Mach Wave Radiation by Mixing Layers. Part II: Analysis of the Source Field

In Part II a large-scale sound source in a time-developing planar free mixing layer is studied using an acoustic analogy approach. It is shown that only the non-compact character of the source resembles well the character of the corresponding source in a spatially developing flow. A model based on a continuous assembly of wave packets is derived and applied to direct numerical simulation results of two supersonic time-developing mixing layers undergoing transition to turbulence. The analysis predicts two distinctive dominant Mach wave sources in agreement with the direct analysis of Part~I. The first dominates during the stage of the Λ-vortex structure and the second just prior to the final breakdown to a fine-scale structure. The convective velocity of the second Mach wave source is higher than the first and thus its Mach angle of radiation is higher. The second source has a reduced strength at the higher free-stream Mach number. Directivity and frequency spectra compare well with the results of Part I, demonstrating that the assumptions inherent in the analogy are quite reasonable. Received 5 August 1997 and accepted 6 April 1998     

Noise radiated by a non-isothermal,temporal mixing layer. Part I: Direct computation and prediction using compressible DNS

Direct numerical simulations (DNSs) are performed in order to study acoustic emissions generated during the transition of isothermal and non-isothermal mixing layers. The sound from temporally evolving mixing layers is computed directly using DNS for a computational domain, which includes both aerodynamic and acoustic fields. Good precision of the computed acoustic field is ensured by using a numerical code based on high-order finite difference schemes of quasi-spectral accuracy. Two- and three-dimensional simulations of mixing layers are performed for various Mach numbers and temperature ratios. For each case, the acoustic radiation of the mixing layer transition is investigated. Comparisons illustrate the importance of the combined effects of temperature and Mach number on the acoustic intensity. Qualitative agreement with existing experimental observations for hot jet flows is observed. It is also found that the appearance of three-dimensional motion leads to a substantial reduction of sound emissions. In the second part of this study, DNS data are used to perform acoustic analogy predictions. Excellent agreement between direct computations and predictions is obtained in all cases. Analysis of the source terms yields a new interpretation of temperature and Mach number effects, based on the predominance of one term over the other.     

基于直接数值模拟的可压缩湍流模型评估和改进

对来流Mach数2.25和6的平板边界层湍流进行了直接数值模拟, 并通过与理论、实验及他人计算结果的对比对数值结果进行了验证. 基于直接数值模拟得到的湍流数据库, 对常用的湍流模型进行了先验评估. 评估的湍流模型有k-εvarepsilon模型(包括标准k-εvarepsilon 模型、可实现的k-εvarepsilon模型及低Reynolds数k-εvarepsilon模型)、SA模型及BL模型. 结果显示, 对于Mach2.25的平板边界层, 可实现的k-εvarepsilon 模型及低Reynolds 数k-εvarepsilon模型具有较好的预测能力, 而标准k-εvarepsilon模型预测的湍流黏性系数偏高; SA模型在边界层内层预测准确度较高, 而在外层预测值偏高. 而对于Mach6的平板边界层, k-εvarepsilon模型及SA模型预测的湍流黏性系数均偏高, 尤其是标准k-εvarepsilon模型. 对于Mach6的平板边界层, BL模型低估了内-外层交界位置, 造成湍流黏性系数预测值严重偏低. 作者通过修改模型系数及内-外层交界位置对BL模型进行了修改, 修改后模型预测的湍流黏性系数与DNS给出的值吻合较好.     

Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows

The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M _t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M _g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M _t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.      

Noise radiated by a non-isothermal, temporal mixing layer

The ability of Lighthill's analogy to predict the sound radiated by a transitional mixing layer is evaluated by means of direct numerical simulation (DNS). The specific case of low Mach number flows with density variations is investigated. In order to limit the global computational cost, the acoustic source information is based on numerical results where the sound waves have been removed. It is shown that the low Mach number approximation coupled with the acoustic analogy can lead to very accurate predictions for the radiated sound if the acoustic sources in Lighthill's equation are taken into account carefully. Results for the acoustic intensity deduced from a repeated use of the Lighthill's analogy over a wide range of Mach numbers allow us to discuss the adequacy of scaling laws proposed by previous authors (J. Sound Vib. 28(3), 563–585, 1973; 31(4), 391–397, 1973; 48(1), 95–111, 1976) for the prediction of noise from hot jets.     

可压缩自由剪切流混合转捩大涡模拟

针对湍流气动光学效应与冲压发动机气体混合机理问题,开展了可压缩混合层流动空间模式大涡模拟和时间模式直接数值模拟研究.通过对流场(包含亚/亚混合、超/亚混合两种情况)失稳、转捩直至完全湍流的空间发展过程的研究表明,对流Mach数0.4状态下流场失稳以二维最不稳定扰动为主;非线性发展中,基频涡对并及展向涡撕裂主控流动转捩,流场发生混合转捩;转捩后脉动流场基本达到局部各向同性,此时,湍流Mach数低于0.3,流动压缩性可近似忽略.     

DNS,LES and RANS of turbulent heat transfer in boundary layer with suddenly changing wall thermal conditions

The objectives of this study are to investigate a thermal field in a turbulent boundary layer with suddenly changing wall thermal conditions by means of direct numerical simulation (DNS), and to evaluate predictions of a turbulence model in such a thermal field, in which DNS of spatially developing boundary layers with heat transfer can be conducted using the generation of turbulent inflow data as a method. In this study, two types of wall thermal condition are investigated using DNS and predicted by large eddy simulation (LES) and Reynolds-averaged Navier–Stokes equation simulation (RANS). In the first case, the velocity boundary layer only develops in the entrance of simulation, and the flat plate is heated from the halfway point, i.e., the adiabatic wall condition is adopted in the entrance, and the entrance region of thermal field in turbulence is simulated. Then, the thermal boundary layer develops along a constant temperature wall followed by adiabatic wall. In the second case, velocity and thermal boundary layers simultaneously develop, and the wall thermal condition is changed from a constant temperature to an adiabatic wall in the downstream region. DNS results clearly show the statistics and structure of turbulent heat transfer in a constant temperature wall followed by an adiabatic wall. In the first case, the entrance region of thermal field in turbulence can be also observed. Thus, both the development and the entrance regions in thermal fields can be explored, and the effects upstream of the thermal field on the adiabatic region are investigated. On the other hand, evaluations of predictions by LES and RANS are conducted using DNS results. The predictions of both LES and RANS almost agree with the DNS results in both cases, but the predicted temperature variances near the wall by RANS give different results as compared with DNS. This is because the dissipation rate of temperature variance is difficult to predict by the present RANS, which is found by the evaluation using DNS results.     

Optimized group velocity control scheme and DNS of decaying compressible turbulence of relative high turbulent Mach number

To overcome the difficulty in the DNS of compressible turbulence at high turbulent Mach number, a new difference scheme called GVC8 is developed. We have succeeded in the direct numerical simulation of decaying compressible turbulence up to turbulent Mach number 0.95. The statistical quantities thus obtained at lower turbulent Mach number agree well with those from previous authors with the same initial conditions, but they are limited to simulate at lower turbulent Mach numbers due to the so‐called start‐up problem. The energy spectrum and coherent structure of compressible turbulent flow are analysed. The scaling law of compressible turbulence is studied. The computed results indicate that the extended self‐similarity holds in decaying compressible turbulence despite the occurrence of shocklets, and compressibility has little effects on relative scaling exponents when turbulent Mach number is not very high. Copyright © 2005 John Wiley & Sons, Ltd.     

HLLE++格式在高马赫数空腔流动模拟中的应用

空腔流动存在剪切层运动、涡脱落与破裂,以及激波与激波、激波与剪切层、激波与膨胀波和激波/涡/剪切层相互干扰等现象,流动非常复杂,特别是高马赫数(M>2)时,剪切层和激波更强,激波与激波干扰更严重,对数值格式的要求更高,既需要格式耗散小,对分离涡等有很高的模拟精度,又需要格式在激波附近具有较大的耗散,可以很好地捕捉激波,防止非物理解的出现。Roe和HLLC等近似Riemann解格式在高马赫数强激波处可能会出现红玉现象,而HLLE++格式大大改善了这种缺陷,在捕捉高超声速激波时避免了红玉现象的发生,同时还保持在光滑区域的低数值耗散特性。本文在结构网格下HLLE++格式的基础上,通过改进激波探测的求解,建立了基于非结构混合网格的HLLE++计算方法,通过无粘斜坡算例,验证了HLLE++格式模拟高马赫数流动的能力,并应用于高马赫数空腔流动的数值模拟,开展了网格和湍流模型影响研究,验证了方法模拟高马赫数空腔流动的可靠性和有效性。     

超声速平面剪切层声辐射涡模态数值分析

对Mc = 1.2二维超声速空间发展平面自由剪切层, 进行了扰动模态及流动结构的数值分析. 采用时空三阶改进MacCormack格式, 差分求解可压缩扰动Navier-Stokes方程, 直接数值模拟入口不同基频谐波扰动的非线性演化特征. 采用空间线性稳定性理论证明, 计算所促发的扰动波是声辐射涡模态. 扰动参数及特征函数分析显示, 声辐射涡模态是弱色散的快／慢两种外部模态, 在扰动对流Mach数为超声速一侧呈膨胀／压缩状辐射. 单频受迫扰动可无相差地促发多模态混合扰动波, 而在自然扰动条件下, 剪切层的稳定性受慢模态主导.     

Interaction of isotropic turbulence with a normal shock wave

The evolution of incompressible and compressible isotropic 2-d turbulent fields interacting with a normal shock wave up to Mach numbers of 2.4 was investigated by means of direct numerical simulation using an ENO scheme. A comparison of statistics with linear analysis results is presented. Vorticity amplification in the DNS agrees well with the linear theory. Energy spectra are enhanced more in the small scales than in the large scales for incoming incompressible turbulence. The amplification rate for initially compressible turbulence is comparatively small.     

Effects of Compressibility and Shock-Wave Interactions on Turbulent Shear Flows

Compressibility effects are present in many practical turbulent flows, ranging from shock-wave/boundary-layer interactions on the wings of aircraft operating in the transonic flight regime to supersonic and hypersonic engine intake flows. Besides shock wave interactions, compressible flows have additional dilatational effects and, due to the finite sound speed, pressure fluctuations are localized and modified relative to incompressible turbulent flows. Such changes can be highly significant, for example the growth rates of mixing layers and turbulent spots are reduced by factors of more than three at high Mach number. The present contribution contains a combination of review and original material. We first review some of the basic effects of compressibility on canonical turbulent flows and attempt to rationalise the differing effects of Mach number in different flows using a flow instability concept. We then turn our attention to shock-wave/boundary-layer interactions, reviewing recent progress for cases where strong interactions lead to separated flow zones and where a simplified spanwise-homogeneous problem is amenable to numerical simulation. This has led to improved understanding, in particular of the origin of low-frequency behaviour of the shock wave and shown how this is coupled to the separation bubble. Finally, we consider a class of problems including side walls that is becoming amenable to simulation. Direct effects of shock waves, due to their penetration into the outer part of the boundary layer, are observed, as well as indirect effects due to the high convective Mach number of the shock-induced separation zone. It is noted in particular how shock-induced turning of the detached shear layer results in strong localized damping of turbulence kinetic energy.     

带展向曲率的超音速混合层中扰动演化的研究

超音速混合层的流动不稳定性较之亚音速或不可压的混合层大大减弱,为了提高混合效率, 通过数值模拟的方法分别研究了展向曲率、展向速度、来流马赫数等因素对混合效率所 起的作用. 计算结果表明：在给定展向速度的情况下,带有展向曲率的三维混合层,曲率越 大三维扰动增长率越大,而且法向的卷起范围也越大. 当展向曲率不为零时,展向速度的增 大也能有效地增强混合能力. 由流场中的高频扰动波产生的涡,在向下游发展过程中会有破 碎、拉伸,低频扰动波没有发现这一现象. 对于有展向曲率和展向速度的混合层,提高来流 马赫数时,流场中最不稳定扰动的增长率仍很大. 因此,这是一种提高混合层混合效率的新 途径.     

Study of a Mixing Layer Formed by a Supersonic Main Stream of Air and a Co-Flowing Supersonic Helium Jet

The development of large-scale organised motions in a compressible mixing layer is studied experimentally using holographic interferometry, pressure and turbulence measurements. The mixing layer was formed behind the base of a parallel strut with a Mach 2 air main stream and a co-flowing two-dimensional slot jet (aspect ratio = 45) of helium at a Mach number of 1.2. The mixing layer exhibited highly organised vortical structures which were elongated and inclined approximately 45–50° to the flow direction. The mixing layer showed a linear growth and the mean velocity data indicated self-similar behaviour at sufficiently downstream distances. The results have shown that the thickness of the primary boundary layer has a strong influence on the growth and structure of the mixing layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Direct Numerical Simulation of the Three-Dimensional Transition to Turbulence in the Transonic Flow around a Wing

The three-dimensional transition to turbulence in the transonic flow around a NACA0012 wing of constant spanwise section has been analysed at zero incidence and within the Reynolds number range [3000, 10000], by performing the direct numerical simulation. The successive stages of the 2D and 3D transition beyond the first bifurcation have been identified. A 2D study has been carried out near the threshold concerning the appearance of the first bifurcation that is the von-Kármán instability. The critical Mach number associated with this flow transition has been evaluated. Three other successive stages have been detected as the Mach number further increases in the range [0.3, 0.99]. Concerning the 3D transition of this nominally 2D flow configuration, the amplification of the secondary instability has been studied within the Reynolds number range of [3000, 5000]. The formation of counter-rotating longitudinal vorticity cells and the consequent appearance of a large-scale spanwise wavelength have been obtained downstream of the trailing-edge shock. A vortex dislocation pattern is developed as a consequence of the shock-vortex interaction near the trailing edge. The subcritical nature of the present 3D transition to turbulence has been proven by means of the DNS amplification signals and the Landau global oscillator model.     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

Periodic oscillation and fine structure of wedge-induced oblique detonation waves

An oblique detonation wave for a Mach 7 inlet flow over a long enough wedge of 30 turning angle is simulated numerically using Euler equation and one-step rection model.The fifth-order WENO scheme is adopted to capture the shock wave.The numerical results show that with the compression of the wedge wall the detonation wave front structure is divided into three sections:the ZND model-like strcuture,single-sided triple point structure and dual-headed triple point strucuture.The first structure is the smooth straight,and the second has the characteristic of the triple points propagating dowanstream only with the same velocity,while the dual-headed triple point structure is very complicated.The detonation waves facing upstream and downstream propagate with different velocities,in which the periodic collisions of the triple points cause the oscillation of the detonation wave front.This oscillation process has temporal and spatial periodicity.In addition,the triple point trace are recorded to obtain different cell structures in three sections.     

High-resolution measurements of two-dimensional instabilities and turbulence transition in plane mixing layers

 High-resolution two-dimensional (2D) measurements on a large plane mixing layer provide new quantitative information of its spatial and temporal evolution to turbulence. Periodic acoustic excitation with three frequencies was used to stabilize the fundamental instability of the mixing layer (roll-up) and its first and second subharmonics (vortex pairings). Phase-locked velocity measurements of the time evolution in 2D space (x, y, t) reveal accurate spatially resolved primary (2D) instabilities of the mixing layer and turbulence transition. The measurements unveil new quantitative details of the initial Kelvin–Helmholtz waves and their spatial and temporal evolution into vortex shedding and the effect of the second subharmonic on the first vortex pairing. The second-subharmonic effect hastens alternate first pairings of the rollers, with the result that pairing is completed at two downstream locations. The pairings that occur closer to the knife-edge are more organized (laminar) than those occurring farther downstream (transitional). This effect is corroborated using Taylor’s hypothesis to compute the vorticity distributions from the measured velocity field and a pseudo-spectral simulation of the temporal evolution of the mixing layer. Received: 26 March 1998/Accepted: 2 March 1999     

Model of non-stationary,inhomogeneous turbulence

We compare results from a spectral model for non-stationary, inhomogeneous turbulence (Besnard et al. in Theor Comp Fluid Dyn 8:1–35, 1996) with direct numerical simulation (DNS) data of a shear-free mixing layer (SFML) (Tordella et al. in Phys Rev E 77:016309, 2008). The SFML is used as a test case in which the efficacy of the model closure for the physical-space transport of the fluid velocity field can be tested in a flow with inhomogeneity, without the additional complexity of mean-flow coupling. The model is able to capture certain features of the SFML quite well for intermediate to long times, including the evolution of the mixing-layer width and turbulent kinetic energy. At short-times, and for more sensitive statistics such as the generation of the velocity field anisotropy, the model is less accurate. We propose two possible causes for the discrepancies. The first is the local approximation to the pressure-transport and the second is the a priori spherical averaging used to reduce the dimensionality of the solution space of the model, from wavevector to wavenumber space. DNS data are then used to gauge the relative importance of both possible deficiencies in the model.     

Modelling of pressure–strain correlation in compressible turbulent flow

Previous studies carried out in the early 1990s conjectured that the main compressible effects could be associated with the dilatational effects of velocity fluctuation. Later, it was shown that the main compressibility effect came from the reduced pressure-strain term due to reduced pressure fluctuations. Although better understanding of the compressible turbulence is generally achieved with the increased DNS and experimental research effort, there are still some discrepancies among these recent findings. Analysis of the DNS and experimental data suggests that some of the discrepancies are apparent if the compressible effect is related to the turbulent Mach number, Mt. From the comparison of two classes of compressible flow, homogenous shear flow and inhomogeneous shear flow （mixing layer）, we found that the effect of compressibility on both classes of shear flow can be characterized in three categories corresponding to three regions of turbulent Mach numbers： the low-Mr, the moderate-Mr and high-Mr regions. In these three regions the effect of compressibility on the growth rate of the turbulent mixing layer thickness is rather different. A simple approach to the reduced pressure-strain effect may not necessarily reduce the mixing-layer growth rate, and may even cause an increase in the growth rate. The present work develops a new second-moment model for the compressible turbulence through the introduction of some blending functions of Mt to account for the compressibility effects on the flow. The model has been successfully applied to the compressible mixing layers.