Over a century of research into the origin of turbulence in wall-bounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow1,2,3,4,5,6. At moderate flow speeds, turbulence is confined to localized patches; it is only at higher speeds that the entire flow becomes turbulent. The origin of the different states encountered during this transition, the front dynamics of the turbulent regions and the transformation to full turbulence have yet to be explained. By combining experiments, theory and computer simulations, here we uncover a bifurcation scenario that explains the transformation to fully turbulent pipe flow and describe the front dynamics of the different states encountered in the process. Key to resolving this problem is the interpretation of the flow as a bistable system with nonlinear propagation (advection) of turbulent fronts. These findings bridge the gap between our understanding of the onset of turbulence7 and fully turbulent flows8,9.
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We thank A. P. Willis for sharing his hybrid spectral finite-difference code, and X. Tu for helping to set up and test the experiment. We acknowledge the Deutsche Forschungsgemeinschaft (Project No. FOR 1182), and the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement 306589 for financial support. B.S. acknowledges financial support from the Chinese State Scholarship Fund under grant number 2010629145. B.S. acknowledges support from the International Max Planck Research School for the Physics of Biological and Complex Systems and the Göttingen Graduate School for Neurosciences and Molecular Biosciences. We acknowledge computing resources from GWDG (Gesellschaft für wissenschaftliche Datenverarbeitung Göttingen) and the Jülich Supercomputing Centre (grant HGU16) where the simulations were performed.
The authors declare no competing financial interests.
Extended data figures and tables
a, b, Speeds as a function of model Reynolds number r both without (a) and with (b) advection. Although strong downstream fronts cannot exist and have no physical meaning below the formation of the upper branch fixed point, the expression for strong front speeds in equation (10) still gives the speed that such a strong downstream front would have; these speeds are shown dashed. The effect of nonlinear advection in b is to mask the nominal critical point for the onset of fully turbulent flow. The neutral speed is naturally displaced from the mean speed U = 1.
Sketch illustrating solutions to the boundary value problem in equation (8) for a downstream front near the critical point. a, Eigenvalue s as a function of uf. uc is the value of uf such that q− = q+. For this value there are infinitely many possible eigenvalues s, indicated by the thin line. b, c, Phase planes (q, q′) showing solutions for the second order differential equation (8). Downstream fronts are heteroclinic connections from the upper fixed point q+ to the lower fixed point q0. When uf = uc and hence q− = q+, the upper fixed point is not hyperbolic and there are infinitely many connections, each corresponding to a value of s. When u > uc, q+ is hyperbolic and there is a unique connection and hence a unique value of s.
Extended Data Figure 3 Determination of corresponding Reynolds numbers and speeds for pipe and duct flow.
a–d, Speeds from pipe (a, b) and duct flow (c, d) are plotted, as in Fig. 1c, but additionally with the upstream front speeds reflected about the neutral speed C0. a, c, Experimental and simulation data only; b, d, model fits to the experimental and simulation data. The determined values for R0, R1, C0 and C1 are: R0 = 1,920, C0 = 1.06, R1 = 2,250 and C1 = 0.92 for pipe flow, and R0 = 1,490, C0 = 1.12, R1 = 2,030 and C1 = 0.90 for duct flow.
Pipe and duct flow are plotted together using different axes. The data are plotted so that the two points (R0, C0) and (R1, C1) align for each data set; for example, (R1, C1) = (2,250, 0.92) for pipe flow is aligned with (R1, C1) = (2,030, 0.90) for duct flow, bringing into alignment the onset of weak fronts.
a, Determination of D. Points are data from pipe and duct flow (as in Extended Data Fig. 4) here plotted in terms of reduced Reynolds number (R − R0)/(R1 − R0) and reduced speed (C − C0)/[2(C0 − C1)]. Dashed curves are asymptotic speed curves (as in Extended Data Fig. 1) plotted in terms of model Reynolds number r and speed c − c0. For D = 0.13 there is very good agreement between the data and the model. This choice of D fixes the asymptotic branches (dashed curves). b, Determination of ζ and ε. Pipe and duct flow are necessarily considered separately. In each case, downstream branches are shown for four values of ε. Smaller values yield more abrupt transitions between weak and strong branches.
a, Control concept illustrated in the model phase plane. Without forcing (that is, without control), there is an upper-branch fixed point (upper intersection of nullclines) corresponding to fully turbulent flow. Applying an additive forcing term to the u equation corresponds to forcing the shear profile and blunting its shape. This can remove the turbulent fixed point thus eliminating fully turbulent flow. b, Proof of concept in a direct numerical simulation of pipe flow at R = 5,000. Without forcing the flow is fully turbulent. A global body force is applied that blunts the velocity profile to a more plug-like form. Subsequently, only localized turbulent patches remain, reminiscent of those at much lower R.
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Barkley, D., Song, B., Mukund, V. et al. The rise of fully turbulent flow. Nature 526, 550–553 (2015). https://doi.org/10.1038/nature15701
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