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NASA NTRS1992

Mixed convection in a horizontal rectangular channel - Experimental and numerical velocity distributions

Mixed convection in a rectangular channel (width/height = 2) with bottom-heated and top-cooled sections is studied by laser Doppler anemometry in nitrogen at Ra = 22,200 and Re = 18.75, 36, and 54. At the lower Re values, symmetry breaking is observed in steady but spatially oscillating flows that prevail over a certain distance from the leading edge of the differentially heated section. Further downstream, unsteady flows are found even for Re = 18.75. Numerical models are used to investigate the effects of adiabatic, conducting (with a conductive-convective heat transfer coefficient), and perfectly conducting side walls; channel tilts and Prandtl number dependence. Good agreement between calculations and experiment is obtained for longitudinal convective roll velocities. The transverse velocities are found to be independent of Re.

Nyce, Thomas A.↗

NASA NTRS2015

Effects of Unsteady Flow Interactions on the Performance of a Highly-Loaded Transonic Compressor Stage

The primary focus of this paper is to investigate the loss sources in an advanced GE transonic compressor design with high reaction and high stage loading. This advanced compressor has been investigated both experimentally and analytically in the past. The measured compressor efficiency is significantly lower than the efficiency calculated with various existing tools based on RANS and URANS. The general understanding is that some important flow physics in this modern compressor design are not represented in the current tools. To pinpoint the source of the efficiency miss, an advanced test with detailed flow traverse was performed for the front one and a half stage at the NASA Glenn Research Center. In the present paper, a Large Eddy Simulation (LES) is employed to determine whether a higher-fidelity simulation can pick up any additional flow physics that can explain past efficiency miss with RANS and URANS. The results from the Large Eddy Simulation were compared with the NASA test results and the GE interpretation of the test data. LES calculates lower total pressure and higher total temperature on the pressure side of the stator, resulting in large loss generation on the pressure side of the stator. On the other hand, existing tools based on the RANS and URANS do not calculate this high total temperature and low total pressure on the pressure side of the stator. The calculated loss through the stator from LES seems to match the measured data and the GE data interpretation. Detailed examination of the unsteady flow field from LES indicates that the accumulation of high loss near the pressure side of the stator is due to the interaction of the rotor wake with the stator blade. The strong rotor wake interacts quite differently with the pressure side of the stator than with the suction side of the stator blade. The concave curvature on the pressure side of the stator blade increases the mixing of the rotor wake with the pressure side boundary layer significantly. On the other hand, the convex curvature on the suction side of the stator blade decreases the mixing and the suction side blade boundary layer remains thin. The jet velocity in the rotor wake in the stator frame seems to magnify the curvature effect in addition to inviscid redistribution of wake fluid toward the pressure side of the blade.

LES↗

NASA NTRS1998

Aerodynamic Parameters of High Performance Aircraft Estimated from Wind Tunnel and Flight Test Data

A concept of system identification applied to high performance aircraft is introduced followed by a discussion on the identification methodology. Special emphasis is given to model postulation using time invariant and time dependent aerodynamic parameters, model structure determination and parameter estimation using ordinary least squares an mixed estimation methods, At the same time problems of data collinearity detection and its assessment are discussed. These parts of methodology are demonstrated in examples using flight data of the X-29A and X-31A aircraft. In the third example wind tunnel oscillatory data of the F-16XL model are used. A strong dependence of these data on frequency led to the development of models with unsteady aerodynamic terms in the form of indicial functions. The paper is completed by concluding remarks.

Klein, Vladislav↗

NASA NTRS1999

Aerodynamic Parameters of High Performance Aircraft Estimated from Wind Tunnel and Flight Test Data

A concept of system identification applied to high performance aircraft is introduced followed by a discussion on the identification methodology. Special emphasis is given to model postulation using time invariant and time dependent aerodynamic parameters, model structure determination and parameter estimation using ordinary least squares and mixed estimation methods. At the same time problems of data collinearity detection and its assessment are discussed. These parts of methodology are demonstrated in examples using flight data of the X-29A and X-31A aircraft. In the third example wind tunnel oscillatory data of the F-16XL model are used. A strong dependence of these data on frequency led to the development of models with unsteady aerodynamic terms in the form of indicial functions. The paper is completed by concluding remarks.

Klein, Vladislav↗

NASA NTRS1999

Towards Integrated Pulse Detonation Propulsion and MHD Power

The interest in pulse detonation engines (PDE) arises primarily from the advantages that accrue from the significant combustion pressure rise that is developed in the detonation process. Conventional rocket engines, for example, must obtain all of their compression from the turbopumps, while the PDE provides additional compression in the combustor. Thus PDE's are expected to achieve higher I(sub sp) than conventional rocket engines and to require smaller turbopumps. The increase in I(sub sp) and the decrease in turbopump capacity must be traded off against each other. Additional advantages include the ability to vary thrust level by adjusting the firing rate rather than throttling the flow through injector elements. The common conclusion derived from these aggregated performance attributes is that PDEs should result in engines which are smaller, lower in cost, and lighter in weight than conventional engines. Unfortunately, the analysis of PDEs is highly complex due to their unsteady operation and non-ideal processes. Although the feasibility of the basic PDE concept has been proven in several experimental and theoretical efforts, the implied performance improvements have yet to be convincingly demonstrated. Also, there are certain developmental issues affecting the practical application of pulse detonation propulsion systems which are yet to be fully resolved. Practical detonation combustion engines, for example, require a repetitive cycle of charge induction, mixing, initiation/propagation of the detonation wave, and expulsion/scavenging of the combustion product gases. Clearly, the performance and power density of such a device depends upon the maximum rate at which this cycle can be successfully implemented. In addition, the electrical energy required for direct detonation initiation can be significant, and a means for direct electrical power production is needed to achieve self-sustained engine operation. This work addresses the technological issues associated with PDEs for integrated aerospace propulsion and MHD power. An effort is made to estimate the energy requirements for direct detonation initiation of potential fuel/oxidizer mixtures and to determine the electrical power requirements. This requirement is evaluated in terms of the possibility for MHD power generation using the combustion detonation wave. Small scale laboratory experiments were conducted using stoichiometric mixtures of acetylene and oxygen with an atomized spray of cesium hydroxide dissolved in alcohol as an ionization seed in the active MHD region. Time resolved thrust and MHD power generation measurements were performed. These results show that PDEs yield higher I(sub sp) levels than a comparable rocket engine and that MHD power generation is viable candidate for achieving self-excited engine operation.

Litchford, Ron J.↗

DOE OSTI2022

On the effects of reactant stratification and wall curvature in non-premixed rotating detonation combustors

The optimization of non-premixed rotating detonation combustors (RDCs) requires improved understanding of the coupled effects of reactant stratification, fluid property gradients, and complex shock-wave interactions on the detonation wave structure within annular geometries. In the current work, simultaneous orthogonal views of chemiluminescence and hydroxyl planar laser-induced fluorescence (PLIF) are utilized to establish the existence of a dual-wave system characterized by leading and trailing detonation waves that are closely coupled by the local flow physics. These features are persistent over a wide range of mass flow rates and are consistent with prior observations of non-premixed rotating detonations in annular geometries. The detailed instantaneous time sequences are compared with a 3D reactive unsteady Reynolds averaged Navier-Stokes (URANS) simulation to more clearly elucidate the in-situ combustion dynamics and the sensitivity to reactant inlet conditions. It is found that the dual-wave system results from unburned reactants that survive the leading detonation wave in the injector near field and are consumed within a trailing azimuthal reflected-shock combustion (ARSC) zone. By contrast, the injector far field is characterized by rapid mixing due to a sudden drop to subsonic conditions, and the bifurcated detonation wave structure collapses into a stronger, single-wave detonation front with higher overall pressure ratio as compared with the dual-wave system. While each RDC will have different inflow, mixing, and combustion characteristics, the underlying interactions between the stratified reactants and azimuthal wave dynamics identified through the combination of advanced MHz-rate diagnostics and 3D numerical simulations have important implications for the study of detonation wave stability, mode transition, and combustion efficiency in non-premixed annular RDCs.

3D URANS↗

NASA NTRS1995

Aerodynamic sound of flow past an airfoil

The long term objective of this project is to develop a computational method for predicting the noise of turbulence-airfoil interactions, particularly at the trailing edge. We seek to obtain the energy-containing features of the turbulent boundary layers and the near-wake using Navier-Stokes Simulation (LES or DNS), and then to calculate the far-field acoustic characteristics by means of acoustic analogy theories, using the simulation data as acoustic source functions. Two distinct types of noise can be emitted from airfoil trailing edges. The first, a tonal or narrowband sound caused by vortex shedding, is normally associated with blunt trailing edges, high angles of attack, or laminar flow airfoils. The second source is of broadband nature arising from the aeroacoustic scattering of turbulent eddies by the trailing edge. Due to its importance to airframe noise, rotor and propeller noise, etc., trailing edge noise has been the subject of extensive theoretical (e.g. Crighton & Leppington 1971; Howe 1978) as well as experimental investigations (e.g. Brooks & Hodgson 1981; Blake & Gershfeld 1988). A number of challenges exist concerning acoustic analogy based noise computations. These include the elimination of spurious sound caused by vortices crossing permeable computational boundaries in the wake, the treatment of noncompact source regions, and the accurate description of wave reflection by the solid surface and scattering near the edge. In addition, accurate turbulence statistics in the flow field are required for the evaluation of acoustic source functions. Major efforts to date have been focused on the first two challenges. To this end, a paradigm problem of laminar vortex shedding, generated by a two dimensional, uniform stream past a NACA0012 airfoil, is used to address the relevant numerical issues. Under the low Mach number approximation, the near-field flow quantities are obtained by solving the incompressible Navier-Stokes equations numerically at chord Reynolds number of 104. The far-field noise is computed using Curle's extension to the Lighthill analogy (Curle 1955). An effective method for separating the physical noise source from spurious boundary contributions is developed. This allows an accurate evaluation of the Reynolds stress volume quadrupoles, in addition to the more readily computable surface dipoles due to the unsteady lift and drag. The effect of noncompact source distribution on the far-field sound is assessed using an efficient integration scheme for the Curle integral, with full account of retarded-time variations. The numerical results confirm in quantitative terms that the far-field sound is dominated by the surface pressure dipoles at low Mach number. The techniques developed are applicable to a wide range of flows, including jets and mixing layers, where the Reynolds stress quadrupoles play a prominent or even dominant role in the overall sound generation.

Wang, Meng↗

DOE OSTI2026

Statorless mixed flow turbine for transonic pulsating inflow

Effective harvesting of power from high speed highly transient inflows such as the outflow of rotating detonation combustors (RDCs) is key to achieving their promised cycle efficiency step jump. To increase power density and efficiency simultaneously, a concept that can directly ingest transonic outflow, addressing the choking is needed, i.e. without additional transition elements. A new statorless design was assessed using a comprehensive approach to quantify all contributions to loss generation in transient flows, locally and globally. The turbine rotor was designed under steady flow conditions with a genetic algorithm. The comparison of power and loss generation between steady and unsteady flow results shows that a design methodology under steady-state conditions is suitable to characterize the performance of different designs. Three design families were further assessed under highly transient transonic conditions including traveling shock waves with a relative total pressure amplitude of 149.6% around the mean value and inflow angle variations from -26.5 deg to +51.8 deg. The oblique shock impinging on the pressure side (PS) of the turbine augments the shaft power extraction. When the oblique shock reflects between the pressure side and suction side (SS), its strength diminishes. This paper provides design guidelines on efficient turbine work extraction from the shocks emanating from detonation combustors.

rotating detonation engines↗

NASA NTRS2007

An Approach to Improved Credibility of CFD Simulations for Rocket Injector Design

Computational fluid dynamics (CFD) has the potential to improve the historical rocket injector design process by simulating the sensitivity of performance and injector-driven thermal environments to. the details of the injector geometry and key operational parameters. Methodical verification and validation efforts on a range of coaxial injector elements have shown the current production CFD capability must be improved in order to quantitatively impact the injector design process.. This paper documents the status of an effort to understand and compare the predictive capabilities and resource requirements of a range of CFD methodologies on a set of model problem injectors. Preliminary results from a steady Reynolds-Average Navier-Stokes (RANS), an unsteady Reynolds-Average Navier Stokes (URANS) and three different Large Eddy Simulation (LES) techniques used to model a single element coaxial injector using gaseous oxygen and gaseous hydrogen propellants are presented. Initial observations are made comparing instantaneous results, corresponding time-averaged and steady-state solutions in the near -injector flow field. Significant differences in the flow fields exist, as expected, and are discussed. An important preliminary result is the identification of a fundamental mixing mechanism, accounted for by URANS and LES, but missing in the steady BANS methodology. Since propellant mixing is the core injector function, this mixing process may prove to have a profound effect on the ability to more correctly simulate injector performance and resulting thermal environments. Issues important to unifying the basis for future comparison such as solution initialization, required run time and grid resolution are addressed.

Tucker, Paul K.↗

DOE OSTI2023

Evaluating Dynamic Stall-Onset Criteria for Mixed and Trailing-Edge Stall

We evaluate two leading-edge-based dynamic stall-onset criteria (namely, the maximum magnitudes of the leading-edge suction parameter and the boundary enstrophy flux) for mixed and trailing-edge stall. These criteria have been shown to successfully predict the onset of leading-edge stall at Reynolds numbers of [Formula: see text], where the leading-edge suction drops abruptly. However, for mixed/trailing-edge stall, leading-edge suction tends to persist even when there is significant trailing-edge reversed flow and stall is underway, necessitating further investigation into the effectiveness of these criteria. Using wall-resolved large-eddy simulations and the unsteady Reynolds-averaged Navier–Stokes method, we simulate one leading-edge stall and three mixed/trailing-edge stall cases at Reynolds numbers of [Formula: see text] and [Formula: see text]. We contrast the progression of flow features such as trailing-edge separation and vortex formation across different stall types and evaluate the stall-onset criteria relative to critical points in the flow. We find that the criteria nearly coincide with the instance of leading-edge suction collapse and are reached in advance of dynamic stall vortex formation and lift stall for all four cases. We conclude that the two criteria effectively signal dynamic stall onset in cases where the dynamic stall vortex plays a prominent role.

Engineering↗

NASA NTRS1999

Effects of Hot Streak Shape on Rotor Heating in a High-Subsonic Single-Stage Turbine

Experimental data have shown that combustor temperature non-uniformities can lead to the excessive heating of first-stage rotor blades in turbines. This heating of the rotor blades can lead to thermal fatigue and degrade turbine performance. The results of recent studies have shown that variations in the circumferential location (clocking) of the hot streak relative to the first-stage vane airfoils can be used to minimize the adverse effects of the hot streak. The effects of the hot streak/airfoil count ratio on the heating patterns of turbine airfoils have also been evaluated. In the present investigation, three-dimensional unsteady Navier-Stokes simulations have been performed for a single-stage high-pressure turbine operating in high subsonic flow. In addition to a simulation of the baseline turbine, simulations have been performed for circular and elliptical hot streaks of varying sizes in an effort to represent different combustor designs. The predicted results for the baseline simulation show good agreement with the available experimental data. The results of the hot streak simulations indicate: that a) elliptical hot streaks mix more rapidly than circular hot streaks, b) for small hot streak surface area the average rotor temperature is not a strong function of hot streak temperature ratio or shape, and c) hot streaks with larger surface area interact with the secondary flows at the rotor hub endwall, generating an additional high temperature region.

Dorney, Daniel J.↗

DOE OSTI2025

Freestream turbulence effects on unsteady wind turbine loads and wakes: An IDDES study

We investigate numerically the effects of freestream turbulence on the unsteady aerodynamics and wakes of the National Renewable Energy Laboratory Phase VI wind turbine rotor for increasing wind speed. Turbulence is modeled using the Improved Delayed Detached-Eddy Simulation (IDDES) method. As a first step, a detailed mesh resolution study is conducted with the decaying freestream turbulence model at turbulence intensity of 0.5%. Our blade-resolved IDDES simulations show that grid-independent average torque and thrust results can be achieved with relatively coarse meshes, whereas dramatically higher mesh resolution is required for grid-independent results for power spectral densities of thrust force, especially in the deep-stall regime. Comparing the loads with the Shear-Stress Transport model demonstrates the superiority of IDDES in predicting massively separated flows. The aerodynamic performance and wake predictions with the decaying freestream turbulence model are compared with the synthetic freestream turbulence model. Both models predict nearly the same loads, spectral energy content, and wake characteristics. The properties of both the near- and far-wake regions are then examined. Furthermore, we show that separated boundary layers accelerate turbulent mixing and entrainment of the external flow, which results in faster wake recovery. The effect of increasing turbulence intensity to 6% is investigated using the synthetic freestream turbulence model. In contrast with the fully attached boundary layer, higher freestream turbulence in deep stall does not significantly affect the loads and vortex-shedding characteristics. However, the turbulent mixing in the wake is enhanced, which further hastens the recovery of the self-similar velocity profile. In general, increasing the wind speed at high turbulence intensity shifts the recovery farther upstream and increases the wake width.

17 WIND ENERGY↗

NASA NTRSDate not supplied

Physics and Modeling of Quasi-Steady and Forced Shock Trains

The dual-mode scramjet is a propulsive concept for contemporary hypersonic vehicles promising flight cruise speeds upwards of Mach 5. Despite decades of development and a recent resurgence in interest, such engines are not routinely deployed for commercial or military purposes. Instead, they have been confined to the realm of ground-test research and limited one-off flight test missions. One of the key stumbling blocks on the critical path to deployment of a dual-mode scramjet is the optimal design and stable operation of the isolator, a typically constant area duct that separates the scramjet inlet from the combustor. During ramjet mode at Mach numbers from 3 to 5 and during the transition to scramjet mode at higher Mach numbers, the isolator contains the pseudo-shock. Necessary to meet the back-pressure in the combustor, the pseudo-shock is typically divided into a shock dominated shock train region followed by a turbulence driven mixing region. Fluctuations in combustor pressure due to vehicle maneuvering, throttling, or combustion instabilities drive the shock train to shift streamwise within the isolator. In response to sharp pressure fluctuations, the shock train is unable to stabilize within the isolator, and the scramjet unstarts by ejecting the shock train out the inlet. This work addresses three key gaps in understanding of the isolator pseudo-shock. The phenomenon is studied in three experimental facilities to investigate the quasi-steady and unsteady fluid dynamics of the shock train in fueled and un-fueled test conditions. The strong link between boundary layer morphology and shock train structure is studied in detail and used as a tool to evaluate the state-of-the-art definitions and models of the pseudo-shock. Identifying the limitations of these models, a new process driven flux-conserved pseudo-shock model is proposed. This provides a computationally efficient model grounded in the fluid dynamic mechanisms of the pseudo-shock with better predictive capability than the current state-of-the-art across all facilities tested. The unsteady behavior of the shock train is further investigated to identify and understand two distinct forced dynamic modes: mechanically induced, low-rate forcing and high-rate combustion driven unsteadiness that sheds upstream propagating shocks. The proposed process driven model is found to be effective in the the limiting case of high-rate combustion driven unsteadiness by evaluating the model in a shock stationary reference frame. Further analysis identifies the approximately rigid chain motion of the shock train structure to mechanically induced forcing. The time delay between the onset of forced dynamics is found to saturate to a constant value and maximum shock train speed along the isolator with respect to the perturbation rate of change. The result of this work provides dual-mode scramjet designers with improved tools and physical understanding to develop more efficient, robust, and capable propulsion systems for hypersonic vehicles.

JAXA↗

NASA NTRS1975

A numerical method for solving the Navier-Stokes equations with application to shock-boundary layer interactions

A numerical method for solving the compressible form of the unsteady Navier-Stokes equations is described. This method was originally presented in 1970 and has since been modified during the development of computer programs at Ames for implementing models that account for the effects of turbulence in shock-induced separated flows. Although this paper does not describe the turbulence models themselves, a complete description of the basic numerical method is given with emphasis on the choice of a computational mesh for high Reynolds number flows, finite-difference approximations for mixed partial derivatives, extension of the Courant-Friedrichs-Lewy stability condition for viscous flows, mesh boundary conditions, and numerical smoothing for strong shock-wave calculations.

Maccormack, R. W.↗

NASA NTRS2008

A Numerical Investigation of Turbine Noise Source Hierarchy and Its Acoustic Transmission Characteristics

Understanding the relative importance of the various turbine noise generation mechanisms and the characteristics of the turbine acoustic transmission loss are essential ingredients in developing robust reduced-order models for predicting the turbine noise signature. A computationally based investigation has been undertaken to help guide the development of a turbine noise prediction capability that does not rely on empiricism. The investigation relies on highly detailed numerical simulations of the unsteady flowfield inside a modern high-pressure turbine (HPT). The simulations are developed using TURBO, which is an unsteady Reynolds-averaged Navier-Stokes (URANS) code capable of multi-stage simulations. The purpose of this study is twofold. First, to determine an estimate of the relative importance of the contributions to the coherent part of the acoustic signature of a turbine from the three potential sources of turbine noise generation, namely, blade-row viscous interaction, potential field interaction, and entropic source associated with the interaction of the blade rows with the temperature nonuniformities caused by the incomplete mixing of the hot fluid and the cooling flow. Second, to develop an understanding of the turbine acoustic transmission characteristics and to assess the applicability of existing empirical and analytical transmission loss models to realistic geometries and flow conditions for modern turbine designs. The investigation so far has concentrated on two simulations: (1) a single-stage HPT and (2) a two-stage HPT and the associated inter-turbine duct/strut segment. The simulations are designed to resolve up to the second harmonic of the blade passing frequency tone in accordance with accepted rules for second order solvers like TURBO. The calculations include blade and vane cooling flows and a radial profile of pressure and temperature at the turbine inlet. The calculation can be modified later to include the combustor pattern factor at the turbine inlet to include that contribution to turbine noise. We shall present preliminary analysis of the results obtained so far in order to assess the validity of such an approach and to seek feedback on improving the approach. This work addresses both Area 1 (Turbine Tone Noise) and Area 5 (Influence of the Turbine on Combustor Noise) topics.

VanZante, Dale↗

NASA NTRS2016

Unsteady Flow Simulation of a Sweeping Jet Actuator Using a Lattice-Boltzmann Method

Active flow control technology is increasingly used in aerospace applications to control flow separation and to improve aerodynamic performance. In this paper, PowerFLOW is used to simulate the flow through a sweeping jet actuator at two different pressure ratios. The lower pressure ratio leads to a high subsonic flow, whereas the high pressure ratio produces a choked flow condition. Comparison of numerical results with experimental data is shown, which includes qualitatively good agreement of pressure histories and spectra. PIV measurements are also available but the simulation overestimates mean and fluctuation quantities outside the actuator. If supply pressure is matched at one point inside the mixing chamber a good qualitative agreement is achieved at all other monitor points.

Duda, B.↗

NASA NTRS2006

Flight Test Results for Uniquely Tailored Propulsion-Airframe Aeroacoustic Chevrons: Shockcell Noise

Azimuthally varying chevrons (AVC) which have been uniquely tailored to account for the asymmetric propulsion-airframe aeroacoustic interactions have recently shown significant reductions in jet-related community noise at low-speed take-off conditions in scale model tests of coaxial nozzles with high bypass ratio. There were indications that such AVCs may also provide shockcell noise reductions at high cruise speeds. This paper describes the flight test results when one such AVC concept, namely, the T-fan chevrons with enhanced mixing near the pylon, was tested at full-scale on a modern large twin-jet aircraft (777-300ER) with focus on shockcell noise at mid-cruise conditions. Shockcell noise is part of the interior cabin noise at cruise conditions and its reduction is useful from the viewpoint of passenger comfort. Noise reduction at the source, in the exhaust jet, especially, at low frequencies, is beneficial from the perspective of reduced fuselage sidewall acoustic lining. Results are shown in terms of unsteady pressure spectra both on the exterior surface of the fuselage at several axial stations and also microphone arrays placed inside the fuselage aft of the engine. The benefits of T-fan chevrons, with and without conventional chevrons on the core nozzle, are shown for several engine operating conditions at cruise involving supersonic fan stream and subsonic or supersonic core stream. The T-fan AVC alone provides up to 5 dB low-frequency noise reduction on the fuselage exterior skin and up to 2 dB reduction inside the cabin. Addition of core chevrons appears to increase the higher frequency noise. This flight test result with the previous model test observation that the T-fan AVCs have hardly any cruise thrust coefficient loss (< 0.05%) make them viable candidates for reducing interior cabin noise in high bypass ratio engines.

Mengle, Vinod G.↗

NASA NTRS1994

Interaction of the sonic boom with atmospheric turbulence

Theoretical research was carried out to study the effect of free-stream turbulence on sonic boom pressure fields. A new transonic small-disturbance model to analyze the interactions of random disturbances with a weak shock was developed. The model equation has an extended form of the classic small-disturbance equation for unsteady transonic aerodynamics. An alternative approach shows that the pressure field may be described by an equation that has an extended form of the classic nonlinear acoustics equation that describes the propagation of sound beams with narrow angular spectrum. The model shows that diffraction effects, nonlinear steepening effects, focusing and caustic effects and random induced vorticity fluctuations interact simultaneously to determine the development of the shock wave in space and time and the pressure field behind it. A finite-difference algorithm to solve the mixed type elliptic-hyperbolic flows around the shock wave was also developed. Numerical calculations of shock wave interactions with various deterministic and random fluctuations will be presented in a future report.

Rusak, Zvi↗