**Problem C2.2.**** Steady Turbulent Transonic flow over an Airfoil**

**Overview
**

This problem is aimed at testing
high-order methods for a two-dimensional turbulent flow under transonic
conditions with weak shock-boundary layer interaction effects. The test case is
the RAE2822 airfoil Case 9, for which an extensive experimental database exists
[CMcDF79]. The test case has also been investigated numerically by many authors
using low order methods. It was also used in the ADIGMA project (test case
MTC5). The target quantities of interest are the lift and drag coefficients and
the skin friction distribution at one free-stream condition, as described
below.

**Governing
Equations**

The governing equation is the 2D
Reynolds-averaged Navier-Stokes equations with a
constant ratio of specific heats of 1.4 and Prandtl
number of 0.71. The dynamic viscosity is also a constant. The choice of turbulence model is left
up to the participants; recommended suggestions are 1) the Spalart
Allmaras model, and 2) the Wilcox k-omega model or EARSM.

**Flow
Conditions**

The flow conditions are based
on the experimental data base provided in [CMcDF79]. Only Case 9 is retained
for this workshop. The original flow conditions in the wind tunnel experiment
are M_{∞ }= 0.730, angle of attack α = 3.19^{o}, Reynolds
number (based on the reference chord) Re = 6.5x10^{6}. However, in
order to take into account the wind tunnel corrections for comparison with
experimental data the computations for the workshop have to
be made with
corrected flow conditions, namely Mach number M_{∞ }= 0.734, angle of
attack α = 2.79^{o}, with the same Reynolds number, see [A1982]. Laminar
to turbulent transition is fixed at 3% of the chord, on both pressure and
suction side. No further wind tunnel effects are to be modeled.

**Geometry**

The airfoil geometry is given in
[CMcDF79]. Originally the geometry is defined with a set of points. These
points are then used to define a high-order geometry,
which will be available online at the workshop web site.

**Boundary
Conditions**

Adiabatic
no-slip wall on the airfoil surface, free-stream at the farfield
(subsonic inflow / outflow). A sensitivity study must be performed
to find a far field boundary location whose effect on the lift and drag
coefficient is less than 0.01 counts, i.e., 1e-6.

**Grids**

Participants may use their own
grids for the convergence study. However, the geometry definition provided at
the workshop web site should be used such that all the participants will use
the same geometry. The workshop will also provide sample high-order
computational meshes.

**Requirements**

1.
Perform
a convergence study of drag and lift coefficient, *c _{l}* and

a.
Uniform
mesh refinement of the coarsest mesh

b.
Quasi-uniform
refinement of the coarsest mesh, in which the meshes are not necessarily nested
but in which the relative grid density throughout the domain is constant.

c.
Adaptive
refinement using an error indicator (e.g. output-based).

Record the degrees of freedom and
the work units for each data point, where the CPU t=0 corresponds to
initialization with free-stream conditions on the coarsest mesh.

2.
Submit
three sets of data to the workshop contact for this case

a. *c _{l}* &

b. *c _{l}* &

c. Skin
friction distribution on pressure and suction side of the airfoil, compared to
the experiments

Include a
description of the coarsest mesh resolution and of the strategy used for
refinement.

**References**

[CMcDF79] Cook, P.H.,
M.A. McDonald, M.C.P. Firmin, "Aerofoil RAE 2822
- Pressure Distributions, and Boundary Layer and Wake Measurements,"
Experimental Data Base for Computer Program Assessment, AGARD Report AR 138,
1979.

[A1982] NACA0012 Oscillating and Transient Pitching, AGARD-R-702, 1982.