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A new energy-consistent discretization of the viscous dissipation function inincompressible flows is proposed. It is implied by choosing a discretization ofthe diffusive terms and a discretization of the local kinetic energy equationand by requiring that continuous identities like the product rule are mimickeddiscretely. The proposed viscous dissipation function has a quadratic, strictlydissipative form, for both simplified (constant viscosity) stress tensors andgeneral stress tensors. The proposed expression is not only useful inevaluating energy budgets in turbulent flows, but also in natural convectionflows, where it appears in the internal energy equation and is responsible forviscous heating. The viscous dissipation function is such that a consistenttotal energy balance is obtained: the 'implied' presence as sink in the kineticenergy equation is exactly balanced by explicitly adding it as source term inthe internal energy equation.  Numerical experiments of Rayleigh-B\'enard convection (RBC) andRayleigh-Taylor instabilities confirm that with the proposed dissipationfunction, the energy exchange between kinetic and internal energy is exactlypreserved. The experiments show furthermore that viscous dissipation does notaffect the critical Rayleigh number at which instabilities form, but it doessignificantly impact the development of instabilities once they occur.Consequently, the value of the Nusselt number on the cold plate becomes largerthan on the hot plate, with the difference increasing with increasing Gebhartnumber. Finally, 3D simulations of turbulent RBC show that energy balances areexactly satisfied even for very coarse grids; therefore, we consider that theproposed discretization forms an excellent starting point for testing sub-gridscale models.

Energy-consistent discretization of viscous dissipation with application to natural convection flow