Influence of nozzle geometry on the near-field structure of a highly underexpanded sonic jet

Detailed near-field structures of highly underexpanded sonic free jets have been investigated with the help of computational fluid dynamics. Two-dimensional, axisymmetric Euler equations have been chosen to predict the underexpanded jets, and the third-order total variation diminishing finite-difference scheme has been applied to solve the system of governing equations numerically. Several different nozzles have been employed to investigate the influence of the nozzle geometry on the near-field structures of highly underexpanded sonic free jets. The results obtained show that the distance from the nozzle exit to the Mach disk is an increasing function of the jet–pressure ratio, which also significantly influences the shape of the jet boundary. The diameter of the Mach disk increases with the jet–pressure ratio, and it is further significantly influenced by the nozzle geometry, unlike the distance of the Mach disk from the nozzle exit. However, such a dependence on the nozzle geometry is no longer found when an effective-diameter concept is taken into account for the flow from a sharp-edged orifice. A good correlation in the diameters of the Mach disk is obtained, so that the near-field structure of highly underexpanded sonic free jets is a unique function of the pressure ratio, regardless of the nozzle geometry.     

A numerical study of turbulent opposed impinging jets issuing from triangular nozzles with different geometries

In present research, two turbulent opposed impinging air jets issuing from triangular nozzles with fixed and variable exit velocity ratios and different nozzle-to-nozzle distances have been studied numerically and then compared with rectangular and circular nozzles. The finite volume method has been applied for solving mass and momentum equations. The turbulence model being used here is k-ε RNG. Distributions of pressure, turbulence, kinetic energy and its dissipation rate in various regions especially on the impingement regions have been obtained with high accuracy. Study of the nozzle geometries has shown the advantage of triangular nozzles over other geometries. First, the triangle’s base in nozzle geometry has an important role in our study case which, mixing two flows and regions with high turbulence intensity, directly depends on it. Second, our results show that circular and rectangular nozzles have less efficiency than triangular nozzles in mixing applications. Third and last, it was found that the radial jet being created by opposed jets has some similarities to free jets. In this investigation, air in standard atmospheric pressure has been applied as working fluid.     

The Problem of an External Circular Crack Under Asymmetric Loadings

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Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Integration of the boussinesq equations for steady channel and jet flows

The equations of long nonlinear waves in round jets and channels of arbitrary cross section are considered with account for the transverse acceleration of the fluid particles (Boussinesq approximation). In the general case of steady flows, the equations in the form of shallow water equations with the pressure expressed in terms of the variational derivative of the kinetic energy of a thin transverse fluid layer, have three first integrals with three arbitrary constants. Examples of solutions of the equations for solitary capillary-gravitation waves in rectangular and triangular channels are presented and compared with the higher approximations. The shape of the free boundary of the round jet is determined. In the case of outflow from a conical nozzle an analytical dependence of the jet contraction ratio on the conicity angle is obtained. The dependence is in agreement with the experimental data for angles of less than 45°.     

Numerical investigations of radial jet reattachment flows

A finite volume computational scheme to solve the Navier-Stokes equations for the laminar flow fields of partially enclosed axial and radial jets impinging on a flat plate has been devised and tested. This scheme is based on the SIMPLEC technique. However, because of the backflow at the ‘outflow’ boundary, the SIMPLEC pressure correction technique has to be modified. The need for this modification, necessitated by the convergence failure, showed the ‘hidden’ pressure boundary condition of SIMPLE-type techniques. Test computations with the present scheme for flows in a channel with a built-in cylinder show that the location of the exit boundary affects very slightly the separated flow behind the cylinder. Computed Squire jet flows compare quite well with the available analytical solution. Finally, impinging radial jets have been computed for different Reynolds numbers. The results show the critical Reynolds number below which a steady solution is obtained and above which periodic and eventually chaotic flows result.     

Relationship between vortex and sound in under‐expanded supersonic impinging jets

The study of an under‐expanded supersonic jet impinging on a flat plate by using large‐eddy simulation is reported. A third‐order upwind compact difference and a fourth‐order symmetric compact scheme are employed to discretize the nondimensional axisymmetric compressible Favre‐filtered Navier–Stokes equations in space, whereas the third‐order Runge–Kutta method with the total variation diminishing property is adopted to deal with the temporal discretization. The numerical simulation successfully captures the shock wave and vortex structures with different scales in the flow field. Waves with high and low frequencies traveling forward and reflecting back, and sound sources in different locations can be observed. By comparison with the frequency of the impinging tone from the experiment, it can be deduced that the change of pressure and swirling strength in the shear layer, pressure change on the impinging plate, and vortex merging in the jet shear layer are interdependent with the impinging tone. The effects of nozzle lip thickness on the impinging jet flow field have been investigated. The results show that the values of pressure fluctuation and vortex swirling strength in the shear layer near the nozzle have an extremum with the variation of the nozzle lip thickness. The results provide a theoretical foundation for the design of supersonic nozzles. Copyright © 2011 John Wiley & Sons, Ltd.     

Special characteristics of the propagation of three-dimensional viscous jets

Numerical investigations were made of the propagation, in a supersonic wake, of uncalculated jets, flowing out of nozzles of square and rectangular cross section, and of lumped jets, made up of from two to nine individual jets; the special characteristics of their flow were investigated in the initial, transitional, and main sections. Specifically, for lumped jets, the possibility of replacing them by a single axisymmetric jet, equivalent in mass-flow rate, is discussed. To calculate a three-dimensional unexpanded supersonic jet, flowing out into a wake, in [1] it was proposed to use a numerical method for solving a simplified system of Navier-Stokes equations for steady-state flow, and numerical investigations were made of the three-dimensional interaction of four jets in a supersonic wake, at small distances from the outlet cross section of the nozzle, i.e., mainly in the initial sections of the jets, where the mixing layers along the boundaries of the jets are still not closed. Here the method of [1] is used to study the special characteristics of three-dimensional viscous jets at large distances from the outlet cross section of the nozzle in the region of the main section, where the mixing layers have come together and a single three-dimensional jet has been formed. The system of equations, the boundary conditions, the numerical method, the system of coordinates, and the nomenclature used are the same as in [1].     

An experimental study on jets issuing from elliptic inclined nozzles

This paper reports on an experimental flow visualisation and digital particle image velocimetry investigation on forced jets exhausting from aspect ratio equal to three elliptic nozzles with exits inclined at 30° and 60°. Flow images show that shear layer instabilities and subsequent vortex roll-ups are formed parallel to the inclined nozzle exits at 30° incline and that rapid re-orientation of the vortex roll-ups occurs at 60° incline. Flow observations also show that strong axis-switching occurs in a non-inclined elliptic nozzle. However, 30° and 60° elliptic inclined nozzles produce significant distortions to and suppression of the axis-switching behaviour, respectively. As a result, flow stresses and turbulent kinetic energy distributions become increasingly asymmetric. Their coherency and magnitudes along the shorter nozzle lengths also vary significantly. This can be attributed to the dissimilar formations of vortex roll-ups and rib structures, as well as unequal mutual interactions between them as the incline-angle increases. Lastly, results also show that unlike circular inclined nozzles, elliptic inclined nozzles do not produce serpentine-shaped jet columns nor lead to significant lateral jet-spread at large incline-angles.     

旋流喷嘴中空旋转射流近区域流动的研究

从理论上分析了旋流喷嘴喷出的中空射流近区域的液膜的运动,在只考虑液膜表面张力的作用下,应用质量守恒和动量定理,建立了描述液膜运动的非线性常微分（积分）方程组,该方程组可以用数值方法方便地求解,结果表明,液体离开旋流喷嘴后在自由空间形成的液膜呈葫芦形状,其速度和液膜厚度等都周期性地变化。本结果是在液厝受拓动失称碎成液滴前的最基本运动状态,可以在射流的近区域内实验观察到,也是进一步从理论液膜破碎雾化过     

Wall jet similarity of impinging planar underexpanded jets

Velocity profiles and wall shear stress values in the wall jet region of planar underexpanded impinging jets are parameterized based on nozzle parameters (stand-off height, jet hydraulic diameter, and nozzle pressure ratio). Computational fluid dynamics is used to calculate the velocity fields of impinging jets with height-to-diameter ratios in the range of 15–30 and nozzle pressure ratio in the range of 1.2–3.0. The wall jet has an incomplete self-similar profile with a typical triple-layer structure as in traditional wall jets. The effects of compressibility are found to be insignificant for wall jets with Ma < 0.8. Wall jet analysis yielded power-law relationships with source dependent coefficients describing maximum velocity, friction velocity, and wall distances for maximum and half-maximum velocities. Source dependency is determined using the conjugate gradient method. These power-law relationships can be used for mapping wall shear stress as a function of nozzle parameters.     

Oscillations of the fluid flow and the free surface in a cavity with a submerged bifurcated nozzle

The free surface dynamics and sub-surface flow behavior in a thin (height and width much larger than thickness), liquid filled, rectangular cavity with a submerged bifurcated nozzle were investigated using free surface visualization and particle image velocimetry (PIV). Three regimes in the free surface behavior were identified, depending on nozzle depth and inlet velocity. For small nozzle depths, an irregular free surface is observed without clear periodicities. For intermediate nozzle depths and sufficiently high inlet velocities, natural mode oscillations consistent with gravity waves are present, while at large nozzle depths long term self-sustained asymmetric oscillations occur.For the latter case, time-resolved PIV measurements of the flow below the free surface indicated a strong oscillation of the direction with which each of the two jets issue from the nozzle. The frequency of the jet oscillation is identical to the free surface oscillation frequency. The two jets oscillate in anti-phase, causing the asymmetric free surface oscillation. The jets interact through a cross-flow in the gaps between the inlet channel and the front and back walls of the cavity.     

Shock wave configurations and flow structures in non-axisymmetric underexpanded sonic jets

An experimental and numerical study of underexpanded free sonic jet flows issuing from rectangular, elliptical and slot nozzles has been undertaken. Aspect ratios (AR) of 1, 2, and 4 are described at pressure ratios (exit plane pressure to ambient pressure), of 2 and 3. There is good qualitative agreement between the experimental observations and the numerical predictions. In the case of rectangular jets, a complex system of shock waves forming the incident shock system is identified. This shock wave system originates at the corners of the nozzle exits, and proceeds downstream. Mach reflections are found to occur on the incident shock wave surface as well as the presence of a Mach disk terminating the first jet cell. This Mach disk has the shape of a square, a hexagon, or an octagon depending on the nozzle shape. For slot and elliptical jets, the formation of the incident shock wave was not observed along the minor axis plane of the nozzle for AR > 2. The incident shock wave was observed to originate downstream of the nozzle exit in the major axis plane. This wave system undergoes a transition to Mach reflection as it propagates downstream of the nozzle exit. In all cases tested, the shape of the jet boundary is significantly distorted. In rectangular jets, the narrowing of the jet boundary along the diagonal axis of the nozzle exit is observed, and in the case of the elliptical and slot jets axis switching is noted.     

横观各向同性材料三维裂纹问题的数值分析

严格从三维横观各向同性材料弹性空间问题的Green函数出发,采用Hadamard有限部积分概念,导出了三维状态下单位位移间断(位错)集度的基本解.在此基础上,将三维任意形状的片状裂纹问题归结为求解-组以未知位移间断表示的超奇异积分方程;并给出了边界元离散形式.对方程中出现的超奇异积分,采用了Had-alnard定义的有限部积分来处理.论文最后给出了若干典型片状裂纹问题的数值算例,数值结果表明了本文方法是非常有效的.     

Asymmetric flow of subsonic and sonic jets over an infinite wedge

A study is made of the flow of subsonic or sonic jets over an infinite wedge when the stagnation streamline bifurcates at the tip of the wedge. This regime can be realized only for a definite (previously unknown) relationship between the geometrical parameters. The problem is solved in the hodograph plane by the numerical method of [1] developed for the problem of a profiled Laval nozzle. A solution to the asymmetric problem obtained in the hodograph plane can be realized physically only for a definite relationship between the boundary values for the flow function. This relationship (which generalizes Prandtl's well-known formula [2] derived for asymmetric flow of incompressible jets over a plate on the basis of the momentum theorem) is obtained by analyzing the asymptotic behavior of the solution near the stagnation point. Examples of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 137–141, March–April, 1980.     

COMPUTATIONAL STUDIES OF IMPINGING JETS USING K-ϵ TURBULENCE MODELS

This paper reports numerical modelling of impinging jet flows using Rodi and Malin corrections to the k–ϵ turbulence model, carried out using the PHOENICS finite volume code. Axisymmetric calculations were performed on single round free jets and impinging jets and the effects of pressure ratio, height and nozzle exit velocity profile were investigated numerically. It was found that both the Rodi and Malin corrections tend to improve the prediction of the hydrodynamic field of free and impinging jets but still leave significant errors in the predicted wall jet growth. These numerical experiments suggest that conditions before impingement significantly affect radial wall jet development, primarily by changing the wall jet's initial thickness.     

PIV Measurements in the Near and Intermediate Field Regions of Jets Issuing from Eight Different Nozzle Geometries

An experimental study was conducted to investigate the effect of nozzle geometry on the mixing characteristics and turbulent transport phenomena in turbulent jets. The nozzle geometry examined were round, square, cross, eight-corner star, six-lobe daisy, equilateral triangle as well as ellipse and rectangle each with aspect ratio of 2. The jets were produced from sharp linear contoured nozzles which may be considered intermediate to the more widely studied smooth contraction and orifice nozzles. A high resolution particle image velocimetry was used to conduct detailed velocity measurements in the near and intermediate regions. It was observed that the lengths of the potential cores and the growth rates of turbulence intensities on the jet centerline are comparable with those of the orifice jets. The results indicate that the decay and spreading rates are lower than reported for orifice jets but higher than results for smooth contoured jets. The jets issuing from the elliptic and rectangular nozzles have the best mixing performance while the least effective mixing was observed in the star jet. The distributions of the Reynolds stresses and turbulent diffusion clearly showed that turbulent transport phenomena are quite sensitive to nozzle geometry. Due to the specific shape of triangular and daisy jets, the profiles of mean velocity and turbulent quantities are close to each other in their minor and major planes while in the elliptic and rectangular jets are considerably different. They also exhibit more isotropic behavior compared to the elliptic and rectangular jets. In spite of significant effects of nozzle geometry on mean velocity and turbulent quantities, the integral length scales are independent of changes in nozzle geometry.     

LES and RANS modelling of under-expanded jets with application to gaseous fuel direct injection for advanced propulsion systems

A density-based solver with the classical fourth-order accurate Runge-Kutta temporal discretization scheme was developed and applied to study under-expanded jets issued through millimetre-size nozzles for applications in high-pressure direct-injection (DI) gaseous-fuelled propulsion systems. Both large eddy simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) turbulence modelling techniques were used to evaluate the performance of the new code. The computational results were compared both quantitatively and qualitatively against available data from the literature. After initial evaluation of the code, the computational framework was used in conjunction with RANS modelling (k-ω SST) to investigate the effect of nozzle exit geometry on the characteristics of gaseous jets issued from millimetre-size nozzles. Cylindrical nozzles with various length to diameter ratios, namely 5, 10 and 20, in addition to a diverging conical nozzle, were studied. This study is believed to be the first to provide a direct comparison between RANS and LES within the context of nozzle exit profiling for advanced high-pressure injection systems with the formation of under-expanded jets. It was found that reducing the length of the straight section of the nozzle by 50% resulted in a slightly higher level of under-expansion (∼2.6% higher pressure at the nozzle exit) and ∼1% higher mass flow rate. It was also found that a nozzle with 50% shorter length resulted in ∼6% longer jet penetration length. At a constant nozzle pressure ratio (NPR), a lower nozzle length to diameter ratio resulted in a noticeably higher jet penetration. It was found that with a diverging conical nozzle, a fairly higher penetration length could be achieved if an under-expanded jet formed downstream of the nozzle exit compared to a jet issued from a straight nozzle with the same NPR. This was attributed to the radial restriction of the flow and consequently formation of a relatively smaller reflected shock angle. With the conical nozzle used in this study and a 30 bar injection pressure, an under-expanded hydrogen jet exhibited ∼60% higher penetration length compared to an under-expanded nitrogen jet at 100 μs after start of injection. Moreover, the former jet exhibited ∼22% higher penetration compared to a nitrogen jet issued through the conical profile with 150 bar injection pressure.     

Study of strongly underexpanded flooded jets

The results are presented on a numerical study of the propagation of strongly underexpanded jets of an ideal gas in a flooded space with Mach numbers at the nozzle cut slightly exceeding unity. In this case the angle of turn of the stream in the vicinity of the rim of the conical nozzle can exceed 90 and the boundary of the jet turns back beyond the cross section of the nozzle cut. It is shown that the square root of the expansion ratio of the discharge is a self-similarity parameter for the geometrical dimensions of the calculated jets. The dependence of the longitudinal dimensions of such jets on the Mach number at the nozzle cut is close to linear. The effect of the adiabatic index on the shape of the jet boundary and the suspended compression shock is illustrated. The integration of the steady system of gasdynamic equations describing the two-dimensional supersonic flow of a nonviscous and thermally nonconducting gas, performed by a monotonie finite-difference system of through calculation of the first order of accuracy, was proposed in [1, 2] and generalized to the case of arbitrary orientation of the velocity vector in the plane of flow in [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 151–154, November–December, 1976.The authors are grateful to A. N. Kraiko for attention to the work.     

Transonic Shocks in Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for a steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity (non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Fréchet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.