Small internal waves in sheared flows: lower bound for the vorticity transient growth and propagation bound in the parameter space
We show that transient kinetic energy growth of small traveling waves inside incompressible viscous wall flows is not a sufficient condition for their enstrophy (the size of vorticity) growth.
Historically, mathematical difficulties related to the wall vorticity boundary conditions left the vorticity perturbation problem open. In the last decades of the 20th century, the discovery of the transient perturbation growth and its link to the subcritical transition to turbulence was still treated by means of energy-based analysis.
By using the non-modal approach, we extend the work of J. L. Synge (1930s) which was an alternative way to attack the flow stability problem based on the flow vorticity analysis instead of the classical kinetic energy analysis. Our calculations leads to the lower bound for the algebraic enstrophy growth as a function of the perturbation wavelength. This highlights how the enstrophy monotonic decay region inside the wavenumber - Reynolds number () map is wider than that of the kinetic energy.
By taking the general point of view of internal wave propagation in a medium, we present and discuss the dispersion variability inside the stability map. By keeping fixed the flow control parameter, we observe that this variability implies that inside a general wave packet, there will be a subset of waves that disperse aside a subset of waves that do not. In association to a situation where wave packets undergo powerful and long-lasting transient growth, this information is key to the understanding of a potential process of nonlinear coupling onset.
The dynamics of a fluid in motion is intrinsically linked to the evolution of the velocity gradient tensor. In recent decades, the local statistical and geometric structure of two and three-dimensional turbulent flows has been often described in terms of the properties of the velocity gradient tensor, in particular of the local strain rates and vorticity. The size of vorticity, the enstrophy, is a velocity gradient tensor invariant, which is commonly considered statistically relevant in turbulence dynamics Frisch (1995); Abramov and Majda (2003); Yeung et al. (2015); Schumacher et al. (2015) and has been correlated to the evolution of the kinetic energy and dissipation of the motion fluctuations Tsinober (2001). Moreover, the enstrophy is not a three-dimensional inviscid invariant, as it is the kinetic energy. This is due to the vortex stretching terms which are responsible for the stretching and tilting of the vorticity. In fact, these terms appear as nonlinear terms in the enstrophy integral evolution equation, and they are responsible for self-amplification of vorticity. Hence, differently from the kinetic energy, the enstrophy growth rate depends on the amplitude of the initial perturbation. These reasons make considering the vorticity a useful tool to gain insight into the physics of fluid flows. However, the level of interest in the enstrophy has not been as high as that in kinetic energy; at least in regards of the literature dedicated to the prodrome of the transition to turbulence, that is the hydrodynamic instability.
One reason for the limited use of the enstrophy variable in hydrodynamic stability theory is certainly due to the lack of knowledge of physical wall boundary conditions on the vorticity, see the discussion in Synge (1936). Unlike the velocity field, for which the boundary conditions for the flows bounded by solid walls (no-slip boundary conditions) are well known for more than a century now, at least for the wide class of wall flows under the continuum hypothesis. Notwithstanding this, our work is focused on the enstrophy of traveling perturbation waves in wall flows. In particular, we are interested in the threshold for possible enstrophy transient growth of perturbations during their temporal and spatial evolution. We consider the two-dimensional plane Poiseuille and Couette flows, which are emblematic problems of the hydrodynamic stability theory. One objective is to highlight the role of the enstrophy as well as its interrelationship with the more commonly considered kinetic energy.
Departures from the base state may appear in the subcritical range below , that is the threshold for unconditional instability. The energy method Joseph (1976) generates a lower bound to the unconditional or global stability threshold, Manneville (2016). This lower bound called specifies the value below which the kinetic energy contained in any perturbation to the base flow decreases to zero in a monotonic way. The condition defining instead stands on the ultimate decay of the perturbations, which may arrive at the end of long transients during where the kinetic energy may range significantly before monotonically decrease.
In this paper, for all possible initial conditions, we obtain the limiting curve in the base flow Reynolds number, , wavenumber map for the monotonic decay of the integral enstrophy and show that it is less restrictive than the limiting curve for the kinetic energy decay. This is accomplished by extending the non-modal approach to a procedure proposed by J. L. Synge long time ago, in 1938, in a proceeding paper of the London Mathematical Society Synge (1938a) that, to our knowledge, has not been further exploited. The impact of this result is that the lower bound for perturbation transient growth is improved if we consider the problem in terms of enstrophy instead of kinetic energy.
In synthesis, Synge’s procedure was aimed to find analytical conditions satisfied by both the vorticity and the stream function in a viscous liquid moving in two dimension between fixed parallel planes. This procedure is based on the deduction of the cross derivative of the flow vorticity by using the Orr-Sommerfeld equation, which is then coupled to an optimization process acting directly of the vorticity integral. An interesting side result is that it is possible to prove that the only terms that can produce a transient growth of the integral enstrophy are boundary terms which represent the vorticity and cross vorticity variation at the walls. This result is contextualized within the structure of the wavenumber- Re maps that describe the maximum transient growth of perturbations and the time necessary to reach it.
I Relationship between enstrophy and energy for small internal waves in parallel flows. The problem of enstrophy growth
Let us introduce the integral enstrophy for the traveling wave perturbation in a two-dimensional parallel flow field:
were is an arbitrary two-dimensional domain and its volume.
is the vorticity of the perturbation velocity field of components (longitudinal) and (wall-normal).
Since we are interested in the evolution equation of the integral enstrophy and kinetic energy , it is convenient to consider the linearized viscous vorticity equation for small disturbances:
where the prime symbol stands for a total y-derivative, is the basic flow state and
is the Reynolds number based on the channel half-height and the reference velocity, the centerline speed for PPF and the wall speed for PCF (figure 1).
The enstrophy equation is then:
In the following, we introduce the stream-function formulation , for the perturbation and adopt the following Fourier representation
where is any generic perturbed quantity, is the imaginary unit, is the wavenumber. We will consider a single wave component at a time.
The evolutive equation for small-amplitude two-dimensional wave perturbations is known as the Orr-Sommerfeld initial-value problem. In terms of the perturbation streamfunction , the aoociated equation can be written as:
According to the refrence frame introduced in figure 1, the plane Poiseuille base flow is
while the plane Couette base flow is
The associated initial-value problem is then formulated by adding the initial condition
see figure 2, and the no-slip boundary conditions
where stand for real and imaginary part, respectively; the integral enstrophy as:
The last expression was obtained by using integration by parts.
It is interesting to observe that the integral enstrophy can be split in two parts
is the integral kinetic energy of the perturbation, and is
a positive quantity related to the streamwise component of the velocity perturbation and its cross-derivative. Notice that, for wavenumbers of order one, the integral enstrophy is always greater than the integral kinetic energy, as can be seen in the example of figure 4. These wavenumbers are typically the most unstable, both asymptotically and in the transient.
The temporal evolution equations for and are derived as follows:
where the bar symbol stands for the complex conjugate and for the real part. The right-hand side of Eq. 16 and Eq. 17 comes directly from Eq. I. At this point, the enstrophy equation for a small wavy perturbation can be obtained as from the two equations above, by considering the base flow expressions 7, 8 and the boundary conditions 10:
Notice that all the convective terms, which are those containing the base flow , do not appear in the above enstrophy evolution equation. In fact, many of them drop out by taking the real part in of Eq. 16 and 17. Others vanish because they are contained in both and with opposite sign. As a consequence, the temporal evolution of the enstrophy is physically determined by the diffusive terms of the motion equation and can be factored out. On top of that, it is of great importance that the only term that can produce a temporal growth is the boundary term associated to the tho wall values of the cross flow variation of the streamwise perturbation.
Our aim is to find a lower bound for the enstrophy transient growth in terms of the Reynolds number, that is:
meaning that for it exists at least one initial condition leading to enstrophy temporal growth in the transient.
It is interesting to focus on the term in Eq. I since, as observed above, it is the only term that can be positive and thus can induce a possible growth. However, boundary conditions on vorticity are notoriously unknown a priori. As an example of discussion, the reader may see Synge 1936 Synge (1936). This fact has represented the main obstacle to the solution of problem 19. The mathematical formulation developed by Synge in 1938 was a peculiar application of the modal temporal theory to the vorticity equation (see Synge, 1938b, Eq.11.28), and Synge (1938a). In synthesis, the method is the following. By multiplying the OS equation by an exponential factor acting on the cross-flow direction, Synge obtained two integral relationships 20 yielding the link between the wall values of the vorticity and its y-derivative (actually, the part of the vorticity associated to the cross-flow momentum variation). By using these equations, he gets un updated integral enstrophy equation which is then optimized to maximize the enstrophy decay as a function of the base flow Reynolds number. At the time, the author was aimed at finding a lower bound for linear asymptotic stability and, ultimately, conditions for linear instability. That is, the focus was on seeking the marginal stability curve and the unconditional instability threshold (showed in figure 3), which justifies the use of the exponential time factor in the perturbative hypothesis. Today we know that Orszag (1971) for PPF, while, for PCF Romanov (1971) . Since at the time, the phenomenon of non-modal transient growth was unknown - it was discovered in the late nineties Trefethen et al. (1993)- Synge could not be aware that his computations would lead instead to a much lower bound for the algebraic transient growth of the vorticity. His calculations worked out for the plane Poiseuille flow but not for the plane Couette flow. In this work, in the place of the exponential time dependence () Synge (1938b, a), we use the non-modal approach () and solve Eq. 19 for both the plane Couette and Poiseuille flows.
Ii Lower bound for the enstrophy transient growth. Results and Discussion
The full mathematical formulation developed in order to solve Eq. 19 is reported in the Supplementary Information Appendix, see sections S2 and S3. Even though our procedure does not impose any temporal dependence, conditions for enstrophy monotonic decay in the case of PPF were formally obtained as done by J. L. Synge in Synge (1938a).
The route to the solution of the problem 19 is made of four steps:
(i) obtain the following conditions (see also ref. Synge (1936)):
(ii) use Eq. 20 in the enstrophy equation I. Get the enstrophy growth rate, , parameterized with all the possible boundary terms ; (iii) by calculus of variations, get a 6th order PDE for the perturbation maximizing the enstrophy growth rate functional; (iv) solve it and obtain - from the corresponding maximal enstrophy functional - the region of the map where transient enstrophy growth is not allowed, i.e. the curve (solved both analytically and via numerical optimization for PPF, just numerically for PCF). The expression for the plane Couette flow, unexpectedly, is apparently more complicated due to an additional term (the last term in Eq. A.2.1).
The minimum value of for which 2D perturbations can experience transient enstrophy growth is named . In the case of plane Couette flow, we found the value , occurring at a wavenumber . For the plane Poiseuille flow instead, at (see figure 3)
Figure 3 compares the enstrophy lower bound for transient growths with the bound for the kinetic energy, that is the curve (note that throughout this discussion we adopt the terminology used by Manneville Manneville (2016)). The kinetic energy problem was first formulated by Orr Orr (1907a), and subsequently by Synge Synge (1938b) and Joseph Joseph (1976), while numerical solutions for the three-dimensional case have been obtained many years later Reddy and Henningson (1993). The monotonic decay threshold for the kinetic energy is considered a lower bound for the global stability threshold , defined as the smallest Reynolds number allowing conditional departure to turbulence. Chapman (2002); Manneville (2015). That is, for Reynolds number values below the flow returns to the laminar state whatever the initial disturbance amplitude. Even if our study is two-dimensional, a brief review of the experimental values of and the related literature, for the three-dimensional problem, is reported in figure 4.
We computed using the well known energy method, based on a variational formulation Orr (1907b); Joseph (1976); Reddy and Henningson (1993). This is shown by the white curves in figure 3, and by the white regions in figure 4 and figure S5, top panels. A relevant outcome is that the threshold for enstrophy monotonic decay for longitudinal waves is greater than the threshold for the kinetic energy , for any wavenumber. This is highlighted by the pink region of figure 3. That is, there is a region in the - space where transient kinetic energy growth can occur, while it is forbidden for the enstrophy, for any initial perturbation.
In support to the results given by the analytical procedure, we performed numerical simulations of the initial-value problem I-10 by using the method described in reference De Santi et al. (2016).
The Reynolds number-wavenumber maps of the maximal kinetic energy and enstrophy, normalized to the initial values, , , are shown in figure 4 .
To our knowledge, enstrophy maps have not yet been presented in the literature. A fortiori, neither were stability maps including both enstrophy and wave dispersion properties. Instead, maps of kinetic energy have been shown Gustavsson (1991); Reddy et al. (1993); Schmid (2007), but they represent the maximum amplification over all possible initial conditions.
Here, the initial condition is smooth and excites both symmetric and antisymmetric Orr-Sommerfeld modes. It was chosen in order to trigger a transient energy growth for almost any above the limit . By an optimization process, we get the perturbation giving the maximal kinetic energy growth rate in the surrounding of , (for PPF , for PCF ). Notice that for both flows, these values fall inside the pink region of figure 3. In this way, we were able to show that the kinetic energy growth is not a sufficient condition for enstrophy growth. Indeed, as predicted by the analytical result, the vorticity starts to experience a transient growth only for , see figure 4 (c), (d).
Comments are now proposed about the map structure. It is possible to observe that the internal structure of both kinetic energy and enstrophy maps reflects the trend of the lower bound for transient growth. This fact can be observed from the iso-lines of and in the low-wavenumber region of figure 4. This translates into the scaling , . The exponents , depend on the initial condition, and for the cases span here for PCF, for PPF; for PCF, for PPF. Considering the non-dimensional time necessary to achieve the maximal growth, we observe the same scaling for the enstrophy and the kinetic energy, for both PCF and PPF, as shown in Figure S5: . This time scale appears therefore to be very mildly dependent on the Reynolds number, in the limit of high and small wavenumber.
Figures 3, 4, and S5, also include information about wave propagation properties.
Firstly, we observe that the parameter space is split in two regions which have different dispersion characteristics. This result has been observed and reported recently by our research group for the plane Poiseuille and wake flows, see ref. Santi et al. (2015); De Santi et al. (2016). For PPF the boundary between the two regions is represented by a curve named, in the following, . Below this boundary, waves travel in a dispersive way which may be mild or intense (see figure 2 in De Santi et al. (2016) and chapter 2 in Fraternale (2017)). Above, the behavior is typically nondispersive, the propagation becomes convective. For PCF, such an abrupt transition between dispersive and non-dispersive behavior does not exist, since small traveling waves always disperse. In general, the dispersion is mild, but becomes intense close to a boundary curve that we call below which waves become stationary (see the shadowed part of figure 3(a) and the orange curves in figures 4 and S5. This threshold was first found by Gallagher & Mercer in 1962 Gallagher and Mercer (1962)). However, above , dispersion in the Couette motion is observed to be intense in a quite broader subregion below . The significance of the above picture is that any spatially localized perturbation (the so called wave packet) - which may contain a broad range of traveling wave components - will present both the dispersive and non-dispersive behavior. Namely, there will be a subset of dispersive waves that will spread information in the surrounding environment. So doing, enhancing the probability to catch possible other similar perturbations propagating in the neighborhood. In case the enstrophy is sufficiently high (see figure 4), this can trigger a nonlinear coupling between two close perturbations. Furthermore, there will be also a nondispersive subset of waves which propagates with the convective speed of the basic flow. Once again, if the enstrophy and kinetic energy content is sufficiently high, since this subset cannot unpack, the onset of a nonlinear coupling is expected. In our vision, we believe that the wave propagation properties, in particular the dispersion or non-dispersion of adjacent wavenumber waves, must play a key role in triggering the nonlinear cascade, and that beside the kinetic energy growths, the enstrophy amplification should be considered.
For small two-dimensional internal vortical waves in the plane Couette and Poiseuille flows, we determined the lower bound for the enstrophy algebraic growth. In particular we explored the wavenumber and Reynolds number parameter space in the region . This result was obtained by extending the nonmodal approach to the analytical modal procedure conceived by J. L. Synge (1938), as an alternative to the perturbation analysis based on the study of the wave kinetic energy.
The lower bound for enstrophy transient growth is actually a surjective function . As far as the monotonic decay is concerned, we found that at all wave-numbers this bound is less restrictive than the kinetic energy one, . That is . This is physically noticeable because an initial vortical traveling perturbation which experiences a fast kinetic energy growth, simply because optimized to do so, will not necessarily experience an enstrophy transient growth. The enstrophy growth may therefore lag until higher Reynolds number are reached. In particular, this can be so when the system evolves in the central region of the parameter space where the nonlinear coupling is expected to show up.
With this study we give information on the scaling laws for the kinetic energy, the enstrophy maximal growth and related times in terms of the Reynolds number-wavenumber product. We highlight that Poiseuille and Couette maps differ more in the enstrophy case rather than the kinetic energy case.
In addition, by building on the results of De Santi et al. (2016), we here underline the notable changes in the dispersion properties within the parameter space. By keeping the Reynolds number constant, if one moves from very low wavenumbers, in the case of Couette motion one can pass from stationary waves to dispersive waves which then becomes progressively less dispersive reaching a quasi-convective propagation. While in the case of Poiseuille motion one can pass abruptly from very dispersive waves to non-dispersive waves.
This means that in general a wave packet will be composed by a dispersive subset of waves which spread out the perturbation information on an always larger portion of the spatial domain and by a non-dispersive subset of waves which travel in a compact fashion. The inference can be made that in turn, or also simultaneously, both these components may contribute to the nonlinear wave coupling when a sufficient enstrophy level, more than a high kinetic energy level, is reached.
Acknowledgements.The authors acknowledge support from the US NSF (grants DMS 1362509 and DMS 1462401) and from the MISTI-Seeds Italy MITOR project “Long-term interaction in fluid systems”, 2012-2014, http://web.mit.edu/mitor/grants/seed.html
- Frisch (1995) U. Frisch, Turbulence: the legacy of A.N. Kolmogorov (Cambridge university press, 1995), ISBN 0-521-45713-0.
- Abramov and Majda (2003) R. V. Abramov and A. J. Majda, PNAS 100, 3841 (2003).
- Yeung et al. (2015) P. K. Yeung, X. M. Zhai, and K. R. Sreenivasan, PNAS 112, 12633 (2015).
- Schumacher et al. (2015) J. Schumacher, P. Goetzfried, and J. D. Scheel, PNAS 112, 9530 (2015).
- Tsinober (2001) A. Tsinober, An informal introduction to turbulence, vol. 63 (Kulver Academic Publishers, 2001), ISBN 1-4020-0110-X.
- Synge (1936) J. L. Synge, P. Lond. Math. Soc. 40, 23 (1936).
- Joseph (1976) D. D. Joseph, Stability of fluid motions vols I and II (Springer-Verlag, 1976).
- Manneville (2016) P. Manneville, Mech. Eng. Rev. 3, 15 (2016).
- Synge (1938a) J. L. Synge, Proc. Fifth Intern. Congress of Appl. Mechanics 2, 326 (1938a).
- Synge (1938b) J. L. Synge, Semicentenn. Publ. Amer. Math. Soc. 2, 227 (1938b).
- Orszag (1971) S. A. Orszag, J. Fluid Mech. 50, 689 (1971).
- Romanov (1971) V. A. Romanov, Doklady Akademi Nauk SSSR 196, 1049 (1971).
- Trefethen et al. (1993) L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Science 261, 578 (1993).
- Orr (1907a) W. M. Orr, Proc. R. Irish. Acad. 27, 9 (1907a).
- Reddy and Henningson (1993) S. C. Reddy and D. S. Henningson, J. Fluid Mech. 252, 209 (1993).
- Chapman (2002) S. J. Chapman, J. Fluid Mech. 451, 35 (2002).
- Manneville (2015) P. Manneville, Eur. J. Mech B-Fluid 49, 345 (2015).
- Gallagher and Mercer (1962) A. P. Gallagher and A. M. Mercer, J. Fluid Mech. 13, 91 (1962).
- De Santi et al. (2016) F. De Santi, F. Fraternale, and D. Tordella, Phys. Rev. E 93 (2016).
- Orr (1907b) W. M. Orr, Proc. R. Irish. Acad. 27, 69 (1907b).
- Gustavsson (1991) L. H. Gustavsson, J. Fluid Mech. 224, 241 (1991).
- Reddy et al. (1993) S. C. Reddy, P. J. Schmid, and D. S. Henningson, Siam J. Appl. Math. 53, 15 (1993).
- Schmid (2007) P. J. Schmid, Annual Rev. Fluid Mech. 39, 129 (2007).
- Bayly et al. (1988) B. J. Bayly, S. A. Orszag, and T. Herbert, Annual Rev. Fluid Mech. 20, 359 (1988).
- Orszag and Kells (1980) S. A. Orszag and L. C. Kells, J. Fluid Mech. 96, 159 (1980).
- Carlson et al. (1982) D. R. Carlson, S. E. Widnall, and M. Peeters, J. Fluid Mech. 121, 487 (1982).
- Lundbladh and Johansson (1991) A. Lundbladh and A. V. Johansson, J. Fluid Mech 229, 499 (1991).
- Daviaud et al. (1992) F. Daviaud, J. Hegseth, and P. Berge, Phys. Rev. Lett. 69, 2511 (1992).
- Waleffe (1995a) F. Waleffe, Physics of Fluids 7, 3060 (1995a).
- Waleffe (1995b) F. Waleffe, Stud. Applied Math. 95, 319 (1995b).
- Bottin et al. (1998) S. Bottin, F. Daviaud, P. Manneville, and O. Dauchot, Europhys. Lett. 43, 171 (1998).
- Bottin and Chate (1998) S. Bottin and H. Chate, Eur. Phys. J. B 6, 143 (1998).
- Duguet et al. (2010) Y. Duguet, P. Schlatter, and D. S. Henningson, J. Fluid Mech. 650, 119 (2010).
- Manneville (2012) P. Manneville, Fluid Dyn. Res. 44 (2012).
- Tsukahara and Ishida (2014) T. Tsukahara and T. Ishida, in EUROMECH Colloquium EC565, Subcritical Transition to Turbulence, May 6th-9th 2014, Corsica, France (2014).
- Santi et al. (2015) F. D. Santi, F. Fraternale, and D. Tordella, in Proceedings of the 15th European Turbulence Conference (Delft, Netherlands, 2015), URL http://www.etc15.nl/proceedings/proceedings/documents/492.pdf.
- Fraternale (2017) F. Fraternale, Ph.D. thesis, Politecnico di Torino, Torino, Italy (2017).
- Charbonneau (1995) P. Charbonneau, Astrophys. J. Suppl. S. 101, 309 (1995).
- Charbonneau and Knapp (1995) P. Charbonneau and B. Knapp (1995), URL http://www.hao.ucar.edu/modeling/pikaia/pikaia.php#sec9.
Appendix A Supplementary Information Appendix
List of Contents:
S1. Time scaling of the maximal transient growth for both the kinetic energy and the enstrophy of traveling perturbation waves.
S2. Mathematical procedure to get the maximum time derivative of perturbation enstrophy. Preamble.
S2.1. Plane Couette flow: limit curve in the Reynolds - wave number plane for the transient growth of traveling wave perturbation. Analytical method and final numerical optimization. Sufficient conditions for no enstrophy growth.
S2.2. Plane Poiseuille flow: limit curve in the Reynolds - wave number plane for the transient growth of traveling wave perturbation. Analytical method and related numerical validation procedure. Sufficient conditions for no enstrophy growth.
S2.3. Mathematica coding for the solution of equation (37) and the computation of the expression of the maximal enstrophy for both the Couette and the Poiseuille flow.
a.1 S1. Time scaling of the maximal transient growth for both the kinetic energy and the enstrophy of traveling perturbation waves
a.2 S2. Mathematical procedure to get the maximum time derivative of perturbation enstrophy. Preamble
The enstrophy equation I is recalled below for the reader’s convenience:
The procedure starts by eliminating ; this was done by multiplying Eq. I by and integrating over . By setting , and , two independent equations were obtained. Then we solved for and . For the reader’s convenience, Eq. I is re-written as follows:
The right-hand side of Eq. 22 requires some passages, since the operator contains both a time derivative and the function :
The terms and were evaluated separately by integrating by parts and using the boundary conditions 10:
The system of equations to find and is the following:
Substituting these expressions in Eq. I and naming:
a new form for was obtained:
Note that depends on the parameters and and the function . In order to get to conditions on and implying non-positivity of , calculus of variations was used to maximize with respect to the function , with and being assigned.
Considering the part of depending on :
introducing the variations on the perturbation, we evaluated:
Calculus of variations led to a sixth-order differential equation for the disturbance which maximizes the enstrophy rate. This particular function will be named in the following:
To express in convenient form the corresponding maximum value of , named in the following, we multiplied Eq. 37 by , integrated over the range and added the complex conjugate. This gave:
and so from the definition of :
where here is the maximizing function, solution of Eq. 37.
The procedure followed up to this point leads to an expression for formally identical to that found by Synge (Eq. 2.12 in Synge (1938a)). The difference is that here having adopted the nonmodal approach is time dependent. In the following, we solve the problem for the PCF and then for PPF.
a.2.1 S2.1. Plane Couette flow: limit curve in the Reynolds - wave number plane for the transient growth of traveling wave perturbation. Analytical method and final numerical optimization. Sufficient conditions for no enstrophy growth
In this section, conditions for no-growth of perturbation enstrophy are derived for the plane Couette flow. In this case , so the equation 37, together with the boundary conditions, reads:
Starting from the homogeneous equation:
where the is for the homogeneous solution. Since the solutions of the characteristic equation are and both with multiplicity of 3, it was possible to write the solution as:
Based on the form of the forcing term and in order to get a simpler computation of the constants when applying the boundary conditions, a different basis was chosen. In particular we wrote the solution as follows:
To show that this is indeed allowed, we proceeded by proving that the two basis
are linearly independent: to do so we write , where
Since is a triangular block matrix the determinant is the product of the determinants of the three matrices on the diagonal. , that implies linear independence. So the general solution of Eq. 45 can be also represented in the form A.2.1.
To solve equation A.2.1, a particular solution was to be found. We considered the forcing term as a superposition of two terms, one containing and the other one with . We first found a particular solution of the equation:
We looked for a solution with the form: