$ and $2000$ (Chan, Schlatter & Chin Reference Chan, Schlatter and Chin2021), respectively. The thick grey line denotes $y^+ \simeq h^+$. Dash-dotted lines represent linear and log-law regions with $\kappa = 0.41$ and $C=3.2$ and $5.2$.

The velocity defect is known to be associated with skin friction drag. Locally, the skin friction is modulated over the high- and low-speed regions along the spanwise direction. On the upwash side of the PVP, low-speed fluid is lifted from the wall, which reduces the streamwise wall shear stress $\tau _w$ and results in substantial skin friction drag reduction. On the other hand, downwash motion induced by PVP transports the high-speed fluid towards the wall and increases the skin friction drag at the centre of a MVG pair. To determine the skin friction drag reduction over the low-speed regions and the increased skin friction drag over the high-speed regions, we computed the local skin friction variation related to the time-averaged wall shear stress

(3.3)\begin{equation} D(x,z^\ast) = \frac{{\bar{\tau}_w}-\bar{\tau}_{w,0}}{\bar{\tau}_{w,0}}, \end{equation}

where $\bar {\tau }_w = \nu (\textrm {d}\bar {u}/{\textrm {d} y})|_{y=0}$ is the time-averaged wall shear stress. The subscript $0$ refers to the smooth case, $D>0$ denotes the local increase and $D<0$ denotes the local drag reduction. The variation is illustrated in figure 8$(a)$ for the spanwise variation and figure 8$(b)$ for the streamwise evolution. At $x^\ast /h =5$, the high-speed region is initially constrained by the spanwise separation between two MVG blades (i.e. $d$). Thus, the downwash motion is relatively strong and induces a higher $D$, which peaks at approximately $D \simeq 5, z^\ast /h \simeq \pm 1$. The high-speed peak value of $D$ is relaxed further downstream to a value similar to that of the low-speed peak value. The location of the low-speed peak shifts to a higher $z^\ast /h$ moving downstream, which is consistent with the streamwise evolution of PVPs as shown in figure 5$(c)$ and is due to mirrored vortices induction (Lögdberg et al. Reference Lögdberg, Fransson and Alfredsson2009). Furthermore, figure 8$(c,d)$ shows the skin friction coefficient $c_f = {\langle {\bar {\tau }_w}\rangle }/{\frac {1}{2}\rho U_\infty ^2}$ and the global skin friction variation rate $R = (c_f-c_{f,0})/c_{f,0}$. The value of the skin friction coefficient is analogous to the value for a smooth wall zero pressure gradient (ZPG) TBL for $Re_\theta > 1600, x^\ast /h > 100$. This suggests that the skin friction drag along the plate (in the streamwise direction) in the MVG flow is nearly identical to that in a smooth case. Further, the skin friction variation rate $R$ is similar to that in a laminar boundary layer (Fransson & Talamelli Reference Fransson and Talamelli2012; Shahinfar et al. Reference Shahinfar, Fransson, Sattarzadeh and Talamelli2013; Camarri, Fransson & Talamelli Reference Camarri, Fransson and Talamelli2014). However, the skin friction drag variation is significantly different from that observed in relatively higher Reynolds number ZPG TBL in which the MVG array may lead to a substantial increase in wall shear drag. Lögdberg et al. (Reference Lögdberg, Fransson and Alfredsson2009) examined the MVG with $h/\delta \simeq 0.22$ in a ZPG TBL for $Re_\theta > 6000$ and reported a $30\,\%$ increase of $c_f$ downstream of the vortex generators at $x^\ast /h \simeq 100$ compared with the smooth wall ZPG TBL.

Figure 8. $(a)$ Spanwise variation of the local wall shear stress at downstream locations: $x^\ast /h \simeq 5$ (black $\diamond$), $25$ (red $\diamond$), $50$ (blue $\diamond$) and $1000$ (purple $\diamond$). $(b)$ Streamwise evolution of the local wall shear stress at $z^\ast /h=0\ (\square ),1\ (\circ )$ and $2\ (\diamond )$. $(c)$ The skin friction coefficient $c_f$ for the cases: MVG ($\diamond$), smooth wall ($\blacksquare$). Solid line represents the empirical formula $c_f = 0.024 Re_\theta ^{-1/4}$ by Smits, Matheson & Joubert (Reference Smits, Matheson and Joubert1983) and dashed line represents $c_f = 2[1/0.384 \ln (Re_\theta ) + 4.08]^{-2}$ (Österlund Reference Österlund1999). $(d)$ The global skin friction variation rate ${R}$ ($\diamond$).

3.3. Turbulence characteristics

3.3.1. Triple decomposition of velocity fluctuations

The presence of the MVGs introduces a strong spanwise modulation effect on the velocity fluctuations. To investigate the spatial variations of velocity fluctuations due to such spanwise modulation, we adopted a similar approach to analyse roughness surface flow by triple decomposition of the velocity components, which reads as

(3.4)\begin{equation} u_i(x,y,z,t) = U_i(x,y) + u'_i(x,y,z,t) + \tilde{u}_i(x,y,z) = \bar{u}_i(x,y,z) + u'_i(x,y,z,t), \end{equation}

where the $u'_i$ and $\tilde {u}_i$ on the right-hand side of (3.4) are the turbulent fluctuation and MVG-induced fluctuation, respectively. The MVG-induced fluctuation $\tilde {u}_i = \bar {u}_i - U_i$ is the spatial variation of the time-averaged flow due to MVG. The total fluctuation, $u_i'' = u_i'+\tilde {u}_i$ is equal to the turbulent fluctuation $(u_i')$ for the smooth wall case since $\tilde {u}_i = 0$. The streamwise total stress $(u''u'')$, Reynolds stress $(u'u')$ and the MVG-induced stress $(\tilde {u}\tilde {u})$ are illustrated in figure 9 at four streamwise locations: $x^\ast /h = 5, 25, 50$ and $500$.

Figure 9. Fluctuations of triple decomposition streamwise velocity. Total fluctuations ($u''u''$) are displayed in black, turbulent fluctuations ($u'u'$) in red and MVG-induced fluctuations ($\tilde {u}\tilde {u}$) in blue. The green dashed lines represent the smooth wall case at (a–c) $Re_\tau \simeq 500$ and $(d)$ $Re_\tau \simeq 1000$ (Chan et al. Reference Chan, Schlatter and Chin2021). The light blue dashed lines represent smooth wall DNS TBL at (a–c) $Re_\tau \simeq 492$ and $(d)$ $Re_\tau \simeq 974$ (Schlatter & Örlü Reference Schlatter and Örlü2010). The vertical dashed lines denote $y^+ = h^+$. Purple $(\circ )$ are features discussed in the text.

Two notable peaks are observed in the MVG-induced fluctuation in the near-wake region ($x^\ast /h = 5$, marked with purple $(\circ )$ in figure 9$a$). The outer peak is centred at $y/h \simeq 0.67$ and the inner peak is very close to the wall at $y^+ \simeq 7$, representing the high- and low-speed streaks generated by the PVP and the induced vortical motions, as previously shown in figure 1$(b)$. The near-wall peak of the streamwise turbulent fluctuation is weakened compared with that of the smooth wall, indicating a disruption of the near-wall cycle (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). In figure 9$(a)$, these curves collapse to that for the smooth wall case for $y^+ > 165$, approximately equal to $y\simeq 2h$. Between the inner and outer regions $(25< y^+<165)$, the turbulent fluctuation is enhanced, resulting from an increase in turbulence production by mean shear and is associated with the generation of low-speed streaks. This is illustrated in figure 10$(a)$ for the turbulence production of the Reynolds-averaged TKE transport, defined as the product of time-averaged Reynolds stress tensor and velocity gradients (Pope Reference Pope2000) (denoted as $P_k = -\overline {u'_iu'_j}{\overline {\partial {u}_i/\partial x_j}}$) at $x^\ast /h = 5$. The location of the excess turbulence production ($25< y^+<165$) coincides with the low-speed regions (as outlined by white solid iso-lines), which are imposed by the PVP and reflect the outer peak in the MVG-induced fluctuation. The inner peak corresponds to the high-speed streaks induced very close to the wall at $y^+ \simeq 7, z^\ast /h \simeq \pm 0.8$ (outlined by solid black iso-lines). Further, the low-speed region coincides with the excess turbulence production at the outer region moving downstream at $x^\ast /h = 25$ and $75< y^+<300$, which is more than three times the blade height ($y/h \simeq 3.8$), as shown in figure 10$(b)$. Based on the observation of the double peaks in figure 9$(a)$, the MVG-induced fluctuations observed in figure 9$(b,c)$ thus represent the transition between vortical motions and the formation of streaks. Further evidence supporting this is seen in the peaks centred at $y^+ \simeq 22$ for $x^\ast /h = 25$ (figure 9$b$), and centred at $y^+ \simeq 45$ for $x^\ast /h = 50$ (figure 9$(c)$ marked with purple $\circ$). The peak locations are marked with blue dashed lines in figure 10$(b,c)$ and are at the same wall-normal heights as the centre of the streamwise streaks. At $x^\ast /h =500$, the MVG-induced fluctuation $\tilde {u} \simeq 0$ and therefore $u' \simeq u''$, and these curves collapse well to the smooth wall data, indicating the onset of equilibrium.

Figure 10. (a,b) TKE production $P_k$ (colour contour). The low-speed streaks intensity: $\bar {u}(x,y,z)-U(x,y) < 0$ (velocity deficit, see also (3.1)) is shown by white iso-lines: (a) $\text {--}0.1[0.025]\text {--}0.025$ and (b) $\text {--}0.095[0.01]\text {--}0.025$. The high-speed streaks intensity: $\bar {u}(x,y,z)-U(x,y) > 0$ is shown by black iso-lines: (a) $0.025[0.01]0.095$ and (b) $0.025[0.01]0.095$. In (a,b), red and blue contour lines illustrate time-averaged streamwise vorticity at (a) $\bar {\omega }_xh/U_\infty = \pm 0.2$ and (b) $\bar {\omega }_xh/U_\infty = \pm 0.04$. Inset: the turbulence production $\langle P_k \rangle$ (in red) compared with the smooth wall case (in green). $(c)$ The colour contour of the high- and low-speed intensities: $\bar {u}(x,y,z)-U(x,y)$, with white iso-lines: $\text {--}0.045[0.005]\text {--}0.025$ and black iso-lines: $0.025[0.01]0.095$. Light blue dashed lines in (b,c) denote $y^+ \simeq 22$ and $y^+\simeq 45$, respectively.

3.3.2. Spectral analysis

To characterise the energetic scales associated with the peaks of the MVG-induced fluctuations and investigate the influences on the length scales of dominant turbulent structures in the flow, i.e. the naturally occurring autonomous near-wall turbulence and the large-scale outer motions, which reflect the inner and outer peaks in the pre-multiplied energy spectra (Hutchins & Marusic Reference Hutchins and Marusic2007a; Lee & Moser Reference Lee and Moser2015), streamwise and spanwise pre-multiplied energy spectra of the streamwise velocity fluctuations by triple decomposition $(u',u'',\tilde {u})$ at $x^\ast /h = 5$ and $25$ are shown in figure 11. Taylor's frozen turbulence hypothesis is utilised to reconstruct the streamwise wavenumber from the temporal frequency where $k_x = 2{\rm \pi} f/U_c$ (del Álamo & Jiménez Reference del Álamo and Jiménez2009) where the convection velocity $U_c(y) = U(y)$ is assumed (i.e. mean streamwise velocity $U(y) = \langle \bar {u} \rangle$).

Figure 11. The one-dimensional streamwise and spanwise pre-multiplied spectra at $x^\ast /h=5$ and $25$. Columns from left to right represent triple decomposition of velocity fluctuations: total $E_{u''u''}$, turbulent $E_{u'u'}$ and MVG-induced fluctuations $E_{\tilde {u}\tilde {u}}$. The symbol $\times$ marks the near-wall peak. The symbol $(\circ )$ denotes the feature discussed in the text.

In figure 11$(a)$, we observe that the near-wall peak, which is commonly observed and centred at $\lambda _x^+ \simeq 1000, y^+=15$ (marked by $\times$ in $k_xE_{u''u''}$) (Hutchins & Marusic Reference Hutchins and Marusic2007a,Reference Hutchins and Marusicb) is diffuse due to influence by the MVG-induced fluctuation, which has two peaks at $y^+ \simeq 6.5, \lambda _x^+ \simeq 200$ and $y^+ \simeq 55, \lambda _x^+ \simeq 300$, as shown in the contour $k_xE_{\tilde {u}\tilde {u}}$. This is consistent with the observations at the double peak locations in figure 9$(a)$ corresponding to the PVP-induced streaks. These structures have limited streamwise extents with streamwise wavelengths of order $\lambda _x^+/L_x^+ \simeq 1\text {--}1.5$, i.e. scale approximately with the blade length. The spanwise spectrum of MVG-induced fluctuation, $k_zE_{\tilde {u}\tilde {u}}$, is relatively simpler, as shown in figure 11$(b)$. The peaks occur at the wavelengths reflecting the spanwise spacing between the MVG pairs and its higher harmonics, i.e. $\lambda _{z,n}^+ = \varLambda _z^+/n$ ($n = 1,2,3\ldots$), similar to those observed by Fransson & Talamelli (Reference Fransson and Talamelli2012). These modes have little effect on the near-wall peak that appears to be located at $\lambda _z^+ \simeq 100, y^+\simeq 15$ (marked with $\times$ in $k_zE_{u''u''}$). This is presumably because the dominant spanwise modes ($\varLambda _z^+/n \approx 800/n$ with $n = 1\text {--}5$) are much larger than the mean spacing of near-wall streaks of $\lambda _z^+ \simeq 100$ (Tomkins & Adrian Reference Tomkins and Adrian2003). In figure 11($c$) at $x^\ast /h =25$, the energy peak in the contour of $k_xE_{\tilde {u}\tilde {u}}$ is centred at $y^+ \simeq 22, \lambda ^+ \simeq 600$, and corresponds to the streamwise characteristic length scale of the regions of high- and low-speed fluids where $\lambda _x^+ \simeq 3L_x^+$ (see figure 10$b$). The inner peak of the total fluctuations spectrum contour ($k_xE_{u''u''}$) is located at $y^+ = 15,\lambda _z^+ \simeq 708$ (marked with $\circ$). The dominant wavelength is at a lower value compared with the classical inner peak, where $y^+=15,\lambda _x^+ = 1000$. This indicates that the near-wall cycle is interrupted. For the spanwise spectra shown in figure 11$(d)$, two energy peaks at $(i)$ the fundamental mode ($\lambda _z^+ = \varLambda _z^+$) and $(ii)$ its first harmonic ($\varLambda _z^+/2$) are observed in the MVG-induced fluctuation ($k_zE_{\tilde {u}\tilde {u}}$) and the total fluctuation ($k_zE_{u''u''}$). Despite the fact that the Reynolds number is still low $(Re_\tau \simeq 468)$, it is interesting that an outer peak is observed in the turbulence fluctuation $k_zE_{u'u'}$ (marked with $\circ$), and is associated with the first harmonic peak as shown in $k_zE_{u''u''}$. The outer peak is centred at $y \simeq 0.25\delta,\lambda _z \simeq 0.9\delta$, which is similar to the well-known outer peak in canonical flow in the spanwise spectrum where $y \simeq 0.2\delta,\lambda _z \simeq \delta$ (Hutchins & Marusic Reference Hutchins and Marusic2007b; Lee & Moser Reference Lee and Moser2015).

The long-term influence of the MVG-induced effects on the turbulent structures is evaluated by inspecting the pre-multiplied energy spectrum at $x^\ast /h = 50$ and $500$ compared with the smooth wall case in figure 12. In figure 12$(a)$, the near-wall peak is at $y^+ \simeq 15$, but more energy resides within the region approximately centred at $y^+ \simeq 100,\lambda _x^+ \simeq 1000$ compared with the smooth wall (marked by red $\circ$), suggesting that more high- and low-speed streaks are propagating at higher wall-normal heights ($y^+ \simeq 100$). A comparison of the MVG-induced streamwise contours ($k_xE_{\tilde {u}\tilde {u}}$) at $x^\ast /h = 25$ and $50$ (figures 11$c$ and 12$b$ respectively) suggests that the MVG-induced streaks with shorter streamwise lengths (i.e. $y^+=22, \lambda _x^+ \simeq 3L_x^+$, as shown in figure 11$c$), develop into much longer streamwise streaks that are quite similar and of the order of the near-wall streaks observed in the smooth wall case (i.e. $\lambda _x^+ \simeq 1000$). The MVG-induced streaks lift up as they propagate downstream, resulting in the energy peak at $y^+\simeq 45, \lambda _x^+ \simeq 1000$, as shown in figure 12$(b)$, so that two contours do not coincide at $y^+\simeq 100,\lambda _x^+\simeq 1000$ in figure 12$(a)$. In the spanwise energy spectra, the peak energy at $\lambda _z^+ \simeq \varLambda _z^+$ is observed and centred at $y^+ \simeq 45$ (marked by $\circ$ in figure 12$c$). The spanwise peak is different from its streamwise component, which clearly scales with $\varLambda _z^+$ (figure 12$d$). Overall, a stronger modulation in the spanwise direction is observed compared with the streamwise direction, which links to the spanwise periodicity of the MVG array. In the streamwise direction, these streaks tend to scale with the size of naturally present near-wall streaks of the order $\lambda _x^+ \simeq 1000$.

Figure 12. The one-dimensional streamwise and spanwise pre-multiplied spectra at $x^\ast /h=50\ (Re_\tau \simeq 500)$ and $500\ (Re_\tau \simeq 950)$. In (a,c,e,f), black lines represent the smooth wall case at $Re_\tau \simeq 500$ and $1000$ (Chan et al. Reference Chan, Schlatter and Chin2021) and light blue contour lines represent the MVG case. Spanwise contour levels are selected at $[0.6, 1.2, 1.8, 3]$. Streamwise contour levels are at $[0.25, 0.5, 1, 1.8]$. The symbols $\times$ mark the near-wall peak or the outer peak. Symbol $(\circ )$ in (a,c) denotes the feature discussed in the text.

Finally, figure 12(e,f) shows the contours of $k_xE_{u''u''}$ and $k_zE_{u''u''}$ at $x^\ast /h = 500$ ($Re_\tau \simeq 950$), which collapse reasonably well to the smooth wall case. The inner peaks are approximately at $y^+ \simeq 15,\lambda _z^+ \simeq 100$ and $\lambda _x^+ \simeq 1000$. The outer peak is at $y \simeq 0.15\delta,\lambda _z \simeq 0.15\delta$ in the spanwise component and at $y \simeq 0.06\delta,\lambda _z \simeq 6\delta$ in the streamwise component.

3.4. Secondary flow and kinetic energy transport

In the previous section, we employed triple decomposition of velocity fluctuations and spectral analysis to investigate the spanwise modulation of the MVG on the velocity fluctuations. Here, we extend the analysis of kinetic energy transport by introducing the triple velocity decomposition (3.4) to the incompressible Navier–Stokes equations. Following a similar approach of Reynolds & Hussain (Reference Reynolds and Hussain1972), the transport equation for the turbulent kinetic energy is obtained by multiplying the turbulent velocity fluctuations $u'_j$ and taking the time average; then interchanging $i$ and $j$ and summing up the resulting equation, the transport equation for TKE is obtained

(3.5)\begin{align} {\frac{\partial{k}}{\partial t}} + \underbrace{U_j{\frac{{\partial {k}}}{\partial x_j}}}_{\mathcal{C}'} &= \underbrace{-\overline{u_i' u_j'}\frac{\partial U_i}{\partial x_j}}_{\mathcal{P}'} \underbrace{-\frac{1}{2} \frac{\partial}{\partial x_j} [\overline{u_i'u_i'u_j'} + \overline{u_i'u_i'\tilde{u}_j} + 2\overline{u_i'\tilde{u}_iu_j'} ]}_{\mathcal{D}'}\nonumber\\ &\quad +\underbrace{{\tilde{u}_i\overline{\frac{\partial u_i' u_j'}{\partial x_j}}}}_\mathcal{T} \underbrace{- \frac{1}{\rho}\overline{\frac{\partial p' u_i'}{\partial x_i}}}_{\mathcal{\varPi}'} + \underbrace{\frac{1}{2}\nu \overline{{\rm \Delta} u_i'u_i'}}_{\mathcal{D'_{\nu}}} - \underbrace{ \nu\overline{\frac{\partial u_i'}{\partial x_j} \frac{\partial u_i'}{\partial x_j}}}_{\mathcal{\epsilon}'}, \end{align}

and the MVG-induced kinetic energy (MKE) transport equation is obtained in a similar manner

(3.6)\begin{align} \underbrace{\tilde{u}_i U_j\frac{\partial U_i}{\partial x_j} + \tilde{u}_iU_j{\frac{\partial \tilde{u}_i}{\partial x_j}}}_{\tilde{\mathcal{C}}} &= \underbrace{-{\tilde{u}_i\tilde{u}_j}\frac{\partial U_i}{\partial x_j}}_{\tilde{\mathcal{P}}} - \underbrace{{\tilde{u}_i\overline{\frac{\partial u_i' u_j'}{\partial x_j}}}}_\mathcal{T}\nonumber\\ &\quad \underbrace{-\frac{1}{2} \frac{\partial}{\partial x_j}\left[{\tilde{u}_i\tilde{u}_i\tilde{u}_j}\right]}_{\tilde{\mathcal{D}}} - \underbrace{\frac{1}{\rho}\left[\frac{\partial P}{\partial x_i}\cdot\tilde{u}_i + {\frac{\partial \tilde{p}}{\partial x_i} \cdot \tilde{u}_i }\right]}_{\tilde{\mathcal{\varPi}}}\nonumber\\ &\quad + \underbrace{\nu {\rm \Delta} U_i \cdot \tilde{u}_i + \frac{1}{2} \nu {{\rm \Delta} \tilde{u}_i\tilde{u}_i}}_{\tilde{\mathcal{D}_\nu}}- \underbrace{\nu{\frac{\partial \tilde{u}_i}{\partial x_j}\frac{\partial \tilde{u}_i}{\partial x_j}}}_{\tilde{\mathcal{\epsilon}}}. \end{align}

Equations (3.5) and (3.6) are rewritten as

(3.7)\begin{gather} \mathcal{C' = {P}' + {D}' + \mathcal{T} + \varPi'+ D'_\nu - \epsilon'}, \end{gather}

(3.8)\begin{gather}\tilde{\mathcal{C}} = \tilde{\mathcal{P}} + \tilde{\mathcal{D}} - \mathcal{T} + \tilde{\varPi}+ \tilde{\mathcal{D}}_\nu - \tilde{\epsilon}, \end{gather}

where, on the right-hand side of (3.7), are the production, spatial transport, inter-component transport, pressure transport, viscous diffusion and dissipation. The terms in the (3.8) are the MVG-induced component decomposed from the turbulent component (3.7). The inter-component transport term $\mathcal {T}$ in (3.7) and (3.8) are of opposite signs, representing a local exchange between turbulent ($\overline {u_i'u_i'}$) and MVG-induced (${\tilde {u}_i\tilde {u}_i}$) kinetic energy, and on the left-hand side of (3.8) is the advection term due to secondary motion. Adding up (3.7) and (3.8) yields the total TKE transport equation

(3.9)\begin{equation} \mathcal{C'' = {P}'' + {D}'' + \varPi'' + D''_\nu - \epsilon'',}\end{equation}

where $\partial (\cdot )/\partial t = 0$, and (3.9) collapses to the Reynolds-averaged TKE transport equation (Pope Reference Pope2000) when $\tilde {u}_i \to 0$.

To simplify the analysis, we considered the dominant terms in the general form of the equations. This is assessed by visual inspecting the spanwise-averaged TKE (3.7) and MKE transports (3.8). Figure 13$(a)$ shows that the (3.7) is dominated by the TKE production and dissipation at regions away from the wall where the viscous effect is small, while other terms are relatively small. The balance between production and dissipation is more prominent at $x^\ast /h = 25$, as shown in figure 13$(b)$, and at very close to the wall $(y < 0.01\delta )$, the viscous diffusion ($\mathcal {D'_\nu }$) dominates and is mainly balanced by the dissipation $( \epsilon ' )$. Figure 13(c,d) shows (3.8) in the spanwise-averaged form. Very close to the wall, the trend is similar to (3.7). The equation forms a balance between viscous diffusion and dissipation. Further away from the wall, we observe that MKE production, MKE advection and inter-component transport are the dominant terms (i.e. $\tilde {\mathcal {P}}, \tilde {\mathcal {C}}$ and $\tilde {\mathcal {T}}$). Based on the visual inspection of the data, (3.7) and (3.8) are written in the approximated form as

(3.10)\begin{gather} {\mathcal{P}' \approx {\epsilon}'}, \end{gather}

(3.11)\begin{gather}\tilde{\mathcal{C}} \approx \tilde{\mathcal{P}} - \mathcal{T}, \end{gather}

where in (3.10), $\mathcal {P'} = \mathcal {P}'_{ij}$ is the production of TKE by mean flow, and $\mathcal {\epsilon }' = \mathcal {\epsilon }'_{ij}$ denotes the viscous dissipation of TKE. In (3.11), $\tilde {\mathcal {C}} = \tilde {\mathcal {C}}_{ij}$ represents the advection of MKE for the existence of secondary motion, $\tilde {\mathcal {P}} = \tilde {\mathcal {P}}_{ij}$ denotes the production of MKE by the mean flow and $\mathcal {T}$ is the local transfer between the TKE and MKE. Figure 14 shows all the TKE production terms $\mathcal {P'} = \mathcal {P'}_{ij}$, averaged in time and periodic spanwise direction ($z^\ast$) at $x^\ast /h = 5$. The TKE production associated with Reynolds shear stress and wall-normal gradient of mean flow ($\mathcal {P'}_{uv} = -\overline {u'v'} {\partial U}/{\partial y}$) dominates over other terms, which may be expected since it is often assumed that the TKE production is defined as the product of Reynolds shear stress and mean flow gradient. The peak TKE production is consistent with the Reynolds-averaged TKE production. The observed arc shape depends on the MVG geometry (i.e. $d$ and $h$), which is clearly related to the formation of low-speed regions at the upwash side of the PVP and the outer peak of the MVG-induced fluctuations (see figures 9$a$ and 10$a$). Figure 15$(a)$ shows all the MKE production terms $\tilde {\mathcal {P}} = \tilde {\mathcal {P}}_{ij}$. Similar to the TKE production, the streamwise and wall-normal components $\tilde {\mathcal {P}}_{uv}$ dominate over the other terms. It is apparently concluded that $\tilde {\mathcal {P}} \approx -{\tilde {u}\tilde {v}} {\partial U}/{\partial y}$. Positive $\tilde {\mathcal {P}}$ is found in two distinct regions. One centred at $z^\ast /h \simeq \pm 1.8, y/h \simeq 0.4$ (marked with white $\times$ in $\tilde {\mathcal {P}}_{uv}$) and another at $z^\ast /h \simeq \pm 1, y/h \simeq 0.15$ (marked with $\circ$). These two regions correspond to the low-speed streaks and the near-wall high-speed streaks as seen in figure 10$(a)$, suggesting that this production term sustains the secondary motion. This is more evident further downstream at $x^\ast /h =25$, as shown in figure 16$(a)$. The production $\tilde {\mathcal {P}}$ is seen to be highly correlated with the generation of the high- and low-speed streaks and is adjoined to the PVP. Another feature we observed in the term $\tilde {\mathcal {P}}_{uv}$ (figure 15$a$) is that there is a non-negligible amount of negative production $\tilde {\mathcal {P}}_{uv}$, located adjacent to the two distinct positive regions. This indicates a reverse transport of MKE to the mean flow. Figure 15$(b)$ shows the nine components that constitute the transport term $\mathcal {T}$, representing the exchange of kinetic energy between TKE and MKE components, where negative region implies that energy is transported from the TKE to the MKE components and vice versa for the positive region. It is observed that $\mathcal {T}$ is dominated by two features: (i) $\mathcal {T}_{uw}$ associated with the spanwise gradient of $\overline {u'w'}$ where the negative and positive regions are roughly aligned with the high- and low-speed regions, respectively, and (ii) $\mathcal {T}_{uv}$ that is associated with the wall-normal gradient of Reynolds shear stress $\overline {u'v'}$. The value of $\mathcal {T}_{uv}$ shows a more complex pattern, with a positive peak approximately centred around the low-speed streaks and surrounded by a slightly negative transport. This complex pattern presumably arises from the collective effect of the short-lived (in the streamwise direction) motions that results in the high- and low-speed streaks (figure 10$a$). The transport term $\mathcal {T}$ becomes much clearer further downstream at $x^\ast /h = 25$, as shown in figure 16$(b)$. The positive transport highly overlaps with the production $\tilde {\mathcal {P}}$, at the position of the high- and low-speed regions (figure 16$a$). The positive transport $\mathcal {T}$ also acts as a sink of MKE because $\tilde {\mathcal {\epsilon }} \simeq 0$. The comparison of the dissipation terms between $\mathcal {\epsilon '}$ and $\tilde {\mathcal {\epsilon }}$ at $x^\ast /h = 5$ is illustrated in figures 16$(c,d)$. Thus, it is required that the gain in energy from the production $\tilde {\mathcal {P}}$ must be balanced by transferring to the TKE component, and dissipate via viscosity ($\mathcal {\epsilon '}$). Another interesting feature of the $\mathcal {T}$ is the regions of negative transport (marked with grey iso-lines in figure 16$b$), especiallyoccurring near the wall and observed only in the high-speed region. This negative patch is attributed to spanwise anisotropy of the Reynolds stress component ($\partial \overline {u'w'}/\partial z$) (not shown), is primarily from the TKE to the MKE components and occurs close to the wall ($y^+ < 20$). Finally, figure 16$(e)$ shows the MKE advection $\tilde {\mathcal {C}}$ at $x^\ast /h = 5$. The advection of MKE largely reflects the trajectory of the secondary motion (i.e. PVP). The PVP encloses the area of negative advection pointing to the negative production observed in figure 15$(a)$. Further downstream (figure 16$f$), positive and negative advections commonly occur at the regions where $\mathcal {P} > \mathcal {T}$ and $\mathcal {P} < \mathcal {T}$, respectively (see figures 16a,b). The advection term represents a spatial redistribution of MKE in order to maintain a balance of the MKE transport between $\mathcal {T}$ and $\tilde {\mathcal {P}}$, consistent with (3.11).

Figure 13. Spanwise-averaged $\langle \cdot \rangle$ of (3.7) and (3.8) the (a,b) turbulent and (c,d) MVG-induced kinetic energy transport equations at (a,c) $x^\ast /h =5$ and (b,d) $x^\ast /h =25$. All terms are scaled in viscous units.

Figure 14. ($\times 10^{-3}$) Contours of $\overline {u'_iu'_j}$ TKE production: $\mathcal {{P'}}^+ = (\mathcal {{P}}'_{ij}\nu )/u_\tau ^4$ at $x^\ast /h=5$. The centre-plane projection of MVG in the $y\text {--}z$ plane is outlined by a rectangular black box.

Figure 15. ($\times 10^{-3}$) $(a)$ Contours of $\tilde {u}_i\tilde {u}_j$ MKE production: $\tilde {\mathcal {P}}^+ = (\tilde {\mathcal {P}}_{ij}\nu )/u_\tau ^4$ at $x^\ast /h=5$. The symbols $\times$ and $\circ$ highlight the local multiple peaks. ($b$) Contours of inter-component transport $\mathcal {T}^+ = (\mathcal {T}_{ij}\nu )/u_\tau ^4$ between turbulent and MVG-induced fluctuations at $x^\ast /h=5$.

Figure 16. ($\times 10^{-3}$) $(a)$ Contour of MKE production $\tilde {\mathcal {P}}^+$ at $x^\ast /h = 25$. $(b)$ Contour of $\mathcal {T}^+$ at $x^\ast /h = 25$. Grey iso-lines are at $\mathcal {T}^+ < 0$: $\text {--}0.005[0.0005]\text {--}0.0005$. In (a,b), the low-speed streaks intensity ($\bar {u}(x,y,z)-U(x,y) < 0$) is shown in white iso-lines: $\text {--}0.095[0.01]\text {--}0.025$. The high-speed streak intensity ($\bar {u}(x,y,z)-U(x,y) > 0$) is shown in black iso-lines: $0.025[0.01]0.095$. The red and blue contour lines illustrate the time-averaged streamwise vorticity at $\bar {\omega }_xh/U_\infty = \pm 0.04$. Contours of dissipations: ($c$) $-\mathcal {{\epsilon }}'^+$ and ($d$) $-\tilde {\mathcal {\epsilon }}^+$ at $x^\ast /h=5$. Contours of MKE advection: $\tilde {\mathcal {C}}^+ = \tilde {\mathcal {C}}^+_{ij}$ at ($e$) $x^\ast /h=5$ and ($f$) $x^\ast /h=25$.