**Problem C1.3.**** Flow over a NACA0012 Airfoil**

**Overview
**

This problem is aimed at testing
high-order methods for the computation of external flow with a high-order
curved boundary representation. Both inviscid and
viscous, subsonic and transonic flow conditions will be simulated. The
transonic problem will also test various methods’ shock capturing ability. The
lift and drag coefficients will be computed, and compared with those obtained
with lower order methods.

**Governing
Equations**

The governing equation is the 2D
Euler and Navier-Stokes equations with a constant
ratio of specific heats of 1.4 and Prandtl number of
0.72. For the viscous flow problem, the viscosity is assumed a constant.

**Flow
Conditions**

Three different flow conditions
are considered:

a)
Subsonic
inviscid flow with _{},
and angle of attack _{}

b)
Inviscid
transonic flow with _{},
and _{}

c)
Subsonic
viscous flow with _{},
and _{ }Reynolds number
(based on the chord length) Re = 5,000.

**Geometry**

The NACA0012 airfoil is defined
in the following equation

_{}

where _{}.
The airfoil defined using this equation has a finite trailing edge of .252%.
Various ways exist in the literature to modify this definition such that the
trailing edge has a zero thickness. We adopt the one which modifies the _{ }coefficient, i.e.,

_{}

The airfoil is shown in the
following figure.

Figure
1.3.
NACA0012 Airfoil

**Boundary
Conditions**

Far field boundary: subsonic
inflow and outflow

Airfoil surface: slip wall for
inviscid flow, or no slip adiabatic wall for viscous flow

**Requirements**

1.
The
results for external flow problems obviously depend on the location of the far
field outer boundary and the corresponding boundary condition. A sensitivity
study should be performed to find a far field boundary location whose effect on
the lift and drag coefficients is less than 0.01 counts, i.e., 1e-6.

2.
Start
the simulation from a uniform free stream everywhere, and monitor the L_{2}
norm of the density residual. Compute the work units needed to achieve a steady
state. Compute the lift and drag coefficients *c _{l}* and

3.
Perform
hp-refinement studies to find “converged” *c _{l}* and

1.
Plot
the *c _{l}*
and

2.
Study
the numerical order of accuracy according to *c _{l}* and

3.
Submit
two sets of data to the workshop contact for this case

a.
*c _{l}* and

b.
*c _{l}* and