Density and compressibility effects in turbulent subsonic jets part 1: mean velocity field

The behavior of compressible jets originated from initially turbulent pipe flows issuing in still air has been investigated at three different subsonic Mach numbers, 0.3, 0.6 and 0.9. Helium, nitrogen and krypton gases were used to generate the jet flows and investigate the additional effects of density on the flow structure. Particle image velocimetry, high-frequency response pressure transducers and thermocouples were used to obtain velocity, Mach number and total temperature measurements inside the flow field. The jets were formed at the exit of an adiabatic compressible frictional turbulent pipe flow, which was developing toward its corresponding sonic conditions inside the pipe, and continued to expand within the first four diameters distance after it exited the pipe. Theoretical considerations based on flow self-similarity were used to obtain the decay of Mach number along the centerline of the jets for the first time. It was found that this decay depends on two contributions, one from the velocity field which is inversely proportional to the distance from the exit and one from the thermal field which is proportional to this distance. As a result, a small non-linearity in the variation of the inverse Mach number with downstream distance was found. The decay of the Mach number at the centerline of the axisymmetric jets increases by increasing the initial Mach number at the exit of the flow for all jets. The decay of mean velocity at the centerline of the jets is also higher at higher exit Mach numbers. However, the velocity non-dimensionalized by the exit velocity seems to decrease faster at low exit Mach numbers, suggesting a reduced mixing with increasing exit flow Mach numbers. Helium jets were found to have the largest spreading rate among the three different gas jets used in the present investigation, while krypton jets had the lowest spreading rate. The spreading rate of each gas decreases with increasing its kinetic energy relatively to its internal energy.     

可压缩各向同性衰减湍流直接数值模拟研究

采用五阶有限差分WENO格式直接模拟了高初始湍流Mach数的可压缩均匀各向同性湍流,主要分析了湍流的统计特性 和压缩性的影响,包括能谱特征、激波串、耗散率、标度律等. 研究表明,湍动能主要来自于速度场螺旋分量的贡献;各向同性湍流的小尺度脉动对压缩性更为敏感,并且压缩性的增强加快了湍流大 尺度脉动向小尺度脉动的湍动能输运;随着湍流Mach数的升高,胀量(压缩)耗散率所占比率也显著增长. 标度律分析表明,强可压缩湍流的横向速度结构函数仍然具有扩展自相似性;当阶数较高(p ≥ 5)时,纵向速度结构函数的扩展自相似性则不再成立. 对于压缩性较弱的湍流,与不可压缩湍流一致,横向湍流脉动的间歇性要强于纵向湍流脉动;而对于强可压缩湍流,纵向湍流脉动的 间歇性要强于横向湍流脉动.     

A Mach‐uniform preconditioner for incompressible and subsonic flows

In this study, a novel Mach‐uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach‐uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two‐dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach‐uniform preconditioner in comparison to the non‐preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well‐known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimate of the correlation between pressure pulsations and the divergence of the velocity in subsonic flows of variable density

In flows with variable density, the turbulence energy equation contains a large number of correlations, about which little is at present known [1]. One of the least studied is the correlation between the pressure and the divergence of the velocity. Usually, this correlation is ignored [2, 3]. The aim of the present paper is to estimate the pulsations of the divergence of the velocity and the correlation with the pressure pulsations in a subsonic turbulent flow with variable density. Three cases are considered: 1) mixing of gases having different densities, 2) diffusion combustion, 3) combustion of a homogeneous mixture. It is assumed that the Mach number is small, the Reynolds number large, and the coefficients of molecular diffusion and thermal diffusivity equal; external forces are absent.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–11, May–June, 1979.     

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.     

Role of the momentum interpolation mechanism of the Roe scheme in shock instability

The shock instability phenomenon is a well‐known problem for hypersonic flow computation by the shock‐capturing Roe scheme. The pressure checkerboard is another well‐known problem for low‐Mach‐number flow computation. The momentum interpolation method (MIM) is necessary for low‐Mach‐number flows to suppress the pressure checkerboard problem, and the pressure‐difference‐driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high‐Mach‐number flows or low‐Mach‐number cells in numerical shock. Therefore, MIM should be retained for low‐Mach‐number flows and be completely removed for high‐Mach‐number flows and low‐Mach‐number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd.     

Direct Numerical Simulation of Self-Similar Turbulent Boundary Layers in Adverse Pressure Gradients

Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.     

A multiblock operator‐splitting algorithm for unsteady flows at all speeds in complex geometries

This paper describes a non‐iterative operator‐splitting algorithm for computing all‐speed flows in complex geometries. A pressure‐based algorithm is adopted as the base, in which pressure, instead of density, is a primary variable, thus allowing for a unified formulation for all Mach numbers. The focus is on adapting the method for (a) flows at all speeds, and (b) multiblock, non‐orthogonal, body‐fitted grids for very complex geometries. Key features of the formulation include special treatment of mass fluxes at control volume interfaces to avoid pressure–velocity decoupling for incompressible (low Mach number limit) flows and to provide robust pressure–velocity–density coupling for compressible (high‐speed) flows. The method is shown to be robust for all Mach number regimes for both steady and unsteady flows; it is found to be stable for CFL numbers of order ten, allowing large time steps to be taken for steady flows. Enhancements to the method which allow for stable solutions to be obtained on non‐orthogonal grids are also discussed. The method is found to be very reliable even in complex engineering applications such as unsteady rotor–stator interactions in turbulent, all‐speed turbomachinery flows. Copyright © 2004 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.     

Theoretical investigation of some thermal effects in turbulence modeling

Fluid compressibility effects arising from thermal rather than dynamical aspects are theoretically investigated in the framework of turbulent flows. The Mach number is considered low and not to induce significant compressibility effects which here occur due to a very high thermal gradient within the flowfield. With the use of the Two-Scale Direct Interaction Approximation approach, essential turbulent correlations are derived in a one-point one-time framework. In the low velocity gradient limit, they are shown to directly depend on the temperature gradient, assumed large. The impact of thermal effects onto the transport equations of the turbulent kinetic energy and dissipation rate is also investigated, together with the transport equation for both the density and the internal energy variance.      

Simultaneous concentration and velocity field measurements in a shock-accelerated mixing layer

A novel technique to obtain simultaneous velocity and concentration measurements is applied to the Richtmyer–Meshkov instability. After acceleration by a Mach 2.2 shock wave, the interface between the two gases develops into a turbulent mixing layer. A time-separated pair of acetone planar laser-induced fluorescence images are processed to yield concentration and, through application of the Advection-Corrected Correlation Image Velocimetry technique, velocity fields. This is the first application of this technique to shock-accelerated flows. We show that when applied to numerical simulations, this technique reproduces the velocity field to a similar quality as particle image velocimetry. When applied to the turbulent mixing layer of the experiments, information about the Reynolds number and anisotropy of the flow is obtained.     

Fully Compressible Low-Mach Number Simulations of Carbon-dioxide at Supercritical Pressures and Trans-critical Temperatures

This work investigates fully developed turbulent flows of carbon-dioxide close to its vapour-liquid critical point in a channel with a hot and a cold wall. Two direct numerical simulations are performed at low Mach numbers, with the trans-critical transition near the channel centre and the cold wall, respectively. An additional simulation with constant transport properties is used to selectively investigate the effect of the non-linear equation of state on turbulence. Compared to the case where the pseudo-critical transition occurs in the channel center, the case with the pseudo-critical transition close to the cold wall reveals that compressibility effects can exist in the near-wall region even at low Mach numbers. An analysis of the velocity streaks near the hot and the cold walls also indicates a greater degree of streak coherence near the cold wall. A comparison between the constant and variable viscosity cases at the same Reynolds number, Mach number and having the same isothermal wall boundary conditions reveals that variable viscosity increases turbulence near the cold wall and also causes higher velocity gradients near the hot wall. We also show that the extended van Driest transformation results in a better agreement of the velocity profile with the log-law of the wall compared to the standard van Driest transformation. The semi-locally scaled turbulent velocity fluctuations and the turbulent kinetic energy budgets on the hot and the cold sides of the channel collapse on top of each other, thereby establishing the validity of Morkovin’s hypothesis.     

Turbulent Flame Speeds of Premixed Supersonic Flame Kernels

It is unclear whether turbulent flame speed scalings established in low speed regimes are applicable to supersonic flames. To investigate this question, the canonical flame kernel is investigated in a scramjet-like channel having a one degree wall divergence. The growth, shape and internal kernel dynamics are investigated. Results are presented for three Mach numbers, four equivalence ratios, and three turbulence generators. Schlieren photography provides flame images for growth rate statistics and particle image velocimetry (PIV) provides turbulence statistics and investigation of internal kernel dynamics. Supersonic flame kernels are self-propagating and respond to the equivalence ratio in a fashion that is similar to low speed flames. However, supersonic flame kernels have features that are not present in subsonic flame kernels. Baroclinicity, resulting from pressure-density misalignment, creates a reacting vortex ring structure. Further, the mean kernel shape has a Mach number dependence and the vortex ring enhances the turbulent flame speed through entrainment of reactants and augmented flame surface growth. Hence, the previously established (low speed) flame speed scalings are inappropriate for supersonic flame kernels. Drawing motivation from vortex ring literature, the ring propagation velocity is used as the characteristic velocity and a new flame speed scaling is proposed.     

Sound and vorticity interactions: transmission and scattering

We review several aspects of the propagation of sound in vortical flows. We restrict ourselves to isothermal, humidity-free flows at low Mach number M. Since vorticity plays a major role in vortex-flow interactions we focus on vortical flows. We consider two main canonical situations. The first concerns the transmission of sound. We analyze the evolution of acoustic wavefronts as they propagate across a single vortex. The second situation addresses the scattering of sound waves by nonstationary vortices. We study the evolution of the acoustic pressure emitted in the far field, at an angle with the initial direction of propagation. In this geometry one performs direct spectroscopy of the flow vorticity field. In each case, we review theoretical results and compare with experimental measurements and numerical simulations when available. We also briefly report how the following new acoustic techniques have recently been used to study complex or turbulent flows: time-resolved acoustic spectroscopy, speckle interferometry and Lagrangian particle tracking. PACS 43.25, 43.28, 47.32, 67.40     

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.     

Continuation methods applied to the 2D Navier–Stokes equations at high Reynolds numbers

Nonlinearities arise in aerodynamic flows as a function of various parameters, such as angle of attack, Mach number and Reynolds number. These nonlinearities can cause the change from steady to unsteady flow or give rise to static hysteresis. Understanding these nonlinearities is important for safety validation and performance enhancement of modern aircraft. A continuation method has been developed to study nonlinear steady state solutions with respect to changes in parameters for two‐dimensional compressible turbulent flows at high Reynolds numbers. This is the first time that such flows have been analysed with this approach. Continuation methods allow the stable and unstable solutions to be traced as flow parameters are changed. Continuation has been carried out on two‐dimensional aerofoils for several parameters: angle of attack, Mach number, Reynolds number, aerofoil thickness and turbulent inflow as well as levels of dissipation applied to the models. A range of results are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Large Eddy Simulation of the sound field of a round turbulent jet

In this paper we will use Large Eddy Simulation (LES) to obtain the flow field of a turbulent round jet at a Reynolds number based on the jet orifice velocity of 11000. In the simulations it is assumed that the flow field is incompressible. The acoustic field of the jet is calculated with help of the Lighthill acoustic analogy. The coupling between the flow solver and the acoustic solver is discussed in detail. The Mach number used in the acoustic calculation was equal to 0.6. It is shown that the decay of the jet centerline velocity and centerline rms are in good agreement with experimental data of [12]. Furthermore, it is shown that the influence of the LES modeling on the acoustic field is very small, if the dynamic subgrid model is used.     

Reconstruction of velocity fields from wall pressure measurements in a shock wave/turbulent boundary layer interaction

We present here experimental results in a shock wave/turbulent boundary layer interaction at Mach number of 2.3 impinged by an oblique shock wave, with a deflection angle of 9.5°, as installed in the supersonic wind tunnel of the IUSTI laboratory, France. For such a shock intensity, strong unsteadiness are developing inside the separated zone involving very low frequencies associated with reflected shock motions.The present work consists in simultaneous PIV velocity fields and unsteady wall pressure measurements. The wall pressure and PIV measurements were used to characterize the pressure distribution at the wall in an axial direction, and the flow field associated. These results give access for the first time to the spatial-time correlation between wall pressure and velocity in a shock wave turbulent boundary layer interaction and show the feasibility of such coupling techniques in compressible flows. Linear Stochastic Estimation (LSE) coupled with Proper Orthogonal Decomposition (POD) has been applied to these measurements, and first results are presented here, showing the ability of these techniques to reproduce both the unsteady breathing of the recirculating bubble at low frequency and the Kelvin–Helmholtz instabilities developing at moderate frequency.