The motivation of these lectures is to give practical examples of
implementation of Direct Numerical Simulation and Large-Eddy Simulation
methods, with emphasis on the physics of the results obtained, in a series
of cases ranging from the most fundamental turbulent flow cases
in cubic or parallelepipedic computational domains, to
compressible simulations in curvilinear and/or multi-block grids
related to aeronautical and aerospace applications.
The starting point will be the pseudo-spectral DNS code developed by
Orszag and Patterson (1971) for homogeneous isotropic turbulence.
This will be an opportunity to brush up reminiscences of numerical
methods that will be needed for the rest of the course. We will
therefore briefly introduce the discrete Fourier transform, its inverse and
the undelying aliasing errors, which affect most numerical methods,
in particular in the treatment of non-linear terms in the presence
of a significant level
of energy in the small scales. The Gibbs phenomenon causing oscillations
in the presence of under-resolved gradients will also be evoked.
Among the ways of dealing with aliasing errors, aside 2/3 truncations
or shifted-grids methods which are practically limited to pseudo-spectral
methods, are the different formulations of the non-linear terms
(namely, the divergence, convective and skew-symmetric forms), which are
of general interest and that will reappear later in the compressible
Navier-Stokes equations and their closure in the sense of LES.
An illustration of DNS performed by means of a pseudo-spectral code will
be shown in the case of temporally-growing mixing layers, in which
the formation and the pairing of Kelvin-Helmoltz vortices will be shown,
thanks to both isosurfaces of local entrophy and low pressures, and
vortex lines. The bundling of vortex lines into vortical tubes and their
subsequent stretching will illustrate nicely the Kelvin theorem,
despite the low Reynolds number of the simulation. Depending on the nature
of the initial forcing of the base hyperbolic-tangent profile flow,
3D instabilities develop, yielding secondary streamwise oriented vortices
and larger-scale obliqueness caused by helical pairings.
A natural way of extending to spatially-growing shear flows is to replace
the Fourier expansions in the streamwise direction by a high-order
finite-difference discretization capable of accommodating non-periodic
inflow/outflow boundary conditions. Schemes that are implicit in space
have become very popular for this purpose, for they can be proved to
be more accurate than explicit schemes of the same formal order accuracy,
which can be seen as a faster-than-algebraic-convergence and grant them
the label ``spectral like".
Originally proposed by Krause,
Hirschel and Kordulla (1976) in a fourth-order formulation
under the name Mehrenstellen schemes, they have come back into fashion as
``compact schemes" since the thorough investigation of Lele (1992) who
used a sixth-order accurate compact scheme to perform the first simulations of
compressible spatially-growing mixing layers. One cannot omit
mentioning the ``effective wavenumber", an original vision of discretization
errors which shows the lack of resolving power of most numerical schemes
in the small scales. In the case of incompressible flows,
abandoning the fully pseudo-spectral discretization in one direction makes
the problem of the enforcement of the continuity equation come back, which
makes it inevitable to describe the projection method used in the
applications that will be shown later on. This will be
the right moment to mention discretization in time and introduce the
reduced-storage Runge-Kutta schemes and discuss their stability.
To conclude on the numerical aspects related to the incompressible DNS and LES
of these lectures, a version suited to mixing layers, jets and wakes
of the Orlansky boundary conditions, widely used
in geophysical flow simulations, will be presented.
The LES formalism will then be tackled in the spectral space, yielding
the Heisenberg-Kraichnan spectral eddy-viscosity and eddy-diffusivity
concepts. Two-point stochastic closures shows these to be wavenumber-dependent,
depending on the type of low-pass filter used to separate the
large scales, to be resolved, from the small scales, to be modelled.
This is one of the arguments is favour of hyperviscosities, widely used
in geophysical CFD but less popular in industrial CFD. Anyway, one of the
key targets is the estimation of energy spectra which are local in space,
for which no theoretically satisfactory solutions seem to be applicable to date
to non strictly academic configurations,
although this might change in a relatively near future
due to the progress of multi-resolution methods and adaptive wavelet
decompositions. However, if a certain level of
empiricism can be tolerated, we will show that simple eddy-viscosity
subgrid-scale models such as the structure-function model
and its filtered and selective variants (see
Lesieur and Métais, Ann. Rev. Fluid Mech., 1996) have given quite
interesting and realistic results in a variety of applications, the review
of which will make up most of the remainder of these two talks.
In spatially-growing incompressible mixing layers, LES results at zero
molecular viscosity are much more reminiscent of the famous CalTech experiments
than DNS at comparable numerical costs. One indirect proof that LES
can yield higher-Reynolds number dynamics than DNS on the same computational
grid is the observation of the multiple-stage roll up and pairing of
streamwise vortices conjectured by Corcos for high Reynolds numbers, in
contrast with the scenario of one-stage roll-up due to hyperbolic instability
near the stagnation line inbetween the Kelvin-Helmholtz billows.
Translative instability and helical pairings are also found.
The counterpart of helical pairings in round jets is usually referred to
as alternated pairings and examples of these will be shown, both
from spectral/compact methods and 3rd-order upwind finite volume methods,
showing that spectral-like accuracy might not be essential provided resolution
is sufficient.
Visualizations of the forced transition to turbulence of spatially-growing
boundary layers will then be shown. The numerical method used for this purpose
is a conservative finite-difference scheme for the compressible Navier-Stokes
equations with a fourth-order accurate discretization of the convective terms
and second-order accuracy in time. This proved to be needed to capture
transition, together with a SubGrid-Scale model smart enough not to dampen the
modes of secondary instability. Both in the subharmonic and fundamental
cases (respectively K-type and H-type transition) the shedding of
hairpin vortices from the tip of the $\Lambda$ vortices that result
from the growth of secondary instability modes is observed, yielding
streamwise streaks with a preferential spanwise spacing of about 100 local
wall units. The novelty lies in the fact that this spacing, well known
in the fully turbulent regime, seems to be observed also
well upstream of the peak of skin-friction coefficient, which might indicate
a contribution of some streamwise-independent instability modes existing
both in the transitional and the fully turbulent regimes.
Before moving to higher Mach numbers and more complex geometries,
the additional difficulties of the LES formalism for the compressible
Navier-Stokes equations will be mentioned, and a straightforward way out
be presented, namely, the macro-temperature closure.
Supersonic boundary-layer and channel flows will then be tackled.
Although their transition scenarios are quite different from
the incompressible regime, the turbulent fluctuations do not seem to be
strongly marked by compressibility, as was observed in the 60's by Morkovin,
who drew analogies between stratified and supersonic boundary layers.
Different scalings for the mean and fluctuation profiles will be presented,
although it is known that universality cannot be expected.
On the other hand, the growth of mixing layers is strongly
inhibited beyond a convective Mach number of 0.6, which was detrimental
to the development of supersonic-combustion air-breathing engines such as
SCRAMJETS. In the case of supersonic compression ramp flows, as around the flaps
of space shuttles during their re-entry in the atmosphere,
intense streamwise-oriented heat-flux gradients have been observed, and
LES in a simplified configuration confirm that they might be due to Görtler
vortices. Such vortices have also been observed in LES, on the nozzle of a model
a solid-propellent rocket engine related to the boosters of Ariane V.
These talks will end on situations involving stronger unsteadiness requiring
phase averaging, namely, pulsating channel flows and high-subsonic cavity
flows. The first case is related to sound propagation,
and features an intriguing frequency range in which attenuation is
reduced by turbulence. In cavity flows, it is shown that LES can not only
reproduce the pressure levels of the low-frequency Rossiter modes measured
experimentally, but also match experimental phase-averaged vortex dynamics in the
detached mixing-layer region over the cavity. Depending on the geometry of the cavities,
the LES can also reproduce symmetry breaking and mode-switching instabilities observed experimentally.
Non-reflecting boundary condition is a critical component in developing Computational Aeroacoustics (CAA) algorithms. This presentation will review recent progresses on the application of the Perfectly Matched Layer (PML) technique to CAA problems.
An emerging theory of formulating PML absorbing boundary conditions for the governing equations of fluid dynamics will be discussed. This theory involves two essential steps.
First, a proper space-time transformation is determined and applied to the governing equations;
second, a PML complex change of variables is utilized for the derivation of time domain absorbing boundary conditions.
The proper space-time transformation ensures the stability of the boundary condition. Since the PML equation is perfectly matched (theoretically) to the physical governing equation, numerical reflections are generally very small which makes the PML technique attractive for CAA calculations.
Two- and three-dimensional examples of PML for the Euler equations will be given in this presentation. They include the solution of the linearized Euler equations with uniform and non-uniform mean flows as well as the absorption of convective vortices in the nonlinear Euler equations. Extensions of the PML to the viscous Navier-Stokes equations will also be discussed.
See abstract on the Euromech site.
See abstract on the Euromech site.
The industrial requirement is for a robust numerical system, with an accuracy of the order
of 2dB over a wide range of directions and frequencies, and the ability to include the following effects:
dual streams, with co-flow; large temperature ratios; imperfectly-expanded supersonic jets,
containing shock cells; and subtle nozzle modifications, such as chevrons.
The system developed to match these demands uses Large-Eddy Simulation and extracts t
he far-field noise from the Ffowcs-Williams/Hawkings equation, with a surface tight around the jet.
Many decisions are discussed: grid design, using two blocks and very slow stretching;
use of low-dissipation hybrid differencing (4th order centered / 5th order upwind-biased);
absence of arbitrary inflow perturbations; alterations of the FWH equation,
particularly to immunize it against turbulence crossing the outflow disk of the surface;
approximate representation of chevrons through the inflow condition.
Simulations covering all the effects listed have been performed without major
failure on personal computers, which allow grids with a few million points.
The principal restriction of the method is an upper limit on frequency,
of about 2 for the Strouhal number, which is 300Hz for large engines.
This limit will loosen very slowly. Nevertheless, most of the sound energy is captured,
and the method often achieves its accuracy target for the overall sound pressure level,
even when confronted with very appreciable physical complexity.
See abstract on the Euromech site.
