A continuous adjoint approach to design optimization in cavitating flow using a barotropic model David A. Boger ⇑ , Eric G. Paterson 1 Computational Mechanics Division, Applied Research Laboratory, The Pennsylvania State University, PO Box 30, State College, PA 16804, USA article info Article history: Received 29 April 2013 Received in revised form 7 May 2014 Accepted 10 June 2014 Available online 19 June 2014 Keywords: Barotropic model Cavitation Continuous adjoint Design optimization Multiphase flow abstract A continuous adjoint method is developed for design optimization in multiphase flow based on a homo- geneous multiphase mixture model. The mixture model consists of variable-density mass and momen- tum equations and uses a barotropic equation of state for the density that depends on only the local static pressure and the vapor pressure of the liquid. In that case, the primary equations are homogeneous, and a conventional hybrid multistage explicit method based on central differencing and second- and fourth-order scalar artificial dissipation can be applied to solve both the primary and adjoint systems. Results are presented for both surface- and volume-based vapor minimization cost functions for a two-dimensional cavitating hydrofoil in which the geometry is parameterized using B-splines. The cost function gradients computed using the adjoint method are shown to compare well with gradients com- puted using the complex-step method. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction Hydropower is generally regarded as the largest source of clean, renewable energy and is receiving renewed interest in terms of both increasing generation capacity and integrating intermittent renew- able sources such as wind and solar power. Within the US, new large-scale development is unlikely, so increased capacity is expected to come from increases in efficiency at existing sites, increasing the use of pumped-storage projects, and encouraging the use of run-of-the-river turbines [1–4]. These goals add new chal- lenges for the designer, who will now be required to improve off- design efficiency and cavitation characteristics without sacrificing maximum power and peak efficiency. Addressing such challenges calls for a ‘‘new approach to hydropower development’’ [1,5]. Multiphase flow effects, including cavitation and air injection, are an important consideration for both expanding the operating range of hydroturbines and making them more environmentally friendly. For example, bulb turbines, which are suitable for large flow rates and low head applications such as in run-of-river power plants, reg- ularly encounter cavitation as a part of their normal operation, mak- ing it necessary to include cavitation effects during the design stage [6]. Kaplan runners typically allow some amount of cavitation dur- ing normal operation primarily as a tradeoff to a deeper setting of the machine. For Francis turbines, cavitation is generally considered intolerable, but in off-design conditions, swirling flow leaving the runner often forms a cavitating vortex rope. Air injection can be used to reduce pressure fluctuations from the vortex rope and other flow sources [7,8]. Hydroturbines also tend to discharge water with low dissolved oxygen content from the bottom of the reservoir, which can severely impact aquatic life downstream, so air injection is also used in that case to improve water quality [9]. Multiphase CFD simulations for hydroturbines have become increasingly common in the last several years [6,9–13]. But even when CFD simulations are able to accurately predict problems with a design, it is usually still incumbent on the designer to correct these problems using traditional design tools and his own experi- ence and intuition. CFD-based design optimization, on the other hand, can provide a direct link between the CFD solution and the required design improvements. Among optimization methods, gra- dient-based methods can be more efficient when the optimum is ‘‘nearby,’’ so they are appropriate given the relative maturity of existing hydroturbines and the constrained design space that often accommodates retrofit operations. Among gradient-based meth- ods, adjoint methods are of interest due to their ability to effi- ciently handle large numbers of design variables. The introduction of continuous adjoint methods for fluid dynamic design is normally attributed to Pironneau, who studied drag minimization for two-dimensional shapes in Stokes [14] and low-Reynolds number flows [15]. Jameson was the first to apply the continuous adjoint method to transonic inviscid flow [16], and his subsequent body of work with his students and colleagues is http://dx.doi.org/10.1016/j.compfluid.2014.06.014 0045-7930/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +1 814 865 1741. E-mail address: boger@psu.edu (D.A. Boger). 1 Present Address: Department of Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall (0203), Blacksburg, VA 24061, USA. Tel.: +1 7176950409. Computers & Fluids 101 (2014) 155–169 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid