Axisymmetric Compressible Navier-Stokes Equations Research Articles

Flow Field Simulations Over Reentry Modules at High Speed

Reentry capsule configurations significantly differ from each other due to entry conditions and mission requirements. This paper describes numerical simulations of the viscous flow over the Beagle and the OREX (Orbital Reentry EXperiments) configurations for freestream Mach numbers in the range of 1.2 - 5.0. The flow fields over the reentry modules are obtained by solving time-dependent axisymmetric compressible laminar Navier-Stokes equations. The fluid mechanics equations are discretized in spatial coordinates employing a finite volume method, which reduces the governing equations to semi-discretized ordinary differential equations. Temporal integration is carried out using a two-stage Runge-Kutta time-stepping scheme. A local time-stepping is employed to get the steady state solution. The numerical simulation is performed on a single-block structured grid. The flow field features around the reentry capsules such as bow shock wave, sonic line, expansion fan and recirculation flow in the base-shell region are well captured by the present numerical computation. The effects of the geometrical parameters of the module, such as aspect ratio, frontal segment bluntness, fore body cone angle and shoulder rounding radius on the wake throat width, distance from the wake throat to the base of the model, flow departure angle and aerodynamic drag are analyzed using the numerically simulated flow field.

Numerical Simulation of Flow Field Over a Sharp-Tipped Double Cone at High Speed

The flowfield over a sharp-tipped double cone axisymmetric configuration is carried out by solving time dependent axisymmetric laminar compressible Navier-Stokes equations for freestream Mach number range of 2.0 - 6.0. The fluid dynamics equations are discretized in spatial coordinates using integral formulation in conjunction with finite volume method which reduces the governing equations to semi-discretized ordinary differential equations. Temporal integration is performed employing multistage Runge-Kutta time-stepping scheme. A local time step is used to achieve steady-state solution. The numerical computation is carried out on a mono-block with structured grids. The flowfield features over the sharp-tipped double cone configuration such as conical shock wave, separation region, separation and reattachment shock wave and bow shock wave in the cone region, expansion fan, and recirculation flow in the base region are well captured at Mach number 3 and 6 which are identical to the Edney VI type shock interaction. The velocity vector, Mach contours plots and variations of surface pressure coefficient along the sharp-tipped double cone configuration are analyzed at various Mach numbers. The fore-body aerodynamic drag is calculated employing computed pressure distribution. The paper presents the influence of the freestream Mach number on the flows with shock interactions over the sharp-tipped double cone geometry.

High Enthalpy Non-Equilibrium Expansion Effects in Turbulent Flow of the Conical Nozzle

High enthalpy stagnation gas can be converted into hypervelocity flow through the contraction—expansion nozzle. The enthalpy flow in the nozzle can be divided into three regions: an equilibrium region, a non-equilibrium region, and a frozen region. The stagnation gas with a total enthalpy of 13.4 MJ/kg is used to analyze the thermochemical non-equilibrium effects. At the selected conditions, the effects of a conical nozzle under different expansion angles of the expansion section, curvature radius of the throat, throat radius, and convergence angle of the convergent section are investigated. Based on the Spalart–Allmaras one-equation turbulence model with the Catris–Aupiox compressibility correction, a multi-block solver for axisymmetric compressible Navier–Stokes equations is applied to simulate the thermochemical non-equilibrium flow in several high enthalpy conical nozzles. The multi-species two-temperature equation is employed in the calculation. The results reveal three interesting characteristics: Firstly, the thermochemical non-equilibrium effects are sensitive to the maximum expansion angle and throat radius but not to the radius of throat curvature and contraction angle. Secondly, as the maximum expansion angle decreases and the throat radius increases, the flow approaches equilibrium state. When the maximum expansion angle decreases from 12° to 4°, the freezing temperature decreases from 2623 K to 2018 K. When the throat diameter increased from 10 mm to 30 mm, the freezing temperature decreased from 2442 K to 2140 K. Finally, the maximum expansion angle and throat radius not only affect the position of the freezing point but also the flow field parameters, such as temperature, Mach number, and species mass fraction.

Computational Fluid Dynamics Analysis and Design of Payload Shroud of Satellite Launch Vehicle

The main focus of the present paper is to computational fluid dynamics analysis and design of payload fairing of satellite launch vehicle at freestream Mach number range of 0.6 - 3.0. Initially, time-dependent compressible three-dimensional Euler equations are solved employing a finite volume discretization method with a multi-stage Runge-Kutta time-stepping scheme to compute surface pressure and aerodynamic coefficients at various payload fairing and at angle of attack up to 5o with an increment of 1o. Payload fairing dimensions are selected that satisfies permissible structure load on satellite launch vehicle. Detailed flowfield simulation is carried out on the selected payload fairing employing axisymmetric compressible Reynolds-average Navier-Stokes equations to assess unsteady flowfield characteristics. The numerical simulations are used to locate terminal shock on the payload fairing at transonic Mach number. Unsteady flow characteristics are used to compute acoustic load. Shock standoff distances at supersonic speeds are tabulated and compared with the analytical solution. Schlieren images and oil flow pictures are compared with experimental results and in good agreement. Aerodynamic shape optimization of satellite launch vehicle payload fairing shape has been performed to satisfy structural load at maximum drag and dynamic pressure.

The thermochemical non-equilibrium scale effects of the high enthalpy nozzle

The high enthalpy nozzle converts the high enthalpy stagnation gas into the hypervelocity free flow. The flow region of the high enthalpy nozzle consists of three parts: an equilibrium region upstream of the throat, a non-equilibrium region near the throat, and a frozen region downstream of the throat. Here we propose to consider the thermochemical non-equilibrium scale effects in the high enthalpy nozzle. By numerically solving axisymmetric compressible Navier-Stokes equations coupling with Park’s two-temperature model, the fully non-equilibrium solution is employed throughout the entire nozzle. Calculations are performed at different stagnation conditions with the different absolute scales and expansion ratio. The results of this study are twofold. Firstly, as the absolute scale and expansion ratio increase, the freezing position is delayed, and the flow approaches equilibrium. Secondly, the vibrational temperature and Mach number decrease with the increase in the nozzle scale and expansion ratio, while the speed of sound, static pressure, and translational temperature increase as the nozzle scale and expansion ratio increase.

Study of Interaction between Unsteady Supersonic Jet and Vortex Rings Discharged from Elliptical Cell

The unsteady supersonic jet formed by a shock tube with a small high-pressure chamber was used as a simple alternative model for pulsed laser ablation. Understanding the vortex ring formed by the shock wave is crucial in clarifying behavior of unsteady supersonic jet discharged from elliptical cell. The purpose of this study has been to reveal behavior of the supersonic jet and the vortex rings. The unsteady behavior of a flow is investigated numerically by solving the axisymmetric two-dimensional compressible Navier-Stokes equations. The system of the calculation is a model of laser ablation of a certain duration followed by discharge through the cell exit. The parameters for the calculations are the pressure ratio of the shock tube and the cell diameter. As a result, it was found that the diameter of the elliptical cell had the effect on the interaction between vortex rings and the jet.

Simulation of high-speed droplet impact against a dry/wet rigid wall for understanding the mechanism of liquid jet cleaning

Physical cleaning techniques are of great concern to remove particulate contamination because of their low environmental impact. One of the promising candidates is based on water jets that often involve fission into droplet fragments. Particle removal is believed to be achieved by droplet-impact-induced wall shear flow. Here, we simulate a high-speed droplet impact on a dry/wet rigid wall to investigate the wall shear flow as well as water hammer after the impact. The problem is modeled by the axisymmetric compressible Navier–Stokes equations and solved by a finite volume method that can capture both shocks and material interface. As an example, we consider the impact of a spherical water droplet (200 µm in diameter) at velocity from 30 to 50 m/s against a dry/wet rigid wall. In our simulation, we can reproduce both acoustic and hydrodynamic events. In the dry wall case, the strong wall shear appears near the moving contact line at the wetted surface. On the other hand, once the wall is covered with the liquid film, the wall shear stress gets weaker as the film thickness increases—a similar trend holds for the water-hammer shock loading at the wall. According to the simulated base flow, we compute hydrodynamic force acting on small particles that are assumed to be attached at the wall, in a one-way-coupling manner. The hydrodynamic force acting on the particles is estimated under Stokes’ assumption and compared to particle adhesion of van der Waals type, enabling us to derive a simple criterion of the particle removal.

Study of Interaction between Unsteady Supersonic Jet and Vortex Rings Discharged from Elliptical Cell

Pulsed laser ablation with an elliptical cell gives well-defined thermodynamic conditions to the growth of high-quality thin films. The unsteady supersonic jet formed by the shock tube with small high-pressure chamber was used as a simple alternative model of pulsed laser ablation. The vortex ring formed by the shock wave is important to reveal behavior of unsteady supersonic jet discharged from elliptical cell. However, there has been little effort to investigate the interaction between the vortex ring and the jet. The purpose of the present study is to investigate the behavior of the vortex rings and the jet. The experiment and numerical calculation were carried out by schlieren method and by solving the axisymmetric two-dimensional compressible Navier-Stokes equations, respectively. The system of the calculation and the experiment is a model of laser ablation of a certain duration followed by a discharging process through the exit. Moreover, a parametric study was performed to demonstrate the effect of pressure ratio on the interaction among vortex rings and the supersonic jet. The interaction between the supersonic jet and the vortex rings increased the velocity of the supersonic jet up to the magnitude of the velocity at the center of the vortex rings. Closing a distance between the vortex ring and the jet is higher interaction between the vortex rings.

Numerical Investigation of Self-Ignition Characteristics of Solid-Fuel Scramjet Combustor

The self-ignition characteristics of a solid fuel under supersonic flow have been investigated theoretically and numerically. Time-dependent two-dimensional axisymmetric compressible Navier–Stokes equations and species transport equations are solved numerically. Turbulence closure is achieved using the shear stress transport model. Polymethylmethacrylate fuel and a global one-step reaction mechanism are used in this study. The reaction rate is determined by finite-rate chemical kinetics with the turbulence–chemistry interaction modeled by an eddy-dissipation model. The numerical results generally agree with the experimental data in the published literature. The flame spread and pressurization during the self-ignition of polymethylmethacrylate in the combustor have been studied. The effect of inlet flow conditions and the geometry of the combustor on self-ignition behavior have been analyzed. Three stages of flame spread (namely, heat accumulation, secondary recirculation zone self-ignition, and orderly flame spreading) are identified during the self-ignition transient. Pressurization in the combustor during the third stage is evident. Self-ignition of the solid fuel can be affected by three kinds of limits: namely, lean oxygen, rich oxygen, and total temperature inlet conditions. A relatively long and deep cavity is beneficial to establish self-ignition. It is suggested that a step-type cavity can enhance self-ignition performance.

Numerical simulations of convergent-divergent nozzle and straight cylindrical supersonic diffuser

The flowfields inside a contour and a conical nozzle exhausting into a straight cylindrical supersonic diffuser are computed by solving numerically axisymmetric turbulent compressible Navier-Stokes equations for stagnation to ambient pressure ratios in the range 20 to 34. The diffuser inlet-to-nozzle throat area ratio and exit-to-throat area ratio are 21.77, and length-to-diameter ratio of the diffuser is 5. The flow characteristics of the conical and contour nozzle are compared with the help of velocity vector and Mach contour plots. The variations of Mach number along the centre line and wall of the conical nozzle, contour nozzle and the straight supersonic diffuser indicate the location of the shock and flow characteristics. The main aim of the present analysis is to delineate the flowfields of conical and contour nozzles operating under identical conditions and exhausting into a straight cylindrical supersonic diffuser.

Vortical flow structure of thermoacoustic oscillations in a closed tube

Spontaneous thermoacoustic oscillations of a gas in a closed cylindrical tube are studied. Numerical simulations of the flow field in the tube on which a temperature gradient along the axis is imposed are performed by solving the axisymmetric compressible Navier–Stokes equations. The wall temperature of the hot part near both ends (300 K) and that of the cold part near the center (20 K) are fixed. The computations are done for various values of the length ratio ξ of the hot part to the cold part between 0.3 and 1.0, and steady oscillatory states are obtained. These states are divided into three groups according to the magnitude of the pressure amplitude. The state in each group has distinguished features of the flow field. We analyze the effect of vortices on the structure of the temperature distribution. The difference of the boundary layer thickness between the hot part and the cold part is shown to play an important role.

Heat Transfer in Turbulent Boundary Layers of Conical and Bell Shaped Rocket Nozzles with Complex Wall Temperature

The objective of this article is to perform detailed analysis of heat transfer in accelerated supersonic nozzle flows with cooled walls. Since most of the heat transfer occurs near the nozzle walls, correct prediction of the boundary layer under strong adverse pressure gradient is therefore required to achieve high fidelity numerical prediction. In this study, a two-equation SST-V turbulence model is used in conjunction with a second-order explicit-implicit method to solve axisymmetric compressible Navier-Stokes equations. First, the effect of the inlet pipe diameter and the associated contraction area on the heat transfer is studied in nozzles having 15° and 30° diverging half-angles. Then, a series of computations are conducted to examine the efficiency of the use of a constant wall temperature as a function of the stagnation temperature in heat transfer calculations. The computations are performed for nominal stagnation pressure of 208 N/cm2 and stagnation temperature of 539 K. The computed heat-transfer coefficients are compared to experimental data and a good agreement is found. A pronounced increase in the throat heat transfer coefficient peak is observed accompanied with a reduction in the contraction area ratio. Also, the peak of the heat transfer coefficient for the pipe inlet diameter of 7.8 cm is found to be 70% higher than the one related to the pipe of 16.51 cm diameter.

Coupled Navier–Stokes/Direct Simulation Monte Carlo Simulation of Multicomponent Mixture Plume Flows

A coupled particle-continuum simulation method is presented to compute the multicomponent mixture plume flow from the attitude-control engines of a satellite through the use of the Navier–Stokes solver and direct simulation Monte Carlo method. In this study, the implicit hybrid flux iteration scheme for the axisymmetric compressible Navier–Stokes equations is constructed to solve the inner flowfield of the nozzle. The Navier–Stokes computing technique is implemented considering slip boundary theory for the near-continuum slip flow near the nozzle exit. Direct simulation Monte Carlo methods for the axisymmetric core plume and three-dimensional far-field plume flows of multispecies mixture gas are developed by using a radial weighted factor, by tracking the molecular motion trajectory, and with secondary Cartesian and unstructured cell processing techniques. A multiregion decomposition and hybrid Navier–Stokes/direct simulation Monte Carlo algorithm with two-way and one-way coupling is established to solve the internal and external flow of the thruster, including the near-field, far-field, backflow, and gas-surface contaminated regions. After constructing the coupled Navier–Stokes/DSMC simulation scheme, the present method is applied to solve the plume flowfield and impinging contamination effects from the satellite attitude-control engine and solar array panel; the simulation results correspond well with the experimental measurements from the low-density wind tunnel and the theoretical predictions. To study the contamination effects produced by the multicomponent mixture plume, the current method is employed to simulate a five-component mixture plume flowfield of tens of meters from two representative attitude-control engines installed in different locations of a satellite in orbit. The results of the nonreacting multicomponent flow with equivalent mole fractions can be achieved with a one-component gas simulation if the species is assigned the properties of the mixture. The deposition rate of the plume flows produced by the two teamwork engines can be superimposed on each other. This method can be used to efficiently predict the impingement contamination effects of the gas-fired mixture plume flow on the satellite and solar array panels.

Study of Unsteady Supersonic Jet Using Shock Tube with Small High-Pressure Chamber with Elliptical Cell

The unsteady supersonic jet formed by the shock tube with small high-pressure chamber was used as a simple alternative model of Pulsed laser ablation. Since the pressure of the shock wave formed by Pulsed laser ablation is very high, the interaction between the shock wave and the plume is important for the formation of nanoparticles especially under high background gas pressure. An elliptical cell is used for controlling the shock wave. The shock wave discharged in one focal point converge another focal point. Numerical calculation was carried out by solving the axisymmetric two-dimensional compressible Navier-Stokes equations. The wall static pressure monitored at the center of substrate is used to evaluate the influence of the shape and the pressure ratio on the behavior of the shock wave, the plume, and their interaction. As a result, it was found that the vortex ring was generated downstream of the cell, which is followed by the discharging shock wave from the cell exit. It was also found that the influence of the vortex ring on the wall static pressure at the center of the substrate.

Numerical heat transfer study around a spiked blunt-nose body at Mach 6

An aerospike attached to a blunt body significantly alters its flowfield and influences aerodynamic drag at high speeds. The dynamic pressure in the recirculation area is highly reduced and this leads to the decrease in the aerodynamic drag. Consequently, the geometry of the aerospike has to be simulated in order to obtain a large conical recirculation region in front of the blunt body to get beneficial drag reduction. Axisymmetric compressible Navier–Stokes equations are solved using a finite volume discretization in conjunction with a multistage Runge–Kutta time stepping scheme. The effect of the various types of aerospike configurations on the reduction of aerodynamic drag is evaluated numerically at a length to diameter ratio of 0.5, at Mach 6 and at a zero angle of incidence. The computed density contours are showing satisfactory agreement with the schlieren pictures. The calculated pressure distribution on the blunt body compares well with the measured pressure data on the blunt body. Flowfield features such as formation of shock waves, separation region and reattachment point are examined for the flat-disc spike and on the hemispherical disc spike attached to the blunt body. One of the critical heating areas is at the stagnation point of a blunt body, where the incoming hypersonic flow is brought to rest by a normal shock and adiabatic compression. Therefore, the problem of computing the heat transfer rate near the stagnation point needs a solution of the entire flowfield from the shock to the spike body. The shock distance ahead of the hemisphere and the flat-disc is compared with the analytical solution and a good agreement is found between them. The influence of the shock wave generated from the spike is used to analyze the pressure distribution, the coefficient of skin friction and the wall heat flux facing the spike surface to the flow direction.

미세 간극을 지나는 축대칭 내부 유동의 압축성 효과 분석

In this study, a simplified axisymmetric model is proposed for the problem of compressible internal flow past a microgap. Using numerical and experimental methods, the phenomena of choked flows are observed; these flows are induced by the acceleration of subsonic flows past the narrow cross-section of an annular shape made by a microgap. The relation between mass flow rate and differential pressure is obtained, and by comparing the result with experimental results, the reliability of the numerical results is discussed. The generation of a supersonic jet flow and its diffraction are visualized by performing the numerical analysis of axisymmetric compressible Navier-Stokes equations. This investigation greatly extends the physical understanding of the axisymmetric compressible flow, which has a wide range of engineering applications, e.g., in the case of valves in automotive power systems.

Numerical simulation of the flow field over conical, disc and flat spiked body at Mach 6

AbstractA forward facing spike attached to a hemispherical body significantly changes its flow field and influences aerodynamic drag and wall heat flux in a high speed flow. The dynamic pressure in the recirculation area is highly reduced and this leads to the decrease in the aerodynamic drag and heat load on the surface. Consequently, the geometry, that is, the length and shape of the spike, has to be simulated in order to obtain a large conical recirculation region in front of the blunt body to get beneficial drag reduction. It is, therefore, a potential candidate for aerodynamic drag reduction for a future high speed vehicle. Axisymmetric compressible laminar Navier-Stokes equations are solved using a finite volume discretisation in conjunction with a multistage Runge-Kutta time stepping scheme. The effect of the spike length and shape, and the spike nose configuration on the reduction of drag is numerically evaluated at Mach 6 at a zero angle-of-attack. The computed density contours agree well with the schlieren images. Additional modification to the tip of the spike to get different types of flow field such as the formation of a shock wave, separation area and reattachment point are examined. The spike geometries include the conical spike, the flat-disk spike and the hemispherical disk spike of different length to diameter ratios attached to the blunt body.

Behavior of shock waves formed by unsteady supersonic jet injected into cell

The unsteady behavior of flow driven by a jet suddenly injected into a cell is numerically studied by solving the axisymmetric two-dimensional compressible Navier-Stokes equations. The system of the calculation is a model of the laser ablation of a certain duration followed by a discharging process through the exit hole at the down-stream end of the cell. In the calculations, the contour of the cell is changed while other parameters such as the Mach number of the jet, its duration, and the diameter of the cell exit are fixed. Monitoring the velocity at the exit hole is used to investigate the influence of the shape on the interaction between the shock wave and the jet. As the result, it was found that the velocity peak value and its arrival time at the downstream end of the cell exit are determined by the diameter of the cell.

Study of Interaction between Unsteady Supersonic Jet and Shock Waves in Elliptical Cell

The unsteady behavior of a flow driven by a jet suddenly injected into cells is investigated numerically by solving the axisymmetric two-dimensional compressible Navier-Stokes equations. The system of the calculation is a model of laser ablation of a certain duration followed by a discharging process through the exit hole at the downstream end of the cell. The parameters for the calculations are the diameter of the cell, the exit diameter of the cell and the duration of the injected jet. The velocity monitored at the exit hole is used to evaluate the influence of the shape and the exit diameter of the cell on the shock wave, the plume, and their interactions. As a result, it was found that the position in which the shock wave converges and the focal point of the elliptical cell are almost the same for various parameters. The spreading shock wave that converged at the geometrical focal point reflects from the cell wall and this shock wave alternate between the jet injection and the cell exit. It was also found that the number of peaks and the maximum velocity downstream of the cell exit are roughly determined by the contour of the walls of the elliptical cell.

Study of the effects of unsteady downstream conditions on the gas flow through a critical nozzle

The present study addresses a computational result of unsteady gas flow through a critical nozzle. The axisymmetric unsteady compressible Navier-Stokes equations are solved using a finite volume method that makes use of the second-order upwind scheme for spatial derivatives and the multi-stage Runge-Kutta integral scheme for time derivatives. The steady solutions of the governing equation system are validated with the previous experimental data to ensure that the present computational method is valid to predict the critical nozzle flows. In order to simulate the effects of back-pressure fluctuations on the critical nozzle flows, an excited pressure oscillation with an amplitude and frequency is assumed downstream of the exit of the critical nozzle. The results obtained show that, for low Reynolds numbers, the unsteady effects of the pressure fluctuations can propagate upstream of the throat of the critical nozzle, thus giving rise to the applicable fluctuations in mass flow rate through the critical nozzle, while, for high Reynolds numbers, the pressure signals occurring at the exit of the critical nozzle do not propagate upstream beyond the nozzle throat. For a low Reynolds number, it is found that the sonic line near the throat of the critical nozzle markedly fluctuates with time, providing an important mechanism for pressure signals to propagate upstream of the nozzle throat, even in choked flow conditions. The present study is the first investigation to clarify the unsteady effects on the critical nozzle flows.