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NASA NTRS1993

A representation for the turbulent mass flux contribution to Reynolds-stress and two-equation closures for compressible turbulence

The turbulent mass flux, or equivalently the fluctuating Favre velocity mean, appears in the first and second moment equations of compressible kappa-epsilon and Reynolds stress closures. Mathematically it is the difference between the unweighted and density-weighted averages of the velocity field and is therefore a measure of the effects of compressibility through variations in density. It appears to be fundamental to an inhomogeneous compressible turbulence, in which it characterizes the effects of the mean density gradients, in the same way the anisotropy tensor characterizes the effects of the mean velocity gradients. An evolution equation for the turbulent mass flux is derived. A truncation of this equation produces an algebraic expression for the mass flux. The mass flux is found to be proportional to the mean density gradients with a tensor eddy-viscosity that depends on both the mean deformation and the Reynolds stresses. The model is tested in a wall bounded DNS at Mach 4.5 with notable results.

Ristorcelli, J. R.↗
NASA NTRS1993

Dynamic localization and second-order subgrid-scale models in large eddy simulations of channel flow

The objective here is to test the Dynamic Localization (DL) model in a wall-bounded channel flow for numerical stability and accuracy of results. Algebraic stress models suggest that the model for the residual subgrid-scale (SGS) Reynolds stress and scalar flux should generally have terms comprising most of the unique products of the resolved strain (S) and rotation (R) tensors with S and the resolved scalar gradient. The standard dynamic SGS model uses a simple (Smagorinsky) base model for the residual Reynolds stress, which is made proportional to S, and down-gradient base models for residual scalar fluxes; these correspond to the lowest, 'first-order' terms in algebraic stress models. Temporal scaling terms in these base models are formed from the magnitude of the resolved strain rate. While this is appropriate for simple shear flows, it may not be appropriate for more complicated flows (relevant to geophysical and astrophysical problems) that include any combination of shear, rotation, buoyancy, etc. On the other hand, the coefficient in the dynamic SGS model readily adjusts itself to different flow conditions and may adequately take account of these effects without the need for more complicated base models. Cabot (1993) has begun to test the dynamic SGS model in buoyant flows (Rayleigh-Benard and internally heated convection) with and without buoyancy terms explicitly included in the scaling terms of the base model; no great differences were found in large eddy simulation (LES) results for the different base model scalings. The second objective in this work is to test base models with additional, 'second-order' terms (e.g., S(sup 2) and RS for the residual Reynolds stress). These terms have been found to improve large-scale flow predictions by kappa-epsilon models in the presence of rotation and shear. Second-order base models will be tested here in the LES of channel flow with and without solid-body rotation and compared with results from the standard first-order base models to determine if there are significant differences or improvements in results that would warrant the added complexity of the second-order base models.

Cabot, William H.↗
NASA NTRS1993

The production of premixed flame surface area in turbulent shear flow

In the present work, we use three-dimensional Direct Numerical Simulation (DNS) of premixed flames in turbulent shear flow to characterize the effect of a mean shear motion on flame surface production. The shear is uniform in the unburnt gas, and simulations are performed for different values of the mean shear rate, S. The data base is then used to estimate and compare the different terms appearing in the Sigma-equation as a function of S. The analysis gives in particular the relative weights f the turbulent flow and mean flow components, a(sub T) and A(sub T), of the flame surface production term. This comparison indicates whether the dominant effects of a mean flow velocity gradient on flame surface area are implicit and scale with the modified turbulent flow parameters, kappa and epsilon, or explicit and scale directly with the rate of deformation.

Trouve, A.↗
NASA NTRS1993

LRAT: Lightning Radiative Transfer

In this report, we extend to cloud physics the work done for single and multiple scattering of electromagnetic waves. We consider the scattering of light, visible or infrared, by a spherical cloud represented by a statistically homogeneous ensemble of configurations of N identical spherical water droplets whose centers are uniformly distributed in its volume V. The ensemble is specified by the average number rho of scatterers in unit volume and by rho f(R) with f(R) as the distribution function for separations R of pairs. The incident light, vector-phi(sub 0) a plane electromagnetic wave with harmonic time dependence, is from outside the cloud. The propagation parameter kappa(sub 0) and the index of refraction eta(sub 0) determine physically the medium outside the distribution of scatterers. We solve the interior problem separately to obtain the bulk parameters for the scatterer equivalent to the ensemble of spherical droplets. With the interior solution or the equivalent medium approach, the multiple scattering problem is reduced to that of an equivalent single scatterer excited from outside illumination. A dispersion relation which determines the bulk propagation parameter K and the bulk index of refraction eta of the cloud is given in terms of the vector equivalent scattering amplitude vector-G and the dyadic scattering amplitude tilde-g of the single object in isolation. Based on this transfer model we will have the ability to consider clouds composed of inhomogeneous distribution of water and/or ice particles and we will be able to take into account particle size distributions within the cloud. We will also be able to study the effects of cloud composition (i.e., particle shape, size, composition, orientation, location) on the polarization of the single or the multiple scattered waves. Finally, this study will provide a new starting point for studying the problem of lightning radiative transfer.

Phanord, Dieudonne D.↗
NASA NTRS1994

A vorticity dynamics based model for the turbulent dissipation: Model development and validation

A new model dissipation rate equation is proposed based on the dynamic equation of the mean-square vorticity fluctuation for large Reynolds number turbulence. The advantage of working with the vorticity fluctuation equation is that the physical meanings of the terms in this equation are more clear than those in the dissipation rate equation. Hence, the model development based on the vorticity fluctuation equation is more straightforward. The resulting form of the model equation is consistent with the spectral energy cascade analysis introduced by Lumley. The proposed model dissipation rate equation is numerically well behaved and can be applied to any level of turbulence modeling. It is applied to a realizable eddy viscosity model. Flows that are examined include: rotating homogeneous shear flows; free shear flows; a channel flow and flat plate boundary layers with and without pressure gradients; and backward facing step separated flows. In most cases, the present model predictions show considerable improvement over the standard kappa-epsilon model.

Shih, Tsan-Hsing↗
NASA NTRS1994

Computational prediction of isolated performance of an axisymmetric nozzle at Mach number 0.90

An improved ability to predict external propulsive performance was incorporated into the three-dimensional Navier-Stokes code PAB3D. The improvements are the ability to account for skin friction and external pressure forces. Performance parameters for two axisymmetric supersonic cruise nozzle configurations were calculated to test the improved methodology. Internal and external flow-field regions were computed using a two-equation kappa-epsilon turbulent viscous-stress model. The computed thrust-minus-drag ratios were within 1 percent of the absolute level of experimental data and the trends of data were predicted accurately. The predicted trend of integrated nozzle pressure drag matched the trend of the integrated experimental pressure drag over a range of nozzle pressure ratios, but absolute drag levels were not accurately predicted.

Carlson, John R.↗
NASA NTRS1994

Computational fluid dynamics analysis in support of the simplex turbopump design

Simplex is a turbopump that is being developed at NASA/Marshall Space Flight Center (MSFC) by an in-house team. The turbopump consists of a single-stage centrifugal impeller, vaned-diffuser pump powered by a single-stage, axial, supersonic, partial admission turbine. The turbine is driven by warm gaseous oxygen tapped off of the hybrid motor to which it will be coupled. Rolling element bearings are cooled by the pumping fluid. Details of the configuration and operating conditions are given by Marsh. CFD has been used extensively to verify one-dimensional (1D) predictions, assess aerodynamic and hydrodynamic designs, and to provide flow environments. The complete primary flow path of the pump-end and the hot gas path of the turbine, excluding the inlet torus, have been analyzed. All CFD analyses conducted for the Simplex turbopump employed the pressure based Finite Difference Navier-Stokes (FDNS) code using a standard kappa-epsilon turbulence model with wall functions. More detailed results are presented by Garcia et. al. To support the team, loading and temperature results for the turbine rotor were provided as inputs to structural and thermal analyses, and blade loadings from the inducer were provided for structural analyses.

Garcia, Roberto↗
NASA NTRS1994

Effects of turbulence compressibility and unsteadiness in compression corner flow

The structure of the separated flow region over a 20 degree compression corner at a free-stream Mach number of 2.84 is investigated computationally using a Reynolds averaged Navier Stokes (R.A.N.S.) solver and kappa-epsilon model. At this Mach number and ramp angle, a steady-state recirculation region of order delta(sub o) is observed, with onset of a 'plateau' in the wall pressure distribution near the corner. At lower ramp angles, separation is negligible, while at an angle of 24 degrees, separation regions of length 2 delta(sub o) are expected. Of interest here is the response of the mathematical model to inclusion of the pressure dilatation term for turbulent kinetic energy. Compared with the experimental data of Smits and Muck (1987), steady-state computations show improvement when the pressure dilatation term is included. Unsteady computations, using both unforced and then forced inlet conditions, did not predict the oscillatory motion of the separation bubble as observed in laboratory experiments. An analysis of the separation bubble oscillation and the turbulent boundary layer (T.B.L.) frequencies for this flow suggests that the bubble oscillations are of nearly the same order as the turbulent frequencies, and therefore difficult for the model to separate and resolve.

Brankovic, A.↗
NASA NTRS1994

Role of pressure diffusion in non-homogeneous shear flows

A non-local model is presented for approximating the pressure diffusion in calculations of turbulent free shear and boundary layer flows. It is based on the solution of an elliptic relaxation equation which enables local diffusion sources to be distributed over lengths of the order of the integral scale. The pressure diffusion model was implemented in a boundary layer code within the framework of turbulence models based on both the kappa-epsilon-(bar)upsilon(exp 2) system of equations and the full Reynolds stress equations. Model computations were performed for mixing layers and boundary layer flows. In each case, the pressure diffusion model enabled the well-known free-stream edge singularity problem to be eliminated. There was little effect on near-wall properties. Computed results agreed very well with experimental and DNS data for the mean flow velocity, the turbulent kinetic energy, and the skin-friction coefficient.

Demuren, A. O.↗
NASA NTRS1994

Direct simulation of a self-similar plane wake

Direct simulations of two time-developing turbulent wakes have been performed. Initial conditions for the simulations were obtained from two realizations of a direct simulation of a turbulent boundary layer at momentum thickness Reynolds number 670. In addition, extra two-dimensional disturbances were added in one of the cases to mimic two-dimensional forcing. The unforced wake is allowed to evolve long enough to attain self-similarity. The mass-flux Reynolds number (equivalent to the momentum thickness Reynolds number in spatially developing wakes) is 2000, which is high enough for a short kappa(exp -5/3) range to be evident in the streamwise one-dimensional velocity spectrum. Several turbulence statistics have been computed by averaging in space and over the self-similar period in time. The growth rate in the unforced flow is low compared to experiments, but when this growth-rate difference is accounted for, the statistics of the unforced case are in reasonable agreement with experiments. However, the forced case is significantly different. The growth rate, turbulence Reynolds number, and turbulence intensities are as much as ten times larger in the forced case. In addition, the forced flow exhibits large-scale structures similar to those observed in transitional wakes, while the unforced flow does not.

Moser, Robert D.↗
NASA NTRS1994

Experimental and numerical study of the intermittency exponent mu

After publication of the Kolmogorov refined similarity hypotheses, the small-scale intermittency of the energy dissipation field became a central problem in fully developed turbulence (FDT). This phenomenon has been studied in many different ways, e.g. by searching for corrections to scaling exponents in the inertial range velocity structure functions. A direct measure of this intermittency is, however, available by studying the local rate of energy dissipation, and it may be quantitatively characterized by the intermittency exponent mu. As far as we know, nobody has posed an obvious question: Is the intermittency exponent mu a unique constant, i.e., are the values mu(sub kappa), mu(sub epsilon), mu(sub r), mu(sub b), and mu(sub e) the same at high Reynolds numbers, or do they create a set of different (and perhaps independent) exponents? This paper addresses the above question using the high Reynolds number experiments.

Praskovsky, Alexander↗
NASA NTRS1992

Block LU factorization

Many of the currently popular 'block algorithms' are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by columns. Moreover, for a general matrix the level of instability in block LU factorization can be founded in terms of the condition number kappa(A) and the growth factor for Gaussian elimination without pivoting. A consequence is that block LU factorization is stable for a matrix A that is symmetric positive definite or point diagonally dominant by rows or columns as long as A is well-conditioned.

Demmel, James W.↗
NASA NTRS1994

Acoustic waves in M dwarfs: Maintaining a corona

We use a time-dependent hydrodynamics code to follow the propagation of acoustic waves into the corona of an M dwarf star. An important qualitative difference between M dwarfs and stars such as the Sun is that the acoustic spectrum in M dwarfs is expected to peak at periods close to the acoustic cutoff P(sub A): this allows more effective penetration of waves into the corona. In our code, radiative losses in the photosphere, chromosphere, and corona are computed using Rosseland mean opacities, Mg II kappa and Ly alpha emission, and optically thin emissivities respectively. We find that acoustic heating can maintain a corona with a temperature of order 0.7-1 x 10(exp 6) K and a surface X-ray flux as large as 10(exp 5)ergs/sq cm/s. In a recent survey of X-rays from M dwarfs, some (20%-30%) of the stars lie at or below this limiting X-ray flux: we suggest that such stars may be candidates for acoustically maintained coronae.

Mullan, D. J.↗
NASA NTRS1994

Thin shell formation in radiative shocks. 1: Supernova remnants in low-density media

This paper explores the onset of thin-shell formation in interstellar shocks associated with supernova explosions. We outline a simple but useful scheme that indicates the time at which thin shell formation begins for supernova remnants (SNRs) evolving in a range of interstellar environments, extending the previous analytical models to arbitrary power-law density media. The result depends on the gas cooling properties and the shock velocity and radius. This is then applied to the specific case of SNRs in low-density media. The procedure for defining the time for the onset of shell formation, t(sub sf), equates the value of the adiabat, kappa = p/rho(exp gamma), to zero using the known time dependence of the shock radius and velocity. For the case of a power-law density ambient medium of the form rho(r) = Br(exp -omega), it is found that shell formation can be prevented when the ambient density drops faster than a critical rate. For a cooling function of the form Lambda = Lambda(sub 0) tau(exp beta), with beta = -0.5 (appropriate for line cooling), shell formation never occurs for omega greater than or equal to 9/5. The shell formation time is then computed for spherical shocks in a power-law density medium. For omega = 0, the onset of shell formation is found to be at t(sub sf) approx. equal to 2.87 x 10(exp 4) E51(exp 3/14) n(sub 0 exp -4.7) yr, which agrees well with previous estimates derived by other means. We compare the analytical shell formation time with the results of detailed numerical models for omega = 0 and three different ambient densities and find good agreement. The extension of the criterion for the onset of thin shell formation using the ratio of cooling to swept-up column density is also described. This method provides a useful approximation for cases when the exact solution is not known.

Franco, Jose↗
NASA NTRS1994

Regulation of the solar wind electron heat fluxfrom 1 to 5 AU: Ulysses observations

In this study we use observations from the three-dimensional electron spectrometer and magnetometer aboard the Ulysses spacecraft to examine the solar wind electron heat flux from 1.2 to 5.4 AU in the ecliptic plane. Throughout Ulusses' transit to 5.4 AU, the electron heat flux decreases more rapidly (approximately R(exp -30)) than simple collisionless expansion along the local magnetic field and is smaller than expected for a thermal gradient heat flux, q(sub parallel e) (r) = - Kappa(sub parallel) del(sub parallel) T(sub e)(r). The radial gradients and magnitudes expected for a number of electron heat flux regulatory mechanisms are examined and compared to the observations. The best agreement is found for heat flux regulation by the whistler heat flux instability. The upper bound and radial scaling for the electron heat flux predicted for the whistler heat flux instability are consistent with the observations.

Scime, Earl E.↗
NASA NTRS1994

Turbulence models for compressible boundary layers

It is shown that to satisfy the general accepted compressible law of the wall derived from the Van Driest transformation, turbulence modeling coefficients must actually be functions of density gradients. The transformed velocity profiles obtained by using standard turbulence model constants have too small a value of the effective von Karman constant kappa in the log-law region (inner layer). Thus, if the model is otherwise accurate, the wake component is overpredicted and the predicted skin friction is lower than the expected value.

Huang, P. G.↗
NASA NTRS1996

Combustion Processes in Hybrid Rocket Engines

In recent years, there has been a resurgence of interest in the development of hybrid rocket engines for advanced launch vehicle applications. Hybrid propulsion systems use a solid fuel such as hydroxyl-terminated polybutadiene (HTPB) along with a gaseous/liquid oxidizer. The performance of hybrid combustors depends on the convective and radiative heat fluxes to the fuel surface, the rate of pyrolysis in the solid phase, and the turbulent combustion processes in the gaseous phases. These processes in combination specify the regression rates of the fuel surface and thereby the utilization efficiency of the fuel. In this paper, we employ computational fluid dynamics (CFD) techniques in order to gain a quantitative understanding of the physical trends in hybrid rocket combustors. The computational modeling is tailored to ongoing experiments at Penn State that employ a two dimensional slab burner configuration. The coordinated computational/experimental effort enables model validation while providing an understanding of the experimental observations. Computations to date have included the full length geometry with and with the aft nozzle section as well as shorter length domains for extensive parametric characterization. HTPB is sed as the fuel with 1,3 butadiene being taken as the gaseous product of the pyrolysis. Pure gaseous oxygen is taken as the oxidizer. The fuel regression rate is specified using an Arrhenius rate reaction, which the fuel surface temperature is given by an energy balance involving gas-phase convection and radiation as well as thermal conduction in the solid-phase. For the gas-phase combustion, a two step global reaction is used. The standard kappa - epsilon model is used for turbulence closure. Radiation is presently treated using a simple diffusion approximation which is valid for large optical path lengths, representative of radiation from soot particles. Computational results are obtained to determine the trends in the fuel burning or regression rates as a function of the head-end oxidizer mass flux, G=rho(e)U(e), and the chamber pressure. Furthermore, computation of the full slab burner configuration has also been obtained for various stages of the burn. Comparisons with available experimental data from small scale tests conducted by General Dynamics-Thiokol-Rocketdyne suggest reasonable agreement in the predicted regression rates. Future work will include: (1) a model for soot generation in the flame for more quantitative radiative transfer modelling, (2) a parametric study of combustion efficiency, and (3) transient calculations to help determine the possible mechanisms responsible for combustion instability in hybrid rocket motors.

Venkateswaran,S.↗
NASA NTRS1996

Numerical simulation of jet aerodynamics using the three-dimensional Navier-Stokes code PAB3D

This report presents a unified method for subsonic and supersonic jet analysis using the three-dimensional Navier-Stokes code PAB3D. The Navier-Stokes code was used to obtain solutions for axisymmetric jets with on-design operating conditions at Mach numbers ranging from 0.6 to 3.0, supersonic jets containing weak shocks and Mach disks, and supersonic jets with nonaxisymmetric nozzle exit geometries. This report discusses computational methods, code implementation, computed results, and comparisons with available experimental data. Very good agreement is shown between the numerical solutions and available experimental data over a wide range of operating conditions. The Navier-Stokes method using the standard Jones-Launder two-equation kappa-epsilon turbulence model can accurately predict jet flow, and such predictions are made without any modification to the published constants for the turbulence model.

Pao, S. Paul↗