Shape Optimization for Drag Minimization Using the Navier-Stokes Equation

• Main Author: Chukwudozie, Chukwudi Paul
• Other Authors: Walker, Shawn
• Format: Others
• Language: en
• Published: LSU 2015
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-10202015-221459/

Fluid drag is a force that opposes relative motion between fluid layers or between solids and surrounding fluids. For a stationary solid in a moving fluid, it is the amount of force necessary to keep the object stationary in the moving fluid. In addition to fluid and flow conditions, pressure drag on a solid object is dependent on the size and shape of the object. The aim of this project is to compute the shape of a stationary 2D object of size 3.5 m2 that minimizes drag for different Reynolds numbers. We solve the problem in the context of shape optimization, making use of shape sensitivity analysis. The state variables are fluid pressure and velocity modeled by the Navier-Stokes equation with cost function given by the fluid drag which depends on the state variables. The geometric constraint is removed by constructing a Lagrangian function. Subsequent application of shape sensitivity analysis on the Lagrangian generates the shape derivative and gradient. Our optimization routine uses a variational form of the sequential quadratic programming (SQP) method with the Hessian replaced by a variational form for the shape gradient. The numerical implementation is done in Python while the open source finite element package, FEniCS, is used to solve all the partial differential equations. Remeshing of the computational domain to improve mesh quality is carried out with the open source 2D mesh generator, Triangle. Final shapes for low Reynolds numbers resemble an american football while shapes for moderate to high Reynolds numbers are more streamlined in the tail end of the object than at the front.