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NASA NTRS1990

Global pressure relaxation for laminar two-dimensional internal flow

This study extends the reduced Navier-Stokes (RNS) global pressure relaxation procedure developed by Rubin and co-workers for external flow to internal flow applications. The streamwise pressure gradient is split into a backward-differenced or initial value component, as in boundary layer marching, and a forward-differenced or boundary value component that represents the elliptic downstream effects. The streamwise convection terms are upwind-differenced and all other streamwise derivatives are backward-differenced. A standard boundary layer marching technique imbedded in a conventional line relaxation technique is obtained. For compressible flow the pressure iteration determines the interior flow interaction as well as the inlet mass flux that is consistent with the outflow pressure boundary condition. Results have been computed for incompressible flow in both rectangular and curved channels, and for subsonic compressible flow in the simulation of an aerofoil in a wind tunnel. Converged solutions were obtained over a range of Reynolds numbers generating small to moderately large separation bubbles.

Rosenbaum, D.↗

NASA NTRS2000

Ultraviolet Galaxy Counts From STIS Observations of The Hubble Deep Fields

We present galaxy counts in the near and far ultraviolet (NUV and FUV) obtained from Space Telescope Imaging Spectrograph (STIS) observations of portions of the Hubble Deep Field North, (HDFN), the Hubble Deep Field South, (HDFS) and a parallel field near the HDFN. All three fields have deep (AB>29) optical imaging, and we determine magnitudes by taking the ultraviolet flux detected within the limiting optical isophote. An analysis of the UV-optical colors of detected objects, combined with a visual inspection of the UV images, indicates that there are no detectable objects in the UV images which are not also detected in the optical. We measure the detection area and completeness as a function of magnitude by taking the size-magnitude distribution of galaxies in the entire HDFN WFPC2 V+I image, applying the measured UV-optical colors from the detected galaxies, and determining the total area over which each galaxy would have been detected in the UV images. The average area for the simulated galaxies in each UV magnitude bin, (including galaxies which would not be detected at all), provides the effective area and completeness for the bin. We test this procedure with Monte Carlo simulations. The galaxy counts reach to AB=29 in both the NUV and FUV; 1 magnitude fainter than the HDF F30OW counts, and 7 magnitudes fainter than balloon-based counts. We compare our measured counts to various models.

Gardner, J. P.↗

NASA NTRS2001

Entropy Splitting for High Order Numerical Simulation of Vortex Sound at Low Mach Numbers

A method of minimizing numerical errors, and improving nonlinear stability and accuracy associated with low Mach number computational aeroacoustics (CAA) is proposed. The method consists of two levels. From the governing equation level, we condition the Euler equations in two steps. The first step is to split the inviscid flux derivatives into a conservative and a non-conservative portion that satisfies a so called generalized energy estimate. This involves the symmetrization of the Euler equations via a transformation of variables that are functions of the physical entropy. Owing to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the second step is to reformulate the split Euler equations in perturbation form with the new unknowns as the small changes of the conservative variables with respect to their large stagnation values. From the numerical scheme level, a stable sixth-order central interior scheme with a third-order boundary schemes that satisfies the discrete analogue of the integration-by-parts procedure used in the continuous energy estimate (summation-by-parts property) is employed.

Mueller, B.↗

NASA NTRS2008

Use of Vertically Integrated Ice in WRF-Based Forecasts of Lightning Threat

Previously reported methods of forecasting lightning threat using fields of graupel flux from WRF simulations are extended to include the simulated field of vertically integrated ice within storms. Although the ice integral shows less temporal variability than graupel flux, it provides more areal coverage, and can thus be used to create a lightning forecast that better matches the areal coverage of the lightning threat found in observations of flash extent density. A blended lightning forecast threat can be constructed that retains much of the desirable temporal sensitivity of the graupel flux method, while also incorporating the coverage benefits of the ice integral method. The graupel flux and ice integral fields contributing to the blended forecast are calibrated against observed lightning flash origin density data, based on Lightning Mapping Array observations from a series of case studies chosen to cover a wide range of flash rate conditions. Linear curve fits that pass through the origin are found to be statistically robust for the calibration procedures.

McCaul, E. W., jr.↗

NASA NTRS1993

Preconditioning for the Navier-Stokes equations with finite-rate chemistry

The extension of Van Leer's preconditioning procedure to generalized finite-rate chemistry is discussed. Application to viscous flow is begun with the proper preconditioning matrix for the one-dimensional Navier-Stokes equations. Eigenvalue stiffness is resolved and convergence-rate acceleration is demonstrated over the entire Mach-number range from nearly stagnant flow to hypersonic. Specific benefits are realized at the low and transonic flow speeds typical of complete propulsion-system simulations. The extended preconditioning matrix necessarily accounts for both thermal and chemical nonequilibrium. Numerical analysis reveals the possible theoretical improvements from using a preconditioner for all Mach number regimes. Numerical results confirm the expectations from the numerical analysis. Representative test cases include flows with previously troublesome embedded high-condition-number areas. Van Leer, Lee, and Roe recently developed an optimal, analytic preconditioning technique to reduce eigenvalue stiffness over the full Mach-number range. By multiplying the flux-balance residual with the preconditioning matrix, the acoustic wave speeds are scaled so that all waves propagate at the same rate, an essential property to eliminate inherent eigenvalue stiffness. This session discusses a synthesis of the thermochemical nonequilibrium flux-splitting developed by Grossman and Cinnella and the characteristic wave preconditioning of Van Leer into a powerful tool for implicitly solving two and three-dimensional flows with generalized finite-rate chemistry. For finite-rate chemistry, the state vector of unknowns is variable in length. Therefore, the preconditioning matrix extended to generalized finite-rate chemistry must accommodate a flexible system of moving waves. Fortunately, no new kind of wave appears in the system. The only existing waves are entropy and vorticity waves, which move with the fluid, and acoustic waves, which propagate in Mach number dependent directions. The nonequilibrium vibrational energies and species densities in the unknown state vector act strictly as convective waves. The essential concept for extending the preconditioning to generalized chemistry models is determining the differential variables which symmetrize the flux Jacobians. The extension is then straight-forward. This algorithm research effort will be released in a future version of the production level computational code coined the General Aerodynamic Simulation Program (GASP), developed by Walters, Slack, and McGrory.

Godfrey, Andrew G.↗

NASA NTRS2000

Influence of the Biosphere on Precipitation: July 1995 Studies with the ARM-CART Data

Ensemble sets of simulation experiments were conducted with a single column model (SCM) using the Goddard GEOS II GCM physics containing a recent version of the Cumulus Scheme (McRAS) and a biosphere based land-fluxes scheme (SSiB). The study used the 18 July to 5 August 1995 ARM-CART (Atmospheric Radiation Measurement-Cloud Atmospheric Radiation Test-bed) data, which was collected at the ARM-CART site in the mid-western United States and analyzed for single column modeling (SCM) studies. The new findings affirm the earlier findings that the vegetation, which increases the solar energy absorption at the surface together with soil and soil-moisture dependent processes, which modulate the surface, fluxes (particularly evapotranspiration) together help to increase the local rainfall. In addition, the results also show that for the particular study period roughly 50% of the increased evaporation over the ARM-CART site would be converted into rainfall with the Column, while the remainder would be advected out to the large-scale. Notwithstanding the limitations of only one-way interaction (i.e., the large-scale influencing the regional physics and not vice versa), the current SCM simulations show a very robust relationship. The evaporation-precipitation relationship turns out to be independent of the soil types, and soil moisture; however, it is weakly dependent on the vegetation cover because of its surface-albedo effect. Clearly, these inferences are prone to weaknesses of the SCM physics, the assumptions of the large-scale being unaffected by gridscale (SCM-scale) changes in moist processes, and other limitations of the evaluation procedures.

Sud, Y. C.↗

NASA NTRS2008

An Analytical Model for the Prediction of a Micro-Dosimeter Response Function

A rapid analytical procedure for the prediction of a micro-dosimeter response function in low Earth orbit (LEO), correlated with the Space Transportation System (STS, shuttle) Tissue Equivalent Proportional Counter (TEPC) measurements is presented. The analytical model takes into consideration the energy loss straggling and chord length distribution of the detector, and is capable of predicting energy deposition fluctuations in a cylindrical micro-volume of arbitrary aspect ratio (height/diameter) by incoming ions through both direct and indirect (ray) events. At any designated (ray traced) target point within the vehicle, the model accepts the differential flux spectrum of Galactic Cosmic Rays (GCR) and/or trapped protons at LEO as input. On a desktop PC, the response function of TEPC for each ion in the GCR/trapped field is computed at the average rate of 30 seconds/ion. The ionizing radiation environment at LEO is represented by O'Neill fs GCR model (2004), covering charged particles in the 1 less than or equal to Z less than or equal to 28. O'Neill's free space GCR model is coupled with the Langley Research Center (LaRC) angular dependent geomagnetic cutoff model to compute the transmission coefficient in LEO. The trapped proton environment is represented by a LaRC developed time dependent procedure which couples the AP8MIN/AP8MAX, Deep River Neutron Monitor (DRNM) and F10.7 solar radio frequency measurements. The albedo neutron environment is represented by the extrapolation of the Atmospheric Ionizing Radiation (AIR) measurements. The charged particle transport calculations correlated with STS 51 and 114 flights are accomplished by using the most recent version (2005) of the LaRC deterministic High charge (Z) and Energy TRaNsport (HZETRN) code. We present the correlations between the TEPC model predictions (response function) and TEPC measured differential/integral spectra in the lineal energy (y) domain for both GCR and trapped protons, with the conclusion that the model correctly accounts for the increase in flux at low y values where energetic ions are the primary contributor. We further discuss that, even with the incorporation of angular dependency in the cutoffs, comparison of the GCR differential/integral flux between STS 51 and 114 TEPC measured data and current calculations indicates that there still exists an underestimation by the simulations at low to mid range y values. This underestimation is partly related the exclusion of the secondary pion particle production from the current version of HZETRN.

Badavi, Francis F.↗

NASA NTRS1989

Monte Carlo simulation of particle acceleration at astrophysical shocks

A Monte Carlo code was developed for the simulation of particle acceleration at astrophysical shocks. The code is implemented in Turbo Pascal on a PC. It is modularized and structured in such a way that modification and maintenance are relatively painless. Monte Carlo simulations of particle acceleration at shocks follow the trajectories of individual particles as they scatter repeatedly across the shock front, gaining energy with each crossing. The particles are assumed to scatter from magnetohydrodynamic (MHD) turbulence on both sides of the shock. A scattering law is used which is related to the assumed form of the turbulence, and the particle and shock parameters. High energy cosmic ray spectra derived from Monte Carlo simulations have observed power law behavior just as the spectra derived from analytic calculations based on a diffusion equation. This high energy behavior is not sensitive to the scattering law used. In contrast with Monte Carlo calculations diffusive calculations rely on the initial injection of supra-thermal particles into the shock environment. Monte Carlo simulations are the only known way to describe the extraction of particles directly from the thermal pool. This was the triumph of the Monte Carlo approach. The question of acceleration efficiency is an important one in the shock acceleration game. The efficiency of shock waves efficient to account for the observed flux of high energy galactic cosmic rays was examined. The efficiency of the acceleration process depends on the thermal particle pick-up and hence the low energy scattering in detail. One of the goals is the self-consistent derivation of the accelerated particle spectra and the MHD turbulence spectra. Presumably the upstream turbulence, which scatters the particles so they can be accelerated, is excited by the streaming accelerated particles and the needed downstream turbulence is convected from the upstream region. The present code is to be modified to include a better description of particle scattering (pitch-angle instead of hard-sphere) and as iterative procedure for treating the self-excitation of the MHD turbulence.

Campbell, Roy K.↗

NASA NTRS1996

Design and Laboratory Validation of a Capacitive Sensor for Measuring the Recession of Thin-Layered Ablator

Flight vehicles are typically instrumented with subsurface thermocouples to estimate heat transfer at the surface using inverse analysis procedures. If the vehicle has an ablating heat shield, however, temperature time histories from subsurface thermocouples no longer provide enough information to estimate heat flux at the surface. In this situation, the geometry changes and thermal energy leaves the surface in the form of ablation products. The ablation rate is required to estimate heat transfer to the surface. A new concept for a capacitive sensor has been developed to measure ablator depth using the ablator's dielectric effect on a capacitor's fringe region. Relying on the capacitor's fringe region enables the gage to be flush mounted in the vehicle's permanent structure and not intrude into the ablative heat shield applied over the gage. This sensor's design allows nonintrusive measurement of the thickness of dielectric materials, in particular, the recession rates of low-temperature ablators applied in thin (0.020 to 0.060 in. (0.05 to 0.15 mm)) layers. Twenty capacitive gages with 13 different sensing element geometries were designed, fabricated, and tested. A two-dimensional finite-element analysis was performed on several candidate geometries. Calibration procedures using ablator-simulating shims are described. A one-to-one correspondence between system output and dielectric material thickness was observed out to a thickness of 0.055 in. (1.4 mm) for a material with a permittivity about three times that of air or vacuum. A novel method of monitoring the change in sensor capacitance was developed. This technical memorandum suggests further improvements in gage design and fabrication techniques.

Gregory K Noffz↗

NASA NTRS2019

Scaling Evaluation of Ice-Crystal Icing on a Modern Turbofan Engine in PSL Using the COMDES-MELT Code

This paper presents preliminary ice-crystal icing (ICI) altitude scaling evaluation results of a Honeywell Uncertified Research Engine (HURE) that was tested in the NASA Glenn Research Center Propulsion Systems Laboratory (PSL) during January of 2018. This engine geometry features a hidden core design to keep the core less exposed. The engine was fitted with internal video cameras to observe various ice buildup processes at multiple selected locations within the engine core flow path covering the fan stator, the splitter-lip/shroud/strut, and the high pressure compressor (HPC) variable inlet guide vane (IGV) regions. The potential ice accretion risk was pre-determined to occur by using NASA’s in-house 1D Engine Icing Risk assessment code, COMDES-MELT. The code was successful in predicting the risk of ice accretion in adiabatic regions like the fan-stator of the HURE at specific engine operating points. However at several operating points during the test, liquid water was observed running along the shroud toward the variable IGV of the HPC regions with an air temperature well below freezing, thus no particle melting could have occurred due to heating from the air alone. It was reasoned that other sources of heat were present in that region. To account for these heat sources the inlet total temperature was adjusted to give a wet bulb temperature of 24 F below the standard minimum wet bulb temperature of 492 R to allow ice to accrete in the splitter-lip/shroud/strut region, which was determined from a reference case where hard ice was observed in that region. With that adjustment the COMDES-MELT code was successful in providing operating points where there was a risk of ice accretion during the test campaign. In addition to calculating possible conditions at different selected lower altitudes, simulations were run to determine potential inlet conditions that could lead to ice-crystal accretion along the prescribed stations where the cameras were available. From there, scaled test conditions were determined by best matching the following three icing related parameters of the reference condition: (1) the local air total wet bulb temperature, (2) the local ice crystal cloud melt ratio and (3) the engine fan face ice/water to air mass flux ratio of the ice crystal cloud. Instantaneous images taken from the time-lapsed movies of ice buildup were used along with the relevant thermodynamic data of air, water vapor and local icing condition to help evaluate how closely the proposed altitude scaling method could be used in ground based test facility to duplicate selected reference ICI features observed at specific location inside this engine at different scale altitudes. Discussions on observed limitation for engine icing scaling application from this test campaign and needed improvement are provided. A scaling test procedure to help identify potential ICI risk conditions and possible ice accretion locations of a new turbofan engine is evaluated in PSL.

Engine Testing in PSL↗

NASA NTRS1998

Enhanced Sea Ice Concentration and Ice Temperature Algorithms for AMSR

Accurate quantification of sea ice concentration and ice temperature from satellite passive microwave data is important because they provide the only long term, spatially detailed and consistent data set needed to study the climatology of the polar regions. Sea ice concentration data are used to derive large-scale daily ice extents that are utilized in trend analysis of the global sea ice cover. They are also used to quantify the amount of open water and thin ice in polynya and divergence regions which together with ice temperatures are in turn needed to estimate vertical heat and salinity fluxes in these regions. Sea ice concentrations have been derived from the NASA Team and Bootstrap algorithms while a separate technique for deriving ice temperature has been reported. An integrated technique that will utilizes most of the channels of AMSR (Advanced Microwave Scanning Radiometer) has been developed. The technique uses data from the 6 GHz and 37 GHz channels at vertical polarization obtain an initial estimate of sea ice concentration and ice temperature. The derived ice temperature is then utilized to estimate the emissivities for the corresponding observations at all the other channels. A procedure for calculating the ice concentration similar to the Bootstrap technique is then used but with variables being emissivities instead of brightness temperatures to minimizes errors associated with spatial changes in ice temperatures within the ice pack. Comparative studies of ice concentration results with those from other algorithms, including the original Bootstrap algorithm and those from high resolution satellite visible and infrared data will be presented. Also, results from a simulation study that demonstrates the effectiveness of the technique in correcting for spatial variations in ice temperatures will be shown. The ice temperature results are likewise compared with satellite infrared and buoy data with the latter adjusted to account for the effects of the snow cover.

Comiso, Josefino C.↗

NASA NTRS1997

Reduced Navier Stokes Relaxation Procedures for Internal Flows

In spite of significant advancement in the field of high speed computing, flow calculations involving complex geometries and/or flow behavior still require large amounts of CPU time and memory. In order to predict such flows without sacrificing grid convergence and accuracy, adaptive gridding techniques that provide optimal resolution are highly desirable. The present work combines multigrid techniques and domain decomposition concepts to provide local, solution adaptive, grid refinement. Several viscous compressible and incompressible, two and three-dimensional, flows with strong inviscid interaction and/or axial flow reversal, are considered with a segmented multigrid domain decomposition (SMGDD) procedure for which uniform meshes result in each domain. A pressure-based form of flux-vector splitting is applied to the Navier-Stokes equations, which are represented by an implicit lowest-order reduced Navier-Stokes (RNS) system and a purely diffusive, higher-order, deferred-corrector. A trapezoidal or box-like form of discretization insures that all mass conservation properties are satisfied at interfacial and outflow boundaries, even for this primitive-variable non-staggered grid computation. The SMGDD technique presented herein has previously been applied for incompressible two dimensional flows. The present work offers improvement in the gridding strategy, by allowing for disjoint subdomains that provide optimal resolution of disparate flow features. It also extends the SMGDD technique to three dimensional compressible flows. Laminar and turbulent flow in a backward facing step channel is considered; although the procedure is applicable to more severe geometries. The standard K-epsilon model is applied for turbulence closure. For Re greater than 400, differences between two-dimensional theory and experiment are resolved through a three dimensional simulation, which confirms the experimentally observed three dimensionality of the recirculation patterns on the upper and lower surfaces.

Rubin, Stanley G.↗

NASA NTRS1989

An embedded grid formulation applied to delta wings

An embedded grid algorithm for the Euler and/or Navier-Stokes equations is developed and applied to delta wings at high angles of attack in low speed flow. The Navier-Stokes code is an implicit, finite volume algorithm, using flux difference splitting for the convective and pressure terms and central differencing for the viscous and heat transfer terms. Calculations are compared with detailed experimental results over an angle of attack range up to and beyond the maximum lift coefficient, corresponding to vortex breakdown at the trailing edge, for a delta wing nominally of unit aspect ratio. The results indicate that the overall flowfield, including surface pressures, surface streamlines, and vortex trajectories, can be simulated accurately with the global grid version of the present algorithm. However, comparison of computed velocities and vorticity with experimentally measured off-body values at an angle of attack of 20.5 deg indicates the core region is substantially more diffuse in the computations than that measured with either a five-hole probe or a laser velocimeter. Embedded grids, used to improve the numerical discretization in the core region, are formulated within the framework of the implicit, upwind-biased multi-grid algorithm. Structured levels of local nested refinements are made. Three-dimensional results for both Euler and Navier-Stokes calculations are shown, with up to 3 levels of embedded refinement. The embedding procedure was effective in eliminating a crossflow secondary separation produced in the Euler solutions on coarse grids.

Thomas, James L.↗

NASA NTRS2022

Experimental Heat Transfer Results and Flow Visualization of Vertical Upflow Boiling in Earth Gravity with Subcooled Inlet Conditions – In Preparation for Experiments Onboard the International Space Station

Since 2012, researchers at the Purdue University Boiling and Two-Phase Flow Laboratory (PU-BTPFL) and NASA Glenn Research Center have been collaborating on a long-term effort to study flow boiling and condensation in microgravity. The ultimate goal has been to develop the Flow Boiling and Condensation Experiment (FBCE) for the International Space Station (ISS). Based on the findings from prior flow boiling experiments both at different orientations in Earth gravity and onboard parabolic flights simulating short durations of microgravity, a final refined experiment design, construction, and operating procedure have been arrived at for long-duration microgravity flow boiling experiments onboard the ISS. This study investigates flow boiling of n-Perfluorohexane with subcooled inlet in a rectangular channel of dimensions 114.6 mm heated length, 2.5 mm width, and 5 mm height. These pre-launch experiments (Mission Sequence Testing) were conducted in vertical upflow orientation in Earth gravity using the same experimental rig that was launched to the ISS in August 2021. The various operating parameters varied are heating configuration (single- and double-sided), mass velocity (180 – 3200 kg/m2s), inlet subcooling (+0 – 32°C, encompassing both highly subcooled and near-saturated inlet conditions), and inlet pressure (119 – 191 kPa). High-speed video flow visualization images are presented to explain the two-phase interfacial physics within the channel’s heated section. Heat transfer results in terms of flow boiling curves, streamwise profiles of wall temperature and heat transfer coefficient, and averaged heat transfer coefficients are analyzed and parametric effects elucidated. Severe temporal thermodynamic equilibrium is observed for near-saturated inlet at very low velocities. Nucleate boiling degradation starts at larger heat fluxes for single-sided heating than double sided at low mass velocities with highly subcooled inlet, and conversely at high mass velocities with near-saturated inlet. Nucleate boiling degradation can be delayed to higher heat fluxes by highly subcooling the inlet and increasing mass velocity. The entire local heat transfer coefficient profiles are degraded at higher heat fluxes for near-saturated inlet, but only the downstream part for highly subcooled inlet. This study also confirmed reliability of the upcoming ISS experimental data for subcooled inlet conditions and the collected Earth-gravity data will be used for comparison against the ISS data.

flow boiling↗

NASA NTRS2013

Parasitic Interference in Nulling Interferometry

Nulling interferometry aims to detect faint objects close to bright stars. Its principle is to produce a destructive interference along the line of sight so that the stellar flux is rejected, while the flux of the off-axis source can be transmitted. In practice, various instrumental perturbations can degrade the nulling performance. Any imperfection in phase, amplitude or polarization produces a spurious flux that leaks to the interferometer output and corrupts the transmitted off-axis flux. One of these instrumental perturbations is the crosstalk phenomenon, which occurs because of multiple parasitic reflections inside transmitting optics, and/or diffraction effects related to beam propagation along finite size optics. It can include a crosstalk of a beam with itself, and a mutual crosstalk between different beams. This can create a parasitic interference pattern, which degrades the intrinsic transmission map - or intensity response - of the interferometer. In this context, we describe how this instrumental effect impairs the performance of a Bracewell interferometer. A simple formalism is developed to derive the corresponding modified intensity response of the interferometer, as a function of the two parameters of interest: the crosstalk level (or contamination rate) and the phase shift between the primary and secondary - parasitic - beams. We then apply our mathematical approach to a few scientific cases, both analytically and using the GENIESIM simulation software, adapted to handle coherent crosstalk. Our results show that a coherent crosstalk level of about 1 per cent implies a 20 per cent drop of the signal-to-noise ratio at most. Careful attention should thus be paid to reduce the crosstalk level inside an interferometric instrument and ensure an instrumental stability that provides the necessary sensitivity through calibration procedures.

circumstellar matter↗

NASA NTRS2004

Role of Turbulent Prandtl Number on Heat Flux at Hypersonic Mach Number

Present simulation of turbulent flows involving shock wave/boundary layer interaction invariably overestimates heat flux by almost a factor of two. One possible reason for such a performance is a result of the fact that the turbulence models employed make use of Morkovin's hypothesis. This hypothesis is valid for non-hypersonic Mach numbers and moderate rates of heat transfer. At hypersonic Mach numbers, high rates of heat transfer exist in regions where shock wave/boundary layer interactions are important. As a result, one should not expect traditional turbulence models to yield accurate results. The goal of this investigation is to explore the role of a variable Prandtl number formulation in predicting heat flux in flows dominated by strong shock wave/boundary layer interactions. The intended applications involve external flows in the absence of combustion such as those encountered in supersonic inlets. This can be achieved by adding equations for the temperature variance and its dissipation rate. Such equations can be derived from the exact Navier-Stokes equations. Traditionally, modeled equations are based on the low speed energy equation where the pressure gradient term and the term responsible for energy dissipation are ignored. It is clear that such assumptions are not valid for hypersonic flows. The approach used here is based on the procedure used in deriving the k-zeta model, in which the exact equations that governed k, the variance of velocity, and zeta, the variance of vorticity, were derived and modeled. For the variable turbulent Prandtl number, the exact equations that govern the temperature variance and its dissipation rate are derived and modeled term by term. The resulting set of equations are free of damping and wall functions and are coordinate-system independent. Moreover, modeled correlations are tensorially consistent and invariant under Galilean transformation. The final set of equations will be given in the paper.

Xiao, X.↗