conditional fourier spectral method for direct numerical simulation of incompressible flows

Journal o f Scientific Computing. VoL I0. No. 2, 1995

Conditional Fourier Spectral Method for Direct Numerical Simulation of Incompressible Flows

lwao H o s o k a w a I and Kiyoshi Y a m a m o t o 2

Received June 1, 1994

A new method of treating incompressible flows with nonslip boundaries is proposed as an extension of the Fourier spectral method. This is characteristic in using the function subspace that is a hyperplane in the Fourier-transformed velocity space, prescribed by the boundary condition, as well as in taking the solenoidal field representation in the Fourier space so that the pressure term need not be involved in the main dynamics and then time-integration can simply be made by the high-order Runge-Kutta scheme. The method can be applied in a more complicated case with an active scalar. As examples, the flow transitions to turbulence in a channel and in a rectangular duct heated from below are treated.

KEY WORDS: Conditional Fourier spectral method; incompressible flow; Boussinesq flow; transition to turbulence; channel; duct; solenoidal field representation.

1. I N T R O D U C T I O N

It is well known [e.g., Canuto etal. (1988)] that the Fourier spectral method in direct numerical simulation is efficient for the problems of turbulent incompressible flows with periodic boundary conditions, while the Chebyshev spectral method and the like are to be used for those with no- slip boundaries. However, in view of simplicity in mathematical treatment it would be convenient if we had any extended Fourier spectral method that could be applied to the latter problems. From this point of view, we shortly reported such an idea in Hosokawa and Yamamoto (1986a, b). Since then, we have applied it to various problems of flows bounded with a simple geometry and made sure that it really works with enough

i University of Electro-Communications, Chofu, Tokyo 182, Japan. 2 National Aerospace Laboratory, Chofu, Tokyo 182, Japan.

271

0885-7474/95/0600-0271507.50/0 (d~ 1995 Plenum Publishing Corporation

272 Hosokawa and Yamamoto

accuracy. Here we would like to explain it more concretely in the name of the conditional Fourier spectral method with some convincing results.

The best advantage of the Fourier spectral method for incompressible flows is the removal of the pressure term in the Navier-Stokes equation from the main dynamics. This is exactly made by the solenoidal representation of the velocity field, which is described in the next section. Then, all numerical complications and errors associated with treatments of the pressure term and incompressibility condition disappear in this method. However, in order for the Fourier components of the velocity field and a possibly existing advected scalar field to satisfy the necessary boundary conditions, there must be some additional mathematical restrictions to them, that will be explained in Section 3. It will be seen there that our method is methodologically different from the so-called Fourier tau method.

To show how the method works practically, we will take up two examples. One is the case of plane Poiseuille flow in Section 4, where a subcritical flow is shown to develop to turbulence if a small initial three- dimensional (3D) disturbance exists. The other is the rather complicated case of a mixed convective flow in a horizontal rectangular duct heated from below in Section 5. In both cases, an essential role of 3D wavenumber modes of the velocity field in the transition to turbulence is observed.

2. SOLENOIDAL FIELD REPRESENTATION

Let us consider the Navier-Stokes equation:

Du/Dt = - V p + (1/Re) zlu (2.1)

with the incompressibility condition:

div u = 0 (2.2)

where u(x) and p(x) are the dimensionless velocity and pressure fields in the dimensionless x space and Re is the Reynolds number, u(x) may be expanded into the Fourier series within the parallelpiped of sides L~, Ly and L_ like

u(x) = Z v(k) exp(ik, x) (2.3) k

with k=(2~zn,./L.,., 2gn.,./Lv, 2nn=/L=); nx, n~., 17=: integers, and then v(k) obeys the incompressibility condition:

k . v(k) = 0 (2.4)

Conditional Fourier Spectral Method for Incompressible Flows 273

Two orthogonal unit vectors to k may be

e,(k) = ( - l /k , kxky/(kl), kxk:/(kl)) (2.5)

e2(k) = (0, -k=/l, ky/l) (2.6)

where l=(ky+k=) - and k=(k:.+l~-) u2, so that k/k, e~, and e2 build a local right-handed orthogonal coordinate system at k in the wavenumber space. [In case 1 (or k ) = 0 , Eqs. (2.5) and (2.6) are indefinite, but we can make a unique setting of et and e2 by defining ky/l= 1, k:/l=O (and kx/k = 1, l/k= 0) there.] Hence we may have the solenoidal expression of v(k) exactly satisfying Eq. (2.4) as

v(k) = vl(k) el(k) + v2(k) e2(k) (2.7)

Since v(k) has no component in the k direction, the number of unknown variables are greatly reduced by this, and moreover these variables v~(k) and v2(k ) a re entirely governed by the two Fourier-component equations of Eq. (2.1) orthogonal to k, which do not involve the pressure term at all. The component equation parallel to k of Eq. (2.1) does not play a role of any dynamics but the equation only to determine the Fourier-component ofp(x) from the knowledge of v(k) at every instant; it is really an equiv- alent to the Poisson equation for p.

Such a Fourier decomposition of Eq. (2.1) simplifies time-integration very much. Any sophisticated technique such as time-step splitting to avoid a numerical instability is unnecessary in this case, where only the well- known Runge-Kutta-Gill scheme is usually recommendable. On this point, the present method is obviously advantageous over the original Orszag and Patterson (1972) algorithm and even the one by Rogallo (1977). Of course, the aliasing error which happens in the collocation process using the FFT technique in calculating the nonlinear terms should be exactly removed, e.g., by the so-called 2/3 rule [e.g., Canuto etaL (1988)]. We have often used this method to obtain various significant results of isotropic turbulence [Yamamoto and Hosokawa (1981, 1988); Hosokawa and Yamamoto (1986a, b)]. It is our motivation that such an advantage may ensue even in treating a shear-flow turbulence.

3. CONDITIONAL FOURIER SPECTRAL METHOD

When there is a nonslip boundary, for example at z= +_L:/2, the condition of u (x)= 0 there places a restriction on v(k), that is

( - 1 )"" v(k,., ky, 2rm__/L._) = 0 (3.1) n :

854/10/2-8


This restricts v~(k) = r/l(k) + iff~(k) and v2(k) = r/~(k) + i~2(k) into the inter- section of the hyperplanes prescribed by Eq. (3.1), which includes really three equations for PlJ and q2 (real part) and the same for ~ and ~2 (imaginary part). How to achieve such conditional Fourier components is explained next.

Let us write one of the equations generally as

2N

~" a,,x,, = b (a,,, b: const) (3.2)

where N is the total number of wavenumber k_. considered, and {x,,} and {xN+,,} represent either {q~} and {r/,} or {~,} and {~2} in Eq. (3.1). First, we note that the normal to the hyperplane has the direction cosines equal

nt " ~ t / 9 to { t l , ; n = 1 ..... 2N} = (al ..... a2N)/f2N where f , , = ( ~ j = l a j l -. Second, 2 N - I orthogonal vectors normal to this are found quite successively from (ala2, - a ~ , O,..., O) to (ala2N, a2a2N ..... a2N_la2N, --(a~ +a~ + . . . + a_~ N_ ~)). Normalization of these vectors gives a set of base vectors in the hyperplane, denoted as { t,,,,} for m = 2,..., 2N, where

t ..... =a, , ,a , , / ( f , ,_ i f , , , ) for n < m

= --f,, ,- 1/f,, for n = m

= 0 for n > m

(3.3)

2 N

x , ,= ~ t , , , , ,y , ,+t l , ,b / f ,u (3.4) m

without any restriction to {y,,,}, which constitute the subspace of the v(k) function space satisfying the boundary condition. By virtue of the orthonormality of t ...... we have the transform:

2 N

y, , = ~ t,,,,x,, for m~>2 (3.5)

Eventually we can obtain the required expression for r]l(k), r/2(k), ~t(k) and ~2(k) in a form similar to Eq. (3.4), together with the new 2 N - 3 essential variables in a form similar to Eq. (3.5) for fixed kx and ky in this case. Hence the basic equations for these variables are easily derived from those for qt , 112, ~ , and ~2. In principle, any number of linear conditions like Eq. (3.2) (even if including the Neumann condition or the mixed-type)

(Here, f , , should not be zero. If it happens to be zero, we have to avoid the difficulty by properly changing the order of {x,}.) Thus, x,, subject to the condition of Eq. (3.2) can be expressed as


may be set, since the same procedure has only to be repeated to obtain further restricted Fourier modes.

In other words, we have fixed an orthogonal function system satisfying the no-slip boundary condition but still expressed in the Fourier series. We are dealing with the dynamics of two vectors corresponding to (r/m , r/z) and (~1, Q,) spanned by this fixed, orthogonal function series, each consisting of 2 N - 3 components. In this point our method is distinct from the so-called Fourier tau method, which has quite a different algorithm; there the dynamics of ( 2 N - 3 ) Fourier components of the two vectors are directly dealt with and the other three components should be solved consistently with the no-slip boundary condition at every time step. But since both methods are very similar in treating the dynamics of (2N--3) degrees of freedom, their mathematical quality may be essentially the same. It is a future problem to compare the efficiency of both methods in practice.

When there is another no-slip boundary condition at y = +_Ly/2, we have also the restriction:

n,,( -- 1 )", v(k.,., 2rm:,/L.v, 2rcn:/L:) = 0 (3.6)

The same idea as before to get Eqs. (3.4) and (3.5) can be developed to this case, too. In this case, n and N in Eq. (3.2) should read as nyn: and N,,N__, so that x,, be the cross wavenumber components. Particularly, we note that there are no longer 3 but 3(N,. + N_) restrictions to x,, for fixed kx. There- fore, a relatively large computer capacity is necessary in this case. A successful result of such a computation is shown in Section 5.

A defect of spectral methods is the Gibbs phenomenon sometimes arising near the boundary. If such a phenomenon happens, a smoothing technique such as the Lanczos method may be effective, but in the cases in Sections 4 and 5 no particular pathologic result relevant to this has appeared. There might be an apprehension that the field obtained by this method is periodic at the opposite boundaries, but a possible effect of the periodicity stays very near the boundary and must decrease with the increasing number of the Fourier modes, as the theory of functional analysis insures. Indeed, as is seen in Figs. 4 and 5, such an effect hardly persists even near the wall in a fully-developed turbulent flow. The behavior of transition to turbulence is fairly well described by this method. A remarkable difference between this and the Chebyshev spectral method [Canuto etal. (1988)] consists in that the former resolves the field uniformly in space, while the latter resolves it nonuniformly in a way that the resolution is evolved near the boundary. In spite of this advantage of the latter, however, it must bear with the complication of the algorithm and the additional errors associated with it.


It is particularly noteworthy that the Fourier spectral method clarifies directly the time-evolution of all the wavenumber modes of th e field, which represent the whole phase space of the system. Attractors or chaos can be investigated from this viewpoint. For example, the evolution of twice the energy of each wave mode, Iv(k)] 2, illustrates the structural change of the system, and sometimes shows steadiness and other times oscillation and further turbulence. A picture of the logarithm of [v(k)] 2 or their partial sum against time may be called an evolutional Fourier spectrograph (EFS). It will play an interesting role in the following sections. Since we have a large number of modes in direct numerical simulation, an EFS involves too many curves to be drawn altogether. Therefore, usually only some leading modes are picked in one figure. Even if so, there are often still many modes to govern a complex system so that the use of several color lines with various symbols is really unavoidable, as is seen later.

4. F L O W TRANSITION IN A C H A N N E L

As the first example, a subcritical plane Poiseuille flow for Re = 5000 is treated by this method. The critical Re is known as 5772 in the linear stability theory [Orszag (1971)]. The Re is defined by the maximum speed of the basic flow as shown and the half channel width L=/2. For con- venience, u(x) in Eq. (2.1) is decomposed into

u(x) = U(z) e,- + u'(x) (4.1)

where U(z)= 1 - z 2 is a usual laminar parabolic solution and e,. is the unit vector in the x direction. Then, the conditional Fourier spectral method is applied to the compensating field u'(x); that is if(x, y, + L = / 2 ) = 0 and the field is assumed to be periodic in the x and y directions. We set here L_ = 2 and L x = L y = 4 ~ / 2 , where 2 ( = 1.2) is the wavenumber of the Tollmien- Schlichting (TS) wave according to Itoh (1974), and use 3 2 x 3 2 x 128 wavenumber modes. The step of time-integration in the Runge-Kutta-Gill method is taken as 0.05.

In Fig. l(a) is shown the EFS of u'(x), which depends very much on the initial condition. Here, the initial condition is given by normal-random disturbances of the order-of-magnitude of 10 -6 to the modes of u'(x) with Inxl ~< 1, In:,l ~<6, and In:l ~<6 except for v ' ( 0 )=0 and the TS wave: v'(k) with n,. = 2 and n,. = 0, which is of 10 -2 and almost on the lower branch of the nonlinear stability curve of Herbert (1975). In this EFS, E(kx, ky) is defined as

E(kx, k,,) = y ' Iv'(k)l z (4.2) k:

x ,7,

c~

O

-4

-6

-8

-10

-12

R=5000

Co

se-

4A

Prim

ory

-14

0 1

00

t

20

0

0 lO

O

t 2

00

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• 0

12

34

n

x

¢3

O

,=)

--.

O .=_

. O

,=1

g)=

o o B

| J O

(a)

(b)

Fig.

1.

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Evo

luti

onal

Fou

rier

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ctro

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ms

of E

(k,.,

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ow f

or R

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he t

ime-

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Fig. 2.

0 X Lx

( a ) t = 120

I I I I I i I I I I I I I I I I I I I I I I I I I I I I I I I

Z

1 ' ' ' ' ' ; . . . . . . . . . . . . . . . . , ' i ....... ; , . . . . .

0 x L~

( b ) t = 140

+1

Z

-1

- . . . . . . . . . . . . . .

o ~ ,~

0 X Lx

( c ) t = 160

Snapshots of the contour of u'(x) in a typical x z plane of a channel flow for Re = 5000. Solid lines indicate positive values and dotted lines negative values.


0

, !!;if: !i,:;!ii ili;i:

.ii..;i!i .i.i!.ii:;!i 0 x Lx

(a) t = 1 0 0

L~ ~. ~.~~~.

~ ~ ~~.--. ,.-~

0 . ~ ~ - ~ - ! : .

0 × Lx

Fig. 3.

(b) t = 2 0 0

Snapshots of the contour of u'(x) near the watI (z = -0.9) of a channel flow for Re = 5000. Solid lines indicate negative values and dotted lines positive values.


the partial sum with respect to k_. The EFS regenerates the secondary instability mechanism of the TS wave (indicated as "primary") due to entanglement of 3D waves, which was first reported by Orszag and Kells (1980), and eventually elucidates the transition to turbulence of the flow in detail. The evolution of only substantially important low-wavenumber modes is recorded in this EFS. The development of the field u',. in a typical vertical plane is visualized by contour lines in Fig. 2 successively at t = 120, 140 and 160. Also the field ul,-(x, y, - 0 . 9 ) is shown in Fig. 3. The solid lines indicate the negative values in this figure. At t = 100 (transition period) we can see the remains of the TS wave and the growth of peaks and valleys, while at t = 200 (turbulent period) the organized streak structure appears.

When we take another initial condition of u'(x) as normal random numbers of the order-of-magnitude of 10 - 6 to the modes with In,.[ ~< I, in,,] = 2 and 3, and In=l ~< 6 plus the TS wave: v'(k) with n+,. = 2 and n:, = 0, which is of 10 -2 , we have another mechanism of transition which is shown in Fig. l(b). The figure illustrates that the 3D modes with ln+,.L = 1 and Inyl = 2 leads throughout transition. This mode is subharmonic of the TS wave. This phenomenon is in contrast to the former case of transition which is associated by the ordered peak-valley structure, and basically in accordance with the transition by resonance of subharmonic modes with the TS wave in the boundary layer of a fiat plate first observed by Saric and Thomas (1984). The flow eventually reaches a fully-developed turbulent state near t = 200, where all modes except the one with 17,. = ny = 0 are mingled up. The structure of turbulence is similar irrespectively of initial conditions.

Z

1.0

0

- I .0

U +,(up

9 0.15

J

/ t = 250

170 8O

So

Fig. 4. The time-development of the profile of the spatially averaged velocity in the x direction, U + (u~,+), of a channel flow for Re = 5000.

U

Z


X

t=iO0.O

Z

X

t = l S O . O

Fig. 5.

Z < t = 2 0 0 . O

X

Snapshots of the velocity profile of u.,. at successive positions of x in a typical vertical plane of a channel flow for Re = 5000.

2 8 2 H o s o k a w a a n d Y a m a m o t o

The transition to turbulence with time for the case of Fig. l(a) is seen in Fig. 4 from the viewpoint of the average velocity profile of the flow, which is calculated as

< u,.> = U(z) + ~[. u'(x, y, z) dx dy/L..Ly (4.3)

Snapshots of u,. in a typical vertical plane are shown in Fig. 5 at successive positions of x at t = 100, 150 and 200. At the fully-developed state of turbulence (t = 250), the wall law of the average velocity near z = - I appears as is seen in Fig. 6, in which U* is <u,.> normalized by the friction velocity v* and z * = (z + 1) v*/v (v: kinematic viscosity); almost the same figure can be gained also in the neighborhood of z = 1. The logarithmic law with a proper Karman constant is well regenerated there, except for the constant 7.0 a little larger than the accepted value 5.5 [e.g., Schlichting (1960)]. The reason of this discrepancy seems to be partly that the Reynolds number is not large enough, and partly that the resolution of the present calculation is insufficient for investigating every detail of the flow very near the wall. This kind of upward deviation of the velocity profile has also been recognized in the recent similar calculations (mainly for investigating the organized structure of a fully-developed turbulence rather than the transition process) of Kasagi (1993) and Huser and Biringen (1993), which are quite different in methodology from the present one [first published in Yamamoto and Hosokawa (1987) ].

Fig. 6.

3O

U*

20

10

R =5000

, ° . . ~ o U 0.41 "~''z 7.0

I I , I t i o J oo Z * i0oo

The profile of the spatially averaged velocity in the x direction normalized by the friction velocity very near the wall at t =250.


5. F L O W IN A H O R I Z O N T A L RECTANGULAR DUCT HEATED FROM BELOW

In this case, the Boussinesq equation is used in place of Eq. (2.1):

Du/Dt = --Vp + (1/Re) zlu + (Gr/Re 2) 0e_ (5.1)

and the equation for the dimensionless temperature 0:

DO/Dt = 1/(PrRe) AO (5.2)

should be added. Here Gr is the Grashof number, Pr the PrandtI number, and e_ the unit vector of the (vertical) z direction. The Re here is referred to the mean flow speed and the hydraulic diameter De. The boundary conditions for u and 0 are set to be

u(x, +.Ly/2, z) =u(x, y, +L_./2) = 0 (5.3)

O(x, +Ly/2, z) = O,.(z), O(x, y, L:/2) = -1/2 , O(x, y, -L._/2) = 1/2 (5.4)

Ow(z) is assumed to be given by a certain temperature control device at the side walls. Both the velocity and temperature fields are assumed to be periodic in x with period L,..

In the same way as before, u(x) and 0(x) are decomposed into the compensating fields u'(x) and 0'(x) plus the basic fields U(z) and O(y, z), which satisfy

0 = -dP/dx + (1/Re)(O2/Oy 2 + OZ/Oz z) U (5.5)

0 = (02/0)22 + O2/OZ 2) O (5.6)

together with Eqs. (5.3) and (5.4). The conditional Fourier spectral method must be applied to both u'(x) and 0'(x) with a more careful use of the function subspace technique based on Eqs.(3.1)-(3.6). We use 3 2 x 6 4 x 3 2 wavenumber modes in this case. We made sure that, in order to reach a fully-developed state in this problem, it is necessary and sufficient to give nonzero initial values to not all but some v'(k), even if 0'(x) starts from zero. We gave normal-random initial disturbances of the order-of- magnitude of 10-6 to the lowest 9 x 19 x 9 wavenumber modes except for v'(0). The duct geometry is given by L,_ = 3n, L;, = 1.5 and L_- = 0.75, which give De = 1. Here the example of calculation for Re = 220 and Pr = 0.7 is presented with O,,.(z)= - z , that is the case of perfectly-conducting side walls. The step of time-integration is varied depending on Gr so as to insure a sufficient accuracy.

In this case, both u' and 0' extinguish for Gr less than 7837, that corresponds to the critical Rayleigh number for aspect ratio 2 by Lee et aL


N .x _;p

(b)

RE= 2 2 0 GR~ 2 4 0 0 0 OT=O.OIO T T P E - I O

~-4

-I0

- 1 2

-14 0 100 200 L

(a) f%~t~O

2 S 4 n~

P h , - 2

5ECOHDRRT ~'LOH

X= 1/33 VHRX O . | g 4 3 E + 0 0 T I H E - 2 0 0 . 0 0

RE 2 2 0 . 0 0 OR 2 4 0 0 0 . 0 0 - T T P E -

T

TEI'~PERFITURE CONTOUR :

X = I / 33 T / H E = 2 0 0 , 0 0 RE - 220.00 GR o 24000.00

CHnX = O. IO00E~0t CHIN = O.OOOOE*O0 LSt'l 0. SO00E'00 5H O. 4999E+Q0 - T'(PE =

-

. . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . '

T

Fig. 7. (a) Evolutional Fourier spectrograph in terms of Iv ' (k)l 2 of the compensating velocity u'{x) of a flow in a horizontal rectangular duct heated from below with the perfectly-conducting side walls for Re = 220, Gr = 24,000 and Pr = 0.7. All curves are distinguished by the symbols shown below right. (b) The secondary flow velocity in a cross section at the steady state. (c) Temperature contour in a cross section at the steady state.

"uoI]o;s sso.~3 [~3!dM t~ ut .tno;uoo o.~nlea~dtuol po}(~e.]~A~-ottl!I (3} 'uo!l~os sso.T3 I~3.1dAl ~ Ul

~O U ,(a~puoo~s p~a~^~-~ua!,L (q) "90I x 8"t~=aD ao 3 }d~ox~ eL "~!d s,~ oua~s ~ql (~) "8 "~!A

(~)

= ddZ~- UU*JEUUS"U = N~ DB+30O0S'0 = N~q 00~]0000"0 ~ NIH~ 10+~0001"0 = XUH3

0"000008~ ~O 00"0~ 3~ 0"00~-D'OE 39UY3AU = ]NI~ EE / I = X

: W~OINO3 ~gQIUW~N~I

~. ,~,.." ............. .,~.k"--"~

.~,, .............. ::': ....

;::::::::: ::: : : : : :: : : :',::: ::::

O'OO~-O'O~ 39UU3^U ~HI.~ O0,3L@~L'O XUH^ ~;E / I = X

HO'I~ ),~UONO3~ ~

:q)

l==U

U t ~ [ ; O

{

r/o O~=U

4 00~ 00[ 0 ! ....

j|-

8-g

I- rrt t-

,- x

;K"

N

J

.;8t: S~OlA alq!sssJdmo~u I zoj poq~alA ~ iga)~odS z~!zno A i~uo!]!puo:)


(1989), Beyond this critical value until Gr x 105, only the 2D components of u' and 0' evolve to a certain steady state, as is seen in the EFS (this time, in terms of Iv(k)l 2) of Fig. 7a for Gr=24,000 at which the steady secondary flow makes a symmetric pair of longitudinal convection rolls, as is seen in Fig. 7b. All the longitudinal modes of u' with k,, ~ 0 are not excited at all but decay. The corresponding temperature contour is shown in Fig. 7c. In contrast to this flow, we can see a turbulent flow for Gr =4.8 x 106, as is seen in Fig. 8a. u' is no longer two-dimensional but many longitudinal modes are excited and entangled with all the other modes. (See green and yellow lines. The other higher modes are omitted in the figure.) Some of the most leading modes with large energy determine the large-scale behavior of the flow. The time-averaged pictures of the secondary flow velocity and temperature contour in Fig. 8b and c suggest such a behavior. More details of this calculation are seen in Tanaka (1992), Hosokawa (1993), and Hosokawa et aL (1993).

6. CONCLUSION

The conditional Fourier spectral method presented here for direct numerical simulation of incompressible flows is simple to treat and clear in logic in comparison with the other methods for the same purpose. Two examples of application have been shown to illustrate how useful and reliable it is in practice. Although the accuracy depends on the number of the wavenumber modes we can take, the result shown here suggests that the method has a considerable degree of reliability in analyzing incompressible unsteady shear flows. The overall feature of the flows is expected to be captured by this method within the reach of the resolution.

REFERENCES

Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1988j. Spectral Methods #3 Fluid D),namics, Springer-Verlag, New York.

Herbert, T. (1975). Finite Amplitude Stability of Plane Parallel Flows. AGARD CP-224, p. 3-1.

Hosokawa, t., and Yamamoto, K. (1986a). On Direct Numerical Simulation of Incom- pressible Shear-Flow Turbulences by the Fourier-Spectral Method. J. Phys. Soc. Japan 55, 1030-1031.

Hosokawa, I., and Yamamoto, K. (1986b). Energy decay of three-dimensional isotropic turbulence. Phys. Fluids 29, 2013-2014.

Hosokawa, L (1993). Numerical Simulation of 3D No-Slip Bounded Flows in a Horisontal Chemical Vapor Deposition Duct. In Nakayama, W., and Yang, K-T. (eds.), Computers and Computhlg 0~ Heat Transfer Science and Enghleering, CRC Press, Boca Raton, pp. 43-58.


Hosokawa, I., Tanaka, Y., and Yamamoto, K. (1993). Mixed convective flow with mass transfer in a horizontal rectangular duct heated from below simulated by the conditional Fourier spectral analysis, bit. J. Heat Mass Transfer 36, 3029-3042.

Huser, A., and Biringen, S. (1993). Direct Numerical Simulation of Turbulent Flow in a Square Duct. A1AA Paper 93-0198.

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