Stability, convergence and optimization of interface treatments in weak and strong thermal fluid-structure interaction

Abstract

This paper presents the stability, convergence and optimization characteristics of interface treatments for steady conjugate heat transfer problems. The Dirichlet-Robin and Neumann-Robin procedures are presented in detail and compared on the basis of the Godunov-Ryabenkii normal mode analysis theory applied to a canonical aero-thermal coupling prototype. Two fundamental parameters are introduced, a "numerical" Biot number that controls the stability process and an optimal coupling coefficient that ensures unconditional stability. This coefficient is derived from a transition of the amplification factor. A comparative study of these two treatments is made in order to implement numerical schemes based on adaptive and local coupling coefficients, with no arbitrary relaxation parameters, and with no assumptions on the temporal advancement of the fluid domain. The coupled numerical test case illustrates that the optimal Dirichlet-Robin interface conditions provide effective and oscillation-free solutions for low and moderate fluidstructure interactions. Moreover, the computation time is slightly shorter than the time required for a CFD computation only. However, for higher fluid-structure interactions, a Neumann interface condition on the fluid side presents good numerical properties so that no relaxation coefficients are required.

A single stable scheme for steady conjugate heat transfer problems

“…The stability condition 1 ) , ( < f z g α applied to (2) leads, after some basic calculus manipulations [16] [18], to a lower stability bound…”

“…This number can also be estimated as a function of a normalized Fourier number D [16] [18] as indicated in (8) Note that the expression of the optimal coefficient given by Eq. (7) contains only "fluid parameters".…”

“…However, a good choice of the coupling coefficient is needed. Figure 2 displays the temperature profile at the leading edge of a flat plate in a CHT computation (see details in [16]). Two coupling coefficients have been used : theoretically leads to a monotone and fast convergence as predicted by the stability analysis.…”

“…On the basis of the above, it is clear that the coupling coefficient must be chosen such that (16) As can be seen from Figure 1, the slope of the right half-curve is very steep at…”

“…Indeed, the Dirichlet-Robin condition is widely used and it is judicious to be able to extend its application. On the other hand, although it is natural to make use of a Neumann condition when the Biot number is high [10] [16], this condition must be implemented according to a criterion that is difficult to estimate. Finally, in the presence of a very heterogeneous surface, for example a metallic wall partially protected by a thermal barrier coating (TBC), one is then confronted with a particularly delicate situation involving a succession of disparate thermal problems much easier to deal with if a single interface condition can be used.…”