# Reduced order models in fluid-thermal problems with variable geometry and efficient generation of multi-parameter data bases

- BACHE, ELLIOTT

- Ángel Velázquez López Director
- Jose Manuel Vega de Prada Co-director

Defence university: Universidad Politécnica de Madrid

Fecha de defensa: 16 December 2011

- José Luis Montañés García Chair
- Carlos Martel Escobar Secretary
- Elena Beatriz Martín Ortega Committee member
- Luis Francisco López Bonilla Committee member
- Mario Sánchez Sanz Committee member

Type: Thesis

## Abstract

Today in age, Computational Fluid Dynamics (CFD) calculations are the main tool companies use to design new producís and optimize existing products. Since fewer calculations are needed in the latter stages of the design cycle, CFD calculations in these stages are allowed to be more complex and expensive (time-wise as well as computational-wise). High fidelity methods are often used for these calculations. On the other hand, the conceptual design phase requires a far greater amount of calculations, which means that they cannot be slow or very demanding. In this phase, low fidelity methods are often used. One of the objectives of this thesis is to reduce the CFD calculation times for conceptual design while retaining the same precisión as in the higher fí-delity methods. To achieve this goal, a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD) has been created. The test problem to which the ROM has been applied is the 2D backward-facing step. Various studies have been previously conducted to créate ROMs for this problem, and their authors have made the first steps towards fast solutions with good accuracy. The ROMs created in the previous studies were able to reduce the calculation time for solutions to about 1% of the CFD calculation time. These ROMs generally treated two parameters such as the Reynolds number of the flow and the wall temperahire just downstream of the step. Unfortunately, they were not able to take into account a geometry parameter. A geometry parameter introduces an additional dificulty when creating the ROM. This stems from the fact that the POD used to créate the ROM requires identical geometries for all of the system states within the param¬eter range. In this thesis, this obstacle is overeóme by means of a virtual mesh. The resulting ROM is able to successfully recréate the solutions to any test point in the parameter range while reducing the calculation time to about 2% of the CFD calculation time. While the reduction in calculation times for this system was quite satisfactory (3 minutes instead of 6 hours), an even smaller calculation time would be preferable. Since the conceptual designer would be constantly varying parameters and performing calculations, it would be most fitting to have calculation times on the order of seconds. With this in mind, a ROM is created for the previous system (this time without the (variable geometry) based on a different method for minimizing the residual on the equations. In the previous (variable geometry) ROM, a Genetic Algorithm (GA) was used to minimize the residual on the governing equations of the ROM, which were chosen to coincide as nearly as possible with the CFD equations. In the new formulación, the residual on the equations is minimized by means of a gradient-like method coupled with a continuation method. The resulting calculations are precise as well as rapid (about 2 seconds are required for each calculation, which is 0.0001 times the original CFD calculation time of 6 hours). The governing equations used in CFD calculations are often modifíed to include extra terms (numerical devices) to help convergence. These terms deviate from the original equations by a small amount although in some localized zones of the flow, the differences can be large. While the ROM that uses a gradient-like method requires specific knowledge of the additional terms to obtain solutions cióse to the CFD solutions, a better residual point selection would allow the ROM to reach the same solution as for CFD. Without explicit knowledge of these terms, the ROM would possibly find a solution to the defíned equations and boundary conditions. On the other hand, it could also possibly find a spurious solution. With the point selec¬tion developed in this thesis, the ROM is consistently able to find the CFD solution with a calculation time of about 2 seconds as before.