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.     

Determination of specific thrust in unchoked regimes and design of a separationless convergent nozzle contour

The variation of the specific thrust R_Y on the angle of inclination of the wall is analyzed within the framework of the ideal gas model using the results of specific impulse and flow rate calculations for conical convergent nozzles. It is shown that in unchoked regimes nozzles with different have almost the same values of R_Y for both subcritical and supercritical pressure ratios _c. On the interval _C < 6 typical of convergent nozzles conical convergent nozzles with =30–90° have almost the same value of the specific thrust, maximal relative to the R_Y of nozzles with < 30°. In the presence of viscosity forces local boundary layer separation may occur in the neighborhood of the entrance section of the convergent nozzle. A method of constructing a separationless convergent nozzle contour with enhanced thrust is developed on the basis of a boundary layer separation criterion. The separationless contour is determined for given values of the flow rate, specific heat ratio, Reynolds number, wall temperature and initial boundary layer displacement thickness.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–164, January–February, 1990.     

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.     

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.     

Variational problem of optimum side wall design for a narrow three-dimensional nozzle

The optimum design of the side walls of the supersonic section of a three-dimensional nozzle with two planes of symmetry is considered in the narrow channel model approximation, which reduces three-dimensional to two-dimensional flow. This nozzle realizes maximum thrust for given sonic or supersonic inlet flow, upper and lower walls, maximum permissible length and pressure outside the nozzle. In general, an approximate solution of the variational problem can be obtained by the indeterminate control contour method [1]. For nozzles with nonexpanding end sections of the upper and lower walls this is a rigorous solution. Numerical algorithms, based on the method of characteristics, for constructing the optimum, side walls and calculating the flow in narrow channels are developed in the formulation adopted using the optimality conditions found, which generalize the wellknown conditions for plane and axisymmetric configurations [1]. In addition, the three-dimensional supersonic flow in the nozzles thus designed has been calculated in accordance with a shock-capturing marching scheme [2], which for the uniform grids employed in the calculations gives a second-order approximation. A rather complex relation is established between the thrust of the optimum configurations constructed and the shape of their inlet cross sections.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 102–112, March–April, 1992.The authors are grateful to L. E. Sternin for drawing their attention to the problem and to V. A. Vostretsova for assisting with the work.     

Experimental investigation of the recovery of the total pressure in a hypersonic wind tunnel with conical and profiled nozzles

Measurements have been made of the coefficient of recovery of the total pressure of a gas flow exhausting from axisymmetric and conical profiled hypersonic nozzles into a cylindrical channel of diameter equal to or greater than the nozzle exit and also in the presence of an Eiffel chamber. The experiments were made at Mach numbers M = 4.83–12.4 in the isentropic core. It is shown that the values of differ slightly (by 5%) from the corresponding value for a normal shock wave at the number M determined for a onedimensional flow by the ratio of the area of the cylindrical channel to the area of the critical section of the nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 170–173, January–February, 1984.     

Two-dimensional transonic vortex flows of an ideal gas

Equations are obtained for two-dimensional transonic adiabatic (nonisoenergetic and nonisoentropic) vortex flows of an ideal gas, using the natural coordinates (=const is the family of streamlines, and =const is the family of lines orthogonal to them). It is not required that the transonic gas flow be close to a uniform sonic flow (the derivation is given without estimates). Solutions are found for equations describing vortex flows inside a Laval nozzle and near the sonic boundary of a free stream.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–109, September–October, 1973.     

Comparison of integrated characteristics and shapes of profiled contours of laval nozzles with “smooth” and “abrupt” contractions

The influence of the selection of the generator of subsonic sections of plane and axisymmetric Laval nozzles on the integrated characteristics and on the shapes of their profiled supersonic parts is investigated in the approximation of an ideal (inviscid and nonheat-conducting) gas. Nozzles with a smooth entrance and with an abrupt contraction are compared for the same flow rates and size restrictions on the whole nozzle, not only to its supersonic part. In such a formulation, in accordance with [1], nozzles with abrupt contractions in the flow of ideal gas in them should be expected to have the best characteristics. This is confirmed by the results of the calculations performed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 129–137, July–August, 1986.The authors are grateful to V. A. Vostretsova for her help in the work.     

Profiling the Supersonic Part of a Plug Nozzle with a Nonuniform Transonic Flow

The effect of transonic flow nonuniformity on the profiling of optimal plug nozzles is studied in the inviscid gas approximation. Sonic and supersonic regions providing maximum thrust for given nozzle dimensions and a given outer pressure are designed for given subsonic contours and calculated nonuniform transonic flows. As in the case of uniform flow on a cylindrical sonic surface, the initial regions of the designed contours satisfy the condition that in these regions the flow Mach number is unity or near-unity. In all the examples calculated, the optimal plug nozzles produce a greater thrust than the optimal axisymmetric and annular nozzles with a near-axial flow for the same lengths and the same gas flow rates through the nozzle. It is established that contouring without regard for transonic flow nonuniformity can result in considerable thrust losses. However, these losses are due only to a decrease in the flow rate, while the specific thrust may even increase slightly.     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Asymptotic Modeling of Nonlinear Wave Processes in Shock‐Loaded Elastoplastic Materials

Nonlinear wave processes in shockloaded elastoplastic materials are modeled. A comparison of the results obtained with experimental data and numerical solutions of exact systems of dynamic equations shows that the model equations proposed qualitatively describe the stressdistribution evolution in both the elasticflow and plasticflow regions and can be used to solve one and twodimensional problems of pulsed deformation and fracture of elastoplastic media.     

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.     

Effect of the Streamwise Component of the Vorticity Formed in a Turbulent Jet Source on the Acoustic Characteristics of the Jet

The results of an experimental study of the effects of different nozzle heads on turbulent jet noise are analyzed. A configuration of four cylindrical heads, tabbed heads, and chevron nozzles are considered and the decreases in the acoustic-mechanical efficiency of the jet (acoustic power reduction) for jets exposed to different modes of action are compared.It is shown that the effects of tabbed and cylindrical heads, as well as of chevrons, share a common property which is associated with the occurrence of vorticity in the jet source and can be described on the basis of a unified criterion characterizing the action on both the jet flow structure and the jet noise.     

Negative wake generation of FENE-CR fluids in uniform and Poiseuille flows past a cylinder

The effect of flow conditions on the negative wake generation (longitudinal velocity overshoot behind a cylinder in the viscoelastic fluid flow along the centerline) has been investigated. FENE-CR model that predicts constant shear viscosity and controlled extensional viscosity was considered as a constitutive equation. The discrete elastic viscous split stress-G/streamline upwind Petrov–Galerkin (DEVSS-G/SUPG) formulation was employed and the high-resolution solutions were obtained with an efficient iterative solver based on the incomplete LU(0)-type preconditioner and BiCGSTAB. We found that the negative wake generation was more obvious in uniform flow conditions than in Poiseuille flow, which suggests that the experimentally unrevealed negative wake generation of Boger fluids could be partially attributed to the geometrical effect of Poiseuille flow. The negative wake generation was more discernable at low extensibility and high value of viscosity ratio, which agrees well with the previous studies. In addition, we could observe an undershoot phenomenon in Poisseuille flow condition, which has never been reported.     

Underexpanded wall jet in a cocurrent flow

The flow of a two-dimensional underexpanded wall jet flowing out of a sonic nozzle along a channel wall has been experimentally investigated. The dependence of the dimension of the first barrel of the jet on the underexpansion is obtained. It is shown that the flow of the jet in the channel is associated with a significant axial pressure gradient on the initial interval of the induced cocurrent flow and that this leads to a substantial change in the geometric dimensions of the jet.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 196–199, January–February, 1993.     

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.     

Deformed particle image pattern matching in Particle Image Velocimetry

The deformation of particle image patterns due to velocity gradients causes errors of velocity measurements and false velocity detections in PIV (Particle Image Velocimetry). A novel technique to overcome those limitations inherent in the conventional PIV by correcting the particle image pattern according to the local velocity gradients in two dimensional flows, i.e. u/x, u/y, v/x and v/y, is proposed and successfully applied to a water flow downstream of a backward facing step.     

Theory of a laval nozzle for a two-phase mixture containing particles of small lag

A two-velocity and two-temperature model is considered for a continuous medium in relation to the flow of a mixture of gas and particles in the subsonic, transsonic, and supersonic parts of a Laval nozzle. It is assumed that the particles are small, and hence that the coefficients f and ^q, which define the interaction with the gas, are large (these coefficients are inversely proportional to the square of the particle radius for a Stokes mode of flow). This means that the velocity or thermal lag of the particles relative to the gas is small. The solution is sought as expansions with respect to the small parameters ₁=1/f and ₂=1/^q.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 89–100, July–August, 1973.     

Analytical Solution for the Effect of Viscous Dissipation on Mixed Convection in Saturated Porous Media

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Ge _x =gx/c _p. The governing parameters are Re, Ra, Pe and Ge _x . The case of Re=0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra=0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows.     

On flow and stability of a Newtonian fluid past a rotating plane

An exact solution is given for the steady flow of a Newtonian fluid occupying the halfspace past the plane z=0 uniformly rotating about a fixed normal axis (Oz). This solution is obtained in a velocity field of the form considered by Berker [2] and can be deduced as a limiting case, as h+, of the solution to the problem relative to the strip 0zh imposing at z=h either the adherence boundary conditions or the free surface conditions. Furthermore, the stability of this flow, subject to periodic disturbances of finite amplitude, is studied using the energy method and the result is compared with those corresponding to stability of flows in the strip 0zh.

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Sommario In questa nota si mostra che-oltre alla calssica soluzione di von Karman [1] — esiste, per opportuni valori del gradiente di pressione all'infinito, una soluzione esatta per il moto stazionario di un fluido Newtoniano posto nel semispazio limitato dal piano z=0 uniformemente rotante attorno ad un asse ad esso perpendicolare (Oz). Tale soluzione, ottenuta sulla scia del lavoro di Berker [2], si può dedurre anche come limite, per h+, della soluzione del problema relativo alla striscia 0zh quando sul piano z=h si assegnano o le condizioni di aderenza o le condizioni di frontiera libera. Si studia poi la stabilità di tale moto rispetto a perturbazioni spazialmente periodiche di ampiezza finita col metodo dell'energia e si confronta il risultato ottenuto con quelli relativi alla stabilità dei moti nella striscia 0zh.