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Stability of unstably stratified shear flow between parallel plates ∗
281
Fluid Dynamics Research 2 (1988) 281-292 North-Holland
Stability of unstably stratified shear flow between parallel plates * Kaoru FUJIMURA
and Robert E. KELLY
Mechanical, Aerospace and Nuclear Engineering Department, Los Angeles, CA 90024, U.S.A.
University of California,
Received 5 October 1987 Abstract. The linear stability of unstably stratified shear flows between two horizontal parallel plates has been investigated. The eigenvalue problem was solved numerically by making use of the expansion method in Chebyshev polynomials, and critical Rayleigh numbers were obtained accurately in the Reynolds number range of [O.Ol, 1001. It was found that the critical Rayleigh number for two-dimensional disturbances increases with an increase of the Reynolds number. The result strongly supports previous stability analyses except for the analysis by Makino and Ishikawa (1985) in which a decrease of the critical Rayleigh number was obtained. For some cases, a discontinuity in the critical wavenumber occurs, due to the development of two extrema in the neutral stability boundary.
1. Introduction An unstably stratified fluid layer between two horizontal parallel plates with different temperatures will be unstable to an external disturbance if the Rayleigh number Ra is greater than its critical value, Ra,. On a linear basis, the unstable disturbance can have various spatial patterns like a roll or a hexagonal cell because the fluid layer has no preferred direction in the horizontal plane. An unstably stratified parallel shear flow, on the other hand, has a special direction corresponding to the basic shear flow. Its stability characteristics therefore strongly depend on the structure of a disturbance in the horizontal plane. Let us denote the direction of the shear flow as x. The coordinate y is selected to be perpendicular to x and perpendicular to the parallel plates. The basic parallel shear flow is expressed as u = [U(r), 0, 01. The stability of stratified shear flows with regard to a longitudinal-roll disturbance whose axis is parallel to the x-direction is known to be independent of the basic shear flow U(y). The critical Rayleigh number Ra, is thus equivalent to that for the pure thermal convection problem without shear (U = 0). The effect of the shear flow becomes essential as the axis of the roll deviates from the x-direction. Many authors have investigated the stability of unstably stratified shear flows: Gage and Reid (1968) and Tveitereid (1974) considered the stability of plane Poiseuille flow and showed that Ra, for a transverse-roll disturbance whose axis is in the spanwise direction is always larger than Ra, for the longitudinal roll. Deardorff (1965), Gallagher and Mercer (1965), Ingersoll (1966), and Asai (1970), investigated the stability of plane Couette flow. They also showed that Ra, with U f 0 is always greater than Ra, with U = 0 for transverse disturbances. Ingersoll obtained Ra,‘s for the transverse roll analytically based on expansions in (cxR)-‘/~ and R for R B- 1 and R -e 1, respectively, where R is the Reynolds number and a! is a * Translated from Nagare, Journal of Japan Society of Fluid Mechanics 6 (1987) 248-257.
0169-5983/88/$4X)0
0 1988, The Japan Society of Fluid Mechanics
282
K. Fujimura,
K.E. Kelly / Stability
of unstably stratified shenr$ow
wavenumber of the transverse roll. He showed that Ra, is an increasing function of R and Ra, (Uf 0) is thus always greater than Ra, (V 0). He noted that the behavior of Ra, for small R is consistent with Ra, for an intermediate Reynolds number which was obtained by Deardorff, Gallagher and Mercer and by himself numerically. Judging from these linear stability results, the longitudinal rolf is the most unstable disturbance in an unstably stratified shear flow. Makino and Ishikawa (1985) analyzed the linear stability of unstabiy stratified plane Poiseuille flow, plane Couette flow and coupled Poiseuille-Couette flaw by making use of an expansion in LYR(-=c 1). They showed that there is a range of small Reynolds numbers in which Ra, (U $0) Q Ra, (U = O), contrary to the previous analyses including Ingersoll’s. They claimed that their results are complementary to those of Deardorff because stability for small Reynolds numbers had not been considered by Deardorff. The result by Makino and Ishikawa is obviously opposed to Ingersoll’s finding, although they did not refer to Ingersoll’s paper. Either the results of Makino and Ishikawa or those of Ingersoll must therefore be in error in the small Reynolds number range. In order to identify whieh one is correct, it is reasonable to make use of a method independent of the expansions in R or in (YR used by Ingersoll or Makino and Ishikawa, respectively. In this paper, we solve disturbance equations numerically by using an expansion method in Chebyshev polynomials and investigate the linear stability of unstably stratified plane Poiseuille flow, plane Couette flow and coupled Poiseuille-Couette flow with respect to the transverse roll in the Reynolds number range of [O.Ol, loo]. We show that the inequality Ra, (U f 0) > Ra, (U = 0) obtained by previous authors including Ingersoll for Couette or Poiseuille flow is supported in the whole Reynolds number range considered here and also holds for the combined case.
Consider the linear stability of the parallel shear flow [U(v), 0, O] between two horizontal parallel plates with different temperatures: T, - AT (AT > 0) at y = h and To + AT at y = -15. We present here only the final form of the disturbance equations because their detailed derivation has been given by the previous authors. All quantities are non-dimensionalized on the basis of the characteristic length h, the characteristic velocity U, which is the maximum velocity in the cross-section of the channel, and the characteristic temperature AT, as done by Makino and Ishikawa. There are three non-~mensional parameters in this problem: Reynolds number, R = &Q%: Prandtl number, P = Y/K Rayleigh number, Ra = g~AT~~~(~K): where g is the gravity acceleration, v is the kinematic viscosity, K is the thermal diffusivity, and x is the thermal expansion coefficient. Values of the physical properties are evaluated at the center temperature of the channel, Tba Deriving the linear equation for a disturbance 8 = (6, 0, $), j, and ? added to the velocity field 0 = [V(y), 0, 01, pressure_ p and temperature T and eliminating ii, Q and 6, we obtain coupled equations for v^ and T. Further adopting the normal mode analysis for the transverse roll:
(2.1)
K. Fujimura, R.E. Kelly / Stability of unstably stratified
we obtain the following well known coupled equations: icu(U-
c)S - ia-
d2U
1
- R-?S2
C#J = - Ra( PR2)-‘c-x28
dy2
shear flow
283
7 (2.2)
where S is a linear operator defined by
+d2_& dy2 and c is a complex phase velocity of the disturbance. We consider the linear stability of coupled plane Poiseuille-Couette flow. Let the ratio of the mean velocity to the velocity of the top wall at y = 1 be k (the bottom wall at y = - 1 is at rest). The normalized velocity profile is then given by qy>
0
=
-w
-Y’>
+
umax
Ml
+r> 9
(2.3)
where k = 4/(6k + 1) and
uInax=
i
(4-3k)2/[16(1-k)],
forO
k,
for:
Although a parameter ‘m ’ was used by Makino and Ishikawa to identify the coupled Poiseuille-Couette flow profile, the definition of ‘m’ has not been presented in their paper. The relation between k and ‘m’ is thus unclear. The boundary conditions
(2.4) are imposed at y = f 1.
3. Numerical method In order to solve the linear disturbance equations (2.2) under the boundary conditions (2.4) we expand $J and 8 in Chebyshev polynomials T,(y), n = 0, 1, 2,. . . , as 9(Y)
= Ecu”@
d(Y) = E &(l n=O
-Y2J2T,(Y)
-Y2)T,(Y)
= 5 %UY), n=O = Z? b,<(Y). n=O
(3.1)
The expansion functions f” and ?n include the factors (1 - y2)’ and (1 - y*), respectively, in order for $I and 8 to satisfy the boundary conditions (2.4) automatically. Substituting (3.1) into (2.2) and making use of the collocation method, we obtain the following coupled algebraic equations for expansion coefficients {a, } and {b,}:
[AA:: :::I[;] =c[Bg:: :::][;].
(3.2)
284
K. Fujimura,
R. E. Kelly / Stability of unstably stratified shear flow
where
[A,2]j, = Ra(PR’)-ta2$y,), L%ll,m
= - <(YJ,
[4,lj,
= [Wy,)
- (PR)-‘S]i;,(y,),
[BIIljm=iaS~(Ym)~ [B121jm=[B2*ljm=o~ [%ljm =ia<(y,), a and
b are (N + l)-vectors: a= [a,,
a,, a, ,...I
a,lT,
b= [ba, b,, b,,...,
b,jT,
and y, denotes the collocation points selected as 2rn7f
Ytn=cos
( 1 N+2
3
m=1,2
,...)
N+ 1,
l-=K INI < t-w.
Equations (3.2) can be written as Ax = CBX, concisely. The complex phase velocity c and the vector x can be solved as an eigenvalue and the corresponding eigenfunction of the matrix B-IA, respectively. The QR method was utilized to solve the eigenvalue problem. In order to evaluate the accurate critical condition for given value of k, R, and P, the following procedure was adopted: the neutral Rayleigh number Ra, for fixed (Y is obtained on the basis of the Newton-Raphson method and then the value of ( LY,Ra,) at which dRa,/dcu = 0 is satisfied is searched again by the Newton-Raphson procedure. The values of (Y and Ra, correspond to the critical wavenumber and the critical Rayleigh number, respectively, if d2Ra,/da2 is positive. We will further discuss this point in the next section. The accuracy of the resulting critical condition ((Ye, Ra,) depends on the convergence criterion of the Newton-Raphson method as well as on the number of terms (N -t 1) at which the polynomial expansions (3.1) are truncated. In order to get the critical condition within eight significant figures, N > 10 was necessary for R < 10; and in order to evaluate the critical condition within this accuracy for one choice of (k, R, P), calculation of the eigenvalues was done more than 288 times. All the results for the critical condition presented in this paper are based on N = 14, and all the figures of the critical value presented in Table 1 are significant.
4. Results and discussions The critical wavenumber ay,, the critical Rayleigh number Ra, and the critical phase velocity c, for the transverse roll in unstably stratified plane Poiseuille-Couette flow are presented in Table 1 for various values of the Reynolds number. (Hereafter the suffix r indicates the real part and the suffix i indicates the imaginary part of the preceding term.) The parameter k in the definition of the basic flow (2.3) was selected to be 0 (plane Poiseuille flow), 0.2, 0.4, 0.6, 0.8, and 1 (plane Couette flow). We fixed the Prandtl number as 0.51 in order to compare the
K. Fujimura, R. E. Kelry / Stability of unstably stratified shear flow
285
Table 1 The critical wavenumber, the critical Rayleigh number and the critical phase velocity for unstably stratified plane Poiseuille-Couette flow. P = 0.51 R
k = 0.2
k=O a,
Ra,
c,
%
Ra,
%
0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 5 10 20 50 100
1.5581619 1.5581619 1.5581623 1.5581635 1.5581685 1.5581769 1.5581886 1.5582036 1.5583290 1.5588300 1.5623177 1.5744903 1.6183148 1.73537 2.68960
106.73512 106.73514 106.73529 106.73584 106.73801 106.74163 106.74671 106.75323 106.80760 107.02507 108.54811 113.99812 135.98804 303.493 1097.47
0.733261 0.733261 0.733261 0.733261 0.733262 0.733262 0.733262 0.733262 0.733265 0.733275 0.733345 0.733594 0.734518 0.735566 0.836267
1.5581618 1.5581619 1.5581621 1.5581629 1.5581661 1.5581714 1.5581789 1.5581884 1.5582682 1.5585871 1.5608054 1.5685294 1.5963677 1.70303 2.12301
106.73512 106.73514 106.73526 106.73571 106.73752 106.74053 106.74475 106.75017 106.79538 106.97612 108.24249 112.77979 131.16317 271.179 860.685
0.760259 0.760259 0.760259 0.760259 0.760259 0.760259 0.760259 0.760260 0.760261 0.760266 0.760299 0.760418 0.760896 0.764519 0.808983
R
k = 0.4
0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 5 10 20 50 100
1.5581618 1.5581618 1.5581618 1.5581615 1.5581606 1.5581591 1.5581569 1.5581541 1.5581310 1.5580383 1.5573844 1.5549828 1.5445706 1.47826 2.34219
106.73512 106.73514 106.73527 106.73575 106.73765 106.74083 106.74527 106.75099 106.79863 106.98924 108.32516 113.12009 132.66378 282.637 670.625
0.784535 0.784535 0.784535 0.784535 0.784535 0.784534 0.784534 0.784534 0.784530 0.784514 0.784403 0.784005 0.782391 0.770637 0.905541
R
k = 0.8
% 0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 5 10 20 50 100
1.5581618 1.5581616 1.5581606 1.5581689 1.5581421 1.5581173 1.5580828 1.5580383 1.5576678 1.5561894 1.5459916 1.5115777 1.3982778 1.02178 0.66545
k = 0.6
106.73512 106.73513 106.73525 106.73567 106.73734 106.74012 106.74401 106.74901 106.79072 106.95760 108.12773 112.33523 129.60043 267.630 620.305
0.783621 0.783621 0.783621 0.783621 0.783621 0.783620 0.783620 0.783620 0.783619 0.783614 0.783578 0.783452 0.782995 0.782508 0.864349
1.5581618 1.5581617 1.5581612 1.5581591 1.5581510 1.5581374 1.5581185 1.5580941 1.5578910 1.5570796 1.5514484 1.5319945 1.4625463 1.14356 2.37093 k=l
Ra, 106.73512 106.13514 106.73519 106.73584 106.73804 106.74170 106.74683 106.75342 106.80833 107.02796 108.56367 114.02607 135.58607 280.231 771.570
cm
%
Ra,
crc
0.683315 0.683315 0.683315 0.683315 0.683315 0.683315 0.683315 0.683314 0.683311 0.683298 0.683208 0.682879 0.681529 0.673479 0.660212
1.5581618 1.5581616 1.5581606 1.5581568 1.5581416 1.5581164 1.5580811 1.5580357 1.5576574 1.5561478 1.5457357 1.5106217 1.3955472 1.025482 0.68565
106.73512 106.73514 106.73528 106.73580 106.73786 106.74129 106.74610 106.75238 106.80380 107.00982 108.44925 113.55460 133.49495 261 A447 671.811
0.5 0.5 0.5 0.5 0.5 0.5 0.5 OS 0.5 0.5 0.5 0.5 0.5 0.5 0.5
286
K. Fujimura, R.L: Kel& / Stability of unstably stratified shear flow
Table 2 The criticalcondition for p&meComtte flow based on eq. (4.1). P = 0.51 R
‘yc
R%
a.01 0.02 0.05 0.1 0.2 0.3 0.4 0.s I 2 5 10 20 50 100
1.55850 1.55850 1.55850 1.55850 l.55848 1.55845 1.55842 1.55837 1.55799 1.55648 1.54587 1.50798 1.35640 0.295 - 3.494
156.735 106.735 106.735 106.736 106.738 106.741 106.746 106.752 106.804 107.01a 108.453 113.607 134.233 273.537 793.945
stability results with those in Makino and Ishikawa’s paper. The table clearly shows that the critical Rayleigh number Ra, increases with an increase of the Reynolds number R for all values of k. The dependence of Ra, on k is so small for R G 10 that Ra, cannot be a decreasing function of R for particular values of k E [Cl, 11. Makino and Ishikawa, on the other hand, showed that Ra, decreases monotonously as R increases from 0 to 0.5 for k = 0, 1, and ‘m = 1’ (coupled Poiseuille-Couette flow). Their conclusion is therefore opposed to the present numerical results. Ingersoll inv~tigat~ the stability of plane Couette flow and obtained Ra, and OL, For Relas Ra, = & 11707.76 + 4R2(0.5598Pt a, = $13.117 - 0.0004R2(2.325P2
+ 0.127OP + 0.06451)5) + 1.503P + l.lSS)j.
(4.0
The first equation of (4.1) indicates that Ra, is an increasing function of R for R K 1. Values of OL,and Ra, based on eqs. (4,l) are presented in Table 2 for P = 0.51. The corresponding values for k = 1 in Table 1 agree well with those in Table 2 for R G 10. The formulae (4.1) obtained by Ingersoll for R -=z 1 were thus reproduced by the present numerical calculation. The validity of the present analysis was also demonstrated by a comparison of our results with those obtained by Deardorff (1965) and Gallagher and Mercer (1965) for Couette flow in the intermediate Reynolds number range although we do not show the comparison here, Equations (4.1) are therefore found to be consistent with the previous results for intermediate R, as Ingersoll claimed. The stability characteristic of plane Poiseuille flow obtained by Gage and Reid is also found to agree with the present one for intermediate R. Judging from the agreement between the present result and the previous one for both k = 0 and 1, we can conclude that all the stability characteristics obtained here are consistent with those obtained by previous authors except for Makino and Isbikawa. We found eventually that the neutral curves obtained by Makino and Ishikawa based on the expansion in rrR are consistent neither with the critical conditions obtained here for 0.01 < R =S ‘100 nor with the linear stability characteristics for intermediate Reynolds number obtained by the previous authors. The conclusion in Makino and Ishikawa’s paper that Ra, for the transverse roll decreases with an increase of R for R -ZC1 is thus found to be in error. Our
K. Fujimura, R. E. KeNy / Stab&y
I
t
I
I
of unstably stratified sheor flow
I
,
I
I
I
1000
500
t
287
Ret
Fig. 1. Transition from singly peaked neutral curve to doubly peaked neutral curve. k = 0.5 and P = 0.5.
numerical results strongly support the previous linear stability prediction at least in the Reynolds number range 0.01 G R G 100: a lon~tudinal roll appears as the most unstable disturbance in unstably stratified, Fully developed shear flows. We have discussed the critical condition for the transverse roll. Careful reading of Table 1 lets us find that the critical wavenumber and the critical phase veiocity for k = 0.4 and 0.6 do not change monotonously for R z 20. This non-monotonous change of EY,and c, is due to the intrinsic nature of a stability problem in which the velocity field couples with the temperature field through disturbance equations. Let us further consider in detail the linear stability with respect to transverse roll disturbances for various values of k, R and P in the remainder of this paper. The neutral stability curves for k = OS and P = 0.5 are depicted in Fig. 1 with various values of R. The curve for R = 50 posesses a single extremal point, and this point corresponds to the critical point. The point at which relations dRa -“=o dcu
and
d”Ra -----““>o daz
are satisfied is called the extremal point hereafter. The extremal point giving the minimum value of Ra, corresponds to the critical point if there are other extremal points on the neutral curve, A second extremal point appears in the higher wavenumber range on the curve in Fig. 1 for R = 70. The new extremal point becomes the critical point because the value of Ra, for the new point is smaller than that for the old one. The new extremal point thus replaces the old critical point. Hereafter we refer to this phenomenon as the replacement of the critical point. The neutral curves for k = 0.5 and R = 100 are depicted in Fig. 2 with various values of the Prandtl number. The replacement of the critical point again occurs as P changes. Change of Ra, and LY,for the extremal point is presented in Figs, 3 and 4, respectively, for k = 0.5 and for P = 0.2, 0.5 and 1. Solid lines in these figures denott: the critical points, and broken Iines indicate the other extremal points. There is only one extremal point corresponding
288
ofunstablystratified
K. Fujimura, R.E. Kelly / Stability
P:O.l
I
I
I
I
0.2 0.5
1.
I
I
2000
0
shear flow
2.
l-l
I
4000
Ra
Fig. 2. Dependence of the second extremal point on the Prandtl number. k = 0.5 and R = 100.
I I
Pzl
Rat
' I
3rr2
0
IR
100
Fig. 3. Replacement of the critical point in Ra-R plane. k = 0.5. Solid line denotes the critical point and broken line denotes the second extremal point.
0
R
100
Fig. 4. Replacement of the critical point on aR plane. k = 0.5. Solid line denotes the critical point and broken line denotes the second extremal point. Dotted line is the smoothly connected curve between solid line in the range 0 < R i 30 and broken line in the range 73 < R < 120 for P = 1.
K. Fujimwa,
shear j&w
R.E. ICeI& / Stability of zmstab& stmtifed
289
2.5
2.
cx.
1.5
I
330
350
400
Ra
430
Fig. 5. Shift of the critical point (closed circle) to higher wavenumber range for k = 0.5 and P = 1.
to the critical point in low Reynolds number range for each value of P. The value of Ra, increases with an increase of R. The second extremal point appears at R = 95 for P = 0.2, at R = 61 for P = 0.5 and at R = 73 for P = 1. The replacement of the critical point occnrs at the crossing of the solid line with the broken line: at R = 111 for P = 0.2 and at R = 65 for P = 0.5. The critical Rayleigh number changes continuously at the replacement point while its derivative changes discontinuously. The solid line never crosses the broken line for P = 1, but they separate as R increases.
I
3I -
I
kcO.5 0.4 0.6
I
I
I o-2
1
1
0.0
a
500
moo
I
I 0.6
i
O-
I
Ra
Fig. 6. Nosed neutral curve for various basic flows. R = 100 and P = 0.5.
290
K. Fujimura, R.E. Kelly / Stability of unstably stratified shear
flow
The critical wavenumber, on the other hand, leaps discontinuously when the replacement of the critical point occurs as is shown in Fig. 4. The relationship between the critical point and the other extremal point is qu~itatively equivalent for P = 0.2 and 0.5. 3ut the relation is different for P = 1. The critical wavenumber of P = 1 changes suddenly but continuously for 40 G R G 45. The shape of the curve for P = 1 is like a smoothed shape of the curve of CY,for P = 0.2 or 0.5. The sudden change of the critical wavenumber comes from the shift of the
PzO.1
4-
‘, .
:
r’ ‘.
,-:* . .’
/=
/’
i
0.05 Ra
Fig. 7. Neutral stability curve for the plane Couette flow (k = 1). Solid tine denotes stationary wave with c, = 0.5 and broken line denotes traveling wave with cr+-0.5.
I
I
I
PzO.1 0.2 0.5
I
I
1
2
I aCi
5
0:
Fig. 8. Transition from singly peaked neutral curve to tripiy peaked neutral curve for the plane Poiseuille flow (k = 0). R = 100. Dotted broken line denotes the calculation range in which the linear growth rate was evaluated for Fig. 9.
Fig. 9, Replacement of different eigenmodes on the Linear growth rate oci for plane Poiseuille flow. R = 100, P = 2 and Ra = 53’75. The q’s were evaluated along the dotted broken line in Fig. 8.
K. Fujimura9 RX. Kelly / Stability of unstably stratified shear flow
291
critical point from the low wavenumber range to the high wavenumber range as is shown in Fig, 5, in which the neutral curves for P = 1 and 40 < R < 45 are enlarged. The dotted line in Fig. 4 shows the smoothly connected line between the sohd line in the range 0 -C R < 30 and the broken line in the range 73 c R < 120 for P = 1. The dotted line passes through a point (CX, R) z (1.2,45) at which the neutral curve for R = 45 has a large value of d2Ra,,fdfuL in Fig. 5, It is concluded that there is no big qualitative difference in the change of the critical point between these three different Prandtl number cases although there is difference in the replacement of the critical point. The neutral curves for P = 0.5 and R = 100 are presented in Fig. 6 with various values of k. Each curve is found to be composed of two qualitatively different parts. A neutral curve having such a “nose” is known to occur also in the stability problem for the boundary layer along the heated vertical flat plate (Gebhart, 1973). The coupling between velocity field and temperature field is considered to be the cause of the shape. Let us show the neutral curve for k = 1 (plane Couette flow] and R = 100 in Fig. 7. The solid lines denote c, = 0.5 and the broken lines indicate c, f 0.5. A disturbance with c, = OS corresponds to the stationary wave with respect to moving frame with velocity 0.5 (the center velocity of the basic flow}, while a disturbance with t, # 0.5 cormsponds to the pair of traveling waves passing through in opposite direction ( f x) with phase veIacity AC,, say. Consider the case P = 1 and Ra = 3250, for example. The phase velocity of the stationary disturbance, c, = OS changes suddenly to c, = OS it AC, (AC, # 0) at a: = 1.36819, as CYincreases. This sudden change from stationary mode to traveling mode has been observed by previous authors (Deardorff, Gallagher and Mercer, 1965; Ingersoll, 1966; and Asai, 1970). It is helpful to consider the linear growth rate aCi in order to understand this qualitative change. The linear growth rate aicti corresponding to the first eigenvalue approaches the linear growth rate aczi corresponding to the second eigenvalue (ocrI > CXC~~) as tyf 1.36819, and two (Y&~‘S coincide with each other for cy>, 1.36819. The coincidence of two LYC;%corresponds to the sudden appearance of the traveling wave. Such merging of two modes on the linear growth rate is usually observed in isothermal plane Couette flow (Gailagher and Mercer, 1962). The change from the stationary wave to the traveling wave is therefore not the replacement of one mode by another but should be interpreted as the bifurcation into qualitatively different modes. The neutral curve for k = 0 (plane Poiseuille flow) and for R = 100 is depicted in Fig. 8. Most interesting is the transition from a singly peaked neutral curve to a triply peaked one. The three linear growth rates corresponding to the first three eigenvalues for P = 2 are plotted in Fig. 9 along the dotted broken line (Ra = 5375) in Fig. 8. Figure 9 shows that two unstable wavenumber bands in the lower wavenumber range are composed of the same eigenmode while the unstable wavenumber band centered round the critical wavenumber in higher waven~ber range is composed of a different eigenmode, which is the third eigenmode in the lower wavenumber range. The unstable wavenumber bands for k = 0, R = 100 and P = I, on the other hand, are composed of the same first eigenmode. Higher eigenmodes never replace the first eigenmode for all cases in Fig. 6. 5. Conclusion The linear stability of unstably stratified plane Poiseuille-Couette flow was investigated. It was found that the critical Rayleigh number for the transverse roll is an increasing function of the Reynolds number in the range 0.01 G R G 100 at least, and the effective critical condition is given by the stability characteristic for the lon~tudinal roll. The neutral stability curve for relatively high Prandtl number and high Reynolds number is found to possess more than one extremal point. The critical condition can change place among
292
K. Fujimura, R. E. Kelly / Stability of unstably stratified shear flow
these extremal points. The critical Rayleigh number changes continuously with an increase of the Reynolds number, while the critical wavenumber leaps discontinuously during the replacement. It is also found that more than two extremal points may be composed of different eigenmodes.
References Asai, T. (1970) Three-dimensional features of thermal convection in a plane Couette flow, J. Meteor. Sot. Jpn. 48, 18-29. Deardorff, J.W. (1965) Gravitational instability between horizontal plates with shear, Phys. Fluids 8, 1027-1030. Gage, K.S. and W.H. Reid (1968) The stability of thermally stratified plane Poiseuille flow, J. Fluid Mech. 33, 21-32. Gallagher, A.P. and A.McD. Mercer (1962) On the behaviour of small disturbances in plane Couette flow, J. Fluid Mech. 13, 91-100. Gallagher, A.P. and A.McD. Mercer (1965) On the behaviour of small disturbances in plane Couette flow with a temperature gradient, Proc. Roy. Sot. London A286, 117-128. Gebhart, B. (1973) Instability, transition and turbulence in buoyancy-induced flows, Ann. Rev. Fluid. Mech. 5, 213-246. Ingersoll, A.P. (1966) Convective instabilities in plane Couette flow, Phys. Fluid 9, 682-689. Makino, M. and Y. Ishikawa (1985) The stability of horizontal two-dimensional shear flow with vertical temperature gradient, J. Jpn. Sot. Fluid. Mech. 4, 148-158 (in Japanese). Platten, J.K. and J.C. Legros (1984) Convection in Liquids (Springer, New York) 529-556. Tveitereid, M. (1974) On the stability of thermally stratified plane Poiseuille flow, ZAMM 54, 533-540.
Fluid Dynamics Research 2 (1988) 281-292 North-Holland
Stability of unstably stratified shear flow between parallel plates * Kaoru FUJIMURA
and Robert E. KELLY
Mechanical, Aerospace and Nuclear Engineering Department, Los Angeles, CA 90024, U.S.A.
University of California,
Received 5 October 1987 Abstract. The linear stability of unstably stratified shear flows between two horizontal parallel plates has been investigated. The eigenvalue problem was solved numerically by making use of the expansion method in Chebyshev polynomials, and critical Rayleigh numbers were obtained accurately in the Reynolds number range of [O.Ol, 1001. It was found that the critical Rayleigh number for two-dimensional disturbances increases with an increase of the Reynolds number. The result strongly supports previous stability analyses except for the analysis by Makino and Ishikawa (1985) in which a decrease of the critical Rayleigh number was obtained. For some cases, a discontinuity in the critical wavenumber occurs, due to the development of two extrema in the neutral stability boundary.
1. Introduction An unstably stratified fluid layer between two horizontal parallel plates with different temperatures will be unstable to an external disturbance if the Rayleigh number Ra is greater than its critical value, Ra,. On a linear basis, the unstable disturbance can have various spatial patterns like a roll or a hexagonal cell because the fluid layer has no preferred direction in the horizontal plane. An unstably stratified parallel shear flow, on the other hand, has a special direction corresponding to the basic shear flow. Its stability characteristics therefore strongly depend on the structure of a disturbance in the horizontal plane. Let us denote the direction of the shear flow as x. The coordinate y is selected to be perpendicular to x and perpendicular to the parallel plates. The basic parallel shear flow is expressed as u = [U(r), 0, 01. The stability of stratified shear flows with regard to a longitudinal-roll disturbance whose axis is parallel to the x-direction is known to be independent of the basic shear flow U(y). The critical Rayleigh number Ra, is thus equivalent to that for the pure thermal convection problem without shear (U = 0). The effect of the shear flow becomes essential as the axis of the roll deviates from the x-direction. Many authors have investigated the stability of unstably stratified shear flows: Gage and Reid (1968) and Tveitereid (1974) considered the stability of plane Poiseuille flow and showed that Ra, for a transverse-roll disturbance whose axis is in the spanwise direction is always larger than Ra, for the longitudinal roll. Deardorff (1965), Gallagher and Mercer (1965), Ingersoll (1966), and Asai (1970), investigated the stability of plane Couette flow. They also showed that Ra, with U f 0 is always greater than Ra, with U = 0 for transverse disturbances. Ingersoll obtained Ra,‘s for the transverse roll analytically based on expansions in (cxR)-‘/~ and R for R B- 1 and R -e 1, respectively, where R is the Reynolds number and a! is a * Translated from Nagare, Journal of Japan Society of Fluid Mechanics 6 (1987) 248-257.
0169-5983/88/$4X)0
0 1988, The Japan Society of Fluid Mechanics
282
K. Fujimura,
K.E. Kelly / Stability
of unstably stratified shenr$ow
wavenumber of the transverse roll. He showed that Ra, is an increasing function of R and Ra, (Uf 0) is thus always greater than Ra, (V 0). He noted that the behavior of Ra, for small R is consistent with Ra, for an intermediate Reynolds number which was obtained by Deardorff, Gallagher and Mercer and by himself numerically. Judging from these linear stability results, the longitudinal rolf is the most unstable disturbance in an unstably stratified shear flow. Makino and Ishikawa (1985) analyzed the linear stability of unstabiy stratified plane Poiseuille flow, plane Couette flow and coupled Poiseuille-Couette flaw by making use of an expansion in LYR(-=c 1). They showed that there is a range of small Reynolds numbers in which Ra, (U $0) Q Ra, (U = O), contrary to the previous analyses including Ingersoll’s. They claimed that their results are complementary to those of Deardorff because stability for small Reynolds numbers had not been considered by Deardorff. The result by Makino and Ishikawa is obviously opposed to Ingersoll’s finding, although they did not refer to Ingersoll’s paper. Either the results of Makino and Ishikawa or those of Ingersoll must therefore be in error in the small Reynolds number range. In order to identify whieh one is correct, it is reasonable to make use of a method independent of the expansions in R or in (YR used by Ingersoll or Makino and Ishikawa, respectively. In this paper, we solve disturbance equations numerically by using an expansion method in Chebyshev polynomials and investigate the linear stability of unstably stratified plane Poiseuille flow, plane Couette flow and coupled Poiseuille-Couette flow with respect to the transverse roll in the Reynolds number range of [O.Ol, loo]. We show that the inequality Ra, (U f 0) > Ra, (U = 0) obtained by previous authors including Ingersoll for Couette or Poiseuille flow is supported in the whole Reynolds number range considered here and also holds for the combined case.
Consider the linear stability of the parallel shear flow [U(v), 0, O] between two horizontal parallel plates with different temperatures: T, - AT (AT > 0) at y = h and To + AT at y = -15. We present here only the final form of the disturbance equations because their detailed derivation has been given by the previous authors. All quantities are non-dimensionalized on the basis of the characteristic length h, the characteristic velocity U, which is the maximum velocity in the cross-section of the channel, and the characteristic temperature AT, as done by Makino and Ishikawa. There are three non-~mensional parameters in this problem: Reynolds number, R = &Q%: Prandtl number, P = Y/K Rayleigh number, Ra = g~AT~~~(~K): where g is the gravity acceleration, v is the kinematic viscosity, K is the thermal diffusivity, and x is the thermal expansion coefficient. Values of the physical properties are evaluated at the center temperature of the channel, Tba Deriving the linear equation for a disturbance 8 = (6, 0, $), j, and ? added to the velocity field 0 = [V(y), 0, 01, pressure_ p and temperature T and eliminating ii, Q and 6, we obtain coupled equations for v^ and T. Further adopting the normal mode analysis for the transverse roll:
(2.1)
K. Fujimura, R.E. Kelly / Stability of unstably stratified
we obtain the following well known coupled equations: icu(U-
c)S - ia-
d2U
1
- R-?S2
C#J = - Ra( PR2)-‘c-x28
dy2
shear flow
283
7 (2.2)
where S is a linear operator defined by
+d2_& dy2 and c is a complex phase velocity of the disturbance. We consider the linear stability of coupled plane Poiseuille-Couette flow. Let the ratio of the mean velocity to the velocity of the top wall at y = 1 be k (the bottom wall at y = - 1 is at rest). The normalized velocity profile is then given by qy>
0
=
-w
-Y’>
+
umax
Ml
+r> 9
(2.3)
where k = 4/(6k + 1) and
uInax=
i
(4-3k)2/[16(1-k)],
forO
k,
for:
Although a parameter ‘m ’ was used by Makino and Ishikawa to identify the coupled Poiseuille-Couette flow profile, the definition of ‘m’ has not been presented in their paper. The relation between k and ‘m’ is thus unclear. The boundary conditions
(2.4) are imposed at y = f 1.
3. Numerical method In order to solve the linear disturbance equations (2.2) under the boundary conditions (2.4) we expand $J and 8 in Chebyshev polynomials T,(y), n = 0, 1, 2,. . . , as 9(Y)
= Ecu”@
d(Y) = E &(l n=O
-Y2J2T,(Y)
-Y2)T,(Y)
= 5 %UY), n=O = Z? b,<(Y). n=O
(3.1)
The expansion functions f” and ?n include the factors (1 - y2)’ and (1 - y*), respectively, in order for $I and 8 to satisfy the boundary conditions (2.4) automatically. Substituting (3.1) into (2.2) and making use of the collocation method, we obtain the following coupled algebraic equations for expansion coefficients {a, } and {b,}:
[AA:: :::I[;] =c[Bg:: :::][;].
(3.2)
284
K. Fujimura,
R. E. Kelly / Stability of unstably stratified shear flow
where
[A,2]j, = Ra(PR’)-ta2$y,), L%ll,m
= - <(YJ,
[4,lj,
= [Wy,)
- (PR)-‘S]i;,(y,),
[BIIljm=iaS~(Ym)~ [B121jm=[B2*ljm=o~ [%ljm =ia<(y,), a and
b are (N + l)-vectors: a= [a,,
a,, a, ,...I
a,lT,
b= [ba, b,, b,,...,
b,jT,
and y, denotes the collocation points selected as 2rn7f
Ytn=cos
( 1 N+2
3
m=1,2
,...)
N+ 1,
l-=K INI < t-w.
Equations (3.2) can be written as Ax = CBX, concisely. The complex phase velocity c and the vector x can be solved as an eigenvalue and the corresponding eigenfunction of the matrix B-IA, respectively. The QR method was utilized to solve the eigenvalue problem. In order to evaluate the accurate critical condition for given value of k, R, and P, the following procedure was adopted: the neutral Rayleigh number Ra, for fixed (Y is obtained on the basis of the Newton-Raphson method and then the value of ( LY,Ra,) at which dRa,/dcu = 0 is satisfied is searched again by the Newton-Raphson procedure. The values of (Y and Ra, correspond to the critical wavenumber and the critical Rayleigh number, respectively, if d2Ra,/da2 is positive. We will further discuss this point in the next section. The accuracy of the resulting critical condition ((Ye, Ra,) depends on the convergence criterion of the Newton-Raphson method as well as on the number of terms (N -t 1) at which the polynomial expansions (3.1) are truncated. In order to get the critical condition within eight significant figures, N > 10 was necessary for R < 10; and in order to evaluate the critical condition within this accuracy for one choice of (k, R, P), calculation of the eigenvalues was done more than 288 times. All the results for the critical condition presented in this paper are based on N = 14, and all the figures of the critical value presented in Table 1 are significant.
4. Results and discussions The critical wavenumber ay,, the critical Rayleigh number Ra, and the critical phase velocity c, for the transverse roll in unstably stratified plane Poiseuille-Couette flow are presented in Table 1 for various values of the Reynolds number. (Hereafter the suffix r indicates the real part and the suffix i indicates the imaginary part of the preceding term.) The parameter k in the definition of the basic flow (2.3) was selected to be 0 (plane Poiseuille flow), 0.2, 0.4, 0.6, 0.8, and 1 (plane Couette flow). We fixed the Prandtl number as 0.51 in order to compare the
K. Fujimura, R. E. Kelry / Stability of unstably stratified shear flow
285
Table 1 The critical wavenumber, the critical Rayleigh number and the critical phase velocity for unstably stratified plane Poiseuille-Couette flow. P = 0.51 R
k = 0.2
k=O a,
Ra,
c,
%
Ra,
%
0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 5 10 20 50 100
1.5581619 1.5581619 1.5581623 1.5581635 1.5581685 1.5581769 1.5581886 1.5582036 1.5583290 1.5588300 1.5623177 1.5744903 1.6183148 1.73537 2.68960
106.73512 106.73514 106.73529 106.73584 106.73801 106.74163 106.74671 106.75323 106.80760 107.02507 108.54811 113.99812 135.98804 303.493 1097.47
0.733261 0.733261 0.733261 0.733261 0.733262 0.733262 0.733262 0.733262 0.733265 0.733275 0.733345 0.733594 0.734518 0.735566 0.836267
1.5581618 1.5581619 1.5581621 1.5581629 1.5581661 1.5581714 1.5581789 1.5581884 1.5582682 1.5585871 1.5608054 1.5685294 1.5963677 1.70303 2.12301
106.73512 106.73514 106.73526 106.73571 106.73752 106.74053 106.74475 106.75017 106.79538 106.97612 108.24249 112.77979 131.16317 271.179 860.685
0.760259 0.760259 0.760259 0.760259 0.760259 0.760259 0.760259 0.760260 0.760261 0.760266 0.760299 0.760418 0.760896 0.764519 0.808983
R
k = 0.4
0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 5 10 20 50 100
1.5581618 1.5581618 1.5581618 1.5581615 1.5581606 1.5581591 1.5581569 1.5581541 1.5581310 1.5580383 1.5573844 1.5549828 1.5445706 1.47826 2.34219
106.73512 106.73514 106.73527 106.73575 106.73765 106.74083 106.74527 106.75099 106.79863 106.98924 108.32516 113.12009 132.66378 282.637 670.625
0.784535 0.784535 0.784535 0.784535 0.784535 0.784534 0.784534 0.784534 0.784530 0.784514 0.784403 0.784005 0.782391 0.770637 0.905541
R
k = 0.8
% 0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 1 2 5 10 20 50 100
1.5581618 1.5581616 1.5581606 1.5581689 1.5581421 1.5581173 1.5580828 1.5580383 1.5576678 1.5561894 1.5459916 1.5115777 1.3982778 1.02178 0.66545
k = 0.6
106.73512 106.73513 106.73525 106.73567 106.73734 106.74012 106.74401 106.74901 106.79072 106.95760 108.12773 112.33523 129.60043 267.630 620.305
0.783621 0.783621 0.783621 0.783621 0.783621 0.783620 0.783620 0.783620 0.783619 0.783614 0.783578 0.783452 0.782995 0.782508 0.864349
1.5581618 1.5581617 1.5581612 1.5581591 1.5581510 1.5581374 1.5581185 1.5580941 1.5578910 1.5570796 1.5514484 1.5319945 1.4625463 1.14356 2.37093 k=l
Ra, 106.73512 106.13514 106.73519 106.73584 106.73804 106.74170 106.74683 106.75342 106.80833 107.02796 108.56367 114.02607 135.58607 280.231 771.570
cm
%
Ra,
crc
0.683315 0.683315 0.683315 0.683315 0.683315 0.683315 0.683315 0.683314 0.683311 0.683298 0.683208 0.682879 0.681529 0.673479 0.660212
1.5581618 1.5581616 1.5581606 1.5581568 1.5581416 1.5581164 1.5580811 1.5580357 1.5576574 1.5561478 1.5457357 1.5106217 1.3955472 1.025482 0.68565
106.73512 106.73514 106.73528 106.73580 106.73786 106.74129 106.74610 106.75238 106.80380 107.00982 108.44925 113.55460 133.49495 261 A447 671.811
0.5 0.5 0.5 0.5 0.5 0.5 0.5 OS 0.5 0.5 0.5 0.5 0.5 0.5 0.5
286
K. Fujimura, R.L: Kel& / Stability of unstably stratified shear flow
Table 2 The criticalcondition for p&meComtte flow based on eq. (4.1). P = 0.51 R
‘yc
R%
a.01 0.02 0.05 0.1 0.2 0.3 0.4 0.s I 2 5 10 20 50 100
1.55850 1.55850 1.55850 1.55850 l.55848 1.55845 1.55842 1.55837 1.55799 1.55648 1.54587 1.50798 1.35640 0.295 - 3.494
156.735 106.735 106.735 106.736 106.738 106.741 106.746 106.752 106.804 107.01a 108.453 113.607 134.233 273.537 793.945
stability results with those in Makino and Ishikawa’s paper. The table clearly shows that the critical Rayleigh number Ra, increases with an increase of the Reynolds number R for all values of k. The dependence of Ra, on k is so small for R G 10 that Ra, cannot be a decreasing function of R for particular values of k E [Cl, 11. Makino and Ishikawa, on the other hand, showed that Ra, decreases monotonously as R increases from 0 to 0.5 for k = 0, 1, and ‘m = 1’ (coupled Poiseuille-Couette flow). Their conclusion is therefore opposed to the present numerical results. Ingersoll inv~tigat~ the stability of plane Couette flow and obtained Ra, and OL, For Relas Ra, = & 11707.76 + 4R2(0.5598Pt a, = $13.117 - 0.0004R2(2.325P2
+ 0.127OP + 0.06451)5) + 1.503P + l.lSS)j.
(4.0
The first equation of (4.1) indicates that Ra, is an increasing function of R for R K 1. Values of OL,and Ra, based on eqs. (4,l) are presented in Table 2 for P = 0.51. The corresponding values for k = 1 in Table 1 agree well with those in Table 2 for R G 10. The formulae (4.1) obtained by Ingersoll for R -=z 1 were thus reproduced by the present numerical calculation. The validity of the present analysis was also demonstrated by a comparison of our results with those obtained by Deardorff (1965) and Gallagher and Mercer (1965) for Couette flow in the intermediate Reynolds number range although we do not show the comparison here, Equations (4.1) are therefore found to be consistent with the previous results for intermediate R, as Ingersoll claimed. The stability characteristic of plane Poiseuille flow obtained by Gage and Reid is also found to agree with the present one for intermediate R. Judging from the agreement between the present result and the previous one for both k = 0 and 1, we can conclude that all the stability characteristics obtained here are consistent with those obtained by previous authors except for Makino and Isbikawa. We found eventually that the neutral curves obtained by Makino and Ishikawa based on the expansion in rrR are consistent neither with the critical conditions obtained here for 0.01 < R =S ‘100 nor with the linear stability characteristics for intermediate Reynolds number obtained by the previous authors. The conclusion in Makino and Ishikawa’s paper that Ra, for the transverse roll decreases with an increase of R for R -ZC1 is thus found to be in error. Our
K. Fujimura, R. E. KeNy / Stab&y
I
t
I
I
of unstably stratified sheor flow
I
,
I
I
I
1000
500
t
287
Ret
Fig. 1. Transition from singly peaked neutral curve to doubly peaked neutral curve. k = 0.5 and P = 0.5.
numerical results strongly support the previous linear stability prediction at least in the Reynolds number range 0.01 G R G 100: a lon~tudinal roll appears as the most unstable disturbance in unstably stratified, Fully developed shear flows. We have discussed the critical condition for the transverse roll. Careful reading of Table 1 lets us find that the critical wavenumber and the critical phase veiocity for k = 0.4 and 0.6 do not change monotonously for R z 20. This non-monotonous change of EY,and c, is due to the intrinsic nature of a stability problem in which the velocity field couples with the temperature field through disturbance equations. Let us further consider in detail the linear stability with respect to transverse roll disturbances for various values of k, R and P in the remainder of this paper. The neutral stability curves for k = OS and P = 0.5 are depicted in Fig. 1 with various values of R. The curve for R = 50 posesses a single extremal point, and this point corresponds to the critical point. The point at which relations dRa -“=o dcu
and
d”Ra -----““>o daz
are satisfied is called the extremal point hereafter. The extremal point giving the minimum value of Ra, corresponds to the critical point if there are other extremal points on the neutral curve, A second extremal point appears in the higher wavenumber range on the curve in Fig. 1 for R = 70. The new extremal point becomes the critical point because the value of Ra, for the new point is smaller than that for the old one. The new extremal point thus replaces the old critical point. Hereafter we refer to this phenomenon as the replacement of the critical point. The neutral curves for k = 0.5 and R = 100 are depicted in Fig. 2 with various values of the Prandtl number. The replacement of the critical point again occurs as P changes. Change of Ra, and LY,for the extremal point is presented in Figs, 3 and 4, respectively, for k = 0.5 and for P = 0.2, 0.5 and 1. Solid lines in these figures denott: the critical points, and broken Iines indicate the other extremal points. There is only one extremal point corresponding
288
ofunstablystratified
K. Fujimura, R.E. Kelly / Stability
P:O.l
I
I
I
I
0.2 0.5
1.
I
I
2000
0
shear flow
2.
l-l
I
4000
Ra
Fig. 2. Dependence of the second extremal point on the Prandtl number. k = 0.5 and R = 100.
I I
Pzl
Rat
' I
3rr2
0
IR
100
Fig. 3. Replacement of the critical point in Ra-R plane. k = 0.5. Solid line denotes the critical point and broken line denotes the second extremal point.
0
R
100
Fig. 4. Replacement of the critical point on aR plane. k = 0.5. Solid line denotes the critical point and broken line denotes the second extremal point. Dotted line is the smoothly connected curve between solid line in the range 0 < R i 30 and broken line in the range 73 < R < 120 for P = 1.
K. Fujimwa,
shear j&w
R.E. ICeI& / Stability of zmstab& stmtifed
289
2.5
2.
cx.
1.5
I
330
350
400
Ra
430
Fig. 5. Shift of the critical point (closed circle) to higher wavenumber range for k = 0.5 and P = 1.
to the critical point in low Reynolds number range for each value of P. The value of Ra, increases with an increase of R. The second extremal point appears at R = 95 for P = 0.2, at R = 61 for P = 0.5 and at R = 73 for P = 1. The replacement of the critical point occnrs at the crossing of the solid line with the broken line: at R = 111 for P = 0.2 and at R = 65 for P = 0.5. The critical Rayleigh number changes continuously at the replacement point while its derivative changes discontinuously. The solid line never crosses the broken line for P = 1, but they separate as R increases.
I
3I -
I
kcO.5 0.4 0.6
I
I
I o-2
1
1
0.0
a
500
moo
I
I 0.6
i
O-
I
Ra
Fig. 6. Nosed neutral curve for various basic flows. R = 100 and P = 0.5.
290
K. Fujimura, R.E. Kelly / Stability of unstably stratified shear
flow
The critical wavenumber, on the other hand, leaps discontinuously when the replacement of the critical point occurs as is shown in Fig. 4. The relationship between the critical point and the other extremal point is qu~itatively equivalent for P = 0.2 and 0.5. 3ut the relation is different for P = 1. The critical wavenumber of P = 1 changes suddenly but continuously for 40 G R G 45. The shape of the curve for P = 1 is like a smoothed shape of the curve of CY,for P = 0.2 or 0.5. The sudden change of the critical wavenumber comes from the shift of the
PzO.1
4-
‘, .
:
r’ ‘.
,-:* . .’
/=
/’
i
0.05 Ra
Fig. 7. Neutral stability curve for the plane Couette flow (k = 1). Solid tine denotes stationary wave with c, = 0.5 and broken line denotes traveling wave with cr+-0.5.
I
I
I
PzO.1 0.2 0.5
I
I
1
2
I aCi
5
0:
Fig. 8. Transition from singly peaked neutral curve to tripiy peaked neutral curve for the plane Poiseuille flow (k = 0). R = 100. Dotted broken line denotes the calculation range in which the linear growth rate was evaluated for Fig. 9.
Fig. 9, Replacement of different eigenmodes on the Linear growth rate oci for plane Poiseuille flow. R = 100, P = 2 and Ra = 53’75. The q’s were evaluated along the dotted broken line in Fig. 8.
K. Fujimura9 RX. Kelly / Stability of unstably stratified shear flow
291
critical point from the low wavenumber range to the high wavenumber range as is shown in Fig, 5, in which the neutral curves for P = 1 and 40 < R < 45 are enlarged. The dotted line in Fig. 4 shows the smoothly connected line between the sohd line in the range 0 -C R < 30 and the broken line in the range 73 c R < 120 for P = 1. The dotted line passes through a point (CX, R) z (1.2,45) at which the neutral curve for R = 45 has a large value of d2Ra,,fdfuL in Fig. 5, It is concluded that there is no big qualitative difference in the change of the critical point between these three different Prandtl number cases although there is difference in the replacement of the critical point. The neutral curves for P = 0.5 and R = 100 are presented in Fig. 6 with various values of k. Each curve is found to be composed of two qualitatively different parts. A neutral curve having such a “nose” is known to occur also in the stability problem for the boundary layer along the heated vertical flat plate (Gebhart, 1973). The coupling between velocity field and temperature field is considered to be the cause of the shape. Let us show the neutral curve for k = 1 (plane Couette flow] and R = 100 in Fig. 7. The solid lines denote c, = 0.5 and the broken lines indicate c, f 0.5. A disturbance with c, = OS corresponds to the stationary wave with respect to moving frame with velocity 0.5 (the center velocity of the basic flow}, while a disturbance with t, # 0.5 cormsponds to the pair of traveling waves passing through in opposite direction ( f x) with phase veIacity AC,, say. Consider the case P = 1 and Ra = 3250, for example. The phase velocity of the stationary disturbance, c, = OS changes suddenly to c, = OS it AC, (AC, # 0) at a: = 1.36819, as CYincreases. This sudden change from stationary mode to traveling mode has been observed by previous authors (Deardorff, Gallagher and Mercer, 1965; Ingersoll, 1966; and Asai, 1970). It is helpful to consider the linear growth rate aCi in order to understand this qualitative change. The linear growth rate aicti corresponding to the first eigenvalue approaches the linear growth rate aczi corresponding to the second eigenvalue (ocrI > CXC~~) as tyf 1.36819, and two (Y&~‘S coincide with each other for cy>, 1.36819. The coincidence of two LYC;%corresponds to the sudden appearance of the traveling wave. Such merging of two modes on the linear growth rate is usually observed in isothermal plane Couette flow (Gailagher and Mercer, 1962). The change from the stationary wave to the traveling wave is therefore not the replacement of one mode by another but should be interpreted as the bifurcation into qualitatively different modes. The neutral curve for k = 0 (plane Poiseuille flow) and for R = 100 is depicted in Fig. 8. Most interesting is the transition from a singly peaked neutral curve to a triply peaked one. The three linear growth rates corresponding to the first three eigenvalues for P = 2 are plotted in Fig. 9 along the dotted broken line (Ra = 5375) in Fig. 8. Figure 9 shows that two unstable wavenumber bands in the lower wavenumber range are composed of the same eigenmode while the unstable wavenumber band centered round the critical wavenumber in higher waven~ber range is composed of a different eigenmode, which is the third eigenmode in the lower wavenumber range. The unstable wavenumber bands for k = 0, R = 100 and P = I, on the other hand, are composed of the same first eigenmode. Higher eigenmodes never replace the first eigenmode for all cases in Fig. 6. 5. Conclusion The linear stability of unstably stratified plane Poiseuille-Couette flow was investigated. It was found that the critical Rayleigh number for the transverse roll is an increasing function of the Reynolds number in the range 0.01 G R G 100 at least, and the effective critical condition is given by the stability characteristic for the lon~tudinal roll. The neutral stability curve for relatively high Prandtl number and high Reynolds number is found to possess more than one extremal point. The critical condition can change place among
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K. Fujimura, R. E. Kelly / Stability of unstably stratified shear flow
these extremal points. The critical Rayleigh number changes continuously with an increase of the Reynolds number, while the critical wavenumber leaps discontinuously during the replacement. It is also found that more than two extremal points may be composed of different eigenmodes.
References Asai, T. (1970) Three-dimensional features of thermal convection in a plane Couette flow, J. Meteor. Sot. Jpn. 48, 18-29. Deardorff, J.W. (1965) Gravitational instability between horizontal plates with shear, Phys. Fluids 8, 1027-1030. Gage, K.S. and W.H. Reid (1968) The stability of thermally stratified plane Poiseuille flow, J. Fluid Mech. 33, 21-32. Gallagher, A.P. and A.McD. Mercer (1962) On the behaviour of small disturbances in plane Couette flow, J. Fluid Mech. 13, 91-100. Gallagher, A.P. and A.McD. Mercer (1965) On the behaviour of small disturbances in plane Couette flow with a temperature gradient, Proc. Roy. Sot. London A286, 117-128. Gebhart, B. (1973) Instability, transition and turbulence in buoyancy-induced flows, Ann. Rev. Fluid. Mech. 5, 213-246. Ingersoll, A.P. (1966) Convective instabilities in plane Couette flow, Phys. Fluid 9, 682-689. Makino, M. and Y. Ishikawa (1985) The stability of horizontal two-dimensional shear flow with vertical temperature gradient, J. Jpn. Sot. Fluid. Mech. 4, 148-158 (in Japanese). Platten, J.K. and J.C. Legros (1984) Convection in Liquids (Springer, New York) 529-556. Tveitereid, M. (1974) On the stability of thermally stratified plane Poiseuille flow, ZAMM 54, 533-540.