Scientific and Technical Aerospace Reports Volume 39 April 6, 2001

20010022648 Stanford Univ., Center for Turbulence Research, Stanford, CA USA
A New Method for Accurate Treatment of Flow Equations in Cylindrical Coordinates Using Series Expansions
Constantinescu, G.S., Stanford Univ., USA; Lele, S. K., Stanford Univ., USA; Annual Research Briefs - 2000: Center for Turbulence
Research; December 2000, pp. 199-210; In English; See also 20010022631
Contract(s)/Grant(s): NAG2-1373; No Copyright; Avail: CASI; A03, Hardcopy; A03, Microfiche
The motivation of this work is the ongoing effort at the Center for Turbulence Research (CTR) to use large eddy simulation
(LES) techniques to calculate the noise radiated by jet engines. The focus on engine exhaust noise reduction is motivated by the
fact that a significant reduction has been achieved over the last decade on the other main sources of acoustic emissions of jet
engines, such as the fan and turbomachinery noise, which gives increased priority to jet noise. to be able to propose methods to
reduce the jet noise based on results of numerical simulations, one first has to be able to accurately predict the spatio-temporal
distribution of the noise sources in the jet. Though a great deal of understanding of the fundamental turbulence mechanisms in
high-speed jets was obtained from direct numerical simulations (DNS) at low Reynolds numbers, LES seems to be the only realistic
available tool to obtain the necessary near-field information that is required to estimate the acoustic radiation of the turbulent
compressible engine exhaust jets. The quality of jet-noise predictions is determined by the accuracy of the numerical method that
has to capture the wide range of pressure fluctuations associated with the turbulence in the jet and with the resulting radiated noise,
and by the boundary condition treatment and the quality of the mesh. Higher Reynolds numbers and coarser grids put in turn a
higher burden on the robustness and accuracy of the numerical method used in this kind of jet LES simulations. As these calculations
are often done in cylindrical coordinates, one of the most important requirements for the numerical method is to provide a
flow solution that is not contaminated by numerical artifacts. The coordinate singularity is known to be a source of such artifacts.
In the present work we use 6th order Pade schemes in the non-periodic directions to discretize the full compressible flow equations.
It turns out that the quality of jet-noise predictions using these schemes is especially sensitive to the type of equation treatment
at the singularity axis. The objective of this work is to develop a generally applicable numerical method for treating the singularities
present at the polar axis, which is particularly suitable for highly accurate finite-differences schemes (e.g., Pade schemes) on
non-staggered grids. The main idea is to reinterpret the regularity conditions developed in the context of pseudo-spectral methods.
A set of exact equations at the singularity axis is derived using the appropriate series expansions for the variables in the original
set of equations. The present treatment of the equations preserves the same level of accuracy as for the interior scheme. We also
want to point out the wider utility of the method, proposed here in the context of compressible flow equations, as its extension
for incompressible flows or for any other set of equations that are solved on a non-staggered mesh in cylindrical coordinates with
finite-differences schemes of various level of accuracy is straightforward. The robustness and accuracy of the proposed technique
is assessed by comparing results from simulations of laminar forced-jets and turbulent compressible jets using LES with similar
calculations in which the equations are solved in Cartesian coordinates at the polar axis, or in which the singularity is removed
by employing a staggered mesh in the radial direction without a mesh point at r = 0.
Author
Large Eddy Simulation; Numerical Analysis; Series Expansion; Cylindrical Coordinates
20010022649 Stanford Univ., Center for Turbulence Research, Stanford, CA USA
An Evaluation of a Conservative Fourth Order DNS Code in Turbulent Channel Flow
Gullbrand, Jessica, Stanford Univ., USA; Annual Research Briefs - 2000: Center for Turbulence Research; December 2000, pp.
211-218; In English; See also 20010022631; No Copyright; Avail: CASI; A02, Hardcopy; A03, Microfiche
Direct numerical simulation (DNS) and large eddy simulation (LES) of turbulent flows require a numerical method that is
able to capture a wide range of turbulent length scales. In DNS, all the turbulent length and time scales are resolved. In LES, the
large energy carrying length scales of turbulence are resolved and the small structures are modeled. The separation of large and
small scales is done using a filtering procedure that is applied to the Navier-Stokes equations. The effect of the small scale turbulence
on the resolved scales is modeled using a subgrid scale (SGS) model. Widely used SGS models are the scale similarity models
by Bardina et al. and Liu et al and the eddy viscosity based dynamic Smagorinsky model proposed by Germano et al.. In general,
scale similarity models do not dissipate enough energy, and eddy viscosity models do not carry enough stress. SGS models tend
to over- and underpredict velocity fluctuations. SGS models typically use information from the smallest resolved length scales
to model the stresses of the unresolved scales. Therefore, it is of great importance that these resolved length scales are captured
accurately. This requires that the numerical error of the scheme is sufficiently small. One approach is to use high order finite difference
schemes. However, high order schemes require that the differentiation and filtering operations commute. This is generally
not the case in inhomogeneous flow fields where the required smallest resolved length scales vary throughout the flow field. For
this situation, the filter width varies introducing commutation errors of O(del(exp 2)) where del represents the filter width. High
order finite difference schemes with good conservation properties and three-dimensional commutative filters have been developed
by Morinishi et al., Vasilyev et al. and Vasilyev. These discretization schemes can conserve energy, which ensures a stable
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