Directive-based programming of graphics processing units (GPUs) has recently appeared as a viable alternative to using specialized low-level languages such as CUDA C and OpenCL for general-purpose GPU programming. This technique, which uses directive or pragma statements to annotate source codes written in traditional high-level languages, is designed to permit a unified code base to serve multiple computational platforms and to simplify the transition of legacy codes to new architectures. This work analyzes the popular OpenACC programming standard, as implemented by the PGI compiler suite, in order to evaluate its utility and performance potential in computational fluid dynamics (CFD) applications. Of particular interest is the handling of stencil algorithms, which are an important component of finite-difference and finite-volume numerical methods. To this end, the process of applying the OpenACC Fortran API to a preexisting finite-difference CFD code is examined in detail, and all modifications that must be made to the original source in order to run efficiently on the GPU are noted. Optimization techniques for OpenACC are also explored, and it is demonstrated that tuning the code for a particular accelerator architecture can result in performance increases of over 30%. There are also some limitations and programming restrictions imposed by the API: it is observed that certain useful features of modern Fortran (2003/8) are effectively disabled within OpenACC regions. Finally, a combination of OpenACC and OpenMP directives is used to create a truly cross-platform Fortran code that can be compiled for either CPU or GPU hardware. The performance of the OpenACC code is measured on several contemporary NVIDIA GPU architectures, and a comparison is made between double and single precision arithmetic showing that if reduced precision can be tolerated, it can lead to significant speedups. To assess the performance gains relative to a typical CPU implementation, the execution time for a standard benchmark case (lid-driven cavity) is used as a reference. The OpenACC version is compared against the identical Fortran code recompiled to use OpenMP on multicore CPUs, as well as a highly-optimized C++ version of the code that utilizes hardware aware programming techniques to attain higher performance on the Intel Xeon platforms being tested. Low-level optimizations specific to these architectures are analyzed and it is observed that the stencil access pattern required by the structured-grid CFD code sometimes leads to performance degrading conflict misses in the hardware managed CPU caches. The GPU code, which primarily uses software managed caching, is found to be free from these issues. Overall, it is observed that the OpenACC GPU code compares favorably against even the best optimized CPU version: using a single NVIDIA K20x GPU, the Fortran+OpenACC code is seen to outperform the optimized C++ version by 20% and the Fortran+OpenMP version by more than 100% with both CPU codes running on a 16-core Xeon workstation.