Problem C2.1. Unsteady Viscous Flow over Tandem Airfoils
Overview
This problem is aimed to test the
unsteady interaction of a vortex with a solid wall. Specifically, several cases
are defined that address both unsteady pressure generation and viscous
separation in a 2D framework by using a pair of airfoils. The geometry remains
relatively simple and provides a test bed for several different types of
analysis. In general, the time history of the lift coefficient on the aft
airfoil will be used as a metric. Other quantities to assess would be the pressure
distribution on the two airfoils and the total circulation in the problem
obtained by integrating the vorticity throughout the
domain.
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. Compressibility is not anticipated
to be a significant player in these problems. Specific conditions will be
provided with each subcase.
Flow
Conditions
Two different problems are defined. In each of these cases, the Mach
number (_{}) is 0.2, the angle of
attack (a) is _{}, and the Reynolds number
based on the chord of one of the airfoils is 10^{4}.
·
Case A examines the evolution of the flow
field from a prescribed initial solution that is C^{1}. In this condition, d is the distance to the closest wall, which is often required for
turbulence models. Density and pressure are initialized to their free stream values.
_{}
with _{}
·
Case B is closer to an impulsive start
condition. Let _{} with _{ }Density and pressure are
again set to uniform free stream values.
_{}
with _{}
Geometry
The basic geometry for this case is two relatively
positioned NACA0012 airfoils. As done in problem C.1.3, the airfoil geometry is
given by
_{}
The leading airfoil is rotated by d about (0.25, 0) while the trailing airfoil is translated by (1 + d_{sep},
-d_{off} ).
For the present case, take _{}, d_{sep} = 0.5, and d_{off} = 0. The far field
boundary can be determined such that the steady state lift coefficient of both
airfoils (based on a reference length of 1 chord) varies less than 0.01 counts.
Figure
2.1 Geometry for Tandem Airfoils
Requirements
1. Perform a sensitivity study for the far field boundary
condition distance required to have
the total
lift and drag on both airfoils vary less than 0.01 counts.
2. Perform the indicated simulation to an elapsed time of
20 convective lengths (_{}), maintaining a time
history of the lift and drag coefficients on the aft airfoil. Perform a time
step/mesh resolution study to ensure the lift coefficient is resolved to within
1 count throughout the time interval. Track the work units for this simulation.
(Optional: repeat at different orders of accuracy)
3. Provide work units, the converged time history of lift
and drag on the aft wing, nDOFs in the solution, and the distance to the far field
boundary for each case. Submit this data to the workshop contact. Use _{} to non-dimensionalize
time.