Control of Euler and Navier-Stokes Equations: Applications to Optimal Shape Design in Aeronautics

Abstract-Recently, shape design has been acquiring increasing relevance within the aeronautical community, both in the industrial sector as well as in research centers. This increasing interest stems from the speed and precision of computations which are being achieved with the use of CFD (Computational Fluid Dynamics) techniques. Optimal shape design aims at finding the minimum of a functional by controlling the PDE modelling the flow using surface (domain boundaries) deformation techniques. As a solution to the enormous computational resources required for classical shape optimization of functionals of aerodynamic interest, the best strategy is to use in a systematic way methods inspired in control theory. To do this one assumes that a given aerodynamic surface (typically a wing) is an element that produces lift or drag by controlling or modifying the flow. In the present paper we will restrict our attention to optimal shape design in systems governed by the Euler or Navier-Stokes equations. We first review some standard facts on control theory applied to optimal shape design, and recall the Euler and Navier-Stokes equations in aerodynamical problems. We then study the adjoint formulation, providing a detailed exposition of how the derivatives of functionals may be obtained. Finally, the application of the Level Set methodology to optimal shape design is reviewed

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Department of Nuclear EngineeringMolten salts are promising heat transfer media with high boiling temperature, which have advantages in single-phase heat transfer. Especially, the energy engineering systems operated in high temperature range like solar energy system or advanced nuclear system are strongly interested in using molten salts as heat transport or heat storage media. Therefore, there has been many researches to investigate the heat transfer behavior of molten salt system. However, still the fundamental knowledge on the single-phase heat transfer of molten salt is insufficient to assess its heat transfer performance. In addition, there are only few studies on the heat transport system using molten salt in both natural and forced circulation. Thus, this study focuses on the study on single-phase heat transfer behavior of molten salt, which is characterized by its high-Prandtl number, in both natural and forced circulation system with numerical and experimental approaches. The first part of this paper includes the feasibility test on passive heat transport system using high-Prandtl number. The sensitivity analysis method is adopted to represent system reliability especially for high-Prandtl number fluid. The reliability assessment of the natural circulation system can give fundamental insight to the design of various passive safety systems for the advanced nuclear reactors. Especially for the passive system, the weak driving force requires accurate assessment of reliability and performance, or it can give large uncertainty during the operation of system. Here the reliability assessment employs one of efficient techniques referred as adjoint-based sensitivity method to test the heat transport system using high-Prandtl number fluid. The conservative governing equations in the natural circulation inside a closed rectangular loop were established, and its adjoint system were developed based on the Lagrangian approach. The developed adjoint system showed reasonable accuracy in the sensitivity analysis with more efficient computational effort as expected. Based on the developed adjoint sensitivity system, the reliability of natural circulation of molten salt simulant inside the closed loop is tested. The general sensitivity analyses are performed with different design parameters which were categorized into fluid property, geometric parameter, heater and cooler conditions, pressure drop parameters, and Nusselt correlation. Three different system conditions are imposed to investigate the effects of implementation of temperature-dependent fluid property, the orientations of heat exchanger, and operating temperature range, on the entire system reliability. It is found that the variation of fluid properties with respect to the temperature gives great effect on the reliability of heat transfer performance in the system. The further assessment on Nusselt correlation also verifies the importance of property variation with respect to the temperature in evaluating heat transfer performance. Thus, the reliable assessment on the heat transfer performance requires the consideration of property variation. Especially, the heat transport system using high-Prandtl number fluid should aware of property variation in estimating heat transfer performance, since the drastic temperature drop takes place near the heat transfer surface due to thin thermal boundary layer. The following parts describe the experimental study on heat transfer behaviors of high-Prandtl number fluid to give reliability to the assessment of high-Prandtl number fluid system. In specific, the second part includes the investigation of the convective heat transfer phenomena of high-Prandtl number oil and the third includes the experimental work using high-Prandtl number salt. In the previous studies of same research group, the distinct heat transfer behavior of high Prandtl number oil was reported in the natural circulation system. The present study extends the experimental work to the forced convective heat transfer and discusses the unique feature of heat transfer behavior of high-Prandtl number oil. Specifically, the convective heat transfer performance of high-Prandtl number oil in transition flow regime isn???t fully understood with the previous correlations. It is suggested that the distinct heat transfer feature of high Prandtl number fluid is attributed to the existence of local natural convection in radial direction, which is induced by the weak thermal diffusivity and resulting large radial temperature difference of high Prandtl number fluid. The theoretical discussion on the local natural convection of high-Prandtl number oil verifies its existence in the heat transfer between fluid and heated wall. Based on the discussion, the newly developed correlation for high Prandtl number takes the local natural convection into consideration by adding Grashof number into the general convective heat transfer correlation. The proposed correlation well agrees with experimental data from the present work as well as the previous work, which demonstrates the effect of local natural convection on the convective heat transfer performance of high Prandtl number fluid. Finally, the experimental facility using heat transport salt is established and natural circulation test is performed giving opportunity to further discussion on distinct heat transfer behavior of high-Prandtl number fluid.clos