Structure of Time-Dependent Transonic Flows in Plane Channels

The evolution and interaction of supersonic zones and the shock waves closing them are considered on the basis of the Lin-Reissner-Tsien equation for flow in plane channels with a local bottleneck.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 168–179.Original Russian Text Copyright © 2005 by Bibik, Duesperov, and Popov.     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Behavior of sonic lines in a shock layer behind a detached shock wave

A numerical investigation is made into the formation of local supersonic zones in the subsonic flow region between a detached shock wave and the surface of the body in the case of supersonic three-dimensional flow over conical bodies with opening angle _k = 120 ° of the cone in the range of Mach numbers M = 2.5–15.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No.4 pp. 143–145, July–August, 1979.We thank G. I. Petrov for suggesting the problem and for helpful advice and O. M. Belotserkovskii for constant interest in the work.     

NUMERICAL SIMULATION OF THREE-DIMENSIONAL SHOCK-SHOCK INTERACTIONS ON A BLUNT BODY

A numerical study is conducted to simulate the effects of extraneous shock impingement on a blunt body in viscous hypersonic flow. The interaction of extraneous shock with the leading-edge shock results in a very complex flow field that contains local regions of high pressure and intense heating. The heating and pressure can be orders of magnitude higher than the peak values in the absence of shock impingement. The flow field is calculated by solving thin-layer Navier-Stokes equations with a finite-volume flux splitting technique developed by van Leer. For a zero or small sweep of the body, a type IV interaction occurs, which produces a lambda shock structure with a supersonic jet embedded in the otherwise subsonic flow; for a moderate sweep of about 25^°, a type V interaction occurs in which a subsonic shear layer sandwiched in supersonic flow is produced with a transmitted shock. In the present study, both type IV and type V interactions are investigated. Results of the present numerical investigation are compared with available experimental results. For the present conditions, the peak pressure is 2.2 times the unimpinged stagnation point pressure and the peak heating is 3 times the unimpinged stagnation point heating. The flow for a type IV interaction is found to be unsteady.     

Existence of supersonic zones in three-dimensional separation flows

Up till now the region of three-dimensional separation flows which occur with supersonic flow past obstacles has received insufficient study. Supersonic flow with a Mach number of 2.5 past a cylinder mounted on a plate was studied in [1]. A local zone with supersonic velocities was found in the reverse subsonic flow region ahead of the cylinder. Its presence is explained by the three-dimensional nature of the flow. Similar supersonic zones are not observed in the case of supersonic flow over plane and axisymmetric steps.The present paper presents the results of experimental studies whose objective was refinement of the flow pattern ahead of a cylinder on a plate and the study of the local supersonic zones.The experiments were performed in a supersonic wind tunnel with a freestream Mach number M₁=3.11. The 24-mm-diameter cylinder with pressure taps along the generating line was mounted perpendicular to the surface of a sharpened plate. The distance from the plate leading edge to the cylinder axis wasl ₀=140 mm. The plate was pressure tapped along the flow symmetry axis. The Reynolds number was Rl ₀=u₀ l ₀/v ₁, Rl ₀=1.87.10⁷, where u₁ andv ₁ are the freestream velocity and the kinematic viscosity, respectively. The pressures were measured using a Pilot probe with internal and external diameters of 0.15 and 0.9 mm, respectively.The probe was displaced in the flow symmetry plane at a distance of 1.6 mm from the plate surface and at a distance of 1.1 mm along the leading generator of the cylinder. The flow on the surface of the plate and cylinder was studied with the aid of a visualization composition and the flow past the model was photographed with a schlieren instrument. Typical patterns of the visualization composition distribution and the pressure distribution curves over the plate surface, and also photographs of the flow past the model, are shown in [1].     

Stabilization of propane combustion in a supersonic air flow using a nonequilibrium longitudinal discharge and a coaxial local low-pressure zone

The results of the experimental investigation of the flow in the vicinity of an electric-discharge module having a low aerodynamic drag and intended for igniting hydrocarbon fuel and stabilizing its burning in a supersonic flow at low initial static temperature and pressure are presented. The distinctive feature of the module is that the combustion zone is not attached to the combustion chamber walls. Due to a certain geometric connection between different regions of the module anode, an interference of shock and expansion waves occurs in its vicinity. This leads to the formation of a local longitudinal low-pressure zone behind the anode, the convergence of individual fuel jetlets injected through orifices in the anode in this zone, the formation precisely there of a longitudinal nonequilibrium discharge, and the intensification of fuel mixing and plasmochemical reactions. The gasdynamic features of supersonic flow past the module are numerically investigated. The dynamics of electrical discharge formation and the combustion zones thus formed are studied under particular conditions. The data on the stagnation temperature distribution in the discharge wake are obtained.     

Calculation of supersonic steady-state flow in axisymmetric nozzles with an upwind difference scheme

This paper is on the application of the upwind difference scheme proposed by the author^[1] to the calculation of supersonic steady-state flow in axisymmetric nozzles. The upwind scheme is conservative (or weakly conservative), it yields results approximating those from the characteristic relations, and it has corresponding boundary difference schemes. The entropy phenomenon in the calculation of shock reflection on boundaries with the shock-capturing method will be discussed and a correction of this phenomenon will be proposed. From numerical experiments on an arbitrary nozzle, it is seen that the upwind difference scheme, its corresponding boundary scheme, and the improved treatment of shock reflection work well for the calculation of supersonic steady-state flow in axisymmetric nozzles.     

Correlation of the separation region length in shock wave/channel boundary layer interaction

 Results of an experimental investigation of the characteristics of a separation region induced by the interaction of an externally generated oblique shock with the turbulent boundary layer formed in a rectangular half channel are discussed. The experiments were carried out in the supersonic wind tunnel of the Institute of Theoretical and Applied Mechanics SB RAS at a free-stream Mach number M _∞=3.01 over a range of Reynolds numbers Re ₁=(9.7–47.5)×10⁶ m^-1 and at zero incidence and zero yaw of the model. Particular attention is paid to the size of the zone of the upstream propagation of disturbances (upstream influence region) under different experimental conditions: with varied values of the shock wave strength, half channel width, and Reynolds number. It is shown, in particular, that the normalized upstream influence region length as a function of inclination angle of the shock generator in a rectangular half channel is readily approximated by a simple exponential function. In support of the known reference data obtained for supersonic numbers M _∞ and moderate Re in other configurations, it is also shown that the upstream influence region length decreases with increasing Reynolds number. Generalization of experimental data on the length of the upstream influence region formed in similar geometric configurations is possible using an additional reference linear scale which is the distance from the leading edge of the shock generator to the exposed surface. A substantial dependence of the reference dimensions of separation region on the half channel width is also established. Received: 20 January 1997/Accepted: 30 June 1997     

Influence of the initial boundary layer on base pressure

The base pressure p_b, for an initial turbulent boundary layer, is determined for supersonic nonisothermal flow about a two-dimensional backward-facing step. This problem has been considered previously. In solving it in [1, 2], use was made of the Korst condition [3], which assumes equality of the total pressure p_j ^* on the line of constant mass to the pressure behind the closing oblique shock. However the pressure at the reattachment section p_* is lower than that behind the closing shock by 30–40% [4], and consequently the Korst condition is inaccurate. Therefore in the references cited only qualitative agreement with experiment was obtained. In contrast with [1, 3], Nash [5] introduces p_*; however, it is defined by an empirical coefficient. In the present study, to find p_b we make use of the condition of conservation of mass in the base region, written in the form of the equality p_j ^*=p_*, where p_* is defined from the assumption of minimum thickness of the dissipative layer at the reattachment section.Satisfactory agreement with the available experimental data is obtained without the use of correction factors. In the simplest case, when the thickness of the oncoming boundary layer ₁=0, the proposed method is no more complex than that of Korst. The determination of the base pressure with ₁=0 is considered in §1, and the determination with ₁>0 is considered in §2.     

Results of an experimental and numerical study of the aerodynamic heating of the undersurface of delta wings with sharp leading edges at mach numbers M∞=6.1 and 8

The heat transfer between a supersonic flow and the undersurface of delta wings with leading-edge sweep angles x=65 and 70° is investigated in a shock tunnel at angles of attack 15°. The supersonic inviscid flow over these wings in regimes in which the bow shock is attached to the sharp leading edges is calculated numerically. The compressible boundary layer problem is solved for the calculated inviscid flow fields in the laminar, transition and turbulent flow zones. The calculations and experimental values of the heat flux on the surface of the wings are compared. The calculations are in satisfactory agreement with the experimental data in the laminar and transition zones, but diverge significantly (by up to 20%) in the turbulent zone.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–188, July–August, 1991.The authors wish to thank A. A. Golubinskii for assisting with the solution of the problem of supersonic inviscid gas flow over a wing.     

Contouring the nozzles producing a uniform supersonic flow or a thrust maximum in the presence of a curvilinear sonic line

Within the framework of the ideal, i.e., inviscid and non-heat conducting, gas model we consider the problem of designing the supersonic section of a two-dimensional or axisymmetric nozzle realizing a uniform supersonic flow limitingly similar with a sonic flow when the choked flow involves a curvilinear sonic line. Emphasis is placed on nozzles with abruptly or steeply converging subsonic sections and a strongly curved sonic line formed by the C ⁻-characteristics of the expansion fan with the focus at the lower bend point of the vertical section of the subsonic contour. In the two-dimensional case, the least possible greater-than-unity Mach number M _em at the nozzle exit corresponds to the flow in which the first intersection of the C ⁺-characteristics originated at the closing C ⁻-characteristic of the expansion fan falls on the unknown contour of its supersonic part. For a uniform flow with M _e < M _em the intersection of C ⁺-characteristics beneath the unknown contour make impossible its construction. A part of the contour realizing a uniform flow with M _em > 1 ensures a limitingly rapid flow acceleration and forms the initial region of the supersonic generator of a maximum-thrust nozzle. For this reason, in the case of a curvilinear sonic line the supersonic generators of these nozzles have two, rather than one, bends, which, however, is interesting only for the theory. At least, in the calculated examples the thrusts of the nozzles with one and two bends differ only by a hundredth or even thousandth fractions of per cent.     

Experimental study of shock waves from the interaction of a supersonic plasma jet with an ambient gas

Preliminary results of the interaction of a supersonic, radiatively cooled plasma jet with an ambient gas are presented. The experimental setup consists of a radial foil, a mum-thick aluminium disc held between two concentric electrodes and subjected to a 1.4 MA, 250-ns current pulse from the MAGPIE generator. The plasma flow, with typical velocities of ~70?C90?km/s, is produced by the J?× B force acting on the plasma ablated from the foil. A jet is formed from the convergence of this ablated plasma on the axis of the system. A new setup allows the jet to interact with an argon ambient (particle density N ~10^16-17 cm^?3) from a supersonic gas nozzle (Mach ~9). First results are characterised by the presence of several (previously unseen) shock structures, which are formed from the interaction of the jet with the argon ambient.     

Modeling unsteady flow in a variable-area channel with distributed pulsed-periodic energy supply

The results of a numerical simulation of the unsteady flow in a variable-area channel with a pulsed-periodic supply of external thermal energy in local zones are presented. It is shown that in the case in which the amount of energy supplied is less than that required for thermal choking of the channel the original supersonic flow restructures itself in such a way that in the vicinity of the channel inlet a normal shock is formed and the energy is subsequently supplied a subsonic flow velocity.     

Formation of Supersonic Zones and Local Separation Zones in Transonic Steady-State Flow Past Surface Roughness in the Free Interaction Regime

Within the framework of free interaction theory numerical methods are used to investigate the occurrence of supersonic zones with shocks in the outer inviscid region for flow past roughness in the lower viscous sublayer, with and without the formation of local separation zones.     

Separated flows induced by the interaction between the shock wave ahead of a conic internal-compression obstacle and the boundary layer

Specific features of three-dimensional separated flows induced by the interaction between a conic convergent shock and the boundary layer are experimentally studied for supersonic flow velocities. The data demonstrate the formation of separated-flow zones whose properties were not registered in earlier studies.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 147–151, January–February, 1995.     

Shock wave standoff distance for a sphere slightly above Mach one

This paper describes a numerical simulation of bow shock formation ahead of a sphere at steady supersonic flow in the Mach number range of 1.025–1.20. Turbulent viscous flow results are presented using the Spalart–Allmaras turbulence model. The purpose of this study is to determine the shock standoff distance for a spherical projectile at slightly supersonic free flight speeds. Results are compared to experimental data, including double exposure holographic interferograms obtained from a 40 mm polycarbonate sphere launched by a light gas gun. The shock standoff distance was determined from the interferograms. The present numerical simulations were found to agree with previously published data, and reached down to M = 1.025—a range where almost no previously published data exists. The computed flow structure and shock wave locations agree well with recently obtained free-flight interferograms.     

Supersonic flow past circular cones at large angles of attack

Flow past sharp-nosed circular cones is investigated for a broad range of freestream Mach numbers M>1 and cone half-angles _c at angles of attack from zero to the value at which conical flow breaks down. Several new results are obtained with regard to the position of the Ferri point, the shape of the local supersonic zones and internal shock wave, and the nonmonotonicity of the windward shock slope as a function of the angle of attack. The existence of flow regimes in which the radial velocity on the windward side is directed toward the apex of the cone is demonstrated. The investigation is carried out numerically with relaxation of the solution in a fictitious time coordinate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 6, pp. 79–84, November–December, 1973.     

Collective Bow Shock Ahead of a Transverse System of Spheres in a Supersonic Flow Behind a Moving Shock Wave

The results of investigating the dynamics and physical conditions of formation of a collective bow shock ahead of a system of spheres with the line of centers normal to the supersonic flow behind a traveling shock wave are presented. Two types of shock-wave patterns that necessarily precede the formation of the collective shock wave and correspond to regular and Mach interaction of the bow shocks were detected experimentally. On the basis of a local gasdynamic-discontinuity interference theory, quantitative criteria of the existence of these regimes and of the formation of a common shock wave are determined. These criteria are confirmed in a series of experiments for the transitional regimes.     

Transient three-dimensional startup side load analysis of a regeneratively cooled nozzle

The objective of this effort is to develop a computational methodology to capture the side load physics and to anchor the computed aerodynamic side loads with the available data by simulating the startup transient of a regeneratively cooled, high-aspect-ratio nozzle, hot-fired at sea level. The computational methodology is based on an unstructured-grid, pressure-based, reacting flow computational fluid dynamics and heat transfer formulation, and a transient inlet history based on an engine system simulation. Emphases were put on the effects of regenerative cooling on shock formation inside the nozzle, and ramp rate on side load reduction. The results show that three types of asymmetric shock physics incur strong side loads: the generation of combustion wave, shock transitions, and shock pulsations across the nozzle lip, albeit the combustion wave can be avoided with sparklers during hot-firing. Results from both regenerative cooled and adiabatic wall boundary conditions capture the early shock transitions with corresponding side loads matching the measured secondary side load. It is theorized that the first transition from free-shock separation to restricted-shock separation is caused by the Coanda effect. After which the regeneratively cooled wall enhances the Coanda effect such that the supersonic jet stays attached, while the hot adiabatic wall fights off the Coanda effect, and the supersonic jet becomes detached most of the time. As a result, the computed peak side load and dominant frequency due to shock pulsation across the nozzle lip associated with the regeneratively cooled wall boundary condition match those of the test, while those associated with the adiabatic wall boundary condition are much too low. Moreover, shorter ramp time results show that higher ramp rate has the potential in reducing the nozzle side loads.

---

     

Numerical modeling of vortex/shock wave interaction and its transformation by localized energy deposition

Three-dimensional unsteady Euler simulations are presented for the interaction of a streamwise vortex with an oblique shock of angle β = 23.3° at Mach 3 and 5. The flowfield features are analyzed for weak, moderate and strong interaction regimes. The details of the free recirculation zone at conditions of subsonic and supersonic flow on the vortex axis are considered. The vortex breakdown under conditions of a subsonic vortex core is characterized by a continuous growth and gradual degeneration of the region, unlike the supersonic core condition wherein a steady recirculation zone is achieved. The possibility of using a localized steady and pulsed periodic energy deposition on the vortex axis for stimulating the breakdown is demonstrated for various interaction regimes. It is shown that the formation of a subsonic wake downstream of an energy source lying on the vortex axis contributes to a more significant growth of the dimensions of the recirculation zone compared to the case when the vortex core remains supersonic. The possibility of achieving the effects similar to the steady case is demonstrated by the effect of a pulsed periodic energy source on the flow under consideration for corresponding equivalence parameters.