The main purpose of engineering applications for fluid with natural and mixed convection is to control or enhance the flow motion and the heat transfer. In this paper, we use mathematical tools based on optimal control theory to show the possibility of systematically controlling natural and mixed convection flows. We consider different control mechanisms such as distributed, Dirichlet, and Neumann boundary controls. We introduce mathematical tools such as functional spaces and their norms together with bilinear and trilinear forms that are used to express the weak formulation of the partial differential equations. For each of the three different control mechanisms, we aim to study the optimal control problem from a mathematical and numerical point of view. To do so, we present the weak form of the boundary value problem in order to assure the existence of solutions. We state the optimization problem using the method of Lagrange multipliers. In this paper, we show and compare the numerical results obtained by considering these different control mechanisms with different objectives.

The study of turbulent heat transfer in liquid metal flows has gained interest because of applications in several industrial fields. The common assumption of similarity between the dynamical and thermal turbulence, namely, the Reynolds analogy, has been proven to be invalid for these fluids. Many methods have been proposed in order to overcome the difficulties encountered in a proper definition of the turbulent heat flux, such as global or local correlations for the turbulent Prandtl number and four parameter turbulence models. In this work we assess a four parameter logarithmic turbulence model for liquid metals based on the Reynolds Averaged Navier-Stokes (RAN) approach. Several simulation results considering fluids with $Pr=0.01$ and $Pr=0.025$ are reported in order to show the validity of this approach. The Kays turbulence model is also assessed and compared with integral heat transfer correlations for a wide range of Peclet numbers.

This paper deals with boundary optimal control problems for the Navier-Stokes equations and Wilcox turbulence model. In this paper we study adjoint optimal control problems for Navier-Stokes equations to improve the advantages of using simulations where turbulence models play a significant role in designing engineering devices. We assess first distributed optimal control problems with the purpose to control the fluid behavior by injecting a flow on the boundary solid region to obtain a desired control over the fluid velocity and the kinetic turbulence energy in specific parts of the domain. Then, with the same purpose, we use lifting functions and boundary control. For this reason we reformulate the boundary optimal control problem into a distributed problem through a lifting function approach. The stronger regularity requirements which are imposed by standard boundary control approaches can then be avoided. The state, adjoint and control equations are derived and the optimality system solved for some simple cases with a finite element. Furthermore, we propose numerical strategies that allow to solve the coupled optimality system in a robust way for a large number of unknowns. The approach presented in this work is general and can be used to assess different objectives and types of control.

Due to their interesting thermal properties, liquid metals are widely studied for heat transfer applications where large heat fluxes occur. In the framework of the Reynolds-Averaged Navier–Stokes (RANS) approach, the Simple Gradient Diffusion Hypothesis (SGDH) and the Reynolds Analogy are almost universally invoked for the closure of the turbulent heat flux. Even though these assumptions can represent a reasonable compromise in a wide range of applications, they are not reliable when considering low Prandtl number fluids and/or buoyant flows. More advanced closure models for the turbulent heat flux are required to improve the accuracy of the RANS models dealing with low Prandtl number fluids. In this work, we propose an anisotropic four-parameter turbulence model. The closure of the Reynolds stress tensor and turbulent heat flux is gained through nonlinear models. Particular attention is given to the modeling of dynamical and thermal time scales. Numerical simulations of low Prandtl number fluids have been performed over the plane channel and backward-facing step configurations.