Shock wave-induced vortex loops emanating from nozzles with singular corners

The focus of the current study is to examine experimentally the diffracted shock wave pattern and the consequent vortex loop formation, propagation, and decay from nozzles having singular corners. Non-intrusive qualitative and quantitative techniques: schlieren, shadowgraphy, and particle image velocimetry (PIV) are employed to analyze the induced flow-fields. Eye-shaped nozzles were used with the corner joints representing singularities. The length of the minor axes are a = 6 and 15 mm, with the major axis b = 30 mm for both cases. The experiments are performed for flow Reynolds numbers in the range 0.8 × 10⁵ and 4.6 × 10⁵. Air is used in both driver and driven sections of the shock tube.     

Calculation of nonequilibrium flows in nozzles

This paper studies in the one-dimensional formulation the flow of a reacting gas with account for the nonequilibrium behavior of the chemical reactions; the pressure distribution along the stream filament is given. Viscosity, heat conduction, diffusion, and ionization are not taken into account. It is assumed that there is equilibrium excitation of the translational, rotational, and vibrational degrees of freedom.Several studies have already been made of nonequilibrium flows in nozzles [1–5]. It is known that in the calculation of nonequilibrium flows considerable difficulty arises in selecting the integration step in those regions where the flow is nearly equilibrium. It is found that with the use for numerical integration of the explicit difference schemes of the type of the Euler, Runge-Kutta, etc., methods the integration step for carrying out a stable calculation must be so small that the calculation becomes practically impossible. The present study proposes a method for calculating nonequilibrium flows using a single implicit difference scheme to calculate with a high degree of accuracy and a quite large step (exceeding the step in the explicit schemes by several orders) both those flow regions which are close to equilibrium and those regions where the flow deviates markedly from equilibrium. A program was compiled using this method for the M-20 electronic digital computer which permitted calculating in the one-dimensional approximation flows in nozzles with account for the nonequilibrium behavior of the chemical reactions for mixtures containing H, O, C, and N atoms.Some qualitative peculiarities of the nonequilibrium flows are demonstrated using as an example nonequilibrium air discharge. A comparison is made with experimental and theoretical results of other authors.The authors wish to thank L. F. Kuz'mina for her assistance in carrying out the present study.     

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.     

Study of spatial mixed flows in nozzles with asymmetrical inlets

The results of some investigations into the behavior of the spatial mixed flows of a nonviscous and nonheat-conducting gas in nozzles deviating from axisymmetrical shape in the subsonic region are presented. The investigation is based on the numerical integration of the transient gasdynamic equations using the Godunov scheme [1, 2] generalized to the spatial case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–169, March–April, 1975.The author wishes to thank M. Ya. Ivanov, A. N., Kraiko, and G. G. Chernyi for help and discussion of the results.     

Some compression flows in nonaxisymmetric ring nozzles

Busemann's problem concerning fully developed conical flow in an axisymmetric nozzle of special type is extended to include certain nonaxisymmetric ring nozzles. The constructed flows contain strong discontinuities in the form of developable surfaces (in Busemann's solution, strong discontinuities have the form of a circular-cone surface).     

Inviscid gas flows in nozzles with a steeply converging contour

Inviscid gas flows in nozzles with a uniform exit flow and contours profiled starting from the lower point of a steeply converging region with an angle θ = −90° are analyzed. It is shown that there exists a class of convergent-divergent contours, within which the flow is characterized by the fact that the line θ = 0 of zero angle of the velocity vector inclination to the x axis consists of two oppositely-directed regions located partially or even completely ahead of the minimum section, while near the minimum sections their regions convex inward the gas stream are in decelerated flow. The minimum sections of the nozzles with M _e → 1 approach the center of the nozzle from the right.     

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.     

Oscillatory flows in straight tubes with an axisymmetric bulge

A laser-Doppler anemometer has been used to study oscillatory flow of a Newtonian viscous fluid in straight circular tube with an axisymmetric bulge of two different sizes. The axial velocities were measured at successive cross-sectional planes for sinusoidal waveforms having Reynolds numbers (based on Stokes layer thickness at the inlet) from 445 to 806 and Womersley numbers ranged from 7.2 to 12.2. The cyclic flow development inside the bulge at different phases within a cycle was determined. Stability analysis obtained by solving the Orr–Sommerfield equation on instantaneous velocity profiles showed instability grows progressively during the acceleration phase and transition to turbulence in the bulge happened shortly after the commencement of the deceleration phase. Depending on the bulge geometry, the turbulent region was initially confined either to the proximal or the distal end of the bulge. This region would spread larger as the deceleration phase furthered and the smaller bulge had a larger spread than the bigger bulge. The differences could be attributed to the vortical structures development inside the bulge. Relaminarisation for the flow appeared in the subsequent acceleration phase. Finally, some comparisons had been made with results obtained from using the physiological waveform.     

Theory of slightly perturbed three-dimensional flows in nozzles

The theory of slightly perturbed flows in conical nozzles is used to determine the transverse force and moment generated in the presence of asymmetric perturbations. A system of ordinary differential equations is derived for finding the transverse force and moment. An approximate analytical solution of this system is constructed and its qualitative features are studied. A comparison is made with a numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1977.     

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.     

An experimental investigation of underexpanded jets from oval sonic nozzles

 Underexpanded jets from oval sonic nozzles were experimentally studied for various pressure ratios up to 20.3. The results revealed that the barrel type of shock structure was present only in the major axis plane except at low aspect ratios. The results also revealed that the jet spreading rate in the minor axis plane of the nozzle was much higher compared to that in the major axis plane, resulting in axis switching of jets. The cross sectional area of these jets were considerably higher compared to the axisymmetric jets indicating higher interface area for viscous mixing in the near field region and increased mixing in the far field region clear of shock structure. Received: 7 January 1997 / Accepted: 7 August 1997     

Investigation of nonequilibrium two-phase flows in axisymmetric laval nozzles

The singularities of two-phase flows in Laval nozzles were investigated within the framework of the model of a two-fluid continuous medium [1, 2] mainly in a quasi-one-dimensional approximation ([3] and the bibliography therein). Two-dimensional computations of such flows were performed only recently by using the method of buildup [4–7]. However, systematic computations to clarify the influence of the second phase on such fundamental nozzle characteristics as the magnitude of the specific impulse, its losses, and discharge coefficient were performed only in the quasi-one-dimensional approximation [8, 9] and only for the supersonic parts of the nozzle in the two-dimensional approximation under the assumption of uniform flow in the throat [10, 3]. Such an investigation is performed in this paper in the two-dimensional case for the nozzle as a whole, including the sub-, trans-, and supersonic flow domains, and a comparative analysis is given of the magnitudes of the loss of a unit pulse obtained in the quasi-one-dimensional approximation [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 86–91, November–December, 1977.     

Secondary flows of viscoelastic liquids in straight tubes

The unsteady flow of the Green–Rivlin fluids in straight tubes of arbitrary cross-section driven by a pulsating pressure gradient is investigated. The non-linear constitutive structure defined by a series of nested integrals over semi-infinite time domains is perturbed simultaneously with the boundary of the base flow through a novel approach to domain mapping. The dominant primary component of the flow, the longitudinal field, and the much weaker transversal field arise at the first and the second orders of the analysis, respectively. The secondary field is driven by first-order terms stemming from the linearly viscoelastic longitudinal flow at the first order. The domain mapping technique employed yields a continuous spectrum of unconventional closed cross-sectional shapes. We present longitudinal velocity profiles and transversal time-averaged, mean secondary flow streamline patterns for a specific fluid and for representative cross-sectional shapes in the spectrum the triangular, square and hexagonal shapes.     

Theory of vortical helical ideal gas flows in laval nozzles

An asymptotic solution is found for the direct problem of the motion of an arbitrarily vortical helical ideal gas flow in a nozzle. The solution is constructed in the form of double series in powers of parameters characterizing the curvature of the nozzle wall at the critical section and the intensity of stream vorticity. The solution obtained is compared with available theoretical results of other authors. In particular, it is shown that it permits extension of the known Hall result for the untwisted flow in the transonic domain [1]. The behavior of the sonic line as a function of the vorticity distribution and the radius of curvature of the nozzle wall is analyzed. Spiral flows in nozzles have been investigated by analytic methods in [2–5] in a one-dimensional formulation and under the assumption of weak vorticity. Such flows have been studied by numerical methods in a quasi-one-dimensional approximation in [6, 7]. An analogous problem has recently been solved in an exact formulation by the relaxation method [8, 9]. A number of important nonuniform effects for practice have consequently been clarified and the boundedness of the analytical approach used in [2–7] is shown.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–137, March–April, 1978.The authors are grateful to A. N. Kraiko for discussing the research and for valuable remarks.     

Global stability of separated flows: subsonic flow past corners

The triple-deck equations for the steady subsonic flow past a convex corner are solved numerically using a novel technique based on Chebychev collocation in the direction normal to the body combined with finite differences in the direction along the flow. The resulting set of nonlinear algebraic equations are solved with Newton linearization and using the GMRES method for the solution of the linear system of equations. The stability of the computed steady flows is then examined using global stability analysis. It is found that for small corner angles, the Tollmien?CSchlichting modes are globally unstable and these persist to larger corner angles. Multiple steady state solutions also exist beyond a critical corner angle but these are globally unstable because of the presence of the Tollmien?CSchlichting modes.     

Use of direct variational methods to optimize axisymmetric Laval nozzles in the case of equilibrium and nonequilibrium two-phase flows

In the construction of the optimal profile of a Laval nozzle when there are subsonic regions in the flow, the use of effective methods such as the general method of Lagrangian multipliers [1] becomes very difficult. In the present paper, direct variational methods are therefore used. For nozzles, these methods were used for the first time to profile the supersonic parts of nozzles in the case of nonequilibrium two-phase flows by Dritov and Tishin [2]. For equilibrium flows, they have been used to optimize supersonic nozzles [3, 4] and in the construction of a profile of the subsonic part of a nozzle ensuring parallel sonic flow in the minimal section of the nozzle [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–183, January–February, 1982.I thank A. N. Kraiko for a number of helpful comments in a discussion of the formulation of the problem.     

Numerical computations of internal flows for axisymmetric and two-dimensional nozzles

MacCormack's explicit time-marching scheme is used to solve the full Navier–Stokes unsteady, compressible equations for internal flows. The requirement of a very fine grid to capture shock as well as separated flows is circumvented by employing grid clustering. The numerical scheme is applied for axisymmetric as well as two-dimensional flows. Numerical predictions are compared with experimental data and the qualitative as well as the quantitative agreement is found to be quite satisfactory. © 1997 John Wiley & Sons, Ltd.