3.1 Comparison with channel flow

We compare our flow control results with the steady streamwise transpiration study of Quadrio et al. (Reference Quadrio, Floryan and Luchini2007) in a channel. The variation of the mean friction coefficient ${\rm\Delta}C_{f}$ with respect to the smooth pipe is employed to assess the drag reduction/increase. The mean friction coefficient is defined as

Given that the mean wall shear stress is fixed, the mean friction coefficient variation can be expressed solely as function of the ratio between the smooth and controlled pipe bulk velocities, hence

in which the definition of ${\rm\Delta}Q$ in (3.4) is employed. Note that negative values of ${\rm\Delta}C_{f}$ correspond to drag reduction. Figure 3 shows the comparison of the variation of the friction coefficient versus transpiration wavelength at fixed amplitude $A^{+}=0.7$ in pipe and channel flow. Low- and high- $Re$ results are presented for the channel case. Similar behaviour is observed, in which a drag reduction can be achieved in both channel and pipe flows. The maximum friction variation is smaller for the pipe than for the channel. In addition, the rate at which the friction coefficient increases with wavelength is higher for the channel than for the pipe. We note that the manner in which parameters are varied differs between the two studies: while Quadrio et al. (Reference Quadrio, Floryan and Luchini2007) fixed the bulk velocity to arrange a constant bulk Reynolds number during the parameter sweep, here we maintained a constant friction Reynolds number $Re_{{\it\tau}}$ . It is also worth noting that because of the obvious difference in geometry, identical results are not expected.

Figure 3.

Comparison of the variation of the friction coefficient versus transpiration wavelength at fixed amplitude $A^{+}=0.7$ . Solid circles, pipe flow at $Re_{{\it\tau}}=314$ (present results); open circles, channel flow at $Re_{{\it\tau}}=400$ (Quadrio et al. Reference Quadrio, Floryan and Luchini2007); squares, channel flow at $Re_{{\it\tau}}=180$ (Quadrio et al. Reference Quadrio, Floryan and Luchini2007).

4.1 Constant transpiration amplitude

The effect of transpiration is evident in the behaviour of the mean velocity characteristics. Figure 4(a) shows the effect of changing the transpiration wavenumber $k_{c}$ at a small constant amplitude $A^{+}=0.7$ on the mean streamwise velocity $\bar{u}_{0}(r)$ compared to the smooth pipe. In addition, dashed lines for the linear sublayer and fitted log law (den Toonder & Nieuwstadt Reference den Toonder and Nieuwstadt1997) are shown as reference. In agreement with figure 2(a), we observe reduced flow rates in figure 4(a) for $k_{c}<8$ with a small increase occurring around $k_{c}=10$ . Two interesting features are noticed. First, the viscous sublayer is only linear for $y^{+}<1$ , instead of $y^{+}<5$ . Since the mean wall-shear ${\it\tau}_{w}$ is constant, all the profiles must collapse as $y^{+}=(R^{+}-r^{+})$ approaches zero. In other words, the value of $\partial u_{0}/\partial y$ at the wall is the same for all cases considered.

Figure 4.

Comparison of profile data for the smooth-wall pipe (solid line) and pipe with transpiration at constant amplitude $A^{+}=0.7$ and different wavenumbers $k_{c}$ : (a) mean flow normalized by $U_{b}$ , with dashed lines for linear sublayer and fitted log law; (b) axial turbulent intensity; (c) radial turbulent intensity; (d) Reynolds shear stress.

Second, the outer-layer of all the profiles are parallel, suggesting that the log layer region can be expressed as

with the transpiration factor, ${\rm\Delta}T$ arising in a similar way as the roughness factor of Hama (Reference Hama1954), or the corrugation factor of Saha et al. (Reference Saha, Klewicki, Ooi and Blackburn2015b ). However, we note that the present Reynolds number may not be sufficiently high for the emergence of a self-similar log law in the overlap region. Although there is not a well-defined log layer at such low $\mathit{Re}$ , we refer to it as log layer for convenience. The present results suggest what will happen at a greater $\mathit{Re}$ . Figure 5 shows a defect velocity scaling of the mean profiles. A collapse of the mean velocity profiles is observed in the outer layer, hence Townsend’s wall similarity hypothesis (Townsend Reference Townsend1976) appears to apply in the present low amplitude transpiration cases.

Figure 4(b,c) show the influence of the transpiration on the turbulent intensities, presented as root mean squares. At this small amplitude, the transpiration mainly affects the location and value of the maximum turbulent intensities and all profiles collapse as $y^{+}$ approaches the centreline, indicating that small amplitude transpiration does not alter the turbulent activity in the outer layer. Note that the peak in the axial turbulence intensity correlates with a reduction in the shear at the same wall-normal location in all the cases, which is apparent in the main velocity profiles in figure 4(a).

Figure 4(d) shows that the influence of small amplitude transpiration on the Reynolds shear stress ${\it\tau}_{uv}=\langle u^{c}v^{c}\rangle$ . We observed that the transpiration influence on ${\it\tau}_{uv}$ is only significant at small wavenumbers and, as with the turbulent intensities, small amplitude transpiration does not change the outer layer of the profile.

Despite the significant changes observed in the mean streamwise velocity profiles in figure 4(a), the variation in Reynolds shear stress in figure 4(d) are small, hence it is inferred that additional effects related to steady streaming and non-zero mean streamwise gradients play a major role in the momentum balance.

Figure 5.

Defect velocity law for profile data of the smooth-wall pipe (solid line) and pipe with transpiration at constant amplitude $A^{+}=0.7$ and different wavenumbers $k_{c}$ . A collapse of all the mean velocity profiles is observed in the outer layer hence small amplitude effects do not alter the turbulent activity in the outer layer.

Figure 6.

Comparison of profile data for the smooth-wall pipe (solid line) and pipe with transpiration at constant wavenumber $k_{c}=2$ and different amplitudes $A^{+}$ : (a) mean flow normalized by $U_{b}$ , with dashed lines for linear sublayer and fitted log law; (b) axial turbulent intensity; (c) radial turbulent intensity; (d) Reynolds shear stress.

4.2 Constant transpiration wavenumber

The effect of changing the amplitude $A^{+}$ at a constant forcing wavenumber $k_{c}=2$ in the mean streamwise velocity $u_{0}$ is shown in figure 6(a). For $A^{+}<2$ , results are similar to those observed for small amplitudes. At $A^{+}=2$ the flow rate is dramatically decreased and the log layer region is reduced compared to the small amplitude cases. At $A^{+}=4$ the parallel log layer region is not existent. This large amplitude can significantly increase the mean velocity in the outer layer. Similarly, we observe that large amplitude transpiration shifts the location of maximum turbulent intensities towards the centreline, as seen in figure 6(b,c). We can state that large amplitudes $A^{+}>2$ can influence the turbulent activity in the outer layer. Figure 6(d) shows that high amplitudes have a similar effect on the shear stress and how a high amplitude $A^{+}=4$ can dramatically reduce the Reynolds shear stress and shift the location of the maximum towards the centreline.

These tendencies in $A^{+}$ and $k_{c}$ suggest that very high values of the transpiration amplitude combined with high wavenumber $k_{c}$ can lead to significant drag-reducing configurations. This has been confirmed through the investigation of a static transpiration case with an amplitude $A^{+}=10$ and wavenumber $k_{c}=10$ , yielding an increase in flow rate of ${\rm\Delta}Q=19.3\,\%$ . This case will be investigated in detail in the next section.

5.1 Representative transpiration configurations

The relative importance of the three terms acting in the transpiration is investigated by inspecting three representative transpiration configurations in terms of drag modification. Table 1 lists the contributions to the change in flow rate of these configurations: (i) the largest drag-reducing case found ( ${\rm\Delta}Q=19.3\,\%$ ) consisting of a large amplitude and transpiration wavenumber $(A^{+},k_{c})=(10,10)$ , (ii) a small drag-reducing case ( ${\rm\Delta}Q=0.4\,\%$ ) induced by a tiny amplitude at a large wavenumber $(A^{+},k_{c})=(0.7,10)$ , and (iii) a large drag-increasing case ( ${\rm\Delta}Q=-36.1\,\%$ ) with small wavenumber and amplitude $(A^{+},k_{c})=(2,2)$ .

We first compare the Reynolds shear stress generated by the fluctuating velocity ${\it\tau}_{uv}$ and the shear stress arising from the deviation velocity ${u_{0}^{c}}^{\prime }{v_{0}^{c}}^{\prime }$ . The radial distributions of the three different cases are showed in figure 7. We observe that the streaming or coherent Reynolds shear stress is dominant close to the wall and opposes the Reynolds stress generated by the fluctuating velocity. Far from the wall, the Reynolds stress is governed by the fluctuating velocity.

Figure 8.

Radial distribution of the different terms in (5.5) to the streamwise momentum balance. Solid line ${\rm\Delta}\bar{u}_{0}(r)$ ; — ⋅ — Reynolds shear stress term $\text{RSS}$ ; – – steady streaming terms $\text{ST}$ ; $\cdots \cdots$  non-homogeneous terms $\text{NH}$ . (a) Case I, large drag decrease $(A^{+},k_{c})=(10,10)$ ; (b) case II, small drag decrease $(A^{+},k_{c})=(0.7,10)$ ; (c) case III, drag increase $(A^{+},k_{c})=(2,2)$ .

Table 1.

Contributions to the change in flow rate of different transpiration configurations based on integration of (5.5) ( $\int$ denotes integration in the wall-normal direction).

Figure 8 shows the radial profile of the different contributions to the mean profile and figure 9 the corresponding mean profiles of the cases considered. Figure 8(a) corresponds to the drag-decrease case $(A^{+},k_{c})=(10,10)$ . It can be observed that the contributions to the increase in flow rate are caused by a large reduction of Reynolds shear stress (28.4 %) and the streamwise gradients produced in the flow at such large amplitude and wavenumber (11.3 %). However, these contributions are mitigated by the opposite flow rate induced by the steady streaming ( $-20.4\,\%$ ). Figure 8(b) corresponds to a low-amplitude static transpiration $A^{+}=0.7$ at the same axial wavenumber as case (b), yielding a very small reduction in flow rate. Similarly, the effect on the Reynolds shear stress (1.5 %) and the non-homogeneity effects (0.6 %) contribute to increase the flow rate while the streaming produced is opposite to the mean flow, but of a lesser magnitude ( $-1.7\,\%$ ). Figure 8(c) represents the drag-increase case $(A^{+},k_{c})=(2,2)$ . As opposed to the two previous cases, the main contribution to the drag is caused by a large steady streaming in conjunction with a large non-zero streamwise gradients opposite to the mean flow. Although the decrease in Reynolds stress is significant and favourable towards a flow rate increase, the sum of the other two terms is much larger. In terms of the variation of mean profile, we observe a similar behaviour in all cases: there is a decrease in the mean streamwise velocity profile close to the wall induced by the streaming term or coherent Reynolds shear stress. Within the buffer layer, the difference in mean velocity decreases until a minimum is achieved and then it increases to a maximum in the log layer region. A smooth decrease towards the centreline is then observed.

Figure 9.

(a) Streamwise mean profile. (b) Mean wall shear along the axial direction.

Streamlines and kinetic energy of the two-dimensional mean velocities $\boldsymbol{u}_{0}(x,r)$ are shown in figure 10. Preceding the analysis of the flow dynamics, we observe that blowing is associated with areas of high kinetic energy while suction areas slow down the flow. Despite the large amplitudes employed, we observe an apparent absence of recirculation areas in these mean flows. Figure 9(b) shows the mean wall shear along the axial direction. Only a very small flow separation occurs in the large drag decrease $(A^{+},k_{c})=(10,10)$ . Hence, flow separation effects are not associated with the observed drag modifications.

Figure 10.

Two-dimensional mean flow: streamlines and normalized kinetic energy for each different case (white to blue): ( $a$ ) reference case, smooth pipe; ( $b$ ) drag-reducing case $(A^{+},k_{c})=(10,10)$ ; ( $c$ ) neutral case $(A^{+},k_{c})=(0.7,10)$ ; ( $d$ ) drag-increasing case $(A^{+},k_{c})=(2,2)$ .

5.2 Flow control energetic performance

As stated by Quadrio (Reference Quadrio2011), a flow control system study must always be accompanied with its respective energetic performance analysis in order to determine its validity in real applications. The control performance indices in the sense of Kasagi, Hasegawa & Fukagata (Reference Kasagi, Hasegawa and Fukagata2009) are employed here to assess the energy effectiveness of the transpiration. We define a drag-reduction rate as

where $P$ is the power required to drive the pipe flow and the subscripts $c$ and $s$ refer again to controlled and smooth pipe. The index $W$ can also be interpreted as the proportional change in the power developed by the constant body force $f_{x}$ as a consequence of the transpiration. Note from (5.12) and (5.13) that the drag-reduction rate $W$ is equivalent to the flow rate variation ${\rm\Delta}Q$ . The power required to drive the flow is given by the product of the body force times the bulk velocity

A net energy saving rate $S$ is defined to take into account the power required to operate the flow control

The mean flow momentum and energy equation are employed to obtain the power required to apply the transpiration control, as also done by Marusic, Joseph & Mahesh (Reference Marusic, Joseph and Mahesh2007) and Mao, Blackburn & Sherwin (Reference Mao, Blackburn and Sherwin2015). The power employed to apply the transpiration control reads

with $v_{0}(x,R)=A\sin (k_{c}x)$ . The first term in the integral represents the rate at which energy is introduced or removed as kinetic energy in the flow through the pipe wall. This term is exactly zero because the change of net mass flux is zero and the velocity is sinusoidal, imposed by the transpiration boundary condition in (2.3). The second term represents the rate of energy expenditure by pumping flow against the local pressure at the wall. Finally, the effectiveness of the transpiration control is defined as the ratio between the change in pumping power and power required to apply the transpiration control, which reads

Note that the net power saving can be alternatively written as

which indicates that an effectiveness $G$ higher than one is required for a positive power gain. As reported by Kasagi et al. (Reference Kasagi, Hasegawa and Fukagata2009), typical maxima for active feedback control systems are in the range of $G\sim 100$ and $S\sim 0.15$ and for predetermined control strategies, such as spanwise wall-oscillation control in a channel (Quadrio & Ricco Reference Quadrio and Ricco2004) or streamwise travelling transpiration in a channel (Min et al. Reference Min, Kang, Speyer and Kim2006), a range of $2\lesssim G\lesssim 6$ and $0.05\lesssim S\lesssim 0.25$ was found.

Table 2.

Flow control energetic performance indices. Here $W$ represents the drag reduction rate, $G$ the effectiveness of the transpiration control and $S$ the net energy saving rate.

The resulting flow control energetic performance indices are listed in table 2. Note that the change in pumping power rate $W$ is equivalent to the flow rate variation ${\rm\Delta}Q$ . Despite the large drag reduction produced by the configuration $(A^{+},k_{c})=(10,10)$ , the net energy saving rate is marginal and the small drag reduction caused at $(A^{+},k_{c})=(0.7,10)$ has an effectiveness less than one, with a net energy expenditure. The case $(A^{+},k_{c})=(2,2)$ is interesting, as it shows a high effectiveness in decreasing the flow rate and has the potential to relaminarize the flow at lower $Re_{{\it\tau}}$ . For instance, a reduction in flow rate ${\rm\Delta}Q=-36\,\%$ at $Re_{{\it\tau}}=115$ could reduce the critical bulk flow Reynolds below $Re<2040$ and thus achieve a relaminarization of the flow. Finally, we remark that the particular case $(A^{+},k_{c})=(10,10)$ is probably not the globally optimal configuration for steady transpiration and net energy saving rates and effectiveness in the range of typical values for the flow control strategies reported by Kasagi et al. (Reference Kasagi, Hasegawa and Fukagata2009) might be possible, if the full parameter space was considered.

6.1 Resolvent analysis

The loss of spatial homogeneity in the axial direction is the main challenge when incorporating transpiration effects into the resolvent model. One solution is to employ the two-dimensional resolvent framework of Gómez et al. (Reference Gómez, Blackburn, Rudman, Mckeon, Luhar, Moarref and Sharma2014). This method is able to deal with flows in which the mean is spatially non-homogeneous. The dependence on the axial coordinate $x$ is retained in the formulation, in contrast with the classical one-dimensional resolvent formulation of McKeon & Sharma (Reference Mckeon and Sharma2010). However, the method leads to a singular value decomposition (SVD) problem with storage requirements of order $O(N_{r}^{2}N_{x}^{2})$ , with $N_{r}$ and $N_{x}$ being the resolution in the radial and axial directions, respectively, as opposed to the original formulation which was of order $N_{r}^{2}$ . Consequently, the two-dimensional method is not practical for a parameter sweep owing to the large computational effort required. In the following we present a computationally cheaper and simple alternative based on a triple decomposition of the total velocity.

Based on the decompositions in (3.1) and (5.9), the total velocity is decomposed as a sum of the axial mean profile, a steady but spatially varying deviation from the axial profile and a fluctuating velocity

The fluctuating velocity is expressed as a sum of Fourier modes. These are discrete since the domain has a fixed periodic length and is periodic in the azimuth:

where $k$ , $n$ and ${\it\omega}$ are the axial, azimuthal wavenumber and the temporal frequency, respectively. Note than a complex-conjugate must be added because $\hat{\boldsymbol{u}}$ is real.

Without loss of generality, we have assumed that the deviation velocity can be expressed as a single Fourier mode with axial wavenumber $l$ . As a consequence of the triadic interaction between the spatial Fourier mode of the deviation velocity and the fluctuating velocity, there is a coupling of the fluctuating velocity at $(k,n,{\it\omega})$ with that at $(k\pm l,n,{\it\omega})$ (see appendix A). Similarly, the nonlinear forcing terms generated by the fluctuating velocity are written as $\boldsymbol{f}_{k,n,{\it\omega}}=(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u})_{k,n,{\it\omega}}$ .

Taking (6.1) into account, it follows that the Fourier-transformed Navier–Stokes equation (2.2) yields the linear relation

for $(k,n,{\it\omega})\neq (0,0,0)$ and $(k,n,{\it\omega})\neq (\pm l,0,0)$ , where $\mathscr{C}_{k,n,{\it\omega}}$ is a coupling operator representing the triadic interaction between deviation and fluctuating velocity. The triadic interaction can be considered as another unknown forcing and it permits lumping all forcing terms as

hence the following linear velocity–forcing relation is obtained

The resolvent operator $\mathscr{H}_{k,n,{\it\omega}}$ acts as a transfer function between the fluctuating velocity and the forcing of the nonlinear terms, thus it provides information on which combination of frequencies and wavenumber are damped/excited by wall transpiration effects. Equation (6.6) shows the resolvent $\mathscr{H}_{k,n,{\it\omega}}$ . The first, second and third rows corresponds to the streamwise, wall-normal and azimuthal momentum equation, respectively:

with

The physical interpretation of the present resolvent formulation is the same as that of the original formulation (McKeon & Sharma Reference Mckeon and Sharma2010). Figure 11 presents the new resolvent formulation (6.5) by means of a block diagram. The mean velocity profile is sustained in the $(k,n,{\it\omega})=(0,0,0)$ equation via the Reynolds stress $f_{0,0,0}$ and interactions with deviation velocity. Similarly, the deviation velocity is also sustained via the forcing $f_{l,0,0}$ and interactions with the mean flow in the mean flow equation corresponding to $(k,n,{\it\omega})=(l,0,0)$ . The deviation velocity drives the triadic interactions generated via the operator $\mathscr{C}_{k,n,{\it\omega}}$ and, closing the loop, the mean profile restricts how the fluctuating velocity responds to the nonlinear forcing via the resolvent operator $\mathscr{H}_{k,n,{\it\omega}}$ . We note that it is relatively straightforward to generalize the block diagram in figure 11 if the deviation velocity is composed of multiple axial wavenumbers, or even frequencies.

Figure 11.

Diagram of the new triple-decomposition-based resolvent model. The mean velocity profile is sustained in the $(k,n,{\it\omega})=(0,0,0)$ equation via the Reynolds stress $f_{0,0,0}$ and the interactions with deviation velocity. Similarly, the deviation velocity is also sustained via the forcing $f_{l,0,0}$ and interactions with the mean flow in the $(k,n,{\it\omega})=(l,0,0)$ equation. The original model of McKeon & Sharma (Reference Mckeon and Sharma2010) is represented by the subset within the dashed border.

We highlight that this formulation represents the exact route for modifying a turbulent flow using the mean profile $u_{0}$ . Despite the non-homogeneity of the flow in our present cases of interest, the operator $\mathscr{H}_{k,n,{\it\omega}}$ is identical to that developed by McKeon & Sharma (Reference Mckeon and Sharma2010) for one-dimensional mean flows and thus it only depends on the axial mean $\bar{u}(r)$ ; the deviation velocity does not appear in the resolvent, but it does appear in the mean flow equation. Also note that the equation of continuity enforces the condition that the mean profile of wall-normal velocity $v_{0}$ is zero in all cases. Hence, the modification of the mean profile $\bar{u}_{0}$ is sufficient to analyse the dynamics of the flow. We recall that McKeon et al. (Reference Mckeon, Sharma and Jacobi2013) used a comparable decomposition in order to assess the effect of a synthetic large-scale motion on the flow dynamics.

Following the analysis of McKeon & Sharma (Reference Mckeon and Sharma2010), a SVD of the resolvent operator

delivers an input–output amplification relation between response modes ${\bf\psi}_{k,n,{\it\omega},m}$ and forcing modes ${\bf\phi}_{k,n,{\it\omega},m}$ through the magnitude of the corresponding singular value ${\it\sigma}_{k,n,{\it\omega},m}$ . Here the subscript $m$ is an index that ranks singular values from largest to smallest. The nonlinear terms $\boldsymbol{g}_{k,n,{\it\omega}}$ can be decomposed as a sum of forcing modes to relate the amplification mechanisms to the velocity fields,

where the unknown forcing coefficients ${\it\chi}_{n,{\it\omega},n}$ represent the unknown mode interactions and Reynolds stresses. The decomposition of the fluctuating velocity field is then constructed as a weighted sum of response modes

in which the low-rank nature of the resolvent, ${\it\sigma}_{k,n,{\it\omega},1}\gg {\it\sigma}_{k,n,{\it\omega},2}$ , is exploited (McKeon & Sharma Reference Mckeon and Sharma2010; Sharma & McKeon Reference Sharma and Mckeon2013; Luhar et al. Reference Luhar, Sharma and Mckeon2014). The low-rank nature of the resolvent operator has been computationally confirmed for each of the base flows considered but omitted for brevity. We also note that this assumption is not required; in principle any number of singular response modes can be taken into account to improve the above resolvent decomposition.

A numerical method similar to that developed by McKeon & Sharma (Reference Mckeon and Sharma2010) is employed for the discretization of the resolvent operator $\mathscr{H}_{k,n,{\it\omega}}$ . Following the approach of Meseguer & Trefethen (Reference Meseguer and Trefethen2003), wall-normal derivatives are computed using Chebychev differentiation matrices, properly modified to avoid the axis singularity. Note that instead of projecting the velocity into a divergence-free basis, here the divergence-free velocity fields are enforced by adding the continuity equation as an additional column and row in the discretized resolvent (Luhar et al. Reference Luhar, Sharma and Mckeon2014). The velocity boundary conditions at the wall are zero Dirichlet. The velocity profile inputs of the four cases subject to study were shown in figure 9.

6.2 Fourier analyses and DMD

It may be observed in (6.10) that the energy associated with each Fourier mode $\boldsymbol{u}_{k,n,{\it\omega}}$ is proportional to its weighting ${\it\chi}_{k,n,{\it\omega},1}{\it\sigma}_{k,n,{\it\omega},1}$ , under the rank-one assumption. The resolvent analysis yields the amplification properties in $(k,n,{\it\omega})$ but it does not provide information on the amplitude of nonlinear forcing ${\it\chi}_{k,n,{\it\omega},1}$ .

As briefly exposed in § 1, a snapshot-based DMD analysis (Rowley et al. Reference Rowley, Mezić, Bagheri, Schlatter and Henningson2009; Schmid Reference Schmid2010) is carried out on the DNS data in order to unveil the unknown amplitudes of the nonlinear forcing terms ${\it\chi}_{k,n,{\it\omega},1}$ . DMD obtains the most energetic flow structures, i.e. the set of wavenumber/frequencies $(k,n,{\it\omega})$ corresponding to the maximum product of amplitude forcing and amplification. As shown by Chen, Tu & Rowley (Reference Chen, Tu and Rowley2012) and Mezić (Reference Mezić2013), the results from a DMD analysis of statistically steady flows such as those considered are equivalent to a discrete Fourier transform of the DNS data once the time mean is subtracted. This is confirmed if values of decay/growth of the DMD eigenvalues are close to zero. Hence, the DMD modes obtained are marginally stable, and can be considered Fourier modes. The norms of these modes indicates the energy corresponding to each set of wavenumber/frequencies and can unveil the value of the unknown forcing coefficients ${\it\chi}_{k,n,{\it\omega},1}$ (Gómez et al. Reference Gómez, Blackburn, Rudman, Mckeon and Sharma2015).

To avoid additional post-processing of the DNS data in the DMD analysis, we directly employ two-dimensional snapshots with Fourier expansions in the azimuthal direction, obtained from the DNS based on (2.8). Given an azimuthal wavenumber $n$ , two matrices of snapshots equispaced in time are constructed as

The size of these snapshot matrices is $N_{r}N_{x}\times N_{s}-1$ with $N_{s}$ being the number of snapshots employed. DMD consists of the inspection of the properties of the linear operator $\boldsymbol{A}$ that relates the two snapshot matrices as

the linear operator $\boldsymbol{A}$ is related to Koopman theory. To obtain the eigenvectors of $\boldsymbol{A}$ , we employ the DMD algorithm based on the SVD of the snapshot matrices developed by Schmid (Reference Schmid2010). This algorithm circumvents any rank-deficiency in the snapshot matrices and it provides a small set of eigenvectors ordered by energy norm. The dataset consist of 1200 DNS snapshots equispaced over $O(40)$ wash-out times $L/u_{0}(R)$ .

We anticipate that because of the discretization employed in the DMD analysis, the Fourier modes obtained are two-dimensional and can contain multiple axial wavenumbers $k$ . A spatial Fourier transform in the axial direction can be carried out in order to identify the dominant axial wavenumber $k$ of a DMD mode.

In the following, we employ the resolvent analysis and DMD to address how the most amplified and energetic flow structures are manipulated by the transpiration. This is the first step in establishing a relation between the changes in flow structures and drag reduction or increase mechanisms.

Figure 12.

Distribution of amplification $\log _{10}({\it\sigma}_{k,6,{\it\omega},1})$ and energy peak frequencies: (a) reference, smooth pipe; (b) case I, large drag decrease $(A^{+},k_{c})=(10,10)$ ; (c) case II, small drag decrease $(A^{+},k_{c})=(0.7,10)$ ; (d) case III, drag increase $(A^{+},k_{c})=(2,2)$ . Symbols denote peak frequencies corresponding to the most energetic two-dimensional DMD modes with $k=0,1,2,3$ dominant wavenumber. Dashed lines indicate the wavespeed corresponding to the centreline velocity $\bar{u}^{c}(R)$ for each case.

Figure 13.

In each of the four blocks, the upper diagram represents each transpiration configuration. Below, isosurfaces of $50\,\%$ of the maximum/minimum streamwise velocity are represented corresponding to (top) the DMD mode with dominant wavenumbers $k=1$ and $n=6$ (red squares in figure 12), (middle) the DMD modes axially filtered at $k=1$ and (bottom) resolvent mode associated with $k=1$ and $n=6$ at the same frequency: (a) reference, smooth pipe; (b) case I, large drag decrease $(A^{+},k_{c})=(10,10)$ ; (c) case II, small drag decrease $(A^{+},k_{c})=(0.7,10)$ ; (d) case III, drag increase $(A^{+},k_{c})=(2,2)$ .

6.3 Amplification and energy

We focus on the effect of transpiration on large-scale motions. Hence, the broad parameter space $(k,n,{\it\omega})$ is reduced by taking into consideration findings from the literature. As discussed in detail by Sharma & McKeon (Reference Sharma and Mckeon2013), VLSMs in pipe flows can be represented with resolvent modes of lengths scales $(k,n)=(1,6)$ and with a convective velocity $c=2/3$ of the centreline streamwise velocity. This representation is based on the work of Monty et al. (Reference Monty, Stewart, Williams and Chong2007) and Bailey & Smits (Reference Bailey and Smits2010), which experimentally investigated the spanwise length scale associated with the VLSM and found to be of the order of the outer length scale, $n=6$ . Hence, we consider only the azimuthal wavenumber $n=6$ .

Reference values of the amplification are provided in contours in figure 12(a), which show the distribution of resolvent amplification $\log _{10}({\it\sigma}_{k,6,{\it\omega},1})$ for the smooth pipe in a continuum set of $(k,{\it\omega})$ wavenumbers for $n=6$ . We observe a narrow band of high amplification caused by a critical-layer mechanism (McKeon & Sharma Reference Mckeon and Sharma2010). This critical-layer mechanism can be described by examining the resolvent operator. The diagonal terms of the inverse of the resolvent matrix in (6.6) read

thus, for a given value of the Laplacian, there is a large amplification if the wavespeed $c={\it\omega}/k$ matches the mean streamwise velocity, i.e.  $c=u_{0}$ . That means that flow structures that travel at the local mean velocity create high amplification. Furthermore, we note that straight lines that pass through the origin in figure 12 correspond to constant wavespeed values. The wavespeed corresponding to the centreline velocity is indicated by a dashed line.

Turning first to figure 12(a), the symbols represent the dominant wavenumber and the frequency corresponding to the four most energetic DMD modes at $n=6$ . The value of the energy corresponding to each the four modes is similar and omitted. As explained by Gómez et al. (Reference Gómez, Blackburn, Rudman, Mckeon, Luhar, Moarref and Sharma2014), a sparsity is observed in energy as a consequence of the critical layer mechanism in a finite length periodic domain; only structures with an integer axial wavenumber, $k_{i}=1,2,\ldots$ can exist in the flow because of the finite length periodic domain with length $L=4{\rm\pi}R$ . This fact creates a corresponding sparsity in frequency. For each integer axial wavenumber $k_{i}$ there is a frequency ${\it\omega}_{i}$ for which the critical layer mechanism occurs, i.e.  ${\it\omega}_{i}=k_{i}c=k_{i}u_{0}(r_{c})$ , with $r_{c}$ being the wall-normal location of the critical layer. This energy sparsity behaviour is clearly observed in the reference case, as shown in figure 12(a), in which the peak frequencies are approximately integral multiples of the fundamental. Thus, they are approximately aligned in a constant wavespeed line.

We also observe that, for a given $k$ , the frequencies corresponding to the peaks of energy differ from the most amplified frequency. As observed by Gómez et al. (Reference Gómez, Blackburn, Rudman, Mckeon and Sharma2015), this is related to the major role that the non-linear forcing ${\it\chi}_{n,{\it\omega},n}$ maintaining the turbulence plays in the resolvent decomposition (6.10). Nevertheless, we observe that the frequency corresponding to the most energetic modes can be found in the proximity of the high amplification band.

Furthermore, the fact there is only one narrow band of amplification indicates a unique critical layer and that each frequency corresponds to only one wavenumber. This result is confirmed through the similar features exhibited by the most energetically relevant flow structures, arising from DMD of the DNS data, and the resolvent modes associated with the same frequency and wavenumber. For instance, figure 13(a) presents a comparison of the DMD and resolvent mode corresponding to $k=1$ (red square in figure 12(a)). We observe that both DMD and resolvent modes present a unique dominant wavenumber $k=1$ with a similar wall-normal location of its maximum/minimum velocity. The DMD analysis makes use of two-dimensional DNS snapshots based on the Fourier decomposition in (2.8) and so it permits multiple axial wavenumbers. Hence, the DMD modes are not constrained to one unique dominant axial wavenumber. This does not apply to the resolvent modes, which only admit one wavenumber $k$ by construction of the model. This uniqueness of a dominant streamwise wavenumber in the most energetic flow structures of a smooth pipe flow has also been observed in the work of Gómez et al. (Reference Gómez, Blackburn, Rudman, Mckeon, Luhar, Moarref and Sharma2014). In addition, figure 13(a) presents DMD modes axially filtered to $k=1$ to ease comparisons with the resolvent modes. We consider it most appropriate to show the axial velocity because it is the energetically dominant velocity component in these flow structures. In addition, the flow structures corresponding to positive and negative azimuthal wavenumber $n=\pm 6$ have been summed; a single azimuthal wavenumber necessarily corresponds to a helical shape.

Figure 12(b) represents the amplification results of case I, corresponding to large drag decrease with $(A^{+},k_{c})=(10,10)$ . The effect of transpiration on the flow dynamics is significant; two constant-wavespeed rays of high amplification may be observed. One of the wavespeeds is similar to that corresponding the critical layer in the reference case while the second is much faster, almost coincident with the centreline velocity. Also the amplification in the area between these two constant wavespeed lines is increased with respect to the reference case. Hence, multiple axial wavenumber $k$ could be amplified at each frequency. This is confirmed in figure 13(b), which shows that the DMD mode features waviness corresponding to multiple wavenumbers. Although the dominant axial wavenumber $k=1$ can be visually identified, other wavenumbers corresponding to interactions of this $k=1$ with the transpiration wavenumber $k_{c}=10$ are also observed. The waviness in figure 13(b) is steady; it does not travel with the $k=1$ flow structure. This has been confirmed via animations of the flow structures and it was expected as a result of the steady transpiration.

As mentioned before, the resolvent modes arise from a one-dimensional model based on a $(k,n,{\it\omega})$ Fourier decomposition hence they only contain one axial wavenumber $k$ . As such, a single resolvent mode cannot directly replicate the waviness. However, it provides a description of the dominant wavenumber flow structure. Nevertheless, the axially filtered DMD mode in figure 13(b) presents a similar structure to the resolvent mode. Similarly to the reference case, the wavespeed based on the dominant wavenumber of the DMD modes are aligned in a constant wavespeed line near the high-amplification regions, as seen in figure 12(b).

Figure 12(c) shows that the small transpiration $(A^{+},k_{c})=(0.7,10)$ does not have a significant influence on the flow dynamics. The main difference with respect to the reference case is that the amplification line is slightly broader in this case. Consistent with this result, the DMD mode in figure 13(c) shows a tiny waviness corresponding to other wavenumbers. Similarly to the previous cases, the resolvent mode replicates the dominant axial wavenumber flow structure.

The amplification and energy results corresponding to the drag-increase case $(A^{+},k_{c})=(2,2)$ are shown in figure 12(d). We observe that, in agreement with the decrease of bulk velocity, the transpiration slows down the flow dynamics; the wavespeed corresponding to the critical layer is slower than in previous cases, i.e. a steeper dashed line in figure 12(d). In addition, the critical layer is broad in contrast to the reference case, indicating that multiple axial wavenumbers could be excited. As such, the DMD mode in figure 13(d) shows a dominant axial wavenumber $k=1$ which interacts with others in order to produce two steady localized areas of fluctuating velocity. These clusters are located slightly after the two blowing sections, which corresponding to the high-velocity areas in figure 10(d). As in previous cases, the resolvent mode at $k=1$ captures the same dynamics of the filtered DMD mode.