1 Introduction
In recent years, many attempts have been made to control fluid flows in order to reduce drag and improve energy efficiency. In the absence of wave drag, the total drag consists of pressure drag and skin friction. To reduce the pressure drag, extensive shape optimization has been performed. This can be seen, for instance, in the historical evolution of the nose shape of bullet trains. In contrast, there are few practical control techniques to reduce skin friction. One notable exception is the polymer injection technique used in the Trans Alaska Pipeline System. While polymer injection has been successful for internal flows with liquids, there are few practical friction reduction techniques for external air flows. This fundamental limitation is especially important given the fact that skin friction contributes significantly to total drag (
${\geqslant}50\,\%$
) for commercial aircraft (Gad-el-Hak Reference Gad-el-Hak1994, Reference Gad-el-Hak2000). As a result, skin friction reduction has been, and continues to be, the subject of extensive research (Kim Reference Kim2003; Kasagi, Suzuki & Fukagata Reference Kasagi, Suzuki and Fukagata2009; Abdulbari et al.
Reference Abdulbari, Yunus, Abdurahman and Charles2013).
Skin friction reduction techniques can be broadly classified into active control, which requires energy input for actuation, and passive control, which involves the use of textured surfaces such as riblets (e.g. Walsh & Weinstein Reference Walsh and Weinstein1979). Active control can be further classified into predetermined control and feedback control, depending on whether the actuation is based on real-time sensor information or determined a priori. Examples of predetermined control include wall oscillation as reviewed, e.g. by Karniadakis & Choi (Reference Karniadakis and Choi2003) and Quadrio (Reference Quadrio2011), travelling-wave-like blowing and suction (Min et al. Reference Min, Kang, Speyer and Kim2006; Lieu, Moarref & Jovanović Reference Lieu, Moarref and Jovanović2010; Mamori, Fukagata & Hoepffner Reference Mamori, Fukagata and Hoepffner2010; Moarref & Jovanović Reference Moarref and Jovanović2010; Mamori, Iwamoto & Murata Reference Mamori, Iwamoto and Murata2014) and travelling-wave-like wall deformation (Hoepffner & Fukagata Reference Hoepffner and Fukagata2009; Nakanishi, Mamori & Fukagata Reference Nakanishi, Mamori and Fukagata2012; Tomiyama & Fukagata Reference Tomiyama and Fukagata2013). Extensive studies have been conducted on the efficacy and drag reduction mechanisms of these predetermined control techniques. For further detail, readers are referred to the references provided above.
1.1 Feedback control for drag reduction
Feedback flow controls have also been studied extensively in the past twenty years. The simplest feedback control scheme is the opposition control technique proposed by Choi, Moin & Kim (Reference Choi, Moin and Kim1994), in which blowing and suction are generated at the wall to counteract the wall-normal velocity component detected at a virtual detection plane above the wall. Choi et al. (Reference Choi, Moin and Kim1994) achieved approximately 25 % drag reduction using opposition control in their direct numerical simulation (DNS) of low Reynolds number turbulent channel flow. The performance of opposition control has been studied using DNS under a range of different conditions, e.g. at moderate Reynolds numbers (Iwamoto, Suzuki & Kasagi Reference Iwamoto, Suzuki and Kasagi2002) for various blowing and suction amplitudes (Chung & Talha Reference Chung and Talha2011) and in pipe flow (Fukagata & Kasagi Reference Fukagata and Kasagi2003). Motivated by the simulation results above, experimental attempts at feedback control have also been made (Kasagi et al.
Reference Kasagi, Suzuki and Fukagata2009). For instance, Yoshino, Suzuki & Kasagi (Reference Yoshino, Suzuki and Kasagi2008) constructed an experimental feedback control system, in which local wall deformations were generated to suppress streamwise vortices based on the streamwise wall shear stress signal. This technique succeeded in reducing drag by
$6\pm 3$
% after taking measurement uncertainty into account.
Although conceptually simple, opposition control is very difficult to put into practice because it requires sensors located above the wall. To overcome this problem, Lee, Kim & Choi (Reference Lee, Kim and Choi1998) proposed theoretical control laws based on suboptimal control theory, which require information measurable at the wall alone, i.e. the wall pressure, the spanwise wall shear stress or the streamwise wall shear stress. DNS conducted by Lee et al. (Reference Lee, Kim and Choi1998) showed a 16 %–22 % drag reduction when the wall pressure or the spanwise shear stress were used as the input signal or control objective. However, drag increased when the streamwise wall shear stress was used. Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) confirmed the results obtained by Lee et al. (Reference Lee, Kim and Choi1998), and found that the efficacy of control deteriorates slightly as the Reynolds number increases. Fukagata & Kasagi (Reference Fukagata and Kasagi2004) extended these suboptimal control techniques by setting a cost function based on the near-wall Reynolds shear stress, which is directly related to the friction drag (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002), and succeeded in reducing drag using only the streamwise wall shear stress as a sensor input. We will introduce the details of these suboptimal control laws in § 2.2.
1.2 Coherent structures in wall turbulence
Many of the active control techniques mentioned above target the suppression of coherent structures known to be dynamically and energetically important in wall-bounded turbulent flows. Since the pioneering experimental work by Kline et al. (Reference Kline, Reynolds, Schraub and Runstadl1967) and DNS by Kim, Moin & Moser (Reference Kim, Moin and Moser1987), such coherent structures have been studied extensively in both experiments and numerical simulations (Robinson Reference Robinson1991; Adrian Reference Adrian2007).
It is widely accepted that, at low Reynolds number, wall turbulence is dominated by the streaks and quasi-streamwise vortices characteristic of the near-wall cycle, which have streamwise length and spanwise spacing of approximately
$10^{3}$
and
$10^{2}$
wall units, respectively. The generation of these near-wall coherent structures is approximately periodic, as explained in the cyclic models proposed by Hamilton, Kim & Waleffe (Reference Hamilton, Kim and Waleffe1995), Waleffe (Reference Waleffe1997) and Schoppa & Hussain (Reference Schoppa and Hussain2002). Kawahara & Kida (Reference Kawahara and Kida2001) have also visualized this periodic process in a low Reynolds number turbulent Couette flow.
Recent efforts show that, as Reynolds number increases, larger coherent structures emerge in the logarithmic and outer regions of the flow (e.g. the so called very-large-scale and large-scale motions) and become increasingly important from an energetic and dynamic point of view (see e.g. Hutchins & Marusic Reference Hutchins and Marusic2007a ; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009; Marusic, Mathis & Hutchins Reference Marusic, Mathis and Hutchins2010; Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011). For instance, Hutchins & Marusic (Reference Hutchins and Marusic2007b ) show that the peak magnitude of the streamwise turbulence intensity increases with the Reynolds number due to the influence of the very-large-scale motions (VLSMs). Further, the VLSMs also appear to modulate the amplitude of the near-wall cycle (Marusic et al. Reference Marusic, Mathis and Hutchins2010). As such, flow control at higher Reynolds number must consider the effect of these larger-scale coherent structures as well. For more information regarding the scaling and structure of VLSMs, readers are referred to Smits et al. (Reference Smits, McKeon and Marusic2011).
Over the past two decades, it has become increasingly clear that linear processes play an important role in dictating the dynamics of these coherent structures, and hence, in determining the efficacy of any control technique. Kim & Lim (Reference Kim and Lim2000) studied the role of linear and nonlinear processes in a turbulent channel flow by carrying out numerical experiments that eliminated either the linear or nonlinear coupling term. These experiments demonstrated that the near-wall turbulence decays without the presence of the linear coupling term (which enhances non-normality of the linearized Navier–Stokes system). At the same time, the near-wall streamwise vortices could not be reproduced with the proper length scales without the nonlinear convective terms. Thus, the formation of the near-wall structures requires the nonlinear term, but the maintenance of these structures relies on linear interactions. Based on these insights, Lim & Kim (Reference Lim and Kim2004) conducted a singular value decomposition (SVD) analysis of the linearized Navier–Stokes system, modified to account for opposition control (Choi et al. Reference Choi, Moin and Kim1994), and obtained results similar to those found in DNS or large eddy simulation (LES). These results, among others, imply that analyses based on the linearized Navier–Stokes equation give useful insights, even if they cannot entirely reproduce the coherent structures found in turbulent flows.
Despite this limitation, such linearized analyses offer significant advantages over DNS in the preliminary design and development of control methods as well as understanding the underlying mechanisms due to their low computational expense. This is especially true at higher Reynolds numbers, for which DNS can be prohibitively expensive.
1.3 Resolvent analysis and the objective of the present study
The resolvent formulation proposed recently by McKeon & Sharma (Reference McKeon and Sharma2010) is conceptually similar to the SVD analysis of Lim & Kim (Reference Lim and Kim2004). However, it has some important differences in that the linearized system is continuously forced by the nonlinear term, and that it enables an evaluation of control in wavenumber–frequency space (i.e. spatio-temporal modes) at low computational cost. More specifically, the resolvent analysis treats the Fourier transformed Navier–Stokes equations as a forcing–response system: the nonlinear convective terms are considered as the forcing (input) to the system and the turbulent velocity is the response (output). The turbulent velocity and pressure fields are expressed as a linear superposition of propagating Fourier modes, identified via a gain-based decomposition (i.e. SVD) of the forcing–response transfer function: the resolvent operator. This decomposition enables targeted analyses of the effects of control on spatio-temporal modes resembling important coherent structures, such as the near-wall (NW) re-generation cycle and VLSMs. Details of the resolvent analysis are provided in § 2.1 below.
Previous studies have demonstrated that the resolvent formulation captures many of the key statistical and structural features of wall turbulence (see e.g. McKeon, Jacobi & Sharma Reference McKeon, Jacobi and Sharma2013; Moarref et al. Reference Moarref, Sharma, Tropp and McKeon2013; Sharma & McKeon Reference Sharma and McKeon2013). In addition, the resolvent formulation is also emerging as a powerful tool for the analysis of specific control techniques for skin friction reduction. For example, Luhar, Sharma & McKeon (Reference Luhar, Sharma and McKeon2014) investigated the effect of opposition control in turbulent pipe flow, while Luhar, Sharma & McKeon (Reference Luhar, Sharma and McKeon2015) and Luhar, Sharma & McKeon (Reference Luhar, Sharma and McKeon2016) studied the effect of compliant surfaces in turbulent channel flow. These studies demonstrated that the resolvent analysis is able to reproduce trends observed in DNS and LES with minimal computation (although there are quantitative differences).
The present work is an extension of the study carried out by Luhar et al. (Reference Luhar, Sharma and McKeon2014): we investigate the effect of suboptimal control on coherent structures. The remainder of this paper is organized as follows. Outlines of the resolvent analysis and the suboptimal control theory proposed by Lee et al. (Reference Lee, Kim and Choi1998) are presented in § 2. The effect of suboptimal control on modes resembling the NW cycle and VLSMs is considered in § 3. Further discussion is presented in § 4. In particular, we compare the present results with those obtained in previous DNS. We also compare and contrast the effects of suboptimal control and opposition control within the resolvent framework. Finally, conclusions are drawn in § 5.
2 Approach
2.1 Resolvent analysis
In this section, we briefly review the resolvent formulation proposed by McKeon & Sharma (Reference McKeon and Sharma2010) and the procedure used by Luhar et al. (Reference Luhar, Sharma and McKeon2015). The problem under consideration is a fully developed incompressible turbulent channel flow governed by the continuity equation and Navier–Stokes equation, i.e.
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D735}p-\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+\frac{1}{Re_{\unicode[STIX]{x1D70F}}}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}, & \displaystyle\end{eqnarray}$$
where
$\boldsymbol{u}=[u~v~w]^{\text{T}}$
represents the velocity with
$u$
,
$v$
and
$w$
being the streamwise (
$x$
), wall-normal (
$y$
) and spanwise (
$z$
) components,
$p$
is the pressure,
$t$
is the time and
$Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}^{\sharp }h^{\sharp }/\unicode[STIX]{x1D708}^{\sharp }$
is the friction Reynolds number, Here, the quantities are made dimensionless using the channel half-width
$h^{\sharp }$
and the friction velocity
$u_{\unicode[STIX]{x1D70F}}^{\sharp }$
. Note that, a superscript
$\sharp$
represents dimensional variables and a superscript
$+$
is used to denote normalization with respect to
$u_{\unicode[STIX]{x1D70F}}^{\sharp }$
and the kinematic viscosity
$\unicode[STIX]{x1D708}^{\sharp }$
.
For a fully developed turbulent channel flow, the velocity and pressure fields can be expressed as a superposition of Fourier modes with streamwise wavenumber
$k_{x}$
, spanwise wavenumber
$k_{z}$
and temporal frequency
$\unicode[STIX]{x1D714}$
:
$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{u}(x,y,z,t)\\ p(x,y,z,t)\end{array}\right]=\int \int \int _{-\infty }^{\infty }\left[\begin{array}{@{}c@{}}\boldsymbol{u}_{\boldsymbol{k}}(y)\\ p_{\boldsymbol{k}}(y)\end{array}\right]\text{e}^{\text{i}(k_{x}x+k_{z}z-\unicode[STIX]{x1D714}t)}\text{d}k_{x}\,\text{d}k_{z}\,\text{d}\unicode[STIX]{x1D714}, & & \displaystyle\end{eqnarray}$$
where
$\text{i}=(-1)^{1/2}$
and
$[\boldsymbol{u}_{\boldsymbol{k}}p_{\boldsymbol{k}}]^{\text{T}}=[u_{\boldsymbol{k}}v_{\boldsymbol{k}}w_{\boldsymbol{k}}p_{\boldsymbol{k}}]^{\text{T}}$
are Fourier coefficients for the velocity and pressure fields, which vary in the non-homogeneous
$y$
direction. Each wavenumber–frequency combination
$\boldsymbol{k}=(k_{x},k_{z},\unicode[STIX]{x1D714})$
(or wavenumber–propagating speed combination
$\boldsymbol{k}=(k_{x},k_{z},c)$
) indicates a flow structure with streamwise wavelength
$\unicode[STIX]{x1D706}_{x}=2\unicode[STIX]{x03C0}/k_{x}$
and spanwise wavelength
$\unicode[STIX]{x1D706}_{z}=2\unicode[STIX]{x03C0}/k_{z}$
propagating downstream at speed
$c=\unicode[STIX]{x1D714}/k_{x}$
.
Based on this Fourier transformation, at each
$\boldsymbol{k}$
, we can express the Navier–Stokes equations for turbulent channel flow as
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{k}}\boldsymbol{\cdot }\boldsymbol{u}_{\boldsymbol{k}}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle (-\text{i}\unicode[STIX]{x1D714}+\text{i}k_{x}U)\boldsymbol{u}_{\boldsymbol{k}}+(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)v_{\boldsymbol{k}}\boldsymbol{e}_{\boldsymbol{x}}+\unicode[STIX]{x1D735}p_{\boldsymbol{k}}-Re_{\unicode[STIX]{x1D70F}}^{-1}\unicode[STIX]{x1D6FB}_{\boldsymbol{k}}^{2}\boldsymbol{u}_{\boldsymbol{k}}=\boldsymbol{f}_{\boldsymbol{k}}. & \displaystyle\end{eqnarray}$$
Here,
$\unicode[STIX]{x1D735}_{\boldsymbol{k}}=[\text{i}k_{x}\,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y\,\text{i}k_{z}]^{\text{T}}$
and
$\unicode[STIX]{x1D6FB}_{\boldsymbol{k}}^{\text{T}}$
represent the gradient and divergence operators, and
$\boldsymbol{f}_{\boldsymbol{k}}=(-u\boldsymbol{\cdot }\unicode[STIX]{x1D735}u)_{\boldsymbol{k}}$
represents the Fourier transformed nonlinear term.
$U(y)$
represents the mean velocity profile.
These equations can be rearranged into the following forcing–response (or input–output) relationship:
$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\boldsymbol{u}_{\boldsymbol{k}}\\ p_{\boldsymbol{k}}\end{array}\right] & = & \displaystyle \left(-\text{i}\unicode[STIX]{x1D714}\left[\begin{array}{@{}cc@{}}\boldsymbol{I} & \\ & 0\end{array}\right]-\left[\begin{array}{@{}cc@{}}\boldsymbol{L}_{\boldsymbol{k}} & -\unicode[STIX]{x1D735}_{\boldsymbol{k}}\\ \unicode[STIX]{x1D6FB}_{\boldsymbol{k}}^{\text{T}} & 0\end{array}\right]\right)^{-1}\left[\begin{array}{@{}c@{}}\boldsymbol{I}\\ 0\end{array}\right]\boldsymbol{f}_{\boldsymbol{k}}\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & = & \displaystyle \underbrace{\boldsymbol{H}_{\boldsymbol{k}}}_{resolvent\,operator}\boldsymbol{f}_{\boldsymbol{k}},\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{L}_{\boldsymbol{k}}=\left[\begin{array}{@{}ccc@{}}-\text{i}k_{x}U+Re_{\unicode[STIX]{x1D70F}}^{-1}\unicode[STIX]{x1D6FB}_{\boldsymbol{k}}^{2} & -(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y) & 0\\ 0 & -\text{i}k_{x}U+Re_{\unicode[STIX]{x1D70F}}^{-1}\unicode[STIX]{x1D6FB}_{\boldsymbol{k}}^{2} & 0\\ 0 & 0 & -\text{i}k_{x}U+Re_{\unicode[STIX]{x1D70F}}^{-1}\unicode[STIX]{x1D6FB}_{\boldsymbol{k}}^{2}\end{array}\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{\boldsymbol{k}}^{2}=-k_{x}^{2}-(\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}y^{2})-k_{z}^{2}. & \displaystyle\end{eqnarray}$$
Here, the nonlinear term
$\boldsymbol{f}_{\boldsymbol{k}}$
is identified as a forcing term to the linear Navier–Stokes system and the resolvent operator
$\boldsymbol{H}_{\boldsymbol{k}}$
, which depends on the linear operator
$\boldsymbol{L}_{\boldsymbol{k}}$
, translates this forcing into a velocity and pressure response:
$[\boldsymbol{u}_{\boldsymbol{k}}~p_{\boldsymbol{k}}]^{\text{T}}$
.
A conceptual illustration of the resolvent analysis.
A singular value decomposition (SVD) of the resolvent operator
$\boldsymbol{H}_{\boldsymbol{k}}$
identifies an ordered set of orthonormal forcing and response modes for each wavenumber–speed combination
$\boldsymbol{k}$
under an
$L^{2}$
energy norm:
$$\begin{eqnarray}\displaystyle \boldsymbol{H}_{\boldsymbol{k}}=\mathop{\sum }_{m}\boldsymbol{u}_{\boldsymbol{k},m}\unicode[STIX]{x1D70E}_{\boldsymbol{k},m}\,{\boldsymbol{f}_{\boldsymbol{k},m}}^{\ast }. & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D70E}_{\boldsymbol{k},1}\gg \unicode[STIX]{x1D70E}_{\boldsymbol{k},2}>\cdots >\unicode[STIX]{x1D70E}_{\boldsymbol{k},m}>\cdots >0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \int _{-1}^{1}\boldsymbol{f}_{\boldsymbol{k},\,l}^{\ast }\,\boldsymbol{f}_{\boldsymbol{k},m}\,\text{d}y=\unicode[STIX]{x1D6FF}_{lm}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \int _{-1}^{1}\boldsymbol{u}_{\boldsymbol{k},l}^{\ast }\boldsymbol{u}_{\boldsymbol{k},m}\,\text{d}y=\unicode[STIX]{x1D6FF}_{lm}. & \displaystyle\end{eqnarray}$$
Here,
$\unicode[STIX]{x1D6FF}$
denotes the Kronecker delta and a superscript of
$\ast$
denotes the complex conjugate.
From (2.10)–(2.13), it is evident that the
$m$
th forcing mode yields the
$m$
th velocity–pressure response mode, amplified by the singular value
$\unicode[STIX]{x1D70E}_{\boldsymbol{k},m}$
, as conceptually illustrated in figure 1. For instance, forcing
$\boldsymbol{f}_{\boldsymbol{k},1}$
leads to a response
$\unicode[STIX]{x1D70E}_{\boldsymbol{k},1}[\boldsymbol{u}_{\boldsymbol{k},1}~p_{\boldsymbol{k},1}]^{\text{T}}$
. It is worth noting that for
$\boldsymbol{k}$
combinations relevant to real flows the resolvent operator is often low rank: the first singular value tends to be much larger than the rest (2.11). In other words, the first singular mode (rank-1 mode) is expected to dominate the turbulent flow field. Indeed, previous studies (McKeon & Sharma Reference McKeon and Sharma2010; McKeon et al.
Reference McKeon, Jacobi and Sharma2013; Moarref et al.
Reference Moarref, Sharma, Tropp and McKeon2013) have shown that models based on this rank-1 approximation yield predictions for turbulent statistics and coherent structure consistent with observations. Following these findings, Luhar et al. (Reference Luhar, Sharma and McKeon2014) and Luhar et al. (Reference Luhar, Sharma and McKeon2015) consider only these rank-1 modes to analyse the effect of control on wall turbulence. Similarly, we only consider rank-1 modes and singular values at each
$\boldsymbol{k}$
in the remainder of this manuscript:
$\displaystyle [\boldsymbol{u}_{\boldsymbol{k}}~p_{\boldsymbol{k}}]^{\text{T}}=[\boldsymbol{u}_{\boldsymbol{k},1}~p_{\boldsymbol{k},1}]^{\text{T}}$
and
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}=\unicode[STIX]{x1D70E}_{\boldsymbol{k},1}$
.
Note that the resolvent analysis procedure presented in (2.6)–(2.13) does not consider just the linearized Navier–Stokes equations. The analysis assumes that the nonlinearity supports the base flow (i.e. the mean velocity profile in the resolvent operator) via the mean Reynolds stress, and forces the fluctuating velocity and pressure responses via the time-varying Reynolds stresses. Consistent with the findings of Kim & Lim (Reference Kim and Lim2000), this means that a complete quantitative description of the turbulent velocity and pressure fields requires knowledge of the nonlinear interaction between modes. However, even without this information, the resolvent framework permits analysis of control techniques on a mode-by-mode basis, in a forcing-normalized sense. This is discussed in greater detail in § 2.3.
Schematic of the suboptimal control method.
2.2 Suboptimal control laws
As shown in figure 2, the opposition control technique proposed by Choi et al. (Reference Choi, Moin and Kim1994) generates blowing and suction at the wall to oppose the wall-normal velocity measured at a ‘virtual detection plane’ located above wall in order suppress the quasi-streamwise vortices associated with the NW cycle. At low Reynolds numbers, this yields more than
$20\,\%$
drag reduction in DNS.
Recognizing that the requirement of sensor information away from the wall may be impractical, Lee et al. (Reference Lee, Kim and Choi1998) sought to develop control laws similar to opposition control that only required sensor information at the wall, i.e. the shear stress or pressure. These control laws were determined by analytical solutions that minimized physically intuitive cost functions for discretized versions of the governing equations. Based on observations that opposition control led to an enhancement of the fluctuating spanwise shear stress at the wall, the following cost function was used for the case where
$(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}y)_{w}$
was used as the sensed quantity,
$$\begin{eqnarray}\displaystyle J_{SP}(\unicode[STIX]{x1D719})=\frac{\ell }{2A\unicode[STIX]{x0394}t}\int _{S}\int _{t}^{t+\unicode[STIX]{x0394}t}\unicode[STIX]{x1D719}^{2}\,\text{d}t\,\text{d}S-\frac{1}{2A\unicode[STIX]{x0394}t}\int _{S}\int _{t}^{t+\unicode[STIX]{x0394}t}\underbrace{\left(\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}y}\right)_{w}^{2}}_{spanwise\,wall{-}shear}\,\text{d}t\,\text{d}S. & & \displaystyle\end{eqnarray}$$
Here,
$\unicode[STIX]{x1D719}$
represents the wall-normal velocity at the wall used as the control input,
$A$
is the area of wall,
$\unicode[STIX]{x0394}t$
is the short time span for optimization and
$\ell$
is the relative cost of the actuation. The derived control law was expressed in the Fourier domain as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{\boldsymbol{k}}=\unicode[STIX]{x1D6FC}\frac{\text{i}k_{z}}{K}\left(\frac{\unicode[STIX]{x2202}w_{\boldsymbol{k}}}{\unicode[STIX]{x2202}y}\right)_{w}, & & \displaystyle\end{eqnarray}$$
where
$K=(k_{x}^{2}+k_{z}^{2})^{1/2}$
and
$\unicode[STIX]{x1D6FC}$
is a coefficient determining the blowing and suction magnitude. Lee et al. (Reference Lee, Kim and Choi1998) carried out DNS of channel flow at
$Re_{\unicode[STIX]{x1D70F}}=110$
, and obtained
$22\,\%$
drag reduction using this control law. For the case where the streamwise wall shear
$(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)_{w}$
was used as the control input, the control law, derived using the same procedure as (2.15), was expressed as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{\boldsymbol{k}}=-\unicode[STIX]{x1D6FC}\frac{\text{i}k_{x}}{K}\left(\frac{\unicode[STIX]{x2202}u_{\boldsymbol{k}}}{\unicode[STIX]{x2202}y}\right)_{w}. & & \displaystyle\end{eqnarray}$$
However, the friction drag was reported to increase with this control law. Note that, for both cases, the magnitude of the blowing and suction (
$\unicode[STIX]{x1D6FC}$
) was tuned to ensure that the root-mean-square value (r.m.s.) of
$\unicode[STIX]{x1D719}$
matched the r.m.s. of the wall-normal velocity at
$y^{+}\approx 10$
.
In this paper, we assess the suboptimal control laws shown in (2.15) and (2.16), which are hereafter denoted as Case SP and Case ST, respectively. The effect of suboptimal control is included in the resolvent operator
$\boldsymbol{H}_{\boldsymbol{k}}$
as a boundary condition for the upper (
$y=+1$
) and lower (
$y=-1$
) walls. The no-slip condition is employed for the uncontrolled case. The complete boundary conditions for the suboptimal control laws at a wavenumber–speed combination
$\boldsymbol{k}$
are expressed as follows:
$$\begin{eqnarray}\displaystyle \text{[Case SP]} & : & \displaystyle ~(u_{\boldsymbol{k}})_{w}=(w_{\boldsymbol{k}})_{w}=0,\quad (v_{\boldsymbol{k}})_{w}=\unicode[STIX]{x1D719}_{\boldsymbol{k}}=\unicode[STIX]{x1D6FC}\frac{\text{i}k_{z}}{K}\left(\frac{\unicode[STIX]{x2202}w_{\boldsymbol{k}}}{\unicode[STIX]{x2202}y}\right)_{w},~~\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \text{[Case ST]} & : & \displaystyle ~(u_{\boldsymbol{k}})_{w}=(w_{\boldsymbol{k}})_{w}=0,\quad (v_{\boldsymbol{k}})_{w}=\unicode[STIX]{x1D719}_{\boldsymbol{k}}=-\unicode[STIX]{x1D6FC}\frac{\text{i}k_{x}}{K}\left(\frac{\unicode[STIX]{x2202}u_{\boldsymbol{k}}}{\unicode[STIX]{x2202}y}\right)_{w}.\,\end{eqnarray}$$
Finally, it is worth mentioning that the blowing and suction magnitude
$\unicode[STIX]{x1D6FC}$
is essentially a length scale (i.e.
$\unicode[STIX]{x1D6FC}^{+}=\unicode[STIX]{x1D6FC}Re_{\unicode[STIX]{x1D70F}}$
).
2.3 Resolvent-based assessment of suboptimal control
As mentioned in §§ 2.1 and 2.2, we follow the procedure used by Luhar et al. (Reference Luhar, Sharma and McKeon2015) to assess how suboptimal control affects resolvent modes that serve as models for the dynamically important coherent structures. In the framework of resolvent mode analysis, there are two primary measures to assess the effectiveness of control. The first measure is the forcing–response gain, i.e. the singular value
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}$
, itself. If the singular value in the controlled case decreases relative to that in the no-control case, the mode is suppressed by the control; if it increases, the mode is amplified. The second measure is the Reynolds shear stress contribution (Luhar et al.
Reference Luhar, Sharma and McKeon2015):
$$\begin{eqnarray}\displaystyle R_{\boldsymbol{k}} & = & \displaystyle \int _{-1}^{1}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{2}~\text{Re}({u_{\boldsymbol{k}}}^{\ast }v_{\boldsymbol{k}})(+y)\,\text{d}y,\end{eqnarray}$$
(where
$\text{Re}(\cdot \cdot )$
denotes the real component), which is based on the Fukagata–Iwamoto–Kasagi (FIK) identity (Fukagata et al.
Reference Fukagata, Iwamoto and Kasagi2002). Note that (2.19) represents a forcing-normalized quantity; it assumes that the velocity response at a wavenumber–propagating speed contribution
$\boldsymbol{k}$
is given by
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}\boldsymbol{u}_{\boldsymbol{k}}$
.
Of course, both these measures neglect the effect of control on the nonlinear forcing term, which arises from mode–mode interactions. As noted in § 1.2, while linear mechanisms are essential for maintaining near-wall turbulence, the nonlinear term plays an important role in the formation of the near-wall cycle (e.g. Kim & Lim Reference Kim and Lim2000). However, due to its conservative nature, the nonlinear term does not dissipate or generate turbulent kinetic energy, but merely serves to transfer energy across spectral space. Turbulent kinetic energy is generated via the action of the vertical fluctuating velocity on the mean shear
$v_{k}(\text{d}U/\text{d}y)$
(e.g. Chernyshenko & Baig Reference Chernyshenko and Baig2005); this is a linear mechanism that is captured by the resolvent operator as shown in (2.6)–(2.9). In other words, energy transfer from the mean flow to the turbulence is reflected in the forcing–response gain singular value
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}$
. Therefore, the conservative nature of the nonlinearity suggests that if a control technique suppresses singular values and Reynolds stress contributions across spectral space, then it is likely to be effective in suppressing turbulence regardless of nonlinear effects.
2.4 Numerical details
The Chebyshev collocation method (Weideman & Reddy Reference Weideman and Reddy2000) is used for discretization in the wall-normal direction. The mean velocity profile
$U(y)$
in the linear operator
$\boldsymbol{L}_{\boldsymbol{k}}$
(in the resolvent operator
$\boldsymbol{H}_{\boldsymbol{k}}$
) is obtained by using the turbulent viscosity model proposed by Reynolds & Tiederman (Reference Reynolds and Tiederman1967). To test whether the exact form of the mean profile led to a substantial difference in results, we also repeated the analysis for a limited subset of modes with uncontrolled and controlled (for Case SP) mean profiles obtained in DNS. The results obtained using the mean profiles from DNS were in good agreement with the results obtained using the mean profile computed using the eddy viscosity model. For example, singular values with and without control changed by less than 6 % for modes resembling the near-wall cycle. These results suggest that the model predictions reported below are not very sensitive to the exact form of the mean velocity profile.
Luhar et al. (Reference Luhar, Sharma and McKeon2015) conducted the resolvent mode analysis in the lower half-channel only to avoid the so-called pairing (Moarref et al. Reference Moarref, Sharma, Tropp and McKeon2013) and to make the computation more efficient. In turn, one must carefully specify the symmetry of each mode. In the present study, in contrast, we compute the whole channel to avoid this complexity. We have confirmed excellent agreement for rank-1 modes (which are of interest in the present work) in the uncontrolled case between the results computed in the whole channel and the half-channel using the user-specified symmetry.
Investigation on the grid number (
$N$
) dependency in the uncontrolled case showed that the relative error of the Reynolds shear stress contribution
$R_{\boldsymbol{k}}$
between the cases with
$N=240~(=2\times 120)$
and
$N=1000~(2\times 500)$
was of the order of
$10^{-4}$
per cent. Further, for the controlled case, the relative difference of
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
between
$N=240$
and
$N=1000$
depends on the blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
: the difference is of the order of
$0.1\,\%$
, approximately
$1\,\%$
and
$2\,\%$
for
$\unicode[STIX]{x1D6FC}^{+}=1$
,
$10$
and 100, respectively. Therefore, we use
$N=240$
and
$N=400$
in the rest of the paper. We use
$N=400$
only when we plot the wall-normal profiles (e.g. figure 4). Note that this is comparable to typical DNS resolutions for
$Re_{\unicode[STIX]{x1D70F}}<O(1000)$
.
3 Results
In the present study, we focus on how control affects resolvent modes resembling the near-wall coherent structure (NW cycle) and VLSMs. As mentioned in § 1.2, the dynamic importance of these structures has been highlighted in many previous studies, which suggests that targeting such modes is central to the development of effective control.
Following prior resolvent-based analyses (McKeon & Sharma Reference McKeon and Sharma2010; McKeon et al.
Reference McKeon, Jacobi and Sharma2013; Sharma & McKeon Reference Sharma and McKeon2013; Luhar et al.
Reference Luhar, Sharma and McKeon2015), we consider the wavenumber–propagating speed combination
$\boldsymbol{k}=(k_{x},k_{z},c^{+})\approx (12,120,10)$
at
$Re_{\unicode[STIX]{x1D70F}}=2000$
or
$\boldsymbol{k}=(k_{x},k_{z},c^{+})\approx (0.66,6.6,10)$
at
$Re_{\unicode[STIX]{x1D70F}}=110$
, corresponding to wavelengths
$\unicode[STIX]{x1D706}_{x}^{+}=2\unicode[STIX]{x03C0}Re_{\unicode[STIX]{x1D70F}}/k_{x}\approx 10^{3}$
and
$\unicode[STIX]{x1D706}_{z}^{+}=2\unicode[STIX]{x03C0}Re_{\unicode[STIX]{x1D70F}}/k_{z}\approx 10^{2}$
, as the modes that represent the near-wall cycle (NW modes). Note that
$k_{z}^{+}>0$
represents oblique waves that propagate in the positive
$z$
direction, while
$k_{z}^{+}<0$
represents oblique waves that propagative in the negative
$z$
direction. Apart from this directionality, both modes are structurally similar due to the spanwise homogeneity of the flow.
To capture the dynamics of VLSMs, we consider the wavenumber–propagating speed combination
$\boldsymbol{k}=(k_{x},k_{z},c^{+})\approx (1,10,16)$
at
$Re_{\unicode[STIX]{x1D70F}}=2000$
. This mode represents flow structures with
$\unicode[STIX]{x1D706}_{x}^{+}\approx 6h^{+}$
,
$\unicode[STIX]{x1D706}_{z}^{+}\approx 0.6h^{+}$
, and a propagation speed approximately two thirds of the centreline velocity. These length and velocity scales were chosen based on previous experimental measurements and modelling results. For example, the boundary layer measurements made by Hutchins & Marusic (Reference Hutchins and Marusic2007a
) showed the presence of a local peak in the streamwise velocity spectrum for structures of length
$\unicode[STIX]{x1D706}_{x}\approx 6h$
near wall-normal location
$y\approx 0.06h$
at
$Re_{\unicode[STIX]{x1D70F}}=1910$
. Resolvent modes tend to localize near their critical layer, which is defined as the location where the mode speed matches the local mean velocity (McKeon & Sharma Reference McKeon and Sharma2010). For propagation speed
$c^{+}=16$
, this corresponds to
$y_{c}^{+}\approx 105$
, or
$y_{c}/h\approx 0.05$
at
$Re_{\unicode[STIX]{x1D70F}}=2000$
.
3.1 Effect of control on the near-wall modes
Relationship between the blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
and the ratio of controlled to uncontrolled cases for NW modes: (a) singular value
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}$
; (b) Reynolds shear stress contribution
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
. Solid lines, Case SP (
$Re_{\unicode[STIX]{x1D70F}}=2000$
); dashed lines, Case ST (
$Re_{\unicode[STIX]{x1D70F}}=2000$
). Triangle, Case SP (
$Re_{\unicode[STIX]{x1D70F}}=110$
); circle, Case ST (
$Re_{\unicode[STIX]{x1D70F}}=110$
).
First, we focus on the NW modes. Figure 3 shows the relationship between the blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
and ratio of controlled to uncontrolled singular values (figure 3
a) and Reynolds shear stress contributions (figure 3
b). Consistent with previous studies (Luhar et al.
Reference Luhar, Sharma and McKeon2014, Reference Luhar, Sharma and McKeon2015), there is close correspondence between the singular value and the Reynolds shear stress contribution. Solid lines in figure 3, which represent Case SP (i.e. control based on spanwise shear) at
$Re_{\unicode[STIX]{x1D70F}}=2000$
, show that the maximum reduction of
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}$
is achieved at
$\unicode[STIX]{x1D6FC}^{+}\approx 2$
(
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}\approx 0.61$
,
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}\approx 0.40$
), maximum reduction of
$R_{\boldsymbol{k}}$
is achieved at
$\unicode[STIX]{x1D6FC}^{+}\approx 4$
(
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}\approx 0.72$
,
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}\approx 0.35$
) and drag increases for
$\unicode[STIX]{x1D6FC}^{+}>10^{1}$
. Case ST (control based on streamwise shear; dashed lines) shows drag reduction too, but there is little dependency on blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
once the effects of control become significant (
$\unicode[STIX]{x1D6FC}^{+}\geqslant 10^{1}$
). Surprisingly, Case ST was reported to increase the drag in DNS at
$Re_{\unicode[STIX]{x1D70F}}=110$
by Lee et al. (Reference Lee, Kim and Choi1998). We will discuss this discrepancy in detail in § 4.1.2.
Note that there is close correspondence between the lines and the markers (triangle or circle) in figure 3, which suggests that the predictions are not very sensitive to Reynolds numbers, at least within the range tested (
$Re_{\unicode[STIX]{x1D70F}}=110{-}2000$
). In other words, the effect of control on the NW coherent structures does not change with increasing
$Re_{\unicode[STIX]{x1D70F}}$
. At first glance, this appears to contradict many previous studies which show that control performance deteriorates with Reynolds number. However, as discussed in later sections (see §§ 4.1.3 and 3.4), the deterioration in performance may be attributed to the fact that as Reynolds number increases, additional regions of spectral space become energetic, where control can have a harmful influence.
For both cases, the ratios
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}$
and
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
approach a constant value as
$\unicode[STIX]{x1D6FC}^{+}$
increases in the figure 3. This is simply because the dynamics associated with the (large) feedback component dominates over the dynamics emerging from the original (uncontrolled) resolvent operator as
$\unicode[STIX]{x1D6FC}^{+}$
increases. In other words, for low
$\unicode[STIX]{x1D6FC}^{+}$
, the blowing and suction interacts with and modifies the natural dynamics of the flow. However, for high
$\unicode[STIX]{x1D6FC}^{+}$
, blowing and suction overwhelms the natural dynamics. The magnitude of
$\unicode[STIX]{x1D6FC}^{+}$
that effectively suppresses NW coherent structures is discussed further in § 4.1.1.
Profiles showing wall-normal variations in structure for resolvent modes resembling the NW cycle at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: (a,c,e,g) amplitude of streamwise velocity (black lines) and wall-normal velocity (grey lines); (b,d,f,h) Reynolds shear stress. Dashed lines, uncontrolled; solid lines, controlled (Case SP). (a,b)
$\unicode[STIX]{x1D6FC}^{+}=0.44$
; (c,d)
$\unicode[STIX]{x1D6FC}^{+}=2.0$
; (e,f)
$\unicode[STIX]{x1D6FC}^{+}=4.0$
; (g,h)
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
. Black horizontal line indicates the location of critical layer,
$y_{c}^{+}\approx 16$
.
Velocity structure for NW modes in the spanwise wall-normal plane at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: (a,c,e,g) wall-normal velocity amplitude; (b,d,f,h) wall-normal and spanwise velocity field. (a,b) no control; (c,d)
$\unicode[STIX]{x1D6FC}^{+}=2.0$
; (e,f)
$\unicode[STIX]{x1D6FC}^{+}=4.0$
; (g,h)
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
. The horizontal dashed lines show the critical layer location,
$y_{c}^{+}\approx 16$
.
Based on the trends observed in figure 3, we focus on values of
$\unicode[STIX]{x1D6FC}^{+}$
for Case SP at
$Re_{\unicode[STIX]{x1D70F}}=2000$
that show salient behaviour (i.e. maximum drag reduction/increment) and consider the effect of control on mode structure. Figure 4 shows the profile of velocity amplitudes and Reynolds shear stress. The effects of control start to appear at
$\unicode[STIX]{x1D6FC}^{+}=0.44$
. For this case, the peak of controlled wall-normal velocity (
$|v_{\boldsymbol{k}c}|$
) is a little larger than that for the uncontrolled case (see figure 4
a,b). At the same time, the well-known ‘virtual wall’ starts to form, which mitigates vertical momentum transport and reduces the effect of linear coupling (Kim & Lim Reference Kim and Lim2000). Here, the presence of virtual wall is indicated by the sudden fall of the magnitude of wall-normal velocity. Maximum suppression of singular value is attained at
$\unicode[STIX]{x1D6FC}^{+}=2.0$
, at which point, we can clearly observe the establishment of a virtual wall. The peak location of streamwise velocity is shifted above the critical layer
$y_{c}^{+}$
, where the mode speed matches the local mean velocity
$U^{+}(y_{c}^{+})=c^{+}$
(figure 4
c). Despite the strong suppression of singular value, the normalized Reynolds shear stress increases relative to the uncontrolled case (figure 4
d). However, the Reynolds shear stress contribution, as defined in (2.19), is suppressed because it is proportional to
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{2}$
. The Reynolds shear stress contribution is suppressed most at
$\unicode[STIX]{x1D6FC}^{+}=4.0$
. For this case, we can observe a virtual wall-like velocity drop at
$y^{+}\approx 10$
(figure 4
$e,f$
) and the wall-normal velocity at the wall (i.e. control input) is as high as the peak value. The peak normalized Reynolds shear stress is also substantially lower. Based on these observations, we suggest that the effect of Case SP is essentially similar to that of opposition control. Note that the controlled streamwise velocity is shifted further toward the wall as the actuation amplitude increases from
$\unicode[STIX]{x1D6FC}^{+}=2.0$
to
$\unicode[STIX]{x1D6FC}^{+}=4.0$
: we discuss this point further in § 4.2. Figure 4(g,h) shows velocity and Reynolds stress profile for the case of drag increment at
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
; the peak locations of controlled streamwise velocity and Reynolds shear stress are shifted toward the wall, from which we can assume more shear stress is generated at the wall when the control is applied. This is also consistent with our earlier observation that, for large blowing and suction, the natural dynamics of the mode is overwhelmed by the actuation.
Figure 5 shows predicted velocity fields in the spanwise wall-normal plane. Here, the velocity structure includes contributions from both oblique modes (
$k_{z}>0$
and
$k_{z}<0$
). Note that these figures show the cross-section where the magnitude of the streamwise velocity is maximum: since the resolvent formulation yields periodic flow structures, maxima and minima occur at intervals of
$\unicode[STIX]{x1D706}_{x}^{+}/2$
in the streamwise direction.
Figure 5(a,b) shows the uncontrolled case. The presence of periodic sweeps and ejections and counter-rotating streamwise vortices is consistent with known features of the NW cycle, confirming that this resolvent mode serves as a useful model for these dynamically important coherent structures. For control with
$\unicode[STIX]{x1D6FC}^{+}=2.0$
(figure 5
c,d), a virtual wall (
$v\approx 0$
) is clearly observed at
$y^{+}\approx 10$
in accordance with figure 4(c). On the other hand, figure 5(e,f) reveals that a virtual wall-like velocity drop observed in figure 4(e) was not a virtual wall, even though this case (
$\unicode[STIX]{x1D6FC}^{+}=4.0$
) corresponds to the maximum reduction of Reynolds shear stress contribution: we discuss this point further in § 4.2. Figure 5(g,h) shows the case where performance deteriorates (
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
). Due to the strong blowing and suction, the magnitude of the wall-normal velocity is highest at the wall and structure in the spanwise wall-normal plane suggests that the quasi-streamwise vortices are partially ‘absorbed’ into the wall.
3.2 Effect of control on the very-large-scale motions
Relationship between the blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
and drag contribution of the mode resembling VLSMs at
$Re_{\unicode[STIX]{x1D70F}}=2000$
. Black lines, Case SP; grey lines, Case ST. Solid lines,
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}$
; dashed lines,
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
.
Next, we investigate the effect of suboptimal control on resolvent modes resembling VLSMs. Figure 6 shows the relationship between the blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
and the two ratios that quantify the effect of control, i.e. the singular value ratio
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}$
and the ratio of Reynolds shear stress contribution
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
. For Case SP (black line),
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
increases after the control effect appears at
$\unicode[STIX]{x1D6FC}^{+}>10^{-1}$
and reaches a maximum value at
$\unicode[STIX]{x1D6FC}^{+}=7.0$
. As
$\unicode[STIX]{x1D6FC}^{+}$
increases further,
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
begins to decrease. Large reduction of
$R_{\boldsymbol{k}}$
(
${\approx}50\,\%$
) is achieved for
$\unicode[STIX]{x1D6FC}^{+}\approx 10^{2}$
. These observations contrast sharply with the
$\unicode[STIX]{x1D6FC}^{+}$
dependence observed for the NW modes. For similar values of
$\unicode[STIX]{x1D6FC}^{+}$
, suboptimal control seems to have the opposite effect on modes resembling the VLSMs compared to modes resembling the NW cycle. This is discussed in greater detail below. Note that Case ST (black line) leads to a monotonic increase in the Reynolds shear stress contribution, and there is little dependence on the magnitude of blowing and suction above
$\unicode[STIX]{x1D6FC}^{+}\geqslant 10^{1}$
.
Wall-normal profiles for VLSM-type modes at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: (a,c) amplitude of streamwise velocity (black lines) and wall-normal velocity (grey lines); (b,d) Reynolds shear stress. Dashed lines, no control; solid lines, controlled. (a,b)
$\unicode[STIX]{x1D6FC}^{+}=7.0$
; (c,d)
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
. Black horizontal line shows the location of critical layer,
$y_{c}^{+}\approx 105$
.
Figure 7 shows wall-normal profiles of each velocity component for VLSM-type modes at
$Re_{\unicode[STIX]{x1D70F}}=2000$
for (i) the uncontrolled case, (ii) Case SP with blowing and suction amplitude
$\unicode[STIX]{x1D6FC}^{+}=7$
, which leads to an increase and drag, and (iii) Case SP with
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
, which leads to a drag decrease. Figure 8 shows the velocity structure associated with this mode in the
$y$
–
$z$
plane (cf. figure 5 for the NW modes). As expected, the peak of the streamwise velocity is located slightly above the critical layer
$y_{c}^{+}\approx 105$
for this mode (uncontrolled case in the figure 7
a,c). As shown in figure 8(a), the flow structure associated with this mode comprises velocity fields that are periodic over the prescribed length scales
$\unicode[STIX]{x1D706}_{x}^{+}\approx 6h^{+}$
and
$\unicode[STIX]{x1D706}_{z}^{+}\approx 0.6h^{+}$
, and give rise to large streamwise roll cells. Although the wall-normal velocity associated with this mode is much smaller than the streamwise velocity, the motions are clearly active in the sense that they contribute Reynolds shear stress (uncontrolled case in figure 7
a,b). All of these characteristics are consistent with previous studies (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007a
; Smits et al.
Reference Smits, McKeon and Marusic2011), confirming that this large-scale resolvent mode can serve as a useful model in the evaluation of flow control for VLSMs.
In figure 7(a,b), for which the
$R_{\boldsymbol{k}}$
increment is maximum (i.e.
$\unicode[STIX]{x1D6FC}^{+}\approx 7$
), the profiles of velocities amplitude and Reynolds shear stress remain nearly unchanged near and above the critical layer, although a virtual wall looks to be formed near the wall (
$y^{+}\approx 50$
). This observation suggests that there is little control effect near and above the critical layer (
$y_{c}^{+}\approx 105$
), which is the region considered important for drag reduction since localization around the critical layer leads to high singular values. Instead, blowing and suction simply generate additional turbulence near the wall. This is also evident in the predicted flow structure shown in figure 8(b). For the case where large
$R_{\boldsymbol{k}}$
reduction is observed (
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
), the peak magnitudes of the streamwise and wall-normal velocity are shifted towards the wall (figure 7
c). Further, the normalized Reynolds shear stress is successfully suppressed near the critical layer, indicating a change in the phase relationship between the streamwise and wall-normal velocity components, but amplified close to the wall (figure 7
d). From these observations, it can be postulated that effective control for VLSMs suppresses the Reynolds shear stress near, or above, the critical layer. Figure 8 shows that, for both
$\unicode[STIX]{x1D6FC}^{+}=7$
and
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
, blowing and suction lead to the generation of a strong wall-normal velocity field that extends to
$y^{+}\leqslant 30$
. For conditions in which drag is reduced (i.e.
$\unicode[STIX]{x1D6FC}^{+}=10^{2}$
), the wall-normal velocity close to the critical layer is suppressed, and the streamwise roll cells are substantially weakened (figure 8
c).
As noted earlier, for a given value of
$\unicode[STIX]{x1D6FC}^{+}$
, suboptimal control has the opposite effect on NW modes and VLSMs. Together, the model predictions from §§ 3.1 and 3.2 suggest the following interpretation. For weak blowing and suction with
$\unicode[STIX]{x1D6FC}^{+}<1$
, control effectively suppresses structures located very close to the wall (figure 4
a) but has little effect on larger-scale structures resembling VLSMs present far from the wall. For moderate values of blowing and suction,
$\unicode[STIX]{x1D6FC}^{+}\approx 1{-}7$
, control suppresses the NW modes further but leads to harmful effects on larger-scale modes, i.e. the generation of additional turbulence near the wall. Finally, powerful blowing and suction (e.g.
$\unicode[STIX]{x1D6FC}^{+}\approx 100$
) successfully suppress modes resembling the VLSMs, but overwhelms structures close to the wall (see figures 5
h and 7
d).
Recall that the design parameter
$\unicode[STIX]{x1D6FC}^{+}$
is essentially a length scale that translates the sensed wall shear stress into a blowing and suction velocity. As a result, it is not altogether surprising that large values of
$\unicode[STIX]{x1D6FC}^{+}\geqslant O(10)$
are required to effectively interact with modes resembling VLSMs whose critical layer is located at
$y_{c}^{+}\approx 105$
, while smaller values of
$\unicode[STIX]{x1D6FC}^{+}=O(1)$
are sufficient for modes resembling the near-wall cycle with critical layer location
$y_{c}^{+}\approx 16$
. Unfortunately, this also suggests that, for control to be effective across spectral space, a value of
$\unicode[STIX]{x1D6FC}^{+}$
that varies with mode speed (or critical layer location) may be required. We consider such mode speed effects in the following section.
Velocity structure for VLSMs in the spanwise wall-normal plane at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: contour, wall-normal velocity amplitude; arrows, wall-normal and spanwise velocity field. (a) no-control; (b)
$\unicode[STIX]{x1D6FC}^{+}=7$
; (c)
$\unicode[STIX]{x1D6FC}^{+}=1\times 10^{2}$
. The horizontal dashed lines show the critical layer location,
$y_{c}^{+}\approx 105$
.
3.3 Effect of propagation speed
The normalized change in the Reynolds shear stress contribution relative to the uncontrolled case as a function of blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
and propagation speed
$c^{+}$
for resolvent modes with length scales comparable to the NW cycle at
$Re_{\unicode[STIX]{x1D70F}}=2000$
. (a) Case SP; (b) Case ST.
In this section, we investigate the effect of propagation speed
$c^{+}$
to provide further insight into the effects of suboptimal control. Since high-gain resolvent modes tend to localize near the critical layer (McKeon & Sharma Reference McKeon and Sharma2010), for modes with identical wavelengths, an increase in propagation speed also indicates an increase in the distance from the wall, where the sensing and actuation take place. Figure 9 shows the normalized change of the Reynolds shear stress contribution relative to the uncontrolled case as a function of blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
, for modes with similar length scales as the near-wall cycle,
$(\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+})\approx (10^{3},10^{2})$
, but different propagation speeds. Since the energetic contribution of structures with convection velocities larger than channel centreline
$U_{CL}^{+}$
is known to be negligible in real turbulent flows, in the present study we focus on the modes with
$0<c^{+}\leqslant U_{CL}^{+}$
. For reference, the centreline velocity obtained using the eddy viscosity model proposed by Reynolds & Tiederman (Reference Reynolds and Tiederman1967) is
$U_{CL}^{+}=23.8$
at
$Re_{\unicode[STIX]{x1D70F}}=2000$
.
For Case SP, figure 9(a) suggests that large drag reduction (
$1-(R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0})\approx 0.5$
) can be achieved at low magnitudes of blowing and suction for modes with low propagation speeds, especially in the ranges
$6\leqslant c^{+}\leqslant 11$
and
$1\times 10^{0}\leqslant \unicode[STIX]{x1D6FC}^{+}\leqslant 4\times 10^{0}$
(A in the figure). In contrast, there is a significant increase in the Reynolds shear stress contribution from slower modes with
$c^{+}\leqslant 12$
when the blowing and suction magnitude becomes larger,
$\unicode[STIX]{x1D6FC}^{+}\geqslant 10^{1}$
. For the fast propagating modes (
$c^{+}\geqslant 15$
), there is little or no control effect.
For Case ST, figure 9(b) shows an approximately
$40\,\%$
drag reduction for slow propagating modes with
$c^{+}\leqslant 12$
when the blowing and suction magnitude is strong (
$\unicode[STIX]{x1D6FC}^{+}\geqslant 10^{1}$
; C in the figure), while drag increases for
$\unicode[STIX]{x1D6FC}^{+}\leqslant 3\times 10^{0}$
and
$c^{+}\leqslant 5$
. In addition, modes with
$c^{+}\approx 12{-}14$
show drag increment over the entire range of blowing and suction magnitude studied in the present study. For even faster propagating modes with
$c^{+}\geqslant 15$
, there is little or no control effect, similar to Case SP (figure 9
a).
For this wavenumber–frequency combination, previous studies (McKeon & Sharma Reference McKeon and Sharma2010; Luhar et al.
Reference Luhar, Sharma and McKeon2014) classify slower-moving modes with
$c^{+}\leqslant 10$
as being ‘attached’ to the wall since the peak wall-normal velocity associated with these modes is located at a roughly fixed position (
$20\leqslant y^{+}\leqslant 30$
). On the other hand, as the propagation speed increases to
$c^{+}\geqslant 10$
, the location of peak wall-normal velocity shifts away from the wall and tracks the critical layer, where
$U^{+}(y_{c}^{+})=c^{+}$
. These faster-moving modes are termed ‘critical’ modes, and can be further classified into ‘attached critical’ or ‘detached critical’ depending on whether they have a substantial velocity signature in the near-wall region. Luhar et al. (Reference Luhar, Sharma and McKeon2014) reported that wall-based control can be effective for attached modes but not necessarily for detached critical modes. Further, the transition region from attachment to criticality often tends to be harmful to control performance.
Wall-normal profiles for length scales comparable to the NW cycle with faster propagating speed
$c^{+}=13.5$
at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: (a) amplitude of streamwise velocity (black) and wall-normal velocity (grey); (b) Reynolds share stress. Dashed lines, no control; solid lines, Case SP at
$\unicode[STIX]{x1D6FC}^{+}\approx 5\times 10^{1}$
. The horizontal dashed lines show the critical layer location,
$y_{c}^{+}\approx 39$
.
With these arguments in mind, the present results can be reviewed as follows. Suboptimal control has little effect on detached critical modes far from the wall (
$c^{+}\geqslant 15$
), while strong effects (both good and harmful) can be seen for attached modes. In addition, similar to Luhar et al. (Reference Luhar, Sharma and McKeon2014), the transition from attached to critical behaviour leads to a reversal in the control effect; sudden drag reduction (figure 8
a) or temporary drag increment (figure 8
b) is observed in the range of mode transition (
$c^{+}\approx 12{-}15$
).
For modes with length scales comparable to the NW cycle, the present analysis shows that large Reynolds shear stress reductions are observed for
$\unicode[STIX]{x1D6FC}^{+}\geqslant 3\times 10^{1}$
and
$c^{+}\approx 13$
–14 (B in figure 9
a). However, the normalized change in singular values,
$1-(\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0})^{2}$
, shows further amplification in this region. In other words, the two measures used to evaluate control efficacy in this study show contradictory behaviour for this range of
$\unicode[STIX]{x1D6FC}^{+}$
and
$c^{+}$
(elsewhere, they are in good agreement). To provide further insight into this inconsistency, we consider how control under Case SP with
$\unicode[STIX]{x1D6FC}^{+}=50$
affects rank-1 mode structure for wavenumber–frequency combination
$(\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+},c^{+})\approx (10^{3},10^{2},13.5)$
. Figure 10 shows that control leads to the development of a large near-wall peak in velocity and Reynolds stress (
$y^{+}<10$
), and a significant suppression around the critical layer,
$y_{c}^{+}\approx 39$
. This observation suggests that, for region B, large-amplitude blowing and suction changes the structure of the rank-1 response mode completely compared to the uncontrolled case. Unlike the results shown in figure 4 for lower-amplitude blowing and suction, the rank-1 mode is not a modified version of the uncontrolled mode. Further, the first singular value under control is not significantly larger than subsequent singular values (i.e.
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c,1}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}c,3}\approx 1.4$
, while
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}0,1}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0,3}\approx 4.1$
). Thus, we suggest that for modes that are transitioning from being attached to detached, large values of
$\unicode[STIX]{x1D6FC}^{+}$
weaken the rank-1 approximation, and trigger a ‘new’ kind of amplified flow structure close to the wall without influencing the original mode. The magnitude of
$\unicode[STIX]{x1D6FC}^{+}$
that leads to effective suppression of NW modes is discussed further in § 4.1.1.
The normalized change in the Reynolds shear stress contribution relative to the uncontrolled case as a function of blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
and propagation speed
$c^{+}$
for modes with length scales similar to VLSMs at
$Re_{\unicode[STIX]{x1D70F}}=2000$
. (a) Case SP; (b) Case ST.
Figure 11 shows the normalized change of the Reynolds shear stress contribution for resolvent modes with similar length scales to the VLSMs,
$(\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+})\approx (0.6h^{+},6h^{+})$
at
$Re_{\unicode[STIX]{x1D70F}}=2000$
, for varying
$c^{+}$
and
$\unicode[STIX]{x1D6FC}^{+}$
. For Case SP, figure 11(a) shows beneficial drag reduction is achieved at slow propagation speeds (
$c^{+}<14$
) for
$10^{-1}\leqslant \unicode[STIX]{x1D6FC}^{+}\leqslant 10^{0}$
(D in the figure). As blowing and suction amplitude increases (
$\unicode[STIX]{x1D6FC}^{+}\geqslant 2\times 10^{0}$
), a significant drag increment is observed. In contrast, for slightly higher speeds (
$c^{+}\approx 14{-}16$
), drag reduction is observed over a wide range of blowing and suction magnitude,
$\unicode[STIX]{x1D6FC}^{+}\geqslant 10^{1}$
(E in the figure). For critical modes with
$16\leqslant c^{+}\leqslant 20$
, significant drag increases are predicted over a wide range of blowing and suction magnitude.
A comparison of figures 9(b) and 11(b) shows that Case ST has a similar effect on both the VLSM- and NW-type modes. The major difference is that the preferable control effect (i.e. drag reduction) is observed over a wider range of propagation speeds:
$c^{+}\leqslant 15$
(F in figure 11
b) compared to
$c^{+}\leqslant 12$
in figure 9(b). This difference is explained by the fact that modes with larger streamwise and spanwise length scales also tend to have a larger wall-normal footprint (Luhar et al.
Reference Luhar, Sharma and McKeon2014). As a result, the transition from attachment to criticality occurs at a higher propagation speed:
$c^{+}\approx 15$
for modes with length scales similar to the VLSMs, compared to
$c^{+}\approx 12$
for modes with length scales comparable to the NW cycle.
3.4 Effect of control in spectral space
Normalized change of squared singular value relative to the uncontrolled case at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: (a) Case SP (
$\unicode[STIX]{x1D6FC}^{+}=2.0$
); (b) Case ST (
$\unicode[STIX]{x1D6FC}^{+}=10$
).
Up to here, we have discussed the effect of suboptimal control on modes resembling typical coherent structures in wall turbulence (§§ 3.1 and 3.2), considered the effect of phase speed
$c^{+}$
(§ 3.3) and confirmed that control can lead to drag increment as well as reduction. Next, we consider the effect of control across wavenumber space. For a given wavenumber combination (
$k_{x},k_{z}$
) corresponding to wavelength (
$\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+}$
), real turbulent flows are likely to involve many frequency components (i.e. many propagation speeds
$c$
). To account for this effect, we assume that the forcing for the resolvent operator is broadband over wave speed (Moarref et al.
Reference Moarref, Sharma, Tropp and McKeon2013). More specifically, we assume that there is unit forcing over the range of mode speeds
$0\leqslant c\leqslant 1$
, where
$c$
is normalized by the centreline velocity, and predict the effect of control on modes with wavelengths in the range of
$10^{1}\leqslant (\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+})\leqslant 10^{5}$
by simply integrating squared singular values:
$$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D70E}}_{\boldsymbol{k}}^{2}(k_{x},k_{z})=\int _{0}^{1}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{2}(k_{x},k_{z},c)\,\text{d}c. & & \displaystyle\end{eqnarray}$$
Note that the broadband forcing assumption does not weight all frequency components equally. The resolvent operator itself acts as a strong filter on what frequency components, or mode speeds, are likely to be energetic in real turbulent flows. For example, only modes around
$c^{+}=10{-}18$
are highly amplified (i.e. have high
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}$
) for modes with wavelength
$\unicode[STIX]{x1D706}_{x}^{+}\approx 10^{3}$
and
$\unicode[STIX]{x1D706}_{z}^{+}\approx 10^{2}$
. The remaining frequency components do not contribute significantly to the integrated measure shown above.
Figure 12(a) shows the result for Case SP at
$\unicode[STIX]{x1D6FC}^{+}=2.0$
(
$Re_{\unicode[STIX]{x1D70F}}=2000$
). Consistent with the results presented above, the resolvent analysis predicts large drag reductions (
$1-(\tilde{\unicode[STIX]{x1D70E}}_{\boldsymbol{k}c}/\tilde{\unicode[STIX]{x1D70E}}_{\boldsymbol{k}0})^{2}\geqslant 0.3$
) for
$\unicode[STIX]{x1D706}_{x}^{+}\geqslant 10^{3}$
and
$\unicode[STIX]{x1D706}_{z}^{+}=3\times 10^{1}$
–
$3\times 10^{2}$
(G in the figure), which is close to the wavenumber set of NW modes, i.e.
$(\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+})\approx (10^{3},10^{2})$
. In contrast, a slight increase in drag is indicated in the spectral region representative of VLSMs (H in the figure). Further, the drag reduction effect deteriorates at shorter spanwise wavelength (
$\unicode[STIX]{x1D706}_{z}^{+}<3\times 10^{1}$
: I in the figure). Although not shown, we have also investigated the effect of Case SP with different blowing and suction magnitudes (
$\unicode[STIX]{x1D6FC}^{+}=0.5$
,
$1.0$
,
$4.0$
and
$10$
). We have found that as the magnitude of blowing and suction increases, the region of drag reduction G diminishes, while the harmful region H widens. Region I, which represents drag increment, does not exist for
$\unicode[STIX]{x1D6FC}^{+}=0.5$
. However, after this region appears, it becomes wider and wider as blowing and suction magnitude increases. In other words, the detrimental effects of suboptimal control at lower spanwise wavelengths increase as the blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
increases above
$\unicode[STIX]{x1D6FC}^{+}>0.5$
. These observations suggest that
$\unicode[STIX]{x1D6FC}^{+}\approx 0.5$
represents a good design choice, since it yields some drag reduction for modes that resemble the NW cycle but avoids detrimental effects elsewhere in spectral space. Below, we show that the magnitude of blowing and suction employed in previous DNS (Lee et al.
Reference Lee, Kim and Choi1998) corresponds closely to this value.
Figure 12(b) shows the effect of control across spectral space for Case ST at
$\unicode[STIX]{x1D6FC}^{+}=10$
(
$Re_{\unicode[STIX]{x1D70F}}=2000$
). Substantial drag reduction (
$1-(\tilde{\unicode[STIX]{x1D70E}}_{\boldsymbol{k}c}/\tilde{\unicode[STIX]{x1D70E}}_{\boldsymbol{k}0})^{2}\geqslant 0.2$
) is indicated in the region corresponding to NW modes (J in the figure). On the other hand, shorter streamwise wavelengths (
$\unicode[STIX]{x1D706}_{x}^{+}<10^{2}$
) are adversely affected by the control (K in the figure). Unlike Case SP, as the magnitude of blowing and suction increases, the spectral region of drag increment (or drag reduction) does not significantly change in Case ST. However, the magnitude of the drag increment or reduction becomes larger. Note that, for modes with relatively long wavelengths (
$\unicode[STIX]{x1D706}_{x}^{+}$
,
$\unicode[STIX]{x1D706}_{z}^{+}>10^{3}$
), Case ST has almost no effect. In other words, Case ST is unlikely to affect modes resembling VLSMs.
In sum, the above results indicate that suboptimal control has a positive effect around the wavelengths of the NW modes for Case SP with
$\unicode[STIX]{x1D6FC}^{+}=2$
and Case ST with
$\unicode[STIX]{x1D6FC}^{+}=10$
, but a detrimental (Case SP) or negligible (Case ST) effect around the wavelengths corresponding to VLSMs. At first glance, the fact that Case ST is predicted to have no detrimental effects on structures resembling VLSMs suggests that suboptimal control based on the streamwise shear stress may be useful for application at high Reynolds number. However, in the following section, we show that Case ST also leads to substantial amplification of structures that are long in the spanwise direction, which is detrimental to the overall performance of control.
4 Discussion
4.1 Comparison with previous DNS studies
In this section, we compare the present resolvent-based predictions with previous DNS studies (Lee et al.
Reference Lee, Kim and Choi1998; Choi & Sung Reference Choi and Sung2002; Iwamoto et al.
Reference Iwamoto, Suzuki and Kasagi2002). To provide further insight into the difference between Case SP and Case ST, we also conduct additional channel flow DNS using an in-house code (Fukagata, Kasagi & Koumoutsakos Reference Fukagata, Kasagi and Koumoutsakos2006) for representative values of
$\unicode[STIX]{x1D6FC}^{+}$
. Briefly, these simulations have following key properties: the code is based on energy conservative finite difference method for the channel flow; the time integration is done by using the third-order Runge–Kutta/Crank–Nicolson scheme; the bulk mean velocity is kept constant and the friction Reynolds number is
$Re_{\unicode[STIX]{x1D70F}}\approx 180$
for uncontrolled flow; the computational domain is
$(L_{x},L_{y},L_{z})\approx (7,2,3.5)$
; finally, grid number dependencies for the time scale and spatial space have been well verified.
4.1.1 Case SP
For a fair comparison between the present predictions and the DNS results, the parameter
$\unicode[STIX]{x1D6FC}^{+}$
should be related to the blowing and suction amplitudes used in the DNS. In the DNS studies, the root-mean-square value of the control input
$\unicode[STIX]{x1D719}_{rms}^{+}$
was kept constant, at a value corresponding to the wall-normal velocity at
$y^{+}\approx 10$
. In contrast, the present analysis considers
$\unicode[STIX]{x1D6FC}^{+}$
to be a design parameter.
As a first step, we estimate a realistic value for the magnitude
$\unicode[STIX]{x1D6FC}^{+}$
for Case SP based on values of
$\unicode[STIX]{x1D719}_{rms}^{+}$
from previous DNS studies. As mentioned in § 2.2, suboptimal control aims to suppress the quasi-streamwise vortices associated NW cycle (this is generally true for most active control techniques). For the relative long modes resembling the NW cycle, the streamwise wavenumber is an order of magnitude smaller than the spanwise wavenumber,
$k_{x}\ll k_{z}$
, such that
$k_{z}/K\approx 1$
(Lee et al.
Reference Lee, Kim and Choi1998). For example,
$k_{z}/K=120/\sqrt{12^{2}+120^{2}}\approx 0.995$
for NW modes at
$Re_{\unicode[STIX]{x1D70F}}=2000$
in the analysis employed here. Assuming
$k_{z}/K\approx 1$
in (2.15), we can derive the following relationship between the amplitudes of the control input and the spanwise wall shear stress:
$$\begin{eqnarray}\displaystyle \left|\unicode[STIX]{x1D719}_{\boldsymbol{k}}^{+}\right|\simeq \unicode[STIX]{x1D6FC}^{+}\left|\left(\unicode[STIX]{x2202}w_{\boldsymbol{k}}^{+}/\unicode[STIX]{x2202}y^{+}\right)_{w}\right|. & & \displaystyle\end{eqnarray}$$
The information of root-mean-square values from previous DNS studies.
Thus, the magnitude of
$\unicode[STIX]{x1D6FC}^{+}$
can be estimated from
$\unicode[STIX]{x1D719}_{rms}^{+}$
and
$\left(\unicode[STIX]{x2202}w_{rms}^{+}/\unicode[STIX]{x2202}y^{+}\right)_{w}$
obtained in DNS. Based on results from the in-house DNS at
$Re_{\unicode[STIX]{x1D70F}}=180$
(Fukagata et al.
Reference Fukagata, Kasagi and Koumoutsakos2006), the magnitude of the spanwise wall shear stress is estimated to be
$$\begin{eqnarray}\displaystyle \left(\unicode[STIX]{x2202}w_{rms}^{+}/\unicode[STIX]{x2202}y^{+}\right)_{w}\approx 0.2, & & \displaystyle\end{eqnarray}$$
although any database of turbulent channel flow at similar low Reynolds numbers should yield similar values. As shown in table 1, the blowing and suction amplitude used in the DNS studies (Lee et al. Reference Lee, Kim and Choi1998; Choi & Sung Reference Choi and Sung2002; Iwamoto et al. Reference Iwamoto, Suzuki and Kasagi2002) is
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{rms}^{+}\approx 0.1. & & \displaystyle\end{eqnarray}$$
Combining these two estimates,
$\unicode[STIX]{x1D6FC}^{+}$
for Case SP in the DNS studies is approximately
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}^{+}\approx \frac{\unicode[STIX]{x1D719}_{rms}^{+}}{\left(\unicode[STIX]{x2202}w_{rms}^{+}/\unicode[STIX]{x2202}y^{+}\right)_{w}}\approx \frac{0.1}{0.2}=0.5. & & \displaystyle\end{eqnarray}$$
The full range of
$\unicode[STIX]{x1D719}_{rms}^{+}$
employed in previous DNS (
$0.075{-}0.3$
) yields
$\unicode[STIX]{x1D6FC}^{+}\approx 0.375{-}1.5$
. As discussed in § 3.4, this range of
$\unicode[STIX]{x1D6FC}^{+}$
corresponds closely to the qualitative optimum where the NW modes are suppressed due to the blowing and suction, but the damaging effects elsewhere in spectral space are limited. Further, figure 3(b) indicates that the Reynolds stress contribution from the NW modes is suppressed by approximately
$25\,\%$
for Case SP with
$\unicode[STIX]{x1D6FC}^{+}\approx 0.5$
. This is in good agreement with the 22 % drag reduction obtained by Lee et al. (Reference Lee, Kim and Choi1998). These observations suggest that
$\unicode[STIX]{x1D6FC}^{+}\approx 0.5$
represents a reasonable estimate for the conditions tested in previous DNS, and that the resolvent analysis is able to successfully reproduce the effects of suboptimal control based on the spanwise shear stress.
Profiles showing wall-normal variations in structure for resolvent modes: (a) amplitude
$|v_{\boldsymbol{k}}|$
and (b) phase
$\angle v_{\boldsymbol{k}}$
of wall-normal velocity. Dashed lines, uncontrolled; circle markers, Case SP (
$\unicode[STIX]{x1D6FC}^{+}=1.0$
); triangle markers, opposition control (
$y_{d}^{+}\approx 10$
). In (b), phase of spanwise wall shear stress is also shown: cross,
$(\unicode[STIX]{x2202}w_{\boldsymbol{k}0}/\unicode[STIX]{x2202}y)_{w}$
; asterisk,
$(\unicode[STIX]{x2202}w_{\boldsymbol{k}c}/\unicode[STIX]{x2202}y)_{w}$
. Each figure considers NW modes at
$Re_{\unicode[STIX]{x1D70F}}=2000$
.
Since the r.m.s. value of
$\unicode[STIX]{x1D719}$
was set to equal that of the wall-normal velocity at
$y^{+}\approx 10$
in the original DNS of Lee et al. (Reference Lee, Kim and Choi1998), as another quantitative comparison, we attempt to find the value of
$\unicode[STIX]{x1D6FC}^{+}$
that leads to
$|\unicode[STIX]{x1D719}_{\boldsymbol{k}}|=|v_{\boldsymbol{k}}(y^{+}=10)|$
for the NW modes. As shown in figure 13(a),
$|\unicode[STIX]{x1D719}_{\boldsymbol{k}}|=|v_{\boldsymbol{k}}(y^{+}\approx 10)|$
when
$\unicode[STIX]{x1D6FC}^{+}=1.0$
. This observation confirms that
$\unicode[STIX]{x1D6FC}^{+}\approx 0.5{-}1.0$
represents a reasonable estimate for the conditions tested in DNS. Importantly, an estimate for
$\unicode[STIX]{x1D6FC}^{+}$
also allows us to compare Case SP with the opposition control of Choi et al. (Reference Choi, Moin and Kim1994). Figure 13(b) shows that there are many similarities between Case SP with
$\unicode[STIX]{x1D6FC}^{+}=1.0$
and opposition control with the detection plane at
$y_{d}^{+}\approx 10$
. The phase of wall-normal velocity is reversed between
$y^{+}=10$
and the wall for both Case SP and the opposition control. Namely,
$\unicode[STIX]{x1D719}_{\boldsymbol{k}}$
and
$v_{\boldsymbol{k}}(y^{+}=10)$
have opposite signs. These features have also been confirmed in figure 5(c,d) (Case SP,
$\unicode[STIX]{x1D6FC}^{+}\approx 2.0$
). The trajectory of the phase from
$y^{+}=10$
to the wall, however, is somewhat different: for the opposition control, the phase decreases towards the wall, while it increases in Case SP. Figure 13(b) also shows that the phase angle between the spanwise wall shear stress and the control input is about
$\unicode[STIX]{x03C0}/2$
, although a slight phase shift is observed from the uncontrolled to controlled, as highlighted by the arrow annotation. In summary, Case SP can essentially be regarded as a modified version of opposition control. We will further compare the resolvent-based predictions of suboptimal control and opposition control in § 4.2.
4.1.2 Case ST
Next, we consider the discrepancy for Case ST between the present study and the previous DNS. Recall that Lee et al. (Reference Lee, Kim and Choi1998) observed an increase in drag for Case ST, while the resolvent-based predictions shown in figure 3 indicate drag reduction. In addition, figures 9(b) and 11(b) indicate that Case ST has positive effects around the wavelengths of both NW modes and VLSMs when we consider modes that are attached (i.e. slow propagating modes).
The key to explaining the discrepancy between DNS and the present analysis for Case ST is the rise of flow structures that are detrimental to control performance. As shown in figure 12(b), resolvent analysis suggests that quasi-two-dimensional structures that are long in the spanwise direction, with streamwise wavelength
$\unicode[STIX]{x1D706}_{x}^{+}\approx 10^{2}$
(region K), are amplified substantially under control. To test whether such structures emerge in real flows, we conducted DNS using the in-house code mentioned earlier (Fukagata et al.
Reference Fukagata, Kasagi and Koumoutsakos2006). Figure 14 shows contours of the control input (i.e. the blowing and suction) at the lower wall for Case SP with
$\unicode[STIX]{x1D6FC}^{+}\approx 1$
and for Case ST with
$\unicode[STIX]{x1D6FC}^{+}\approx 5$
. Here, the control input is computed using the uncontrolled velocity field, which corresponds to the control input at the initial time instant when the control is applied. For Case SP (figure 14
a), the velocity contours are long in the streamwise direction and exhibit alternating regions of positive velocity (i.e. blowing) and negative velocity (i.e. suction) in the spanwise direction. These qualitative observations confirm that Case SP targets the quasi-streamwise vortices associated with the NW cycle. On the other hand, for Case ST (figure 14
b), the blowing and suction contours are more elongated in the spanwise direction and much shorter in the streamwise direction. In other words, the control input does not appear to be targeting or interacting with the NW cycle. Snapshots taken at regular intervals show that these quasi-two-dimensional structures that are long in the spanwise direction persist under control based on the streamwise velocity.
Contours show the snapshot of the control input
$\unicode[STIX]{x1D719}^{+}$
at the lower wall calculated by in-house DNS code (
$Re_{\unicode[STIX]{x1D70F}}=180$
): (a) Case SP; (b) Case ST.
As an additional quantitative test, we also generated wavenumber spectra for the control input (for 1000 snapshots acquired at intervals of
$t^{+}\approx 100$
). Note that the distributions of control input, used for the time-average operation, are computed from different snapshots of uncontrolled (i.e. no-slip condition) DNS field. These spectra (not shown here) indicated the presence of an energetic peak in the range of
$\unicode[STIX]{x1D706}_{x}^{+}\approx 1.5\times 10^{2}$
–
$4\times 10^{2}$
and
$\unicode[STIX]{x1D706}_{z}^{+}\approx 3\times 10^{2}$
–
$6\times 10^{2}$
for Case ST. In contrast, there was a peak near length scales corresponding closely to the NW cycle for Case SP, i.e.
$(\unicode[STIX]{x1D706}_{x}^{+},\unicode[STIX]{x1D706}_{z}^{+})\approx (10^{3},10^{2})$
. The length scales for the structures that are highly actuated initially under Case ST do not correspond exactly to the region of highest amplification (K) in figure 12(b). However, these results suggest that streamwise shear-based control sets up a detrimental feedback loop whereby the actuation preferentially targets structures that are short in the streamwise direction and long in the spanwise direction (i.e. high
$k_{x}/k_{z}$
), which corresponds to the region of spectral space that exhibits very high amplification under control.
In summary, while suboptimal control based on the streamwise velocity (Case ST) may suppress the NW cycle, it also leads to the emergence of energetic structures that are long in the spanwise direction. Such energetic spanwise structures could be responsible for the drag increase reported in previous studies (Lee et al. Reference Lee, Kim and Choi1998). Further, the present study suggests that it may be possible to reduce drag under Case ST if the control is applied selectively in spectral space. Specifically, actuation must be limited to regions of spectral space where the resolvent analysis predicts a reduction in gain or Reynolds stress (e.g. for wavenumber–frequency combinations corresponding to the near-wall cycle, as shown in figures 9(b) and 12(b)). Further simulations that test this hypothesis are currently underway.
4.1.3 Interpretation of Reynolds number effect
Finally, we consider the deterioration of control effect at high Reynolds numbers. Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) investigated the effect of suboptimal control (Case SP) at several friction Reynolds numbers (
$Re_{\unicode[STIX]{x1D70F}}=110,~150,~300,~400~\text{and}~650$
) and reported that the drag reduction rate decreases with increasing
$Re_{\unicode[STIX]{x1D70F}}$
. Assuming that the near-wall velocity fluctuations are ideally damped, Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005) also derived a theoretical expression capturing the Reynolds number dependence of the drag reduction rate. According to this theory, the drag reduction rate mildly decreases with the Reynolds number even if the large-scale structures are not taken into account. As we have observed in figure 3, the Reynolds number has little influence on the singular values and the Reynolds shear stress contributions for NW modes in the range where the drag is predicted to decrease, i.e.
$10^{0}\leqslant \unicode[STIX]{x1D6FC}^{+}\leqslant 5\times 10^{0}$
. On the other hand, as illustrated in figure 6, the singular value ratios and Reynolds stress contributions for VLSMs in the same range of
$\unicode[STIX]{x1D6FC}^{+}$
indicate drag increase. Therefore, the deterioration of control at high Reynolds numbers is likely due to a combination of the fundamental Reynolds number effect considered by Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005) and the detrimental effect of control on longer wavelength structures.
4.2 Comparison with the opposition control
Here, we compare resolvent-based predictions for suboptimal control and opposition control in turbulent channel flow, focusing on modes resembling the NW cycle at
$Re_{\unicode[STIX]{x1D70F}}=2000$
. As mentioned previously, the suboptimal control of Lee et al. (Reference Lee, Kim and Choi1998) stems from the opposition control of Choi et al. (Reference Choi, Moin and Kim1994). Previous DNS and LES studies of opposition-controlled channel and pipe flows (Hammond, Bewley & Moin Reference Hammond, Bewley and Moin1998; Chang, Collis & Ramakrishnan Reference Chang, Collis and Ramakrishnan2002; Fukagata & Kasagi Reference Fukagata and Kasagi2003) report that maximum drag reduction is achieved when the detection plane is located at
$y_{d}^{+}\approx 15$
, and drag increases significantly above
$y_{d}^{+}>25$
. This trend was successfully reproduced by Luhar et al. (Reference Luhar, Sharma and McKeon2014) for a turbulent pipe flow.
In the present study, we have also performed resolvent analyses of opposition-controlled channel flow. Although the results are not reproduced here for brevity, these analyses show that opposition control for NW modes achieves significant suppression of
$R_{\boldsymbol{k}}$
near
$y_{d}^{+}\approx 20$
. Consistent with the pipe flow predictions of Luhar et al. (Reference Luhar, Sharma and McKeon2014), the drag contribution (
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}$
) is minimized at
$y_{d}^{+}\approx 16$
. We also observe that the drag contribution from VLSMs increases under opposition control, regardless of the detection plane location. Hereafter, we focus on the opposition control for NW-type modes with
$y_{d}^{+}\approx 16$
and
$y_{d}^{+}\approx 20$
in order to compare with the results obtained for suboptimal control. Note that
$y_{d}^{+}\approx 16$
corresponds to the case where the detection plane and the critical layer are co-located.
Wall-normal profiles for modes resembling the NW cycle at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: (a) amplitude of streamwise velocity; (b) amplitude of wall-normal velocity; (c) Reynolds shear stress. Dashed lines, uncontrolled; solid line, suboptimal control (
$\unicode[STIX]{x1D6FC}^{+}=2.0$
); circle marker, opposition control (
$y_{d}^{+}\approx 16$
). The solid horizontal lines indicate the critical layer location
$y_{c}^{+}$
. The detection plane is located at the same
$y$
value as the critical layer.
Wall-normal profiles for modes resembling the NW cycle at
$Re_{\unicode[STIX]{x1D70F}}=2000$
: (a) amplitude of streamwise velocity; (b) amplitude of wall-normal velocity; (c) Reynolds shear stress. Dashed lines, uncontrolled; solid line, suboptimal control (
$\unicode[STIX]{x1D6FC}^{+}=4.0$
); circle marker, opposition control (
$y_{d}^{+}\approx 20$
). The solid horizontal lines show the critical layer location
$y_{c}^{+}$
. The dashed horizontal lines indicate the detection plane location for opposition control.
Figure 15 shows profiles of velocity amplitude and Reynolds shear stress for both suboptimal and opposition control. The blowing and suction magnitude is set at
$\unicode[STIX]{x1D6FC}^{+}=2.0$
for suboptimal control (Case SP), while the detection plane for opposition control is assumed to be
$y_{d}^{+}\approx 16$
. These profiles show substantial similarities between the two control methods. Further, the present results for opposition control (circle markers) are very similar to the those of pipe flow (Luhar et al.
Reference Luhar, Sharma and McKeon2014).
Figure 16 compares the predictions for Case SP at
$\unicode[STIX]{x1D6FC}^{+}=4.0$
, which maximizes suppression of the Reynolds shear stress contribution, with predictions for opposition control with
$y_{d}^{+}\approx 20$
. The streamwise velocity profiles are quite similar for both controlled cases, particularly near the wall. However, the wall-normal velocity above the critical layer shows some important differences;
$|v_{\boldsymbol{k}}|$
is slightly reduced under suboptimal control but enhanced under opposition control (figure 16
b). Interestingly, despite the significant suppression of Reynolds shear stress, figure 16(a) shows that the peak location of controlled streamwise velocity moves between the critical layer and the wall for Case SP with
$\unicode[STIX]{x1D6FC}^{+}=4.0$
. This is in contrast to the results shown in figure 15(a) for
$\unicode[STIX]{x1D6FC}^{+}=2.0$
. Luhar et al. (Reference Luhar, Sharma and McKeon2014) reported that the displacement of peak of
$|u_{\boldsymbol{k}}|$
away from the critical layer may weaken the critical layer mechanism.
Now, we define a new measure which focuses only on the Reynolds stress contribution in the region between the wall and the critical layer:
$$\begin{eqnarray}\displaystyle R_{\boldsymbol{k},\mathit{attached}} & = & \displaystyle \int _{-1}^{-1+y_{c}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{2}~\text{Re}({u_{\boldsymbol{ k}}}^{\ast }v_{\boldsymbol{ k}})(+y)\,\text{d}y+\int _{1-y_{c}}^{1}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{2}~\text{Re}({u_{\boldsymbol{ k}}}^{\ast }v_{\boldsymbol{ k}})(+y)\,\text{d}y.\end{eqnarray}$$
Here, the first term in (4.5) considers the region attached to the lower wall and the second term considers the region close to the upper wall. As shown in rightmost column of table 2, this near-wall Reynolds shear stress contribution exhibits different trends compared to the total contribution.
$R_{\boldsymbol{k},\mathit{attached}}$
is strongly suppressed for Case SP at
$\unicode[STIX]{x1D6FC}^{+}=2.0$
and for opposition control with
$y_{d}^{+}\approx 16$
. However, for Case SP with
$\unicode[STIX]{x1D6FC}^{+}=4.0$
and the opposition control with
$y_{d}^{+}\approx 20$
, the ratios are recovered toward unity.
Figure 17 shows
$(R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0})_{\mathit{attached}}$
as a function of blowing and suction magnitude
$\unicode[STIX]{x1D6FC}^{+}$
, which helps further clarify the drag reduction mechanism for NW modes under suboptimal control based on spanwise wall shear stress (Case SP). The slope is negative (i.e. towards reduced drag contribution) up to
$\unicode[STIX]{x1D6FC}^{+}=2.0$
and positive (i.e. towards increased drag contribution) above this value. The channel-integrated Reynolds shear stress contribution
$R_{\boldsymbol{k}}$
still decreases between
$2.0\leqslant \unicode[STIX]{x1D6FC}^{+}\leqslant 4.0$
. However, the Reynolds shear stress contribution in the near-wall region
$R_{\boldsymbol{k},\mathit{attached}}$
begins to increase at that point.
The ratio of controlled to uncontrolled
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}}$
and
$R_{\boldsymbol{k}}$
for each control method.
Evaluating the turbulent drag contribution in different ways. Case SP at
$Re_{\unicode[STIX]{x1D70F}}=2000$
(NW modes).
Wall-normal profiles for modes resembling the NW cycle at
$Re_{\unicode[STIX]{x1D70F}}=2000$
for varying blowing and suction magnitudes (Case SP): Grey, uncontrolled; solid lines,
$\unicode[STIX]{x1D6FC}^{+}=2.0$
; dashed lines,
$\unicode[STIX]{x1D6FC}^{+}=4.0$
; cross markers,
$\unicode[STIX]{x1D6FC}^{+}=10^{3}$
. The arrows indicate the directions of velocity components movements, which depend on blowing and suction magnitude.
Further explanation is provided in figure 18. The peak location of controlled streamwise velocity shifts up above the critical layer up to
$\unicode[STIX]{x1D6FC}^{+}=2.0$
. For larger
$\unicode[STIX]{x1D6FC}^{+}$
, the peak in streamwise velocity shifts downwards and localizes between the critical layer and the wall when blowing and suction magnitude is
$\unicode[STIX]{x1D6FC}^{+}\geqslant 3.0$
. In addition, the peak value of wall-normal velocity is larger than that for the uncontrolled case in the range of
$\unicode[STIX]{x1D6FC}^{+}\leqslant 2.0$
, while smaller in the range
$\unicode[STIX]{x1D6FC}^{+}\geqslant 3.5$
. These trends confirm that, beyond a certain threshold, an increase in the magnitude of blowing and suction only serves to generate additional turbulence near the wall. This increase in near-wall velocities with increasing
$\unicode[STIX]{x1D6FC}^{+}$
is also evident in figure 5(e,f).
In summary,
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}$
and
$R_{\boldsymbol{k},\mathit{attached}}$
are minimized for
$\unicode[STIX]{x1D6FC}^{+}=2.0$
, which resembles opposition control with
$y_{d}^{+}\approx 16$
. The channel-integrated Reynolds shear stress contribution
$R_{\boldsymbol{k}}$
is minimized at a slightly larger value of
$\unicode[STIX]{x1D6FC}^{+}$
due to the suppression above the critical layer.
5 Conclusions
Following previous studies (Luhar et al. Reference Luhar, Sharma and McKeon2014, Reference Luhar, Sharma and McKeon2015, Reference Luhar, Sharma and McKeon2016), the results presented above show that the resolvent formulation has the potential to serve as a powerful predictive tool for the design and evaluation of (linear) control laws for wall-bounded turbulent flows. In addition, the formulation also provides fundamental physical insights into the effects of control on dynamically important features of wall turbulence.
For the suboptimal control laws proposed by Lee et al. (Reference Lee, Kim and Choi1998), resolvent analysis predicts that the effect of control across spectral space depends strongly on whether the streamwise shear (Case ST) or spanwise shear (Case SP) is used as the control input. Relative to the uncontrolled flow, Case SP suppresses modes resembling the NW cycle and further amplifies modes resembling the VLSMs when the blowing and suction parameter ranges from
$1\leqslant \unicode[STIX]{x1D6FC}^{+}\leqslant 5$
. As the blowing and suction magnitude increases such that
$\unicode[STIX]{x1D6FC}^{+}\geqslant 10$
, the effects of control on these structures are reversed. For Case ST, the analysis predicts that NW-modes are suppressed across all
$\unicode[STIX]{x1D6FC}^{+}$
, while modes resembling the VLSMs are further amplified. However, the maximum suppression possible for the NW modes is lower under Case ST compared to Case SP, and requires larger
$\unicode[STIX]{x1D6FC}^{+}$
. In other words, Case SP leads to a larger reduction in gain, and for less intense actuation. An increase in Reynolds number from
$Re_{\unicode[STIX]{x1D70F}}=110$
to
$Re_{\unicode[STIX]{x1D70F}}=2000$
does not appreciably change the above trends.
We also investigated the effects of mode speed on control efficacy (§ 3.3). Consistent with previous resolvent-based predictions for opposition control (Luhar et al.
Reference Luhar, Sharma and McKeon2014), suboptimal control has the largest effect on slower-moving modes that are localized closer to the wall (i.e. attached modes). The gain and structure of faster-moving modes (i.e. critical or detached modes) found farther from the wall do not change significantly under control. Physically, this can be explained as following: faster-moving detached modes do not have an appreciable signature at the wall, and so any control technique that employs wall-based sensing and actuation cannot interact such modes. Importantly, note that attached large-scale structures such as VLSMs do interact with and influence the NW region (Marusic et al.
Reference Marusic, Mathis and Hutchins2010). As a result, it may be possible to design wall-based control that suppresses such structures. There are also close similarities between the mode structures predicted for opposition control and for suboptimal control based on spanwise shear. In particular, the predicted mode structures suggest that suboptimal control is essentially a modified version of opposition control (Choi et al.
Reference Choi, Moin and Kim1994), with a phase lag that depends on mode speed and the magnitude of blowing and suction (i.e. the design parameter
$\unicode[STIX]{x1D6FC}^{+}$
).
In addition to considering the gain and structure for individual modes resembling the NW cycle and VLSMs, we also compared the effect of suboptimal control across spectral space. Both Case SP and Case ST favourably affect (i.e. suppress) structures with length scales comparable to the NW cycle. Case SP is predicted to detrimentally affect (i.e. further amplify) structures with length scales comparable to the VLSMs, while Case ST is predicted to have little or no effect on these larger structures. For Case SP, control performance is very sensitive to the magnitude of blowing and suction. As
$\unicode[STIX]{x1D6FC}^{+}$
increases, the spectral region over which modes are suppressed diminishes, while the region over which modes are further amplified widens.
To allow for a quantitative comparison between resolvent-based predictions and previous DNS, we estimated the design parameter
$\unicode[STIX]{x1D6FC}^{+}$
from previous databases. Specifically, a comparison of r.m.s. values for the turbulent spanwise shear and the wall-based blowing and suction suggests that
$\unicode[STIX]{x1D6FC}^{+}\approx 0.5{-}1.0$
in previous DNS for suboptimal control (Lee et al.
Reference Lee, Kim and Choi1998; Choi & Sung Reference Choi and Sung2002; Iwamoto et al.
Reference Iwamoto, Suzuki and Kasagi2002). For this range of
$\unicode[STIX]{x1D6FC}^{+}$
, resolvent analysis suggests that Case SP reduces the Reynolds shear stress contribution from NW modes by approximately
$25\,\%$
, which is consistent with the
$22\,\%$
reduction in drag observed in DNS.
For Case ST, resolvent analysis predicts that control leads to a small, but meaningful, reduction in gain for modes resembling the NW cycle. However, Lee et al. (Reference Lee, Kim and Choi1998) observed an increase in drag in their DNS. The results presented in § 4.1.2 provide a clear explanation for this discrepancy. Specifically, resolvent analysis reveals that Case ST can lead to a substantial increase in amplification for structures that are relatively long in the spanwise direction. High actuation of such energetic spanwise structures was confirmed in a limited DNS investigation at
$Re_{\unicode[STIX]{x1D70F}}\approx 180$
, both via instantaneous snapshots of the flow field and time-averaged statistics.
Finally, we developed a new measure which focuses only on the Reynolds stress contribution very close to the wall,
$R_{\boldsymbol{k},\mathit{attached}}$
. This measure provides insight into why the resolvent analysis occasionally yields differing predictions for changes in gain (
$\unicode[STIX]{x1D70E}_{\boldsymbol{k}c}/\unicode[STIX]{x1D70E}_{\boldsymbol{k}0}$
) and the Reynolds shear stress contribution (
$R_{\boldsymbol{k}c}/R_{\boldsymbol{k}0}$
) due to control.
Acknowledgement
The authors are grateful to Dr J. Kim (UCLA) and Dr B. J. McKeon (Caltech) for sowing a seed for this collaborative work at IPAM Workshop on Mathematics of Turbulence, Workshop IV: Turbulence in Engineering Applications, UCLA, CA, USA, November 17–21, 2014. S.N. and K.F. are also grateful to Drs S. Obi and K. Ando (Keio University) and Dr H. Mamori (Tokyo University of Science) for fruitful discussion. This work was supported through Grant-in-Aid for Scientific Research (C) (no. 25420129) by Japan Society for the Promotion of Science (JSPS). This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0142 (M.L.).
