Problem C2.4. Laminar Flow around a Delta Wing
Overview
A laminar flow
at high angle of attack around a delta wing with a sharp leading edge and a
blunt trailing edge. As the flow passes the leading edge it rolls up and
creates a vortex together with a secondary vortex. The vortex system remains
over a long distance behind the wing. This problem is aimed at testing
highorder and adaptive methods for the computation of vortex dominated
external flows. Note, that methods which show highorder on smooth solutions
will show about 1^{st} order only on this test case because of reduced
smoothness properties (e.g. at the sharp edges) of the flow solution. Finally,
also hadaptive, and hpadaptive computations can be
submitted for this test case.
Governing
Equations
The governing equation is the 3D NavierStokes equations with a constant ratio of specific
heats of 1.4 and Prandtl number of 0.72. The
viscosity is assumed a constant.
Flow
Conditions
Subsonic viscous flow with M_{∞}= 0.3, and α =
12.5°, Reynolds number (based on a mean cord length of 1) Re=4000.
Geometry
The geometry is a delta wing with a sloped and sharp
leading edge and a blunt trailing edge. The geometry can be seen from Fig. 5
which shows the top, bottom and side view of the half model of the delta wing.
Half
model:
Figure 4: Left: Top, bottom and side view of the half
model of the delta wing. The grid has been provided by NLR within the ADIGMA
project. Right: Streamlines and Mach number isosurfaces
of the flow solution over the left half of the wing and Mach number slices over
the right half. The figures are taken from [LH10].
Reference values
Reference
area: 0.133974596
(half model)
Reference
moment length: 1.0
Moment
line: Quarter chord
Boundary
Conditions
Far field boundary: Subsonic
inflow and outflow
Wing surface: no slip
isothermal wall with _{}.
Requirements
1.
Start
the simulation from a uniform free stream everywhere, and monitor the L_{2}
norm of the density residual. Track the work units needed to achieve steady
state. Compute the drag and lift coefficients c_{d} and c_{l.}
2.
Perform
grid and order refinement studies to find “converged” c_{d} and c_{l}
values.
3.
Plot
the c_{d}
and c_{l}
errors vs. work units.
4.
Study
the numerical order of accuracy according to c_{d} and c_{l}
errors vs. _{}.
(Note, that due to the locally
nonsmooth solution, e.g. at the sharp edges, globally highorder methods will
show about 1^{st} order only.)
5.
Submit
two sets of data to the workshop contact for this case
a)
c_{d} and c_{l}
error vs. work units
b)
c_{d} and c_{l}
error vs _{}
6.
The
following data sets can also be submitted
a.
for
sequences of locally refined meshes (hadaptive mesh refinement) and
b.
for sequences of meshes with locally varying mesh
size and order of convergence (hpadaptive mesh refinement), possibly including
improved data based on a posteriori
error estimation results.
Note,
that here the errorvsworkunit data sets should
take account of the additional work units possibly required

for
auxiliary problems (like e.g. adjoint problems),

for the
evaluation of refinement indicators or mesh metrics,

and for the actual mesh refinement or mesh
regeneration procedure.
References
[LH10] T. Leicht and R. Hartmann. Error
estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow
simulations. J. Comput. Phys.,
229(19), 73447360, 2010.