- Email: [email protected]
Observations on a thermally induced instability between rotating cylinders
ANNALS
OF
PHYSICS:
31, 314-324
Observations
(1965)
on a Thermally between Rotating
Induced Cylinders
Instability
S. K. F. KARLSSON Division
of Engineering,
Brown
University,
Providence,
Rhode
Island
AND H. A. SNYDER Department
of Physics,
Blown
University,
Providence,
Rhode
Island
The wavy disturbance which is induced by a radial thermal gradient in a Taylor stability apparatus has been studied experimentally. This type of secondary flow occurs at a lower Taylor number than the toroidal vortex motion. It appears that there is no sharp transition associated with the thermally induced disturbance when only the inner cylinder rotates. When the outer cylinder also rotates a necessary condition for the appearance of the wavy motion is that !&/Q1 < 1. The wave form of the wavy motion is similar to that of Tollmien-Schlichting waves observed in a frame moving with the velocity of propagation. The disturbance does not move axially but drifts azimuthally at approximately ~1/2. The wave form is spiral for gradients exceeding about 5”C/cm. I. INTRODUCTION The experiments reported here deal with the stability of Couette motion between concentric rotating cylinders when a thermal gradient is maintained between the cylinders. If the axis of rotation is vertical, the convective force generated by the density gradient sets up an axial flow. In this case a new facet is added to the Taylor problem. The first mode of secondary flow no longer consists of pairs of Taylor vorticies but of a wavy disturbance on the critical layer. However, the toroidal cells of Taylor’s experiment occur as the second mode of instability. An account of the transition to Taylor cells has been published; it will be referred to as (I) (1) . Additional details on the Taylor type transition may be found in the preceding article by Snyder which will be referred to as (II) (a). The purpose of this paper is to report some observations on the wavy disturbance. In these experiments the gap between the cylinders is sufficiently small and the length of the cylinders is sufficiently long so that the narrow gap approximation is applicable and the cylinders may be considered infinitely long. The fluid in all 314
THERMALLY
IKDUCED
INWABILITY
3 1,-I
cases is water or nlixtures of glycerine and water. The u~aximm gradient that has been investigated is f2S°C/cnl. Both cylinders can be rotated and t,hr rrlative angular velocity L$,/B, ranges from - 2..5 to 0.92 for the data presented here. In Se&ion II the theoretical background of the problem is presented. This is followed by a description of the apparatus and the procedure for taking dat’a ill Section III. The results of the measurenlents arc set dowu in Section IV and the) are discussed in Section 1’. II. THEOKETICAL
BACKGROliNI)
The stability problem for iufinite concentric cylinders with a narrow gap and arbitrary rotation has been investigated rather thoroughly both theoretically and experimentally (3) (see (II) for references to work described in this settion). When an axial pressure gradient is applied and a parabolic axial profile results, the problem has also been treated successfully. Another variation COW sists of adding a radial thermal gradient to the Taylor problem. Again t,he prob10111 has been solved and verified for the case whcu convection can be neglectrd (the cylinders horizontal). In the calculation relatiug to all these problems t,hcl resulting disturbance has the form of toroids. The observed cell t,ype has also beeu toroidal except for the case of axial flows of sufficient strength whcll thcb toroids become spirals. There appear to be 110ealculatious for the problem cot\sidercd here when convection currents are important. The only circunlstanc(~:: under which cells of the type found in these experiments have beet1predictt&d 01 observed are for plane Pouiseuille and boundary layer flom~. For the experiment in baud the axis of the cylinder is v&ical alld fret conv(‘c tion geueratex an unperturbed flow in the axial dir&oil :
where g is the acceleration of gravity, (Y the tJhermal expansion coefficient, v,, the Kinematic viscosity, d the gap width between the cylinders, and A7’ is thck total temperature difference between the cylinders. The variable is defined as E = (1. - Z&)/d where Ro is the radial distance to the midpoint of the gap. 111 addition to the axial flow the thermal gradient also produces a density gradient and a viscosity gradient. For a narrow gap these gradient,s are linear: T = T,, + [A7’
t L’a J
p = po+ &A2
f L’t, i
v = vu + s$h,‘A~
loA2’
I “c 1
The equations of small disturbances which apply to the problem considered in this paper are Eqs. ( 1)) (2 ) , and (3) of (II). These equations apply to axisynm
316
KARLSSON
AND
SNYDER
metric disturbances when m = 0 and this is the observed wave form up to about AT/d = &Y/cm. For larger gradients the wave forms are spiral; the same equations still apply but now m # 0. There is no reason to doubt that these equations have a solution for P < 1 with a Taylor number either equal to or near to zero and with a wave form of the wavy type. In analyzing the data it is useful to bear in mind the following facts: (i) If p = 1 (solid body rotation) and if there were no convective flow, instability would occur when the Rayleigh number based on the angular acceleration RzQz became equal to 1708. Here Rz is the radius of the outer cylinder and Dz is the angular velocity of the outer cylinder. For the maximum 02 considered below and the maximum adverse thermal gradient, (R,, = 1300. (ii) By theorem (3) of (II) the flow may be completely unstable due to the point of inflection in the velocity profile. Note, however, that the velocity of the profile at the point of inflection is zero, so that this theorem holds only in a limiting sense.We may then expect the location of the point of inflection and the velocity of the profile in this vicinity to affect the result strongly. The motivation for adding a net axial flow to the convective velocity field arises from the inflection point theorem (3) of (II). (iii) When viscous effects are neglected the flow is always unstable according to theorem (5) of (II) since the Richardson number never exceeds 44 throughout the gap. The Richardson number which is applicable is R&?[l (1 - p) (?,d + .$)I” daAT/(tlW/dE)‘. (iv) The Rayleigh criterion for Couette motion is exceeded whenever p < R12/Rz2= 7’ = 0.92. Thus in the absence of density gradients and viscosity we may expect complete stability for p > 0.92. It is known that viscosity always stabilizes this type of flow. If secondary flow is observed for P > 0.92 we may relate the instability to the density gradient effect of (i) or to some other cause. III.
APPARATUS
AND
PROCEDURE
The apparatus has been described in detail in (I). The outer cylinder has been changed since the data of the previous paper was taken and now d = 0.258 cm. Special consideration was given to the selection and preparation of the ink used to mark the streamlines. Three different types were used in various parts of this experiment: methylene blue, analine blue, and negrosine. There does not seem to be any effect on the results which are dependent upon the ink. The ink may be balanced against water for equal density or equal viscosity-not both. Experimental results with slightly lighter and slightly heavier ink do not seem to differ. The ink used for all the data presented here is balanced for density equal to water. The viscosity of methylene blue at the concentration injected into the stream was measured to be 20% greater than water. The ink becomes greatly diluted by the time it is spread along the cylinder and the effect of the larger viscosity probably becomes negligible. The procedure of (I) and (II) was followed throughout.
THERMALLY
INDUCED
lNSTr\RILIT~
:',1i
The wave form of the disturbance is shown in Fig. 1. It is the wavy disturbarms conl~nonlg used in the analysis of plaue parallel flows aud viewed in a franlc nioving with the speed of propagation (‘, 5). 111 these experinlents t,he disturbauce does riot propagate in the axial direction-it is in fact viewed in the fralllr of propagation. For mall thermal gradient’s the boundaries of the cells arc horizontal but when AT/d exceeds about, f:i”C’, ~111the wave form becon~s spiral. AH t,he gradieut is iucreased further, the number of starts of the spiral increases. I’or the largest gradient cousidered hrare ( f2;5Y~,/cm) the wave forrrl has 6 start,s for p = 0.
I:sing the lncthods of Section III it has proved impossible to fiud a curv(l of the crit,ical velocity for transition to the wavy disturbance similar to the curves for the Taylor transition. The difficulty is due to an apparent absence of a sharp tratmitiou; perhaps this can be clarified by referring to Fig. 2. The solid line cross hatching is the region where cells are always definitely visible. There is a sharp transition to the themally induced iustability along the line p = 1. The wavs disturbance has never been observed for P > 1. However, irregular and no11 reproducible ink patterns have been observed in this region with a nAxturc> of l,:i glyceritle and ?’3 water. These patterns do uot have the regular structure of t,he wa~vy disturbance. At sufficiently high values of Ii1 t#he disturbance undcl cliscussionundergoes a transition to Taylor cells. The latter transition is sharp attd forms auothcr boundary of the region of wavy cells. Data on t,he Taylor trarlsit,ion Itlay be found in (I) and (II). E’or p < 1 aud below the curve fog transitioii to Taylor cells we have becu unable t)o tied a dcfitlite value of 11,for tllc onset of the wavy disturbance. For a given Q: t)hc sharpnessof the cell pattcrtl dmreases as 6j1decreases.It also takes longer for the cells to appear (for the ink t’o redistribute), as Q, decreasesfor fixed 6ts. This seemsto indicate that that circulation in t,he cells decreasesas fil decreases.I’ventually the time recluired for cells to appear approaches the diffusion rate of the iuk and it is difficult to dctcrllliile if cells are present. The region whcrc dificult,y is cn~ountc~retl in
318
KARLSSON
-60
FIG.
the
area
2.
-40
-20
The stability diagram for of solid line cross hatching.
AND
0
the wavy
SNYDER
20
disturbance.
40
The
60
cells
are always
visible
in
producing ink patterns is indicated by a cross hatching of dashed lines. The lack of definite cells along the line p = - 1 is due to another mechanism-it is known that Taylor cells do not persist in this area of the diagram. It is possible that there is a curve for the onset of the thermally induced instability below the cross hatched area of Fig. 2. The wavy disturbance does not occur when Qr = 0. We wish to emphasize that the stability criterion M = 1 does not appear to be an asymptotic limit in the same sense as Rayleigh’s criterion p = 7’. Here, the wavy disturbance appears whenever p falls below one and the transition falls
THERMALLY
INDUCED
INSTABILITY
Xl!)
accurately on this line even for values of R:‘“O1 d3’2/~ as low as 20. Along t,he line p = 1 the time for cells to appear increases the smaller the deviation of ,.L from one. The criterion used in determining if cells exist are that (i) the pattern bc regular over a large portion of the cylinder, (ii) that the pattern persist, and (iii) that it be reproducible from run to run. ('.
THE
WAVE
KUMBER
OF
THE
DISTURBANCE
The wave number of the disturbance is usually reported in dimensionlessform : a = B?rd/X where X is the wavelength in the axial direction and d is the gap width. The value of a is a function of the position OII bhe stability diagram. Along the curve for transition to Taylor cells the value of a just before transition agrees with the value after transition-one cell of the wavy disturbance becomesa pail of Taylor cells without change of wavelength. For fixed Qta, as !& is decreased from the value for the transition to Taylor cells (designated Qtl,,), the wavelength increases and therefore a decreases.It has not been possible to tell whether t.hc change is continuous or not. The error in determining a is at least 5 % and is due primarily to actual changes in a from one run to another and to variations along t’he cylinder in each run. To measure the dependence of a photographs were taken for fixed values of L$ and with & set at, 5 %, 10 %, and 25 % below !ll,,. . In all the data, which ranged from -0.5 < P < +0.6 and AT/d = fij”(‘lj,cn~, t.he difference between a for Q1,,, 0.95 01,, , and 0.90 01,, are less than 5Y and cannot be measured. For all data a is about 10 % lower than its value at btl,~for 6& = 0.75 Q1,,.. It is not possible to report accurate values of a because of the large scatter from run to run. Apparently these variations are inherent in the mechanism of instability. See (I) and (II) for values of a along the curve of ItI,, vs. /.l. 1).
IN(!LINA4TION
OF
THE
CELL
WALLS
TO
THE
HORIZONT~~L
The cells walls are horizontal for small temperature differences. When the gradient becomes sufficiently large (the amount depends upon the position in the stability diagram) the wave form becomes spiral and the number of starts of the spiral increases with increasing gradient. The senseof inclination of the wavy cells is always the same as the Taylor cells which develop from them. Src~ Fig. 9 of (I) where the senseof inclination is catalogued for all possible cases. From the photographs mentioned in the last subsection taken at 0.95 Q1,,, 0.90 %c , and 0.75 G1,, it appears that the number of starts of the spiral increases as btl decreasesfor fixed L!, . Also the number of starts increaseswith P for 6&equal to a fixed fraction of fil,, . When the sharp transition occurs at K = 1 the cell walls first appear nearly vertical. As L$ is increased to 111,,the angle decreasesto the appropriate value for Taylor cells. For large values of II? , Qtl,, lies along the
320
KARLSSON
AND
SNYDER
line p = 0.92. Consequently in this region the range of 01 between the line p = 1 and Q;tl,,is small and the angle of inclination changes rapidly with a1 . All these results which were obtained with water as the working fluid have been confirmed with a solution of >$ glycerine and N water. In the latter case the axial flow is strongly inhibited by the large viscosity and the Prandtl number is increased considerably. Special care was taken for the data showing the transition at p = 1 to assure that the effect is not due to transients. The wave form of the disturbance is not very reproducible even though it is possible to regulate the conditions of flow and the thermal gradient to a high degree (see (I), section II). When the number of starts of the spiral m is small it is reproducible to f 1. However when m becomes larger it may vary by 20 % from one determination to the next. It is also possible to have two or more fields of cells with different inclinations along the cylinder at the same time. This behavior is not entirely unexpected in experiments of this type. For flows in a laminar boundary layer Schubauer and Skramstad (6) showed that the disturbance which first sets in can be imposed externally by an extremely small driving force. The first mode to set in will grow although it can be shown that other modes are less stable and have a larger growth rate. Recent experiments
0 0
0 75Q,,c
0
0
0 0.9oi-L,,c
0 0 0
0 0.95n,,c 0 0
I,
,
-6
1
I
-4
,
,
0
,
,
,
,
,
,
-2 AT
FIG.
various
3. The number of starts fractions of the angular
,
0
,
,
,
2
I
I
4
I
6
"C
of spirals vs. the temperature difference AT for velocity for the transition to Taylor cell QI., .
p = 0 at
THERK4LLY
INDUCED
INSTABILITY
+
m +
IO
AT = t5”C
x 8
0
0 x
4
2 $
0
0
0
0
0 0
x
+
t t FIG. 4. The number x, 0.90 fi1.c ; +, Q1.c.
of starts
of spirals
x
+ vs. fi at various
fractions
of Sll.,
: 0,
0.95 611 ,. :
by Donnelly (7) on the stability of jets has led to the same conclusion. It appears that there are various possiblewave functions lying close together in the vicinity of minimum stability. Small external sources of disturbance may cause any one of these to set in first and become the mode of secondary flow. Data on the number of starts of the spirals as a function of the thermal gradient at p = 0 are shown in Fig. 3 and for various values of p in Fig. 4. These data should be interpreted as statistical average values in light of the discussion of the previous paragraph.
322
KARLSSON
E. DRIFT
AND
SNYDER
VELOCITY
The thermally induced instability does not drift in the axial direction. Perhaps this is reasonable since there is no net axial flow. When the disturbance has a spiral structure the cells pattern drifts azimuthally with a velocity close to the average velocity of the unperturbed flow. Quantitative measurements were made for the case CL= 0 with the photocell equipment described in (I). In all casesfor CL= 0 the azimuthal velocity is found to equal fi1/2 to within 2 %. When
(a)
(b)
LI B
m
n
(e)
(d) FIG.
5. The
sequence
Taylor vorticies.
of cell
a !4
patterns during the transition from the
(f)
wavy
disturbanceto
THERMALLY
IKDUCED
323
INSTABILITY
p # 0 a few measurements were made by eye and stopwatch with the result t,hat the drift is about (C& + &J/2. The latter data is only approximate. E'.
TRANSITION
OF
THE
WAVT
DISTURBANCE
TO TAYLOR
CELLS
The sequenceof forms which the wave form takes on as it changes front t,he wavy disturbance to Taylor cells can be studied readily with this apparatus. By sweeping through the critical value of L+ very slowly it is possible to slow the progress of the transition so that a complete change over takes Ti minutes 01 nlore. The various shapes which occur are illustrated in Fig. 5. At about 0.98 Q1,, the cusps of 5(b) appear. Recall that the Taylor vorticies which occur in these experiments are the uneven type shown in :i( f) and in Fig. 5 of (I ). Thr wavy disturbance rolls up into the larger cell of the Taylor pair as shown in Fig. 5( b)-5( e). After the larger member of the pair is well established, t#he srnallcr cell appears with opposite circulation. There is a great similarity hetwec>n the results of this subsection and the computed streamlines of a similar problcnl treated by Amsden and Harlow (8). In the latter work the slip instability is considered from the standpoint of an initial value problem for a simple sheer layer separating oppositely directed parallel streams. IGgurc 4 of ref. 8 mar bc> compared with Fig. 5 presented here. (+. EFFECT
OF
AN AXIAL
lkow
0N THE WAVY
I~ISTURBA~XE
The inflection point theorem suggeststhat the addition of a net axial flow t’o the convective flow might produce interesting results. In (II j the effect on Taylor vorticies of a net axial flow is discussed.See Figs. 10 and 12 of (II). The same procedure was carried out for the wavy disturbance. It is found that when the inflection point is displaced nearly out of the region of the gap the wavy dist’urbance no longer appears. As B/A increases (see ( II ) for an explanation of these symbols), the cells become less distinct and for B/.4 > 1 definite cells are not observed. ‘\-. 1~ISCUSSION
The best clue about the mechanism causing the thermally induced instability is probably the sharp transition along the line p = 1. Observe that at large values of DzTaylor cells occur at p = 7’ = 0.92. This instability is associated with the centrifugal potential. The wavy disturbance occurs at larger values of P and thus viola&Rayleigh’s criterion; it is not due primarily to the cetrifugal term. The t,hc~rmally induced instability sets in precisely at p = 1. It occurs for both negative and positive gradients and the only effect of decreasing the magnitude of t.hc gradient is: (i ) to lengthen the time required for the cell patterns to form and also (ii 1 to change the number of starts of the spiral. In fact all the data rcportcd in this article appears to be nearly symmetrical in negative and positJivegradient’s,
324
KARLSSON
AND
SNYDER
Thus, the effect cannot be explained in terms of instabilities measured by the Richardson number or density gradients. The only obvious change in passing through p = 1 is the appearance of a sheer; however, there is also a sheer for p > 1. Apparently the mechanism is also associated with the centrifugal potential; the wavy disturbance does not occur for Q, = 0. Another clue is the fact that net axial flows (with a parabolic profile) with no point of inflection do not produce instability unless p < q2 even when the thermal gradient is present. This shows that the shape of the axial profile is significant and density gradients are unimportant. At the low axial flows set up by free convection the axial Reynolds number is very small and the motion is controlled by viscosity. It is difficult to see how the inviscid inflection point theorem for fluids would apply. It appears that a cubic profile is unstable even with viscosity present. This is the conclusion reached by Amsden and Harlow (8) on the basis of their computations. In conclusion we may summarize what the data implies about the mechanism of instability: (i) the shape of the axial profile is important and the cubic profile is unstable; (ii) the centrifugal potential must be present; and (iii) there must be sheer in the azimuthal direction in the presence of an increasing centrifugal field. ACKNOWLEDGMENTS The original arrangements to Reid. This research was supported tion, NSF GP333. RECEIVED:
carry out primarily
these experiments were made by Professor W. H. by a grant from the National Science Founda-
June 22, 1964 REFERENCES
I. H. A. SNYDER AND S. K. F. KARLSSON, Phys. Fluids 7, 1696 (19G4) 2. H. A. SNYDER, Ann. Phys. (N.Y.), 31, 292-313 (1965). 3. S. CHANDRASEKHAR, “Hydrodynamic and Hydromagnetic Stability.” The Clarendon Press, Oxford, 1961. 4. C. C. LIN, Quart. Appl. Math. 3, 117, 218, 277 (1945). 5. C. C. LIN, “The Theory of Hydrodynamic Stability.” Cambridge Univ. Press, London, 1955. 6. G. B. SCHAUB.IUER AND H. K. SKRAMSTAD, J. Aero. Sci. 14,69 (1947); J. Res. N&Z. Bur. Stand. 38, 251 (1947). 7. R. T. DONNELLY, private communication. 8. A. A. AMSDEN AND F. H. HARLOW, Phys. Fluids 7, 327 (1964).
OF
PHYSICS:
31, 314-324
Observations
(1965)
on a Thermally between Rotating
Induced Cylinders
Instability
S. K. F. KARLSSON Division
of Engineering,
Brown
University,
Providence,
Rhode
Island
AND H. A. SNYDER Department
of Physics,
Blown
University,
Providence,
Rhode
Island
The wavy disturbance which is induced by a radial thermal gradient in a Taylor stability apparatus has been studied experimentally. This type of secondary flow occurs at a lower Taylor number than the toroidal vortex motion. It appears that there is no sharp transition associated with the thermally induced disturbance when only the inner cylinder rotates. When the outer cylinder also rotates a necessary condition for the appearance of the wavy motion is that !&/Q1 < 1. The wave form of the wavy motion is similar to that of Tollmien-Schlichting waves observed in a frame moving with the velocity of propagation. The disturbance does not move axially but drifts azimuthally at approximately ~1/2. The wave form is spiral for gradients exceeding about 5”C/cm. I. INTRODUCTION The experiments reported here deal with the stability of Couette motion between concentric rotating cylinders when a thermal gradient is maintained between the cylinders. If the axis of rotation is vertical, the convective force generated by the density gradient sets up an axial flow. In this case a new facet is added to the Taylor problem. The first mode of secondary flow no longer consists of pairs of Taylor vorticies but of a wavy disturbance on the critical layer. However, the toroidal cells of Taylor’s experiment occur as the second mode of instability. An account of the transition to Taylor cells has been published; it will be referred to as (I) (1) . Additional details on the Taylor type transition may be found in the preceding article by Snyder which will be referred to as (II) (a). The purpose of this paper is to report some observations on the wavy disturbance. In these experiments the gap between the cylinders is sufficiently small and the length of the cylinders is sufficiently long so that the narrow gap approximation is applicable and the cylinders may be considered infinitely long. The fluid in all 314
THERMALLY
IKDUCED
INWABILITY
3 1,-I
cases is water or nlixtures of glycerine and water. The u~aximm gradient that has been investigated is f2S°C/cnl. Both cylinders can be rotated and t,hr rrlative angular velocity L$,/B, ranges from - 2..5 to 0.92 for the data presented here. In Se&ion II the theoretical background of the problem is presented. This is followed by a description of the apparatus and the procedure for taking dat’a ill Section III. The results of the measurenlents arc set dowu in Section IV and the) are discussed in Section 1’. II. THEOKETICAL
BACKGROliNI)
The stability problem for iufinite concentric cylinders with a narrow gap and arbitrary rotation has been investigated rather thoroughly both theoretically and experimentally (3) (see (II) for references to work described in this settion). When an axial pressure gradient is applied and a parabolic axial profile results, the problem has also been treated successfully. Another variation COW sists of adding a radial thermal gradient to the Taylor problem. Again t,he prob10111 has been solved and verified for the case whcu convection can be neglectrd (the cylinders horizontal). In the calculation relatiug to all these problems t,hcl resulting disturbance has the form of toroids. The observed cell t,ype has also beeu toroidal except for the case of axial flows of sufficient strength whcll thcb toroids become spirals. There appear to be 110ealculatious for the problem cot\sidercd here when convection currents are important. The only circunlstanc(~:: under which cells of the type found in these experiments have beet1predictt&d 01 observed are for plane Pouiseuille and boundary layer flom~. For the experiment in baud the axis of the cylinder is v&ical alld fret conv(‘c tion geueratex an unperturbed flow in the axial dir&oil :
where g is the acceleration of gravity, (Y the tJhermal expansion coefficient, v,, the Kinematic viscosity, d the gap width between the cylinders, and A7’ is thck total temperature difference between the cylinders. The variable is defined as E = (1. - Z&)/d where Ro is the radial distance to the midpoint of the gap. 111 addition to the axial flow the thermal gradient also produces a density gradient and a viscosity gradient. For a narrow gap these gradient,s are linear: T = T,, + [A7’
t L’a J
p = po+ &A2
f L’t, i
v = vu + s$h,‘A~
loA2’
I “c 1
The equations of small disturbances which apply to the problem considered in this paper are Eqs. ( 1)) (2 ) , and (3) of (II). These equations apply to axisynm
316
KARLSSON
AND
SNYDER
metric disturbances when m = 0 and this is the observed wave form up to about AT/d = &Y/cm. For larger gradients the wave forms are spiral; the same equations still apply but now m # 0. There is no reason to doubt that these equations have a solution for P < 1 with a Taylor number either equal to or near to zero and with a wave form of the wavy type. In analyzing the data it is useful to bear in mind the following facts: (i) If p = 1 (solid body rotation) and if there were no convective flow, instability would occur when the Rayleigh number based on the angular acceleration RzQz became equal to 1708. Here Rz is the radius of the outer cylinder and Dz is the angular velocity of the outer cylinder. For the maximum 02 considered below and the maximum adverse thermal gradient, (R,, = 1300. (ii) By theorem (3) of (II) the flow may be completely unstable due to the point of inflection in the velocity profile. Note, however, that the velocity of the profile at the point of inflection is zero, so that this theorem holds only in a limiting sense.We may then expect the location of the point of inflection and the velocity of the profile in this vicinity to affect the result strongly. The motivation for adding a net axial flow to the convective velocity field arises from the inflection point theorem (3) of (II). (iii) When viscous effects are neglected the flow is always unstable according to theorem (5) of (II) since the Richardson number never exceeds 44 throughout the gap. The Richardson number which is applicable is R&?[l (1 - p) (?,d + .$)I” daAT/(tlW/dE)‘. (iv) The Rayleigh criterion for Couette motion is exceeded whenever p < R12/Rz2= 7’ = 0.92. Thus in the absence of density gradients and viscosity we may expect complete stability for p > 0.92. It is known that viscosity always stabilizes this type of flow. If secondary flow is observed for P > 0.92 we may relate the instability to the density gradient effect of (i) or to some other cause. III.
APPARATUS
AND
PROCEDURE
The apparatus has been described in detail in (I). The outer cylinder has been changed since the data of the previous paper was taken and now d = 0.258 cm. Special consideration was given to the selection and preparation of the ink used to mark the streamlines. Three different types were used in various parts of this experiment: methylene blue, analine blue, and negrosine. There does not seem to be any effect on the results which are dependent upon the ink. The ink may be balanced against water for equal density or equal viscosity-not both. Experimental results with slightly lighter and slightly heavier ink do not seem to differ. The ink used for all the data presented here is balanced for density equal to water. The viscosity of methylene blue at the concentration injected into the stream was measured to be 20% greater than water. The ink becomes greatly diluted by the time it is spread along the cylinder and the effect of the larger viscosity probably becomes negligible. The procedure of (I) and (II) was followed throughout.
THERMALLY
INDUCED
lNSTr\RILIT~
:',1i
The wave form of the disturbance is shown in Fig. 1. It is the wavy disturbarms conl~nonlg used in the analysis of plaue parallel flows aud viewed in a franlc nioving with the speed of propagation (‘, 5). 111 these experinlents t,he disturbauce does riot propagate in the axial direction-it is in fact viewed in the fralllr of propagation. For mall thermal gradient’s the boundaries of the cells arc horizontal but when AT/d exceeds about, f:i”C’, ~111the wave form becon~s spiral. AH t,he gradieut is iucreased further, the number of starts of the spiral increases. I’or the largest gradient cousidered hrare ( f2;5Y~,/cm) the wave forrrl has 6 start,s for p = 0.
I:sing the lncthods of Section III it has proved impossible to fiud a curv(l of the crit,ical velocity for transition to the wavy disturbance similar to the curves for the Taylor transition. The difficulty is due to an apparent absence of a sharp tratmitiou; perhaps this can be clarified by referring to Fig. 2. The solid line cross hatching is the region where cells are always definitely visible. There is a sharp transition to the themally induced iustability along the line p = 1. The wavs disturbance has never been observed for P > 1. However, irregular and no11 reproducible ink patterns have been observed in this region with a nAxturc> of l,:i glyceritle and ?’3 water. These patterns do uot have the regular structure of t,he wa~vy disturbance. At sufficiently high values of Ii1 t#he disturbance undcl cliscussionundergoes a transition to Taylor cells. The latter transition is sharp attd forms auothcr boundary of the region of wavy cells. Data on t,he Taylor trarlsit,ion Itlay be found in (I) and (II). E’or p < 1 aud below the curve fog transitioii to Taylor cells we have becu unable t)o tied a dcfitlite value of 11,for tllc onset of the wavy disturbance. For a given Q: t)hc sharpnessof the cell pattcrtl dmreases as 6j1decreases.It also takes longer for the cells to appear (for the ink t’o redistribute), as Q, decreasesfor fixed 6ts. This seemsto indicate that that circulation in t,he cells decreasesas fil decreases.I’ventually the time recluired for cells to appear approaches the diffusion rate of the iuk and it is difficult to dctcrllliile if cells are present. The region whcrc dificult,y is cn~ountc~retl in
318
KARLSSON
-60
FIG.
the
area
2.
-40
-20
The stability diagram for of solid line cross hatching.
AND
0
the wavy
SNYDER
20
disturbance.
40
The
60
cells
are always
visible
in
producing ink patterns is indicated by a cross hatching of dashed lines. The lack of definite cells along the line p = - 1 is due to another mechanism-it is known that Taylor cells do not persist in this area of the diagram. It is possible that there is a curve for the onset of the thermally induced instability below the cross hatched area of Fig. 2. The wavy disturbance does not occur when Qr = 0. We wish to emphasize that the stability criterion M = 1 does not appear to be an asymptotic limit in the same sense as Rayleigh’s criterion p = 7’. Here, the wavy disturbance appears whenever p falls below one and the transition falls
THERMALLY
INDUCED
INSTABILITY
Xl!)
accurately on this line even for values of R:‘“O1 d3’2/~ as low as 20. Along t,he line p = 1 the time for cells to appear increases the smaller the deviation of ,.L from one. The criterion used in determining if cells exist are that (i) the pattern bc regular over a large portion of the cylinder, (ii) that the pattern persist, and (iii) that it be reproducible from run to run. ('.
THE
WAVE
KUMBER
OF
THE
DISTURBANCE
The wave number of the disturbance is usually reported in dimensionlessform : a = B?rd/X where X is the wavelength in the axial direction and d is the gap width. The value of a is a function of the position OII bhe stability diagram. Along the curve for transition to Taylor cells the value of a just before transition agrees with the value after transition-one cell of the wavy disturbance becomesa pail of Taylor cells without change of wavelength. For fixed Qta, as !& is decreased from the value for the transition to Taylor cells (designated Qtl,,), the wavelength increases and therefore a decreases.It has not been possible to tell whether t.hc change is continuous or not. The error in determining a is at least 5 % and is due primarily to actual changes in a from one run to another and to variations along t’he cylinder in each run. To measure the dependence of a photographs were taken for fixed values of L$ and with & set at, 5 %, 10 %, and 25 % below !ll,,. . In all the data, which ranged from -0.5 < P < +0.6 and AT/d = fij”(‘lj,cn~, t.he difference between a for Q1,,, 0.95 01,, , and 0.90 01,, are less than 5Y and cannot be measured. For all data a is about 10 % lower than its value at btl,~for 6& = 0.75 Q1,,.. It is not possible to report accurate values of a because of the large scatter from run to run. Apparently these variations are inherent in the mechanism of instability. See (I) and (II) for values of a along the curve of ItI,, vs. /.l. 1).
IN(!LINA4TION
OF
THE
CELL
WALLS
TO
THE
HORIZONT~~L
The cells walls are horizontal for small temperature differences. When the gradient becomes sufficiently large (the amount depends upon the position in the stability diagram) the wave form becomes spiral and the number of starts of the spiral increases with increasing gradient. The senseof inclination of the wavy cells is always the same as the Taylor cells which develop from them. Src~ Fig. 9 of (I) where the senseof inclination is catalogued for all possible cases. From the photographs mentioned in the last subsection taken at 0.95 Q1,,, 0.90 %c , and 0.75 G1,, it appears that the number of starts of the spiral increases as btl decreasesfor fixed L!, . Also the number of starts increaseswith P for 6&equal to a fixed fraction of fil,, . When the sharp transition occurs at K = 1 the cell walls first appear nearly vertical. As L$ is increased to 111,,the angle decreasesto the appropriate value for Taylor cells. For large values of II? , Qtl,, lies along the
320
KARLSSON
AND
SNYDER
line p = 0.92. Consequently in this region the range of 01 between the line p = 1 and Q;tl,,is small and the angle of inclination changes rapidly with a1 . All these results which were obtained with water as the working fluid have been confirmed with a solution of >$ glycerine and N water. In the latter case the axial flow is strongly inhibited by the large viscosity and the Prandtl number is increased considerably. Special care was taken for the data showing the transition at p = 1 to assure that the effect is not due to transients. The wave form of the disturbance is not very reproducible even though it is possible to regulate the conditions of flow and the thermal gradient to a high degree (see (I), section II). When the number of starts of the spiral m is small it is reproducible to f 1. However when m becomes larger it may vary by 20 % from one determination to the next. It is also possible to have two or more fields of cells with different inclinations along the cylinder at the same time. This behavior is not entirely unexpected in experiments of this type. For flows in a laminar boundary layer Schubauer and Skramstad (6) showed that the disturbance which first sets in can be imposed externally by an extremely small driving force. The first mode to set in will grow although it can be shown that other modes are less stable and have a larger growth rate. Recent experiments
0 0
0 75Q,,c
0
0
0 0.9oi-L,,c
0 0 0
0 0.95n,,c 0 0
I,
,
-6
1
I
-4
,
,
0
,
,
,
,
,
,
-2 AT
FIG.
various
3. The number of starts fractions of the angular
,
0
,
,
,
2
I
I
4
I
6
"C
of spirals vs. the temperature difference AT for velocity for the transition to Taylor cell QI., .
p = 0 at
THERK4LLY
INDUCED
INSTABILITY
+
m +
IO
AT = t5”C
x 8
0
0 x
4
2 $
0
0
0
0
0 0
x
+
t t FIG. 4. The number x, 0.90 fi1.c ; +, Q1.c.
of starts
of spirals
x
+ vs. fi at various
fractions
of Sll.,
: 0,
0.95 611 ,. :
by Donnelly (7) on the stability of jets has led to the same conclusion. It appears that there are various possiblewave functions lying close together in the vicinity of minimum stability. Small external sources of disturbance may cause any one of these to set in first and become the mode of secondary flow. Data on the number of starts of the spirals as a function of the thermal gradient at p = 0 are shown in Fig. 3 and for various values of p in Fig. 4. These data should be interpreted as statistical average values in light of the discussion of the previous paragraph.
322
KARLSSON
E. DRIFT
AND
SNYDER
VELOCITY
The thermally induced instability does not drift in the axial direction. Perhaps this is reasonable since there is no net axial flow. When the disturbance has a spiral structure the cells pattern drifts azimuthally with a velocity close to the average velocity of the unperturbed flow. Quantitative measurements were made for the case CL= 0 with the photocell equipment described in (I). In all casesfor CL= 0 the azimuthal velocity is found to equal fi1/2 to within 2 %. When
(a)
(b)
LI B
m
n
(e)
(d) FIG.
5. The
sequence
Taylor vorticies.
of cell
a !4
patterns during the transition from the
(f)
wavy
disturbanceto
THERMALLY
IKDUCED
323
INSTABILITY
p # 0 a few measurements were made by eye and stopwatch with the result t,hat the drift is about (C& + &J/2. The latter data is only approximate. E'.
TRANSITION
OF
THE
WAVT
DISTURBANCE
TO TAYLOR
CELLS
The sequenceof forms which the wave form takes on as it changes front t,he wavy disturbance to Taylor cells can be studied readily with this apparatus. By sweeping through the critical value of L+ very slowly it is possible to slow the progress of the transition so that a complete change over takes Ti minutes 01 nlore. The various shapes which occur are illustrated in Fig. 5. At about 0.98 Q1,, the cusps of 5(b) appear. Recall that the Taylor vorticies which occur in these experiments are the uneven type shown in :i( f) and in Fig. 5 of (I ). Thr wavy disturbance rolls up into the larger cell of the Taylor pair as shown in Fig. 5( b)-5( e). After the larger member of the pair is well established, t#he srnallcr cell appears with opposite circulation. There is a great similarity hetwec>n the results of this subsection and the computed streamlines of a similar problcnl treated by Amsden and Harlow (8). In the latter work the slip instability is considered from the standpoint of an initial value problem for a simple sheer layer separating oppositely directed parallel streams. IGgurc 4 of ref. 8 mar bc> compared with Fig. 5 presented here. (+. EFFECT
OF
AN AXIAL
lkow
0N THE WAVY
I~ISTURBA~XE
The inflection point theorem suggeststhat the addition of a net axial flow t’o the convective flow might produce interesting results. In (II j the effect on Taylor vorticies of a net axial flow is discussed.See Figs. 10 and 12 of (II). The same procedure was carried out for the wavy disturbance. It is found that when the inflection point is displaced nearly out of the region of the gap the wavy dist’urbance no longer appears. As B/A increases (see ( II ) for an explanation of these symbols), the cells become less distinct and for B/.4 > 1 definite cells are not observed. ‘\-. 1~ISCUSSION
The best clue about the mechanism causing the thermally induced instability is probably the sharp transition along the line p = 1. Observe that at large values of DzTaylor cells occur at p = 7’ = 0.92. This instability is associated with the centrifugal potential. The wavy disturbance occurs at larger values of P and thus viola&Rayleigh’s criterion; it is not due primarily to the cetrifugal term. The t,hc~rmally induced instability sets in precisely at p = 1. It occurs for both negative and positive gradients and the only effect of decreasing the magnitude of t.hc gradient is: (i ) to lengthen the time required for the cell patterns to form and also (ii 1 to change the number of starts of the spiral. In fact all the data rcportcd in this article appears to be nearly symmetrical in negative and positJivegradient’s,
324
KARLSSON
AND
SNYDER
Thus, the effect cannot be explained in terms of instabilities measured by the Richardson number or density gradients. The only obvious change in passing through p = 1 is the appearance of a sheer; however, there is also a sheer for p > 1. Apparently the mechanism is also associated with the centrifugal potential; the wavy disturbance does not occur for Q, = 0. Another clue is the fact that net axial flows (with a parabolic profile) with no point of inflection do not produce instability unless p < q2 even when the thermal gradient is present. This shows that the shape of the axial profile is significant and density gradients are unimportant. At the low axial flows set up by free convection the axial Reynolds number is very small and the motion is controlled by viscosity. It is difficult to see how the inviscid inflection point theorem for fluids would apply. It appears that a cubic profile is unstable even with viscosity present. This is the conclusion reached by Amsden and Harlow (8) on the basis of their computations. In conclusion we may summarize what the data implies about the mechanism of instability: (i) the shape of the axial profile is important and the cubic profile is unstable; (ii) the centrifugal potential must be present; and (iii) there must be sheer in the azimuthal direction in the presence of an increasing centrifugal field. ACKNOWLEDGMENTS The original arrangements to Reid. This research was supported tion, NSF GP333. RECEIVED:
carry out primarily
these experiments were made by Professor W. H. by a grant from the National Science Founda-
June 22, 1964 REFERENCES
I. H. A. SNYDER AND S. K. F. KARLSSON, Phys. Fluids 7, 1696 (19G4) 2. H. A. SNYDER, Ann. Phys. (N.Y.), 31, 292-313 (1965). 3. S. CHANDRASEKHAR, “Hydrodynamic and Hydromagnetic Stability.” The Clarendon Press, Oxford, 1961. 4. C. C. LIN, Quart. Appl. Math. 3, 117, 218, 277 (1945). 5. C. C. LIN, “The Theory of Hydrodynamic Stability.” Cambridge Univ. Press, London, 1955. 6. G. B. SCHAUB.IUER AND H. K. SKRAMSTAD, J. Aero. Sci. 14,69 (1947); J. Res. N&Z. Bur. Stand. 38, 251 (1947). 7. R. T. DONNELLY, private communication. 8. A. A. AMSDEN AND F. H. HARLOW, Phys. Fluids 7, 327 (1964).