Prandtl-Meyer Expansion Research Articles

Impact of Laval nozzle structure on the flow characteristics of supersonic gas-solid two-phase flow

This study introduces a mathematical coupling model within the Euler-Lagrange framework to conduct numerical simulations for supersonic gas-particles flow. After validating the model, the gas flow characteristics and particle dynamics within supersonic gas-particle flow under various nozzle configurations are investigated. The findings are as follows: (1) A Laval nozzle with a smoothly connected throat suppresses the generation of fan-shaped Prandtl-Meyer expansion waves and other complex wave systems, effectively mitigating significant changes in gas pressure at the throat and enhancing the stability of supersonic jets. (2) The presence of a particle-free zone in the nozzle divergent segment reduces the collision frequency between particles and the walls, thereby restricting the radial expansion of particles. (3) Utilizing an internally smooth-lined nozzle and expanding the radial divergence angle to 2.5–2.8 times, while ensuring particles maintain consistent axial penetration performance, increases the reflective contact area. This leads to an improvement in the utilization efficiency of particles. These findings provide valuable insights for optimizing nozzle configurations and enhancing the performance of supersonic gas-particle systems.

Transonic flow field in critical flow Venturi nozzle

In this paper, theoretical and numerical analysis of a~transonic flow field of critical flow Venturi nozzles according to the ISO~9300 standard is performed. Deviations of the flow field from an estimate based on one dimensionality are clarified. While the theoretical analysis allows prediction of these deviations, the numerical analysis allows quantification of their influence. The main studied phenomena include the local supersonic compression in transonic expansion and the Prandtl-Meyer expansion. The tendency of the flow field to spatial oscillations is shown, alongside the ability of the system to damp these oscillations.

Computational study of lateral jet interaction in hypersonic thermochemical non-equilibrium flows using nonlinear coupled constitutive relations

The present study reports the numerical analyses of lateral jet interaction around a Terminal High Altitude Area Defense-type (THAAD-type) model in hypersonic rarefied flows, with the real gas effect incorporated. The computation approach employed is the recently developed thermochemical non-equilibrium nonlinear coupled constitutive relations (NCCR) model. Regarding the simulation conditions, the flight velocity and height are set to 20 Ma and 80 km, respectively. To disclose the flow mechanism of lateral jet interaction, the complex flowfield characteristics and surface pressure distributions are discussed at length. Additionally, the research explores the impact of two key factors, namely, the jet pressure ratio and the jet Mach number, on the control performance of an in-flight vehicle's reaction control system (RCS). The results demonstrate that the complicated flowfield structures in lateral jet interaction are successfully reproduced by the NCCR model. With an increase in either the jet pressure ratio or the jet Mach number, the force and moment amplification factors decrease, while the absolute value of the normal force coefficient increases. Notably, it is found that the rarefied gas effect captured by the NCCR model against the Navier–Stokes–Fourier solution affects the lateral jet interaction flowfield, e.g., weakening the compressibility of the barrel shock and the expansibility of the Prandtl–Meyer expansion fan, as well as strengthening the jet wraparound effect. Importantly, the rarefied gas effect also exerts a prominent influence on the performance of RCS, with the degree of influence diminishing as the jet Mach number or the jet pressure ratio increases.

Assessment and application of optical flow in background-oriented schlieren for compressible flows

Optical flow provides an opportunity to elevate the resolution and sensitivity of deflection sensing in background-oriented schlieren (BOS). Despite extensive relevant literature within the field of computer vision, there is a lack of proper quantification of its abilities and limitations with regard to the state-of-the-art BOS experiments. Thus, this study performs an assessment of accuracy and resolution limits in different flow field scenarios utilizing background patterns generated with random dot and wavelet noise distributions. Accordingly, a synthetic assessment over a theoretically generated Prandtl–Meyer expansion fan is conducted with variations introduced in the background patterns and operational parameters of optical flow. A clear superiority of accuracy and resolvable range of density gradient amplitudes over cross-correlation is demonstrated. Moreover, an experimental assessment of supersonic flow features over multiple wind tunnel models is performed. The influence of experimental constraints, limitations and uncertainties related to the application of optical flow in BOS and its comparative performance against the block-matching counterpart is characterized.Graphical abstract

Influence of isolator section on the shock augmented mixing in SCRAMJET engine

The paper aims to numerically study the effect of isolator design on shock-augmented mixing in a generic supersonic Ramjet (SCRAMJET) engine and reduce what would be a very complicated thermodynamics+aerodynamic optimization in the design to a simpler aerodynamic one. Changing the length of the isolator section effectively varies the thickness of the boundary layer that is formed in the isolator, which then enters into the SCRAMJET combustor. The study is conducted on a three-dimensional geometry using the Menter's SST model for turbulence on an in-house unstructured grid RANS solver. In this paper, the authors quantify the effect of the incoming boundary layer thickness on the shock-augment air-fuel mixing in the SCRAMJET combustor. It is observed that for the maximum variation in the isolator length, the normalized mixing volume only improves by 7%, which is only a marginally better mixing of air and fuel obtained as with an increase in the isolator length. The authors conclude that the Prandtl-Meyer expansion fan (PMEF) formed at the flame-holder effectively isolates the combustor from the isolator section to a degree that the isolator dimensions can therefore be chosen entirely on considerations of flow homogenization of inlet air before it enters the combustor section of the SCRAMJET engine. Although, the paper analyses a generic SCRAMJET with specific dimensions, the results are more generally applicable than for the specific design actually considered in the paper. Thus this study is also relevant to SCRAMJET engines that may differ radically from our considered design in form and dimensions.

Mathematical Models of Transonic Flatter of Aerody-namic Control Surfaces of Supersonic Aircraft

In the article, the joint analysis of the Bernoulli equations for compressed gas, variations of the supersonic flow parameters of the Prandtl-Meyer expansion fan and the hypothesis of aerofoil dynamic curvature were used to develop a linear and a nonlinear mathematical models describing the occurrence of transonic flutter of aerodynamic control surfaces of supersonic aircraft. The analysis of the obtained mathematical models confirms a theoretical possibility of the occurrence of transonic flutter of aerodynamic control surfaces of supersonic aircraft that is due to the peculiarities of the interaction of shock waves with the angular velocity of elastic bending oscillations of aerodynamic control surfaces.

Novel Simplified Numerical Simulation Method for Modeling Bleed Holes in Supersonic Inlets

A novel simplified numerical simulation method for bleed holes in supersonic inlets was developed to estimate the bleed mass flow rate and simulate the flowfield structure in the bleed region. First, the Prandtl–Meyer expansion theory was employed to calculate the location of the barrier shock and reconstruct the flow structure inside the bleed hole. Next, a novel discrimination algorithm was developed to distinguish the grids connected to the bleed hole in the bleed zone instead of directly modeling the hole opening and plenum chamber. Finally, the bleed mass flow rate was obtained based on the boundary-layer properties in the bleed region and the fitting scaled coefficients. This method was also evaluated by performing numerical simulations for a series of different bleeding situations. For all the examined cases, the simulation results were in good agreement with the experimental data, and the number of grid points’ requirement for modeling the bleed hole as compared with the fully resolved method was considerably reduced. Moreover, the proposed method was successfully applied to simulate porous bleed systems in a mixed-compression supersonic inlet. This illustrates that the proposed method is expected to be a useful engineering tool for designing the bleed system of the supersonic inlets.

A second-order-accurate immersed boundary ghost-cell method with hybrid reconstruction for compressible flow simulations

This study presents an improved ghost-cell immersed boundary method for geometrically complex boundaries in compressible flow simulations. A bilinearly complete extrapolation scheme is developed for the reconstruction of the ghost-cell. The second-order accuracy of the improved ghost-cell method (GCM) is shown in unit test cases and is also theoretically proven. A hybrid GCM based on both baseline GCM and improved GCM is proposed and constructed. The hybrid GCM applied in compressible flow is validated against five test cases: (a) Stationary rotating vortex, (b) Prandtl–Meyer expansion flow, (c) Double Mach reflection, (d) Moving-shock/obstacle interaction, (e) Blunt body shock-induced combustion. This paper provides a comprehensive comparison of their performance in terms of various accuracy and computation time measurements. The simulation results demonstrate that the hybrid GCM has higher accuracy and convergence than the remaining two GCMs in all cases. By directly comparing the primitive variables along the boundary, it can be concluded that the hybrid GCM has significant advantages in compressible flow simulations. The results of CPU time show that the hybrid GCM can provide more accurate results while ensuring the efficiency of the calculation.

Effects of velocity shear layer on detonation propagation in a supersonic expanding combustor

This study investigates the mechanism of detonation propagation in a stoichiometric hydrogen–oxygen mixture with non-uniform flow velocity entering an expanding combustor. For simulation of the detonation propagation, the Navier–Stokes equations with a one-step two-species chemistry model are solved by employing the hybrid sixth-order weighted essentially non-oscillatory centered difference scheme. The self-sustaining mechanism of detonation propagation in an expanding combustor under the action of non-uniform supersonic flow with a velocity shear layer is revealed. The results show that under the influence of velocity shear layer, two different unburned jets are produced behind the detonation front. These jets are induced by the velocity shear layers and the Prandtl–Meyer expansion fan. The two jets interact and mix gradually. The interaction between the mixed unburned jets and highly unstable shear layers creates large-scale vortices that intensify the turbulent mixing of the unburned jets. Meanwhile, the baroclinic mechanism generates numerous vortices on the boundary of the unburned jet. These vortices promote the mixing of the burned and unburned gases, which eventually leads to the rapid consumption of the unburned pockets. The heat released due to the burning of the unreacted pockets behind the detonation wave supports a self-sustaining propagation of the detonation wave. When the velocity difference among the shear layers increases, the surface fluctuation of the detonation wave increases.

Математична модель фюзеляжних коливань на трансзвукових швидкостях польоту

Ensuring the safety of supersonic aircraft flights and aerospace systems in the transonic range of M flight numbers still remains an urgent scientific and applied problem. This is caused by the peculiarities of the aerodynamic surfaces flow by inhomogeneous (transonic) air and is due to the emergence of various aeroelastic phenomena in these flight modes and the current lack of a generally accepted model of transonic flutter, even for aerodynamic control surfaces. Based on a joint analysis of the conditions for the formation of shock waves on the surface of the aerodynamic profile, changes in the parameters of supersonic flow across the Prandtl-Meyer expansion fan, and the hypothesis of "dynamic curvature of the aerodynamic profile", the approximate laws of interaction of elastic bending vibrations of the fuselage with fluctuations in shock waves were obtained. The obtained regularities are used to substantiate a mathematical model for estimating excited forces and excited bending moments of the fuselage. The analysis of the obtained mathematical model confirms the theoretical possibility of the appearance of fuselage forms of transonic flutter in supersonic aircraft, which was observed in the flight experiment and which is due to the interaction of shock waves with the angular velocity of the fuselage elastic bending vibrations. With the accepted in the article input geometrical data of a fuselage aerodynamic surfaces’ profile, the maximum possible values of fuselage bending moments are obtained using the developed mathematical model. The obtained mathematical model can be used for a preliminary approximate assessment of the transonic flutter fuselage forms characteristics in supersonic aircraft and aerospace systems.

Mechanism of detonation stabilization in a supersonic model combustor

Abstract

Aerodynamic Study of the NASA’s X-43A Hypersonic Aircraft

A 2D aerodynamic study of the NASA’s X-43A hypersonic aircraft is developed using two different approaches. The first one is analytical and based on the resolution of the oblique shock wave and Prandtl–Meyer expansion wave theories supported by an in-house program and considering a simplified aircraft’s design. The second approach involves the use of a Computational Fluid Dynamics (CFD) package, OpenFOAM and the real shape of the aircraft. The aerodynamic characteristics defined as the lift and drag coefficients, the aerodynamic efficiency and the pitching moment coefficient are calculated for different angles of attack. Evaluations are made for an incident Mach number of 7 and an altitude of 30 km. For both methodologies, the required angles of attack to achieve a Vertical Force Balance (VFB) and a completely zero pitching moment conditions are considered. In addition, an analysis to optimise the nose configuration of the aircraft is performed. The mass flow rate throughout the scramjet as a function of the angle of attack is also presented in the CFD model in addition to the pressure, density, temperature and Mach fields. Before presenting the corresponding results, a comparison between the aerodynamic coefficients in terms of the angle of attack of both models is carried out in order to properly validate the CFD model. The paper clarifies the requirements needed to make sure that both oblique shock waves originating from the leading edge meet just at the scramjet inlet clarifying the advantages of fulfilling such condition.

Study of the aerodynamic forces on a simplified shape X-43 aircraft under supersonic conditions, design optimization

The paper focuses on the shape optimization, based on the resultant aerodynamic forces, of a simplified X-43 aircraft. The equations of oblique shock waves and Prandtl-Meyer expansion waves, along with the improvement of the aircraft geometry using a genetic algorithm optimisation, will allow to shape and optimize the external dimensions of the bi-dimensional simplified aircraft design. The main purpose of the optimisation is the reduction of the drag force while maintaining a sufficient lift to hold the weight of the aircraft. The optimization is undertaken at four different altitudes, obtaining for each altitude the most favourable aircraft shape to maximize the aircraft efficiency. The study can be extended to various possible aircraft weight configurations.

Numerical investigations on detonations in a condensed-phase explosive and oblique shock waves in surrounding fluids

In this work, we studied detonation in condensed-phase explosives of PETN and the oblique shock waves in the surrounding fluid. The surrounding fluid was modeled as an ideal gas equation of state using the specific heat ratio as a parameter. Depending on the specific heat ratio, four types of flow structures were observed behind the oblique shock wave in the ideal gas. We measured the detonation/shock angles and the contact angle between the detonation products and the ideal gas. When the specific heat ratio was greater than the critical value, the oblique shock wave was detached from the detonation front at the PETN/ideal gas interface. When attached, three types of waves are observed, depending on the specific heat ratio: a strong oblique shock wave, strong and weak oblique shock waves which meet at the triple point as well as a weak oblique shock wave. To understand the properties of the flow near the detonation and the oblique shock waves, they were modeled as planar Chapman–Jouguet (CJ) detonation in PETN and as oblique shock waves in ideal gas. They were theoretically estimated as (1) Prandtl–Meyer expansion of the detonation products from the CJ state, and (2) oblique shock waves around a wedge using oblique shock theory or around a cone using the Taylor–Maccoll equation. From the flow models, we obtained a solution for the pressure equilibrium and parallel flows between the detonation products and ideal gas under the assumption that the wedge and cone angles correspond to the contact angle between them. For the attached cases, the solution was consistent with the simulated observations at the PETN/ideal gas interface. From the flow models, the maximum deflection angles for the detonation products and the ideal gas were obtained, and their magnitude correlations used to classify the four types of flow.

Investigating focusing of Mach waves by Prandtl-Meyer expansion fan as an explanation for some spatiospectral lobe phenomena

Recent studies of high-performance aircraft have noted the presence of spatiospectral lobes: regions of physical and frequency space where the sound field exhibits increased amplitude not otherwise captured by typical jet noise models. This phenomenon results in multiple spectral peaks at particular positions and multiple spatial peaks at particular frequencies. These peaks have been variously explained as resulting from indirect combustion noise or shock-turbulence interactions of some type. This paper proposes to explain some aspects of the spatiospectral lobe phenomenon in terms of the focusing of Mach waves by the Prandtl-Meyer expansion fans that, with oblique shocks, comprise shock cells. Briefly, a higher convective velocity leads to an angle of propagation farther from the jet axis. Because the flow in an expansion fan is increasing in velocity, Mach waves produced within the latter part of a single expansion fan are directed at an angle farther to the sideline than those produced at an upstream position within the same fan, leading to converging directions of travel and focusing. A simple model is developed and used to demonstrate the explanatory power of this mechanism with input from a prior computational study.

Oscillations of aerodynamic constructions: safe speed band

The article deals with the shock waves regularities interaction with the flutter angular speed of the aerodynamic surfaces control are established basing on the analysis of Bernoulli’s equation for compressible gas, the evaluation of changes of the supersonic flow characteristics in the Prandtl-Meyer expansion fan and the aerodynamic profile “dynamic curvature” hypothesis. The established regularities explain the cause of the disturbing forces and hinge moments of the supersonic aircraft aerodynamic surfaces control and aerospace systems at transonic flight speeds.The mathematical models of estimation of the disturbing forces and the disturbing hinges magnitudes of the aerodynamic surfaces control are obtained basing on the established regularities of shock waves with interaction the flutter angular velocity of the aerodynamic surfaces control.The dangerous range of Mach number flight is determined basing on a quantitative analysis of the obtained mathematical models and it is shown that the genesis of transonic flutter of supersonic aircraft and aerospace systems is possible in a narrow range of the Mach numbers flight.Moreover, the range of the Mach numbers flight, in which there is an increase in the disturbing hinge moments of the aerodynamic surfaces control and in which the planes stay in the phase of flight speed reducing from supersonic to subsonic, is more than three times the range of the Mach numbers flight, in which the planes stay in the phase of the flight speed increasing from subsonic to supersonic. This result permits to explain one of the possible reasons why the transonic flutter occurrence of the supersonic aircraft is more often observed in the phase of flight speed reducing from supersonic to subsonic.The obtained results can be used for a preliminary assessment of the transonic flutter characteristics of the supersonic aircraft aerodynamic surfaces control and aerospace systems, as well as the recommendations developing for flight personnel on the piloting features of supersonic aircraft at transonic flight speeds.

Formation and investigation of nanocomposite gradient coating with upper layer of hafnia

Nanostructured hafnia films and gradient coatings NiCoCrAlY – (7YSZ + NiCoCrAlY) – 7YSZ – (7YSZ + 9YSH) – 9YSH were deposited by low pressure plasma spraying using a supersonic nozzle creating Prandtl-Meyer expansion fan. The coatings have been analyzed by scanning electron microscopy, nuclear backscattering spectrometry (NBS), XRD analysis, scratch testing, thermal analysis and thermal insulation testing. The coating structure are characterized by the mixture of lamellar and nanostructured regions composed of nanoparticle with sizes of 0.02–0.04 μm. Profiling of the coatings elemental composition by NBS method has indicated the presence of transition zones of the mixed composition at the layers boundaries. Synchronous thermal analysis has showed stability of the 9YSH coating structure at temperatures up to 1200 °C and a tiny exothermic process around 1400 °C. In thermal insulation test the thermal conductivity of gradient coatings ceramic layers was estimated as the range of 0.2–0.7 W/m·K at temperatures of 1200–2700 °C. The gradient coating with thickness of 150 μm reduced the heat flow to the cooling system to 0.34 from the initial one.

1-D model for finding geometry of a single phase ejector

This paper presents a novel 1-D mathematical model to determine complete dimensions of an ejector component of Ejector Refrigeration System (ERS). The concepts of Prandtl's mixing length, Prandtl-Meyer expansion wave, Kelvin-Helmholtz instability and Baroclinic effect are introduced in the model to precisely determine the various diameters, mixing length, nozzle exit position etc. for the given conditions of the primary & secondary fluid, cooling capacity and critical condenser pressure. The area ratios obtained using the mathematical model are compared with the experimental/numerical results available in open literature for the same operating conditions and are found to be in good agreement. Moreover, ejector geometry determined from the proposed model is analyzed using CFD for the same input conditions. Average deviation in the entrainment ratio obtained using CFD and that given to the model is found to be less than 2.48% and thus model is validated again. The experimental test rig of ejector refrigeration system is also fabricated and the performance is evaluated while operating at critical condenser pressure. The deviation in the ejector geometry used in the experiment is found to be less than 7% in comparison to the ejector dimensions calculated by the numerical model for the same input conditions.

Plasma Sputtering of a Nanostructured Hafnium-Oxide Coating by Means of a Prandtl–Meyer Flow

The geometric characteristics of the supersonic nozzle of a plasmatron intended for vacuum plasma sputtering, which creates a Prandtl–Meyer expansion fan, are estimated. The nozzle facilitates nanoparticle condensation from the vapor phase of the sputtered material due its cooling in the vicinity of the nozzle cross section rotated by 6° and 27°. During the process of sputtering, a coating containing mainly large-scale particles and their fragments and a film consisting of nanoparticles are formed if the central part of the nozzle and its peripheral regions pass over the sample, respectively. Investigation into the surface structure of a HfO2–9% Y2O3 coating 30 μm thick indicates the existence of deformed particles and nanostructured regions in the form of thin film fragments.

Stagnation Temperature Effect on the Supersonic Flow Around Pointed Airfoils with Application for Air

The aim of this work is to develop a new numerical calculation program to determine the effect of the stagnation temperature on the calculation of the supersonic flow around a pointed airfoils using the equations for oblique shock wave and the Prandtl–Meyer expansion, under the model at high temperature, calorically imperfect and thermally perfect gas, lower than the dissociation threshold of the molecules. The specific heat at constant pressure does not remain constant and varies with the temperature. The new model allows making corrections to the perfect gas model designed for low stagnation temperature, low Mach number, low incidence angle and low airfoil thickness. The stagnation temperature is an important parameter in our model. The airfoil should be pointed at the leading edge to allow an attached shock solution to be seen. The airfoil is discretized into several panels on the extrados and the intrados, placed one adjacent to the other. The distribution of the flow on the panel in question gives a compression or an expansion according to the deviation of the flow with respect to the old adjacent panel. The program determines all the aerodynamic characteristics of the flow and in particular the aerodynamic coefficients. The calculation accuracy depends on the number of panels considered on the airfoil. The application is made for high values of stagnation temperature, Mach number and airfoil thickness. A comparison between our high temperature model and the perfect gas model is presented, in order to determine an application limit of the latter. The application is for air.