A Coupled Prediction Scheme for Solving the Navier--Stokes and Convection-Diffusion Equations

Abstract

This paper presents a new algorithm for the numerical solution of the Navier-Stokes equations coupled with the convection-diffusion equation. After establishing convergence of the semidiscrete formulation at each time step, we introduce a new iterative scheme based on a projection method called the coupled prediction scheme. We show that even though the predicted temperature is advected by a velocity prediction which is not necessarily divergence free, the theoretical time accuracy of the global scheme is conserved. From a numerical point of view, this new approach gives a faster and more efficient algorithm compared to the usual fixed-point approaches. Introduction.Heat transfer is an important factor in many fluid dynamics applications. Whenever there is a temperature difference between the fluid and the confining area, heat will be transferred and the flow will be affected in nontrivial ways. Natural convection is such an example in which the driving forces are density variations and gravity (see Jiji [28], for instance). Natural convection flows are observed in different situations such as geophysics, weather, ocean movement and are also exploited in numerous applications such as double-glazed windows, cooling in electronic devices, building insulation, etc.The model is generally described using the Boussinesq approximation. In this approximation, the density of the fluid is assumed to be constant and the gravitational source force (the buoyancy term in the momentum equation) depends on the temperature (Martynenko and Khramtsov [34]).Typically, in the Boussinesq approximation, the coupling between the fluid and the temperature appears through two terms: a source term depending linearly on the temperature, and a convective term based on the velocity of the fluid (see system (2.1)). In this paper we propose a reinforcement of this coupling by adding an explicit dependency to the temperature for the viscosity and the diffusion coefficients. Moreover, since the assumptions on the source term for the momentum equation are not essential (Remark 2.1), we will consider a more general source term. Owing to this departure from the usual Boussinesq equations, the proposed model can be viewed more generally as a thermally coupled Navier-Stokes problem.Thermally coupled incompressible flow problems present two major difficulties requiring special attention: solving the incompressible Navier-Stokes equations on very fine three-dimensional meshes in a reasonable computational time is a difficult task; the strong coupling between the Navier-Stokes and convection-diffusion equations often leads to very complex time dependent dynamics requiring efficient solvers.

Stability of a second‐order method for phase change in porous media flow

“…Apart from proposing a conforming Galerkin method for the space discretisation of the governing equations (momentum, mass, energy, and enthalpy), our contribution focuses also on deriving stability bounds for the discrete solutions. This result generalises other studies, so far focused on the natural or thermal convection of fluids without phase change, see for instance [1,[4][5][6]8]. In particular, this note summarises the recent results reported in [2] and presents three new numerical tests.…”

Shear rate projection schemes for non-Newtonian fluids

“…Finally, as in [36], these shear rate projections could be used to produce new coupled projection methods applicable to generic convection-diffusion phenomenon coupled with Newtonian/Non Newtonian flows (natural convection, coextrusion, etc.). Following the analysis in [36] these methods should offer important gain in performance and accuracy.…”

Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

“…In fact this condition appears when one analyses the term . In fact that regularity was already required in the works by , where Dirichlet condition is applied on the boundary. The condition (2.27) required for uniqueness of solutions obtained in Proposition is restrictive, but for nonlinear problems, it is very rare to obtain uniqueness of solutions without restrictions.…”

“…The literature of heat convection in a liquid medium whose motion is described by the Navier–Stokes or Darcy equations coupled with the heat equation under Dirichlet boundary condition is rich and we refer the reader among others to .…”