UMEKI, Makoto

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Title Bifurcation Structures of Two-Dimensional PoiseuilleFlow(Mathematical Fluid Mechanics and Modeling)Author(s)UMEKI, MakotoCitation数理解析研究所講究録 (1994), 888: 193-198Issue Date1994-10URL http://hdl.handle.net/2433/84338RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityBifurcation Structures of Two-dimensional Poiseuille FlowMakoto UMEKIDepartment of Physics, University of Tokyo 11 IntroductionBifurcations of two-dimensional progressive waves (Tollmien-Schlichting waves) \’in planePoiseuille flows are studied numerically for many sets of parameters , i.e. Reynolds numberand the fundamental wavenumber. It is confirmed that Hopf bifurcations occur (at least)twice on TS wave and successive instabilities lead to chaotic slates in all examined casesof the fundamental wavenumber, as Reynolds number increases. The global bifurcationdiagram is made based on results of the numerical simulation.2 Methods of numerical simulationThe method of numerical simulation is described in detail in Umeki (1993). The evolutionequation for the disturbance of vorticity is solved in the spectral space. The nonlinear termis computed by the pseudo-spectral method. The second-order Crank-Nicolson scheme isinvoked for the normal $(y-)$ derivative in the viscous term and the second-order Adams-Bashforth method is used for the remaining terms except for the terms linear to the Fourier-Chebysohev coefficients of vorticity, which can be integrated exactly. We imposed the non-slip boundary condition at the wall and the constant-flux condition in order to determinethe streamfunction (and the velocity) from the vorticity.The number of modes in the streamwise $(x-)$ and normal directions is chosed as $(N_{x}, N_{y})=$(16,64). Since we use the two-third law in the pseudo-spectral method, the number oflThe affiliation during the period between 2 Aug. and 21 Oct. 1994: Departamento de F\’isica,Faculultad de Ciencias Fisicas $y$ Matem\’aticas, Universidad de Chile888 1994 193-198 193effective modes is ac$t$ually $(M_{x}, M_{y})=$ (10,42). (Here we define $M_{x}=[2N_{x}/3]$ and$M_{y}=[2(N_{y}+1)/3]$ . $[]$ is the notation of Gauss. ) The time step $\triangle t$ is 1/60. Previ-ously, the simulation for the case $(N_{x}, N_{y})=(16,32)$ was performed. Comparing with this,there exists a small shift of $bif\iota ircation$ curves although qualitative features are common.The fortran program is coded so that cases of many pairs of parameters $(Re$ : Reynoldsnumber and $\alpha$ : the fundamental wavenumber) can be simulated in the single program.This enables us to make use of the ability of the vector supercomputer in the maximalway and to study the bifurcation problems of Navier-Stokes equation. The fundamentalwavenumber $\alpha$ is varied from 0.8 to 1.7 with the step size $\triangle\alpha=0.1$ and Reynolds numberis from 2000 to 12000, depending on $\alpha$ , with $\triangle Re=300$ or 100.In order to determine the range of Reynolds niunber before starting the numerical simu-lation, branches of TS waves are computed by sloving algebraic equations of a progressivesolution in the spectral space with the reduced number of modes as $(M_{x}, M_{y})=(2,42)$ . Athree-dimensional view of the TS wave in the parameter space $(\alpha, Re)$ versus the energyintegral of disturbance $E_{d}$ is shown in Figure 1. (For the precise definition of $E_{d}$ , seeUmeki(1994). $)$In the present simulation, the initial condition is chosen as$\hat{\psi}_{init}=0.3(1-y^{2})^{2}\cos\alpha x+0.01(1-y^{2})^{2}\cos 2\alpha x$so that we can observe growth or decay of the component of symmetry different ffom TSwaves.3 ResultsFigure 2 shows power spectra of (x-directional) averages of vorticity at the upper wall(denoted by $<\omega>y=1$ ). The sequence is visible from the first Hopf bifurcation to thechaotic state. In this paper we use a term”periodic state” for the time-periodic modulationof TS wave. Then the quantity $<\omega>y=1$ is constant for the progressive TS wave, and194periodic for the periodic states. It should be noted that the quantity (e.g. vorticity) atone point is time-periodic for the progressive TS wave. Similarly, we use ”quasi-periodic”and “choatic” for the quasi-periodic and chaotic modulations. Thus if the flow is doubly-periodic in $<\omega>y=1$ , it implies triply-periodic in a one-point quantity.Figure 3 shows the bifurcation diagram in the parameter $(Re, \alpha)$ space obtained by thenumerical $sim\iota ilation$ . There exists a relatively wide region of a stable TS wave from $\alpha=$1.1 to 1.6. In addition, a chaotic state does not appear for $\alpha\geq 1.5$ and $Rei$ 12000.To see how the component different from TS wave increases as the flow becomes turbu-lent, we decompose the energy integral of disturbance into two integrals; one is the sameas TS wave (denoted by $E_{d}^{A}$ ) and the other is different from it $(E_{d}^{B})$ .Figure 4 shows the temporal evolution of $E_{d}^{A}$ and $E_{d}^{B}$ for three cases $(\alpha, Re)=(1.1$ , 5000 $)$ ,(1.1, 6000) and (1.1, 9000), the former two of which correspond to periodic states and thelast to a chaotic state. It is observed that $E_{d}^{B}$ has a non-vanishing value, although it isvery small, at the periodic state. It indicates that there is also a bifurcation from a stateof symmetry with TS wave to a non-symmetric state in the periodic state. In the chaoticcase of $Re=9000$ , the both components $E_{d}^{A}$ and $E_{d}^{B}$ are in the same order.4 ConclusionThe previous studies 1-4) on bifurcations in plane Poiseuille flows are extended to clarifythe global bifurcation structure in the parameter space by means of numerical simulation.Bifurcation diagrams are expected to be produced for the three-dimensional flow or thesubharmonic instability (i.e. the long channel $[\alpha\ll 1]$ ), which will be modified based onthe diagram presented here.1955 References1) Jim\’enez, J. 1990 Transition to turbulence in two-dimensional Poiseuille flow J. FluidMech. 218265-297.2) Mizushima, J, Fujimura, K. and Yanase S. 1988 Nonlinear evolutions of two-dimensionaldisturbances in plane Poiseuille flow Nagare 7259-270. (in Japanese)3) Pugh, J. D. &Saffman, P. G. 1988 Two-dimensional superharmonic stability of finite-amplitude waves in plane Poiseuille flow J. Fluid Mech. 194295-307.4) Soibelman, I., &Meiron, D. I. 1991 Finite-amplitude bifurcations in plane Poiseuilleflow: two-dimensional Hopf bifurcation J. Fluid Mech. 229389-416.5) Umeki, M. 1994 Numerical simulation of plane Poiseuille turbulence Flu. Dyn. Res.1367-79.196Figure 1. Three-dimensional view of TS wave of plane Poiseuilleflow in the $(\alpha, Re.E_{d})$ space. A dashed line denotes the neutral$cur\backslash \cdot e$ of the linear stability.Figure 2. Power spectra of av-erages of vorticity at the upperwall $(y=1)$ in the case $\alpha=1.4$ .A left column shows the sequenceof two Hopf bifurcations. $(Re=$6800, 7100, 7400, 7700 $hom$ theabove.) The right column denotesa fully quasi-periodic state $(Re$$=8600)$ and a chaotic state $(Re$$=10700)$ .1978Figure 3. A global bifurcationdiagram predicted by the numer-ical simulation. Boundaries aredenoted by crosses (between lami-nar Poiseuille flow and TS wave),solid circles (TS wave and peri-odic state), open circles (periodicmd quasi-periodic states), and squares(quasi-periodic and chaotic states).${\rm Re}$Figure 4. Temporal evolutionof disturbance energy of symme-try which is the same as TS waves(the upper) and different $hom$ it(the lower). Solid, dashed anddotted curves denote respectively$Re=5000$ , 6000 and 9000. $\alpha=1.1$ .$t$$t$198

Bifurcation Structures of Two-Dimensional Poiseuille Flow(Mathematical Fluid Mechanics and Modeling)

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Bifurcation Structures of Two-Dimensional Poiseuille Flow(Mathematical Fluid Mechanics and Modeling)

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