Calculating the Thrust Characteristics of Nozzles from Their Outlet Cross-Section Parameters

Within the framework of boundary-layer theory, simple formulas are presented for finding the viscous loss from the gas parameters in the nozzle outlet cross-section. The analysis is performed for ordinary Laval nozzles and spike nozzles. It is found that for nozzles with large expansion ratios the viscous loss is almost independent of the outlet cross-section parameters and is determined only by the parameter values on the nozzle contour. The effect of the longitudinal nozzle curvature on this loss is investigated. It is shown that the viscous losses calculated from the nozzle outlet parameters and by integrating along the nozzle contour with account for the longitudinal curvature fully coincide.     

Extremal nozzle contours for gas flows with particle lag

The determination of the extremal nozzle contour for gas flow without foreign particles has been carried out in several studies [1–6], based on the calculation of the flow field using the method of characteristics.In [7, 8] the equations are derived for the characteristics and the relations along the streamlines which are required for calculating two-dimensional gas flow with foreign particles. The variational problem for two-phase flow in the two-dimensional formulation may be solved by the method of Guderley and Armitage [9] with the use of equations given in [7] or [8]; however this method is very tedious, even with the use of high-speed computers.In [10, 11] studies are made of two-phase one-dimensional flows by expanding the unknown functions in series in a small parameter, defined by the particle dimensions. In [12] a solution is given for the variational problem (in the one-dimensional formulation) of designing the contour of a nozzle with maximal impulse. However that study does not take account of the static term appearing in the impulse and the solution is obtained in relative cumbersome form. Moreover, the question of account for the losses due to nonparallelism and nonuniformity of the discharge was not considered.The present paper considers in the one-dimensional formulation the flow of a two-phase medium in a Laval nozzle with small particle lags (in velocity and temperature). The variational problem of determining the maximal nozzle impulse is formulated along the nozzle contour for fixed geometric expansion ratio. The impulse losses due to nonparallelism of the discharge are simulated by a function which depends on the ordinates which are variable along the contour and on the slope of the tangent to the contour.The author wishes to thank Yu. D. Shmyglevskii and A. N. Kraiko for helpful discussions and V. K. Starkov for carrying out the calculations on the computer.     

Gas-dynamic design of an axisymmetric tunnel air inlet with isentropic compression

A method of designing a supersonic axisymmetric tunnel air inlet based on the problem of an inverted flow in an annular nozzle with isentropic expansion is considered. The nozzle contour is constructed by the method of characteristics. Parameters of one inlet for viscous and inviscid gas flows are calculated.     

Supersonic ideal-gas flow of in the throat region of a nozzle

The problem of profiling a supersonic nozzle with uniform outlet flow is considered when the contour is constructed from a point on a given convergent section of the nozzle. It is shown that there are contours such that the flow in the throat is supersonic in choked regimes (as distinct from common notions of mixed or uniform sonic flow). The influence of flow nonuniformity in the throat region on the thrust of a supersonic nozzle is analyzed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 132–139, May–June, 1996.     

Investigation of lateral forces and moments in the case of asymmetric gas flows in nozzles

Supersonic three-dimensional flows of a gas in a nozzle are studied. The main attention is devoted to the study of lateral forces and moments resulting from asymmetric perturbations of the contour of an axisymmetric conical or profiled nozzle or due to asymmetric perturbations of the parameters at some section of the nozzle. The approach is comprehensive — through a finite-difference numerical method, the method of small perturbations, and experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 70–85, March–April, 1980.     

Shock pattern in the plume of rocket nozzles: needs for design consideration

For ideal nozzles, basically two different types of shock structures in the plume may appear for overexpanded flow conditions, a regular shock reflection or a Mach reflection at the nozzle centreline. Especially for rocket propulsion, other nozzle types besides the ideal nozzles are often used, including simple conical, thrust-optimized or parabolic contoured nozzles. Depending on the contour type, another shock structure may appear: the so-called cap-shock pattern. The exact knowledge of the plume pattern is of importance for mastering the engine operation featuring uncontrolled flow separation inside the nozzle, appearing during engine start-up and shut-down operation. As consequence of uncontrolled flow separation, lateral loads may be induced. The side-load character strongly depends on the nozzle design, and is a key feature for the nozzle’s mechanical structure layout. It is shown especially for the VULCAIN and VULCAIN 2 nozzle, how specific shock patterns evolve during transients, and how - by the nozzle design - undesired flow phenomena can be avoided.     

Choking of ideal-gas flow in convergent nozzles and integral nozzle characteristics

The results of a numerical and theoretical investigation of the local and integral characteristics of convergent nozzles are presented. It is shown that self-similar (choked) nozzle flow, when the gas flow rate does not depend on the external pressure, may occur at subcritical values of the pressure ratio _c . If the nozzle contour consists of the contour of the conical nozzle and the convergent part corresponds to the boundary of the emerging jet, then on a certain interval of _c this nozzle will have a higher thrust coefficient than the initial conical nozzle.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 149–157, November–December, 1994.     

Loss of Specific Impulse Due to Friction in Spike Nozzles

The class of nozzles with a central body, so-called spike nozzles, is considered for axisymmetric and plane central body geometries. A method of constructing the nozzle contour is outlined. The boundary layer is calculated using a three-parameter turbulence model and the loss of specific impulse due to friction in both spike nozzles and a Laval nozzle with the same expansion ratio are determined. A comparative analysis of the calculation results obtained, which makes it possible to determine the advantages and limitations of the nozzles considered, is carried out.     

One method of profiling short plane nozzles

One of the possible methods is considered for profiling short plane nozzles for aerodynamic tubes. The nozzle has a straight sonic line, which allows the subsonic and supersonic sections to be constructed separately. The problem is solved numerically in the plane of a hodograph. In the subsonic region, Dirichlet's problem is formulated for Chaplygin's equation in a rectangle, one side of which is the sonic line. At the present time, two approaches have been defined in papers on calculations of a Laval nozzle, associated with the solution of the so-called “direct” and “inverse” problems (one has in mind a study of the flow in the interconnected region of sub- and supersonic flow). The direct problem determines the flow field in the case of a previously specified contour of the channel wall, the shape of which from technical considerations is obtained with certain geometry conditions. The direct problem can be applied in the construction of the Laval nozzle, if the contour of the inlet section of the channel (generally speaking, quite arbitrary) is chosen so successfully that neither shock compressions nor breakaway zones result in the flow. Although a strictly mathematical theory of the direct problem of the Laval nozzle is only being developed at present, there are still very effective numerical methods for its solution [1, 2]. In the inverse problem (which, by definition, is a problem of profiling), the contour of the nozzle is found with respect to a specified velocity distribution on the axis of symmetry. It is assumed that this quite arbitrary dependence can be selected from the condition of the absence of breakaway zones and shock compressions in the nozzle. By its formulation, the inverse problem is Cauchy's problem which, as is well-known, is incorrect in the classical sense in the ellipticity region — the subsonic section of the nozzle. At present, there are also efficient methods of solving the inverse nozzle problem [3], by interpreting it as an arbitrarily correct problem. Difficulties can arise in the inverse problem, in the provision of short (and, consequently, steep) nozzles because of the sharp increase of the error in the calculation. Together with the stated problems, a procedure can be evolved which is associated with the solution of the correctly posed problem for Chaplygin's equation in the plane of the hodograph. This approach is convenient in that it succeeds a priori in fulfilling the important condition of monotonicity of the velocity at the wall, ensuring (in the absence of shock compressions) nonseparability of the streamline flow at any Reynold's numbers.     

Two-way coupled simulation of a flow laden with metallic particulates in overexpanded TIC nozzle

A simulation of non-reacting dilute gas–solid flow in a truncated ideal contour nozzle with consideration of external stream interactions is performed. The Eulerian–Lagrangian approach involving a two-way momentum and thermal coupling between gas and particles phases is also adopted. Of interests are to investigate the effects of particles diameter and mass flow fraction on the flow pattern, Mach number, pressure and temperature contours and their distributions along the nozzle centerline and wall. The main goal is to determine the separation point quantitatively when the particles characteristics change. Particles sample trajectories are illustrated throughout the flow field and a qualitative discussion on the way that physical properties of the nozzle exit flow and particles trajectories oscillate is prepared. The existence of solid particulates delays the separation prominently in the cases studied. The bigger particles and the higher particles mass flow fractions respectively advance and delay the separation occurrence. The particles trajectories oscillate when they expose to the crisscrossing (or diamond-shape) shock waves generated outside the nozzle to approach the exit jet conditions to the ambient. The simulation code is validated and verified, respectively, against a one-phase 2D convergent–divergent nozzle flow and a two-phase Jet Propulsion Laboratory nozzle flow, and acceptable agreements are achieved.     

Numerical and experimental investigation of a laval nozzle with a cylindrical throat

The results are reported of experimental and numerical investigation of mixed flow and of the parameters of heat transfer in the transonic region of an axisymmetric Laval nozzle whose throat is formed by a cylindrical surface, i.e., the nozzle contour near the minimum cross section contains two bends.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 189–192, September–October, 1984.     

An exact solutin for incompressible flow through a two-dimensional Laval nozzle

A careful examination of the variation of the velocity along the centerline and the contour of a Laval nozzle in the physical plane shows that either the upper or the lower half of the Laval nozzle assumes the same form of a slitted thick airfoil with tandem trailing edges. These two airfoils lie on different Riemann sheets in the hodograph plane. The interior of the airfoil is then mapped onto an infinite strip in the complex potential plane. Making use of these results, we obtained an exact solution for the incompressible potential flow through a two-dimensional Laval nozzle. The solution is applicable for nozzles with any given contraction ratio mexpansion rations, and throat wall radius R^*. As examples of the method, various nozzle contours, the velocity distribution of the flow, and the locations of the fluid particles at different time intervals are presented.     

收缩段线型对超音速喷管氟原子气相复合的影响

使用移轴维氏曲线和双三次方曲线作为收缩段线型分别设计了两种环形 HYLTE主喷管，给出了环形HYLTE喷管三维的多组分有反应湍流场的控制方程、边界条件和 模拟区域的选择. 计算结果表明，这两种线型的喷管均能避免边界层的分离，获得均匀的流 场分布及相同的出口马赫数分布，但移轴维氏曲线在收缩段前部收缩快速，气流速度相应较 快，因此F原子的气相复合程度弱于另外一种收缩段设计的喷管，相应地，使用移轴维氏曲 线设计的喷管F原子流量比另外一种喷管高5%左右, 因此使用移轴维氏曲线作为环形HYLTE主喷管收缩段的设计有助于提高化学激光器的效率.     

Investigation of a Gas Flow with Solid Particles in a Supersonic Nozzle

A two-phase flow with high Reynolds numbers in the subsonic, transonic, and supersonic parts of the nozzle is considered within the framework of the Prandtl model, i.e., the flow is divided into an inviscid core and a thin boundary layer. Mutual influence of the gas and solid particles is taken into account. The Euler equations are solved for the gas in the flow core, and the boundary-layer equations are used in the near-wall region. The particle motion in the inviscid region is described by the Lagrangian approach, and trajectories and temperatures of particle packets are tracked. The behavior of particles in the boundary layer is described by the Euler equations for volume-averaged parameters of particles. The computed particle-velocity distributions are compared with experiments in a plane nozzle. It is noted that particles inserted in the subsonic part of the nozzle are focused at the nozzle centerline, which leads to substantial flow deceleration in the supersonic part of the nozzle. The effect of various boundary conditions for the flow of particles in the inviscid region is considered. For an axisymmetric nozzle, the influence of the contour of the subsonic part of the nozzle, the loading ratio, and the particle diameter on the particle-flow parameters in the inviscid region and in the boundary layer is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 65–77, November–December, 2005.     

Experimental study of boundary layer separation in truncated ideal contour nozzles

DLR Lampoldshausen carried out a cold flow test series to study the boundary layer separation and the related flow field in a truncated ideal contour nozzle. A special focus was set on low nozzle pressure ratios to identify the origin of a locally re-attached flow condition that was detected in previous test campaigns. A convex shaped Mach disc was found for nozzle pressure ratios less than 10 and a slight concave one for nozzle pressure ratios more than 20. Due to boundary layer transition at low nozzle pressure ratios the convex Mach disc is temporary tilted and redirects the flow towards the nozzle wall. A simple separation criterion for turbulent nozzle flows is presented that fits well for both cold and hot flows. It is shown that the oblique separation shock recompresses the flow to 90% of the ambience. The separation zone of the presented film cooled nozzle is compared with a conventional one around 40% longer. Furthermore a relation between shear layer shape and forced side loads is described.      

Construction of asymmetric nozzles of maximum moment given additional conditions on the geometrical and force characteristics

A solution is given to the variation problem of constructing asymmetric plane nossles which realize the maximum moment relative to some point. The contours of the nozzle are assumed to be noninteracting. The method of the undetermined control contour is used [1]. The solution of this problem contains as a special case the solution to the problem of constructing a nozzle of maximum thrust, including also the case of a given lifting force [1–3]. It is shown that the construction of a nozzle of maximum moment under additional conditions on the thrust and the lifting force, or on the moment relative to another point, reduces to the construction of a nozzle of maximun moment relative to some auxiliary point.Translated from Izvestiya Akademi Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 148–152, September–October, 1984.The author thanks A. N. Kraiko for useful discussions and for his appraisal of the study.     

A variational problem of supersonic gasdynamics

A large number of studies have been published which define bodies of minimal drag and nozzles of maximal thrust. A study has been made of the region of existence of the various solutions in planar flow, and it has been shown that the optimal profile in the presence of a length limitation may have a blunt end, which is a segment of a boundary extremum. The results of these studies and also the required bibiliography are contained in [1–3].In the following we present a joint solution of the problems of constructing the optimal afterbody and nozzle contour, which are the boundaries of a single planar body or of a ducted body. The investigation is limited to cases in which the form of the external contour (i.e., the afterbody contour) ensures the absence of shock waves in its influence region. The necessary conditions are obtained which determine the optimal configuration, and computational examples are given.     

Optimum nozzle configurations for two-phase flows

The structure of the optimum supersonic contour of an axisymmetric Laval nozzle is investigated within the framework of the nonequilibrium polydisperse two-phase flow model. In formulating the variational problem attention is focused on taking into account a restriction that makes it possible to construct an optimum contour with no or limited particle fall out. Tomsk. Translated from Izvesriya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 36–45, March-April, 1994.     

Optimal Profiling of the Supersonic Part of a Plug Nozzle Contour

The problem of the optimal profiling of the supersonic part of a plug nozzle contour is solved within the framework of the ideal (inviscid and non-heat-conducting) gas model. The contours obtained provide a thrust maximum for given uniform sonic flow in the radial critical section of the nozzle, given constraints on the nozzle dimensions, and a given outer pressure (counterpressure). The initial sonic regions of the optimal contours are profiled on the basis of the condition that there the flow Mach number is unity. Varying the initial sonic region length makes it possible to construct nozzles of different sizes. The possibilities of the computational programs developed are demonstrated with reference to the example of plug nozzles, optimal when operated in a vacuum. It is shown that low thrust losses are obtained even for moderate nozzle dimensions. In the examples calculated, the optimal plug nozzles provide a greater thrust than the optimal axisymmetric and two-dimensional nozzles with an axial sonic flow for the same lengths and gas flow rates.     

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.