Wall Effect on the Convective–Absolute Boundary for the Compressible Shear Layer

The linear stability of inviscid compressible shear layers is studied. When the layer develops at the vicinity of a wall, the two parallel flows can have a velocity of the same sign or of opposite signs. This situation is examined in order to obtain first hints on the stability of separated flows in the compressible regime. The shear layer is described by a hyperbolic tangent profile for the velocity component and the Crocco relation for the temperature profile. Gravity effects and the superficial tension are neglected. By examining the temporal growth rate at the saddle point in the wave-number space, the flow is characterized as being either absolutely unstable or convectively unstable. This study principally shows the effect of the wall on the convective–absolute transition in compressible shear flow. Results are presented, showing the amount of the backflow necessary to have this type of transition for a range of primary flow Mach numbers M ₁ up to 3.0. The boundary of the convective–absolute transition is defined as a function of the velocity ratio, the temperature ratio and the Mach number. Unstable solutions are calculated for both streamwise and oblique disturbances in the shear layer. Received 9 May 2001 and accepted 21 August 2001     

Flow stability in a channel with porous walls

We study the stability of the flow which forms in a plane channel with influx of an incompressible viscous fluid through its porous parallel walls. Under certain assumptions the study of the stability reduces to the solution of modified Orr-Sommerfeld equation accounting for the transverse component of the main-flow velocity. As a result of numerical integration of this equation we find the dependence of the local critical Reynolds number on the blowing Reynolds number R₀, which may be defined by two factors: the variation of the longitudinal velocity profile with R₀ and the presence of the transverse velocity component. A qualitative comparison is made of the computational results with experimental data on transition from laminar to turbulent flow regimes in channels with porous walls, which confirms that it is necessary to take into account the effect of the transverse component of the main-flow velocity on the main-flow stability in the problem in question.Flows in channels with porous walls are of interest for hydrodynamic stability theory in view of the fact that they can be described by the exact solutions of the Navier-Stokes equations by analogy with the known Poiseuille and Couette flows. However, in contrast with the latter, the flows in channels with porous walls (studies in [1], for example) will be nonparallel.The theory of hydrodynamic stability of parallel flows has frequently been applied to nonparallel flows (in the boundary layer, for example). In so doing the nonparallel nature of the flow has been taken into account only by varying the longitudinal velocity component profiles. A study was made in [2, 3] of the effect of the transverse component of the main flow on its stability. In the case of the boundary layer in a compressible gas, a considerable influence of the transverse velocity component on the critical Reynolds number was found in [2] and confirmed experimentally. A strong influence of the transverse velocity component on the instability region was also found in [3] in a study of the flow stability in a boundary layer with suction for an incompressible fluid.     

Mixing Measurements in a Supersonic Expansion-Ramp Combustor

This paper reports results on molecular mixing for injection via an expansion-ramp into a supersonic freestream with M ₁ = 1.5. This geometry produces a compressible turbulent shear layer between an upper, high-speed “air” stream and a lower, low-speed “fuel” stream, injected through an expansion-ramp at α = 30° to the high-speed freestream. Mass injection is chosen to force the shear layer to attach to the lower guide wall. This results in part of the flow being directed upstream, forming a recirculation zone. Employing the hypergolic hydrogen-fluorine chemical reaction and pairs of “flip” experiments, molecular mixing is quantified by measuring the resulting temperature rise. Initial experiments established the fast-chemistry limit for this flow in terms of a Damköhler number (Da). For Da ≥ 1.4, molecularly mixed fluid effectively reacts to completion. Parameters varied in these experiments were the measurement station location, the injection velocity of the (lower) “fuel” stream, the stoichiometry for the flip experiments, and the density ratio of the fuel and air streams. As expected, mixing increases with increasing distance from the injection surface. The mixed fluid fraction increases by 12% when changing the fuel-to-air stream density ratio from 1 to 0.2. Comparisons with measurements at subsonic (high-speed) “air” stream velocities show that the trend of decreasing mixing with increasing speed documented in free-shear layer flows is also encountered in these flows. The current geometry produces higher mixing levels than do free shear layers.     

Turbulent flow over a rough backward-facing step

This work characterizes the impacts of the realistic roughness due to deposition of foreign materials on the turbulent flows at surface transition from elevated rough-wall to smooth-wall. High resolution PIV measurements were performed in the streamwise-wall-normal (x–y) planes at two different spanwise positions in both smooth and rough backward-facing step flows. The experiment conditions were set at a Reynolds number of 3450 based on the free stream velocity U_∞ and the mean step height h, expansion ratio of 1.01, and the ratio of incoming boundary layer thickness to the step height, δ/h, of 8. The mean flow structures are observed to be modified by the roughness and they illustrate three-dimensional features in rough backward-facing step flows. The mean reattachment length X_r is significantly reduced by the roughness at one PIV measurement position while is slightly increased by the different roughness topography at the other measurement position. The mean velocity profiles at the reattachment point indicate that the studied roughness weakens the perturbation of the step to the incoming turbulent flow. Comparisons of Reynolds normal and shear stresses, productions of normal stresses, quadrant analysis of the instantaneous shear-stress contributing events, and mean spanwise vorticity reveal that the turbulence in the separated shear layer is reduced by the studied roughness. The results also indicate an earlier separation of the turbulent boundary layer over the current rough step, probably due to the adverse pressure gradient produced by the roughness topography even before the step.     

Direct numerical simulation of supersonic pipe flow at moderate Reynolds number

We study compressible turbulent flow in a circular pipe at computationally high Reynolds number. Classical related issues are addressed and discussed in light of the DNS data, including validity of compressibility transformations, velocity/temperature relations, passive scalar statistics, and size of turbulent eddies. Regarding velocity statistics, we find that Huang’s transformation yields excellent universality of the scaled Reynolds stresses distributions, whereas the transformation proposed by Trettel and Larsson (2016) yields better representation of the effects of strong variation of density and viscosity occurring in the buffer layer on the mean velocity distribution. A clear logarithmic layer is recovered in terms of transformed velocity and wall distance coordinates at the higher Reynolds number under scrutiny (Re_τ ≈ 1000), whereas the core part of the flow is found to be characterized by a universal parabolic velocity profile. Based on formal similarity between the streamwise velocity and the passive scalar transport equations, we further propose an extension of the above compressibility transformations to also achieve universality of passive scalar statistics. Analysis of the velocity/temperature relationship provides evidence for quadratic dependence which is very well approximated by the thermal analogy proposed by Zhang et al. (2014). The azimuthal velocity and scalar spectra show an organization very similar to canonical incompressible flow, with a bump-shaped distribution across the flow scales, whose peak increases with the wall distance. We find that the size growth effect is well accounted for through an effective length scale accounting for the local friction velocity and for the local mean shear.     

The stability of two-dimensional linear flows of an Oldroyd-type fluid

A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter λ which ranges from λ = 0 for simple shear flow to λ = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter β varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t → ∞ of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < λ ? 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber α₃ in the direction normal to the basic flow plane are unstable. However, for certain values of β, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber α₃.     

Energy Estimate of the Critical Reynolds Numbers in a Compressible Couette Flow. Effect of Bulk Viscosity

A variational problem of determining the critical Reynolds number of the laminar-turbulent transition is numerically solved within the framework of the nonlinear energy theory of stability of compressible flows. Stability of various modes in the Couette flow of a compressible gas is estimated by the method of collocations. It is demonstrated that the minimum critical Reynolds numbers in the range of the ratio of the bulk viscosity ηb to the shear viscosity η, which is realistic for diatomic gases, are reached for modes of streamwise disturbances. The critical Reynolds numbers increase as the bulk viscosity is increased in the interval η_b = 0-2η, with the maximum increase in the limit being approximately 30%.     

Comparison of magnetic resonance concentration measurements in water to temperature measurements in compressible air flows

Magnetic resonance imaging (MRI) measurements in liquid flows provide highly detailed 3D mean velocity and concentration data in complex turbulent mixing flow applications. The scalar transport analogy is applied to infer the mean temperature distribution in high speed gas flows directly from the MRI concentration measurements in liquid. Compressibility effects on turbulent mixing are known to be weak for simple flows at high subsonic Mach number, and it was not known if this would hold in more complex flows characteristic of practical applications. Furthermore, the MRI measurements are often done at lower Reynolds number than the compressible application, although both are generally done in fully turbulent flows. The hypothesis is that the conclusions from MRI measurements performed in water are transferable to high subsonic Mach number applications. The present experiment is designed to compare stagnation temperature measurements in high speed airflow (M = 0.7) to concentration measurements in an identical water flow apparatus. The flow configuration was a low aspect ratio wall jet with a thick splitter plate producing a 3D complex downstream flow mixing the wall-jet fluid with the mainstream flow. The three-dimensional velocity field is documented using magnetic resonance velocimetry in the water experiment, and the mixing is quantified by measuring the mean concentration distribution of wall-jet fluid marked with dissolved copper sulfate. The airflow experiments are operated with a temperature difference between the main stream and the wall jet. Profiles of the stagnation temperature are measured with a shielded thermocouple probe. The results show excellent agreement between normalized temperature and concentration profiles after correction of the temperature measurements for the effects of energy separation. The agreement is within 1 % near the edges of the mixing layer, which suggests that the mixing characteristics of the large scale turbulence structures are the same in the two flows.     

超声速尾涡的稳定性分析

从Ｎ－Ｓ方程出发,通过正则模方法,研究了超声速尾涡的绝对／对流不稳定性性质．计算了流动的稳定性特征随马赫数Ｍ,周向波数ｎ．,轴向自由流速度Ｗ０和旋转度ｑ等流动参数的变化规律,找到了绝对／对流不稳定区域的边界．通过比较发现,马赫数的增加使流动由绝对不稳定向对流不稳定乃至稳定转化．在所计算的参数范围,周向波数的增加加速了这一转化过程,而且,轴向速度的增加,同样使流动向着稳定的方向转化．同时还分析了不同旋拧程度的流动受可压缩影响的不同．这些结果对于了解旋拧流动稳定性的物理机理以及进行流动控制都有着重要意义．     

A plane turbulent wall jet on a fully rough surface

The mean velocity field and skin friction characteristics of a plane turbulent wall jet on a smooth and a fully rough surface were studied using Particle Image Velocimetry. The Reynolds number based on the slot height and the exit velocity of the jet was Re = 13,400 and the nominal size of the roughness was k = 0.44 mm. For this Reynolds number and size of roughness element, the flow was in the fully rough regime. The surface roughness results in a distinct change in the shape of the mean velocity profile when scaled in outer coordinates, i.e. using the maximum velocity and outer half-width as the relevant velocity and length scales, respectively. Using inner coordinates, the mean velocity in the lower region of the inner layer was consistent with a logarithmic profile which characterizes the overlap region of a turbulent boundary layer; for the rough wall case, the velocity profile was shifted downward due to the enhanced wall shear stress. For the fully rough flow, the decay rate of the maximum velocity of the wall jet is increased, and the skin friction coefficient is much larger than for the smooth wall case. The inner layer is also thicker for the rough wall case. The effects of surface roughness were observed to penetrate into the outer layer and slightly enhance the spread rate for the outer half-width, which was not observed in most other studies of transitionally rough wall jet flows.     

A computational study on robust prediction of transition point over NACA0012 aerofoil surfaces from laminar to turbulent flows

Flowtransition from laminar toturbulent is prerequisite todecide whereabouts to apply surface flowcontrol techniques. This appears missing in a number of works in which thecontrol effects were merelyinvestigated without getting insight into alteration of transition position. The aim of this study is to capture the correctposition of transition overNACA0012 aerofoil at different angles of attack. Firstly, an implicit, time marching, highresolution total variation diminishing (TVD) scheme was developed to solve the governingNavier—Stokes equations forcompressible fluid flows around aerofoil sections to obtain velocity profiles around the aerofoilsurfaces. Secondly, the linear instability solver based on the Orr—Sommerfeld equations and the e^N methods were developed to calculate the onset of transition over the aerofoil surfaces. Forthe low subsonic Mach number of 0.16, the accuracy of the compressible solutions was assessed bysome available experimental results of low speed incompressible flows. In allcases, transition positionswere accurately predicted which shows applicability and superiority of the present work to beextended for higher Mach number compressible flows. Here, transition prediction methodology is described and the results of this analysiswithout active flow controlor separation are presented.     

超声速边界层/混合层组合流动的稳定性分析

利用可压缩线性稳定性理论研究了超声速混合层考虑壁面影响流动时的失稳特性. 基本流场选取了具有不同速度特征的2 股均匀来流,进入存在上下壁面的流道中. 混合层与边界层的距离为1~3 个边界层厚度,其中壁面取为绝热壁. 分析了该流动在超声速情况下的稳定性特征,同时还讨论了不同波角下的三维扰动波的演化特点,并与二维扰动波进行了比较和分析. 研究结果表明,在此流动情况下,边界层流动和混合层流动的稳定性特征同时存在,并互有影响,其流动稳定性特征既有别于单纯的平板边界层,也有别于单纯的平面混合层,呈现出了新的稳定性特征.      

Heat transfer by laminar flow of an elastico-viscous fluid in posttreatment analysis of wire coating with linearly varying temperature along the coated wire

The present study focuses on the heat transfer by the laminar flow of an elastico-viscous fluid in posttreatment of wire coating analysis with linearly varying temperature on the surface of coated wire. The surface of wire (uncoated) and the surface of coated wire were subjected to two thermal boundary conditions. The constitutive equation of motion and equation of energy have been solved by using perturbation theory for velocity, pressure distribution along the radial direction and temperature distribution. The theoretical analysis of flow rate, average velocity, shear stress, thickness of coated wire, and force on the total wire were also derived. Moreover, the flow phenomenon has been studied under the influence of elastic number R _e velocity ratio U and the dimensionless number S in the ranges 0?≤?R _e ?≤?20, 0.2?≤?U?≤?1.4 and 0?≤?S?≤?20. We noticed that with the increase in elastic number R _e velocity decreases whereas thickness of the coated wire and force on the total wire increases. Also temperature profile decreases with the increase of non-dimensional parameter S.     

Assessment of global linear stability analysis using a time‐stepping approach for compressible flows

Global linear stability analysis combined with computational fluid dynamics (CFD) is considered useful for understanding the physics of fluid flows. However, the numerical techniques of global linear stability analysis for compressible flows have not been well established in comparison with those for incompressible flows. In this study, we develop and assess a set of appropriate numerical techniques required to conduct a global linear stability analysis for compressible flows. For the eigensystem analysis, the Arnoldi method combined with time integration is in effect to preserve the memory (RAM) size of the computer. The compact difference scheme is used for the CFD analysis from the viewpoints of computing accurate global modes and saving memory by reducing the number of grid points to obtain the necessary spatial resolution. To assess the proposed method, two‐dimensional compressible flow problems, including regularized cavity flow, flow around a square cylinder, and the compressible mixing layer, are analyzed, and it is confirmed that the proposed method can obtain accurate mode shapes, growth rate, and frequency of the corresponding global modes. In addition, influences and an appropriate formulation of the outflow boundary conditions are investigated. Results reveal that the outflow boundary causes spurious unstable modes in the global linear stability analysis, and the radiation and outflow boundary condition and the extension of the computational domain with grid stretching keep the spurious unstable modes to a minimum. Copyright © 2015 John Wiley & Sons, Ltd.     

Large-eddy Simulation of Film Cooling Flows with Variable Density Jets

The present paper investigates the impact of the velocity and density ratio on the turbulent mixing process in gas turbine blade film cooling. A cooling fluid is injected from an inclined pipe at α=30° into a turbulent boundary layer profile at a freestream Reynolds number of Re_∞ = 400,000. This jet-in-a-crossflow (JICF) problem is investigated using large-eddy simulations (LES). The governing equations comprise the Navier–Stokes equations plus additional transport equations for several species to simulate a non-reacting gas mixture. A variation of the density ratio is simulated by the heat-mass transfer analogy, i.e., gases of different density are effused into an air crossflow at a constant temperature. An efficient large-eddy simulation method for low subsonic flows based on an implicit dual time-stepping scheme combined with low Mach number preconditioning is applied. The numerical results and experimental velocity data measured using two-component particle-image velocimetry (PIV) are in excellent agreement. The results show the dynamics of the flow field in the vicinity of the jet hole, i.e., the recirculation region and the inclination of the shear layers, to be mainly determined by the velocity ratio. However, evaluating the cooling efficiency downstream of the jet hole the mass flux ratio proves to be the dominant similarity parameter, i.e., the density ratio between the fluids and the velocity ratio have to be considered.     

Time-dependent drag reduction and ageing in aqueous solutions of a cationic surfactant

The turbulent pipe flow of a highly dilute aqueous cationic surfactant solution is investigated by means of a pulsed ultrasound Doppler method with special emphasis on the wall boundary layer. The velocity profiles are recorded for several Reynolds numbers at varying ages of the solution. The wall shear stress velocities u _τ used for the normalization of the velocity profiles are determined by fitting the measured profiles to the universal linear velocity profile in the viscous sublayer. The theoretical pressure loss is then calculated from the numerical values of u _τ and compared to the experimental values. Two different scaling methods are discussed for the velocity fluctuations concerning the correlation of the root-mean square values with the effect and the amount of drag reduction. It is shown that outer scaling with the mean velocity is appropriate for the detection of drag reduction in surfactant solutions, rather than inner scaling with the wall shear stress velocity, which is common practice in investigations of 'usual' turbulent flows.     

超声速边界层/混合层组合流动的稳定性分析简

利用可压缩线性稳定性理论研究了超声速混合层考虑壁面影响流动时的失稳特性. 基本流场选取了具有不同速度特征的2 股均匀来流,进入存在上下壁面的流道中. 混合层与边界层的距离为1~3 个边界层厚度,其中壁面取为绝热壁. 分析了该流动在超声速情况下的稳定性特征,同时还讨论了不同波角下的三维扰动波的演化特点,并与二维扰动波进行了比较和分析. 研究结果表明,在此流动情况下,边界层流动和混合层流动的稳定性特征同时存在,并互有影响,其流动稳定性特征既有别于单纯的平板边界层,也有别于单纯的平面混合层,呈现出了新的稳定性特征.     

Three-dimensional vortex wake structure of a flapping-wing micro aerial vehicle in forward flight configuration

The unsteady flow field past a backward-facing step in a rectangular duct is investigated by adopting time-resolved particle image velocimetry (PIV) in the Reynolds number range of 2,640–9,880 based on step height and the inlet average velocity. The PIV realizations are subjected to post-processing techniques, namely, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). At low Reynolds numbers, the second spatial POD modes indicate the presence of the shear layer mode, whereas this feature shifts to higher modes at higher Reynolds numbers. The corresponding temporal modes are Fourier-transformed to obtain the dominant frequency, whose Strouhal number corroborates the above observation. Short-time windows in the transverse velocity component along the shear layer are selected to investigate the temporal stability of the flow field by DMD to quantify the growth rate of the shear layer mode. The higher harmonics of this mode are also observed to grow, albeit at lesser rate. By relating to POD analysis, the most energetic structures were found to correspond to the unstable modes. The correlation between these unstable DMD modes and the Fourier-filtered flow fields for the same frequencies indicate better match for the lower operating Reynolds number case as compared to higher ones. The spatial stability analysis demonstrates the growth of the shear layer vortices, which is combined with the temporal stability analysis to evaluate the phase velocity of the identified shear layer structures. The calculated phase velocity magnitude of the shear layer is found to be reasonably below the local velocity as expected.     

Prediction of resolvent mode shapes in supersonic turbulent boundary layers

This work applies resolvent analysis to compressible zero-pressure-gradient turbulent boundary layers with freestream Mach numbers between 2 and 4, focusing exclusively on large scale motions in the outer region of the boundary layer. We investigate the effects of Mach number on predicted flow structures, and in particular, look at how such effects may be attributed to changes in mean properties. By leveraging the similarity between the compressible and incompressible resolvent operators, we show that the shape of the streamwise velocity and temperature components of resolvent response modes in the compressible regime can be approximated by applying ideas from wavepacket pseudospectral theory to a simple scalar operator. This gives a means of predicting the shape of resolvent mode components for compressible flows without requiring the singular value decompositions of discretized operators. At a Mach number of 2, we find that accurate results are obtained from this approximation when using the compressible mean velocity profile. At Mach numbers of 3 and 4, the quantitative accuracy of these predictions is improved by also considering a local effective Reynolds number based on the local mean density and viscosity.     

平面可压基频涡卷非线性演化行为数值研究

采用高精度迎风/对称紧致混合差分算法,对可压自由剪切层转捩区中的几种典型展向大尺度涡作用型态进行了直接数值模拟,通过施加给定来流条件下的线性最不稳定黏性基频扰动及其亚谐扰动,以被动守恒标量技术给出了基频涡卷的饱和、撕裂、融合以及三涡对并等细节结构。分析显示,亚谐振动相差是促生基频涡卷不同非线性演化过程的重要因素之一,可对扰动量的发展变化,以及剪切层厚度和混合效率产生直接影响,计算结果同实验流动显示图像十分相似,表明了主导线性扰动的非线性耦合效应与一些实际涡作用行为间的内联系。