Compressible Flow and Transonic Shock in a Diverging Nozzle

The paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible Euler system. Consider a nozzle having a shape as a diverging truncated sector with generic opening angle: if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow, and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned, then a transonic shock appears in the nozzle. To determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation. We prove that the receiver pressure can uniquely determine the location of the transonic shock, as well as the flow behind the shock. Such a conclusion was conjectured by Courant and Friedrichs, and is confirmed theoretically in this paper for the divergent nozzles. The main advantage of this paper compared with the previous studies on this subject is that the section of the nozzle is allowed to vary substantially, while the transonic shock is not assumed to pass a fixed point. The situation coincides with the requirement in Courant-Friedrichs’ conjecture. To describe the compressible flow we use the full Euler system, which is purely hyperbolic in the supersonic region and is elliptic-hyperbolic in the subsonic region. Solving the free boundary value problem of an elliptic-hyperbolic problem forms the main part of this paper. In our demonstration some new approaches, including the introduction of a pseudo-free boundary problem and the corresponding relaxation, design of a delicate double iteration scheme, are developed to overcome the difficulties caused by the divergence of the nozzle.     

导弹超声速尾退分离干扰特性试验

超/高超声速尾退分离在防热、保形、隐身、多次投放、回收等方面具有明显优势,有望成为高超声速飞行器载荷投放的优选方案。由此面临一类新的多体分离问题:超/高超声速尾退分离问题（aft super/hypersonic ejection separation,ASES）。超/高超声速尾退分离问题本质上是带空腔底部流动与多体分离构成的耦合问题,具有流场结构复杂、气动非定常非线性非对称效应显著的特点。针对超声速尾退分离问题,采用网格测力和轨迹捕获（captive trajectory system,CTS）风洞试验方法探索了尾退分离干扰流场的结构,发现可根据流场结构和舵效变化分为低速-亚声速无激波、高亚声速-跨声速弱激波、超声速激波和准自由流弱干扰4种典型干扰特征,揭示了尾流场影响后不同区域的全弹气动特性和舵效特性以及控制律、攻角、高度和Mach数对分离位移和姿态的影响规律。相关结论将有助于增强对尾退分离问题的认识,对尾退分离技术的工程实践具有参考价值。      

Analytical and numerical prediction of harmonic sound power in the inlet of aero-engines with emphasis on transonic rotation speeds

Tone noise radiated through the inlet of a turbofan is mainly due to rotor-stator interactions at subsonic regimes (approach flight), and to the shock waves attached to each blade at supersonic helical tip speeds (takeoff). The axial compressor of a helicopter turboshaft engine is transonic as well and can be studied like turbofans at takeoff. The objective of the paper is to predict the sound power at the inlet radiating into the free field, with a focus on transonic conditions because sound levels are much higher. Direct numerical computation of tone acoustic power is based on a RANS (Reynolds averaged Navier–Stokes) solver followed by an integration of acoustic intensity over specified inlet cross-sections, derived from Cantrell and Hart equations (valid in irrotational flows). In transonic regimes, sound power decreases along the intake because of nonlinear propagation, which must be discriminated from numerical dissipation. This is one of the reasons why an analytical approach is also suggested. It is based on three steps: (i) appraisal of the initial pressure jump of the shock waves; (ii) 2D nonlinear propagation model of Morfey and Fisher; (iii) calculation of the sound power of the 3D ducted acoustic field. In this model, all the blades are assumed to be identical such that only the blade passing frequency and its harmonics are predicted (like in the present numerical simulations). However, transfer from blade passing frequency to multiple pure tones can be evaluated in a fourth step through a statistical analysis of irregularities between blades. Interest of the analytical method is to provide a good estimate of nonlinear acoustic propagation in the upstream duct while being easy and fast to compute. The various methods are applied to two turbofan models, respectively in approach (subsonic) and takeoff (transonic) conditions, and to a Turbomeca turboshaft engine (transonic case). The analytical method in transonic appears to be quite reliable by comparison with the numerical solution and with available experimental data.     

On the outflow conditions for spectral solution of the viscous blunt-body problem

The purpose of this paper is to study and identify suitable outflow boundary conditions for the numerical simulation of viscous supersonic/hypersonic flow over blunt bodies, governed by the compressible Navier–Stokes equations, with an emphasis motivated primarily by the use of spectral methods without any filtering. The subsonic/supersonic composition of the outflow boundary requires a dual boundary treatment for well-posedness. All compatibility relations, modified to undertake the hyperbolic/parabolic behaviour of the governing equations, are used for the supersonic part of the outflow. Regarding the unknown downstream information in the subsonic region, different subsonic outflow conditions in the sense of the viscous blunt-body problem are examined. A verification procedure is conducted to make out the distinctive effect of each outflow condition on the solution. Detailed comparisons are performed to examine the accuracy and performance of the outflow conditions considered for two model geometries of different surface curvature variations. Numerical simulations indicate a noticeable influence of pressure from subsonic portion to supersonic portion of the boundary layer. It is demonstrated that two approaches for imposing subsonic outflow conditions namely (1) extrapolating all flow variables and (2) extrapolation of pressure along with using proper compatibility relations are more suitable than the others for accurate numerical simulation of viscous high-speed flows over blunt bodies using spectral collocation methods.     

A pressure-based algorithm for multi-phase flow at all speeds

A new finite volume-based numerical algorithm for predicting incompressible and compressible multi-phase flow phenomena is presented. The technique is equally applicable in the subsonic, transonic, and supersonic regimes. The method is formulated on a non-orthogonal coordinate system in collocated primitive variables. Pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density, which become very small at low Mach numbers. The pressure equation is derived from overall mass conservation. The performance of the new method is assessed by solving the following two-dimensional two-phase flow problems: (i) incompressible turbulent bubbly flow in a pipe, (ii) incompressible turbulent air–particle flow in a pipe, (iii) compressible dilute gas–solid flow over a flat plate, and (iv) compressible dusty flow in a converging diverging nozzle. Predictions are shown to be in excellent agreement with published numerical and/or experimental data.     

A third-order finite-volume residual-based scheme for the 2D Euler equations on unstructured grids

A residual-based (RB) scheme relies on the vanishing of residual at the steady-state to design a transient first-order dissipation, which becomes high-order at steady-state. Initially designed within a finite-difference framework for computations of compressible flows on structured grids, the RB schemes displayed good convergence, accuracy and shock-capturing properties which motivated their extension to unstructured grids using a finite volume (FV) method. A second-order formulation of the FV–RB scheme for compressible flows on general unstructured grids was presented in a previous paper. The present paper describes the derivation of a third-order FV–RB scheme and its application to hyperbolic model problems as well as subsonic, transonic and supersonic internal and external inviscid flows.     

E-H Type Mach Configuration and its Stability

This paper is devoted to the study of the classification and stability of Mach configurations. In the Mach reflection of a shock wave by a rigid wall the flow behind the reflected shock front can be supersonic or subsonic. Correspondingly, the Mach configuration can be classified as E-E type and E-H type. In this paper we proved the stability of E-H type Mach configurations provided the reflected shock is weak. The result is the complement of the stability of E-E type Mach configurations in our earlier work.The stability of E-H Mach configurations is reduced to a generalized Tricomi problem of a nonlinear mixed type equation. The linearization of this problem is a generalized Tricomi problem of a linear Lavrentiev-Bitsadze’s mixed type equation. In order to prove the existence of such a problem we first use the equation in the hyperbolic region and a condition on the reflected shock to establish a relation of the unknown function and its normal derivative on the sonic line. Then the generalized Tricomi problem is reduced to a nonlocal boundary value problem in the elliptic region. Careful analysis gives necessary estimates and the solvability of the linearized problem. It leads to the existence of the solution to the corresponding nonlinear problem, which implies the required stability of E-H Mach configurations.     

Acoustic radiation from a simple structure in supersonic flow

It is well established that fluid flow can have significant effects on structural acoustic behavior, as is the fact that induced coupling between discrete modes of vibration becomes significant as flow velocity increases. To date, work in this area has been confined to subsonic flows, with the effects on sound radiation efficiency and sound power radiation quantified and compared for various subsonic flow speeds. The purpose of this work is to study the effects that supersonic flow has on these structural acoustic phenomena, along with an investigation of the uncoupled behavior of single modes in the transonic region. Theoretical development of the equations governing the vibration of a simply supported plate in an infinite baffle and an aerodynamic system that models a semi infinite flowing medium along with the method for coupling these systems is included. Computational results are presented illustrating the behavior of the uncoupled modes in the transonic region and the uncoupled and coupled effects on the structural response and sound power radiation as well as a study of the radiation efficiency of the coupled system.     

Temperature of a thermally insulated surface in a gas flow

An equation is derived for determining the temperature of a thermally insulated surface in a gas flow. The equation does not contain any empirical coefficients. The derivation is based on allowance for the work done by a jet arrested at an obstructing surface on the surrounding flow layers. The application of the equation to subsonic and supersonic flows is discussed. Zh. Tekh. Fiz. 68, 134–135 (April 1998)     

Modelling atmospheric flows with adaptive moving meshes

An anelastic atmospheric flow solver has been developed that combines semi-implicit non-oscillatory forward-in-time numerics with a solution-adaptive mesh capability. A key feature of the solver is the unification of a mesh adaptation apparatus, based on moving mesh partial differential equations (PDEs), with the rigorous formulation of the governing anelastic PDEs in generalised time-dependent curvilinear coordinates. The solver development includes an enhancement of the flux-form multidimensional positive definite advection transport algorithm (MPDATA) — employed in the integration of the underlying anelastic PDEs — that ensures full compatibility with mass continuity under moving meshes. In addition, to satisfy the geometric conservation law (GCL) tensor identity under general moving meshes, a diagnostic approach is proposed based on the treatment of the GCL as an elliptic problem. The benefits of the solution-adaptive moving mesh technique for the simulation of multiscale atmospheric flows are demonstrated. The developed solver is verified for two idealised flow problems with distinct levels of complexity: passive scalar advection in a prescribed deformational flow, and the life cycle of a large-scale atmospheric baroclinic wave instability showing fine-scale phenomena of fronts and internal gravity waves.     

Micronozzle Chocking under Diffusion Combustion of Hydrogen

The results of experimental investigations of the micronozzle-chocking phenomenon under diffusion combustion of a hydrogen microjet at a high outflow velocity in the case of ignition of hydrogen near the nozzle cut are presented. It is found that the cause of micronozzle chocking is the heating of the nozzle walls from the flame-neck region retained up to transonic velocities and preventing nozzle cooling and the passage of the hydrogen jet to the supersonic-flow velocity. It is shown that hydrogen ignition far from the nozzle cut with a developed hydrogen supersonic flow into the flooded space leads to the disappearance of the flameneck region, flame detachment from the nozzle cut, and, correspondingly, termination of the nozzle heating and the possibility of the microjet coming out at the supersonic-flow velocity for the hydrogen jet. It is established that the flame-neck region is a stabilizing factor for the subsonic combustion of a hydrogen microjet up to transonic velocities. In the second case, the presence of supersonic cells stabilizes the supersonic diffusion combustion of the hydrogen microjet.     

超音速等离子体炬的数值模拟

介绍了描述大气压下超音速等离子体炬内等离子体特性的磁流体力学模型，在二维近似下，对会聚-扩展型喷口等离子体炬进行了数值模拟，获得了等离子体炬内等离子体速度、温度、压力以及马赫数的分布.结果表明超音速等离子体炬内的流场特性可以分为亚音速、跨音速和超音速三个明显的区域. 关键词： 等离子体炬 磁流体力学 数值模拟     

On the Global Existence and Stability of a Multi-Dimensional Supersonic Conic Shock Wave

We establish the global existence and stability of a three-dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle. The flow is described by a 3-D steady potential equation, which is multi-dimensional, quasilinear, and hyperbolic with respect to the supersonic direction. Making use of the geometric properties of the pointed shock surface together with the Rankine–Hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone, we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new Hardy-type inequality on the shock surface. Based on this, we prove that a multi-dimensional conic shock attached to the vertex of the cone exists globally when the Mach number of the incoming supersonic flow is sufficiently large. Moreover, the asymptotic behavior of the 3-D supersonic conic shock solution, which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone, is also explicitly given.     

An unsteady pseudoshock model for barotropic gas flow

A mathematical unsteady pseudoshock model describing the continuous transition from supersonic to subsonic flow is constructed for a barotropic gas flow in a long flat channel or a nozzle. The model is based on a two-layer scheme of flow with mass transfer including a potential supersonic core and a turbulent boundary layer.     

The dynamic characters of excitations in aone-dimensional Frenkel--Kontorova model

A one-dimensional （1D） Frenkel-Kontorova （FK） model is studied numerically in this paper, and two new analytical solutions （a supersonic kink and a nonlinear periodic wave） of the Sine-Gordon （SG） equation （continuum limit approximation of the FK model） are obtained by using the Jacobi elliptic function expansion method. Taking these new solutions as initial conditions for the FK model, we numerically find there exist the supersonic kink and the nonlinear periodic wave in these systems and obtain a lot of interesting and significant results. Moreover, we also investigate the subsonic kink and the breather in these systems and obtain some new feature.     

Theoretical and Experimental Investigations of the Flow Field for Asymmetric Dual-Flow Interrupter Nozzles

Experimental investigations have indicated that electrode vapor can have a significant negative effect in thermal interruption speed for the gas-blast circuit breakers. This electrode vapor contamination can be minimized by the use of asymmetric dual-flow nozzle configuration. A computer program was developed to design the nozzle and electrode geometries of the asymmetric dual-flow interrupter and to calculate both the subsonic and supersonic cold flow fields. The Variational Principle of the finite element method, together with a Newton-Raphson iterative scheme, was used to solve the continuity equation for compressible flow. The supersonic flow field in the conical nozzle was calculated by the one-dimensional flow relationship. Two asymmetric dual-flow nozzle models were constructed to investigate the effects of orifice opening and nozzle divergent angle. The cold flow experiments were conducted in the Rensselaer Polytechnic Institute (RPI) Transonic and Supersonic Wind Tunnel Laboratory. Various upstream-to-downstream nozzle pressure ratios were used to obtain the subsonic and supersonic experimental flow-field data. The experimental flow measurements were correlated with the calculated values to validate the computer program.     

基于熵条件二阶差分格式的嵌套网格分区算法

以带熵条件的二阶TVD格式为基本格式,设计了三维曲线坐标下的Euler方程数值计算方法.对多个物体使用分区嵌套网格设计了相应的分区算法.所建立的算法适用于处理物体相互分离以及相互接触的定常与非定常流动问题.给出了绕两圆柱的超声速流动在两圆柱处于不同位置时的详细结果,并模拟了装配鸭舵火箭弹在超声速旋转飞行时的气动特性.     

A quadrature-based LES/transported probability density function approach for modeling supersonic combustion

The joint-scalar probability density function (PDF) approach provides a comprehensive framework for large eddy simulation (LES) based combustion modeling. However, currently available stochastic approaches for solving the high-dimensional PDF transport equation can be error prone and numerically unstable in highly compressible shock-containing flows. In this work, a novel Eulerian approach called the direct quadrature method of moments (DQMOM) is developed for evolving the PDF-based supersonic combustion model. The DQMOM technique uses a set of scalar transport equations with specific source terms to recover the PDF. The new technique is coupled to a compressible LES solver through the energy equation. The DQMOM approach is then used to simulate two practical flow configurations: a supersonic reacting jet and a cavity-stabilized supersonic combustor. Comparisons with experimental data demonstrate the predictive accuracy of the method.     

A preconditioned Krylov technique for global hydrodynamic stability analysis of large-scale compressible flows

The combination of iterative Krylov-based eigenvalue algorithms and direct numerical simulations (DNS) has proven itself an effective and robust tool in solving complex global stability problems of compressible flows. A Cayley transformation is required to add flexibility to our stability solver and to allow access to specific parts of the full global spectrum which would be out of reach without such a transformation. In order to robustify the overall global stability solver an efficient ILU-based preconditioner has been implemented. With this Cayley-transformed DNS-based Krylov method two flow cases were successfully investigated: (i) a compressible mixing layer, a rather simple but well-known problem, which served as a test case and (ii) a supersonic flow about a swept parabolic body, a challenging large-scale flow configuration.     

Supersonic flow in ablative discharge capillaries

The Mach number equation for plasma flow in ablative discharge capillaries was derived from a set of 1-D flow equations which incorporate the ablation and Ohmic dissipation processes. By examining this equation it is concluded that, while in the case of a capillary with a constant cross section the flow is subsonic and the sound velocity can be reached only at the open end, there is a possibility to reach supersonic plasma velocities inside a capillary with a monotonically increasing cross-sectional area.