Upper bound on angular momentum transport in Taylor-Couette flow

We investigate the upper bound on angular momentum transport in Taylor-Couette flow theoretically and numerically by a one-dimensional background field method. The flow is bounded between a rotating inner cylinder of radius Ri and a fixed outer cylinder of radius Ro. A variational problem is formulated and solved by a pseudo-time-stepping method up to a Taylor number Ta=109. The angular momentum transport, characterized by a Nusselt number Nu, is bounded by Nu≤cTa1/2, where the prefactor c depends on the radius ratio η=Ri/Ro. Three typical radius ratios are investigatedi.e., η=0.5,0.714,and0.909, and the corresponding prefactors c=0.0049,0.0075,and0.0086 are found to improve (lower) the rigorous upper bounds by Doering and Constantin [C. Doering and P. Constantin, Phys. Rev. Lett. 69, 1648 (1992)PRLTAO0031-900710.1103/PhysRevLett.69.1648] and Constantin [P. Constantin, SIAM Rev. 36, 73 (1994)SIREAD0036-144510.1137/1036004] by at least one order of magnitude. Furthermore, we show, via an inductive bifurcation analysis, that considering a three-dimensional background velocity field is unable to lower the bound.