Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse's problem and the Constantin–Doering–Hopf problem with one-dimensional background field

We present an improved upper bound on the energy dissipation rate in plane Couette flow. This is achieved through the numerical solution of the ‘background field’ variational problem formulated by Constantin and Doering with a one-dimensional unidirectional background field. The upper bound presented here both exhausts the bounding potential of the one-dimensional background field problem and also solves the provably equivalent problem formulated by Busse. The solution is calculated up to asymptotically large Reynolds number where we can estimate that the energy dissipation rate $\e \le 0.008553$ as $Re \rightarrow \infty$ (in units of $V^3/d$ where $V$ is the velocity difference across the plates separated by a distance $d$ and $Re = V d /\nu$, with $\nu$ kinematic viscosity). This represents a 21% improvement over the previous best value due to Nicodemus et al. A comparison is drawn between this numerical solution and the so-called multi-alpha asymptotic solutions discovered by Busse.