数理解析研究所講究録 1454 巻 2005 年 161-166

161

鉛直チャンネル中の自然対流におけるスパン方向運動量生成について
板野智昭 (Tomoaki Itano),
中村亮介 (Ryosuke Nakamura),

京都大学・工学研究科

永田 雅人 (Masato Nagata)

Graduate School of Engineering,
Kyoto University
概要

We investigate the bifurcation of three-dimensional tertiary flows numerically in a simple shear
layer with a cubic velocity profile when secondary flow loses its stability to oscillatory perturbations.
It is found that the bifurcating motion is either of periodic nature or of traveling-wave nature,depending on the spanwise symm etry of disturbances. Furthermore, it turns out that the travellingwave propagating in the spanwise direction generates tl le spanwise mean flow

1Introduction

It is of considerable importance to applications in engineering and geophysics, among
to understand the mech anism of the transition from laminar flow to early stages of turbulence in plane parallel
shear layers. As a simple example of such shear layers we consider flows with a cubic velocity profile.
These flows with an inflectional velocity profile can be realized between two parallel vertical plates which
are kept at constant different temperatures under the gravity field. The flows are characterised by aupward motion near a hotter plate and by a downward motion near a colder plate, so that the momentum for the undisturbed state is only in the vertical direction. It is well known that Squire’s theorem isapplicable in this case, so that it is sufficient to analyse the stability of the basic state with respect totwo-dimensional spanwise-independent) perturbations In fact, a spanwise-independent secondary flow
characterized by cats’ eyelike transverse vortices sets in as the shear gets stronger (Vest &Arpaci ).
The stability analysis on the secondary flow indicates that the secondary flow becomes unstable to threedimensional perturbations with either a monotone subh armonic nature or an oscillatory harmonic nature(Nagata &Busse ), In the present paper we investigate the nonlinear development of the perturbationsin the oscillatory harmonic case using two numerical schemes: a direct numerical simulation to integrate
the time evolution of the primitive variables for Navier-Stokes equations and a Newton-Raphson iterative
scheme to solve the nonlinear algebraic equations for the disturbance amplitudes in an equilibrium state.
$\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{S}_{\rangle}$$\mathrm{n}\mathrm{t}$2Mathematical formulation

162$\mathrm{T}^{2}=\mathrm{T}-$$\mathrm{T}^{*}=\mathrm{T}+$

$\xi$

$1$

$\mathrm{g}$$\ovalbox{\tt\small REJECT}_{1}|_{\oint}||$$\mathrm{z}^{*}=- \mathrm{d}|$$\dot{\mathrm{z}=}0|$$\mathrm{z}^{*}=+\mathrm{d}|$$1$$\mathrm{H}$: Configurationin (1) is the Grashof number defined by $G_{r}=\gamma gd^{3}(T_{+}-T_{-})/(2\nu^{2})$ where is the coefficient ofthermal expansion, is the kinematic viscosity and is the acceleration due to gravity.
The equations which govern disturbances deviated from the basic state (1) are written by$G_{r}$$\gamma$

$g$

$\nu$

$\nabla$. at$\partial_{t}u+$$(u. \nabla)u$$d_{t}’\theta+$$(u \cdot\nabla)\theta$

(2)

$=$

0,$=$

$-\nabla p+\theta$$\hat{i}+\nabla^{2}u$$=$$\frac{1}{Pr}\nabla^{2}\theta$

(3)

,

(4)

,

where is the velocity disturbance, the pressure disturbance, the temperature disturbance, $Pr=\nu/\kappa$
on thethe Prandtl number. The no slip boundary condition and the fixed temperatures areplates:(5)$u=0$ andat $z=\pm 1$ .
$\theta$$P$$\mathrm{u}$$\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}^{\backslash }\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{e}A$$\theta=0$

It may be easily found that temperature disturbnce becomes identically zero in the vanishing Prandtl
number limit.
The stability of the basic state is governed by(6),$\partial_{t}\nabla^{2}\triangle_{2}\tilde{\phi}+\{U_{B}(z)\partial_{x}-\nabla^{2}\}\nabla^{2}\triangle_{2}\tilde{\phi}-U_{B}’(z)\partial_{x}\triangle_{2}\tilde{\phi}=0$

and$\partial_{C}\triangle_{2}\tilde{\psi}+\{U_{B}(z)\partial_{x}-\nabla^{2}\}\triangle_{2}\tilde{\psi}-U_{B}’(z)\partial_{y}\triangle_{2}\tilde{\phi}=0$

where$\tilde{\phi}$and$\tilde{\psi}$

are the poloidal and toroidal components of an infinitesimal velocity perturbation$\tilde{u}=\nabla \mathrm{x}$superimposed on the basic flows (1)$\nabla \mathrm{x}$$(\tilde{\phi}k)+\nabla \mathrm{x}$$(\tilde{\psi}k)$

.

(7)

,
$\overline{u}$

:
(8)

183

We express$\tilde{\phi}$and$\tilde{\psi}$as$\tilde{\phi}=\sum_{\ell=0}^{\infty}\tilde{a}_{l}(1-z^{2})^{2}T_{l}(z)\exp\{\mathrm{i}\mathrm{c}\mathrm{y}x+\mathrm{i}\beta y+\sigma t\}$

(9)

,(10)

$\overline{\psi}=\sum_{\mathrm{e}ll=0}^{\infty}\tilde{b}\iota(1-z^{2})T_{l}(z)\exp\{\mathrm{i}\alpha x+\mathrm{i}\beta y+\sigma t\}_{?}$awhere a and are the wavenumbers in the streamwise and the spanwise directions, respectively.
is the r-th Chebyshev polynomial.
In order to analyse the nonlinear development of the perturbation we consider a velocity deviationand the\^u from the laminar state and for convenience separate it into the average part
residual , so that the total velocity is given by$T\ell(z)$$\check{U}(z)i+\check{V}(z)j$$u$$\dot{u}$(11)$u=U(z)i+\check{V}(z)j+\check{u})$where $U(z)=U_{B}(z)+\check{U}$ $(z)$ The residual tt is further decomposed into the poloidal and toroidal parts:$\check{u}=\nabla \mathrm{x}\nabla \mathrm{x}$$(\phi k)+\nabla \mathrm{x}$$(\psi k)$

.

(12)

The nonlinear state is governed by$\partial_{t}\nabla^{2}\triangle_{2}\phi$$\partial_{t}\triangle_{2}\psi$$+$$\{U(z)\partial_{x}-\nabla^{2}\}\nabla^{2}\triangle_{2}\phi-U’(z)\partial_{x}\triangle_{2}\phi$$+$$\check{V}(z)\partial_{y}\nabla^{2}\triangle_{2}\phi-\check{V}’(z)\partial_{y}\triangle_{2}\phi$$+$$\delta[(\check{u}\cdot\nabla)\check{u}]=0$

(13)

,$+$

$\{U(z)\partial_{x}-\nabla^{2}\}\triangle_{2}\psi-U’(z)\partial_{y}\triangle_{2}\phi$$+$$\check{V}(z)\partial_{y}\triangle_{2}\psi+V’(z)\partial_{x}\triangle_{2}\phi \mathrm{v}$$+$$\epsilon[(\dot{u}.

\nabla)\overline{u}]=0$

(14)

,

,

(15)

,

(16)

$\check{U}^{Jl}+\partial_{z}\overline{\triangle_{2}\phi(\partial_{xz}^{2}\phi+\partial_{y}\psi)}=\partial_{\mathrm{g}}\check{U}$$\dot{V}’+\partial_{z}\overline{\triangle_{2}\phi(\partial_{yz}^{2}\phi-\partial_{x}\psi)}=\partial_{t}\check{V}$where the differential operators$\epsilon$in (11) and 6 in (12) are defined by$\epsilon\equiv k(\nabla \mathrm{x}$and(17)$\mathit{5}\equiv k(\nabla\cross\nabla \mathrm{x}.$and the overline in (15) or (16) stands for the , -average. In the above, the possibility of induced
and as follows:in the spanwise direction is incorporated. We express ,average fiow
$x$

$y$

$\phi$$\check{V}(z)$$\phi$$\psi,\check{U}$$\check{V}$$=$$\sum_{t=0}^{\infty}\sum_{(m,n)\neq\langle \mathrm{O}.0)}^{\infty}\sum_{=m=-\infty n-\infty}^{\infty}a_{lmn}(1-z^{2})^{2}T_{l}(z)$$\mathrm{x}\exp(\mathrm{i}ma(x-cxt) +inj3\{y -c_{y}t))$$(1\mathrm{S})$164$\sum\infty b_{lmn}(1-z^{2})T_{l}(z)$$\sum\infty$$\sum\infty$$\psi$$=$$\iota=0m=-\infty n=-\infty$$(m,n)\neq\langle \mathrm{O},0)$0$\check{V}$(19)$\exp(\mathrm{i}m\alpha(x-c_{X}t)+\mathrm{i}n\beta(y-\mathrm{c}_{y}t))$$\mathrm{X}$(20)

Cg $(1-z^{2})T_{f}(z)$$=$$\sum_{l=0}^{\infty}$$=$$\sum_{l=0}^{\infty}d_{l}(1-z^{2})T_{l}(z)$

(21)

.

In the expressions (18)-(19) the phase velocities, and , are included in order to deal with a travellingwave tyPe of nonlinear equilibrium states. For a steady state time-derivatives are omitted in the equations(13)- (16) and . and . are both zero. As a measure of nonl nearity we choose the momentum transport
on the plates normalised by its value for the basic state:$c_{x}$$r_{x}$$c_{y}$$r_{y}$$\tau$(22)$\tau=U’(z)/U_{B}’(z)|_{z=\pm 1}$

In order to investigate the stability of the equilibrium state, we superimpose arbitrary three-dimensional
withequations
infinitesimal perturbbaLions 011 the nonlinear state (18) (21) The
respect to the perturbations are given by$1\mathrm{i}11\mathrm{e}\mathrm{a}1^{\backslash }\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{d}$$\mathrm{s}\mathrm{t}\mathrm{a},\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$-$\partial_{t}\nabla^{2}\triangle_{2}\tilde{\phi}$$\{U(z)\partial_{x}-\nabla^{2}\}\nabla^{2}\triangle_{2}\tilde{\phi}-U’(z)\partial_{x}\triangle_{2}\tilde{\phi}$$+$$\overline{V}(z)\partial_{y}\nabla^{2}\triangle_{2}\tilde{\phi}$

$+$

5

$+$

$-\check{V}’(\approx)\partial_{y}\triangle_{2}\tilde{\phi}$(23)$[(\dot{u}\cdot\nabla)\tilde{u}+(\tilde{u}\cdot \nabla)\check{u}]=0\}$$\partial_{t}’\triangle_{2}\tilde{\sqrt)}+\{U(z)c9_{x}-\nabla^{2}\}\triangle_{2}\overline{\psi}-U’(z)\partial_{y}\triangle_{2}\tilde{\phi}$$+\check{V}(z)\partial_{y}\triangle_{2}\tilde{\psi}+\check{V}’(\approx)\partial_{x}\triangle_{2}\tilde{\phi}$$+$$\tilde{\phi}$and$\tilde{\psi}$$\epsilon$[$(\check{u}\cdot\nabla)\overline{u}+$(tz .

$\nabla)\check{u}$(24)] $=0$ .

are expressed by$\overline{\phi}=\sum\infty$$\sum\infty$$\sum\infty\overline{a}\iota_{mn}(1-z^{2})^{2}T_{I}(z)$$\mathit{1}=0m=-\infty n=-\infty$$\mathrm{x}$$\exp\{i(m\alpha+d)(x-c_{x}t)+\mathrm{i}(n\beta+b)(y-\mathrm{c}_{y}t) + \sigma t\}$$\tilde{\psi}=\sum\infty\sum\infty$

,

(25)

,

(26)

$\sum\infty\overline{b}_{lmn}(1-z^{2})T_{l}(z)$$t=0m=-\infty n=-\infty$$\mathrm{x}$

where

$d$

and

$\mathrm{b}$$\exp\{\mathrm{i}(n\mathrm{z}\alpha +d)(x-\mathrm{c}_{x}t)+\mathrm{i}(n\beta +b)(y-c_{y}t) + \sigma t\}$are Floquet parameters.

3

Numerical methods

3,1

Stability and bifurcation analysis

Substitution of the expansions (18) (21) into the basic equations (13) (16) reduces to a set ofnonlinear algebraic equations for the expansion coefficients with the aid of the Chebyshev collocation
method. We employ the Newton-Raphson method to solve the set of equations
--

165

For the stability of an nonlinear equilibrium state we superimpose the general form of a three dimensional
perturbation (25) and (26) on the equilibrium state and we obtain the eigenvalue problem for the growthrate aof the perturbation as the eigenvalue with the aid of Floquet’s theorem.

3.2

Direct numerical simulation

We choose the wall-normal components of the velocity and the vorticity in addition to the mean velocity
components in the streamwise and the spanwise directions as the flow field. The flow field is expanded by
the Fourier series in the streamwise and spanwise directions and the Chebyshev polynomials in the wallnormal direction The time-development of the flow is followed by the method of the Adams-Bashforth
scheme for convective terms and the Crank-Nicolson scheme for viscous terms.

$\mathrm{G}\mathrm{r}$152: The bifurcation diagram

4Results

The bifurcation diagram in thespa, is shown in Fig. 2. The basic flow becomes unstable atwith the streamwisetransverse vortex flow
$G_{r}=500$ to a two-dimensional perturbation andbecomes unstable firstas a secondary flow bifurcates supercritically. The
wavenumber asubharmonic three-dimensional perturbation with the Floquet parameters
at $G_{r}=534$ to a steady$(d, b)=(0625=\alpha/2,1.0)$ , and later at $G_{r}=545$ to a harmonic three dimensional perturbation with$(d, b)=(0,1.0)$ . The real eogenvalues are associated with the subharmonic perturbation and the thesteady subharcomplex conjugate pair of eigenvalues with the harmonic perturbation. We obtain awhich bifurcates al $G_{r}=534$ as a tertiary solution as shown by the thick cueve inmonic flow
Fig. 2. Time-dependent solutions as a tertiary state may occur at $G_{r}=545$ . For DNS we restricted
the wavenumber pair (or, ) for the computation domain $(L_{x}=2\pi/\alpha L_{y}=2\pi/\beta)$ to (1.25, 1.00) for theharmonic case and to (0.625, 1.0) for the subha rmonic case. Since the three-dimensional subharmonic
pesolution cannot manifest itself by DNS in the harmonic domain we expect some three-dim ensional
at $G_{r}=545$ in the harmonic case. However, it turns
riodic flows to bifurcate directly from the$(G_{r}, \tau)$$\mathrm{e}\mathrm{e}$$2\mathrm{D}$$(2\mathrm{D}\mathrm{T}\mathrm{V})$$2\mathrm{D}\mathrm{T}\mathrm{V}$$=1.2\overline{\tau},$$3\mathrm{D}$$3\mathrm{D}$$(3\mathrm{D}\mathrm{S}\mathrm{b}^{\backslash })$$\beta$$2\mathrm{D}\mathrm{T}\mathrm{V}$out that the periodic flows exist only as a transient state and the solution in the final state is$\mathrm{a}\{^{\backslash }\mathrm{t}\mathrm{u}\mathrm{a}11\mathrm{y}$1GGdoes not change its flow pattern in
instead. The
travelling-wave
a three-dimand keeps a constant momentum transport on thea frame moving with the spanwise phase speedi‘s also c.onfirmed
plates as indicated by the dashed curve in Fig. 2. The existence of thehas a non-zero
the calculation by Newton-Raphson method. It is interesting to note that thein the spanwise direction It should be noted that the solutions in the harmonic
average velocity$\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{I}\mathrm{l}\mathrm{a}4$$3\mathrm{D}\mathrm{T}\mathrm{W}$$(3\mathrm{D}\mathrm{T}\mathrm{W})$$c_{y}$$3\mathrm{D}\mathrm{T}\mathrm{W}$$\mathrm{b}_{\iota}\mathrm{y}$$3\mathrm{D}\mathrm{T}\mathrm{W}$$\check{V}(_{\sim}\mathit{7})$case constitute a subset of the solutions in the subharmonic case.

5Conclusions

In the present paper we have investigated the nonlinear development of the perturbations in the oscillatory harmonic case using two numerical schemes, a direct numerical simulation and a Newton-Raphson
iterative scheme. Both numerical schemes have indicated that the bifurcating three-dimensional flow
when the secondary flow loses its stability to an oscillatory disturbance is periodic for the case where thespanwise symmetry is retained, whereas it is of travelling-wave tyPe travelling with a constant phase in
the spanwise direction when the spanwise symmetry is broken. The mean flow produced by nonlinear
interactions of oscillatory perturbations has only the streamwise component for the three-dimensional
periodic flow. It turns out that the mean flow has an additional spanwise component, thus generating thespanw ise momentum, for the three-dimensional travelling-wave solution. We will also show the generationof the spanwise mom entum by means of symmetry arguments.
In experiments the vertical fluid layer betw een two plates must be confined by the side walls, in general,
which ought to inhibit the total mass flux in the horizontal direction Therefore, in order to detect ahorizontal mass flux we plan to carry out an experiment on na natural convection in a vertical annulus
The mass flux sould be generated in the azimuthal direction either clockwise or anti-clockwise dependingon the form of an initial disturbance.

参考文献[1] Vest, C. M. and Arpaci, V. S., “Stability of natural convection in a vertical slot,” J. Fluid Mech.,
36, (1969),. 1-15,$\mathrm{P}\mathrm{P}$[2] Nagata, M. and Busse, F. H., “Three-dimensional tertiary motions in a plane shear layer,” J. Fluid
Mech., 135, (1983), PP. 1-26