一种三阶有限体积法及其在欠膨胀射流激波结构数值模拟中的应用

以喷管出口欠膨胀射流为研究对象,在Lagrange坐标系下建立欠膨胀射流二维积分形式的流动方程。通过在单元交接面处进行三阶ENO(essentially nonoscillatory)格式插值,构造得到一种适用于求解该方程的三阶ENO有限体积法。采用该格式对一维Sod激波管算例和喷管出口欠膨胀射流进行数值计算。计算结果表明,该方法具有高精度、基本无振荡的特点,能很好地捕捉包含激波、滑移线以及三波交点等复杂流场波系结构。计算得到的波系结构中马赫盘的位置与实验结果吻合很好,相对误差小于1.1%。     

Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.     

聚合物熔体三维挤出胀大的数值模拟

采用有限元方法分析K-BKZ本构方程描述的聚合物熔体的三维挤出胀大.对于本构方程中偏应力张量的计算,首先给出质点的运动轨迹,分段求出局部的变形梯度张量,再求出整体的变形梯度、Cauchy-Green应变张量和 Finger应变张量,沿轨迹采用分段高斯积分计算应力.把应力作为方程的右端项,给出迭代方法,求解非线性方程组.并根据自由面处的边界条件,迭代得出出口处自由面的最终位置.对轴对称流道和矩形流道进行分析计算,并把结果与二维分析和实验结果进行了比较,显示方法是可行的.     

Massively separated flows due to transverse sonic jet in laminar hypersonic stream

A numerical simulation of flow interactions due to a transverse sonic jet ejected from a two-dimensional slot into a hypersonic stream is carried out to examine the capability of Navier–Stokes solutions in predicting a massively separated flow upstream of the jet exit. Grid sensitivity has been studied using gradually refined meshes to address the numerical accuracy of the discretised solution of the governing equations. Comparison has been made with published experimental data regarding the separation and reattachment points and the pressure distribution in the separated region. Flow field visualisation provides further insight into the interaction region and reveals a small clockwise vortex immediately ahead of the jet exit, which is found to be responsible for the second peak in the surface pressure distribution. Received 30 June 1998 / Accepted 27 September 1998     

Flowfield characteristics of a transverse jet into supersonic flow with pseudo-shock wave

We performed an experimental investigation of the flowfield of a transverse jet into supersonic flow with a pseudo-shock wave (PSW). In this study, we injected compressed air as the injectant, simulating hydrocarbon fuel. A back pressure control valve generated PSW into Mach 2.5 supersonic flow and controlled its position. The positions of PSW were set at nondimensional distance from the injector by the duct height (x/H) of ?1.0, ?2.5, and ?4.0. Particle image velocimetry (PIV) gave us the velocity of the flowfield. Mie scattering of oil mist only with the jet was used to measure the spread of the injectant. Furthermore, gas sampling measurements at the exit of the test section were carried out to determine the injectant mole fraction distributions. Gas sampling data qualitatively matched the intensity of Mie scattering. PIV measurements indicated that far-upstream PSW decelerated the flow speed of the main stream and developed the boundary layer on the wall of the test section. The flow speed deceleration at the corner of the test section was remarkable. The PSW produced nonuniformity in the main stream and reduced the momentum flux of the main stream in front of the injector. The blowing ratio, defined as the square root of the momentum flux ratio, of the jet and the main stream considering the effect of the boundary layer thickness was shown to be a useful parameter to explain the jet behavior.     

A pseudo-time integral method for non-isothermal viscoelastic flows and its application to extrusion simulation

A pseudo-time integral scheme based on a finite streamline element method is developed to combine variable temperature with viscoelasticity. A specific KBKZ integral model for isothermal flow is transformed to its non-isothermal version by introducing a pseudo-time and applying the Morland-Lee hypothesis. The coupling between momentum and energy equations is through the time-temperature shifting factor by which the pseudo-time is defined. The observer time and the pseudo-time are simultaneously calculated when tracing the strain history for the stress calculation in a non-homogeneous temperature field. Using this scheme, a full non-isothermal numerical simulation of some IUPAC extrusion experiments is carried out. Results show that while the temperature distribution near the die exit plane is an important factor controlling extrudate swell, either self-heating inside the die tube or external cooling on the free surface dominantly determines the temperature distribution near the die exit when the wall temperature is kept constant, depending on whether the Péclet number is large or small. The hot layer effect predicted by the inelastic swell mechanism is confirmed and well illustrated by the computation. Calculations with reasonable thermal boundary conditions further convince us that the isothermal assumption in our earlier numerical simulation is a good approximation in this particular case.     

A finite element method for computing the flow of multi-mode viscoelastic fluids: comparison with experiments

The numerical computation of viscoelastic fluid flows with differential constitutive equations presents various difficulties. The first one lies in the numerical convergence of the complex numerical scheme solving the non-linear set of equations. Due to the hybrid type of these equations (elliptic and hyperbolic), geometrical singularities such as reentrant corner or die induce stress singularities and hence numerical problems. Another difficulty is the choice of an appropriate constitutive equation and the determination of rheological constants. In this paper, a quasi-Newton method is developed for a fluid obeying a multi-mode Phan-Thien and Tanner constitutive equation. A confined convergent geometry followed by the extrudate swell has been considered. Numerical results obtained for two-dimensional or axisymmetric flows are compared to experimental results (birefringence patterns or extrudate swell) for a linear low density polyethylene (LLDPE) and a low density polyethylene (LDPE).     

Concentration fluctuations in a turbulent jet

Using Spalding's model of turbulence in a turbulent shear flow, we have calculated the root-mean-square value of the concentration fluctuations inside a turbulent jet. Although we used the same equations and the same solution technique as Spalding, we have not been able to find precisely his numerical results derived for a jet issuing into a fluid at rest with the same density as the jet. The differences between our numerical results, Spalding's numerical results and the experimental data of Becker, Hottel and Williams are fairly small only if the initial values of the turbulence energy and the mixing length inside the jet and the turbulence in the ambient fluid are taken into account in the model. For a turbulent jet issuing into a turbulently flowing surrounding stream of different density, we found that the relative concentration fluctuations can increase considerably. This brings out the importance of taking into account property variables in analysing turbulent mixing processes.     

A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

Variational boundary integral approach for asymmetric impinging jets of arbitrary two‐dimensional nozzle

We consider asymmetric impinging jets issuing from an arbitrary nozzle. The flow is assumed to be two‐dimensional, inviscid, incompressible, and irrotational. The impinging jet from an arbitrary nozzle has a couple of separated infinite free boundaries, which makes the problem hard to solve. We formulate this problem using the stream function represented with a specific single layer potential. This potential can be extended to the surrounding region of the jet flow, and this extension can be proved to be a bounded function. Using this fact, the formulation yields the boundary integral equations on the entire nozzle and free boundary. In addition, a boundary perturbation produces an extraordinary boundary integral equation for the boundary variation. Based on these variational boundary integral equations, we can provide an efficient algorithm that can treat with the asymmetric impinging jets having arbitrarily shaped nozzles. Particularly, the proposed algorithm uses the infinite computational domain instead of a truncated one. To show the convergence and accuracy of the numerical solution, we compare our solutions with the exact solutions of free jets. Numerical results on diverse impinging jets with nozzles of various shapes are also presented to demonstrate the applicability and reliability of the algorithm.     

Calculation of spatial flow past an inclined flat plate with a detached shock wave

The results of a numerical analysis of a supersonic underexpanded jet impinging on an inclined flat plate are presented. The effects of the angle between the plate and the jet symmetry axis, the distance from the nozzle exit section, the exit Mach number, and the off-design conditions on the distribution of the gasdynamic parameters in the jet flowfield and on the plate surface are demonstrated. Specific features of the compressed layer and obstacle surface flows are revealed. The three-dimensional flow is simulated using the large particle method on the basis of the nonstationary Euler equations written in the cylindrical coordinate system. The calculated results are compared with experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 31–35, January–February, 1997.     

Investigation of interaction between a plane transverse gas jet and a supersonic stream

The problem of interaction between a two-dimensional transverse gas jet emerging from a slot on a flat plate and a supersonic stream is considered. Several theoretical methods based on various approaches and physical models have been proposed to determine the characteristics of such a flow. The following fundamental directions can be isolated: a quasi-one-dimensional method [1], use of blast theory [2], and methods based on the equivalence of the effect of the jet and some solid on the external flow [3, 4]. However, the listed computational schemes [1–4] do not permit any clarification of the flow configuration in the jet and in the outer stream (the shock configuration, the jet boundaries, the distribution of the gasdynamic parameters in the flow field, etc.). Extensive experimental investigations of this phenomenon have been carried out simultaneously with the development of the theoretical methods, wherein the flow picture was determined, the pressure distribution was measured in the interaction domain, etc. [5, 6]. A computation method is proposed in this paper which will permit a detailed investigation of the flow structure in the jet and in the outer flow outside the separation region. Underlying the method is a hypothesis verified experimentally: The separating streamline in the mixing layer of the separated boundary layer (the “constant mass∝ line) intersects the jet boundary at the point of maximum jet standoff from the nozzle exit towards the incoming stream.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

A short note on the entrainment and exit boundary conditions

An attempt is made to find out the suitable entrainment and exit boundary conditions in laminar flow situations. Streamfunction vorticity formulation of the Navier–Stokes equations are solved by ADI method. Two‐dimensional laminar plane wall jet flow is used to test different forms of the boundary conditions. Results are compared with the experimental and similarity solution and the proper boundary condition is suggested. The Kind 1 boundary condition is recommended. It consists of zero first derivative condition for velocity variable and for streamfunction equation, mixed derivative at the entrainment and exit boundaries. Copyright © 2005 John Wiley & Sons, Ltd.     

Magnetogasdynamic generation of electrical energy in models of aircraft with a high-speed jet engine

The possibility of generating electric power in a plane model of an integral high-speed hydrogen-burning jet engine by mounting a magnetogasdynamic (MHD) generator at the combustion chamber exit is discussed. Attention is concentrated on clarifying the effect of MHD energy extraction from the stream on the aircraft’s thrust characteristics. The internal and external flows are simulated numerically. The two-dimensional supersonic gasdynamic flow inside the engine (in the air-intake, combustion chamber, MHD generator, and nozzle) and the supersonic flow past the aircraft are described on the basis of the complete averaged system of Navier-Stokes equations (in the presence of turbulence), which includes MHD force and heat sources, a one-parameter turbulence model, the electrodynamic equations for an ideal segmented MHD generator, and the equations of the detailed chemical kinetics of hydrogen burning in air. The numerical solution is obtained by means of a computer program that uses a relaxation scheme and an implicit higher-order version of the Godunov method. It is shown that MHD electric power generation can be realized without disturbing the positive balance in the relation between the thrust and the drag of the aircraft with the engine operating with allowance for the MHD drag, but with some loss of effective thrust.     

Extrudate swell of Boger fluids

Boger fluids are dilute polymer solutions exhibiting high elasticity at low apparent shear rates, which leads to high extrudate swell. Numerical simulations have been undertaken for the flow of three Boger fluids (including benchmark Fluid M1), obeying an integral constitutive equation of the K-BKZ type, capable of describing the behavior of dilute polymer solutions. Their rheology is well captured by the integral model. The flow simulations are performed for planar and axisymmetric geometries without or with gravity. The results provide the extrudate swell and the excess pressure losses (exit correction), as well as the shape and extent of the free surface. All these quantities increase rapidly and monotonically with increasing elasticity level measured by the stress ratio, S_R. It was found that the main reason for the high extrudate swelling is high normal stresses exhibited in shear flow (namely, the first normal-stress difference, N₁). Surprisingly, the elongational parameter of the model or a second normal-stress difference N₂ do not affect the results appreciably. Gravity serves to lower the swelling considerably, and makes the simulations easier and in overall agreement with previous experiments.     

A comprehensive analysis of the viscous incompressible flow in quasi-three-dimensional aerofoil cascades

A rigorous model of the fully elliptic flow over the blade-to-blade stream surface in an annular aerofoil cascade is developed. The model accuracy stems from its precise simulation of the meridional hub-to-casing flow effects, including those of the shear stress components that are created by the spanwise velocity gradients. These stresses are unprecedentedly introduced in the flow-governing equations in the form of source terms and are modelled as such. The final set of flow-governing equations are solved using the Galerkin weighted residual method coupled with a biquadratic finite element of the Lagrangian type. The flow solution is verified against the numerical results of a fully three-dimensional flow model and a set of experimental data, both concerning a low-aspect-ratio stator of an axial flow turbine under a low Reynolds number and subsonic flow operation mode. The numerical results in this case show well predicted aerofoil loading and pitch-averaged exit flow conditions. Also evident is a substantial capability of the analysis in modelling such critical regions as the wake subdomain. It is further proven that the new terms in the governing equations enhance the quality of the numerical predictions in this class of flow problems.     

A CORRELATION EQUATION FOR CALCULATING INCLINED JET PENETRATION LENGTH IN A GAS-SOLID FLUIDIZED BED

Numerical simulation of gas-solid flow in a two-dimensional fluidized bed with an inclined jet was performed. The numerical model is based on the two-fluid model of gas and solids phase in which the solids constitutive equations are based on the kinetic theory of granular flow. The improved ICE algorithm, which can be used for both low and high-velocity fluid flow, were used to solve the model equations. The mechanism of jet formation was analyzed using both numerical simulations and experiments. The emergence and movement of gas bubbles were captured numerically and experimentally. The influences of jet velocity, nozzle diameter, nozzle inclination and jet position on jet penetration length were obtained. A semi-empirical expression was derived and the parameters were correlated from experimental data. The correlation equation, which can be easily used to obtain the inclined jet penetration length, was compared with our experimental data and published correlation equations.     

Numerical treatment of the steady flow of a liquid compound jet

A numerical treatment of the outflow of a two-layer laminar jet into a non-viscous continuous phase is performed. The dispersed phases (i.e. the central core and the concentric layer) are immiscible, incompressible and Newtonian fluids. The method of solution allows for the simultaneous determination of the shape of both interfaces, as well as of the corresponding velocity profiles. The equations of motion of both phases are obtained in a boundary layer approximation. The pressure jump in the radial direction, owing to forces of interfacial tension, is taken into account. Also studied is how the initial velocity profiles at the nozzle exit and some dimensionless parameters affect the interaction between the primary and secondary flow. Numerical results agree qualitatively with some experimental evidence. The approach can also be employed to predict the flow within a viscous continuous phase.     

Investigation of the microstructure a turbulent jet in a wake

The study of the characteristics of the turbulence in the boundary layer and in free jets is one of the most important problems of the aerodynamics of viscous fluids. The accumulation of information on the pulsation characteristics of jet flows and the establishment of the corresponding governing laws may serve to verify the basic hypotheses of the semiempirical theories of turbulence, and also for the development of more advanced computational methods. In many cases the measurement of the pulsation characteristics of turbulent jets is of practical interest.The studies made up till now [1–5] of the microstructure of turbulent flow in the primary region of submerged axisymmetric jets have made it possible to obtain several interesting results. In particular, in addition to the average velocity profiles, hot-wire anemometric equipment has been used to measure the normal and tangential Reynolds stresses and also the intermittency factor in cross sections of the jet, the distribution of the intensity of the longitudinal and lateral velocity pulsations along the axis, the correlation coefficients and the corresponding integral turbulence scales, etc. These measurements have made it possible to draw several important conclusions on the mechanism of turbulent exchange, on the order of the terms omitted in the equation of motion, and on the semiempirical theories of turbulence [6–9].The common deficiency of the studies mentioned above is that near the boundary of a submerged jet, where the average velocity is practically equal to zero, the intensity of the pulsations is so great that it makes the reliability of the results obtained by means of the hotwire anemometer questionable. In this connection Townsend [6] indicated the advisability of studying the microstructure of a turbulent jet issuing into a low-velocity ambient flow.The present study had as its objective the investigation of the microstructure of the primary region of an axisymmetric jet in a wake flow over quite a broad range of the flow ratio parameter m=u/u₀;here u₀ is the average velocity at the nozzle exit, u is the velocity of the ambient stream. For various values of the parameter m in the primary region of the jet measurements were made of the profiles of the three components of the pulsation velocity and the Reynolds shear stresses, and also the values of the average velocity and two components of the pulsation velocity at a large number of points on the jet axis. The measured profiles of the Reynolds shear stresses were compared with the corresponding profiles calculated on the basis of the boundary layer equations from the experimentally determined average velocity profiles. For two values of the parameter m, in one of the sections of the jet measurements were made of the correlation coefficients of the longitudinal components of the pulsation velocity and the variation across the jet of the integral turbulence scale was determined.The results obtained give an idea of the influence of the parameter m on the characteristics of the turbulent jet in an ambient stream.