A single stable scheme for steady conjugate heat transfer problems

Abstract

A B S T R A C TThe 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. Stability analysisThe Godunov-Ryabenkii (G-R) stability analysis [27] is very similar to the standard Fourier stability method, although unlike the Fourier method, the G-R method takes into account the boundary conditions. A normal mode solution is thus applied to the case defined by the discrete model problem [17][18], and after elementary transformations, we obtain the following temporal amplification factor

Steady-State Fluid-Solid Mixing Plane to Replace Transient Conjugate Heat Transfer Computations during Design Phase

“…As a consequence, to support the design of efficient impingement cooling within the compressor development design-cycle, a procedure to compute the time-average effect of the cooling on the rotating disk would be sufficient. Numerical algorithms for steady state CHT coupling can be found, e.g., in Reference [17]. This algorithm uses a weak coupling of the interface heat fluxes, i.e., the interface heat fluxes are only balanced at steady state and the solution is then advanced.…”

A numerical predictive model for conjugate heat transfer with radiation

“…Second, if continuity of flux and temperature is mathematically ensured at convergence, as just mentioned, it is far more complex in a discrete formulation and multiple solutions depending on the coupling coefficients can be obtained. This issue was largely discussed in [37] without radiation. With radiation, we have decided to strengthen the continuity, at every step of the process by imposing a single temperature value at the interface…”

“…in (9) which corresponds to a "perfect" conduction in the solid. When the solid conduction departs significantly from that assumption, in a solid ceramic material for instance, effective changes can be made to extend the scope of the D-R conditions so as to continue to retain this single interface treatment [37] [38].…”

“…, it can be easily shown that ( 22) tends toward a finite value (see [37] for more details) given by…”