1.2 Transition to turbulence in the 20th century
1.2.1 Rotta’s pipe flow experiments
Almost 70 years after the experiments by Osborne Reynold, Julius C. Rotta (1956) quan-tified how turbulent patches spread while traveling downstream. In his experiments in Göttingen he used glass pipes of different lengths and water as working fluid, but in con-trast to Reynolds’ experiments the water at the exit of the pipe was ejected as a free jet.
The angle of this jet is different for laminar and turbulent flows, which can be explained by the different mean profiles of the velocity (see Fig. 1.7). Rotta used this effect to determine the turbulent fraction of the intermittent flow by placing two containers at dif-ferent positions (see Fig. 1.5). One container was placed close to the pipe exit thereby collecting the water from a turbulent jet, the other one was located downstream of the first thereby collecting the water from a laminar jet. The ratio of the water volumes of the two containers corresponds to the turbulent fraction at a certain downstream position which corresponds here to the length of the pipe.
From measurements at different pipe length Rotta could infer the corresponding (mean) spreading velocities of the turbulence in terms of the averaged flow velocity for different Re. These are shown in Fig. 1.6. It is clearly visible, that the spreading velocities are quickly decreasing for decreasing Re.
‘Die Reynoldszahl, bei der sie den Wert Null erreicht, muss als die wirkliche kritische Reynoldszahl angesehen werden, unterhalb derer sich die turbulente Rohrströmung nicht mehr aufbauen kann. Der genaue Wert dieser kritischen Reynoldszahl kann aus vorliegenden Versuchen nicht bestimmt werden; er mag etwa bei Re=2000 liegen’ (Rotta 1956).
[freely translated: The Re where the spreading velocity becomes zero, must be regarded as the real critical value. Below it the turbulence cannot be regenerated. The exact value cannot be determined from present experiments, it may be around Re=2000.]
laminar
stromende Mengeturbulent Elektrode
Versuchsrohr
stromende Menge
Figure 1.5: Measuring the turbulent fraction. The angle of the free jet of water at the exit of the the pipe is different for laminar and turbulent flows. The left (right) container collects the water from the laminar (turbulent) free jet. The ratio of the measured fluid volumes corresponds to the turbulent fraction. The number of turbulent patches is deter-mined by an electrode, which is positioned in the laminar free jet. The figure is taken from Rotta (1956).
Credit to Ingenieur-Archiv, vol. 24, issue 4, 1965, ‘Experimenteller Beitrag zur Entstehung turbulenter Strömung im Rohr’, Rotta, page 262, Fig. 4. Reused with kind permission from Springer Science and Business Media.
Despite this clear interpretation, Rotta has been wrongly cited in the literature to have determined the transition to turbulence in pipe flow at Re=2300. Textbooks and even spe-cialized publications (Moxey & Barkley 2010) commonly give this number by referring to his work. Most likely it is due to his last measurement point in Fig. 1.6 at Re=2300 with a spreading velocity of only 2%.
Another rather common misinterpretation of Rotta’s work concerns the flow state that would develop in an infinitely long pipe, called by him ‘stationäre Endzustand’ [final equilibrium state]. He assumed that the onset of turbulence and the onset of the spreading of turbulence to fill the pipe appears at the same critical Re. Once turbulence has set in it would spread with the velocities shown in Fig. 1.6 until the flow is completely turbulent, i.e. without any laminar gaps (‘stationäre Endzustand’). This would apply for anyRe >
Rec.
‘Die Partien turbulenter Strömung werden mit der Strömung fortgeführt; dabei wachsen sie mit einer von der Reynoldszahl abhängigen Ausbreitungsgeschwindigkeit und schmelzen nach und nach zusammen, his schliesslich ein voll
turbulen-ter Strömungszustand hergestellt ist. [. . . ] Dieser Vorgang erstreckt sich über sehr grosse Rohrlängen, die für kleine Reynoldszahlen sogar nach Tausenden
2000 2200 2400 2600
Reynolds number Re
0 0.03 0.06 0.09 0.12
Spreading rate of turbulence
Figure 1.6: Spreading rate of turbulence. Rotta analyzed the turbulent fraction of the flow depending onRe and pipe length. By assuming that the velocity of the turbulent fronts is similar to the mean velocity of the flow the data shown here can be interpreted as the mean spreading rate of turbulence. The line is to guide the eyes. Data are taken from Rotta (1956).
Figure 1.7: Laminar and turbulent velocity profile in pipe flow. The laminar flow profile was mathematically derived by Hagen (1839) and Poiseuille (1840). Hot-wire measure-ments by Rotta confirmed this ‘Poiseuille profile’ for laminar flow and showed in addition that turbulence leads (in average) to a plug-like profile. The figure is reproduced from Rotta (1956).
Credit to Ingenieur-Archiv, vol. 24, issue 4, 1965, ‘Experimenteller Beitrag zur Entstehung turbulenter Strömung im Rohr’, Rotta, page 266, Fig. 9. Reused with kind permission from Springer Science and Business Media.
von Durchmessern zählen’ (Rotta 1956).
[freely translated: Turbulent patches are swept downstream with the mean flow; at the same time these patches spread into the laminar gaps with a spreading velocity depending on the Reynolds number. The final flow state is fully turbulent, meaning that turbulent patches have merged, leaving no laminar gaps in between. [. . . ] This process requires large pipe length, at small Re it can take several thousand pipe diameters.]
To sum up, Rotta was estimating the critical Reynolds number for the onset of turbu-lence from his extrapolation astonishingly well, but he missed the intermittent character of the turbulence that is intrinsic to pipe flow at lowRe.
At that time many researchers have tried to derive a mathematical description that can predict the onset of turbulence. For a flow between two plane plates shearing in opposite directions (plane Couette system)
‘it seemed probable that the mathematical analysis might prove compara-tively simple; but . . . it has actually proved very complicated and difficult’
(Taylor 1923).
Taylor considered also to investigate pipe flow, but from the experiments of Reynolds he concluded that it is too difficult, because of the finite amplitude perturbations that are required to trigger turbulence. He therefore decided to
‘examine the stability of liquid contained between concentric rotating cylin-ders. If instability is found for infinitesimal disturbances in this case it will be possible to examine the matter experimentally’ (Taylor 1923).
The setup used by Taylor was originally invented for a completely different purpose, which is explained in the next section.