A single stable scheme for steady conjugate heat transfer problems

Highlights: • First, in the framework of the Dirichlet-Robin interface condition, it is shown that the normalized distance between the stability bound and the optimal coefficient is equal to the inverse of the numerical Biot number. • Then, we demonstrate that this optimal coefficient is directly connected to two fundamental thermal quantities: the penetration depth and the thermal effusivity. • A thorough examination of the amplification factor (and its derivative) of the coupled thermal problem highlights a small-varying zone. This establishes a reasonable range of values for the coupling coefficient for high thermal fluid-structure interactions. • An example shows that such a coefficient is able to stabilize high thermal fluid-structure interactions (unstable with the optimal coefficient) but also the reasonable value of this coefficient avoids significantly impairing the accuracy of a CHT solution. -- Abstract: The goal of this paper is to propose a single interface treatment, based on the Dirichlet-Robin interface condition to deal with all steady CHT scenarios. These scenarios depend on the so-called numerical Biot number that controls the stability process and the optimal coefficient that ensures, in theory, unconditional stability. It is shown that this coefficient is closely related to fundamental thermal quantities. For very large thermal fluid-solid interactions, the Dirichlet-Robin condition may result in profound stability issues. A thorough examination of the stability behavior has highlighted a narrow and slow-varying stable zone located around the optimal coefficient. This allows us to determine coupling coefficients valid in any case and the reasonable value of these coefficients avoids significantly impairing the accuracy of CHT solutions. A flat plate, partially protected by a thermal barrier coating, is presented as a test case.