Sanjiva K. Lele

Compact finite difference schemes with spectral-like resolution

Abstract Finite difference schemes providing an improved representation of a range of scales (spectral-like resolution) in the evaluation of first, second, and higher order derivatives are presented and compared with well-known schemes. The schemes may be used on non-uniform meshes and a variety of boundary conditions may be imposed. Schemes are also presented for derivatives at mid-cell locations, for accurate interpolation and for spectral-like filtering. Applications to fluid mechanics problems are discussed.

Boundary conditions for direct simulations of compressible viscous flows

Abstract Procedures to define boundary conditions for Navier-Stokes equations are discussed. A new formulation using characteristic wave relations through boundaries is derived for the Euler equations and generalized to the Navier-Stokes equations. The emphasis is on deriving boundary conditions compatible with modern non-dissipative algorithms used for direct simulations of turbulent flows. These methods have very low dispersion errors and require precise boundary conditions to avoid numerical instabilities and to control spurious wave reflections at the computational boundaries. The present formulation is an attempt to provide such conditions. Reflecting and non-reflecting boundary condition treatments are presented. Examples of practical implementations for inlet and outlet boundaries as well as slip and no-slip walls are presented. The method applies to subsonic and supersonic flows. It is compared with a reference method based on extrapolation and partial use of Riemann invariants. Test cases described include a ducted shear layer, vortices propagating through boundaries, and Poiseuille flow. Although no mathematical proof of well-posedness is given, the method uses the correct number of boundary conditions required for well-posedness of the Navier-Stokes equations and the examples reveal that it provides a significant improvement over the reference method.

Motion of particles with inertia in a compressible free shear layer

The effects of inertia of a particle on its flow tracking accuracy and particle dispersion are studied using direct numerical simulations of two‐dimensional compressible free shear layers in convective Mach number (Mc) range of 0.2 to 0.6. The results show that particle response is well characterized by τ, the ratio of particle response time to the flow time scale (Stokes’ number). The slip between particle and fluid imposes a fundamental limit on the accuracy of optical measurements such as LDV and PIV. The error is found to grow like τ up to τ=1 and taper off at higher τ. For τ=0.2 the error is about 2%. In the flow visualizations based on Mie scattering, particles with τ>0.05 are found to grossly misrepresent the flow features. These errors are quantified by calculating the dispersion of particles relative to the fluid. The trend in lateral dispersion of particles is similar to that of incompressible flows reported by previous investigators. Overall, the effect of compressibility does not seem to be si...

Sound generation in a mixing layer

The sound generated by vortex pairing in a two-dimensional compressible mixing layer is investigated. Direct numerical simulations (DNS) of the Navier-Stokes equations are used to compute both the near-field region and a portion of the acoustic field. The acoustic analogy due to Lilley (1974) is also solved with acoustic sources determined from the near-field data of the DNS. It is shown that several commonly made simplifications to the acoustic sources can lead to erroneous predictions for the acoustic field. Predictions based on the quadrupole form of the source terms derived by Goldstein (1976a, 1984) are in excellent agreement with the acoustic field from the DNS. However, despite the low Mach number of the flow, the acoustic far field generated by the vortex pairings cannot be described by considering compact quadrupole sources. The acoustic sources have the form of modulated wave packets and the acoustic far field is described by a superdirective model (Crighton & Huerre 1990). The presence of flow-acoustic interactions in the computed source terms causes the acoustic field predicted by the acoustic analogy to be very sensitive to small changes in the description of the source.

Boundary conditions for direct computation of aerodynamic sound generation

Computation of the sound field of a free shear flow requires that the Navier Stokes equations be solved using accurate numerical differentiation and time-marching schemes, with nonreflecting boundary conditions. These conditions are modified for use with nonlinear Navier Stokes computations of open flow problems. At an outflow, vortical structures are found to produce large reflections due to non linear effects. An exit zone just upstream of an outflow where disturbances are significantly attenuated through grid stretching and filtering is developed for use with the nonreflecting boundary conditions; Reflections from vortical structures are decreased by 3 orders of magnitude

Current status of jet noise predictions using large-eddy simulation

A survey of the current applications of large-eddy simulation for the prediction of noise from single stream turbulent jets is given. After summarizing the numerical techniques used, the data predicted by the simulations are given at conditions from subsonic, heated jets to supersonic, unheated jets. Mach numbers between 0.3 and 2.0 are considered. Following the data presentation, an analysis of the trends exhibited by the data is given, with special attention paid to relationship between numerical and/or modeling choices and the prediction accuracy. The data support the conclusion that the most limiting factor in current large-eddy simulations is the thickness of the initial shear layer, which is commonly one order of magnitude thicker than what is found experimentally. There is also a large amount of uncertainty regarding the influence of the subgrid scale model on the predictions. The influence of inflow conditions is discussed in depth. Uncertainties in the inflow conditions currently prohibit the simulations from reliably predicting the potential core length. The centerline evolution of the mean and fluctuating axial velocity is strongly coupled to the resolution of the initial shear layers, but can be made to agree within experimental uncertainty when sufficiently thin initial shear layers are used. The maximum achieved Strouhal number of the sound in the acoustic far field is 1.5-3.0, depending on flow condition; this limit is due to numerical resources. A listing of some of the open questions and future directions concerning jet noise predictions using large-eddy simulation concludes the survey.

Simulation of spatially evolving turbulence and the applicability of Taylor's hypothesis in compressible flow

For the numerical simulation of inhomogeneous turbulent flows, a method is developed for generating stochastic inflow boundary conditions with a prescribed power spectrum. Turbulence statistics from spatial simulations using this method with a low fluctuation Mach number are in excellent agreement with the experimental data, which validates the procedure. Turbulence statistics from spatial simulations are also compared to those from temporal simulations using Taylor’s hypothesis. Statistics such as turbulence intensity, vorticity, and velocity derivative skewness compare favorably with the temporal simulation. However, the statistics of dilatation show a significant departure from those obtained in the temporal simulation. To directly check the applicability of Taylor’s hypothesis, space‐time correlations of fluctuations in velocity, vorticity, and dilatation are investigated. Convection velocities based on vorticity and velocity fluctuations are computed as functions of the spatial and temporal separations. The profile of the space‐time correlation of dilatation fluctuations is explained via a wave propagation model.

A robust high-order compact method for large eddy simulation

We present a high-order compact method for large eddy simulation (LES) of compressible turbulent flows: Numerical solution of Navier-Stokes equations with high-order compact methods has been limited by numerical instabilities caused by ill-resolved features of the flow. This problem is alleviated by a staggered arrangement of conserved variables. Simulations of decaying isotropic turbulence at high Reynolds numbers demonstrate the superiority of the present method over the collocated method. Furthermore, the present method is applicable to the conservative form of the governing equations, thereby allowing total energy conservation, a property usually sacrificed in LES with the collocated method. Boundary schemes that extend the present methodology to non-periodic domains are also presented.

On using large-eddy simulation for the prediction of noise from cold and heated turbulent jets

The results of a series of large-eddy simulations of heated and unheated jets using approximately 106 grid points are presented. The computations were performed on jets at operating conditions originally investigated by Tanna in the late 1970s [H. K. Tanna, “An experimental study of jet noise Part I: Turbulent mixing noise,” J. Sound Vib., 50, 405 (1977)]. Three acoustic Mach numbers are investigated (Uj∕a∞=0.5, 0.9, and 1.5) at cold (constant stagnation temperature) and heated conditions (Tj∕T∞=1.8, 2.7, and 2.3, respectively). The jets’ initial annular shear layers are thick relative to experimental jets and are quasi-laminar with superimposed disturbances from linear instability theory. It is observed that qualitative changes in the jets’ mean- and turbulent field structure with Uj and Tj are consistent with previous experimental data. However, the jets exhibit a faster centerline mean velocity decay rate relative to the existing data, with a corresponding 3–4 % over-prediction of the peak root-mean-sq...

Direct numerical simulation of isotropic turbulence interacting with a weak shock wave

Interaction of isotropic quasi-incompressible turbulence with a weak shock wave was studied by direct numerical simulations. The effects of the fluctuation Mach number M t of the upstream turbulence and the shock strength M 2 1 — 1 on the turbulence statistics were investigated. The ranges investigated were 0.0567 ≤ M t ≤ 0.110 and 1.05 ≤ M 1 ≤ 1.20. A linear analysis of the interaction of isotropic turbulence with a normal shock wave was adopted for comparisons with the simulations. Both numerical simulations and the linear analysis of the interaction show that turbulence is enhanced during the interaction with a shock wave. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. The predictions of the linear analysis compare favourably with simulation results for flows with M 2 t a ( M 2 1 — 1) with a ≈ 0.1, which suggests that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behaviour not reproduced by the linear analysis. Investigation of the budget of the turbulent kinetic energy transport equation shows that this behaviour can be attributed to the pressure transport term. Shock waves were found to be distorted by the upstream turbulence, but still had a well-defined shock front for M 2 t a ( M 2 1 — 1) with a ≈ 0.1). In this regime, the statistics of shock front distortions compare favourably with the linear analysis predictions. For flows with M 2 t > a ( M 2 1 — 1 with a ≈ 0.1, shock waves no longer had well-defined fronts: shock wave thickness and strength varied widely along the transverse directions. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly.