McTaggart, F. K., “Plasma Chemistry in Electrical Discharges,” Elsevier Publishing Co., Amsterdam, 1967. Mellor, J. W., “Alellor’s Comprehensive Treatise on Inorganic and Theoretical Chemistry,” Vol. 8, p 367, Longmans, Green & Co., London, 1940. Potapov, A. V., High Temperature,4,48,(1966). Rose, D. J., Brown, S. C., Phys. Rev. 98, 310 (1955). Thrush, B. A,, Progr. React. Kinet. 3, 1 (1965). Timmins, R. S., Ammann, P. R., “The Application of Plasmas to Chemical Processing,” Chapter 7, p 99, R. F. Baddour and R. S. Timmins Ed., AIIT Press, Cambridge, Mass., 1967. Townsend, J. S., Phil. Mag. 13,745 (1932).
Trotmsn-Dickenson, A. F., Milne, G. S., “Tables of Bimolecular Gas Reactions,” U. S. Printing Office, Washington, D. C., 1967.
Wechsberg, H. E., Webber, J. E., Mod. Plast. 36, 101 (1959). Westman, H. P., Ed., “Reference Data for Radio Engineers,” 4th ed, pp 600-603, IT&T, New York, N. Y., 1956. for review March 29, 1972 RECEIVED ACCEPTEDApril 12, 1973
This work was supported by a grant from the National Science Foundation.
Comments on Transition from Laminar to Turbulent Flow H. Dennis Spriggs @’est Virginia College of Graduate Studies, Institute, W . Va. 26112
Observations have been made concerning various aspects of transition to turbulence in single-phase and two-phase flow. For single-phase flow, a simple model i s proposed which allows calculation of the friction factor in the transition regime. This result is, in turn, used to interpret the point of transition. For two-phase flow, Hanks’ stability criterion i s used in an attempt to predict transition. It i s shown that this criterion fails and that stabilizing forces are operative which postpone transition in such systems.
I t is a well established fact that transition from laminar to turbulent flow is not a n abrupt phenomenon. Rather, as flow rates are increased, the fluid passes from a stable, rectilinear, or laminar state through various intermediate regimes and finally into a turbulent state (Seneca1 and Rothfus, 1953). True laminar flow exists for Reynolds numbers up to about 1000 (for circular pipes). -1bove 1000 a stable or laminar sinuous motion begins which grows in amplitude as flow rates are increased to a Reynolds number of about 2000. I n the range extending approximately from 2000 to 3000 transition between the various laminar states and various turbulent 5tates occurs. The transition regime is characterized by a n intermittent series of laminar and turbulent patches whose appearance marks the point of departure from the usual laminar flow relationships. Turbulent flow relationships are not valid until the end of the transition regime and even then these relationships vary v ith increasing flow indicating a change in the nature of the turbulent flow. For purposes of simplification i t is often convenient to characterize this complex progression of flow regimes as being merely from laminar to transition to turbulent, ignoring the more subtle changes in the flow. Whereas methods exist for deriving relationships such as those for friction factors for both laminar and turbulent flow, the transition regime has proven to be more elusive. Friction Factor in Transition Regime
-1quantity called the intermittency factor defined as the fraction of time the flow is turbulent is one convenient char286 Ind.
Eng. Chem. Fundam., Vol.
12, NO. 3, 1973
acterization of the transition regime. It has been found that a probability plot of the intermittency factor vs. Reynolds number yields a straight line; Le., there is a normal distribution (Spriggs, 1968; Stellmach 1967). Construction of a simple model of transition is possible by visualizing this regime as being just a combination of laminar flow and turbulent flow. The intermittency factor is used as a weighting factor. For example, the friction factor for transition would be (for circular pipes)
s = (lin,> - ( l - - Y ) + ( s ) Y where y is the intermittency factor. To test this model, it would be desirable to have friction factor and intermittency factor data from the same system, say for flow in circular pipes. Rotta has measured the intermittency factor for flow in pipes using a hot wire anemometer (Rotta, 1956; Schlichting, 1968). He found that this quantity is dependent upon both the radial and axial positions in the pipe as well as the Reynolds number
r X
Y = Y
(5’0,
For example, see the data in Table I. These observations suggest two problems regarding the direct application of intermittency factor data to eq 1.
0.1
0.04
I
1
c
0
Ddfd
O f
I
SOneCd/ m d
Rothfus
60
‘O
1
“ 8
0.004 403
2000
600 800KM3
3
m
5ow
10000
l5
t I
Figure 1. Friction factors vs. Reynolds number for flow in circular pipes. Comparison of experimental data and values calculated using eq 1
1
1500
I
I
2OOO
3000
I 5000
NRe
Figure 2. Theoretical fNR, vs. N R curve ~ using eq 1
Table I r/D
XID
NRe
Center region Center region Center region Wall region
100 500 322 322
2600 2600 2550 2550
r 0 0 0 0
56 98 8-0 9 5-0 6
1. What is the relative importance of main stream and wall region turbulence on the skin friction as measured by the friction factor? I n other words, which values of intermittency factor should be used? 2. The friction factor is measured over some finite length of pipe in which the intermittency factor varies. As indicated by the data of Rotta, the turbulent patches seem to grow as they move along the pipe axis. This suggests t h a t some average or integral intermittency factor should be used. I n keeping with the simplicity of the model, these difficulties can be overcome by noting that: (1) most of the change in fraiction factor occurs in the Reynolds number range of 1900 to 2900 (Senecal and Rothfus, 1953); ( 2 ) this is also the approximate range in which the intermittency factor increases from near zero to near unity. Therefore, eq 1 was tested using the friction factor data of Senecal and Rothfus with the assumption that 7 = 0.01
(1VR.e =
1900)
0.99
(1YRe =
2900)
y
=
Values of y for Reynolds numbers between 1900 and 2900 using ~ the were found from a probability plot of y us. ~ Y R observation that such a plot is linear. Figure 1 s h o w a comparison of calculated and measured values of friction factor. This figure sliovvs excellent agreement between the two which, in part, justifies the simplifications made. One can also see from these results that other flow regimes have a n effect on the friction factor. For example, between A r ~ e= 1000 and
Ruo
1500
2000
3000
4000
3000
4000
NRC
I
1500
2000
NRe
Figure 3. Different interpretations of pressure drop data showing how a proper consideration of the shape of the fNR, curve alters the point of transition. Data are from Allen and Grundberg ( 1 937)for a rectangular channel of 3.92: 1 aspect ratio
XR~ = 2000 most data points lie above the laminar flow curve. This indicate.. that the sinuous motion in this range increases the friction factor slightly. -1more precis? model could take these facts into consideration. T h e Point of Transition to Turbulence. T h e preceding development raises an interesting question as to the “point” Ind. Eng. Chern. Fundarn., Vol.
12, No. 3, 1973 287
-
LAMINAR WATER TURBULENT KEROSENE $
4an
-
--
Lrflmrted
X
x
w
x
X
X
X
X
X
X
PredlctedBoth n/rbulent
w
*
TURBULENT WATER LAMINAR KEROSENE
83
x
0
loa: Bo(
z‘
0
4
6iX
-
Water T u r b u l e n t Kerosene L r m l n a r
4oc 0
20
100
I
X
0
401
\
2ax
x , “ x@
LAMINAR WATER
1
TURBULENT WATER AND KEROSENE
0 0 0
Hanks’ c r t f s r l o n
Exportmentat
0
200
urtng
ff 20c I
4W
I
6W
I loo0
8 2ow
hedlctedB o t h Lamtnar
I 4oa,
6ocO
NRc WATER Figure 4. Flow regimes for two-phase stratified flow in a rectangular channel. Here X indicates transition to turbulence in the kerosene phase and 0 indicates transition in the water phase. Data are from Spriggs ( 1 968)
of transition to turbulence. It has been shown that the appearance of the first turbulent patches in a fluid corresponds closely to deviations from laminar flow relationships such as the relationship between friction factor and Reynolds number (Spriggs, 1968). However, as reported earlier, the appearance of these turbulent bursts follow a normal distribution. This means there is some chance of these bursts occurring even a t very low Reynolds numbers. For practical purposes one must define some point, Le., a Reynolds number, a t nhich deviations become significant. I n keeping with the convention used in boundary layer theory, a convenient definition for the point of transition might be where y = 0.01. Using this convention it can be concluded that transition in circular pipes occurs a t 1900 and not 2100 as usually reported. This concept can also be used to demonstrate a n error often committed when determining the point of transition. Transition is sometimes found by constructing a plot of ~ X us. R ~ S R When ~ , the flon- is laminar, the product f A Y ~ will e be constant (equalling 16 for circular pipes, for example). Deviation from this constant signifies the beginning of transition to turbulence, Le., the “point” of transition. Consideration, however, is usually not given to the proper shape of such a plot. Based on the closeness of fit found in Figure 1, the theoretically correct shape of a n f L Y ~us. e S Rplot ~ can be found using eq 1. This is shown in Figure 2. Figure 3 shows one example found in the literature where the f A Y ~us. e S Rplot ~ was used. As can be seen, a consideration of the proper shape of this curve significantly alters the point of transition. Two-Phase Flow Regimes
Transition to turbulence in two-phase flow has received much less attention than in single-phase flow. Fairly recently, 288
Ind. Eng. Chem. Fundarn., Vol. 12, NO. 3, 1973
however, several investigators have mapped flow regimes for the stratified two-phase flow of various oil and water pairs (Charles, 1963; Spriggs, 1968; Stellmach, 1967). A typical plot is shown in Figure 4. Attempting to Predict Transition. Various criteria have been used to predict the point of transition in single-phase flow. One of the more successful is that of Hanks (Hanks, 1963, 1969). This criterion is based on a local ratio of inertial and viscous force terms selected from the equation of motion. I n its simplest form, this criterion is
where K is the stability parameter. It was postulated t h a t when K exceeds some critical value a t any point in the fluid transition will begin. Hanks found this critical value to be K = 404. This criterion was applied to several different geometries and even to a non-Newtonian fluid and in all cases proved satisfactory (Cope and Hanks, 1972; Hanks, 1963, 1967; Hanks and Ruo, 1966; Hanks and Song, 1967). Because of the success of this criterion with single-phase flow, it was applied to the two-phase stratified flow systems described earlier. It was assumed that whenever the parameter K exceeded 404 in either phase, then that phase would begin transition to turbulence. The results are shown in Figure 5. I n all cases the fluid with the lower viscosity was predicted to become turbulent first. Also, in all cases there was little correspondence between the measured and predicted flow regimes. The significant deviation seemed to be that the fluids were much more stable than predicted. It seemed as though the second phase dissipated disturbances before they could grow into turbulent patches. The interwting conclusion is that a model which was so very successful for single-phase flow failed quite dramatically when applied to tn-0-phase flow.
