Design of supersonic asymmetric nozzles

Problems which are associated with the design of the optimum contours of asymmetric plane nozzles are considered. In [1], in the case of the minimum possible length of the lower wall, this problem was solved exactly by the controlled contour method. For greater lengths of the lower wall, this method is inapplicable and it is necessary to use the so-called general method of Lagrange factors [2, 3]. On the basis of this method, the necessary conditions of optimality of the configurations proposed (but not investigated) in [1] are obtained below, a solution which contains a section of the two-sided extremum is constructed, and configurations which do not contain this section are analyzed. The specifics of the problem considered lead to situations which are different from those encountered earlier in the variational problems of supersonic gasdynamics. The latter is associated both with the specail features of formulation of the combined problem for determining the Lagrange factors and with the disposition of the lines of discontinuity of these factors. In the concluding part of the paper, which is a continuation of [4], the effect of the intensity of the compression wave on the thrust of a shortened asymmetric nozzle is investigated. It is shown that the substitution of the centered wave by a compression wave of very moderate intensity significantly reduces the thrust of the nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–94, May–June, 1977.The author is grateful to A. N. Kraiko for valuable advice and constant attention.     

Optimal supersonic nozzles with power-law generators

The problem of designing supersonic nozzles that provide maximum exhaust thrust is considered. An analytical solution is derived on the basis of a local analysis of the aerodynamic force distribution. It is shown that near-optimal thrust characteristics are realized by nozzles with a power-law dependence of the radius on the longitudinal coordinate with an exponent 2/3. The study is performed within the framework of the Euler model of an inviscid, non-heat-conducting gas.     

Investigation of supersonic flows in conical nozzles

The article gives the results of an investigation of flows in supersonic axisymmetric conical nozzles with the presence of shock waves in the flow. The method of straight-through calculation [1] and the method of small perturbations [2] are used. An investigation is made of the effect of various geometric parameters and of the adiabatic index on the flow of a gas in conical nozzles. A comparison is made with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–107, May–June, 1974.In conclusion, the authors thank N. V. Drozdov for his participation in carrying out the calculations.     

Two-layer flows of gas in supersonic axisymmetric nozzles

This article describes two methods for calculating two-layer flows. The first is a generalization of a numerical method for solving the inverse problem [1] for the case of two-layer flows, without taking mixing into account. The second is a method of characteristics, for calculating a two-layer flow in a supersonic nozzle. In this case, the usual method of characteristics is changed in such a way that it is possible to calculate a point on the interface between two layers having different adiabatic indices, and different total pressures and temperatures. This article also gives the results of calculation of two-layer flows in nozzles with different adiabatic indices and different ratios of the mass flow rates of the gas in the layers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 76–81, July–August, 1970.The calculations were programmed and carried out by G. D. Vladimirova and M. F. Tamarovskii, to whom the author expresses his thanks.     

Design of universal supersonic nozzles with conical flow

The problem of designing the supersonic part of nozzles creating source type flows self-similar with respect to the isentropic exponent is solved. A function that approximates the geometry of nozzle channels with cone angles up to 15° is derived. The results of solving the direct nozzle problem, which confirm the accuracy of the approximation obtained, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–186, July–August, 1989.The authors are grateful to A. N. Ganzhelo for supplying the supersonic flow calculation programs.     

Construction of compact nonsymmetric nozzles of maximal thrust for given lift

The variational problem of the construction of supersonic nonsymmetric plane nozzles of maximal thrust for given lift and given restrictions on the size is solved. According to the restrictions on the size, the contours are fairly short, so that they do not interact. By virtue of this, the method of an undetermined control contour [1] can be used for the solution. It is shown that the problem is equivalent to a problem of maximal thrust (in the absence of a condition on the lift) in a coordinate system rotated through a certain angle relative to the original coordinate system and with unchanged restrictions on the size. The influence of these restrictions on the solution to the problem is investigated. It is shown that in some cases the control contour method is inapplicable, in particular, due to the possible presence of shock waves in the domain of influence of the contour. Numerical calculations illustrate the influence of the lift on the geometry and thrust. Special cases of this problem have been considered earlier. In [2], the problem of a plane contour of maximal thrust for given lift was solved and in [3] the problem of a nonsymmetric nozzle of maximal thrust without condition on the lift.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 132–136, October–December, 1981.I thank A. N. Kraiko for a helpful discussion of the work.     

Structure of supersonic inviscid nonsymmetric conical flow around a V-wing

The shock wave structure of flow around a V-wing and its properties determining the conical flow topology are numerically investigated within the framework of the inviscid gas model on a wide range of the angles of attack and yaw when in the disturbed supersonic flow either nonsymmetric Mach interaction between the shocks attached to the leading edges of the wing or a shockless flow in the compressed layer on the windward cantilever is realized. The subranges of the angles of attack and yaw with the disturbed flow properties characteristic of the wing of the given geometry are determined. It is found that at high angles of attack, when the branching point of the bow shock beneath the leeward cantilever generates an intense contact discontinuity, the structure of the conical flow in the shock layer on the windward cantilever involves a singularity of a new type which can be characterized as a “vortical” Ferri singularity. It is located above the point of convergence of the streamlines proceeding from the leading edges of the wing, at the vertex of the corresponding contact discontinuity. Flow patterns with the point of convergence of the streamlines proceeding from the leading edges located in the elliptical flow region, which is placed at a local maximum of the pressure distribution over the surface are also found. The range of the angles of attack and yaw on which this new property of supersonic conical flows is realized in the presence of a branched shock system is determined.     

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.     

Mixed boundary-value problems of profiling of supersonic nozzles and channels

In the framework of the inverse method of numerical profiling of the supersonic part of nozzles and channels, two new mixed boundary-value problems of gas dynamics are formulated and solved with boundary conditions specified both along and across the flow. A modification of the grid-characteristic method with respect to layers formed by streamlines is constructed. The solution of the posed mixed problems makes it possible to profile a class of flat and axisymmetric nozzles and channels larger than the known class. Transition channels, which transform one expanding flow into another for which the flow at the exit is close to that from a source, are constructed Channel profiles are obtained with a bend in the generator that suppress shock waves on the wall. The results are presented of profiling of exit channels with contact discontinuity, and also channels that realize discontinuities in the exit sections, these being made isentropic by means of compression characteristics focused at the points of the discontinuity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 112–118, July–August, 1983.We thank U. G. Pirumov, M. Ya. Ivanov, and A. N. Kraiko for helpful discussions.     

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.