Variational bound on energy dissipation in plane Couette flow

The compound matrix method, which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k = 2, 3 dimensional subspaces of C n , by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, k (C n ).This formulation leads to a general framework for studying systems of differential equations on k-dimensional subspaces. The framework requires the development of several new ideas: the role of Hodge duality and the Hodge star operator in the construction, an efficient strategy for constructing the induced differential equations on k (C n ), general formulation of induced boundary conditions, the role of geometric integrators for preserving the manifold of k−dimensional subspaces -the Grassmann manifold, G k (C n ), and a formulation for induced systems on an unbounded interval.The numerical exterior algebra framework is most advantageous for numerical solution of differential eigenvalue problems on unbounded domains, where there are significant difficulties in setting up matrix discretizations.The formulation is presented for k-dimensional subspaces of systems on C n with k and n arbitrary, and examples are given for the cases of k = 2 and n = 4, and k = 3 and n = 6, with an indication of implementation details for systems of larger dimension.The theory is illustrated by application to four differential eigenvalue problems on unbounded intervals: hydrodynamic stablity of boundary-layer flow past a compliant surface, the eigenvalue problem associated with the stability of solitary waves, the stability of Bickley jet in oceanography, and

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“…Davey [14], Straughn and Walker [34], Nicodemus et al [33]). In this approach compound matrices are used as coordinates for integrating (1.2).…”

Section: Mathematics Subject Classification (1991): 65l99; 76e99mentioning

confidence: 95%

Numerical exterior algebra and the compound matrix method

Allen

Bridges

2002

Numerische Mathematik

143

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“…They realized that there was still room for improvements. By introducing a weight parameter into Doering-Constantin's formulation, they refined the derivation and obtained a more tight upper bound [3,10] . Meanwhile, Kerswell adopted a different observation angle.…”

Section: Introductionmentioning

confidence: 98%

“…In addition, important nonlinear phenomena, such as global attractors and vortex stretching, also have close ties with this quantity [2] . Unfortunately, full numerical simulations of turbulent flows at high Reynolds numbers are still out of reach today [3] . Therefore, rigorous estimates of the dissipation rate derived directly from the Navier-Stokes equations have received a significant amount of attention from the research community.…”

Section: Introductionmentioning

confidence: 99%

Minimax principle on energy dissipation of incompressible shear flow

Chen

Liu

2010

Appl. Math. Mech.-Engl. Ed.

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The energy dissipation rate is an important concept in the theory of turbulence. Doering-Constantin's variational principle characterizes the upper bounds (maximum) of the time-averaged rate of viscous energy dissipation. In the present study, an optimization theoretical point of view was adopted to recast Doering-Constantin's formulation into a minimax principle for the energy dissipation of an incompressible shear flow. Then, the Kakutani minimax theorem in the game theory is applied to obtain a set of conditions, under which the maximization and the minimization in the minimax principle are commutative. The results explain the spectral constraint of Doering-Constantin, and confirm the equivalence between Doering-Constantin's variational principle and HowardBusse's statistical turbulence theory.

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“…Typical examples include computing bounds for the energy dissipation in shear flows [2][3][4][5][6] and for the net turbulent heat transport in Rayleigh-Bénard convection (e.g. [7,8]).…”

Section: Introductionmentioning

confidence: 99%

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

Fantuzzi

Wynn

2016

Phys. Rev. E

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We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler-Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times, and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface. Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress.

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Variational bound on energy dissipation in plane Couette flow

Cited by 23 publications

References 22 publications

Numerical exterior algebra and the compound matrix method

Numerical exterior algebra and the compound matrix method

Minimax principle on energy dissipation of incompressible shear flow

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

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