Jean Berlamont
Hydraulics Laboratory, K.U.Leuven
de Croylaan 2, 3001 Leuven, Belgium
Tel. + 32 16 321660, fax. + 32 16 321989,
E-mail: Jean.Berlamont@bwk.kuleuven.ac.be
Abstract: Drag reduction is reported in literatures in which the measured friction factors in pipes is smaller than that one would expect on the basis of state of the art knowledge for a perfectly smooth pipe. This occurs in particular at high Re numbers and in large diameter pipes. This apparently impossible results can not be explained by errors of the measurements.
Both from experiments and numerical simulation it is known that rotation causes a certain degree of laminarization of an initially turbulent pipe flow and, accordingly a reduction of the friction factor up to 50%.
The reported “smoother than smooth” conditions could indicate that for some reason or another the flow in the pipe has obtained some degree of rotation.
Using the “minimum rate of energy dissipation N” principle it can be shown that the most probable flow situation is a flow without any rotation. If however a flow would have obtained for some reason (e.g. the passage of a bend, a flow regulator,…) a certain amount of rotation, it may persist for a long time c.q. for a long distance along the pipe since the corresponding N- value is hardly higher than Nmin which corresponds to a flow without rotation.
This explains why in a number of circumstances (unintentionally, and unknown to the user or the experimenter) a moderate amount of rotation (N < 0.7 to 1.0) may be present in a straight pipe and why friction factors can be measured which are up to 10% lower than the values for a perfectly smooth pipe according to the White Colebrook Nikuradse (Blasius) formula. Since in a large diameter pipe a small amount of rotation (') produces a relatively large value of N, “smoother than smooth” conditions have been detected preferably in large diameter pipes.
If a certain
circulation would exist or be imposed in a pipe (e.g. N = 1.0) the reduction of
f with, say 10 % would result in an increase of the discharge Q of the order of
5 %. When maintaining the same discharge, a diameter 1 to 2 % less could
suffice, which is, unfortunately of no practical relevance for design purposes.
Keywords: pipe flow, friction factor, laminarization, entropy principle
Since the use of calculators and PC’s has been widely spread, friction losses in pipes are more and more often calculated using the White - Colebrook (Nikuradse, Moody, Prandtl, von Karman) formula instead of using the empirical formulae by Chézy, Manning, Hazen Williams, Flammant, etc. Although the White – Colebrook formula (for smooth pipes) has been validated by different large data sets, very few or none of them has been obtained in large diameter pipes (more than one meter) nor for large Reynolds numbers. There are some indications in literature that one may question the unconditional validity of the White - Colebrook equation for modern (very) large diameter “smooth” pipes (plastic linings) at high Re – numbers (up to 108). Such very large pipes are used e.g. as pressurized trunk sewers, pennstocks, filling and emptying devices for navigation locks, etc… It appears indeed from measurements that sometimes, in particular for large Re - numbers smooth pipes may have smaller friction factors than the ones following from the White – Colebrook formula. In a way they seem to be “smoother than smooth” or even have a negative roughness! Since the White – Colebrook equation is generally accepted it seems not “wise” to publish such results and rather blame measurement errors for these “deviations”. However, such measurements have been reported by notorious researchers like P. Ackers (Ackers, 1961), D. Barr (Barr, 1973), L. Levin (Levin, 1972) and (Burke, 1953) So far, this question has never been elucidated in literature.
Recently both experimental and numerical experiments (direct integration of the instantaneous NS equations (Orlandi, 1997 and Orlandi and Fatica, 1997)) have been carried out to study flows in a pipe rotating around its axis. It has been found that in an axially rotating pipe flow so called “laminarization” occurs: i.e. the turbulent flow becomes, in particular close to the (rotating wall), more and more similar to a laminar flow when the rotation rate increases resulting in the deformation of the axial velocity profile into a shape similar to the laminar one, drag reduction and subsequent decrease of the friction factor up to 50% (Nishibori, Kikuyama and Murakami, 1987 and Reich and Beer, 1989).
It apparently does not make any difference whether the rotation is imposed by a rotating pipe wall or as an initial condition. Therefore this explains why in a pipe lower friction factors can be found if the flow would be somehow rotating. According to Beer the effect is more pronounced for the same degree of rotation for the higher Re numbers. All this is in accordance with the experimental evidence which was mentioned. From that we may conclude that when smoother than smooth friction factors are reported, it may be an indication that for some reason or another the flow has obtained some degree of rotation.
To explain the measurements of “smoother than smooth” friction factors it was hypothesized by the author that flow in a long straight pipe is not “stable” but, on the contrary, becomes unstable e.g. from a certain Re- number on. Then, a helical flow pattern could develop which is the superposition of the regular longitudinal flow and a secondary (single or double) circulatory flow pattern, similar to the secondary flows in open channels. Secondary flow of the first type may be induced by the unavoidable disturbances due to the presence of bends and/ or regulators in the pipe . This would be analogous to the development of bedforms in rivers and on a seabed, the development of roll waves in supercritical open channel flow (Berlamont, 1976 and 1977) and even the meandering of a river pattern (Yang, 1971; Berlamont, 1995).
So, it can be understood that if rotation is generated and maintained in a pipe flow, the friction factor (defined and measured in the conventional way i.e. using the mean axial velocity) can be (much) lower that given by the Blasius formula.
The only question that remains is: “Is the flow in a (straight) pipe indeed unstable and does rotation indeed develop, apparently preferably in large diameter pipes?”.If yes, one should be able to determine which rotation will ultimately persist.
The entropy principle has been used successfully during the last 10 to 20 years in hydraulics and river mechanics to predict e.g. river meandering and braiding (Yang, 1971; Berlamont, 1989 and 1995), river regime width and sediment transport capacity, velocity and concentration profiles in open channels (Yang and Song, 1986), the development of roll waves in supercritical open channel flow (Berlamont, 1976 and 1977), and shear stress distributions in sewer pipes (Sterling, 1997).
As is the case e.g. with meandering river
patterns or roll-waves, of all different equilibrium situations which may be a
priori theoretically possible (solutions of the equations of motion), the
particular one which eventually really exists is the solution which corresponds
to the minimum “rate of energy dissipation per unit mass and per unit time”
(or the minimum increase of entropy per unit mass) for an irreversible system
without energy input (Berlamont, 1995,
Yang and Song, 1986).
In the case of pipe flow one could imagine that for a given amount of energy dissipation different flow modes are possible: one could for example have either a straight flow or a helical flow which dissipates extra energy due to the secondary flow, but which “saves” energy by the reduction of the friction factor due to the change in the structure of turbulence due to the centrifugal forces (laminarization). Possibly a particular value of the “rotation” N (= tangential velocity at the pipe wall/ mean axial velocity) exists for which the “rate of energy dissipation per unit mass and time is minimum. This would be the rotation which eventually prevails and the corresponding friction factor would be the one measured, most probably lower than the value for straight pipe flow according to White Colebrook, Blasius,…
In order to clarify this, two different approaches have been used:
The simplest expression for the “rate of
energy dissipation per unit mass and time”, which will be called N,
for pipe flow is:
N = Sf . U0 in which: Sf = f U02 / 2gD (definition of f)
Minimizing N for given values of U0 , D and Q would imply to minimize f, which would mean to maximize N. That means that any circulation which would occur accidentally e.g. due to disturbances (bends) would reduce f and thus N and would thus correspond to a more “probable” situation and persist.
N = ' D / 2 U0 ' is the circulation (s-1)
Fig. 1 shows N as a function of N for a helical flow in pipe (superposition of axial flow and circulatory flow). Different friction laws have been tried. The friction losses are calculated with the true velocity.
It is seen from fig. 1 that the most probable flow condition (Nmin) in a straight smooth (infinitely long) pipe is a flow without circulation: N = 0. However, N remains very close to Nmin as long as N < 0.7 to 1.0. This means that, if a (not too strong) secondary circulation would exist, generated by irregularities, bends,… it may subsist for a long time c.q. a long distance along the pipe to die out eventually provided that the pipe is long enough. (This is to be compared with a system in neutral equilibrium). Since experiments are never carried out in infinitely long pipes one may very well have observed rotational flows in straight pipes and, accordingly, have measured friction factors “smoother than smooth”. With the initial presumptions (about the validity of the White Colebrook formula in rotational flows provided that the wall velocity is introduced instead of the mean axial velocity) a rotation rate 0.7 < N < 1.0 corresponds to a reduction of the friction factor of the order of 5 to 10 %. Reich and Beer found 10 to 20 % for 0.5 < N < 1.0 and 104 < Re < 105 (Reich & Beer, 1989) for flow in rotating pipes. In a large diameter pipe a small amount of rotation (') produces a relatively large value of N, which may lead to a measurable reduction of f. This may explain why “smoother than smooth” conditions have been detected preferably in large diameter pipes. If a certain circulation would exist or be imposed in a pipe (e.g. N = 1.0) the reduction of f with, say 10 % would result in an increase of the discharge Q of the order of 5 %. When maintaining the same discharge a diameter 1 to 2 % less could suffice, which, unfortunately, is of no practical relevance.
A finite element software which allows the description of the turbulent flow in pipes in great detail using a sophisticated turbulence model (anisotropic k - , turbulence model) (Koelling, 1994) has been used to verify the results of the simple model. In fact using the detailed description of turbulent flow, one can calculate the friction factor and the energy losses without further assumptions.
The model was driven by applying a double circulation as a boundary condition. It was hoped that simulations for different pipe diameters and with different (imposed) secondary flows would have shown the effect of an (imposed) helical flow pattern on the energy losses, and ideally allow to find an “optimum” circulation which minimizes the energy losses per unit time and unit mass , N. This would then represent the “equilibrium situation” which naturally develops, if the pipe is sufficiently long. Possibly the “optimum circulation” would depend on the Reynolds number.
It is seen that all estimates for f when N > 0 show an increase in f . The model apparently is not capable of reproducing the reduction of f due to the presence of rotation. The reason is most probably the limitations of the turbulence model used: “Calculations applying a conventional k - , two- equation turbulence model with and without modification of the Richardson number cannot predict the tendency of the experimental results : laminarization phenomena and the characteristic behaviors due to the swirl (driven by the pipe wall rotation) because the terms containing the swirl effect in the transport equations of k and , become negligibly small when the tangential velocity profile becomes solid rotational profile (v 2 ~ r)” (Shuichiro Hirai et al., 1988).
It is however interesting to note that N calculated from , / k shows the same trend as found from the simplified 1 D model : it shows that N increases hardly with N as long as N < 2, which means that N . N min for 0 < N < 2. Which confirms the previously mentioned results that a flow with a moderate amount of circulation may persist in a straight pipe and consequently that “smoother than smooth” friction factor may have been measured.
(1) There is ample evidence in literature that
sometimes friction factors in pipes are measured which are smaller than one
would expect on the basis of state of the art knowledge for a perfectly smooth
pipe. This happens especially at high Re numbers and in large diameter pipes.
These apparently impossible results can not be explained by errors of the
measurements.
(2) There is evidence, both from experiments and numerical simulation that rotation causes a certain degree of laminarization of an initially turbulent pipe flow and, accordingly, a reduction of the friction factor up to 50%.
(3) Since apparently it does not make any difference whether the rotation is imposed by a rotating pipe wall or as an initial (and/or boundary) condition the reported “smoother than smooth” conditions could indicate that for some reason or another the flow in the pipe has obtained some degree of rotation.
(4) Using the “minimum rate of energy dissipation N” principle it can be shown with different models that the most probable flow situation is the one without any rotation. However if for some reason or another (e.g. the passage of a bend, a flow regulator,…) the flow in a pipe has obtained a certain amount of rotation it may persist for a long time c.q. for a long distance along the pipe since the corresponding N- value is hardly higher than Nmin which corresponds to a flow without rotation.
(5) This explains why in a number of circumstances (unintentionally, and unknown to the user or the experimenter) a moderate amount of rotation (N < 0.7 to 1.0) may be present in a straight pipe and why friction factors can be measured which are up to 10% lower than the values for a perfectly smooth pipe according to White Colebrook Nikuradse (Blasius).
(6) Since in a large diameter pipe a small amount of rotation (') produces a relatively large value of N, which may lead to measurable reduction of f, “smoother than smooth” conditions have been detected preferably in large diameter pipes.
(7) If a certain circulation would exist or be imposed in a pipe (e.g. N = 1.0) the reduction of f with, say 10 % would result in an increase of the discharge Q of the order of 5 %. When maintaining the same discharge a diameter 1 to 2 % less could suffice, which unfortunately is of no practical relevance for design purposes.
(8) The model by (Reich & Beer, 1989) seems to be capable of reproducing the
reduction of f by a rotation both qualitatively and even to some extend
quantitatively.
(9) The model SIMK (Kölling) apparently is not
able to reproduce the laminarization of turbulent pipe flow due to the presence
of circulation (centrifugal forces).
Acknowledgements
Part of this research has been funded by the Flemish Research Council F.W.O. Krediet aan navorsers S 2/5 – AV. – D 12126 (1996)
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