Controlling the onset of turbulence by streamwise traveling waves. Part 1: Receptivity analysis

Controlling the onset of turbulence by streamwise traveling waves.
Part 1: Receptivity analysis

Rashad Moarref    Mihailo R. Jovanović
Abstract

We examine the efficacy of streamwise traveling waves generated by a zero-net-mass-flux surface blowing and suction for controlling the onset of turbulence in a channel flow. For small amplitude actuation, we utilize weakly nonlinear analysis to determine base flow modifications and to assess the resulting net power balance. Receptivity analysis of the velocity fluctuations around this base flow is then employed to design the traveling waves. Our simulation-free approach reveals that, relative to the flow with no control, the downstream traveling waves with properly designed speed and frequency can significantly reduce receptivity which makes them well-suited for controlling the onset of turbulence. In contrast, the velocity fluctuations around the upstream traveling waves exhibit larger receptivity to disturbances. Our theoretical predictions, obtained by perturbation analysis (in the wave amplitude) of the linearized Navier-Stokes equations with spatially periodic coefficients, are verified using full-scale simulations of the nonlinear flow dynamics in companion paper, Lieu, Moarref & Jovanović (2010).

1 Introduction

The problem of turbulence suppression in a channel flow using feedback control with wall-mounted arrays of sensors and actuators has recently received a significant attention. This problem is viewed as a benchmark for turbulence suppression in a variety of geometries, including boundary layers. Also, there has been mounting evidence that the linearized Navier-Stokes (NS) equations represent a good control-oriented model for the dynamics of transition. Recent research suggests that, in wall-bounded shear flows, one must account for modeling imperfections in the linearized NS equations since they are exceedingly sensitive to external excitations and unmodelled dynamics (see, for example, Trefethen et al. 1993; Farrell & Ioannou 1993; Jovanović & Bamieh 2005; Schmid 2007). This has motivated several research groups to use the linearized NS equations for model-based design of estimators and controllers in a channel flow (Bewley & Liu 1998; Lee et al. 2001; Kim 2003; Högberg et al. 2003a, b; Hœpffner et al. 2005; Chevalier et al. 2006; Kim & Bewley 2007; Vazquez & Krstic 2007a, b; Cochran & Krstic 2009). These results suggest that the proper turbulence suppression design paradigm is that of disturbance attenuation or robust stabilization rather than modal stabilization.

An alternative approach to feedback flow control relies on the understanding of the basic flow physics and the open-loop implementation of controls (i.e., without measurement of the relevant flow quantities and disturbances). Examples of sensorless strategies include: wall geometry deformation such as riblets, transverse wall oscillations, and control of conductive fluids using the Lorentz force. Although several numerical and experimental studies show that properly designed sensorless strategies may yield significant drag reduction, an obstacle to fully utilizing these physics-based approaches is the absence of a theoretical framework for their design and optimization.

An enormous potential of sensorless strategies was exemplified by Min et al. (2006), where direct numerical simulations (DNS) were used to show that a surface blowing and suction in the form of an upstream traveling wave (UTW) results in a sustained sub-laminar drag in a fully developed turbulent channel flow. The underlying mechanism for obtaining drag smaller than in a laminar flow is the generation of the wall region Reynolds shear stresses of the opposite signs compared to what is expected based on the mean shear. By assuming that a wall actuation only influences the velocity fluctuations, Min et al. (2006) determined an explicit solution to the two dimensional NS equations linearized around parabolic profile; they further used an expression for skin-friction drag in fully developed channel flows (Fukagata, Iwamoto & Kasagi 2002; Bewley & Aamo 2004), and showed that the drag is increased with the downstream traveling waves (DTWs) and decreased with the upstream traveling waves.

A comparison of laminar and turbulent channel flows with and without control was presented by *marjosmah07, where a criterion for achieving sub-laminar drag was derived. This study considered effectiveness of streamwise traveling waves at high Reynolds numbers and discussed why such controls can achieve sub-laminar drag. Another recent study, Hœpffner & Fukagata (2009), emphasized that the UTWs introduce a larger flux compared to the uncontrolled flow which motivated the authors to characterize the observed mechanism as a pumping rather than as a drag reduction. It was shown that, even with no driving pressure gradient, blowing and suction along the walls induces pumping action in a direction opposite to that of the wave propagation. By considering flows in the absence of velocity fluctuations Hœpffner & Fukagata (2009) showed that it costs more to drive a fixed flux with wall-transpiration type of actuation than with standard pressure gradient type of actuation. A fundamental limitation on the balance of power in a channel flow was recently examined by Bewley (2009); this study showed that any transpiration-based control strategy that results in a sub-laminar drag necessarily has negative net efficiency compared to the laminar flow with no control. Furthermore, *fuksugkas09 showed that a lower bound on the net driving power in a duct flow with arbitrary constant streamline curvature is determined by the power required to drive the Stokes flow. It was thus concluded that the flow has to be relaminarized in order to be driven with the smallest net power. However, since the difference between the turbulent and laminar drag coefficients grows quadratically with the Reynolds number, Marusic et al. (2007) argued that relaminarization may not be possible in strongly inertial flows. An alternative approach is to design a controller that reduces skin-friction drag in turbulent flows; provided that the control power is less than the saved power, a positive net efficiency can still be achieved.

In this paper, we show that a positive net efficiency can be achieved in a channel flow subject to streamwise traveling waves if the controlled flow stays laminar while the uncontrolled flow becomes turbulent. Starting from this observation, we develop a framework for design of the traveling waves that are capable of (i) improving dynamical properties of the flow; and (ii) achieving positive net efficiency. We quantify receptivity of the NS equations linearized around UTWs and DTWs to stochastic disturbances by computing the ensemble average energy density of the statistical steady-state. Motivated by our desire to have low cost of control we confine our study to small amplitude blowing and suction along the walls. This also facilitates derivation of an explicit formula for energy amplification (in flows with control) using perturbation analysis techniques. Our simulation-free design reveals that the UTWs are poor candidates for preventing transition; conversely, we demonstrate that properly designed DTWs are capable of substantially reducing receptivity of three dimensional fluctuations (including streamwise streaks and Tollmien-Schlichting (TS) waves). This indicates that the DTWs can be used as an effective means for controlling the onset of turbulence. Moreover, we show the existence of DTWs that result in a positive net efficiency compared to the uncontrolled flow that becomes turbulent. Our theoretical predictions are verified in Part 2 of this paper (Lieu et al. 2010) using DNS of the NS equations. Thus, our work (i) demonstrates that the theory developed for the linearized equations with uncertainty has considerable ability to capture full-scale phenomena; and (ii) exhibits the predictive power of the proposed perturbation-analysis-based method for designing traveling waves.

This paper represents an outgrowth of the study performed during the 2006 Center for Turbulence Research Summer Program (Jovanović, Moarref & You 2006). While Jovanović et al. (2006) only focused on receptivity of UTWs with large wavelength, our current study does a comprehensive analysis of the influence of both UTWs and DTWs on the fluctuations’ kinetic energy and the overall efficiency. We also note that linear stability and transient growth of traveling waves were recently examined by Lee, Min & Kim (2008). For selected values of parameters, it was shown that the UTWs destabilize the laminar flow for control amplitudes as small as of the centerline velocity; on the other hand, the DTWs with phase speeds larger than the centerline velocity remain stable even for large wave amplitudes. Moreover, the UTWs (DTWs) exhibit larger (smaller) transient growth relative to the uncontrolled flow. Our study confirms all of these observations; it also extends them at several different levels. First, we pay close attention to a net efficiency by computing the net power gained (positive efficiency) or lost (negative efficiency) in the presence of wall-actuation. Second, we conduct much more detailed study of the influence of traveling waves on velocity fluctuations; this is done by a thorough analysis of the influence of the wave speed, frequency, and amplitude on receptivity of full three dimensional fluctuations. Third, we confirm all of our theoretical predictions in Part 2 of this study, and highlight remaining research challenges.

Our presentation is organized as follows: in § 2, we formulate the governing equations in the presence of traveling wave wall-actuation. The influence of control on the nominal bulk flux and the nominal net efficiency is also discussed in this section. A frequency representation of the NS equations linearized around base velocity induced by traveling waves is presented in § 3. We further discuss a notion of the ensemble average energy density of the statistical steady-state and describe an efficient method for determining this quantity in flows subject to small amplitude traveling waves. In § 4, we employ perturbation analysis to derive an explicit formula for energy amplification. This formula is used to identify the values of wave frequency and speed that reduce receptivity of the linearized NS equations; we show that the essential trends are captured by perturbation analysis up to a second order in traveling wave amplitude. We also discuss influence of amplitude on energy of velocity fluctuations and reveal physical mechanisms for energy amplification. A brief summary of the main results along with an overview of remaining research challenges is provided in § 5.

2 Steady-state analysis

2.1 Governing equations

Consider a channel flow governed by the non-dimensional incompressible NS equations

(2.0)

with the Reynolds number defined in terms of the centerline velocity of the parabolic laminar profile and channel half-height , . The kinematic viscosity is denoted by , the velocity vector is given by , is the pressure, is the body force, is the gradient, and is the Laplacian. The spatial coordinates and time are represented by and , respectively.

Figure 1: A pressure driven channel flow with blowing and suction along the walls.

In addition to a constant pressure gradient, , the flow is exposed to a zero-net-mass-flux surface blowing and suction in the form of a streamwise traveling wave (see figure 1 for illustration). In the absence of the nominal body force, , base velocity represents the steady-state solution to (2.1) subject to

(2.0)

where , , and , respectively, identify frequency, speed, and amplitude of the traveling wave. Positive values of define a DTW, whereas negative values of define a UTW. The time dependence in can be eliminated by the Galilean transformation, This change of coordinates does not influence the spatial differential operators, but it transforms the time derivative to which adds an additional convective term to the NS equations

(2.0)

In new coordinates, i.e. in the frame of reference that travels with the wave, the wall-actuation (2.1) induces a two dimensional base velocity, which represents the steady-state solution to (2.1). Note that the spatially periodic wall actuation, induces base velocity which is periodic in .

The equations describing dynamics (up to a first order) of velocity fluctuations around base velocity, , are obtained by decomposing each field in (2.1) into the sum of base and fluctuating parts, i.e., , , , and by neglecting the quadratic term in

(2.0)

Note that the boundary conditions (2.1) are satisfied by base velocity and, thus, velocity fluctuations acquire homogeneous Dirichlet boundary conditions.

2.2 Base flow

Let us first consider a surface blowing and suction of a small amplitude . In this case, a weakly nonlinear analysis can be employed to solve (2.1) subject to (2.1) and determine the corrections to base parabolic profile; similar approach was previously used by Jovanović et al. (2006); Hœpffner & Fukagata (2009). Up to a second order in control amplitude , and can be represented as

where denotes base velocity in Poiseuille flow and (see Appendix A)

(2.0)
Figure 2: Second order correction to the nominal flux, , for (a) upstream waves; and (b) downstream waves in Poiseuille flow with . Note: the level sets are obtained using a sign-preserving logarithmic scale; e.g., and should be interpreted as and , respectively.

Hœpffner & Fukagata (2009) recently showed that, in the absence of driving pressure gradient, the traveling waves induce nominal bulk flux (i.e., pumping) in the direction opposite to the direction in which the wave travels. While the first order of correction to the base velocity is purely oscillatory, the quadratic interactions in the NS equations introduce mean flow correction at the level of . The nominal bulk flux is determined by where the overline denotes averaging over horizontal directions. In the presence of a pressure gradient, the nominal flux in flow with no control is , and the second order correction (in ) to is given by . Figure 2 shows as a function of wave frequency, , and wave speed, , in Poiseuille flow with . Except for a narrow region in the vicinity of , the upstream and downstream waves increase and reduce the nominal flux, respectively. Furthermore, for a given wave speed , the magnitude of the induced flux increases as the wave frequency is decreased.

Figure 3: (a) The nominal flux, ; and (b) the nominal skin-friction drag coefficient, , for a pair of UTWs and a pair of DTWs in Poiseuille flow with . The results are obtained by solving (2.1) subject to (2.1), in the steady-state, using Newton’s method; and of the uncontrolled laminar and turbulent flows are also shown for comparison.

Figure 3 is obtained by finding the steady-state solution of (2.1) subject to (2.1) using Newton’s method. Originally, we have used base flow resulting from the weakly nonlinear analysis to initialize Newton iterations; robustness of our computations is confirmed using initialization with many different incompressible base flow conditions. The nominal flux and its associated nominal drag coefficient for a UTW with and , and a DTW with and are shown in this figure. The flux and drag coefficient of both laminar and turbulent flows with no control are also given for comparison. The nominal skin-friction drag coefficient is defined as (McComb 1991)

where is the nondimensional average wall-shear stress. For the fixed pressure gradient, , the nominal skin-friction drag coefficient is inversely proportional to square of the nominal flux and, in uncontrolled laminar flow with , we have . The UTWs produce larger nominal flux (and, consequently, smaller nominal drag coefficient) compared to both laminar and turbulent uncontrolled flows. On the other hand, the DTWs yield smaller nominal flux (and, consequently, larger nominal drag coefficient) compared to uncontrolled laminar flow. In situations where flow with no control becomes turbulent, however, the DTWs with amplitudes smaller than a certain threshold value may have lower nominal drag coefficient than the uncontrolled turbulent flow; e.g., for a DTW with and this threshold value is given by (cf. figure 3).

2.3 Nominal net efficiency

For the fixed pressure gradient, the difference between the flux of the controlled and the uncontrolled flows results in production of a driving power (per unit horizontal area of the channel)

where and are the nominal flux of the controlled and uncontrolled flows, respectively. On the other hand, the required control power exerted at the walls (per unit horizontal area of the channel) is given by (Currie 2003)

(2.0)

The control net efficiency is determined by the difference of the produced and required powers (Quadrio & Ricco 2004)

where signifies the net power gained (positive ) or lost (negative ), in the presence of wall-actuation.

For small control amplitudes, the produced power can be represented as

where

The nominal required control power can be determined from (2.0) by evaluating the horizontal average of the product between base pressure, , and base wall-normal velocity, , at the walls. Since, at the walls, the nonzero component of contains only first harmonic in (cf. (2.0)), we need to determine the first harmonic (in ) of to compute . Base pressure can be obtained by solving the two dimensional Poisson equation

(2.0)

where satisfies the following Neumann boundary conditions

These are determined by evaluating the -momentum equation at the walls. For small values of , weakly nonlinear analysis, in conjunction with the expressions for and given in § 2.2, can be employed to solve (2.0) for base pressure

where and are determined from

Here, the prime denotes the partial derivative with respect to , and the required power can be represented as

Since the second order correction to the nominal produced power, , is directly proportional to , is positive for UTWs and negative for DTWs. It turns out that smaller choices of result in larger produced (for UTWs) or lost (for DTWs) power. One of the main points of this paper, however, is to show that it may be misleading to rely on the produced power as the only criterion for selection of control parameters; in what follows, we demonstrate that the required control power as well as the dynamics of velocity fluctuations need to be taken into account when designing the traveling waves.

Figure 4: Second order correction to the nominal net efficiency, , for (a) upstream waves; and (b) downstream waves in Poiseuille flow with . Note: the level sets are obtained using a sign-preserving logarithmic scale; e.g., should be interpreted as .
Figure 5: The steady-state net efficiency, , of laminar controlled flows as a function of control amplitude for a UTW with and a DTW with at . The results are obtained by assuming that the uncontrolled flow (a) remains laminar; and (b) becomes turbulent.

2.4 Nominal efficiency of laminar controlled flows

We next examine the nominal efficiency of laminar controlled flows. Since we are interested in expressing the nominal efficiency relative to the power required to drive flow with no control, we provide comparison with both laminar and turbulent uncontrolled flows. The net efficiency in fraction of the power required to drive the uncontrolled laminar flow is determined by

(2.0)

where and . It can be shown that the second order correction to , , is negative for all choices of and (see figure 4). This is because the required power for maintaining the traveling wave grows faster than the produced power as is increased. In addition, figure 4 shows that is minimized for small wave speeds and for . Formula (2.0) demonstrates that the control net efficiency is negative whenever the uncontrolled flow stays laminar (cf. figure 5). This is a special case of more general results by Bewley (2009) and Fukagata et al. (2009) which have established that any transpiration-based control strategy necessarily has negative net efficiency compared to the laminar uncontrolled flow.

On the other hand, the net efficiency of the laminar controlled flow in fraction of the power required to drive the uncontrolled turbulent flow is determined by

(2.0)

where . Since the bulk flux of the uncontrolled turbulent flow is smaller than that of the uncontrolled laminar flow (i.e., ), it is possible to obtain a positive net efficiency for sufficiently small values of . Note that formula (2.0) is derived under the assumption that the controlled flow stays laminar while the uncontrolled flow becomes turbulent. Clearly, this formula represents an idealization since it assumes that laminar flow can be maintained by both UTWs and DTWs even with infinitesimal control amplitudes. It also indicates that increasing the control amplitude always decreases the nominal net efficiency. In a nutshell, the control amplitude needs to be large enough to maintain a laminar flow but increasing the control amplitude beyond certain value brings the efficiency down and eventually leads to negative efficiency. If the efficiency is negative, maintaining a laminar flow does not lead to any net benefit in the presence of control. This is further illustrated in figure 5 where Newton’s method is used to show that a positive net efficiency can be achieved for control amplitudes smaller than a certain threshold value (e.g., for the DTW with and ). In addition, the net efficiency monotonically decreases as is increased, as predicted by the weakly nonlinear analysis up to a second order in (cf. (2.0)).

Figure 6: The wave amplitude, , for which the nominal net efficiency, , is positive at for (a) a pair of UTWs; and (b) a pair of DTWs. The solid curves are computed using Newton’s method, and the dotted curves are computed using (2.0). The results are obtained by assuming that the controlled flow stays laminar while the uncontrolled flow becomes turbulent.

An estimate for the maximum value of for which a positive net efficiency is attainable can be obtained by solving the following equation (obtained using weakly nonlinear analysis)

(2.0)

Figure 6 shows as a function of for different values of . The dotted curves denote the approximation for obtained using (2.0). The values of (solid curves) obtained using Newton’s method are also shown for comparison; we see that the predictions based on the second order correction capture the essential trends and provide good estimates for (especially for large wave speeds and for wave frequencies between and ). Figures 5 and 6 are obtained by assuming that the flow with control stays laminar while the flow with no control becomes turbulent. Whether or not the traveling waves can control the onset of turbulence depends on the velocity fluctuations; addressing this question requires analysis of the dynamics, which is a topic of § 3 and § 4, where we examine receptivity of velocity fluctuations around UTWs and DTWs to stochastic disturbances.

3 Dynamics of fluctuations around traveling waves

3.1 Evolution model with forcing

A standard conversion of (2.1) to the wall-normal velocity ()/vorticity () formulation removes the pressure from the equations and yields the following evolution model with forcing

(3.0)

This model is driven by the body force fluctuation vector , which can account for flow disturbances. We refer the reader to a recent review article (Schmid 2007) and a monograph (Schmid & Henningson 2001) for a comprehensive discussion explaining why it is relevant to study influence of these excitations on velocity fluctuations. The internal state of (3.0) is determined by , with Cauchy (both Dirichlet and Neumann) boundary conditions on and Dirichlet boundary conditions on . All operators in (3.0) are matrices of differential operators in three coordinate directions , , and . Operator in (3.0) captures a kinematic relation between and , operator describes how forcing enters into the evolution model, whereas operators and determine internal properties of the linearized NS equations (e.g., modal stability). While operators , , and do not depend on base velocity, operator is base-velocity-dependent and, hence, it determines changes in the dynamics owing to changes in (see Appendix B). Moreover, for base velocity of § 2.2, inherits spatial periodicity in from and it can be represented as

where and are spatially invariant operators in the streamwise and spanwise directions and signifies that takes the values . This expansion isolates spatially invariant and spatially periodic parts of operator , which is well-suited for representation of (3.0) in the frequency domain.

Figure 7: A Bloch wave defined in (3.0) is obtained by the superposition of weighted Fourier exponentials with frequencies , with weights determined by .

3.2 Frequency representation of the linearized model

Owing to the structure of the linearized NS equations, the differential operators , , and are invariant with respect to translations in horizontal directions. On the other hand, operator is invariant in and periodic in . Thus, the Fourier transform in can be applied to algebraize the spanwise differential operators. In other words, the normal modes in are the spanwise waves, , where denotes the spanwise wavenumber. On the other hand, the appropriate normal modes in are given by the so-called Bloch waves (Odeh & Keller 1964; Bensoussan et al. 1978), which are determined by a product of and the periodic function in , with . Based on the above, each signal in (3.0) (for example, ) can be expressed as

where only real parts are to be used for representation of physical quantities. Expressing in Fourier series yields (see figure 7 for an illustration)

(3.0)

where are the coefficients in the Fourier series expansions of .

The frequency representation of the linearized NS equations is obtained by substituting (3.0) into (3.0)

(3.0)

This representation is parameterized by and and denotes a bi-infinite column vector, The same definition applies to and . On the other hand, for each and , , , and are bi-infinite matrices whose elements are one dimensional integro-differential operators in . The structure of these operators depends on frequency representation of , , , and in (3.0). In short, and are block-diagonal operators and

where and are structured operators (see Appendix B for more details). The particular structure of and is exploited in perturbation analysis of the energy amplification for small control amplitudes in § 3.4.

3.3 Energy density of the linearized model

Frequency representation (3.0) contains a large amount of information about linearized dynamics. For example, it can be used to assess stability properties of the base flow. However, since the early stages of transition in wall-bounded shear flows are not appropriately described by the stability properties of the linearized equations (see, for example, Schmid & Henningson 2001; Schmid 2007), we perform receptivity analysis of stochastically forced model (3.0) to assess the effectiveness of the proposed control strategy. Namely, we set the initial conditions in (3.0) to zero and study the responses of the linearized dynamics to uncertain body forces. When the body forces are absent, the response of stable flows decays asymptotically to zero. However, in the presence of stochastic body forces, the linearized NS equations are capable of maintaining high levels of the steady-state variance (Farrell & Ioannou 1993; Bamieh & Dahleh 2001; Jovanović & Bamieh 2005). Our analysis quantifies the effect of imposed streamwise traveling waves on the asymptotic levels of variance and describes how receptivity changes in the presence of control. We note that there are substantial differences between the problem considered here and in Jovanović & Bamieh (2005); these differences arise from lack of homogeneity in the streamwise direction which introduces significant computational challenges which we discus below. Furthermore, even though our study is similar in spirit to Jovanović (2008), current work studies dynamics of fluctuations around spatially periodic base velocity, whereas Jovanović (2008) considered dynamics of fluctuations around time periodic base velocity. Theoretical framework for quantifying receptivity in these two conceptually different cases was developed by Fardad, Jovanović & Bamieh (2008) and Jovanović & Fardad (2008), respectively.

Let us assume that a stable system (3.0) is subject to a zero-mean white stochastic process (in and ), . Then, for each and , the ensemble average energy density of the statistical steady-state is determined by

where denotes the inner product and averaging in time, i.e.,

(3.0)

and is the tensor product of with itself. We note that determines the asymptotic level of energy (i.e., variance) maintained by a stochastic forcing in (3.0). Typically, this quantity is computed by running DNS of the NS equations until the statistical steady-state is reached. However, for linearized system (3.0), the energy density can be determined using the solution to the following operator Lyapunov equation (Fardad et al. 2008)

(3.0)

as

Here, denotes the autocorrelation operator of , that is

Since is an identity operator, we have

(3.0)

where denotes the elements on the main diagonal of operator . We note that also has an interesting deterministic interpretation; namely, if denotes the impulse response of (3.0), then

Thus, the same quantity can be used to assess receptivity of the linearized NS equations to exogenous disturbances of either stochastic or deterministic origin.

3.4 Perturbation analysis of energy density

Solving (3.0) is computationally expensive; a discretization of the operators (in ) and truncation of the bi-infinite matrices convert (3.0) into a large-scale matrix Lyapunov equation. Our computations suggest that in order to obtain convergence of

a choice of between ten (for ) and a few thousands (for ) is required. Since we aim to conduct a detailed study of the influence of streamwise traveling waves on dynamics of velocity fluctuations, determining energy density for a broad range of traveling wave parameters, and still poses significant computational challenges.

Instead, we employ an efficient perturbation analysis based approach introduced by Fardad & Bamieh (2008) for solving equation (3.0). For our problem, this approach turns out to be at least times faster than the truncation approach. This method is well-suited for systems with small amplitude spatially periodic terms and it converts (3.0) into a set of conveniently coupled system of operator-valued Lyapunov and Sylvester equations. A finite dimensional approximation of these equations yields a set of algebraic matrix equations whose order is determined by the product between the number of fields in the evolution model (here , the wall-normal velocity and vorticity) and the size of discretization in . While consideration of small wave amplitudes simplifies analysis by providing an explicit expression for energy density, it is also motivated by our earlier observation that large values of introduce high cost of control which is not desirable from a physical point of view.

It can be shown (see Appendix C for details) that the energy density of system (3.0) can be represented as

(3.0)

Thus, only terms with even powers in contribute to , which in controlled flow depends on six parameters. Since our objective is to identify trends in energy density, we confine our attention to a perturbation analysis up to a second order in . We briefly comment on the influence of higher order corrections in § 4.3 where it is shown that the essential trends are correctly predicted by the second order of correction.

4 Energy amplification in Poiseuille flow with

In this section, we study energy amplification of stochastically forced linearized NS equations in Poiseuille flow controlled with streamwise traveling waves. Equation (3.0) reveals the dependence of the energy density on traveling wave amplitude , for . However, since the operators in (3.0) depend on the spatial wavenumbers ( and ), , , and , the energy density is also a function of these parameters. Finding the optimal triple that maximally reduces the energy of the velocity fluctuations is outside the scope of the current study; instead, we identify the values of and that are capable of reducing receptivity in the presence of small amplitude streamwise traveling waves. Since we are interested in energy amplification of the transitional Poiseuille flow, we choose in all of our subsequent computations. This value is selected because it is between the critical Reynolds number at which linear instability takes place, , and the value at which transition is observed in experiments and DNS, . The same Reynolds number was used by Min et al. (2006) in their DNS study.

Figure 8: Energy density of the uncontrolled Poiseuille flow with . The plot is given in the log-log-log scale.

4.1 Energy density of flow with no control

We briefly comment on the energy density in uncontrolled Poiseuille flow with ; for an in-depth treatment see Jovanović & Bamieh (2005). The appropriate normal modes in the uncontrolled flow are purely harmonic streamwise and spanwise waves, , where denotes the streamwise wavenumber. Figure 8 illustrates the energy density of the uncontrolled flow as a function of and , which we denote by . The streamwise constant fluctuations with spanwise wavenumbers carry most energy in flow with no control. Namely, the largest value of occurs at (, ), which means that the most amplified flow structures (the streamwise streaks) are infinitely elongated in the streamwise direction and have the spanwise length scale of approximately , where is the channel half-height. We note that these input-output resonances do not correspond to the least-stable modes of the linearized NS equations. Rather, they arise because of the coupling from the wall-normal velocity to the wall-normal vorticity . Physically, this coupling is a product of the vortex tilting (lift-up) mechanism (Landahl 1975); the base shear is tilted in the wall-normal direction by the spanwise changes in , which lead to a nonmodal amplification of . This mechanism does not take place either when the base shear is zero (i.e., ), or when there are no spanwise variations in (i.e., ). On the other hand, the least-stable modes (TS waves) of uncontrolled flow create a local peak in around (, ), with a magnitude significantly lower compared to the magnitude achieved by the streamwise constant flow structures. Finally, we note that the uncontrolled energy density as appeared in (3.0) can be obtained from using the following expression

In other words, for fixed and , represents the energy density of velocity fluctuations that are composed of all wavenumbers . In comparison, is the energy density of velocity fluctuations composed of a single wavenumber (see figure 9 for an illustration).

Figure 9: For fixed and , represents energy density of fluctuations composed of all wavenumbers ;

4.2 Energy amplification of flow with control

We next consider energy amplification of velocity fluctuations in Poiseuille flow with in the presence of both UTWs and DTWs. As shown in § 3.4, for small amplitude blowing and suction along the walls, the perturbation analysis yields an explicit formula for energy amplification (cf. (3.0)),

Thus, for small wave amplitudes the influence of control can be assessed by evaluating function that quantifies energy amplification up to a second order in . Sign of determines whether energy density is increased or decreased in the presence of control; positive (negative) values of identify wave speed and frequency that increase (decrease) receptivity. Since function is sign-indefinite with vastly different magnitudes, it is advantageous to visualize using a sign-preserving logarithmic scale

For example, or , respectively, signify or . Since depends on four parameters, for visualization purposes, we confine our attention to cross-sections of by fixing two of the four parameters. We first study energy amplification of the modes with and as a function of and ; these spanwise wavenumbers are selected in order to capture influence of control on streamwise streaks and TS waves, respectively. Since, in uncontrolled flow, streamwise streaks (respectively, TS waves) occur at (respectively, ), fluctuations with (respectively, ) are considered; these values of are chosen to make sure that streamwise streaks (respectively, TS waves) represent modes of the controlled flow as well. (Here, denotes the largest integer not greater than .) We then analyze the energy amplification of disturbances with different values of and for a fixed set of control parameters and . Our analysis illustrates the ability of properly designed traveling waves to weaken the intensity of both most energetic and least stable modes of the uncontrolled flow. Direct numerical simulations of Part 2 show that this can be done with positive net efficiency.

Since most amplification in flow with no control occurs for fluctuations with , , it is relevant to first study the influence of controls on these most energetic modes. In flow with control, the streamwise-constant flow structures are imbedded in the fundamental mode, i.e. fluctuations with (cf. § 3.2). As the plots of in figures 10 and 10 reveal, the values of and determine whether these structures are amplified or attenuated by the traveling waves. Up to a second order in , the control parameters associated with the blue regions in these two figures reduce the energy amplification of the uncontrolled flow. As evident from figure 10, only a narrow range of UTWs with is capable of reducing the energy amplification. However, since the required power for maintaining the nominal flow for such low frequency controls is prohibitively large (cf. figure 10), the choice of UTWs for transition control is not favorable from efficiency point of view (receptivity reduction by these UTWs is further discussed in § 4.3). On the other hand, a large range of DTW parameters with and is capable of making the controlled flow less sensitive to stochastic excitations (cf. figure 10). Moreover, figure 10 shows that the region contains the smallest required power for sustaining the DTWs. These two features identify properly designed DTWs as suitable candidates for controlling the onset of turbulence with positive net efficiency (as confirmed by DNS in Part 2).

upstream: upstream:
downstream: downstream:
Figure 10: (a) and (c): Second order correction to the energy amplification, , of the modes with , in the presence of (a) UTWs; and (c) DTWs in Poiseuille flow with . (b) and (d): Second order correction to the nominal required power, , for (b) UTWs; and (d) DTWs. The dot and the square, respectively, denote (as selected in Min et al. (2006)) and . Note: the color plots are obtained using a sign-preserving logarithmic scale; e.g., and should be interpreted as and , respectively.

It is noteworthy that traveling waves with parameters considered in Min et al. (2006) (i.e., and ) increase amplification of the most energetic modes of the uncontrolled flow (cf. figure 10). This is in agreement with a recent study of Lee et al. (2008) where a transient growth larger than that of the laminar uncontrolled flow was observed for UTWs with and . Furthermore, it is shown in Part 2 that such UTWs promote turbulence even for initial conditions for which the uncontrolled flow stays laminar.

The above analysis illustrates the ability of the DTWs to weaken the intensity of the most energetic modes of the uncontrolled flow; this is achieved by reducing receptivity to stochastic disturbances. However, an important aspect in the evaluation of any control strategy is to consider the influence of controls on all of the system’s modes. In view of this, we next discuss how control affects the full three dimensional fluctuations. Since for a given the energy amplification is symmetric around , it suffices to only consider the modes with . Figure 11 shows for a UTW with , , and three DTWs with , , , , and , . As evident from figure 10, the selected UTW increases amplification of the fundamental mode with ; on the other hand, all three DTWs reduce energy amplification of modes with , . Figure 11 further reveals that the largest change in amplification for all of these traveling waves takes place at , , which is precisely where the uncontrolled flow contains most energy. This observation suggests presence of resonant in