4.1.1. Time-averaged flow statistics

When flow goes past a permeable disk, part of the inflow MKE, TKE and the potential energy in pressure are extracted. In the wake of the disk, the pressure restores, the MKE recovers and the TKE first increases and then decays. To investigate how inflow TI affects the above process, the downstream variations of the normalized temporal and disk-averaged (along wake centreline with radius of rotor radius $R$) MKE difference $\langle \Delta {K_m}\rangle _D / U_{\infty }^2$, TKE difference $\langle \Delta {k}\rangle _D / U_{\infty }^2$ and pressure difference $\langle \Delta {\bar {p}}\rangle _D / U_{\infty }^2$ relative to the corresponding non-permeable disk case are plotted in figure 8 for the considered inflow turbulence intensities.

Figure 8.

Time- and disk-averaged MKE difference $\langle \Delta {K_m}\rangle _D / U_{\infty }^{2}$ (a i,ii), TKE difference $\langle \Delta {k}\rangle _D / U_{\infty }^{2}$ (b i,ii) and pressure difference $\langle \Delta {\bar {p}}\rangle _D$ (c i,ii). The dashed black line represents the zero reference line. Here $C_T= 0.2$ (solid lines) and $C_T= 0.7$ (dashed lines).

It is seen that the energy extracted by the permeable disk is mainly converted from MKE and potential energy in pressure, with a small percentage from TKE. With the increase of the inflow TI, the part converted from MKE decreases and the TKE part increases. It is interesting to notice that the pressure drops are almost the same for different inflow TIs for both $C_T$. As expected, the MKE recovers at a faster rate for a higher inflow TI. As for $\langle \Delta {k}\rangle _D$, the position of the maximum peaks is located closer to the permeable disk for a higher inflow TI.

After showing the energy conversion process for different inflow TIs, in the following we examine the statistics of the mean wake. Figure 9 shows contours of the time-averaged streamwise velocity deficit ($\Delta {\bar {u}}$) on the $x\unicode{x2013}y$ plane passing through the rotor centre. The downstream variations of the velocity deficit along the wake centreline are shown in figure 10 for the simulated cases. It is seen that the evolution of maximum velocity deficit approximately follows a power law starting from a certain disk downstream position. The inflow TI affects the starting position of flow recovery in the wake and the recovery rate, being closer to the disk and of a higher value for a higher inflow TI. An overall higher wake recovery rate is observed when $C_T$ is larger. This is because the momentum entrainment due to the wake shear layer is higher for a larger $C_T$, with that due to the ambient turbulence being the same.

Figure 9.

Contours of time-averaged streamwise velocity deficits ($\Delta {\bar {u}} / U_\infty$) in the spanwise plane passing through the permeable disk centre at $C_T=0.7$: (a) $I_\infty =2.5\,\%$; (b) $I_\infty =10\,\%$; (c) $I_\infty =25\,\%$. The solid black lines represent the permeable disk.

Figure 10.

Streamwise variations of the streamwise velocity deficits along disk centreline for (a) $C_T= 0.2$ (solid lines) and (b) $C_T= 0.7$ (dashed lines) for different inflow turbulence intensities. The solid ($\sim x^{-2/3}$) and dashed ($\sim x^{-2}$) black lines represent the two limit cases: (i) the ambient flow is non-turbulent, and the wake shear layer is the cause for flow recovery, and (ii) the wake is considered as a passive scalar with the ambient turbulence as the cause for flow recovery, respectively.

Figure 11 shows the variations of wake half-width ($R_{1/2}$) along the downstream direction, which is defined as the distance from the wake centre where velocity deficit is half of the maximum value. As seen, the wake expands from a position closer to the permeable disk when the inflow TI is higher. A larger wake expansion rate is not observed for a higher inflow TI, possibly because the intensity of momentum entrainment is not always positively correlated with the inflow TI, as reported by Kankanwadi & Buxton (Reference Kankanwadi and Buxton2020). As for the effect of $C_T$, the wake's half-width is greater and the wake starts expansion earlier for $C_T=0.7$ when compared with $C_T=0.2$. To better understand the trends demonstrated in figures 10 and 11, the reference scaling from the two limiting cases, i.e. (i) the wake shear layer serves as the cause for the flow recovery, with the ambient flow being non-turbulent, and (ii) the wake considered as a passive scalar advected by the ambient turbulence are added. The scaling laws can be obtained using the dimensional analysis and the conservation of momentum (Lissaman Reference Lissaman1979; Schlichting & Gersten Reference Schlichting and Gersten2016). It is seen that the trends from the LES cases are located between the scaling laws from the two limit cases. The goodness of the fitting of the exponents for the velocity deficit and wake width is shown in Appendix A.

Figure 11.

Streamwise variations of the time-averaged wake half-width $R_{1/2}$ for different inflow turbulence intensities. $C_T= 0.2$, solid lines; $C_T= 0.7$, dashed lines. The solid ($\sim x^{1/3}$) and dashed ($\sim x$) black lines represent the two limit cases: (i) the ambient flow is non-turbulent, and the wake shear layer is the cause for flow recovery, and (i) the wake is considered as a passive scalar with the ambient turbulence as the cause for flow recovery, respectively.

To probe into the mechanism for flow recovery in the wake, various terms in the MKE budget equation are analysed as in the literature (Meyers & Meneveau Reference Meyers and Meneveau2013; Yang et al. Reference Yang, Howard, Guala and Sotiropoulos2015a; Cortina, Calaf & Cal Reference Cortina, Calaf and Cal2016; Ge et al. Reference Ge, Zhang, Meng and Ma2020). Specifically, the MKE budget equation integrated over the $y\unicode{x2013}z$ plane is employed to analyse the variations of various terms in the downstream direction. The integration starts from $y_1 = y_c - R$ to $y_2 = y_c + R$ and $z_1 = z_c - R$ to $z_2 = z_c + R$ in the transverse directions, with $y_c$ and $z_c$ the coordinates of the wake centreline. The employed MKE budget equation is in the following form:

(4.3)\begin{align} &\underbrace{\int_{y_{1}}^{y_{2}} \int_{z_{1}}^{z_{2}}{\frac{\partial {\overline{u_{i}}}\,{\overline{u_{i}}} / 2}{\partial t}}{\,\mathrm{d}}y{\,\mathrm{d}}z}_{{=}0} = \int_{y_{1}}^{y_{2}} \int_{z_{1}}^{z_{2}} \left\{ \underbrace{-{\overline{u_{j}}} \frac{\partial {\overline{u_{i}}}\, \mathop{\overline{u_{i}}} / 2}{\partial x_{j}}}_{\text{ mean convection }(C_{m})} \right.\nonumber\\ &\qquad \underbrace{-\frac{\partial\left({\bar{p}} \,{\overline{u_{j}}} / \rho\right)}{\partial x_{j}} }_{\text{ pressure transport }(P_{t})} \underbrace{-\frac{\partial\left({\overline{u_{i}^{\prime} u_{j}^{\prime}}}\, {\overline{u_{i}}}\right)}{\partial x_{j}} }_{\text{ turbulence convection }(C_{t})} \underbrace{+2 \frac{\partial\left(v+v_{t}\right) S_{i j}\overline{u_{i}}}{\partial x_{j}}}_{\text{ diffusion }(D_{f})}\nonumber\\ &\qquad \underbrace{+ \overline{u_{i}^{\prime} u_{j}^{\prime}} \frac{\partial\overline{u_{i}}}{\partial x_{j}}}_{\text{turbulence production}(T_{p})} \left. \underbrace{-2\left(v+v_{t}\right) S_{i j} \frac{\partial \overline{u_{i}}}{\partial x_{j}} }_{\text{dissipation}(D_{p})} \underbrace{+ {{\overline{f_{i}}}\,{\overline{u_{i}}}}}_{W_D} \right\}{\,\mathrm{d}}y{\,\mathrm{d}}z. \end{align}

Since the considered case is statistically steady, the time derivative term on the left-hand side is zero after averaging the momentum equation in time. The various terms on the right-hand side are the mean convection ($C_m$), pressure transport ($P_t$), turbulence convection ($C_t$), diffusion ($D_f$), turbulence production ($T_p$), dissipation ($D_p$) and the work done by the permeable disk ($W_D$) (Wu & Porté-Agel Reference Wu and Porté-Agel2012; Abkar & Porté-Agel Reference Abkar and Porté-Agel2015). The integration operation is included for various terms in the above equation in the presented results, although each term is labelled in the curly braces with the integration operation placed outside.

In figure 12, we focus on the analysis of the turbulence production ($P_{t}$), pressure transport ($T_{p}$), turbulence convection ($C_{t}$) and mean convection ($C_{m}$) terms. The $W_D + \text {Res}$ term ($Res$ stands for residual) is shown in figure 12(e), which is negligible at most locations except near the disk where the magnitude of the $W_D$ term is large. It is seen that the inflow TI mainly affects the $P_{t}$, $C_{t}$ and $C_{m}$ terms, in a way that the magnitudes of the peaks become larger with the corresponding streamwise locations located closer to the disk for higher inflow TIs. The $T_{p}$ terms, on the other hand, are approximately the same for different inflow TIs. From the MKE analysis, it is clear that the faster recovery of MKE for higher inflow TI is achieved by allocating the region with a high $C_{t}$ term to near-wake locations and by enhancing the intensity of the $C_{t}$ term.

Figure 12.

(a i,ii) Turbulence production ($T_{p}$), (b i,ii) pressure transport ($P_{t}$), (c i,ii) turbulence convection ($C_{t}$), (d i,ii) mean convection ($C_{m}$) and (e i,ii) $W_D + \text {Res}$ terms in the MKE budget equation for different inflow TIs, in which subpanels (i) are for $C_T = 0.2$ (solid lines) and subpanels (ii) are for $C_T = 0.7$ (dashed lines). The MKE budget terms are normalized using $U_\infty$ and $D$.

After showing the downstream variation of MKE, in the following, we examine wake-added TKE ($\Delta {k}$). In figure 13, we show contours of $\Delta {k}$ on the $x\unicode{x2013}y$ plane pass through the wake centreline. As seen, the TKE is increased along the wake edge because of the existence of the wake shear layer. With the increase of inflow TI, the location for the maximum $\Delta {k}$ is located closer to the disk, with its magnitude decreasing especially for TI from 10 % to 25 %. One important observation is that $\Delta {k}$ becomes negative in the near-wake region for high TIs, indicating that a part of TKE is extracted by the permeable disk. For $C_T=0.2$, $\Delta {k}$ is negative for a long range of wake when $I_\infty =25\,\%$. A quantitative comparison of $\Delta {k}$ is shown in figure 14 for the two values of $C_T$.

Figure 13.

Contours of wake-added TKE ($\Delta {k}$) on the $x\unicode{x2013}y$ plane passing through wake centreline for different inflow TIs: (a i,ii) $I_\infty =2.5\,\%$, (b i,ii) $I_\infty =10\,\%$, (c i,ii) $I_\infty =25\,\%$. The solid black lines represent the permeable disk. The black dots indicate the positions with the maximum wake-added TKE at the edge region of the permeable disk. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$.

Figure 14.

Comparison of transverse profiles of wake-added TKE ($\Delta {k}$) at various disk downstream locations for different inflow TIs for (a) $C_T= 0.2$ (solid lines), (b) $C_T= 0.7$ (dashed lines). The horizontal black dotted lines at $y/D = \pm 0.5$ are the tip locations.

4.1.2. Spectral analysis

To understand how turbulent fluctuations of different scales are affected by a permeable disk and its wake, we carry out spectral analysis in both frequency space and scale space. We first show the premultiplied spectra of TKE ($\,f \varPhi _{k}$) along the edge of the disk to probe into how a disk adds turbulence to ambient flow in figure 15. The Strouhal number ($St$) is defined based on the disk diameter and the inflow wind speed. The most important observation is that the range of frequencies affected by a disk wake depends on the inflow TI, being wider for lower inflow TI. Specifically, a significant amount of TKE can be added to $St<2.0$, $St<1.5$ and $0.1< St<1.0$ for $I_\infty = 2.5\,\%$, $10\,\%$ and $25\,\%$, respectively. Dependence on $C_T$ is observed as well – that the disk becomes almost incapable of adding TKE to the wake when $C_T$ is small and the inflow TI is high.

Figure 15.

Premultiplied PSD of TKE along disk edge at various disk downstream locations for different inflow TIs. Here (a) $I_{\infty } = 2.5\,\%$; (b) $I_{\infty } = 10\,\%$; (c) $I_{\infty } = 25\,\%$. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$; grey lines, cases without a disk.

The premultiplied spectra of TKE ($\,f \varPhi _{k}$) along the disk centreline are examined in figure 16. As expected, the downstream variations of $f \varPhi _{k}$ are different from that along the disk edge, because of the time taken for the shear layer around the edge to expand and meet at the centre. One focus here is to examine the range of frequencies at which TKE is extracted by the disk. It is observed from $f \varPhi _{k}$ at a $0.5D$ disk downstream that such TKE reduction mostly happens at relatively low frequencies with $St<0.15$ and relatively high frequencies with $St>1.5$, respectively.

Figure 16.

Premultiplied PSD of TKE along wake centreline at various disk downstream locations for different inflow TIs. Here (a) $I_{\infty } = 2.5\,\%$; (b) $I_{\infty } = 10\,\%$; (c) $I_{\infty } = 25\,\%$. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$; grey lines, cases without a disk.

After the examination of TKE variations in frequency space, we analyse how TKE of different scales is extracted by the disk and added by the wake by decomposing velocity fluctuations in scale space using the method proposed in our previous work (Zhang, Yang & He Reference Zhang, Yang and He2023). In the employed approach, the decomposition of velocity fluctuations in scale space is carried out using a sphere-averaged filter. In this study, the sphere-averaged filtering is conducted on the $x\unicode{x2013}y$ plane at the disk centre, that the velocity fluctuations larger than and smaller than the filter scale ($s$), i.e. $u_{i}^{\prime, s>}$ and ${u}_{i}^{\prime, s<}$, are computed as follows:

(4.4)\begin{gather} {u}_{i}^{\prime, s>}(x, y) = {f}*u_{i}^{\prime}(x, y) = {\int}_{-\infty}^{+ \infty}{\int}_{-\infty}^{+ \infty}{{f}(x - x^{\prime}, y - y^{\prime})u_{i}^{\prime}(x^{\prime}, y^{\prime})\,\mathrm{d}\kern0.7pt x^{\prime}\,\mathrm{d} y^{\prime}}, \end{gather}

(4.5)\begin{gather} {u}_{i}^{\prime, s<}(x, y) = (1 - {f})*u_{i}^{\prime}(x, y) = u_i^{\prime}(x, y) - {u}_{i}^{\prime, s>}(x, y), \end{gather}

where ${f}(r_{d}, s)$ is the kernel function defined as

(4.6)\begin{equation} {f}(r_{d}, s) = \frac{{\epsilon}\left(\dfrac{s}{2} - r_d\right)}{C_{d}s^{d}}, \end{equation}

in which $r_{d} = \sqrt {{\sum _{i=1}^d}(x_i - x_i^{\prime })^2}$.

With the velocity decomposition in scale space, the corresponding energy density can be computed as follows:

(4.7)\begin{equation} E(x, s) ={-} \frac{{\partial} {Q}}{{\partial}s}, \end{equation}

where ${Q} = \tfrac {1}{2} \langle {u}_{i}^{\prime, s>}{u}_{i}^{\prime, s>} \rangle$ is the TKE calculated using the filtered velocity fluctuations.

Since the inflow TKE decays in the downstream direction, we plot the energy density difference between the cases with and without a disk. Figure 17 shows the contours of energy density difference $\Delta E$ on the $x\unicode{x2013}s$ plane for different inflow TIs. The extraction of TKE in the near wake is clearly demonstrated. At low inflow TI (i.e. $I_\infty =2.5\,\%$ for this work), no TKE extraction happens. For the two high inflow TIs (i.e. $I_\infty =10\,\%, 25\,\%$), similar patterns are observed indicating two types of TKE extraction. The first one is governed by the disk. The second one happens in the near wake as the pressure restores to the inflow level. The first one mainly happens at large scales, while the second one happens at a much wider range of scales. Increase in TKE occurs at farther downstream locations. It is seen that changing inflow TI influences the distribution of $\Delta E$ over $s$. For low inflow TI (e.g. $I_\infty =2.5\,\%$), $\Delta E$ is concentrated at small scales (e.g. $s/D<0.5$ for $I_\infty =2.5\,\%$), which is spread over a wide range of scales for high inflow TI (e.g. $s/D<1.5$ for $I_\infty =25\,\%$). One interesting observation is that, the TKE of large scales arrives the peak earlier than that of small scales, being consistent with the energy cascade process. The critical scale separating the large and small scales is approximately in the range of $0.3D$ to $0.6D$, being larger for $I_\infty =25\,\%$.

Figure 17.

Contours of energy density difference ${\Delta }E(x,s)$ on the $x\unicode{x2013}s$ plane for (a) $I_\infty = 2.5\,\%$, (b) $I_\infty = 10\,\%$ and (c) $I_\infty = 25\,\%$, respectively, for $C_T = 0.7$ and $L_\infty / D= 1.0$ cases. The white dashed line corresponds to 0.25 times of the maximum of TKE reduction. The black dashed line is the peak of energy density in the streamwise location for each scale. The vertical black dotted line indicates the permeable disk position.

The premultiplied energy density differences at various disk streamwise positions are examined in figure 18. It is seen that the disk is capable of extracting inflow TKE for scales greater than approximately $0.5D$, with slight changes in the critical scale when the inflow TI increases from $10\,\%$ to $25\,\%$, and the peak is located around $2.0D$. The TKE added by the wake, on the other hand, is mainly located for scales less than $3D$, with slight decreases in the critical scale when the inflow TI increases from $2.5\,\%$ to $25\,\%$, and the peak is located around $1.0D$.

Figure 18.

Premultiplied energy density difference (${\Delta }sE$) in scale space ($s$) at different streamwise locations for cases with different inflow TIs for (a) $I_{\infty } = 2.5\,\%$, (b) $I_{\infty } = 10\,\%$ and (c) $I_{\infty } = 25\,\%$ with $L_{\infty }=1.0D$ and $C_T=0.7$.

4.1.3. Quadrant analysis

The quadrant analysis (Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972; Wallace Reference Wallace2016) is carried out to probe into the influences of inflow on the interaction of wake with the ambient flow. The events in the four quadrants are outward interactions in quadrant 1 ($Q1$), ejections in quadrant 2 ($Q2$), inward interactions in quadrant 3 ($Q3$) and sweeps in quadrant 4 ($Q4$). The ejection and sweep events in $Q2$ and $Q4$ (where velocity fluctuations in the streamwise and the radial directions are of opposite signs), represent the gradient-type motions, being the main contribution to the Reynolds shear stress, $\overline {u_a^{\prime }u_r^{\prime }}$ (where the subscripts ${{\cdot }}_a$ and ${{\cdot }}_r$ denote the axial and radial directions, respectively). They are often counteracted by the interaction quadrants in $Q1$ and $Q3$ (where velocity fluctuations in the two directions are of the same sign) for shear flows. The analyses are focused on their contributions to the Reynolds shear stress and are conducted for cases with and without the permeable disk, such that the wake effects can be examined. The contributions from the four quadrants to the Reynolds shear stress are denoted as $S_1$, $S_2$, $S_3$ and $S_4$ and are calculated according to (4.9). They correspond to events $Q1$, $Q2$, $Q3$ and $Q4$, respectively, with ${\Delta }S_1, {\Delta }S_2, {\Delta }S_3, {\Delta }S_4$ being the wake-added contributions. The ${\Delta }S_1, {\Delta }S_2, {\Delta }S_3, {\Delta }S_4$ are examined in two ways: (i) normalized using the incoming velocity for comparisons between various cases; (ii) normalized using $\sum _{i=1}^{4}{S_{i}}$ to show their relative contributions to the Reynolds shear stress as (4.9). The definitions for $S_i$ and $\Delta S_i$ are given as follows:

(4.8)\begin{equation} {S_{i} = \left(\overline{u_a^{\prime}u_r^{\prime}}\right)_i}, \end{equation}

where $i=1,2,3,4$ for quadrants $Q_{1,2,3,4}$, respectively, and

(4.9)\begin{equation} {\Delta S_{i} = \left( S_{i} \right)_{D} - \left( S_{i} \right)_{noD},} \end{equation}

where the subscripts ${D}$ and ${noD}$ are for the results with and without a disk, as defined.

The quadrant analysis results for cases with different inflow TIs are shown in figures 19 and 20. As seen in figure 19(i), all the four quadrants contribute positively to the Reynolds shear stress, being different from those observed in the wall-bounded turbulent flows for which the $S_1$ and $S_3$ are negative. This is understandable because of the isotropic turbulent inflow, for which the four quadrants contribute equally. In figure 19(ii), it is seen that the $\Delta S_1$ and $\Delta S_3$ are negative, indicating that the wake reduces the $Q1$ and $Q3$ contributions to the Reynolds shear stress. Such reductions are greater in the near wake and increases with the increase in inflow TI. As expected, the wake enhances the ejections and sweeps in $Q2$ and $Q4$, making them the dominant events for the generation of Reynolds shear stress and the flow recovery in the wake. With the increase in inflow TI, the maximum of $\Delta S_2$ and $\Delta S_4$ increases, with the corresponding position moving upstream to the permeable disk. As for the relative contributions, it is seen that the relative contributions in $Q2$ and $Q4$ are comparable with each other, and their sum decreases with the increase in inflow TI.

Figure 19.

Quadrant analysis for cases with different inflow TIs for (a i,b i,c i,d i) contributions in each quadrant ($S_{i} / U_{\infty }^2$), (a ii,b ii,c ii,d ii) wake-added difference in each quadrant ($\Delta {S}_{i} / U_{\infty }^2$). The inflow integral length scale is $L_{\infty } / D = 1.0$. The thrust coefficient is $C_T = 0.7$.

Figure 20.

(a–c) Wake-added relative contributions in each quadrant ($\Delta {S}_{i} / \sum {\Delta {S}_{i}}$); (d) wake-added relative contributions in $Q1$ and $Q3$ events; (e) wake-added relative contributions in $Q2$ and $Q4$ events with different inflow TIs. The inflow integral length scale is $L_{\infty } / D = 1.0$. The thrust coefficient is $C_T = 0.7$.

4.2.1. Time-averaged flow statistics

The time- and disk-averaged $\langle \Delta {K_m}\rangle _D$, $\langle \Delta {k}\rangle _D$ and $\Delta P_D$ are first examined in figure 21. It is seen that the inflow turbulence length scale affects both the peaks and the corresponding positions of $\langle \Delta {K_m}\rangle _D$ and $\langle \Delta {k}\rangle _D$. Compared with inflow TI, the impacts of inflow $L_\infty$ are mainly found on the magnitudes of peaks. When increasing the inflow $L_\infty$, the maximum MKE reduction decreases, with the differences between cases with different $L_\infty$ more pronounced for higher inflow TI. The inflow turbulence length scale $L_\infty$ influences the TKE peaks in the other way, that the maximum TKE reduction and the maximum TKE addition increase with $L_\infty$. This is reasonable as the disk is more capable of extracting TKE at larger scales. Being the same as the observation for different inflow TIs, changing inflow $L_\infty$ does not affect the streamwise variation of pressure.

Figure 21.

Time- and disk-averaged $\langle \Delta {K_m}\rangle _D / U_{\infty }^{2}$ (a i–iv), $\langle \Delta {k}\rangle _D / U_{\infty }^{2}$ (b i–iv) and $\Delta {P_D}$ (c i–iv), with (a i,ii,b i,ii,c i,ii) for $I_\infty =10\,\%$ and (a iii,iv,b iii,iv,c iii,iv) for $I_\infty = 25\,\%$, respectively. The dashed black line represents the disk position. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$.

The downstream evolution of mean wake is examined in this part.

Figure 22 displays the contours of time-averaged streamwise velocity deficit ($\Delta {\bar {u}}$) contours on the $x\unicode{x2013}y$ plane passing through the wake centreline. It is seen that the streamwise velocity in flow in the wake recovers faster when the inflow $L_\infty$ is larger.

Figure 22.

Contours of time-averaged streamwise velocity deficits ($\Delta {\bar {u}}$) for different inflow integral length scales for (a) $L_\infty / D = 0.5$, (b) $L_\infty / D = 1.0$ and (c) $L_\infty / D = 1.5$, respectively, with $I_\infty =10\,\%$ and $C_T=0.7$. The solid black lines represent the permeable disk.

Figure 23 shows the streamwise velocity deficit along wake centreline for different inflow turbulence length scales. It is seen that the inflow $L_\infty$ affects both the location where the velocity starts to recover and the recovery rate, especially the recovery rate. The inflow turbulence TI, on other hand, mainly affects the starting position of velocity recovery as demonstrated in § 4.1. A monotonic trend of the velocity recovery rate with the inflow TI is not observed for the wake half-width (figure 11). With the increase of $L_\infty$, on the other hand, a monotonic increase in the velocity recovery rate is observed.

Figure 23.

Variations of streamwise velocity deficits ($\Delta {\bar {u}}$) along wake centreline for different inflow turbulence length scales for (a i,ii) $I_\infty = 10\,\%$ and (b i,ii) $I_\infty = 25\,\%$, respectively. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$. The solid ($\sim x^{-2/3}$) and dashed ($\sim x^{-2}$) black lines represent the two limit cases: the ambient flow is non-turbulent, and the wake shear layer is the cause for flow recovery, and the wake is considered as a passive scalar with the ambient turbulence as the cause for flow recovery, respectively.

The wake half-width ($R_{1/2} / D$) is examined in figure 24. It is seen that the wake starts expansion at a location closer to the disk and expands at a higher rate when the inflow $L_\infty$ is larger. When increasing $C_T$, the wake half-width becomes larger for low inflow $L_\infty$. Comparing the impacts of the two inflow statistics shows that the inflow TI mainly affects the starting position for wake expansion, while the inflow $L_\infty$ mainly affects the expansion rate. The wake recovery rate and the expansion rate vary with integral scale and TI, but both fall within the scaling ranges of the two models (George Reference George1989; Stein & Kaltenbach Reference Stein and Kaltenbach2019).

Figure 24.

Streamwise variations of the time-averaged wake half-width $R_{1/2}$ for different inflow turbulence length scales for (a) $I_\infty = 10\,\%$, (b) $I_\infty = 25\,\%$, respectively. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$. The solid ($\sim x^{1/3}$) and dashed (${\sim }x$) black lines represent the two limit cases: (i) the ambient flow is non-turbulent, and the wake shear layer is the cause for flow recovery, and (ii) the wake is considered as a passive scalar with the ambient turbulence as the cause for flow recovery, respectively.

The dominant terms in the MKE budget equation are analysed in figure 25 to probe into the influence of inflow turbulence length scale $L_\infty$ on MKE transport. It is seen that changing $L_\infty$ mainly influences $P_{t}$, $C_{t}$ and $C_{m}$ terms, with negligible impacts on the $T_{p}$ term, being consistent with changing inflow TI. Increasing inflow turbulence length scale increases the production of turbulence and enhances the MKE transport due to turbulence. Although changing $L_\infty$ affects the locations of the peaks of the $P_{t}$, $C_{t}$ and $C_{m}$ terms, it mainly impacts their magnitude when compared with results with different inflow TIs. The $W_D + Res$ term has virtually no impact downstream of the permeable disk.

Figure 25.

(a i,ii, f i,ii) Turbulence production ($T_{p}$), (b i,ii,g i,ii) pressure transport ($P_{t}$), (c i,ii,h i,ii) turbulence convection ($C_{t}$), (d i,ii,i i,ii) mean convection ($C_{m}$) and (e i,ii, j i,ii). The $W_D + {\rm Res}$ terms of the MKE budget equation for different incoming integral length scales for subpanels (i) $I_\infty = 10\,\%$ and subpanels (ii) $I_\infty = 25\,\%$. Panels (a–e) with solid lines, $C_T = 0.2$; ( f–j) with dashed lines, $C_T = 0.7$. The MKE budget terms are normalized using $U_\infty$ and $D$.

In the following, we investigate the effects of inflow turbulence length scales on the wake-added TKE ($\Delta {k}$). Figure 26 shows the contours of $\Delta {k}$ on the $x\unicode{x2013}y$ plane passing through the wake centreline. Figure 27 shows the transverse profiles of $\Delta {k}$ at various downstream locations from the disk. It is seen that increasing the inflow turbulence length scale $L_\infty$ increases the TKE extracted by the disk and the TKE added by the wake, with the corresponding maximum locations closer to the disk.

Figure 26.

Contours of wake-added TKE ($\Delta {k}$) on the $x\unicode{x2013}y$ plane passing through the wake centreline for different inflow turbulence length scales for (a i,ii) $I_\infty =10\,\%$ and (b i,ii) $I_\infty =25\,\%$, respectively. The solid black lines represent the permeable disk. The black dots indicate the positions with the maximum wake-added TKE at the edge region of the permeable disk. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$.

Figure 27.

Transverse profiles of wake-added TKE ($\Delta {k}$) at different disk downstream locations for different inflow turbulence length scales for (a i,ii) $I_\infty = 10\,\%$ and (b i,ii) $I_\infty = 25\,\%$, respectively. The horizontal black dotted lines at $y/D = \pm 0.5$ denote the tip locations. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$.

4.2.2. Spectral analysis

In this section, we investigate how inflow turbulence length scale ($L_\infty$) affects the TKE extraction by the disk and the TKE addition by wake by carrying out spectral analysis.

We first examine the premultiplied PSD of TKE ($\,f \varPhi _{k}$) along the disk edge in figures 28 and 29 for inflow $I_\infty =10\,\%, 25\,\%$, respectively, to examine how inflow $L_\infty$ affects wake-added TKE. It is seen in figure 28 for $I_\infty =10\,\%$ that the wake cannot only add TKE to frequencies less than a critical one, which is approximately $St=1.5$ for $L_\infty / D = 0.5$ and becomes $St=2.5$ for $L_\infty / D = 1.5$. Further increasing inflow TI not only affects the intensity of wake-added TKE, but also the range of frequencies being affected, which becomes much narrower (around $0.1< St<1.0$), especially for high inflow $L_\infty$ (e.g. $L_\infty / D = 1.0, 1.5$). When the inflow TI is high (i.e. $I_\infty =25\,\%$), the ambient turbulence dominates. For relatively low inflow TI (i.e. $I_\infty =10\,\%$), the ambient turbulence and the wake-added turbulence competes. When the peak of $f \varPhi _{k}$ is in the range of $St\in (0.1,1)$ (i.e. $L_\infty / D = 1.0$), the wake amplifies the peak. When the peak of $f \varPhi _{k}$ is not within the range (on the right-hand side for $L_\infty / D = 0.5$, while on the left-hand side for $L_\infty / D = 1.5$), the wake relocates the peak (to the left-hand side for $L_\infty / D = 0.5$, while to the right-hand side for $L_\infty / D = 1.5$), with a larger relocation observed when the inflow $L_\infty$ is smaller (i.e. $L_\infty / D = 0.5$) (for which the wake is more capable of changing because of the overall energy cascade direction).

Figure 28.

Premultiplied PSD of TKE at streamwise locations along the disk edge for different inflow length scales with inflow $I_{\infty } = 10\,\%$: (a) $L_\infty /D = 0.5$; (b) $L_\infty / D = 1.0$; (c) $L_\infty / D = 1.5$. Here solid lines, $C_T= 0.2$; dashed lines: $C_T= 0.7$. The grey lines represent the data for background flow.

Figure 29.

Premultiplied PSD of TKE at streamwise locations along the disk edge for different inflow length scales with inflow $I_\infty = 25\,\%$: (a) $L_\infty /D = 0.5$; (b) $L_\infty / D = 1.0$; (c) $L_\infty / D = 1.5$. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$. The grey lines represent the data for background flow.

We then examine how inflow turbulence length scale affects the disk in extracting inflow TKE in figure 30, which shows $f \varPhi _{k}$ at $0.5D$ disk downstream and the wake centreline. It is seen that the disk is capable of extracting TKE at higher frequencies (i.e. around $St>1.5$) for all three inflow turbulence length scales. It is capable of extracting TKE at lower frequencies (i.e. $St<0.15$), however, it depends on the inflow turbulence length scale. For small inflow $L_\infty$ (i.e. $L_\infty / D = 0.5$), the energy at lower frequencies barely changes. For large inflow $L_\infty$ (i.e. $L_\infty / D = 1.5$), on the other hand, a significant amount of energy at lower frequencies is extracted by the disk. The TKE extraction at medium frequencies (i.e. $0.1< St<1.0$) is negligible for all the inflow turbulence length scales.

Figure 30.

Premultiplied PSD of TKE at $0.5D$ disk downstream and wake centreline for different inflow length scales with inflow $I_\infty = 10, 25\,\%$: (a,d) $L_\infty / D = 0.5$; (b,e) $L_\infty / D = 1.0$; (c, f) $L_\infty / D = 1.5$. Here solid lines, $C_T= 0.2$; dashed lines, $C_T= 0.7$. The grey lines represent the data for background flow.

The TKE extraction at the disk and the TKE generation in the wake are influenced in scale space by inflow turbulence length scales ($L_\infty$) which are examined in figure 31, which shows the energy density difference $\Delta E$ on the $x\unicode{x2013}s$ plane for different inflow $L_\infty$. As seen, changing the inflow $L_\infty$ does not change the overall energy distribution, but influences the extents of energy extraction and addition, which are greater with the maximum $\Delta E$ located closer to the disk when the inflow $L_\infty$ is larger. Figure 32 shows the curves of premultiplied energy density difference $\Delta (sE)$ at several streamwise locations. The impact of inflow turbulence length scale ($L_\infty$) is demonstrated more clearly. It is seen that changing the inflow $L_\infty$ does not influence the shape of the $\Delta (sE)$ distribution with the peak for TKE extraction and TKE addition located at approximately $s=2.0D$ and $s=1.0D$, respectively, and the starting scale with energy extraction around $s=0.5D$. The major effect of increasing inflow $L_\infty$ is on the enhancement of TKE extraction and addition.

Figure 31.

Contours of energy density difference ${\Delta }E(x,s)$ on the $x\unicode{x2013}s$ plane for (a) $L_\infty / D = 0.5$, (b) $L_\infty / D = 1.0$ and (c) $L_\infty / D = 1.5$ with $I_\infty =10\,\%$ and $C_T=0.7$. The white dashed line corresponds to 0.25 times the maximum TKE loss value. The black dashed line is the peak of energy density in the streamwise location for each scale. The vertical black dotted line indicates the location of permeable disk.

Figure 32.

Premultiplied energy density difference (${\Delta }sE$) in scale space ($s$) at different streamwise locations for (a) $L_\infty / D = 0.5$, (b) $L_\infty / D = 1.0$ and (c) $L_\infty / D = 1.5$ with $I_\infty = 10\,\%$ and $C_T=0.7$.

4.2.3. Quadrant analysis

The quadrant analysis results from the cases with different inflow length scales are shown in figures 33 and 34. Impacts of the wake on the quadrant contributions similar to the cases with varying inflow TI are observed, that the intensities of the events are enhanced in quadrants $Q2$ and $Q4$, while are weakened in $Q1$ and $Q3$, making $S_2$ and $S_4$ dominant over $S_1$ and $S_3$. Moreover, it is seen that the reductions in $Q1$ and $Q3$ events (i.e. $\Delta S_1$ and $\Delta S_3$) and the enhancements in $Q2$ and $Q4$ events (i.e. $\Delta S_2$ and $\Delta S_3$) are larger at $x/D\sim 1.0$ and $x/D \sim 3.0$, respectively, and increase with the inflow integral length scale $L_{\infty }$. Two differences can be identified for the effects of inflow $L_{\infty }$ and $I_{\infty }$: (i) varying $I_{\infty }$ affects the locations of wake-induced $\Delta S_j(j=1 - 4)$, while varying $L_{\infty }$ mainly affects the magnitude; (ii) varying $I_{\infty }$ affects the relative contributions from $Q2$ and $Q4$ events, while varying $L_{\infty }$ barely does.

Figure 33.

Quadrant analysis for cases with different inflow length scales for (a i,b i,c i,d i) contributions in each quadrant ($S_{i} / U_{\infty }^2$), (a ii,b ii,c ii,d ii) wake-added difference in each quadrant ($\Delta {S}_{i} / U_{\infty }^2$). The inflow integral TI is $I_{\infty } = 10\,\%$. The thrust coefficient is $C_T = 0.7$.

Figure 34.

(a–c) Wake-added relative contributions in each quadrant ($\Delta {S}_{i} / \sum {\Delta {S}_{i}}$), (d) wake-added relative contributions in $Q1$ and $Q3$ events and (e) wake-added relative contributions in $Q2$ and $Q4$ events with different inflow length scales. The inflow integral TI is $I_{\infty } = 10\,\%$. The thrust coefficient is $C_T = 0.7$.