3.1. Mean and turbulence statistics

The mean streamwise velocity distributions normalized by the wall units (i.e. $U_\tau$ and $\nu /U_\tau$) at different $x$ locations (figure 3) reveal that the effects of PGs are similar for both smooth-wall (not shown here) and rough-wall TBLs in the outer region. There is an upward shift when APG is applied and a downward shift when FPG is applied, which is consistent with deceleration and acceleration, respectively. Similar flow behaviours under PGs are also observed in previous smooth-wall studies at various $Re$ (Spalart & Watmuff Reference Spalart and Watmuff1993; Harun et al. Reference Harun, Monty, Mathis and Marusic2013; Vila et al. Reference Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2020). It is observed though that at the stations closer to the inlet (I to III) (figure 3b), the effect of PGs for the rough-wall TBL penetrates in the logarithmic region $200\leq y^{+} \leq 800$; this influence is, however, modulated by the Reynolds number which shows a less impacted logarithmic region.

Figure 3.

Inner-normalized mean streamwise velocity profiles for different PGs: (a) $U_{i,l}$; (b) $U_{i,h}$. Dashed black lines have a slope of 1/0.4. See table 1 for symbols.

In the present study, we carry out a SP analysis for the rough-wall TBL with PGs. The equation for the streamwise mean velocity, $U$, in the case of the fully rough-wall TBL, where there is no viscous sublayer (Schultz & Flack Reference Schultz and Flack2007; Krogstad & Efros Reference Krogstad and Efros2012), is

(3.1)$$\begin{gather} \langle \bar{U}\rangle \frac{\partial \langle \bar{U}\rangle}{\partial x}+\langle \bar{V}\rangle\frac{\partial \langle \bar{U}\rangle}{\partial y}={-}\frac{\partial (\langle \overline{{u^{\prime}}{v^{\prime}}}\rangle+(\langle \overline{u^{\prime\prime}}\cdot\overline{v^{\prime\prime}}\rangle)}{\partial y} -\frac{\partial(\langle \overline{{u^{\prime}}^{2}}\rangle+\langle \overline{{u^{\prime\prime}}^{2}}\rangle)-(\langle \overline{{v^{\prime}}^{2}}\rangle+\langle \overline{{v^{\prime\prime}}^{2}}\rangle)}{\partial x}\nonumber\\ -\frac{1}{\rho}\frac{\mathrm{d}\langle p\rangle}{\mathrm{d}\kern0.06em x}+F, \end{gather}$$

where $V$ is the mean velocity component in $y$, the direction normal to the wall, and $u$ and $v$ are the $x$ and $y$ components of the velocity fluctuations. The time and spatial fluctuating velocity components are denoted by a prime and double prime, respectively. An overbar denotes the time-averaged quantities. Note that a spatial averaging operation $\langle \,\rangle$ over a thin horizontal slap is also applied in (3.1) which allows accounting for the heterogeneity of the roughness elements (see Raupach, Antonia & Rajagopalan (Reference Raupach, Antonia and Rajagopalan1991) and Finnigan (Reference Finnigan2000) for more details). As the flow is irrotational with a constant total head, the PG can be written as

(3.2)\begin{equation} \frac{1}{\rho}\frac{\mathrm{d}p_1}{\mathrm{d}\kern0.06em x}={-}U_1\frac{\mathrm{d}U_1}{\mathrm{d}\kern0.06em x} \end{equation}

and the streamwise component of form drag $F$ can be expressed as (see Raupach et al. (Reference Raupach, Antonia and Rajagopalan1991) for further details)

(3.3)\begin{equation} F=\frac{C_d}{k}{{\langle \bar{U}\rangle}^{2}}, \end{equation}

where $C_d$ is the drag coefficient. Following Townsend (Reference Townsend1980) and George (Reference George1995), we assume the following SP forms for the various quantities in (3.1):

(3.4)\begin{gather} U_1-\langle \bar{U}\rangle=U_0f(\eta), \end{gather}

(3.5)\begin{gather} \langle \overline{{u^{\prime}}{v^{\prime}}}\rangle+\langle \overline{u^{\prime\prime}}\cdot\overline{v^{\prime\prime}}\rangle=U_{uv}^{2}g_{uv}(\eta), \end{gather}

(3.6)\begin{gather} \langle \overline{{u^{\prime}}^{2}}\rangle+\langle \overline{{u^{\prime\prime}}^{2}}\rangle=U_{u}^{2}g_u(\eta), \end{gather}

(3.7)\begin{gather} \langle \overline{{v^{\prime}}^{2}}\rangle+\langle \overline{{v^{\prime\prime}}^{2}}\rangle=U_{v}^{2}g_v(\eta), \end{gather}

where $\eta =y/l$, and $l$ is a length scale; $U_0$, $U_u$, $U_v$ and $U_{uv}$ are velocity scales and not necessarily equal. The unknown functions $f$, $g_{uv}$, $g_u$ and $g_v$ are functions of $\eta$. Substituting these SP distributions in (3.1), using the continuity equation, then multiplying by $l/U_{uv}^{2}$ yield after some trivial manipulations

(3.8)\begin{align} &-\left[\frac{U_1 l}{{U_{uv}}^{2}}\frac{\mathrm{d}U_{0}}{\mathrm{d}\kern0.06em x}+\frac{U_0 l }{{U_{uv}}^{2}}\frac{\mathrm{d}U_1}{\mathrm{d}\kern0.06em x}\right]f+\frac{U_0 l }{U_{uv}^{2}}\frac{\mathrm{d}U_{0}}{\mathrm{d}\kern0.06em x}{f^{2}}+\left[\frac{U_1 U_0}{U_{uv}^{2}}\frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}-\frac{U_0 l}{U_{uv}^{2}}\frac{\mathrm{d}U_1}{\mathrm{d}\kern0.06em x}\right]\eta f'+g'_{uv}\nonumber\\ &\qquad -\left[\frac{U_0 l}{U_{uv}^{2}}\frac{\mathrm{d}U_{0}}{\mathrm{d}\kern0.06em x}+\frac{{U_{0}}^{2}}{U_{uv}^{2}}\frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}\right]f'\int_{0}^{\eta_1} f \,\mathrm{d}\eta+\frac{l}{U_{uv}^{2}}\frac{\mathrm{d}U_{u}^{2}}{\mathrm{d}\kern0.06em x}g_u-\frac{l}{U_{uv}^{2}}\frac{\mathrm{d}U_{v}^{2}}{\mathrm{d}\kern0.06em x}g_v-\frac{{U_{u}^{2}}}{U_{uv}^{2}}\frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}\eta g'_u\nonumber\\ &\qquad +\frac{{U_{v}^{2}}}{U_{uv}^{2}}\frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}\eta g'_v= \frac{C_d}{k}\left[l\frac{{U_1}^{2}}{U_{uv}^{2}}+2l\frac{U_1 U_0}{U_{uv}^{2}}f+\frac{{{U_{0}}^{2}} l}{U_{uv}^{2}}f^{2}\right]. \end{align}

Self-preservation imposes that the coefficients ($C_1$–$C_{12}$) of the various terms in this equation are constant since the coefficient of $g'_{uv}$ is equal to one:

(3.9)\begin{equation} \left. \begin{gathered} \left(C_1=\frac{U_1 l}{{U_{uv}}^{2}}\frac{\mathrm{d}U_{0}}{\mathrm{d}\kern0.06em x}+\frac{U_0 l}{U_{uv}^{2}}\frac{\mathrm{d}U_1}{\mathrm{d}\kern0.06em x}\right) ,\quad \left(C_2=\frac{U_0 l}{U_{uv}^{2}}\frac{\mathrm{d}U_{0}}{\mathrm{d}\kern0.06em x}\right) ,\quad \left(C_3=\frac{U_1 U_0}{U_{uv}^{2}}\frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}-\frac{U_0 l}{U_{uv}^{2}}\frac{\mathrm{d}U_1}{\mathrm{d}\kern0.06em x}\right),\\ \left(C_4=1\right) ,\quad \left(C_5=\frac{U_0 l}{U_{uv}^{2}}\frac{\mathrm{d}U_0}{\mathrm{d}\kern0.06em x}+\frac{U_0 ^{2}}{U_{uv}^{2}} \frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x} \right) ,\quad \left(C_6=\frac{l}{U_{uv}^{2}}\frac{\mathrm{d}U_{u}^{2}}{\mathrm{d}\kern0.06em x}\right) ,\quad \left(C_{7}=\frac{l}{U_{uv}^{2}}\frac{\mathrm{d}U_{v}^{2}}{\mathrm{d}\kern0.06em x}\right),\\ \left(C_{8}=\frac{{U_{u}^{2}}}{U_{uv}^{2}}\frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}\right) ,\quad \left(C_{9}=\frac{{U_{v}^{2}}}{U_{uv}^{2}}\frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}\right),\\ \left(C_{10}=\frac{l}{k}\frac{{U_1}^{2}}{U_{uv}^{2}}C_d \right),\quad \left(C_{11}=\frac{l}{k} \frac{U_1 U_0}{U_{uv}^{2}}C_d \right) , \left(C_{12}=\frac{l}{k}\frac{U_0^{2}}{U_{uv}^{2}}C_d\right). \end{gathered} \right\} \end{equation}

The ratio $C_{11}/C_{12}$ shows immediately the following:

(3.10)\begin{equation} U_0 \sim U_1 , \end{equation}

while the ratio $C_8/C_9$ yields

(3.11)\begin{equation} U_u \sim U_v. \end{equation}

Condition (3.10) indicates that the free-stream velocity can be an appropriate scaling velocity for the rough-wall TBLs under PGs. Also, note that as the ratio $U_\tau /U_1$ reaches a constant value in the present study (see table 1), the velocity $U_\tau$ is then also a valid scaling velocity. Using (3.10) in the ratio of $C_3/C_2$ leads to

(3.12)\begin{equation} -\frac{l}{U_1\dfrac{\mathrm{d}{l}}{\mathrm{d}{x}}}\frac{\mathrm{d}{U_1}}{\mathrm{d}{x}}=\frac{l}{\rho U_1^{2}\dfrac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}}\dfrac{\mathrm{d}p_1}{\mathrm{d}\kern0.06em x}=\varLambda, \end{equation}

where $\varLambda$ is often defined as a pressure parameter, which must be constant under SP, as also shown by Cal & Castillo (Reference Cal and Castillo2008) for smooth TBLs under PGs. Integration of (3.12) yields $l\sim U_1^{-1/\varLambda }$ for non-zero value of $\varLambda$. The Reynolds stress scale $U_{uv}$ is obtained by the difference $C_5-C_2$, which leads to

(3.13)\begin{equation} {U_{uv}^{2} = \frac{U_0^{2}}{C_5-C_2} \frac{\mathrm{d}l}{\mathrm{d}\kern0.06em x}}, \end{equation}

(3.14)\begin{equation} U_u \sim U_v \sim U_1. \end{equation}

If the flow is self-preserving, one expects that the coefficients for the turbulent kinetic energy expressed in a self-preserving form (Townsend Reference Townsend1980) must also satisfy SP conditions (for brevity, the equation is not presented here). One can then easily show that the ratio between the SP coefficients associated with energy dissipation and Reynolds stress terms (see equation (6.4.4) in Townsend (Reference Townsend1980)) leads to the condition

(3.15)\begin{equation} \frac{U_{uv}}{U_0} = {\rm const.} \end{equation}

Thus, (3.13) immediately leads to

(3.16)\begin{equation} \frac{{\rm d}l}{{\rm d} x} = C_l, \end{equation}

where $C_l$ is a constant that can be either positive or negative depending on whether the TBL is decelerating or accelerating. Further, (3.16) leads to the following relation between the various scaling velocities:

(3.17)\begin{equation} U_0 \sim U_1 \sim U_u \sim U_v \sim U_{uv}. \end{equation}

Finally, the constant $C_{10}$ (or $C_{11}$ and $C_{12}$) imposes the constraint $k \sim C_dl$ on the roughness height. Notice the appearance of $C_d$ in this constraint. If the rough-wall TBL is under ZPG, $C_d$ is then constant and one recovers the SP result of Rotta (Reference Rotta1962) and Talluru et al. (Reference Talluru, Djenidi, Kamruzzaman and Antonia2016), that is, $k \sim l$.

Some comments are warranted regarding the behaviour of $l$ with $x$ according to (3.16). When $C_l$ is positive, $l$ increases with $x$. This corresponds to the case of a TBL evolving under either ZPG or APG. The latter is often referred to as a source flow (Perry, Marušić & Li Reference Perry, Marušić and Li1994). When $C_l$ is negative, the flow corresponds to a sink flow (Perry et al. Reference Perry, Marušić and Li1994) in which $l$ decreases with increasing $x$. Note that, conversely to the scaling velocity which, as shown above, can be either $U_1$ or $U_\tau$, $l$ is yet to be identified. In a ZPG rough-wall TBL, $l$ is identified with $\delta$ when the roughness height $k$ increases linearly with $x$ (Kameda et al. Reference Kameda, Mochizuki, Osaka and Higaki2008; Talluru et al. Reference Talluru, Djenidi, Kamruzzaman and Antonia2016). In order to determine how $\delta$ behaves with $x$ under PGs we report the streamwise variation of $\delta$ in figure 4(a) for the three cases of PG; we also report the streamwise variations of $\delta ^{*}$ and $\theta$ in figures 4(b) and 4(c), respectively. In all three configurations of PG, these thicknesses increase with $x$; the rate of increase is larger for APG and lower for FPG. Further, the rate of increase is relatively well represented by a linear variation, indicating that the TBL is evolving in accordance with SP. Words of caution are warranted here. Self-preservation cannot be strictly achieved in the present study because $k$ does not vary with $x$; however, it is well approximated over the streamwise fetch of the measurements, as reflected by the constancy of $C_{D,p}$ (see table 1). While this is not too surprising for ZPG and APG, the results are rather remarkable if not surprising in the case of FPG. Indeed, SP analysis indicates that $\delta$ should decrease when the flow is accelerated (see also Townsend Reference Townsend1980). However, this applies to an already fully developed TBL. In the present experiment, the boundary layer generated at the entrance of the wind tunnel develops with a growing boundary layer thickness, regardless of the PG. This is illustrated in figure 5 which shows a schematic representation of the TBL in the wind tunnel. Under ZPG and APG, the boundary layer grows continuously. In the case of FPG, the boundary layer growth is generated, but one expects that with increasing $x$ the layer should eventually start to decrease under the ever-increasing action of free-stream velocity, resulting in a sink flow, as represented by the dashed line in the figure; see also Perry et al. (Reference Perry, Marušić and Li1994). As seen above, the SP analysis predicts that $\delta$ decreases linearly with increasing $x$, if $\delta$ satisfies (3.16) with $C_l <0$. However, the SP analysis does not inform as to how $\delta$ should behave while the boundary layer is growing in an accelerated flow as in the present case. The fact that $\delta$ increases linearly with $x$ during this stage of development suggests that the TBL may evolve in a SP state. The above results indicate that one can use $\delta$ as an adequate scaling length.

Figure 4.

Variation of $\delta$ (a), $\delta ^{*}$ (b) and $\theta$ (c) with $x$ for different PGs at two different inlet velocities: $U_{i,l}$ (open symbols with solid lines) and $U_{i,h}$ (filled symbols with dashed lines). Lines are spline fits to a linear interpolation of the data. Red, black and blue colours correspond to FPG, ZPG and APG cases, respectively.

Figure 5.

Schematic showing the evolution of the rough-wall TBL subjected to different PGs. Dashed red line represents the sink flow case.

Now that the behaviour of $\delta$ with $x$ is established we may proceed to determine how $U_1$, which was shown to be an appropriate scaling velocity according to SP, behaves with $x$. Using $\delta$ as a scaling length, then substituting (3.16) into (3.12) and solving for $U_1$, one obtains

(3.18)\begin{equation} U_1 \sim x^{-\varLambda}, \end{equation}

where $\varLambda$ should be constant. Such constancy is well verified as seen in figure 6(a), which shows that $\varLambda \simeq 0.3$ for the APG case and $-0.5$ for the FPG case. The variations of $U$ with $x$ are shown in a log–log representation in figure 6(b) for the APG and FPG cases. If $U_1$ behaves according (3.18) then one should observe a linear variation in the figure, where the slopes of the lines are practically equal to $\varLambda$. This is indeed seen in figure 6(b), indicating that $U_1$ follows (3.18) relatively well. We now use $U_1$ and $\delta$ as scaling velocity and scaling length, respectively, to normalize the various statistical quantities in the rest of the paper.

Figure 6.

Variation of (a) $\varLambda$ and (b) local free-stream velocity with $x$ for different PGs at two different inlet velocities. Here $U_i$ and $x_i$ are the free-stream velocity and the location just prior to introducing the PGs, respectively. Lines are spline fits to linear interpolations of the data. Open symbols and dashed lines: $U_{i,l}$; filled symbols and lines: $U_{i,h}$. Symbols and colours are the same as in figure 4.

Figure 7 shows the $U$-profiles for all three PG cases at all streamwise measurement locations and two inlet velocities. There is a good collapse of the data for each case of PG. Such collapse indicates that SP and Reynolds number similarity achieved in ZPG are not impacted by non-zero PG. However, comparison between the PG cases, as seen in figure 8, reveals some difference in the $U$-profiles. In particular, there is a stronger difference between FPG and ZPG than between APG and ZPG. Such differences indicate that the rough TBL under a non-zero PG will evolve in SP controlled by the PG. In other words, there should be no universal SP state for a non-zero PG.

Figure 7.

Mean streamwise velocity profiles $U/U_1$ at different locations: (a) $U_{i,l}$; (b) $U_{i,h}$. See table 1 for symbols.

Figure 8.

Mean streamwise velocity profiles $U/U_1$ for all PGs at locations III and IV: (a) $U_{i,l}$; (b) $U_{i,h}$. See table 1 for symbols.

Examples of distributions of ${\overline {u^{2}}}/U_1^{2}$ as a function of $y/\delta$ for all three PG cases are shown in figure 9 for all $x$ locations and two inlet velocities. There is a general good collapse between profiles for a given PG, particularly in the region outside the ‘roughness sublayer’ defined as the region where the (local) statistical quantities exhibit streamwise variations. Note that the collapse appears better at the higher inlet velocity, suggesting a slight residual Reynolds number effect affecting the establishment of SP. However, this latter Reynolds number effect is evident in the impact that the PG has on the TBL as seen in figure 10 which shows examples of distributions of ${\overline {u^{2}}}/U_1^{2}$ for the two inlet velocities at the second-from-last streamwise position (IV). The difference in the PG effects on the TBL is more accentuated at higher Reynolds number.

Figure 9.

Distributions of $\overline {u^{2}}/U_1^{2}$ at all measurement locations for ZPG (black), FPG (red) and APG (blue): (a–c) $U_{i,l}$; (d–f) $U_{i,h}$. The vertical dashed lines mark the position $y=k$. See table 1 for symbols.

Figure 10.

Distributions of ${\overline {u^{2}}}/U_1^{2}$ for different PGs at location IV: (a) $U_{i,l}$; (b) $U_{i,h}$. See table 1 for symbols and colours.

A practical way to assess the behaviour of the TBL under PG without reference to the distance to the wall, which for a rough wall strictly requires a virtual origin, is the use of the diagnostic plot proposed by Alfredsson & Örlü (Reference Alfredsson and Örlü2010). There is a remarkable good collapse of the diagnostic plots for each PG at the last two measurements stations (IV and V) (not shown here). However, when comparing the diagnostic plots between the PGs, as shown in figure 11, we note, as in we did in figure 10, that plots do not collapse in the region $y/\delta < 0.7$, particularly at the largest Reynolds number; the APG distribution is systematically lower than the ZPG distribution which in turn is lower than that for FPG.

Figure 11.

Diagnostic plot at location IV for the three PGs: (a) $U_{i,l}$; (b) $U_{i,h}$. See table 1 for symbols and colours.

3.2. Spectral analysis

Contours of the power spectral density of the streamwise velocity fluctuation ($k_x\phi _{uu}/{U_1^{2}}$) are plotted in figure 12 in terms of normalized wall distance ($y/\delta$) and wavelength ($\lambda _x/\delta$). We use Taylor's hypothesis of frozen turbulence (Taylor Reference Taylor1938) to obtain the wavenumber $k_x=2{{\rm \pi} }f/U_c$, where the convection velocity $U_c$ is approximated by the local mean velocity and $f$ is the frequency. It is generally accepted that this hypothesis is valid for both smooth- and rough-wall-bounded flows when the root mean square of fluctuating streamwise velocity $(u')$ is less than approximately 30 % of the local mean velocity (Lee, Lele & Moin Reference Lee, Lele and Moin1992; Romano Reference Romano1995; Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016). In the present study, the value of the ratio $u'/U$ is approximately 0.3 at a distance $y=k$ and decreases continuously as $y$ increases, as shown in figure 13 reporting the distribution of $u'/U$. Accordingly, we focus the spectral analysis on the region delimited $y=k$ (the vertical line in figure 12) where the Taylor hypothesis is appropriate (further discussions on the Taylor hypothesis for the present rough-wall TBL can be found in Ghanadi & Djenidi (Reference Ghanadi and Djenidi2021a)). Note that for clarity, the spectral analysis is carried out for location IV only; similar results are obtained for the other locations.

Figure 12.

Premultiplied energy spectra of streamwise velocity fluctuation $(k_x\phi _{uu}/{U_1}^{2})$ at location IV for the two different inlet velocities: (a–c) $U_{i,l}$; (d–f) $U_{i,h}$. Results are shown for FPG (a,d), ZPG (b,e) and APG (c,f). The red vertical line denotes the coordinate $y=k$ and the region $0 \leq y/k \leq 1$ is shaded. The dashed horizontal line indicates $\lambda _x=\delta$.

Figure 13.

Profiles of $u'/U_1$ as a function of $y/k$ at two streamwise locations, IV and V: (a–c) $U_{i,l}$; (d–f) $U_{i,h}$. See table 1 for symbols and colours.

In figure 12 one can observe the dynamic link between the large-scale structures in the outer region of the boundary layer and the regions closer to the roughness element. Comparing the velocity spectra between FPG (figure 12b) and APG (figure 12c) cases reveals a strong footprint of large-scale structures near the roughness element in rough-wall TBLs under APG; however, the interaction weakens with increasing $Re$ (figure 12f). In the FPG case, the inner peak disappears or is fully submerged below the crest of the roughness elements (figure 12b,e). This indicates that flow accelerations in rough-wall TBLs under FPG assist the roughness elements to fully disrupt the viscous-dominated region near the wall. It can also be seen that the outer peak location in the FPG case occurs at a lower location compared with the APG case. This could be due to increasing the flow deceleration in the APG case leading to reducing the convection speed of the turbulent structures. We also notice first that, despite the the contours presenting similar shapes, there is, with respect to the line $\lambda _x/\delta =1$, a general ‘upward shift’ of the contours for the largest inlet velocity regardless of the PG; this is seen in the location (or locus) of the largest values of $(k_x\phi _{uu}/U_1^{2})$ (gold to yellow contours) in figure 12. This certainly reflects a Reynolds number effect. We also notice that this locus is at slightly lower values of $\lambda _x/\delta$ in the APG case than the ZPG and FPG cases. This latter feature suggests that APG affects the most energetic scales more than FPG when comparing with the ZPG case. In figure 14, the self-similar behaviour of the rough TBLs with PG through contour lines of velocity spectra can also be seen at the two last measurement locations. The results clearly demonstrate the self-similar behaviour of velocity spectra. Most importantly, the location of the outer peak as represented by the cross symbol in the figure does not change at these locations.

Figure 14.

Comparison of contour lines of velocity spectra $(k_x\phi _{uu}/{U_1}^{2})$ at locations IV and V for $U_{i,h}$: (a) FPG, (b) ZPG and (c) APG. The red vertical line denotes the coordinate $y=k$.

To ascertain possible differences between the effects of FPG and APG on the spectra we show in figure 15 the difference contours ($k_x\phi ^{+}_{uu|\,PG}-k_x\phi ^{+}_{uu|\,ZPG}$) where the spectral contours for ZPG are subtracted from those for FPG and APG, respectively (the superscript $+$ represents normalization by $U_1$). In reference to the ZPG case, negative difference contours reveal a reduced energy level, while positive contours mark a higher energy level. Let us first focus on the lower inlet velocity, $U_{i,l}$ (figure 15a,b). There is no significant difference in the contours between FPG and APG in the region below $y/\delta \simeq 0.08$. In that region, the contours change sign from negative to positive; the ‘separation line’ between the negative and positive contours exhibits a positive slope ($\lambda _x/\delta$ increases with $y/\delta$) before it becomes practically horizontal. Notice the narrowing trend of the contours as the separation line is approached from below and above. The similarity of the difference contours between the PG cases suggests that the PG effect is not significant in that region and that the flow behaviour/dynamic is mostly controlled by the roughness. Above this roughness-controlled region (i.e. $y/\delta \geq 0.1$), one can observe a significant change between the FPG and APG cases. For the FPG case, the contours are now all negative, while there is still a change of sign for the APG case. This difference suggests that the TBL in this flow region is more receptive to PG than close to the wall. However, when the inlet velocity is increased to $U_{i,h}$ (figure 15c,d) one can observe marked differences between the two inlet velocity cases and the two PGs suggesting different dynamical behaviours of the TBL under the different PGs. For FPG, the negative contours initially observed in the region ($y/\delta \geq 0.1$) for $U_{i,l}$ are replaced by positive contours. For APG, the negative contours move up to larger values of $\lambda _x/\delta$ for $U_{i,h}$ (figure 15d) than for $U_{i,l}$ (figure 15b) in the region $y/\delta \leq 0.3$, while the positive contours almost vanish.

Figure 15.

Contours of energy difference between APG/FPG and ZPG at location IV: (a,c) $k_x\phi ^{+}_{uu|\,FPG}-k_x\phi ^{+}_{uu|\,ZPG}$ (dashed contours indicate negative values); (b,d) $k_x\phi ^{+}_{uu|\,APG}-k_x\phi ^{+}_{uu|\,ZPG}$. Results are shown for (a,b) $U_{i,l}$ and (c,d) $U_{i,h}$. The red and blue vertical lines denote the coordinates $y=k$ and $y=0.1 \delta$. Some contour values are shown for clarity.

Overall, figure 15 indicates that the differences between FPG and APG effects on the rough-wall TBL appear to be more pronounced with increasing Reynolds numbers, reflecting the results of figure 11. This is perhaps not too surprising considering the opposite effects that APG (deceleration) and FPG (acceleration) have on the flow. Clearly more studies at larger Reynolds numbers than here and over a longer streamwise distance (to achieve a clear sink flow in the case of FPG) are required to confirm or not this observation.

3.3. Proper orthogonal decomposition analysis

We turn our attention now to another approach for investigating the energy distribution between scales of motion in turbulent flows. This approach is based on the proper orthogonal decomposition (POD) method (Lumley Reference Lumley1967; Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993). In this method spatiotemporal streamwise velocity function $u(x,t)$ is decomposed into spatial and temporal characteristics as

(3.19)\begin{equation} u(x,t)=\sum_{n}^{N_m} a_n(t) \psi_n(x), \end{equation}

where $\psi _n(x)$ corresponds to the spatial orthonormal basis function (or eigenmodes), $a_n(t)$ are the time-dependent coefficients (often called random coefficients) and $N_m$ is the number of basis functions (or modes). A POD analysis, which is a generalization of power spectral analysis, decomposes the complex turbulent boundary layer flow into simpler modes, and thus complements the spectral analysis reported above. Indeed, while spectral analysis helps assess how the energy distribution among scales is affected by the PGs, the POD analysis, which contains information about the structure of the turbulent eddies (via the eigenvalues and eigenfunctions), helps determine how the individual features/events, particularly the most energetic ones, are affected. It is commonly agreed that the low-order modes are representative of large-scale motions and higher-order modes are associated with small-scale ones. Further, individual POD modes are also associated with particular events such as sweep and ejection events. A POD analysis is mostly carried out using velocity fields obtained using particle image velocimetry (Kruse, Kuhn & von Rohr Reference Kruse, Kuhn and von Rohr2006; Djenidi et al. Reference Djenidi, Antonia, Amielh and Anselmet2010; Druault, Bouhoubeiny & Germain Reference Druault, Bouhoubeiny and Germain2012; Shehzad et al. Reference Shehzad, Sun, Jovic, Ostovan, Cuvier, Foucaut, Willert, Atkinson and Soria2021). It has been shown, however, that POD can also be carried out using temporal velocity signals acquired via single hot-wire measurements (Tang et al. Reference Tang, Djenidi, Antonia and Zhou2014). This latter variant of POD is used in the present study. In such POD variant method the velocity signal is used to obtain a correlation matrix, written as

(3.20)\begin{equation} U = \begin{bmatrix} u(t_1) & u(t_2) & \cdots & u(t_n) \\ u(t_2) & u(t_3) & \cdots & u(t_n+1)\\ : & : & \cdots & : \\ u(t_{N_m}) & u(t_{{N_m}+1}) & \cdots. & u(t_{{N_m}+n-1}) \end{bmatrix}, \end{equation}

where $n$ is the number of velocity samples, $N_m > > n$ and each column has approximately identical root-mean-square values with similar velocity spectra. The time separation between two consecutive velocity samples is selected such that the correlation $\overline {u(t)u(t+\tau )}$ drops to zero to ensure that the two velocities are uncorrelated (see Tang et al. (Reference Tang, Djenidi, Antonia and Zhou2014) for more details). Then the POD modes can be calculated as

(3.21)\begin{equation} \psi_n(x)=\frac{1}{\sqrt{\zeta_n}}U\upsilon_n, \end{equation}

where $\zeta$ and $\upsilon$ are the eigenvalue and eigenvector of $U^{T}U$, respectively. Figure 16 shows the fractional contribution of the POD modes to the total kinetic energy, $\zeta _n/\sum _{n=1}^{N_t} \zeta _n$, at two wall-normal locations, $y/\delta = 0.1$ and 0.6 (for completeness we also report the cumulative energy contribution). Parameter $N_t$ is the total number of basis functions. The first mode contains the largest energy contribution, which varies between 40 % and 60 % for $U_{i,l}$ and approximately 25 %–50 % for $U_{i,h}$. That contribution drops to approximately 20 % for the second mode for both inlet velocities, while the third mode contributes approximately 10 % to the total energy. Interestingly, the contribution of the first mode is impacted by the PG. For example, that contribution is the largest for APG and the lowest for ZPG. Notably, the contribution of the second mode appears much less affected by the PG; it is practically the same for all three PGs. However, the third mode (and higher for that matter) does show a PG dependence, which is the reverse of that shown by the first mode. This behaviour difference between the modes under different PGs is yet to be physically explained. It nevertheless indicates not only the sensitivity of the large-scale motion to the PG, but also that the response of the TBL varies with the scales of motion as indicated by the non-trivial behaviour of the modes as described above. Indeed, in comparison with the ZPG case, an APG would strengthen the flow structures associated with the first mode as illustrated by the increase of their energy contribution and weaken the flow structures associated with third and higher modes, while leaving the structures represented by the second mode apparently unaffected. This non-trivial behaviour is perhaps best illustrated in figure 17 which shows the fractional contributions for the first three modes as a function of $y/\delta$. First, notice that the energy contribution of the second mode across the TBL is not much affected by the PGs. It is as if the flow structures associated with this mode are independent of PG. The situation for the first and third modes is different. For APG, the first- and third-mode contributions behave in opposite trends when compared with the other PGs across the TBL; the first mode is the largest and the third mode the smallest. These two modes have similar distributions of $\zeta$ across the TBL for the ZPG and FPG cases.

Figure 16.

The POD fractional and cumulative energy contributions at location IV: (a,c) $y/\delta = 0.1$; (b,d) $y/\delta = 0.6$. Open symbols are associated with $U_{i,l}$ and filled symbols are associated with $U_{i,h}$. Black, blue and red symbols represent ZPG, APG and FPG data, respectively. The black arrows indicate the axis corresponding to the cumulative energy plots.

Figure 17.

The POD fractional contribution, $\zeta _n/\sum _{n=1}^{N_t} \zeta _n$, as a function of $y/\delta$ at location IV for the first three modes for both inlet velocities: (a–c) $U_{i,l}$; (d–f) $U_{i,h}$. (a,d) First mode, (b,e) second mode and (c,f) third mode. Symbols are the same as in figure 16.

Figure 18 shows a log–log representation of the POD fractional energy reported in figure 16. This representation allows us to determine whether the decay of the fractional energy with an increasing mode order follows a power law of the form $n^{\alpha }$. If such a power law exists, it should be reflected in a ‘linear’ decreasing trend in the data when reported in a log–log plot. Such a trend is clearly visible for the more or less first 15 modes when the inlet velocity is $U_{i,l}$, but not for $U_{i,h}$. Beyond these modes and while such a trend is less evident, one can nevertheless approximate the decay of the fractional energy by a power-law form with different exponents, $\alpha$, as represented by the lines tentatively drawn in figure 18; this is particularly more discernible for $U_{i,h}$ than for $U_{i,l}$. Notice that the magnitude of the decay exponent $\alpha$ is smaller for $U_{i,h}$ than for $U_{i,l}$. It is tempting to posit that the fractional energy decays in a ‘self-preserving’ manner with an increasing mode order. Note that the difference in the effects of the PGs on the decay is more visible at the lower inlet velocity than at the higher one. It is worthwhile commenting on the power-law trend. Indeed, it is difficult at this stage to link the power law (and its exponent $n$) to the physical nature of the TBL. One tentative interpretation would be that $n$ ‘measures’ the rate at which the energy is lost as the mode order increases. The higher the magnitude of $n$, the faster the decay of energy. This would suggest that a TBL with a large decay rate is predominantly composed of ‘energetic’ structures. Further analysis is required on this issue.

Figure 18.

Log–log representation of the POD fractional energy at location IV: (a,c) $y/\delta = 0.1$; (b,d) $y/\delta$= 0.6. Results are shown for (a,b) $U_{i,l}$ and (c,d) $U_{i,h}$. Black, blue and red symbols represent ZPG, APG and FPG data, respectively.

We close this POD analysis by showing in figure 19 examples of power spectral density, $E_{a_n}(f)$, of the random coefficients $a_n(t)$ for modes 1, 2, 3, 5, 10 and 50 for all PGs at $y/\delta =0.1$ and both inlet velocities. One can observe that these power spectral densities exhibit a similar feature: a peak at a very low frequency. This peak indicates that the coefficients $a_n(t)$ draw their contents mostly from the frequency range 1–10 Hz. However, the peak magnitude decreases as $n$ increases; the peak has significantly reduced for $n=50$. This behaviour is consistent with the idea that the higher the mode order, the lower the contribution to the total energy, as illustrated in the fractional energy plot (see figure 16). There are some differences between the PGs. For example, the magnitude of the peak of mode 1 is larger for the ZPG case than for the APG and FPG cases. Notice also differences between the FPG and APG cases. Altogether the figure indicates that the features captured by the modes ‘oscillate’ at low frequencies and they respond to the PGs.

Figure 19.

Power spectral density of the coefficients $a_n(t)$ for modes 1, 2, 3, 5, 10 and 50 at location IV and $y/\delta$=0.1: (a–c) $U_{i,l}$; (d–f) $U_{i,h}$. Results are shown for (a,d) ZPG, (b,e) FPG and (c,f) APG.