**Problem C3.4.**** Heaving and Pitching Airfoil in Wake**

**Overview
**

This problem is aimed at testing
the accuracy and performance of high-order flow solvers for problems with
deforming domains. An oscillating cylinder produces vortices that interact with
an airfoil performing a typical flapping motion. The time histories of the drag
coefficient on both the cylinder and on the airfoil are used as metrics, and two
test cases corresponding to different stream-wise positions of the airfoil are
studied.

**Governing
Equations**

The governing equations for this
problem are the 2D compressible Navier-Stokes
equations with a constant ratio of specific heats equal to 1.4 and a Prandtl number of 0.72.

**Flow
Conditions**

The free-stream has a mach number
(_{}) of 0.2 and an angle of
attack (a) of _{}. The Reynolds number based
on the chord of the airfoil is 1000 (or, equivalently, 500 based on the
diameter of the cylinder).

**Geometry**

The geometry consists of a cylinder of diameter centered
at the origin, and a NACA0012 airfoil with chord length positioned
downstream from the cylinder. The geometry is given by the modification of the _{ }coefficient to
give zero trailing edge thickness:

The center of the cylinder
and the 1/3 chord of the airfoil are separated by a distance . Both bodies are heaving in an oscillating motion , where and . In addition, the airfoil it pitching about its 1/3
chord by an angle , where the amplitude and the
phase shift .

**Requirements**

1. Perform the indicated simulation to an elapsed time of
100 time units (or 20 periods), for the two test cases **(a)** and **(b) **. Maintain a time history of the drag coefficients on
the cylinder and on the airfoil, and calculate the time-averaged drag
coefficients for the last (10^{th}) period only. Perform a grid/timestep convergence study to get the time-averaged values
accurate to within 1 drag count. Record the work units.

2. Provide the work units, the converged time history of
lift and drag on the wing, *nDOF*s in the solution, and the distance to the far field
boundary for each case. Submit this data to the workshop contact.