On profiling supersonic nozzles to achieve uniform flow in the annular exit section

The problem of profiling supersonic nozzles with a central body in order to achieve uniform supersonic flow with Mach number M _e >1 in the annular exit section is studied. Equal lengths of the profiled inner (central body) and outer nozzle walls can be achieved by displacing downstream the initial bend in the inner contour, i.e., in the central body wall. A displacement of this kind was proposed and tested in [1, 2] for annular nozzles with circular exit sections.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 204–206, March–April, 1995.     

Investigation of supersonic viscous flow past a sphere in the presence of subsonic and sonic injection

Supersonic flow past a sphere with a given rate of gas injection along the generator is investigated numerically on the range Re=10²–10⁴. Calculations have been made on the interval 0 90°, where is the angle between the axis of symmetry and the normal to the surface. It is shown that for high subsonic and sonic injection rates it is possible to observe qualitatively new features in the flow structure and in the distribution of the local supersonic flow characteristics around the perimeter of the sphere not previously noted in [9]. In the case of sonic injection the changes in flow structure occur only in the supersonic zone. In the neighborhood of the transition from a subsonic to sonic injection velocity the heat flux has a local maximum, which in absolute value does not exceed the heat flux in the absence of injection. It is shown that there may be qualitative differences in the pressure distribution over the surface of the body with increase in the injection parameter depending on the distribution and value of the injected gas flow rate and, moreover, the number Re.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 83–89, January–February, 1988.     

Designing three-dimensional nozzles to achieve near-uniform supersonic or hypersonic flow in the rectangular exit section

The problem of profiling three-dimensional nozzles in order to achieve supersonic flow in the given rectangular exit section is studied. The question of designing an essentially three-dimensional nozzle with a variable cross-section shape is discussed. A technique for constructing three-dimensional nozzles with near-uniform flow in the exit section is proposed and realized.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 171–180, September–October, 1995.     

LES of supersonic turbulent flow in nozzles and diffusers

Large-eddy simulations of supersonic nozzle and diffuser flows with circular cross-sections using high-order compact schemes and an explicit filtering version of the approximate deconvolution method are presented in this paper. Two flow cases each for nozzle and diffuser having different outlet to inlet area ratios are presented. The effect of the geometry variations on the Reynolds stresses as well as on the production and pressure-strain terms in their transport equations is demonstrated. A Green’s function analysis of the Poisson equation for pressure fluctuations using LES data is presented and the results show similar trends as found in previous analyses using DNS data. The effects of geometry changes on the rapid and slow parts of pressure-strain correlations is also demonstrated.     

Use of flows with straight sonic line in nozzles with corners

The optimal scheme of a Laval nozzle is discussed. In the case of a profiled nozzle with a corner it is possible to use in the region of mixed flow both flows of general form with curvilinear sonic line as well as the special case when the sonic line is straight. It is shown that the latter alternative is preferable: when the supersonic part of the profile is determined by the simple wave method, the velocity at the wall increases monotonically and the flow does not contain shock waves. In contrast, in nozzles with curvilinear sonic line, a section in which the velocity decreases is formed immediately behind the corner, which can lead to boundary layer separation. In addition, for values of the supersonic velocity at the nozzle exit near the velocity of sound it is proved that the characteristics of the simple wave intersect in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 168–170, January–February, 1981.     

Calculation of supersonic separation flow in circular nozzles with abrupt expansion

A study is made of the interaction of a circular supersonic jet with a turbulent layer of the near-wake kind formed behind a circular step on the end part of a nozzle with abrupt expansion. The flow in the viscous layer is calculated by the integral method, and in the inviscid flow by a through-computation method using a monotonic implicit difference scheme of first order of accuracy. The interaction between the inviscid and turbulent flows is determined by the displacement thickness of the viscous layer. The initial conditions for the flow in the layer are determined from the integral conditions of its matching to the isobaric mixing flow in the base region behind the step. The computed interaction flows are determined as a function of the length of the end part and the counter pressure simultaneously or separately by the boundary condition that the pressure at the end of the end part be equal to the pressure of the external medium and by a singular solution of the equations passing through a saddle singular point — the throat of the wake. In a conical nozzle with profiled attachments of different lengths, calculations were made of separation flows with open and closed base regions and with allowance for secondary separation. The obtained solutions are multivalued in a certain range of lengths of the end part and values of the counter pressure. The solutions realized in reality are selected by analyzing the regimes of operation of the nozzles, and the calculated regions of hysteresis and associated low-frequency nonstationary separation flows are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 120–128, July–August, 1979.We are grateful to M. Ya. Ivanov for consultation on the use of the program of [6] and to Z. A. Donskovaya for assistance in preparing the paper.     

Conical wing of maximum lift-drag ratio in a supersonic gas flow

The calculation of supersonic flow past three-dimensional bodies and wings presents an extremely complicated problem, whose solution is made still more difficult in the case of a search for optimum aerodynamic shapes. These difficulties made it necessary to simplify the variational problems and to use the simplest dependences, such as, for example, the Newton formula [1–3]. But even in such a formulation it is only possible to obtain an analytic solution if there are stringent constraints on the thickness of the body, and this reduces the three-dimensional problem for the shape of a wing to a two-dimensional problem for the shape of a longitudinal profile. The use of more complicated flow models requires the restriction of the class of considered configurations. In particular, paper [4] shows that at hypersonic flight velocities a wing whose windward surface is concave can have the maximum lift-drag ratio. The problem of a V-shaped wing of maximum lift-drag ratio is also of interest in the supersonic velocity range, where the results of the linear theory of [5] or the approximate dependences of the type of [6] can be used.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–133, May–June, 1986.We note in conclusion that this analysis is valid for those flow regimes for which there are no internal shock waves in the shock layer near the windward side of the wing.     

The flow of polymer thickened motor oils in convergent jet thrust nozzles

Jet nozzles are described in which liquids are subjected to uniaxial and biaxial stretching motions, shear also being present near the nozzle walls. Jet thrust measurements show that the behaviour of polymer-thickened oils can best be studied by comparison with the behaviour of Newtonian base oils at equivalent Reynolds numbers. These data compare well with measurements previously made in long, straight tubes.Jet thrusts for polymer-thickened oils are lower than those of base oils and show evidence of a threshold point at which the two forms of behaviour deviate. The axial stress in the oil is almost independent of the strain rate, at high strain rates, but is related to axial strain in a simple manner. For 3.5 per cent polymer dissolved in Iranian light base oil, the axial stress is shown to be τ₁₁ = 2000 γ_G dyn cm^?2 for uniaxial strain and τ₁₁ = 2800 γ^1.30_G dyn cm^?2 for biaxial strain, where γ_G is the measure of strain defined by Green. For a standard Shell Multigrade 20W/50 motor oil the equivalent equation for biaxial strain is τ₁₁ = 3200 γ^1.02_G dyn cm^?2. The stresses developed during biaxial extension are higher than those developed during uniaxial extension.The presence of fully elastic behaviour in polymer-thickened oils at extensional strain rates of the order 500 s^?1 and principal extension ratios of only 8 is of potential significance to lubrication technology. The stress rises with the square of the extension ratio and should become large as lubricated surfaces approach. Polymer degradation will commence as the stress rises; it is suggested that further tests be carried out at higher stresses and temperatures.     

Change in the parameters of a supersonic flow in the presence of transverse blowing

A study has been made of the flow formed in a supersonic nozzle when gas is blown in a transverse jet into an expanding supersonic flow. Measurements were made of the total and static pressures of the flow at several sections of the nozzle. It was established that, depending on the relative flow rate = m_j/(m_j+ m₀) of the blown gas (m_jand m₀ are the flow rates of the blown gas and the main flow, respectively), there exist two flow regimes with different dependences of the Mach number of the flow. At small , the experimental flow parameters correspond satisfactorily to the parameters calculated in a one-dimensional model with a narrow mixing layer near the blowing section. Agreement was observed at flow rates less than a certain ^*, this critical value being determined in the model as the flow rate at which the flow after mixing becomes sonic. In the experiments at large flow rates of the blown gas, ^* < < 1, the value of M for the flow hardly depends on and corresponds to the calculated value of M for a supersonic flow having the velocity of sound near the blowing section. A scheme is proposed for calculating the flow in a nozzle with transverse blowing in the supersonic part; it describes satisfactorily the experimental results in the complete range of blown-gas-main-flow flow rate ratios (0 1) over the complete length of the nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 188–192, May–June, 1984.     

Solution of the variational problem of shaping a nozzle to achieve a uniform supersonic flow

A solution is given to the problem of constructing a supersonic flat unsymmetrical nozzle achieving a given uniform flow at the outlet with a minimal length of one of the contours and a given length (or ordinate of the end point) of the other contour.     

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.     

Vibrations of a cylinder in a uniform flow in the presence of a no-slip side-wall

A circular cylinder placed in a uniform flow, and that spans the entire length between two side walls, may experience either parallel or oblique vortex shedding depending on the end conditions. It was shown by Mittal and Sidharth (2014) that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by the cylinder. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. The free vibrations of a circular cylinder, in the presence of a wall, are investigated. For comparison, computations with end walls, where a slip condition on velocity is specified, are also carried out. The Reynolds number, based on the diameter of the cylinder and free-stream speed of the flow, is Re=100. The initial condition for the free vibrations is the fully developed unsteady flow past a stationary cylinder with oblique shedding. It is found that as the amplitude of vibration of the cylinder builds up, the vortices shed from the cylinder align with its axis leading to parallel shedding. The response of the cylinder is associated with two branches: initial and lower. On the lower branch, the response of the cylinder is virtually identical from two- and three-dimensional computations. The flow as well as the response is different on the initial branch and outside the synchronization regime. Forced vibrations confirm the phenomena.     

Turbulent flow of gas in a curvilinear channel in the presence of suction from a separation region

Integral parameters that characterize reversible and irreversible changes in the flux of the total pressure in channels with perforated walls are introduced. An experimental investigation was made of subsonic gas flow in curvilinear channels of rectangular cross section in the presence of suction of gas from a separation region of the flow formed on an internal (convex) strongly curved wall of the channel. The optimal position of the suction slit was determined and it was shown to be possible to reduce appreciably the loss in the channel and improve its gas-dynamic characteristics. Two-dimensional turbulent flow of an incompressible fluid in curvilinear channels in the presence of suction was simulated numerically. The mathematical model is based on the complete system of Navier-Stokes equations, additional differential equations for the energy of the turbulence and the rate of its dissipation, special correction equations to take into account the curvature of the streamlines, and model boundary conditions for the sections of the walls through which the suction of the fluid takes place.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 4, pp. 72–80, July–August, 1984.     

Similarity properties with the flow of supersonic uniform and nonuniform flows of gas around a sphere

The article proposes an approximation formula for the velocity gradient at the critical point for ellipsoids of revolution, suitable for both an equilibrium and a perfect gas. It is shown that an approximate solution to the problem of the flow of a stream arriving from a supersonic source around a sphere can be obtained, having the solution of the problem of uniform flow around a sphere.     

Calculation of supersonic viscid flow near stagnation line of a blunt body

A numerical study is made of supersonic flow of a viscous gas in the vicinity of the stagnation line of plane and axisymmetric blunt bodies (cylinder, sphere). As in [1–5], which consider the compressed layer of a viscous gas in the vicinity of the stagnation point, use is made of the locally self-similar approximation, which is used to transform the Navier-Stokes equations into a system of ordinary differential equations. In the present paper the solution is sought with the simplifications of [5] and with more general conditions, which makes it possible to study a broad class of flows. The proposed numerical algorithm permits obtaining the structure of the compressed layer near the stagnation line, including the shock wave and the boundary layer. The calculations made on a computer for different flow conditions are illustrated by graphs.The author wishes to thank G. I. Petrov, G. F. Telenin, and L. A., Chudov for their interest in the study and for their helpful discussions. discussions.