Bounds on dissipation in stress-driven flow

Abstract

We calculate the optimal upper and lower bounds, subject to the assumption of streamwise invariance, on the long-time-averaged mechanical energy dissipation rate ε within the flow of an incompressible viscous fluid of constant kinematic viscosity ν and depth h driven by a constant surface stress τ = ρu 2 , where u is the friction velocity. We show that ε 6 ε max = τ 2 /(ρ 2 ν), i.e. the dissipation is bounded above by the dissipation associated with the laminar solution u = τ (z + h)/(ρν)î, whereî is the unit vector in the streamwise x-direction.By using the variational 'background method' (due to Constantin, Doering and Hopf) and numerical continuation, we also generate a rigorous lower bound on the dissipation for arbitrary Grashof numbers G, where G = τ h 2 /(ρν 2 ). Under the assumption of streamwise invariance as G → ∞, for flows where horizontal mean momentum balance and total power balance are imposed as constraints, we show numerically that the best possible lower bound for the dissipation is ε > ε min = 7.531u 3 /h, a bound that is independent of the flow viscosity. This scaling (though not the best possible numerical coefficient) can also be obtained directly by applying the same imposed constraints and restricting attention to a particular, analytically tractable, class of mean flows. IntroductionForced turbulent flows occur in a wide range of natural and industrial fluid flows. Identifying scaling laws for the mechanical energy dissipation rate within such flows has been widely studied, in an effort to capture how the turbulent small-scale motions dissipate the energy input by the forcing. Naturally, the properties of the mechanical energy dissipation rate are strongly dependent on the characteristics of the forcing of the flow. One particular type of forcing that commonly occurs is the application of stress at the upper surface of a layer of fluid over a solid lower surface as happens, for example, when wind blows over a body of water. It is reasonable to suppose that this injection of energy at the upper surface will determine both the bulk and small-scale properties of the flow in the forced fluid layer, with the no-slip lower surface boundary condition also playing a critically important role. In this paper we address the fundamental question of how the forcing affects the flow by determining both upper and lower bounds on the mechanical energy dissipation rate for flows that are statistically stationary, thus defining a range of possible behaviours of such a flow.We concentrate on generating bounds for the mechanical energy dissipation rate, as such bounds can be derived rigorously without introducing heuristic modelling

Optimal bounds with semidefinite programming: An application to stress-driven shear flows

“…[12,13]). Our numerical results improve previous analytical bounds by more than 10 times at large Gr, and agree with the approximate computations carried out by Tang et al [17]. This confirms that flows driven by a surface stress are similar to those driven by a localized body force not only in terms of their energy stability boundaries [12], but also as far as bounds on the dissipation coefficient are concerned.…”

“…Moreover, we could not compare our bounds to values of C ε extracted from experiments or direct numerical simulations because, to the best of our knowledge, such data are not available. Although it is unlikely that there exists a solution to the Navier-Stokes equation whose associated dissipation coefficient equals our bounds [17], performing this comparison remains in the interest of future research.…”

“…Flows of this kind arise, for example, in physical oceanography, when wind blows over a body of water. The three-dimensional flow was first studied by Tang et al [17], who used the background method to estimate C ε ≤ Gr (7.531Gr 0.5 − 20.3) −2 for large Gr, where the Grashoff number Gr represents the nondimensional forcing. Strictly speaking, however, these bounds apply to a different flow, where the imposed stress is approximated by a body force localized near the boundary.…”

A framework for input–output analysis of wall-bounded shear flows

“…and bounds on c 1 and c 2 were obtained for different flows (Doering & Foias (2002); Childress et al (2001); Alexakis & Doering (2006); Rollin et al (2011);Tang et al (2004)). In order to find bounds on the time-averaged mechanical energy dissipation, we can consider the following dissipation inequality…”

Bounds on dissipation in stress-driven flow in a rotating frame

“…By assuming that the horizontal dimensions are much larger than the vertical dimension of the channel, and restricting our attention to particular, analytically tractable, classes of Lagrange multipliers imposing mean horizontal momentum balance analogous to the ones used in Tang, Caulfield & Young (2004), we show that ε 6 ε max = u 4 /ν − 2.93u 2 f , an improved upper bound from the Stokes dissipation, and ε > ε min = 2.795u 3 /h, a lower bound which is independent of the kinematic viscosity ν.…”